Line data Source code
1 : /* Copyright (C) 2016 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : /*************************************************************************/
16 : /* */
17 : /* Modular forms package based on trace formulas */
18 : /* */
19 : /*************************************************************************/
20 : #include "pari.h"
21 : #include "paripriv.h"
22 :
23 : #define DEBUGLEVEL DEBUGLEVEL_mf
24 :
25 : enum {
26 : MF_SPLIT = 1,
27 : MF_EISENSPACE,
28 : MF_FRICKE,
29 : MF_MF2INIT,
30 : MF_SPLITN
31 : };
32 :
33 : typedef struct {
34 : GEN vnew, vfull, DATA, VCHIP;
35 : long n, newHIT, newTOTAL, cuspHIT, cuspTOTAL;
36 : } cachenew_t;
37 :
38 : static void init_cachenew(cachenew_t *c, long n, long N, GEN f);
39 : static long mf1cuspdim_i(long N, GEN CHI, GEN TMP, GEN vSP, long *dih);
40 : static GEN mfinit_i(GEN NK, long space);
41 : static GEN mfinit_Nkchi(long N, long k, GEN CHI, long space, long flraw);
42 : static GEN mf2init_Nkchi(long N, long k, GEN CHI, long space, long flraw);
43 : static GEN mf2basis(long N, long r, GEN CHI, GEN *pCHI1, long space);
44 : static GEN mfeisensteinbasis(long N, long k, GEN CHI);
45 : static GEN mfeisensteindec(GEN mf, GEN F);
46 : static GEN initwt1newtrace(GEN mf);
47 : static GEN initwt1trace(GEN mf);
48 : static GEN myfactoru(long N);
49 : static GEN mydivisorsu(long N);
50 : static GEN Qab_Czeta(long k, long ord, GEN C, long vt);
51 : static GEN mfcoefs_i(GEN F, long n, long d);
52 : static GEN bhnmat_extend(GEN M, long m,long l, GEN S, cachenew_t *cache);
53 : static GEN initnewtrace(long N, GEN CHI);
54 : static void dbg_cachenew(cachenew_t *C);
55 : static GEN hecke_i(long m, long l, GEN V, GEN F, GEN DATA);
56 : static GEN c_Ek(long n, long d, GEN F);
57 : static GEN RgV_heckef2(long n, long d, GEN V, GEN F, GEN DATA);
58 : static GEN mfcusptrace_i(long N, long k, long n, GEN Dn, GEN TDATA);
59 : static GEN mfnewtracecache(long N, long k, long n, cachenew_t *cache);
60 : static GEN colnewtrace(long n0, long n, long d, long N, long k, cachenew_t *c);
61 : static GEN dihan(GEN bnr, GEN w, GEN k0j, long m, ulong n);
62 : static GEN sigchi(long k, GEN CHI, long n);
63 : static GEN sigchi2(long k, GEN CHI1, GEN CHI2, long n, long ord);
64 : static GEN mflineardivtomat(long N, GEN vF, long n);
65 : static GEN mfdihedralcusp(long N, GEN CHI, GEN vSP);
66 : static long mfdihedralcuspdim(long N, GEN CHI, GEN vSP);
67 : static GEN mfdihedralnew(long N, GEN CHI, GEN SP);
68 : static GEN mfdihedral(long N);
69 : static GEN mfdihedralall(long N);
70 : static long mf1cuspdim(long N, GEN CHI, GEN vSP);
71 : static long mf2dim_Nkchi(long N, long k, GEN CHI, ulong space);
72 : static long mfdim_Nkchi(long N, long k, GEN CHI, long space);
73 : static GEN charLFwtk(long N, long k, GEN CHI, long ord, long t);
74 : static GEN mfeisensteingacx(GEN E,long w,GEN ga,long n,long prec);
75 : static GEN mfgaexpansion(GEN mf, GEN F, GEN gamma, long n, long prec);
76 : static GEN mfEHmat(long n, long r);
77 : static GEN mfEHcoef(long r, long N);
78 : static GEN mftobasis_i(GEN mf, GEN F);
79 :
80 : static GEN
81 37324 : mkgNK(GEN N, GEN k, GEN CHI, GEN P) { return mkvec4(N, k, CHI, P); }
82 : static GEN
83 15176 : mkNK(long N, long k, GEN CHI) { return mkgNK(stoi(N), stoi(k), CHI, pol_x(1)); }
84 : GEN
85 8827 : MF_get_CHI(GEN mf) { return gmael(mf,1,3); }
86 : GEN
87 21147 : MF_get_gN(GEN mf) { return gmael(mf,1,1); }
88 : long
89 19943 : MF_get_N(GEN mf) { return itou(MF_get_gN(mf)); }
90 : GEN
91 15456 : MF_get_gk(GEN mf) { return gmael(mf,1,2); }
92 : long
93 7224 : MF_get_k(GEN mf)
94 : {
95 7224 : GEN gk = MF_get_gk(mf);
96 7224 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
97 7224 : return itou(gk);
98 : }
99 : long
100 245 : MF_get_r(GEN mf)
101 : {
102 245 : GEN gk = MF_get_gk(mf);
103 245 : if (typ(gk) == t_INT) pari_err_IMPL("integral weight");
104 245 : return itou(gel(gk, 1)) >> 1;
105 : }
106 : long
107 15358 : MF_get_space(GEN mf) { return itos(gmael(mf,1,4)); }
108 : GEN
109 4466 : MF_get_E(GEN mf) { return gel(mf,2); }
110 : GEN
111 21427 : MF_get_S(GEN mf) { return gel(mf,3); }
112 : GEN
113 1876 : MF_get_basis(GEN mf) { return shallowconcat(gel(mf,2), gel(mf,3)); }
114 : long
115 5621 : MF_get_dim(GEN mf)
116 : {
117 5621 : switch(MF_get_space(mf))
118 : {
119 721 : case mf_FULL:
120 721 : return lg(MF_get_S(mf)) - 1 + lg(MF_get_E(mf))-1;
121 140 : case mf_EISEN:
122 140 : return lg(MF_get_E(mf))-1;
123 4760 : default: /* mf_NEW, mf_CUSP, mf_OLD */
124 4760 : return lg(MF_get_S(mf)) - 1;
125 : }
126 : }
127 : GEN
128 7273 : MFnew_get_vj(GEN mf) { return gel(mf,4); }
129 : GEN
130 686 : MFcusp_get_vMjd(GEN mf) { return gel(mf,4); }
131 : GEN
132 6895 : MF_get_M(GEN mf) { return gmael(mf,5,3); }
133 : GEN
134 4858 : MF_get_Minv(GEN mf) { return gmael(mf,5,2); }
135 : GEN
136 10619 : MF_get_Mindex(GEN mf) { return gmael(mf,5,1); }
137 :
138 : /* ordinary gtocol forgets about initial 0s */
139 : GEN
140 2387 : sertocol(GEN S) { return gtocol0(S, -(lg(S) - 2 + valser(S))); }
141 : /*******************************************************************/
142 : /* Linear algebra in cyclotomic fields (TODO: export this) */
143 : /*******************************************************************/
144 : /* return r and split prime p giving projection Q(zeta_n) -> Fp, zeta -> r */
145 : static ulong
146 1246 : QabM_init(long n, ulong *p)
147 : {
148 1246 : ulong pinit = 1000000007;
149 : forprime_t T;
150 1246 : if (n <= 1) { *p = pinit; return 0; }
151 1225 : u_forprime_arith_init(&T, pinit, ULONG_MAX, 1, n);
152 1225 : *p = u_forprime_next(&T);
153 1225 : return Flx_oneroot(ZX_to_Flx(polcyclo(n, 0), *p), *p);
154 : }
155 : static ulong
156 8534960 : Qab_to_Fl(GEN P, ulong r, ulong p)
157 : {
158 : ulong t;
159 : GEN den;
160 8534960 : P = Q_remove_denom(liftpol_shallow(P), &den);
161 8534960 : if (typ(P) == t_POL) { GEN Pp = ZX_to_Flx(P, p); t = Flx_eval(Pp, r, p); }
162 8399335 : else t = umodiu(P, p);
163 8534960 : if (den) t = Fl_div(t, umodiu(den, p), p);
164 8534960 : return t;
165 : }
166 : static GEN
167 38164 : QabC_to_Flc(GEN x, ulong r, ulong p)
168 8341333 : { pari_APPLY_long( Qab_to_Fl(gel(x,i), r, p)); }
169 : static GEN
170 595 : QabM_to_Flm(GEN x, ulong r, ulong p)
171 38759 : { pari_APPLY_same(QabC_to_Flc(gel(x, i), r, p);) }
172 : /* A a t_POL */
173 : static GEN
174 1484 : QabX_to_Flx(GEN A, ulong r, ulong p)
175 : {
176 1484 : long i, l = lg(A);
177 1484 : GEN a = cgetg(l, t_VECSMALL);
178 1484 : a[1] = ((ulong)A[1])&VARNBITS;
179 233023 : for (i = 2; i < l; i++) uel(a,i) = Qab_to_Fl(gel(A,i), r, p);
180 1484 : return Flx_renormalize(a, l);
181 : }
182 :
183 : /* FIXME: remove */
184 : static GEN
185 1106 : ZabM_pseudoinv_i(GEN M, GEN P, long n, GEN *pv, GEN *den, int ratlift)
186 : {
187 1106 : GEN v = ZabM_indexrank(M, P, n);
188 1106 : if (pv) *pv = v;
189 1106 : M = shallowmatextract(M,gel(v,1),gel(v,2));
190 1106 : return ratlift? ZabM_inv_ratlift(M, P, n, den): ZabM_inv(M, P, n, den);
191 : }
192 :
193 : /* M matrix with coeff in Q(\chi)), where Q(\chi) = Q(X)/(P) for
194 : * P = cyclotomic Phi_n. Assume M rational if n <= 2 */
195 : static GEN
196 1638 : QabM_ker(GEN M, GEN P, long n)
197 : {
198 1638 : if (n <= 2) return QM_ker(M);
199 420 : return ZabM_ker(row_Q_primpart(liftpol_shallow(M)), P, n);
200 : }
201 : /* pseudo-inverse of M. FIXME: should replace QabM_pseudoinv */
202 : static GEN
203 1351 : QabM_pseudoinv_i(GEN M, GEN P, long n, GEN *pv, GEN *pden)
204 : {
205 : GEN cM, Mi;
206 1351 : if (n <= 2)
207 : {
208 1169 : M = Q_primitive_part(M, &cM);
209 1169 : Mi = ZM_pseudoinv(M, pv, pden); /* M^(-1) = Mi / (cM * den) */
210 : }
211 : else
212 : {
213 182 : M = Q_primitive_part(liftpol_shallow(M), &cM);
214 182 : Mi = ZabM_pseudoinv(M, P, n, pv, pden);
215 : }
216 1351 : *pden = mul_content(*pden, cM);
217 1351 : return Mi;
218 : }
219 : /* FIXME: delete */
220 : static GEN
221 1092 : QabM_pseudoinv(GEN M, GEN P, long n, GEN *pv, GEN *pden)
222 : {
223 1092 : GEN Mi = QabM_pseudoinv_i(M, P, n, pv, pden);
224 1092 : return P? gmodulo(Mi, P): Mi;
225 : }
226 :
227 : static GEN
228 10486 : QabM_indexrank(GEN M, GEN P, long n)
229 : {
230 : GEN z;
231 10486 : if (n <= 2)
232 : {
233 9289 : M = vec_Q_primpart(M);
234 9289 : z = ZM_indexrank(M); /* M^(-1) = Mi / (cM * den) */
235 : }
236 : else
237 : {
238 1197 : M = vec_Q_primpart(liftpol_shallow(M));
239 1197 : z = ZabM_indexrank(M, P, n);
240 : }
241 10486 : return z;
242 : }
243 :
244 : /*********************************************************************/
245 : /* Simple arithmetic functions */
246 : /*********************************************************************/
247 : /* TODO: most of these should be exported and used in ifactor1.c */
248 : /* phi(n) */
249 : static ulong
250 110159 : myeulerphiu(ulong n)
251 : {
252 : pari_sp av;
253 110159 : if (n == 1) return 1;
254 91140 : av = avma; return gc_ulong(av, eulerphiu_fact(myfactoru(n)));
255 : }
256 : static long
257 65709 : mymoebiusu(ulong n)
258 : {
259 : pari_sp av;
260 65709 : if (n == 1) return 1;
261 54194 : av = avma; return gc_long(av, moebiusu_fact(myfactoru(n)));
262 : }
263 :
264 : static long
265 3017 : mynumdivu(long N)
266 : {
267 : pari_sp av;
268 3017 : if (N == 1) return 1;
269 2891 : av = avma; return gc_long(av, numdivu_fact(myfactoru(N)));
270 : }
271 :
272 : /* N\prod_{p|N} (1+1/p) */
273 : static long
274 399609 : mypsiu(ulong N)
275 : {
276 : pari_sp av;
277 : GEN P;
278 : long j, l, a;
279 399609 : if (N == 1) return 1;
280 313761 : av = avma; P = gel(myfactoru(N), 1); l = lg(P);
281 746403 : for (a = N, j = 1; j < l; j++) a += a / P[j];
282 313761 : return gc_long(av, a);
283 : }
284 : /* write n = mf^2. Return m, set f. */
285 : static ulong
286 72 : mycore(ulong n, long *pf)
287 : {
288 72 : pari_sp av = avma;
289 72 : GEN fa = myfactoru(n), P = gel(fa,1), E = gel(fa,2);
290 72 : long i, l = lg(P), m = 1, f = 1;
291 275 : for (i = 1; i < l; i++)
292 : {
293 203 : long j, p = P[i], e = E[i];
294 203 : if (e & 1) m *= p;
295 462 : for (j = 2; j <= e; j+=2) f *= p;
296 : }
297 72 : *pf = f; return gc_long(av,m);
298 : }
299 :
300 : /* fa = factorization of -D > 0, return -D0 > 0 (where D0 is fundamental) */
301 : static long
302 5567588 : corediscs_fact(GEN fa)
303 : {
304 5567588 : GEN P = gel(fa,1), E = gel(fa,2);
305 5567588 : long i, l = lg(P), m = 1;
306 18429403 : for (i = 1; i < l; i++)
307 : {
308 12861815 : long p = P[i], e = E[i];
309 12861815 : if (e & 1) m *= p;
310 : }
311 5567588 : if ((m&3L) != 3) m <<= 2;
312 5567588 : return m;
313 : }
314 : static long
315 7042 : mubeta(long n)
316 : {
317 7042 : pari_sp av = avma;
318 7042 : GEN E = gel(myfactoru(n), 2);
319 7042 : long i, s = 1, l = lg(E);
320 14616 : for (i = 1; i < l; i++)
321 : {
322 7574 : long e = E[i];
323 7574 : if (e >= 3) return gc_long(av,0);
324 7574 : if (e == 1) s *= -2;
325 : }
326 7042 : return gc_long(av,s);
327 : }
328 :
329 : /* n = n1*n2, n1 = ppo(n, m); return mubeta(n1)*moebiusu(n2).
330 : * N.B. If n from newt_params we, in fact, never return 0 */
331 : static long
332 7714544 : mubeta2(long n, long m)
333 : {
334 7714544 : pari_sp av = avma;
335 7714544 : GEN fa = myfactoru(n), P = gel(fa,1), E = gel(fa,2);
336 7714544 : long i, s = 1, l = lg(P);
337 15482382 : for (i = 1; i < l; i++)
338 : {
339 7767838 : long p = P[i], e = E[i];
340 7767838 : if (m % p)
341 : { /* p^e in n1 */
342 6596955 : if (e >= 3) return gc_long(av,0);
343 6596955 : if (e == 1) s *= -2;
344 : }
345 : else
346 : { /* in n2 */
347 1170883 : if (e >= 2) return gc_long(av,0);
348 1170883 : s = -s;
349 : }
350 : }
351 7714544 : return gc_long(av,s);
352 : }
353 :
354 : /* write N = prod p^{ep} and n = df^2, d squarefree.
355 : * set g = ppo(gcd(sqfpart(N), f), FC)
356 : * N2 = prod p^if(e==1 || p|n, ep-1, ep-2) */
357 : static void
358 1903283 : newt_params(long N, long n, long FC, long *pg, long *pN2)
359 : {
360 1903283 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
361 1903283 : long i, g = 1, N2 = 1, l = lg(P);
362 5063458 : for (i = 1; i < l; i++)
363 : {
364 3160175 : long p = P[i], e = E[i];
365 3160175 : if (e == 1)
366 2763376 : { if (FC % p && n % (p*p) == 0) g *= p; }
367 : else
368 396799 : N2 *= upowuu(p,(n % p)? e-2: e-1);
369 : }
370 1903283 : *pg = g; *pN2 = N2;
371 1903283 : }
372 : /* simplified version of newt_params for n = 1 (newdim) */
373 : static void
374 42147 : newd_params(long N, long *pN2)
375 : {
376 42147 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
377 42147 : long i, N2 = 1, l = lg(P);
378 105217 : for (i = 1; i < l; i++)
379 : {
380 63070 : long p = P[i], e = E[i];
381 63070 : if (e > 2) N2 *= upowuu(p, e-2);
382 : }
383 42147 : *pN2 = N2;
384 42147 : }
385 :
386 : static long
387 21 : newd_params2(long N)
388 : {
389 21 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
390 21 : long i, N2 = 1, l = lg(P);
391 56 : for (i = 1; i < l; i++)
392 : {
393 35 : long p = P[i], e = E[i];
394 35 : if (e >= 2) N2 *= upowuu(p, e);
395 : }
396 21 : return N2;
397 : }
398 :
399 : /*******************************************************************/
400 : /* Relative trace between cyclotomic fields (TODO: export this) */
401 : /*******************************************************************/
402 : /* g>=1; return g * prod_{p | g, (p,q) = 1} (1-1/p) */
403 : static long
404 36869 : phipart(long g, long q)
405 : {
406 36869 : if (g > 1)
407 : {
408 19670 : GEN P = gel(myfactoru(g), 1);
409 19670 : long i, l = lg(P);
410 40194 : for (i = 1; i < l; i++) { long p = P[i]; if (q % p) g -= g / p; }
411 : }
412 36869 : return g;
413 : }
414 : /* Set s,v s.t. Trace(zeta_N^k) from Q(zeta_N) to Q(\zeta_N) = s * zeta_M^v
415 : * With k > 0, N = M*d and N, M != 2 mod 4 */
416 : static long
417 84756 : tracerelz(long *pv, long d, long M, long k)
418 : {
419 : long s, g, q, muq;
420 84756 : if (d == 1) { *pv = k; return 1; }
421 65618 : *pv = 0; g = ugcd(k, d); q = d / g;
422 65618 : muq = mymoebiusu(q); if (!muq) return 0;
423 47173 : if (M != 1)
424 : {
425 37828 : long v = Fl_invsafe(q % M, M);
426 37828 : if (!v) return 0;
427 27524 : *pv = (v * (k/g)) % M;
428 : }
429 36869 : s = phipart(g, M*q); if (muq < 0) s = -s;
430 36869 : return s;
431 : }
432 : /* Pi = polcyclo(i), i = m or n. Let Ki = Q(zeta_i), initialize Tr_{Kn/Km} */
433 : GEN
434 34048 : Qab_trace_init(long n, long m, GEN Pn, GEN Pm)
435 : {
436 : long a, i, j, N, M, vt, d, D;
437 : GEN T, G;
438 :
439 34048 : if (m == n || n <= 2) return mkvec(Pm);
440 16555 : vt = varn(Pn);
441 16555 : d = degpol(Pn);
442 : /* if (N != n) zeta_N = zeta_n^2 and zeta_n = - zeta_N^{(N+1)/2} */
443 16555 : N = ((n & 3) == 2)? n >> 1: n;
444 16555 : M = ((m & 3) == 2)? m >> 1: m; /* M | N | n */
445 16555 : a = N / M;
446 16555 : T = const_vec(d, NULL);
447 16555 : D = d / degpol(Pm); /* relative degree */
448 16555 : if (D == 1) G = NULL;
449 : else
450 : { /* zeta_M = zeta_n^A; s_j(zeta_M) = zeta_M <=> j = 1 (mod J) */
451 15281 : long lG, A = (N == n)? a: (a << 1), J = n / ugcd(n, A);
452 15281 : G = coprimes_zv(n);
453 150276 : for (j = lG = 1; j < n; j += J)
454 134995 : if (G[j]) G[lG++] = j;
455 15281 : setlg(G, lG); /* Gal(Q(zeta_n) / Q(zeta_m)) */
456 : }
457 16555 : T = const_vec(d, NULL);
458 16555 : gel(T,1) = utoipos(D); /* Tr 1 */
459 140140 : for (i = 1; i < d; i++)
460 : { /* if n = 2N, zeta_n^i = (-1)^i zeta_N^k */
461 : long s, v, k;
462 : GEN t;
463 :
464 123585 : if (gel(T, i+1)) continue;
465 84756 : k = (N == n)? i: ((odd(i)? i + N: i) >> 1);
466 84756 : if ((s = tracerelz(&v, a, M, k)))
467 : {
468 56007 : if (m != M) v *= 2;/* Tr = s * zeta_m^v */
469 56007 : if (n != N && odd(i)) s = -s;
470 56007 : t = Qab_Czeta(v, m, stoi(s), vt);
471 : }
472 : else
473 28749 : t = gen_0;
474 : /* t = Tr_{Kn/Km} zeta_n^i; fill using Galois action */
475 84756 : if (!G)
476 19138 : gel(T, i + 1) = t;
477 : else
478 370874 : for (j = 1; j <= D; j++)
479 : {
480 305256 : long z = Fl_mul(i,G[j], n);
481 305256 : if (z < d) gel(T, z + 1) = t;
482 : }
483 : }
484 16555 : return mkvec3(Pm, Pn, T);
485 : }
486 : /* x a t_POL modulo Phi_n */
487 : static GEN
488 80255 : tracerel_i(GEN T, GEN x)
489 : {
490 80255 : long k, l = lg(x);
491 : GEN S;
492 80255 : if (l == 2) return gen_0;
493 80255 : S = gmul(gel(T,1), gel(x,2));
494 283290 : for (k = 3; k < l; k++) S = gadd(S, gmul(gel(T,k-1), gel(x,k)));
495 80255 : return S;
496 : }
497 : static GEN
498 253855 : tracerel(GEN a, GEN v, GEN z)
499 : {
500 253855 : a = liftpol_shallow(a);
501 253855 : a = simplify_shallow(z? gmul(z,a): a);
502 253855 : if (typ(a) == t_POL)
503 : {
504 80255 : GEN T = gel(v,3);
505 80255 : long degrel = itou(gel(T,1));
506 80255 : a = tracerel_i(T, RgX_rem(a, gel(v,2)));
507 80255 : if (degrel != 1) a = gdivgu(a, degrel);
508 80255 : if (typ(a) == t_POL) a = RgX_rem(a, gel(v,1));
509 : }
510 253855 : return a;
511 : }
512 : static GEN
513 6944 : tracerel_z(GEN v, long t)
514 : {
515 6944 : GEN Pn = gel(v,2);
516 6944 : return t? pol_xn(t, varn(Pn)): NULL;
517 : }
518 : /* v = Qab_trace_init(n,m); x is a t_VEC of polmodulo Phi_n; Kn = Q(zeta_n)
519 : * [Kn:Km]^(-1) Tr_{Kn/Km} (zeta_n^t * x); 0 <= t < [Kn:Km] */
520 : GEN
521 0 : Qab_tracerel(GEN v, long t, GEN a)
522 : {
523 0 : if (lg(v) != 4) return a; /* => t = 0 */
524 0 : return tracerel(a, v, tracerel_z(v, t));
525 : }
526 : GEN
527 16170 : QabV_tracerel(GEN v, long t, GEN x)
528 : {
529 : GEN z;
530 16170 : if (lg(v) != 4) return x; /* => t = 0 */
531 6944 : z = tracerel_z(v, t);
532 260799 : pari_APPLY_same(tracerel(gel(x,i), v, z));
533 : }
534 : GEN
535 147 : QabM_tracerel(GEN v, long t, GEN x)
536 : {
537 147 : if (lg(v) != 4) return x;
538 105 : pari_APPLY_same(QabV_tracerel(v, t, gel(x,i)));
539 : }
540 :
541 : /* C*zeta_o^k mod X^o - 1 */
542 : static GEN
543 2247504 : Qab_Czeta(long k, long o, GEN C, long vt)
544 : {
545 2247504 : if (!k) return C;
546 1485582 : if (!odd(o))
547 : { /* optimization: reduce max degree by a factor 2 for free */
548 1434475 : o >>= 1;
549 1434475 : if (k >= o) { k -= o; C = gneg(C); if (!k) return C; }
550 : }
551 1137486 : return monomial(C, k, vt);
552 : }
553 : /* zeta_o^k */
554 : static GEN
555 200753 : Qab_zeta(long k, long o, long vt) { return Qab_Czeta(k, o, gen_1, vt); }
556 :
557 : /* Operations on Dirichlet characters */
558 :
559 : /* A Dirichlet character can be given in GP in different formats, but in this
560 : * package, it will be a vector CHI=[G,chi,ord], where G is the (Z/MZ)^* to
561 : * which the character belongs, chi is the character in Conrey format, ord is
562 : * the order */
563 :
564 : static GEN
565 3874416 : gmfcharorder(GEN CHI) { return gel(CHI, 3); }
566 : long
567 3816281 : mfcharorder(GEN CHI) { return itou(gmfcharorder(CHI)); }
568 : static long
569 2709 : mfcharistrivial(GEN CHI) { return !CHI || mfcharorder(CHI) == 1; }
570 : static GEN
571 1616874 : gmfcharmodulus(GEN CHI) { return gmael3(CHI, 1, 1, 1); }
572 : long
573 1616874 : mfcharmodulus(GEN CHI) { return itou(gmfcharmodulus(CHI)); }
574 : GEN
575 598843 : mfcharpol(GEN CHI) { return gel(CHI,4); }
576 :
577 : /* vz[i+1] = image of (zeta_o)^i in Fp */
578 : static ulong
579 313040 : Qab_Czeta_Fl(long k, GEN vz, ulong C, ulong p)
580 : {
581 : long o;
582 313040 : if (!k) return C;
583 205982 : o = lg(vz)-2;
584 205982 : if ((k << 1) == o) return Fl_neg(C,p);
585 179053 : return Fl_mul(C, vz[k+1], p);
586 : }
587 :
588 : static long
589 2556092 : znchareval_i(GEN CHI, long n, GEN ord)
590 2556092 : { return itos(znchareval(gel(CHI,1), gel(CHI,2), stoi(n), ord)); }
591 :
592 : /* n coprime with the modulus of CHI */
593 : static GEN
594 14238 : mfchareval(GEN CHI, long n)
595 : {
596 14238 : GEN Pn, C, go = gmfcharorder(CHI);
597 14238 : long k, o = go[2];
598 14238 : if (o == 1) return gen_1;
599 7385 : k = znchareval_i(CHI, n, go);
600 7385 : Pn = mfcharpol(CHI);
601 7385 : C = Qab_zeta(k, o, varn(Pn));
602 7385 : if (typ(C) != t_POL) return C;
603 5327 : return gmodulo(C, Pn);
604 : }
605 : /* d a multiple of ord(CHI); n coprime with char modulus;
606 : * return x s.t. CHI(n) = \zeta_d^x] */
607 : static long
608 3674594 : mfcharevalord(GEN CHI, long n, long d)
609 : {
610 3674594 : if (mfcharorder(CHI) == 1) return 0;
611 2545011 : return znchareval_i(CHI, n, utoi(d));
612 : }
613 :
614 : /* G a znstar, L a Conrey log: return a 'mfchar' */
615 : static GEN
616 378490 : mfcharGL(GEN G, GEN L)
617 : {
618 378490 : GEN o = zncharorder(G,L);
619 378490 : long ord = itou(o), vt = fetch_user_var("t");
620 378490 : return mkvec4(G, L, o, polcyclo(ord,vt));
621 : }
622 : static GEN
623 5824 : mfchartrivial()
624 5824 : { return mfcharGL(znstar0(gen_1,1), cgetg(1,t_COL)); }
625 : /* convert a generic character into an 'mfchar' */
626 : static GEN
627 4046 : get_mfchar(GEN CHI)
628 : {
629 : GEN G, L;
630 4046 : if (typ(CHI) != t_VEC) CHI = znchar(CHI);
631 : else
632 : {
633 889 : long l = lg(CHI);
634 889 : if ((l != 3 && l != 5) || !checkznstar_i(gel(CHI,1)))
635 7 : pari_err_TYPE("checkNF [chi]", CHI);
636 882 : if (l == 5) return CHI;
637 : }
638 3976 : G = gel(CHI,1);
639 3976 : L = gel(CHI,2); if (typ(L) != t_COL) L = znconreylog(G,L);
640 3976 : return mfcharGL(G, L);
641 : }
642 :
643 : /* parse [N], [N,k], [N,k,CHI]. If 'joker' is set, allow wildcard for CHI */
644 : static GEN
645 9233 : checkCHI(GEN NK, long N, int joker)
646 : {
647 : GEN CHI;
648 9233 : if (lg(NK) == 3)
649 728 : CHI = mfchartrivial();
650 : else
651 : {
652 : long i, l;
653 8505 : CHI = gel(NK,3); l = lg(CHI);
654 8505 : if (isintzero(CHI) && joker)
655 4116 : CHI = NULL; /* all character orbits */
656 4389 : else if (isintm1(CHI) && joker > 1)
657 2373 : CHI = gen_m1; /* sum over all character orbits */
658 2016 : else if ((typ(CHI) == t_VEC &&
659 217 : (l == 1 || l != 3 || !checkznstar_i(gel(CHI,1)))) && joker)
660 : {
661 133 : CHI = shallowtrans(CHI); /* list of characters */
662 952 : for (i = 1; i < l; i++) gel(CHI,i) = get_mfchar(gel(CHI,i));
663 : }
664 : else
665 : {
666 1883 : CHI = get_mfchar(CHI); /* single char */
667 1883 : if (N % mfcharmodulus(CHI)) pari_err_TYPE("checkNF [chi]", NK);
668 : }
669 : }
670 9219 : return CHI;
671 : }
672 : /* support half-integral weight */
673 : static void
674 9240 : checkNK2(GEN NK, long *N, long *nk, long *dk, GEN *CHI, int joker)
675 : {
676 9240 : long l = lg(NK);
677 : GEN T;
678 9240 : if (typ(NK) != t_VEC || l < 3 || l > 4) pari_err_TYPE("checkNK", NK);
679 9240 : T = gel(NK,1); if (typ(T) != t_INT) pari_err_TYPE("checkNF [N]", NK);
680 9240 : *N = itos(T); if (*N <= 0) pari_err_TYPE("checkNF [N <= 0]", NK);
681 9240 : T = gel(NK,2);
682 9240 : switch(typ(T))
683 : {
684 5859 : case t_INT: *nk = itos(T); *dk = 1; break;
685 3374 : case t_FRAC:
686 3374 : *nk = itos(gel(T,1));
687 3374 : *dk = itou(gel(T,2)); if (*dk == 2) break;
688 7 : default: pari_err_TYPE("checkNF [k]", NK);
689 : }
690 9233 : *CHI = checkCHI(NK, *N, joker);
691 9219 : }
692 : /* don't support half-integral weight */
693 : static void
694 133 : checkNK(GEN NK, long *N, long *k, GEN *CHI, int joker)
695 : {
696 : long d;
697 133 : checkNK2(NK, N, k, &d, CHI, joker);
698 133 : if (d != 1) pari_err_TYPE("checkNF [k]", NK);
699 133 : }
700 :
701 : static GEN
702 4872 : mfchargalois(long N, int odd, GEN flagorder)
703 : {
704 4872 : GEN G = znstar0(utoi(N), 1), L = chargalois(G, flagorder);
705 4872 : long l = lg(L), i, j;
706 113526 : for (i = j = 1; i < l; i++)
707 : {
708 108654 : GEN chi = znconreyfromchar(G, gel(L,i));
709 108654 : if (zncharisodd(G,chi) == odd) gel(L,j++) = mfcharGL(G,chi);
710 : }
711 4872 : setlg(L, j); return L;
712 : }
713 : /* possible characters for nontrivial S_1(N, chi) */
714 : static GEN
715 1729 : mf1chars(long N, GEN vCHI)
716 : {
717 1729 : if (vCHI) return vCHI; /*do not filter, user knows best*/
718 : /* Tate's theorem */
719 1659 : return mfchargalois(N, 1, uisprime(N)? mkvecsmall2(2,4): NULL);
720 : }
721 : static GEN
722 3255 : mfchars(long N, long k, long dk, GEN vCHI)
723 3255 : { return vCHI? vCHI: mfchargalois(N, (dk == 2)? 0: (k & 1), NULL); }
724 :
725 : /* wrappers from mfchar to znchar */
726 : static long
727 68460 : mfcharparity(GEN CHI)
728 : {
729 68460 : if (!CHI) return 1;
730 68460 : return zncharisodd(gel(CHI,1), gel(CHI,2)) ? -1 : 1;
731 : }
732 : /* if CHI is primitive, return CHI itself, not a copy */
733 : static GEN
734 81676 : mfchartoprimitive(GEN CHI, long *pF)
735 : {
736 : pari_sp av;
737 : GEN chi, F;
738 81676 : if (!CHI) { if (pF) *pF = 1; return mfchartrivial(); }
739 81676 : av = avma; F = znconreyconductor(gel(CHI,1), gel(CHI,2), &chi);
740 81676 : if (typ(F) == t_INT) set_avma(av);
741 : else
742 : {
743 7861 : CHI = leafcopy(CHI);
744 7861 : gel(CHI,1) = znstar0(F, 1);
745 7861 : gel(CHI,2) = chi;
746 : }
747 81676 : if (pF) *pF = mfcharmodulus(CHI);
748 81676 : return CHI;
749 : }
750 : static long
751 397663 : mfcharconductor(GEN CHI)
752 : {
753 397663 : pari_sp av = avma;
754 397663 : GEN res = znconreyconductor(gel(CHI,1), gel(CHI,2), NULL);
755 397663 : if (typ(res) == t_VEC) res = gel(res, 1);
756 397663 : return gc_long(av, itos(res));
757 : }
758 :
759 : /* Operations on mf closures */
760 : static GEN
761 63630 : tagparams(long t, GEN NK) { return mkvec2(mkvecsmall(t), NK); }
762 : static GEN
763 1197 : lfuntag(long t, GEN x) { return mkvec2(mkvecsmall(t), x); }
764 : static GEN
765 56 : tag0(long t, GEN NK) { retmkvec(tagparams(t,NK)); }
766 : static GEN
767 10346 : tag(long t, GEN NK, GEN x) { retmkvec2(tagparams(t,NK), x); }
768 : static GEN
769 36918 : tag2(long t, GEN NK, GEN x, GEN y) { retmkvec3(tagparams(t,NK), x,y); }
770 : static GEN
771 16184 : tag3(long t, GEN NK, GEN x,GEN y,GEN z) { retmkvec4(tagparams(t,NK), x,y,z); }
772 : static GEN
773 0 : tag4(long t, GEN NK, GEN x,GEN y,GEN z,GEN a)
774 0 : { retmkvec5(tagparams(t,NK), x,y,z,a); }
775 : /* is F a "modular form" ? */
776 : int
777 19362 : checkmf_i(GEN F)
778 19362 : { return typ(F) == t_VEC
779 18529 : && lg(F) > 1 && typ(gel(F,1)) == t_VEC
780 13755 : && lg(gel(F,1)) == 3
781 13594 : && typ(gmael(F,1,1)) == t_VECSMALL
782 37891 : && typ(gmael(F,1,2)) == t_VEC; }
783 233905 : long mf_get_type(GEN F) { return gmael(F,1,1)[1]; }
784 185864 : GEN mf_get_gN(GEN F) { return gmael3(F,1,2,1); }
785 140805 : GEN mf_get_gk(GEN F) { return gmael3(F,1,2,2); }
786 : /* k - 1/2, assume k in 1/2 + Z */
787 441 : long mf_get_r(GEN F) { return itou(gel(mf_get_gk(F),1)) >> 1; }
788 120631 : long mf_get_N(GEN F) { return itou(mf_get_gN(F)); }
789 72961 : long mf_get_k(GEN F)
790 : {
791 72961 : GEN gk = mf_get_gk(F);
792 72961 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
793 72961 : return itou(gk);
794 : }
795 62489 : GEN mf_get_CHI(GEN F) { return gmael3(F,1,2,3); }
796 24472 : GEN mf_get_field(GEN F) { return gmael3(F,1,2,4); }
797 19425 : GEN mf_get_NK(GEN F) { return gmael(F,1,2); }
798 : static void
799 588 : mf_setfield(GEN f, GEN P)
800 : {
801 588 : gel(f,1) = leafcopy(gel(f,1));
802 588 : gmael(f,1,2) = leafcopy(gmael(f,1,2));
803 588 : gmael3(f,1,2,4) = P;
804 588 : }
805 :
806 : /* UTILITY FUNCTIONS */
807 : GEN
808 9114 : mftocol(GEN F, long lim, long d)
809 9114 : { GEN c = mfcoefs_i(F, lim, d); settyp(c,t_COL); return c; }
810 : GEN
811 2128 : mfvectomat(GEN vF, long lim, long d)
812 : {
813 2128 : long j, l = lg(vF);
814 2128 : GEN M = cgetg(l, t_MAT);
815 10423 : for (j = 1; j < l; j++) gel(M,j) = mftocol(gel(vF,j), lim, d);
816 2128 : return M;
817 : }
818 :
819 : static GEN
820 4655 : RgV_to_ser_full(GEN x) { return RgV_to_ser(x, 0, lg(x)+1); }
821 : /* TODO: delete */
822 : static GEN
823 665 : mfcoefsser(GEN F, long n) { return RgV_to_ser_full(mfcoefs_i(F,n,1)); }
824 : static GEN
825 833 : sertovecslice(GEN S, long n)
826 : {
827 833 : GEN v = gtovec0(S, -(lg(S) - 2 + valser(S)));
828 833 : long l = lg(v), n2 = n + 2;
829 833 : if (l < n2) pari_err_BUG("sertovecslice [n too large]");
830 833 : return (l == n2)? v: vecslice(v, 1, n2-1);
831 : }
832 :
833 : /* a, b two RgV of the same length, multiply as truncated power series */
834 : static GEN
835 8869 : RgV_mul_RgXn(GEN a, GEN b)
836 : {
837 8869 : long n = lg(a)-1;
838 : GEN c;
839 8869 : a = RgV_to_RgX(a,0);
840 8869 : b = RgV_to_RgX(b,0); c = RgXn_mul(a, b, n);
841 8869 : c = RgX_to_RgC(c,n); settyp(c,t_VEC); return c;
842 : }
843 : /* divide as truncated power series */
844 : static GEN
845 399 : RgV_div_RgXn(GEN a, GEN b)
846 : {
847 399 : long n = lg(a)-1;
848 : GEN c;
849 399 : a = RgV_to_RgX(a,0);
850 399 : b = RgV_to_RgX(b,0); c = RgXn_div_i(a, b, n);
851 399 : c = RgX_to_RgC(c,n); settyp(c,t_VEC); return c;
852 : }
853 : /* a^b */
854 : static GEN
855 112 : RgV_pows_RgXn(GEN a, long b)
856 : {
857 112 : long n = lg(a)-1;
858 : GEN c;
859 112 : a = RgV_to_RgX(a,0);
860 112 : if (b < 0) { a = RgXn_inv(a, n); b = -b; }
861 112 : c = RgXn_powu_i(a,b,n);
862 112 : c = RgX_to_RgC(c,n); settyp(c,t_VEC); return c;
863 : }
864 :
865 : /* assume lg(V) >= n*d + 2 */
866 : static GEN
867 8834 : c_deflate(long n, long d, GEN v)
868 : {
869 8834 : long i, id, l = n+2;
870 : GEN w;
871 8834 : if (d == 1) return lg(v) == l ? v: vecslice(v, 1, l-1);
872 574 : w = cgetg(l, typ(v));
873 11123 : for (i = id = 1; i < l; i++, id += d) gel(w, i) = gel(v, id);
874 574 : return w;
875 : }
876 :
877 : static void
878 14 : err_cyclo(void)
879 14 : { pari_err_IMPL("changing cyclotomic fields in mf"); }
880 : /* Q(zeta_a) = Q(zeta_b) ? */
881 : static int
882 616 : same_cyc(long a, long b)
883 616 : { return (a == b) || (odd(a) && b == (a<<1)) || (odd(b) && a == (b<<1)); }
884 : /* need to combine elements in Q(CHI1) and Q(CHI2) with result in Q(CHI),
885 : * CHI = CHI1 * CHI2 or CHI / CHI2 times some character of order 2 */
886 : static GEN
887 2723 : chicompat(GEN CHI, GEN CHI1, GEN CHI2)
888 : {
889 2723 : long o1 = mfcharorder(CHI1);
890 2723 : long o2 = mfcharorder(CHI2), O, o;
891 : GEN T1, T2, P, Po;
892 2723 : if (o1 <= 2 && o2 <= 2) return NULL;
893 623 : o = mfcharorder(CHI);
894 623 : Po = mfcharpol(CHI);
895 623 : P = mfcharpol(CHI1);
896 623 : if (o1 == o2)
897 : {
898 21 : if (o1 == o) return NULL;
899 14 : if (!same_cyc(o1,o)) err_cyclo();
900 0 : return mkvec4(P, gen_1,gen_1, Qab_trace_init(o1, o, P, Po));
901 : }
902 602 : O = ulcm(o1, o2);
903 602 : if (!same_cyc(O,o)) err_cyclo();
904 602 : if (O != o1) P = (O == o2)? mfcharpol(CHI2): polcyclo(O, varn(P));
905 602 : T1 = o1 <= 2? gen_1: utoipos(O / o1);
906 602 : T2 = o2 <= 2? gen_1: utoipos(O / o2);
907 602 : return mkvec4(P, T1, T2, O == o? gen_1: Qab_trace_init(O, o, P, Po));
908 : }
909 : static GEN
910 49 : inflatemod(GEN f, long o, GEN P)
911 : {
912 49 : f = lift_shallow(f);
913 49 : return gmodulo(typ(f)==t_POL? RgX_inflate(f,o): f, P);
914 : }
915 : static GEN
916 7 : RgV_inflatemod(GEN x, long o, GEN P)
917 56 : { pari_APPLY_same(inflatemod(gel(x,i), o, P)); }
918 : /* *F a vector of cyclotomic numbers */
919 : static void
920 651 : chicompatlift(GEN T, GEN *F, GEN *G)
921 : {
922 651 : long o1 = itou(gel(T,2)), o2 = itou(gel(T,3));
923 651 : GEN P = gel(T,1);
924 651 : if (o1 != 1) *F = RgV_inflatemod(*F, o1, P);
925 651 : if (o2 != 1 && G) *G = RgV_inflatemod(*G, o2, P);
926 651 : }
927 : static GEN
928 651 : chicompatfix(GEN T, GEN F)
929 : {
930 651 : GEN V = gel(T,4);
931 651 : if (typ(V) == t_VEC) F = gmodulo(QabV_tracerel(V, 0, F), gel(V,1));
932 651 : return F;
933 : }
934 :
935 : static GEN
936 637 : c_mul(long n, long d, GEN S)
937 : {
938 637 : pari_sp av = avma;
939 637 : long nd = n*d;
940 637 : GEN F = gel(S,2), G = gel(S,3);
941 637 : F = mfcoefs_i(F, nd, 1);
942 637 : G = mfcoefs_i(G, nd, 1);
943 637 : if (lg(S) == 5) chicompatlift(gel(S,4),&F,&G);
944 637 : F = c_deflate(n, d, RgV_mul_RgXn(F,G));
945 637 : if (lg(S) == 5) F = chicompatfix(gel(S,4), F);
946 637 : return gc_GEN(av, F);
947 : }
948 : static GEN
949 112 : c_pow(long n, long d, GEN S)
950 : {
951 112 : pari_sp av = avma;
952 112 : long nd = n*d;
953 112 : GEN F = gel(S,2), a = gel(S,3), f = mfcoefs_i(F,nd,1);
954 112 : if (lg(S) == 5) chicompatlift(gel(S,4),&F, NULL);
955 112 : f = RgV_pows_RgXn(f, itos(a));
956 112 : f = c_deflate(n, d, f);
957 112 : if (lg(S) == 5) f = chicompatfix(gel(S,4), f);
958 112 : return gc_GEN(av, f);
959 : }
960 :
961 : /* F * Theta */
962 : static GEN
963 448 : mfmultheta(GEN F)
964 : {
965 448 : if (typ(mf_get_gk(F)) == t_FRAC && mf_get_type(F) == t_MF_DIV)
966 : {
967 154 : GEN T = gel(F,3); /* hopefully mfTheta() */
968 154 : if (mf_get_type(T) == t_MF_THETA && mf_get_N(T) == 4) return gel(F,2);
969 : }
970 294 : return mfmul(F, mfTheta(NULL));
971 : }
972 :
973 : static GEN
974 42 : c_bracket(long n, long d, GEN S)
975 : {
976 42 : pari_sp av = avma;
977 42 : long i, nd = n*d;
978 42 : GEN F = gel(S,2), G = gel(S,3), tF, tG, C, mpow, res, gk, gl;
979 42 : GEN VF = mfcoefs_i(F, nd, 1);
980 42 : GEN VG = mfcoefs_i(G, nd, 1);
981 42 : ulong j, m = itou(gel(S,4));
982 :
983 42 : if (!n)
984 : {
985 14 : if (m > 0) { set_avma(av); return mkvec(gen_0); }
986 7 : return gc_GEN(av, mkvec(gmul(gel(VF, 1), gel(VG, 1))));
987 : }
988 28 : tF = cgetg(nd+2, t_VEC);
989 28 : tG = cgetg(nd+2, t_VEC);
990 28 : res = NULL; gk = mf_get_gk(F); gl = mf_get_gk(G);
991 : /* pow[i,j+1] = i^j */
992 28 : if (lg(S) == 6) chicompatlift(gel(S,5),&VF,&VG);
993 28 : mpow = cgetg(m+2, t_MAT);
994 28 : gel(mpow,1) = const_col(nd, gen_1);
995 56 : for (j = 1; j <= m; j++)
996 : {
997 28 : GEN c = cgetg(nd+1, t_COL);
998 28 : gel(mpow,j+1) = c;
999 245 : for (i = 1; i <= nd; i++) gel(c,i) = muliu(gcoeff(mpow,i,j), i);
1000 : }
1001 28 : C = binomial(gaddgs(gk, m-1), m);
1002 28 : if (odd(m)) C = gneg(C);
1003 84 : for (j = 0; j <= m; j++)
1004 : { /* C = (-1)^(m-j) binom(m+l-1, j) binom(m+k-1,m-j) */
1005 : GEN c;
1006 56 : gel(tF,1) = j == 0? gel(VF,1): gen_0;
1007 56 : gel(tG,1) = j == m? gel(VG,1): gen_0;
1008 56 : gel(tF,2) = gel(VF,2); /* assume nd >= 1 */
1009 56 : gel(tG,2) = gel(VG,2);
1010 518 : for (i = 2; i <= nd; i++)
1011 : {
1012 462 : gel(tF, i+1) = gmul(gcoeff(mpow,i,j+1), gel(VF, i+1));
1013 462 : gel(tG, i+1) = gmul(gcoeff(mpow,i,m-j+1), gel(VG, i+1));
1014 : }
1015 56 : c = gmul(C, c_deflate(n, d, RgV_mul_RgXn(tF, tG)));
1016 56 : res = res? gadd(res, c): c;
1017 56 : if (j < m)
1018 56 : C = gdiv(gmul(C, gmulsg(m-j, gaddgs(gl,m-j-1))),
1019 28 : gmulsg(-(j+1), gaddgs(gk,j)));
1020 : }
1021 28 : if (lg(S) == 6) res = chicompatfix(gel(S,5), res);
1022 28 : return gc_upto(av, res);
1023 : }
1024 : /* linear combination \sum L[j] vecF[j] */
1025 : static GEN
1026 3010 : c_linear(long n, long d, GEN F, GEN L, GEN dL)
1027 : {
1028 3010 : pari_sp av = avma;
1029 3010 : long j, l = lg(L);
1030 3010 : GEN S = NULL;
1031 10738 : for (j = 1; j < l; j++)
1032 : {
1033 7728 : GEN c = gel(L,j);
1034 7728 : if (gequal0(c)) continue;
1035 6972 : c = gmul(c, mfcoefs_i(gel(F,j), n, d));
1036 6972 : S = S? gadd(S,c): c;
1037 : }
1038 3010 : if (!S) return zerovec(n+1);
1039 3010 : if (!is_pm1(dL)) S = gdiv(S, dL);
1040 3010 : return gc_upto(av, S);
1041 : }
1042 :
1043 : /* B_d(T_j Trace^new) as t_MF_BD(t_MF_HECKE(t_MF_NEWTRACE)) or
1044 : * t_MF_HECKE(t_MF_NEWTRACE)
1045 : * or t_MF_NEWTRACE in level N. Set d and j, return t_MF_NEWTRACE component*/
1046 : static GEN
1047 83601 : bhn_parse(GEN f, long *d, long *j)
1048 : {
1049 83601 : long t = mf_get_type(f);
1050 83601 : *d = *j = 1;
1051 83601 : if (t == t_MF_BD) { *d = itos(gel(f,3)); f = gel(f,2); t = mf_get_type(f); }
1052 83601 : if (t == t_MF_HECKE) { *j = gel(f,2)[1]; f = gel(f,3); }
1053 83601 : return f;
1054 : }
1055 : /* f as above, return the t_MF_NEWTRACE component */
1056 : static GEN
1057 32648 : bhn_newtrace(GEN f)
1058 : {
1059 32648 : long t = mf_get_type(f);
1060 32648 : if (t == t_MF_BD) { f = gel(f,2); t = mf_get_type(f); }
1061 32648 : if (t == t_MF_HECKE) f = gel(f,3);
1062 32648 : return f;
1063 : }
1064 : static int
1065 4004 : ok_bhn_linear(GEN vf)
1066 : {
1067 4004 : long i, N0 = 0, l = lg(vf);
1068 : GEN CHI, gk;
1069 4004 : if (l == 1) return 1;
1070 4004 : gk = mf_get_gk(gel(vf,1));
1071 4004 : CHI = mf_get_CHI(gel(vf,1));
1072 27377 : for (i = 1; i < l; i++)
1073 : {
1074 25718 : GEN f = bhn_newtrace(gel(vf,i));
1075 25718 : long N = mf_get_N(f);
1076 25718 : if (mf_get_type(f) != t_MF_NEWTRACE) return 0;
1077 23373 : if (N < N0) return 0; /* largest level must come last */
1078 23373 : N0 = N;
1079 23373 : if (!gequal(gk,mf_get_gk(f))) return 0; /* same k */
1080 23373 : if (!gequal(gel(mf_get_CHI(f),2), gel(CHI,2))) return 0; /* same CHI */
1081 : }
1082 1659 : return 1;
1083 : }
1084 :
1085 : /* vF not empty, same hypotheses as bhnmat_extend */
1086 : static GEN
1087 7035 : bhnmat_extend_nocache(GEN M, long N, long n, long d, GEN vF)
1088 : {
1089 : cachenew_t cache;
1090 7035 : long l = lg(vF);
1091 : GEN f;
1092 7035 : if (l == 1) return M? M: cgetg(1, t_MAT);
1093 6930 : f = bhn_newtrace(gel(vF,1)); /* N.B. mf_get_N(f) divides N */
1094 6930 : init_cachenew(&cache, n*d, N, f);
1095 6930 : M = bhnmat_extend(M, n, d, vF, &cache);
1096 6930 : dbg_cachenew(&cache); return M;
1097 : }
1098 : /* c_linear of "bhn" mf closures, same hypotheses as bhnmat_extend */
1099 : static GEN
1100 2303 : c_linear_bhn(long n, long d, GEN F)
1101 : {
1102 : pari_sp av;
1103 2303 : GEN M, v, vF = gel(F,2), L = gel(F,3), dL = gel(F,4);
1104 2303 : if (lg(L) == 1) return zerovec(n+1);
1105 2303 : av = avma;
1106 2303 : M = bhnmat_extend_nocache(NULL, mf_get_N(F), n, d, vF);
1107 2303 : v = RgM_RgC_mul(M,L); settyp(v, t_VEC);
1108 2303 : if (!is_pm1(dL)) v = gdiv(v, dL);
1109 2303 : return gc_upto(av, v);
1110 : }
1111 :
1112 : /* c in K, K := Q[X]/(T) vz = vector of consecutive powers of root z of T
1113 : * attached to an embedding s: K -> C. Return s(c) in C */
1114 : static GEN
1115 84658 : Rg_embed1(GEN c, GEN vz)
1116 : {
1117 84658 : long t = typ(c);
1118 84658 : if (t == t_POLMOD) { c = gel(c,2); t = typ(c); }
1119 84658 : if (t == t_POL) c = RgX_RgV_eval(c, vz);
1120 84658 : return c;
1121 : }
1122 : /* return s(x) in C[X] */
1123 : static GEN
1124 14203 : RgX_embed1(GEN x, GEN vz)
1125 42042 : { pari_APPLY_pol(Rg_embed1(gel(x,i), vz)); }
1126 : /* return s(x) in C^n */
1127 : static GEN
1128 798 : vecembed1(GEN x, GEN vz)
1129 39858 : { pari_APPLY_same(Rg_embed1(gel(x,i), vz)); }
1130 : /* P in L = K[X]/(U), K = Q[t]/T; s an embedding of K -> C attached
1131 : * to a root of T, extended to an embedding of L -> C attached to a root
1132 : * of s(U); vT powers of the root of T, vU powers of the root of s(U).
1133 : * Return s(P) in C^n */
1134 : static GEN
1135 13328 : Rg_embed2(GEN P, long vt, GEN vT, GEN vU)
1136 : {
1137 13328 : P = liftpol_shallow(P);
1138 13328 : if (typ(P) != t_POL) return P;
1139 13300 : if (varn(P) == vt) return Rg_embed1(P, vT);
1140 13293 : return Rg_embed1(RgX_embed1(P, vT), vU); /* varn(P) == vx */
1141 : }
1142 : static GEN
1143 42 : vecembed2(GEN x, long vt, GEN vT, GEN vU)
1144 1050 : { pari_APPLY_same(Rg_embed2(gel(x,i), vt, vT, vU)); }
1145 : static GEN
1146 532 : RgX_embed2(GEN x, long vt, GEN vT, GEN vU)
1147 3724 : { pari_APPLY_pol(Rg_embed2(gel(x,i), vt, vT, vU)); }
1148 : /* embed polynomial f in variable 0 [ may be a scalar ], E from getembed */
1149 : static GEN
1150 1687 : RgX_embed(GEN f, GEN E)
1151 : {
1152 : GEN vT;
1153 1687 : if (typ(f) != t_POL || varn(f) != 0) return mfembed(E, f);
1154 1645 : if (lg(E) == 1) return f;
1155 1407 : vT = gel(E,2);
1156 1407 : if (lg(E) == 3)
1157 875 : f = RgX_embed1(f, vT);
1158 : else
1159 532 : f = RgX_embed2(f, varn(gel(E,1)), vT, gel(E,3));
1160 1407 : return f;
1161 : }
1162 : /* embed vector, E from getembed */
1163 : GEN
1164 1743 : mfvecembed(GEN E, GEN v)
1165 : {
1166 : GEN vT;
1167 1743 : if (lg(E) == 1) return v;
1168 840 : vT = gel(E,2);
1169 840 : if (lg(E) == 3)
1170 798 : v = vecembed1(v, vT);
1171 : else
1172 42 : v = vecembed2(v, varn(gel(E,1)), vT, gel(E,3));
1173 840 : return v;
1174 : }
1175 : GEN
1176 70 : mfmatembed(GEN E, GEN x)
1177 : {
1178 70 : if (lg(E) == 1) return x;
1179 168 : pari_APPLY_same(mfvecembed(E, gel(x,i)));
1180 : }
1181 : /* embed vector of polynomials in var 0 */
1182 : static GEN
1183 98 : RgXV_embed(GEN x, GEN E)
1184 : {
1185 98 : if (lg(E) == 1) return x;
1186 1358 : pari_APPLY_same(RgX_embed(gel(x,i), E));
1187 : }
1188 :
1189 : /* embed scalar */
1190 : GEN
1191 100845 : mfembed(GEN E, GEN f)
1192 : {
1193 : GEN vT;
1194 100845 : if (lg(E) == 1) return f;
1195 13587 : vT = gel(E,2);
1196 13587 : if (lg(E) == 3)
1197 4459 : f = Rg_embed1(f, vT);
1198 : else
1199 9128 : f = Rg_embed2(f, varn(gel(E,1)), vT, gel(E,3));
1200 13587 : return f;
1201 : }
1202 : /* vector of the sigma(f), sigma in vE */
1203 : static GEN
1204 364 : RgX_embedall(GEN f, GEN vE)
1205 : {
1206 364 : long i, l = lg(vE);
1207 : GEN v;
1208 364 : if (l == 2) return RgX_embed(f, gel(vE,1));
1209 35 : v = cgetg(l, t_VEC);
1210 105 : for (i = 1; i < l; i++) gel(v,i) = RgX_embed(f, gel(vE,i));
1211 35 : return v;
1212 : }
1213 : /* matrix whose colums are the sigma(v), sigma in vE */
1214 : static GEN
1215 350 : RgC_embedall(GEN v, GEN vE)
1216 : {
1217 350 : long j, l = lg(vE);
1218 350 : GEN M = cgetg(l, t_MAT);
1219 875 : for (j = 1; j < l; j++) gel(M,j) = mfvecembed(gel(vE,j), v);
1220 350 : return M;
1221 : }
1222 : /* vector of the sigma(v), sigma in vE */
1223 : static GEN
1224 4907 : Rg_embedall_i(GEN v, GEN vE)
1225 : {
1226 4907 : long j, l = lg(vE);
1227 4907 : GEN M = cgetg(l, t_VEC);
1228 14735 : for (j = 1; j < l; j++) gel(M,j) = mfembed(gel(vE,j), v);
1229 4907 : return M;
1230 : }
1231 : /* vector of the sigma(v), sigma in vE; if #vE == 1, return v */
1232 : static GEN
1233 95154 : Rg_embedall(GEN v, GEN vE)
1234 95154 : { return (lg(vE) == 2)? mfembed(gel(vE,1), v): Rg_embedall_i(v, vE); }
1235 :
1236 : static GEN
1237 833 : c_div_i(long n, GEN S)
1238 : {
1239 833 : GEN F = gel(S,2), G = gel(S,3);
1240 : GEN a0, a0i, H;
1241 833 : F = mfcoefs_i(F, n, 1);
1242 833 : G = mfcoefs_i(G, n, 1);
1243 833 : if (lg(S) == 5) chicompatlift(gel(S,4),&F,&G);
1244 833 : F = RgV_to_ser_full(F);
1245 833 : G = RgV_to_ser_full(G);
1246 833 : a0 = polcoef_i(G, 0, -1); /* != 0 */
1247 833 : if (gequal1(a0)) a0 = a0i = NULL;
1248 : else
1249 : {
1250 602 : a0i = ginv(a0);
1251 602 : G = gmul(ser_unscale(G,a0), a0i);
1252 602 : F = gmul(ser_unscale(F,a0), a0i);
1253 : }
1254 833 : H = gdiv(F, G);
1255 833 : if (a0) H = ser_unscale(H,a0i);
1256 833 : H = sertovecslice(H, n);
1257 833 : if (lg(S) == 5) H = chicompatfix(gel(S,4), H);
1258 833 : return H;
1259 : }
1260 : static GEN
1261 833 : c_div(long n, long d, GEN S)
1262 : {
1263 833 : pari_sp av = avma;
1264 833 : GEN D = (d==1)? c_div_i(n, S): c_deflate(n, d, c_div_i(n*d, S));
1265 833 : return gc_GEN(av, D);
1266 : }
1267 :
1268 : static GEN
1269 35 : c_shift(long n, long d, GEN F, GEN gsh)
1270 : {
1271 35 : pari_sp av = avma;
1272 : GEN vF;
1273 35 : long sh = itos(gsh), n1 = n*d + sh;
1274 35 : if (n1 < 0) return zerovec(n+1);
1275 35 : vF = mfcoefs_i(F, n1, 1);
1276 35 : if (sh < 0) vF = shallowconcat(zerovec(-sh), vF);
1277 35 : else vF = vecslice(vF, sh+1, n1+1);
1278 35 : return gc_GEN(av, c_deflate(n, d, vF));
1279 : }
1280 :
1281 : static GEN
1282 175 : c_deriv(long n, long d, GEN F, GEN gm)
1283 : {
1284 175 : pari_sp av = avma;
1285 175 : GEN V = mfcoefs_i(F, n, d), res;
1286 175 : long i, m = itos(gm);
1287 175 : if (!m) return V;
1288 175 : res = cgetg(n+2, t_VEC); gel(res,1) = gen_0;
1289 175 : if (m < 0)
1290 49 : { for (i=1; i <= n; i++) gel(res, i+1) = gdiv(gel(V, i+1), powuu(i,-m)); }
1291 : else
1292 2457 : { for (i=1; i <= n; i++) gel(res, i+1) = gmul(gel(V,i+1), powuu(i,m)); }
1293 175 : return gc_upto(av, res);
1294 : }
1295 :
1296 : static GEN
1297 14 : c_derivE2(long n, long d, GEN F, GEN gm)
1298 : {
1299 14 : pari_sp av = avma;
1300 : GEN VF, VE, res, tmp, gk;
1301 14 : long i, m = itos(gm), nd;
1302 14 : if (m == 0) return mfcoefs_i(F, n, d);
1303 14 : nd = n*d;
1304 14 : VF = mfcoefs_i(F, nd, 1); VE = mfcoefs_i(mfEk(2), nd, 1);
1305 14 : gk = mf_get_gk(F);
1306 14 : if (m == 1)
1307 : {
1308 7 : res = cgetg(n+2, t_VEC);
1309 56 : for (i = 0; i <= n; i++) gel(res, i+1) = gmulsg(i, gel(VF, i*d+1));
1310 7 : tmp = c_deflate(n, d, RgV_mul_RgXn(VF, VE));
1311 7 : return gc_upto(av, gsub(res, gmul(gdivgu(gk, 12), tmp)));
1312 : }
1313 : else
1314 : {
1315 : long j;
1316 35 : for (j = 1; j <= m; j++)
1317 : {
1318 28 : tmp = RgV_mul_RgXn(VF, VE);
1319 140 : for (i = 0; i <= nd; i++) gel(VF, i+1) = gmulsg(i, gel(VF, i+1));
1320 28 : VF = gsub(VF, gmul(gdivgu(gaddgs(gk, 2*(j-1)), 12), tmp));
1321 : }
1322 7 : return gc_GEN(av, c_deflate(n, d, VF));
1323 : }
1324 : }
1325 :
1326 : /* Twist by the character (D/.) */
1327 : static GEN
1328 168 : c_twist(long n, long d, GEN F, GEN D)
1329 : {
1330 168 : pari_sp av = avma;
1331 168 : GEN v = mfcoefs_i(F, n, d), z = cgetg(n+2, t_VEC);
1332 : long i;
1333 994 : for (i = 0; i <= n; i++)
1334 : {
1335 : long s;
1336 826 : GEN a = gel(v, i+1);
1337 826 : if (d == 1) s = krois(D, i);
1338 : else
1339 : {
1340 266 : pari_sp av2 = avma;
1341 266 : s = kronecker(D, muluu(i, d)); set_avma(av2);
1342 : }
1343 826 : switch(s)
1344 : {
1345 259 : case 1: a = gcopy(a); break;
1346 252 : case -1: a = gneg(a); break;
1347 315 : default: a = gen_0; break;
1348 : }
1349 826 : gel(z, i+1) = a;
1350 : }
1351 168 : return gc_upto(av, z);
1352 : }
1353 :
1354 : /* form F given by closure, compute T(n)(F) as closure */
1355 : static GEN
1356 1232 : c_hecke(long m, long l, GEN DATA, GEN F)
1357 : {
1358 1232 : pari_sp av = avma;
1359 1232 : return gc_GEN(av, hecke_i(m, l, NULL, F, DATA));
1360 : }
1361 : static GEN
1362 140 : c_const(long n, long d, GEN C)
1363 : {
1364 140 : GEN V = zerovec(n+1);
1365 140 : long i, j, l = lg(C);
1366 140 : if (l > d*n+2) l = d*n+2;
1367 189 : for (i = j = 1; i < l; i+=d, j++) gel(V, j) = gcopy(gel(C,i));
1368 140 : return V;
1369 : }
1370 :
1371 : /* m > 0 */
1372 : static GEN
1373 525 : eta3_ZXn(long m)
1374 : {
1375 525 : long l = m+2, n, k;
1376 525 : GEN P = cgetg(l,t_POL);
1377 525 : P[1] = evalsigne(1)|evalvarn(0);
1378 7245 : for (n = 2; n < l; n++) gel(P,n) = gen_0;
1379 525 : for (n = k = 0;; n++)
1380 : {
1381 2891 : if (k + n >= m) { setlg(P, k+3); return P; }
1382 2366 : k += n;
1383 : /* now k = n(n+1) / 2 */
1384 2366 : gel(P, k+2) = odd(n)? utoineg(2*n+1): utoipos(2*n+1);
1385 : }
1386 : }
1387 :
1388 : static GEN
1389 539 : c_delta(long n, long d)
1390 : {
1391 539 : pari_sp ltop = avma;
1392 539 : long N = n*d;
1393 : GEN e;
1394 539 : if (!N) return mkvec(gen_0);
1395 525 : e = eta3_ZXn(N);
1396 525 : e = ZXn_sqr(e,N);
1397 525 : e = ZXn_sqr(e,N);
1398 525 : e = ZXn_sqr(e,N); /* eta(x)^24 */
1399 525 : settyp(e, t_VEC);
1400 525 : gel(e,1) = gen_0; /* Delta(x) = x*eta(x)^24 as a t_VEC */
1401 525 : return gc_GEN(ltop, c_deflate(n, d, e));
1402 : }
1403 :
1404 : /* return s(d) such that s|f <=> d | f^2 */
1405 : static long
1406 56 : mysqrtu(ulong d)
1407 : {
1408 56 : GEN fa = myfactoru(d), P = gel(fa,1), E = gel(fa,2);
1409 56 : long l = lg(P), i, s = 1;
1410 140 : for (i = 1; i < l; i++) s *= upowuu(P[i], (E[i]+1)>>1);
1411 56 : return s;
1412 : }
1413 : static GEN
1414 1911 : c_theta(long n, long d, GEN psi)
1415 : {
1416 1911 : long lim = usqrt(n*d), F = mfcharmodulus(psi), par = mfcharparity(psi);
1417 1911 : long f, d2 = d == 1? 1: mysqrtu(d);
1418 1911 : GEN V = zerovec(n + 1);
1419 8414 : for (f = d2; f <= lim; f += d2)
1420 6503 : if (ugcd(F, f) == 1)
1421 : {
1422 6496 : pari_sp av = avma;
1423 6496 : GEN c = mfchareval(psi, f);
1424 6496 : gel(V, f*f/d + 1) = gc_upto(av, par < 0? gmulgu(c,2*f): gmul2n(c,1));
1425 : }
1426 1911 : if (F == 1) gel(V, 1) = gen_1;
1427 1911 : return V; /* no GC needed */
1428 : }
1429 :
1430 : static GEN
1431 203 : c_etaquo(long n, long d, GEN eta, GEN gs)
1432 : {
1433 203 : pari_sp av = avma;
1434 203 : long s = itos(gs), nd = n*d, nds = nd - s + 1;
1435 : GEN c;
1436 203 : if (nds <= 0) return zerovec(n+1);
1437 182 : c = RgX_to_RgC(eta_product_ZXn(eta, nds), nds); settyp(c, t_VEC);
1438 182 : if (s > 0) c = shallowconcat(zerovec(s), c);
1439 182 : return gc_GEN(av, c_deflate(n, d, c));
1440 : }
1441 :
1442 : static GEN
1443 77 : c_ell(long n, long d, GEN E)
1444 : {
1445 77 : pari_sp av = avma;
1446 : GEN v;
1447 77 : if (d == 1) return gconcat(gen_0, ellan(E, n));
1448 7 : v = vec_prepend(ellan(E, n*d), gen_0);
1449 7 : return gc_GEN(av, c_deflate(n, d, v));
1450 : }
1451 :
1452 : static GEN
1453 21 : c_cusptrace(long n, long d, GEN F)
1454 : {
1455 21 : pari_sp av = avma;
1456 21 : GEN D = gel(F,2), res = cgetg(n+2, t_VEC);
1457 21 : long i, N = mf_get_N(F), k = mf_get_k(F);
1458 21 : gel(res, 1) = gen_0;
1459 140 : for (i = 1; i <= n; i++)
1460 119 : gel(res, i+1) = mfcusptrace_i(N, k, i*d, mydivisorsu(i*d), D);
1461 21 : return gc_GEN(av, res);
1462 : }
1463 :
1464 : static GEN
1465 1918 : c_newtrace(long n, long d, GEN F)
1466 : {
1467 1918 : pari_sp av = avma;
1468 : cachenew_t cache;
1469 1918 : long N = mf_get_N(F);
1470 : GEN v;
1471 1918 : init_cachenew(&cache, n == 1? 1: n*d, N, F);
1472 1918 : v = colnewtrace(0, n, d, N, mf_get_k(F), &cache);
1473 1918 : settyp(v, t_VEC); return gc_GEN(av, v);
1474 : }
1475 :
1476 : static GEN
1477 7504 : c_Bd(long n, long d, GEN F, GEN A)
1478 : {
1479 7504 : pari_sp av = avma;
1480 7504 : long a = itou(A), ad = ugcd(a,d), aad = a/ad, i, j;
1481 7504 : GEN w, v = mfcoefs_i(F, n/aad, d/ad);
1482 7504 : if (a == 1) return v;
1483 7504 : n++; w = zerovec(n);
1484 213220 : for (i = j = 1; j <= n; i++, j += aad) gel(w,j) = gcopy(gel(v,i));
1485 7504 : return gc_upto(av, w);
1486 : }
1487 :
1488 : static GEN
1489 5579 : c_dihedral(long n, long d, GEN F)
1490 : {
1491 5579 : pari_sp av = avma;
1492 5579 : GEN CHI = mf_get_CHI(F);
1493 5579 : GEN w = gel(F,3), V = dihan(gel(F,2), w, gel(F,4), mfcharorder(CHI), n*d);
1494 5579 : GEN Tinit = gel(w,3), Pm = gel(Tinit,1);
1495 5579 : GEN A = c_deflate(n, d, V);
1496 5579 : if (degpol(Pm) == 1 || RgV_is_ZV(A)) return gc_GEN(av, A);
1497 1043 : return gc_upto(av, gmodulo(A, Pm));
1498 : }
1499 :
1500 : static GEN
1501 343 : c_mfEH(long n, long d, GEN F)
1502 : {
1503 343 : pari_sp av = avma;
1504 : GEN v, M, A;
1505 343 : long i, r = mf_get_r(F);
1506 343 : if (n == 1)
1507 14 : return gc_GEN(av, mkvec2(mfEHcoef(r,0),mfEHcoef(r,d)));
1508 : /* speedup mfcoef */
1509 329 : if (r == 1)
1510 : {
1511 70 : v = cgetg(n+2, t_VEC);
1512 70 : gel(v,1) = sstoQ(-1,12);
1513 83258 : for (i = 1; i <= n; i++)
1514 : {
1515 83188 : long id = i*d, a = id & 3;
1516 83188 : gel(v,i+1) = (a==1 || a==2)? gen_0: uutoQ(hclassno6u(id), 6);
1517 : }
1518 70 : return v; /* no GC needed */
1519 : }
1520 259 : M = mfEHmat(n*d+1,r);
1521 259 : if (d > 1)
1522 : {
1523 35 : long l = lg(M);
1524 119 : for (i = 1; i < l; i++) gel(M,i) = c_deflate(n, d, gel(M,i));
1525 : }
1526 259 : A = gel(F,2); /* [num(B), den(B)] */
1527 259 : v = RgC_Rg_div(RgM_RgC_mul(M, gel(A,1)), gel(A,2));
1528 259 : settyp(v,t_VEC); return gc_upto(av, v);
1529 : }
1530 :
1531 : static GEN
1532 11333 : c_mfeisen(long n, long d, GEN F)
1533 : {
1534 11333 : pari_sp av = avma;
1535 11333 : GEN v, vchi, E0, P, T, CHI, gk = mf_get_gk(F);
1536 : long i, k;
1537 11333 : if (typ(gk) != t_INT) return c_mfEH(n, d, F);
1538 10990 : k = itou(gk);
1539 10990 : vchi = gel(F,2);
1540 10990 : E0 = gel(vchi,1);
1541 10990 : T = gel(vchi,2);
1542 10990 : P = gel(T,1);
1543 10990 : CHI = gel(vchi,3);
1544 10990 : v = cgetg(n+2, t_VEC);
1545 10990 : gel(v, 1) = gcopy(E0); /* E(0) */
1546 10990 : if (lg(vchi) == 5)
1547 : { /* E_k(chi1,chi2) */
1548 8897 : GEN CHI2 = gel(vchi,4), F3 = gel(F,3);
1549 8897 : long ord = F3[1], j = F3[2];
1550 509467 : for (i = 1; i <= n; i++) gel(v, i+1) = sigchi2(k, CHI, CHI2, i*d, ord);
1551 8897 : v = QabV_tracerel(T, j, v);
1552 : }
1553 : else
1554 : { /* E_k(chi) */
1555 26285 : for (i = 1; i <= n; i++) gel(v, i+1) = sigchi(k, CHI, i*d);
1556 : }
1557 10990 : if (degpol(P) != 1 && !RgV_is_QV(v)) return gc_upto(av, gmodulo(v, P));
1558 8057 : return gc_GEN(av, v);
1559 : }
1560 :
1561 : /* N^k * (D * B_k)(x/N), set D = denom(B_k) */
1562 : static GEN
1563 2016 : bern_init(long N, long k, GEN *pD)
1564 2016 : { return ZX_rescale(Q_remove_denom(bernpol(k, 0), pD), utoi(N)); }
1565 :
1566 : /* L(chi_D, 1-k) */
1567 : static GEN
1568 28 : lfunquadneg_naive(long D, long k)
1569 : {
1570 : GEN B, dS, S;
1571 28 : long r, N = labs(D);
1572 : pari_sp av;
1573 28 : if (k == 1 && N == 1) return gneg(ghalf);
1574 28 : B = bern_init(N, k, &dS);
1575 28 : dS = mul_denom(dS, stoi(-N*k));
1576 28 : av = avma;
1577 7175 : for (r = 0, S = gen_0; r < N; r++)
1578 : {
1579 7147 : long c = kross(D, r);
1580 7147 : if (c)
1581 : {
1582 5152 : GEN t = ZX_Z_eval(B, utoi(r));
1583 5152 : S = c > 0 ? addii(S, t) : subii(S, t);
1584 5152 : S = gc_INT(av, S);
1585 : }
1586 : }
1587 28 : return gdiv(S, dS);
1588 : }
1589 :
1590 : /* Returns vector of coeffs from F[0], F[d], ..., F[d*n] */
1591 : static GEN
1592 38402 : mfcoefs_i(GEN F, long n, long d)
1593 : {
1594 38402 : if (n < 0) return gen_0;
1595 38402 : switch(mf_get_type(F))
1596 : {
1597 140 : case t_MF_CONST: return c_const(n, d, gel(F,2));
1598 11333 : case t_MF_EISEN: return c_mfeisen(n, d, F);
1599 882 : case t_MF_Ek: return c_Ek(n, d, F);
1600 539 : case t_MF_DELTA: return c_delta(n, d);
1601 1645 : case t_MF_THETA: return c_theta(n, d, gel(F,2));
1602 203 : case t_MF_ETAQUO: return c_etaquo(n, d, gel(F,2), gel(F,3));
1603 77 : case t_MF_ELL: return c_ell(n, d, gel(F,2));
1604 637 : case t_MF_MUL: return c_mul(n, d, F);
1605 112 : case t_MF_POW: return c_pow(n, d, F);
1606 42 : case t_MF_BRACKET: return c_bracket(n, d, F);
1607 3010 : case t_MF_LINEAR: return c_linear(n, d, gel(F,2), gel(F,3), gel(F,4));
1608 2303 : case t_MF_LINEAR_BHN: return c_linear_bhn(n, d, F);
1609 833 : case t_MF_DIV: return c_div(n, d, F);
1610 35 : case t_MF_SHIFT: return c_shift(n, d, gel(F,2), gel(F,3));
1611 175 : case t_MF_DERIV: return c_deriv(n, d, gel(F,2), gel(F,3));
1612 14 : case t_MF_DERIVE2: return c_derivE2(n, d, gel(F,2), gel(F,3));
1613 168 : case t_MF_TWIST: return c_twist(n, d, gel(F,2), gel(F,3));
1614 1232 : case t_MF_HECKE: return c_hecke(n, d, gel(F,2), gel(F,3));
1615 7504 : case t_MF_BD: return c_Bd(n, d, gel(F,2), gel(F,3));
1616 21 : case t_MF_TRACE: return c_cusptrace(n, d, F);
1617 1918 : case t_MF_NEWTRACE: return c_newtrace(n, d, F);
1618 5579 : case t_MF_DIHEDRAL: return c_dihedral(n, d, F);
1619 : default: pari_err_TYPE("mfcoefs",F); return NULL;/*LCOV_EXCL_LINE*/
1620 : }
1621 : }
1622 :
1623 : static GEN
1624 385 : matdeflate(long n, long d, GEN x)
1625 1575 : { pari_APPLY_same(c_deflate(n,d,gel(x,i))); }
1626 : static int
1627 6069 : space_is_cusp(long space) { return space != mf_FULL && space != mf_EISEN; }
1628 : /* safe with flraw mf */
1629 : static GEN
1630 2611 : mfcoefs_mf(GEN mf, long n, long d)
1631 : {
1632 2611 : GEN MS, ME, E = MF_get_E(mf), S = MF_get_S(mf), M = MF_get_M(mf);
1633 2611 : long lE = lg(E), lS = lg(S), l = lE+lS-1;
1634 :
1635 2611 : if (l == 1) return cgetg(1, t_MAT);
1636 2499 : if (typ(M) == t_MAT && lg(M) != 1 && (n+1)*d < nbrows(M))
1637 21 : return matdeflate(n, d, M); /*cached; lg = 1 is possible from mfinit */
1638 2478 : ME = (lE == 1)? cgetg(1, t_MAT): mfvectomat(E, n, d);
1639 2478 : if (lS == 1)
1640 455 : MS = cgetg(1, t_MAT);
1641 2023 : else if (mf_get_type(gel(S,1)) == t_MF_DIV) /*k 1/2-integer or k=1 (exotic)*/
1642 364 : MS = matdeflate(n,d, mflineardivtomat(MF_get_N(mf), S, n*d));
1643 1659 : else if (MF_get_k(mf) == 1) /* k = 1 (dihedral) */
1644 : {
1645 308 : GEN M = mfvectomat(gmael(S,1,2), n, d);
1646 : long i;
1647 308 : MS = cgetg(lS, t_MAT);
1648 1589 : for (i = 1; i < lS; i++)
1649 : {
1650 1281 : GEN f = gel(S,i), dc = gel(f,4), c = RgM_RgC_mul(M, gel(f,3));
1651 1281 : if (!equali1(dc)) c = RgC_Rg_div(c,dc);
1652 1281 : gel(MS,i) = c;
1653 : }
1654 : }
1655 : else /* k >= 2 integer */
1656 1351 : MS = bhnmat_extend_nocache(NULL, MF_get_N(mf), n, d, S);
1657 2478 : return shallowconcat(ME,MS);
1658 : }
1659 : GEN
1660 4116 : mfcoefs(GEN F, long n, long d)
1661 : {
1662 4116 : if (!checkmf_i(F))
1663 : {
1664 42 : pari_sp av = avma;
1665 42 : GEN mf = checkMF_i(F); if (!mf) pari_err_TYPE("mfcoefs", F);
1666 42 : return gc_GEN(av, mfcoefs_mf(mf,n,d));
1667 : }
1668 4074 : if (d <= 0) pari_err_DOMAIN("mfcoefs", "d", "<=", gen_0, stoi(d));
1669 4074 : if (n < 0) return cgetg(1, t_VEC);
1670 4074 : return mfcoefs_i(F, n, d);
1671 : }
1672 :
1673 : /* assume k >= 0 */
1674 : static GEN
1675 455 : mfak_i(GEN F, long k)
1676 : {
1677 455 : if (!k) return gel(mfcoefs_i(F,0,1), 1);
1678 294 : return gel(mfcoefs_i(F,1,k), 2);
1679 : }
1680 : GEN
1681 301 : mfcoef(GEN F, long n)
1682 : {
1683 301 : pari_sp av = avma;
1684 301 : if (!checkmf_i(F)) pari_err_TYPE("mfcoef",F);
1685 301 : return n < 0? gen_0: gc_GEN(av, mfak_i(F, n));
1686 : }
1687 :
1688 : static GEN
1689 126 : paramconst() { return tagparams(t_MF_CONST, mkNK(1,0,mfchartrivial())); }
1690 : static GEN
1691 84 : mftrivial(void) { retmkvec2(paramconst(), cgetg(1,t_VEC)); }
1692 : static GEN
1693 42 : mf1(void) { retmkvec2(paramconst(), mkvec(gen_1)); }
1694 :
1695 : /* induce mfchar CHI to G */
1696 : static GEN
1697 311731 : induce(GEN G, GEN CHI)
1698 : {
1699 : GEN o, chi;
1700 311731 : if (typ(CHI) == t_INT) /* Kronecker */
1701 : {
1702 300776 : chi = znchar_quad(G, CHI);
1703 300776 : o = ZV_equal0(chi)? gen_1: gen_2;
1704 300776 : CHI = mkvec4(G,chi,o,cgetg(1,t_VEC));
1705 : }
1706 : else
1707 : {
1708 10955 : if (mfcharmodulus(CHI) == itos(znstar_get_N(G))) return CHI;
1709 10304 : CHI = leafcopy(CHI);
1710 10304 : chi = zncharinduce(gel(CHI,1), gel(CHI,2), G);
1711 10304 : gel(CHI,1) = G;
1712 10304 : gel(CHI,2) = chi;
1713 : }
1714 311080 : return CHI;
1715 : }
1716 : /* induce mfchar CHI to znstar(N) */
1717 : static GEN
1718 42364 : induceN(long N, GEN CHI)
1719 : {
1720 42364 : if (mfcharmodulus(CHI) != N) CHI = induce(znstar0(utoipos(N),1), CHI);
1721 42364 : return CHI;
1722 : }
1723 : /* *pCHI1 and *pCHI2 are mfchar, induce to common modulus */
1724 : static void
1725 11172 : char2(GEN *pCHI1, GEN *pCHI2)
1726 : {
1727 11172 : GEN CHI1 = *pCHI1, G1 = gel(CHI1,1), N1 = znstar_get_N(G1);
1728 11172 : GEN CHI2 = *pCHI2, G2 = gel(CHI2,1), N2 = znstar_get_N(G2);
1729 11172 : if (!equalii(N1,N2))
1730 : {
1731 8715 : GEN G, d = gcdii(N1,N2);
1732 8715 : if (equalii(N2,d)) *pCHI2 = induce(G1, CHI2);
1733 1568 : else if (equalii(N1,d)) *pCHI1 = induce(G2, CHI1);
1734 : else
1735 : {
1736 154 : if (!equali1(d)) N2 = diviiexact(N2,d);
1737 154 : G = znstar0(mulii(N1,N2), 1);
1738 154 : *pCHI1 = induce(G, CHI1);
1739 154 : *pCHI2 = induce(G, CHI2);
1740 : }
1741 : }
1742 11172 : }
1743 : /* mfchar or charinit wrt same modulus; outputs a mfchar */
1744 : static GEN
1745 301861 : mfcharmul_i(GEN CHI1, GEN CHI2)
1746 : {
1747 301861 : GEN G = gel(CHI1,1), chi3 = zncharmul(G, gel(CHI1,2), gel(CHI2,2));
1748 301861 : return mfcharGL(G, chi3);
1749 : }
1750 : /* mfchar or charinit; outputs a mfchar */
1751 : static GEN
1752 1106 : mfcharmul(GEN CHI1, GEN CHI2)
1753 : {
1754 1106 : char2(&CHI1, &CHI2); return mfcharmul_i(CHI1,CHI2);
1755 : }
1756 : /* mfchar or charinit; outputs a mfchar */
1757 : static GEN
1758 147 : mfcharpow(GEN CHI, GEN n)
1759 : {
1760 : GEN G, chi;
1761 147 : G = gel(CHI,1); chi = zncharpow(G, gel(CHI,2), n);
1762 147 : return mfchartoprimitive(mfcharGL(G, chi), NULL);
1763 : }
1764 : /* mfchar or charinit wrt same modulus; outputs a mfchar */
1765 : static GEN
1766 10066 : mfchardiv_i(GEN CHI1, GEN CHI2)
1767 : {
1768 10066 : GEN G = gel(CHI1,1), chi3 = znchardiv(G, gel(CHI1,2), gel(CHI2,2));
1769 10066 : return mfcharGL(G, chi3);
1770 : }
1771 : /* mfchar or charinit; outputs a mfchar */
1772 : static GEN
1773 10066 : mfchardiv(GEN CHI1, GEN CHI2)
1774 : {
1775 10066 : char2(&CHI1, &CHI2); return mfchardiv_i(CHI1,CHI2);
1776 : }
1777 : static GEN
1778 56 : mfcharconj(GEN CHI)
1779 : {
1780 56 : CHI = leafcopy(CHI);
1781 56 : gel(CHI,2) = zncharconj(gel(CHI,1), gel(CHI,2));
1782 56 : return CHI;
1783 : }
1784 :
1785 : /* CHI mfchar, assume 4 | N. Multiply CHI by \chi_{-4} */
1786 : static GEN
1787 980 : mfchilift(GEN CHI, long N)
1788 : {
1789 980 : CHI = induceN(N, CHI);
1790 980 : return mfcharmul_i(CHI, induce(gel(CHI,1), stoi(-4)));
1791 : }
1792 : /* CHI defined mod N, N4 = N/4;
1793 : * if CHI is defined mod N4 return CHI;
1794 : * else if CHI' = CHI*(-4,.) is defined mod N4, return CHI' (primitive)
1795 : * else error */
1796 : static GEN
1797 35 : mfcharchiliftprim(GEN CHI, long N4)
1798 : {
1799 35 : long FC = mfcharconductor(CHI);
1800 : GEN CHIP;
1801 35 : if (N4 % FC == 0) return CHI;
1802 14 : CHIP = mfchartoprimitive(mfchilift(CHI, N4 << 2), &FC);
1803 14 : if (N4 % FC) pari_err_TYPE("mfkohnenbasis [incorrect CHI]", CHI);
1804 14 : return CHIP;
1805 : }
1806 : /* ensure CHI(-1) = (-1)^k [k integer] or 1 [half-integer], by multiplying
1807 : * by (-4/.) if needed */
1808 : static GEN
1809 2821 : mfchiadjust(GEN CHI, GEN gk, long N)
1810 : {
1811 2821 : long par = mfcharparity(CHI);
1812 2821 : if (typ(gk) == t_INT && mpodd(gk)) par = -par;
1813 2821 : return par == 1 ? CHI : mfchilift(CHI, N);
1814 : }
1815 :
1816 : static GEN
1817 4074 : mfsamefield(GEN T, GEN P, GEN Q)
1818 : {
1819 4074 : if (degpol(P) == 1) return Q;
1820 721 : if (degpol(Q) == 1) return P;
1821 630 : if (!gequal(P,Q)) pari_err_TYPE("mfsamefield [different fields]",mkvec2(P,Q));
1822 623 : if (T) err_cyclo();
1823 623 : return P;
1824 : }
1825 :
1826 : GEN
1827 455 : mfmul(GEN f, GEN g)
1828 : {
1829 455 : pari_sp av = avma;
1830 : GEN T, N, K, NK, CHI, CHIf, CHIg;
1831 455 : if (!checkmf_i(f)) pari_err_TYPE("mfmul",f);
1832 455 : if (!checkmf_i(g)) pari_err_TYPE("mfmul",g);
1833 455 : N = lcmii(mf_get_gN(f), mf_get_gN(g));
1834 455 : K = gadd(mf_get_gk(f), mf_get_gk(g));
1835 455 : CHIf = mf_get_CHI(f);
1836 455 : CHIg = mf_get_CHI(g);
1837 455 : CHI = mfchiadjust(mfcharmul(CHIf,CHIg), K, itos(N));
1838 455 : T = chicompat(CHI, CHIf, CHIg);
1839 455 : NK = mkgNK(N, K, CHI, mfsamefield(T, mf_get_field(f), mf_get_field(g)));
1840 448 : return gc_GEN(av, T? tag3(t_MF_MUL,NK,f,g,T): tag2(t_MF_MUL,NK,f,g));
1841 : }
1842 : GEN
1843 77 : mfpow(GEN f, long n)
1844 : {
1845 77 : pari_sp av = avma;
1846 : GEN T, KK, NK, gn, CHI, CHIf;
1847 77 : if (!checkmf_i(f)) pari_err_TYPE("mfpow",f);
1848 77 : if (!n) return mf1();
1849 77 : if (n == 1) return gcopy(f);
1850 77 : KK = gmulsg(n,mf_get_gk(f));
1851 77 : gn = stoi(n);
1852 77 : CHIf = mf_get_CHI(f);
1853 77 : CHI = mfchiadjust(mfcharpow(CHIf,gn), KK, mf_get_N(f));
1854 77 : T = chicompat(CHI, CHIf, CHIf);
1855 70 : NK = mkgNK(mf_get_gN(f), KK, CHI, mf_get_field(f));
1856 70 : return gc_GEN(av, T? tag3(t_MF_POW,NK,f,gn,T): tag2(t_MF_POW,NK,f,gn));
1857 : }
1858 : GEN
1859 28 : mfbracket(GEN f, GEN g, long m)
1860 : {
1861 28 : pari_sp av = avma;
1862 : GEN T, N, K, NK, CHI, CHIf, CHIg;
1863 28 : if (!checkmf_i(f)) pari_err_TYPE("mfbracket",f);
1864 28 : if (!checkmf_i(g)) pari_err_TYPE("mfbracket",g);
1865 28 : if (m < 0) pari_err_TYPE("mfbracket [m<0]",stoi(m));
1866 28 : K = gaddgs(gadd(mf_get_gk(f), mf_get_gk(g)), 2*m);
1867 28 : if (gsigne(K) < 0) pari_err_IMPL("mfbracket for this form");
1868 28 : N = lcmii(mf_get_gN(f), mf_get_gN(g));
1869 28 : CHIf = mf_get_CHI(f);
1870 28 : CHIg = mf_get_CHI(g);
1871 28 : CHI = mfcharmul(CHIf, CHIg);
1872 28 : CHI = mfchiadjust(CHI, K, itou(N));
1873 28 : T = chicompat(CHI, CHIf, CHIg);
1874 28 : NK = mkgNK(N, K, CHI, mfsamefield(T, mf_get_field(f), mf_get_field(g)));
1875 56 : return gc_GEN(av, T? tag4(t_MF_BRACKET, NK, f, g, utoi(m), T)
1876 28 : : tag3(t_MF_BRACKET, NK, f, g, utoi(m)));
1877 : }
1878 :
1879 : /* remove 0 entries in L */
1880 : static int
1881 1918 : mflinear_strip(GEN *pF, GEN *pL)
1882 : {
1883 1918 : pari_sp av = avma;
1884 1918 : GEN F = *pF, L = *pL;
1885 1918 : long i, j, l = lg(L);
1886 1918 : GEN F2 = cgetg(l, t_VEC), L2 = cgetg(l, t_VEC);
1887 11473 : for (i = j = 1; i < l; i++)
1888 : {
1889 9555 : if (gequal0(gel(L,i))) continue;
1890 4620 : gel(F2,j) = gel(F,i);
1891 4620 : gel(L2,j) = gel(L,i); j++;
1892 : }
1893 1918 : if (j == l) set_avma(av);
1894 : else
1895 : {
1896 588 : setlg(F2,j); *pF = F2;
1897 588 : setlg(L2,j); *pL = L2;
1898 : }
1899 1918 : return (j > 1);
1900 : }
1901 : static GEN
1902 6839 : taglinear_i(long t, GEN NK, GEN F, GEN L)
1903 : {
1904 : GEN dL;
1905 6839 : L = Q_remove_denom(L, &dL); if (!dL) dL = gen_1;
1906 6839 : return tag3(t, NK, F, L, dL);
1907 : }
1908 : static GEN
1909 2793 : taglinear(GEN NK, GEN F, GEN L)
1910 : {
1911 2793 : long t = ok_bhn_linear(F)? t_MF_LINEAR_BHN: t_MF_LINEAR;
1912 2793 : return taglinear_i(t, NK, F, L);
1913 : }
1914 : /* assume F has parameters NK = [N,K,CHI] */
1915 : static GEN
1916 490 : mflinear_i(GEN NK, GEN F, GEN L)
1917 : {
1918 490 : if (!mflinear_strip(&F,&L)) return mftrivial();
1919 490 : return taglinear(NK, F,L);
1920 : }
1921 : static GEN
1922 770 : mflinear_bhn(GEN mf, GEN L)
1923 : {
1924 : long i, l;
1925 770 : GEN P, NK, F = MF_get_S(mf);
1926 770 : if (!mflinear_strip(&F,&L)) return mftrivial();
1927 763 : l = lg(L); P = pol_x(1);
1928 3465 : for (i = 1; i < l; i++)
1929 : {
1930 2702 : GEN c = gel(L,i);
1931 2702 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1)
1932 665 : P = mfsamefield(NULL, P, gel(c,1));
1933 : }
1934 763 : NK = mkgNK(MF_get_gN(mf), MF_get_gk(mf), MF_get_CHI(mf), P);
1935 763 : return taglinear_i(t_MF_LINEAR_BHN, NK, F,L);
1936 : }
1937 :
1938 : /* F vector of forms with same weight and character but varying level, return
1939 : * global [N,k,chi,P] */
1940 : static GEN
1941 3227 : vecmfNK(GEN F)
1942 : {
1943 3227 : long i, l = lg(F);
1944 : GEN N, f;
1945 3227 : if (l == 1) return mkNK(1, 0, mfchartrivial());
1946 3227 : f = gel(F,1); N = mf_get_gN(f);
1947 45255 : for (i = 2; i < l; i++) N = lcmii(N, mf_get_gN(gel(F,i)));
1948 3227 : return mkgNK(N, mf_get_gk(f), mf_get_CHI(f), mf_get_field(f));
1949 : }
1950 : /* do not use mflinear: mflineardivtomat rely on F being constant across the
1951 : * basis where mflinear strips the ones matched by 0 coeffs. Assume k and CHI
1952 : * constant, N is allowed to vary. */
1953 : static GEN
1954 1211 : vecmflinear(GEN F, GEN C)
1955 : {
1956 1211 : long i, t, l = lg(C);
1957 1211 : GEN NK, v = cgetg(l, t_VEC);
1958 1211 : if (l == 1) return v;
1959 1211 : t = ok_bhn_linear(F)? t_MF_LINEAR_BHN: t_MF_LINEAR;
1960 1211 : NK = vecmfNK(F);
1961 4494 : for (i = 1; i < l; i++) gel(v,i) = taglinear_i(t, NK, F, gel(C,i));
1962 1211 : return v;
1963 : }
1964 : /* vecmflinear(F,C), then divide everything by E, which has valuation 0 */
1965 : static GEN
1966 427 : vecmflineardiv0(GEN F, GEN C, GEN E)
1967 : {
1968 427 : GEN v = vecmflinear(F, C);
1969 427 : long i, l = lg(v);
1970 427 : if (l == 1) return v;
1971 427 : gel(v,1) = mfdiv_val(gel(v,1), E, 0);
1972 1631 : for (i = 2; i < l; i++)
1973 : { /* v[i] /= E */
1974 1204 : GEN f = shallowcopy(gel(v,1));
1975 1204 : gel(f,2) = gel(v,i);
1976 1204 : gel(v,i) = f;
1977 : }
1978 427 : return v;
1979 : }
1980 :
1981 : /* Non empty linear combination of linear combinations of same
1982 : * F_j=\sum_i \mu_{i,j}G_i so R = \sum_i (\sum_j(\la_j\mu_{i,j})) G_i */
1983 : static GEN
1984 2016 : mflinear_linear(GEN F, GEN L, int strip)
1985 : {
1986 2016 : long l = lg(F), j;
1987 2016 : GEN vF, M = cgetg(l, t_MAT);
1988 2016 : L = shallowcopy(L);
1989 18522 : for (j = 1; j < l; j++)
1990 : {
1991 16506 : GEN f = gel(F,j), c = gel(f,3), d = gel(f,4);
1992 16506 : if (typ(c) == t_VEC) c = shallowtrans(c);
1993 16506 : if (!isint1(d)) gel(L,j) = gdiv(gel(L,j),d);
1994 16506 : gel(M,j) = c;
1995 : }
1996 2016 : vF = gmael(F,1,2); L = RgM_RgC_mul(M,L);
1997 2016 : if (strip && !mflinear_strip(&vF,&L)) return mftrivial();
1998 2016 : return taglinear(vecmfNK(vF), vF, L);
1999 : }
2000 : /* F nonempty vector of forms of the form mfdiv(mflinear(B,v), E) where E
2001 : * does not vanish at oo, or mflinear(B,v). Apply mflinear(F, L) */
2002 : static GEN
2003 2016 : mflineardiv_linear(GEN F, GEN L, int strip)
2004 : {
2005 2016 : long l = lg(F), j;
2006 : GEN v, E, f;
2007 2016 : if (lg(L) != l) pari_err_DIM("mflineardiv_linear");
2008 2016 : f = gel(F,1); /* l > 1 */
2009 2016 : if (mf_get_type(f) != t_MF_DIV) return mflinear_linear(F,L,strip);
2010 1708 : E = gel(f,3);
2011 1708 : v = cgetg(l, t_VEC);
2012 17059 : for (j = 1; j < l; j++) { GEN f = gel(F,j); gel(v,j) = gel(f,2); }
2013 1708 : return mfdiv_val(mflinear_linear(v,L,strip), E, 0);
2014 : }
2015 : static GEN
2016 476 : vecmflineardiv_linear(GEN F, GEN M)
2017 : {
2018 476 : long i, l = lg(M);
2019 476 : GEN v = cgetg(l, t_VEC);
2020 1918 : for (i = 1; i < l; i++) gel(v,i) = mflineardiv_linear(F, gel(M,i), 0);
2021 476 : return v;
2022 : }
2023 :
2024 : static GEN
2025 1022 : tobasis(GEN mf, GEN F, GEN L)
2026 : {
2027 1022 : if (checkmf_i(L) && mf) return mftobasis(mf, L, 0);
2028 1015 : if (typ(F) != t_VEC) pari_err_TYPE("mflinear",F);
2029 1015 : if (!is_vec_t(typ(L))) pari_err_TYPE("mflinear",L);
2030 1015 : if (lg(L) != lg(F)) pari_err_DIM("mflinear");
2031 1015 : return L;
2032 : }
2033 : GEN
2034 1064 : mflinear(GEN F, GEN L)
2035 : {
2036 1064 : pari_sp av = avma;
2037 1064 : GEN G, NK, P, mf = checkMF_i(F), N = NULL, K = NULL, CHI = NULL;
2038 : long i, l;
2039 1064 : if (mf)
2040 : {
2041 721 : GEN gk = MF_get_gk(mf);
2042 721 : F = MF_get_basis(F);
2043 721 : if (typ(gk) != t_INT)
2044 42 : return gc_GEN(av, mflineardiv_linear(F, L, 1));
2045 679 : if (itou(gk) > 1 && space_is_cusp(MF_get_space(mf)))
2046 : {
2047 455 : L = tobasis(mf, F, L);
2048 455 : return gc_GEN(av, mflinear_bhn(mf, L));
2049 : }
2050 : }
2051 567 : L = tobasis(mf, F, L);
2052 567 : if (!mflinear_strip(&F,&L)) return mftrivial();
2053 :
2054 560 : l = lg(F);
2055 560 : if (l == 2 && gequal1(gel(L,1))) return gc_GEN(av, gel(F,1));
2056 301 : P = pol_x(1);
2057 945 : for (i = 1; i < l; i++)
2058 : {
2059 651 : GEN f = gel(F,i), c = gel(L,i), Ni, Ki;
2060 651 : if (!checkmf_i(f)) pari_err_TYPE("mflinear", f);
2061 651 : Ni = mf_get_gN(f); N = N? lcmii(N, Ni): Ni;
2062 651 : Ki = mf_get_gk(f);
2063 651 : if (!K) K = Ki;
2064 350 : else if (!gequal(K, Ki))
2065 7 : pari_err_TYPE("mflinear [different weights]", mkvec2(K,Ki));
2066 644 : P = mfsamefield(NULL, P, mf_get_field(f));
2067 644 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1)
2068 126 : P = mfsamefield(NULL, P, gel(c,1));
2069 : }
2070 294 : G = znstar0(N,1);
2071 924 : for (i = 1; i < l; i++)
2072 : {
2073 637 : GEN CHI2 = mf_get_CHI(gel(F,i));
2074 637 : CHI2 = induce(G, CHI2);
2075 637 : if (!CHI) CHI = CHI2;
2076 343 : else if (!gequal(CHI, CHI2))
2077 7 : pari_err_TYPE("mflinear [different characters]", mkvec2(CHI,CHI2));
2078 : }
2079 287 : NK = mkgNK(N, K, CHI, P);
2080 287 : return gc_GEN(av, taglinear(NK,F,L));
2081 : }
2082 :
2083 : GEN
2084 42 : mfshift(GEN F, long sh)
2085 : {
2086 42 : pari_sp av = avma;
2087 42 : if (!checkmf_i(F)) pari_err_TYPE("mfshift",F);
2088 42 : return gc_GEN(av, tag2(t_MF_SHIFT, mf_get_NK(F), F, stoi(sh)));
2089 : }
2090 : static long
2091 49 : mfval(GEN F)
2092 : {
2093 49 : pari_sp av = avma;
2094 49 : long i = 0, n, sb;
2095 : GEN gk, gN;
2096 49 : if (!checkmf_i(F)) pari_err_TYPE("mfval", F);
2097 49 : gN = mf_get_gN(F);
2098 49 : gk = mf_get_gk(F);
2099 49 : sb = mfsturmNgk(itou(gN), gk);
2100 70 : for (n = 1; n <= sb;)
2101 : {
2102 : GEN v;
2103 63 : if (n > 0.5*sb) n = sb+1;
2104 63 : v = mfcoefs_i(F, n, 1);
2105 119 : for (; i <= n; i++)
2106 98 : if (!gequal0(gel(v, i+1))) return gc_long(av,i);
2107 21 : n <<= 1;
2108 : }
2109 7 : return gc_long(av,-1);
2110 : }
2111 :
2112 : GEN
2113 2163 : mfdiv_val(GEN f, GEN g, long vg)
2114 : {
2115 : GEN T, N, K, NK, CHI, CHIf, CHIg;
2116 2163 : if (vg) { f = mfshift(f,vg); g = mfshift(g,vg); }
2117 2163 : N = lcmii(mf_get_gN(f), mf_get_gN(g));
2118 2163 : K = gsub(mf_get_gk(f), mf_get_gk(g));
2119 2163 : CHIf = mf_get_CHI(f);
2120 2163 : CHIg = mf_get_CHI(g);
2121 2163 : CHI = mfchiadjust(mfchardiv(CHIf, CHIg), K, itos(N));
2122 2163 : T = chicompat(CHI, CHIf, CHIg);
2123 2156 : NK = mkgNK(N, K, CHI, mfsamefield(T, mf_get_field(f), mf_get_field(g)));
2124 2156 : return T? tag3(t_MF_DIV, NK, f, g, T): tag2(t_MF_DIV, NK, f, g);
2125 : }
2126 : GEN
2127 49 : mfdiv(GEN F, GEN G)
2128 : {
2129 49 : pari_sp av = avma;
2130 49 : long v = mfval(G);
2131 49 : if (!checkmf_i(F)) pari_err_TYPE("mfdiv", F);
2132 42 : if (v < 0 || (v && !gequal0(mfcoefs(F, v-1, 1))))
2133 14 : pari_err_DOMAIN("mfdiv", "ord(G)", ">", strtoGENstr("ord(F)"),
2134 : mkvec2(F, G));
2135 28 : return gc_GEN(av, mfdiv_val(F, G, v));
2136 : }
2137 : GEN
2138 182 : mfderiv(GEN F, long m)
2139 : {
2140 182 : pari_sp av = avma;
2141 : GEN NK, gk;
2142 182 : if (!checkmf_i(F)) pari_err_TYPE("mfderiv",F);
2143 182 : gk = gaddgs(mf_get_gk(F), 2*m);
2144 182 : NK = mkgNK(mf_get_gN(F), gk, mf_get_CHI(F), mf_get_field(F));
2145 182 : return gc_GEN(av, tag2(t_MF_DERIV, NK, F, stoi(m)));
2146 : }
2147 : GEN
2148 21 : mfderivE2(GEN F, long m)
2149 : {
2150 21 : pari_sp av = avma;
2151 : GEN NK, gk;
2152 21 : if (!checkmf_i(F)) pari_err_TYPE("mfderivE2",F);
2153 21 : if (m < 0) pari_err_DOMAIN("mfderivE2","m","<",gen_0,stoi(m));
2154 21 : gk = gaddgs(mf_get_gk(F), 2*m);
2155 21 : NK = mkgNK(mf_get_gN(F), gk, mf_get_CHI(F), mf_get_field(F));
2156 21 : return gc_GEN(av, tag2(t_MF_DERIVE2, NK, F, stoi(m)));
2157 : }
2158 :
2159 : GEN
2160 28 : mftwist(GEN F, GEN D)
2161 : {
2162 28 : pari_sp av = avma;
2163 : GEN NK, CHI, NT, Da;
2164 : long q;
2165 28 : if (!checkmf_i(F)) pari_err_TYPE("mftwist", F);
2166 28 : if (typ(D) != t_INT) pari_err_TYPE("mftwist", D);
2167 28 : Da = mpabs_shallow(D);
2168 28 : CHI = mf_get_CHI(F); q = mfcharconductor(CHI);
2169 28 : NT = glcm(glcm(mf_get_gN(F), mulsi(q, Da)), sqri(Da));
2170 28 : NK = mkgNK(NT, mf_get_gk(F), CHI, mf_get_field(F));
2171 28 : return gc_GEN(av, tag2(t_MF_TWIST, NK, F, D));
2172 : }
2173 :
2174 : /***************************************************************/
2175 : /* Generic cache handling */
2176 : /***************************************************************/
2177 : enum { cache_FACT, cache_DIV, cache_H, cache_D, cache_DIH };
2178 : typedef struct {
2179 : const char *name;
2180 : GEN cache;
2181 : ulong minself, maxself;
2182 : void (*init)(long);
2183 : ulong miss, maxmiss;
2184 : long compressed;
2185 : } cache;
2186 :
2187 : static void constfact(long lim);
2188 : static void constdiv(long lim);
2189 : static void consttabh(long lim);
2190 : static void consttabdihedral(long lim);
2191 : static void constcoredisc(long lim);
2192 : static THREAD cache caches[] = {
2193 : { "Factors", NULL, 50000, 50000, &constfact, 0, 0, 0 },
2194 : { "Divisors", NULL, 50000, 50000, &constdiv, 0, 0, 0 },
2195 : { "H", NULL, 100000, 10000000, &consttabh, 0, 0, 1 },
2196 : { "CorediscF",NULL, 100000, 10000000, &constcoredisc, 0, 0, 0 },
2197 : { "Dihedral", NULL, 1000, 3000, &consttabdihedral, 0, 0, 0 },
2198 : };
2199 :
2200 : static void
2201 516 : cache_reset(long id) { caches[id].miss = caches[id].maxmiss = 0; }
2202 : static void
2203 9450 : cache_delete(long id) { guncloneNULL(caches[id].cache); }
2204 : static void
2205 530 : cache_set(long id, GEN S)
2206 : {
2207 530 : GEN old = caches[id].cache;
2208 530 : caches[id].cache = gclone(S);
2209 530 : guncloneNULL(old);
2210 530 : }
2211 :
2212 : /* handle a cache miss: store stats, possibly reset table; return value
2213 : * if (now) cached; return NULL on failure. HACK: some caches contain an
2214 : * ulong where the 0 value is impossible, and return it (typecast to GEN) */
2215 : static GEN
2216 451454363 : cache_get(long id, ulong D)
2217 : {
2218 451454363 : cache *S = &caches[id];
2219 451454363 : const ulong d = S->compressed? D>>1: D;
2220 : ulong max, l;
2221 :
2222 451454363 : if (!S->cache)
2223 : {
2224 391 : max = maxuu(minuu(D, S->maxself), S->minself);
2225 391 : S->init(max);
2226 391 : l = lg(S->cache);
2227 : }
2228 : else
2229 : {
2230 451453972 : l = lg(S->cache);
2231 451453972 : if (l <= d)
2232 : {
2233 357 : if (D > S->maxmiss) S->maxmiss = D;
2234 357 : if (DEBUGLEVEL >= 3)
2235 0 : err_printf("miss in cache %s: %lu, max = %lu\n",
2236 : S->name, D, S->maxmiss);
2237 357 : if (S->miss++ >= 5 && D < S->maxself)
2238 : {
2239 15 : max = minuu(S->maxself, (long)(S->maxmiss * 1.2));
2240 15 : if (max <= S->maxself)
2241 : {
2242 15 : if (DEBUGLEVEL >= 3)
2243 0 : err_printf("resetting cache %s to %lu\n", S->name, max);
2244 15 : S->init(max); l = lg(S->cache);
2245 : }
2246 : }
2247 : }
2248 : }
2249 451454363 : return (l <= d)? NULL: gel(S->cache, d);
2250 : }
2251 : static GEN
2252 70 : cache_report(long id)
2253 : {
2254 70 : cache *S = &caches[id];
2255 70 : GEN v = zerocol(5);
2256 70 : gel(v,1) = strtoGENstr(S->name);
2257 70 : if (S->cache)
2258 : {
2259 35 : gel(v,2) = utoi(lg(S->cache)-1);
2260 35 : gel(v,3) = utoi(S->miss);
2261 35 : gel(v,4) = utoi(S->maxmiss);
2262 35 : gel(v,5) = utoi(gsizebyte(S->cache));
2263 : }
2264 70 : return v;
2265 : }
2266 : GEN
2267 14 : getcache(void)
2268 : {
2269 14 : pari_sp av = avma;
2270 14 : GEN M = cgetg(6, t_MAT);
2271 14 : gel(M,1) = cache_report(cache_FACT);
2272 14 : gel(M,2) = cache_report(cache_DIV);
2273 14 : gel(M,3) = cache_report(cache_H);
2274 14 : gel(M,4) = cache_report(cache_D);
2275 14 : gel(M,5) = cache_report(cache_DIH);
2276 14 : return gc_GEN(av, shallowtrans(M));
2277 : }
2278 :
2279 : void
2280 1890 : pari_close_mf(void)
2281 : {
2282 1890 : cache_delete(cache_FACT);
2283 1890 : cache_delete(cache_DIV);
2284 1890 : cache_delete(cache_H);
2285 1890 : cache_delete(cache_D);
2286 1890 : cache_delete(cache_DIH);
2287 1890 : }
2288 :
2289 : /*************************************************************************/
2290 : /* a odd, update local cache (recycle memory) */
2291 : static GEN
2292 2243 : update_factor_cache(long a, long lim, long *pb)
2293 : {
2294 2243 : const long step = 16000; /* even; don't increase this: RAM cache thrashing */
2295 2243 : if (a + 2*step > lim)
2296 205 : *pb = lim; /* fuse last 2 chunks */
2297 : else
2298 2038 : *pb = a + step;
2299 2243 : return vecfactoroddu_i(a, *pb);
2300 : }
2301 : /* assume lim < MAX_LONG/8 */
2302 : static void
2303 52 : constcoredisc(long lim)
2304 : {
2305 52 : pari_sp av2, av = avma;
2306 52 : GEN D = caches[cache_D].cache, CACHE = NULL;
2307 52 : long cachea, cacheb, N, LIM = !D ? 4 : lg(D)-1;
2308 52 : if (lim <= 0) lim = 5;
2309 52 : if (lim <= LIM) return;
2310 52 : cache_reset(cache_D);
2311 52 : D = zero_zv(lim);
2312 52 : av2 = avma;
2313 52 : cachea = cacheb = 0;
2314 5567640 : for (N = 1; N <= lim; N+=2)
2315 : { /* N odd */
2316 : long i, d, d2;
2317 : GEN F;
2318 5567588 : if (N > cacheb)
2319 : {
2320 678 : set_avma(av2); cachea = N;
2321 678 : CACHE = update_factor_cache(N, lim, &cacheb);
2322 : }
2323 5567588 : F = gel(CACHE, ((N-cachea)>>1)+1); /* factoru(N) */
2324 5567588 : D[N] = d = corediscs_fact(F); /* = 3 mod 4 or 4 mod 16 */
2325 5567588 : d2 = odd(d)? d<<3: d<<1;
2326 5567588 : for (i = 1;;)
2327 : {
2328 7423432 : if ((N << i) > lim) break;
2329 3711733 : D[N<<i] = d2; i++;
2330 3711733 : if ((N << i) > lim) break;
2331 1855844 : D[N<<i] = d; i++;
2332 : }
2333 : }
2334 52 : cache_set(cache_D, D);
2335 52 : set_avma(av);
2336 : }
2337 :
2338 : static void
2339 187 : constfact(long lim)
2340 : {
2341 : pari_sp av;
2342 187 : GEN VFACT = caches[cache_FACT].cache;
2343 187 : long LIM = VFACT? lg(VFACT)-1: 4;
2344 187 : if (lim <= 0) lim = 5;
2345 187 : if (lim <= LIM) return;
2346 159 : cache_reset(cache_FACT); av = avma;
2347 159 : cache_set(cache_FACT, vecfactoru_i(1,lim)); set_avma(av);
2348 : }
2349 : static void
2350 152 : constdiv(long lim)
2351 : {
2352 : pari_sp av;
2353 152 : GEN VFACT, VDIV = caches[cache_DIV].cache;
2354 152 : long N, LIM = VDIV? lg(VDIV)-1: 4;
2355 152 : if (lim <= 0) lim = 5;
2356 152 : if (lim <= LIM) return;
2357 152 : constfact(lim);
2358 152 : VFACT = caches[cache_FACT].cache;
2359 152 : cache_reset(cache_DIV); av = avma;
2360 152 : VDIV = cgetg(lim+1, t_VEC);
2361 7131957 : for (N = 1; N <= lim; N++) gel(VDIV,N) = divisorsu_fact(gel(VFACT,N));
2362 152 : cache_set(cache_DIV, VDIV); set_avma(av);
2363 : }
2364 :
2365 : /* n > 1, D = divisors(n); sets L = 2*lambda(n), S = sigma(n) */
2366 : static void
2367 13999514 : lamsig(GEN D, long *pL, long *pS)
2368 : {
2369 13999514 : pari_sp av = avma;
2370 13999514 : long i, l = lg(D), L = 1, S = D[l-1]+1;
2371 50422992 : for (i = 2; i < l; i++) /* skip d = 1 */
2372 : {
2373 51013120 : long d = D[i], nd = D[l-i]; /* nd = n/d */
2374 51013120 : if (d < nd) { L += d; S += d + nd; }
2375 : else
2376 : {
2377 14589642 : L <<= 1; if (d == nd) { L += d; S += d; }
2378 14589642 : break;
2379 : }
2380 : }
2381 13999514 : set_avma(av); *pL = L; *pS = S;
2382 14705505 : }
2383 : /* table of 6 * Hurwitz class numbers D <= lim */
2384 : static void
2385 153 : consttabh(long lim)
2386 : {
2387 153 : pari_sp av = avma, av2;
2388 153 : GEN VHDH0, VDIV, CACHE = NULL;
2389 153 : GEN VHDH = caches[cache_H].cache;
2390 153 : long r, N, cachea, cacheb, lim0 = VHDH? lg(VHDH)-1: 2, LIM = lim0 << 1;
2391 :
2392 153 : if (lim <= 0) lim = 5;
2393 153 : if (lim <= LIM) return;
2394 153 : cache_reset(cache_H);
2395 153 : r = lim&3L; if (r) lim += 4-r;
2396 153 : cache_get(cache_DIV, lim);
2397 153 : VDIV = caches[cache_DIV].cache;
2398 153 : VHDH0 = cgetg(lim/2 + 1, t_VECSMALL);
2399 153 : VHDH0[1] = 2;
2400 153 : VHDH0[2] = 3;
2401 699685 : for (N = 3; N <= lim0; N++) VHDH0[N] = VHDH[N];
2402 153 : av2 = avma;
2403 153 : cachea = cacheb = 0;
2404 7635575 : for (N = LIM + 3; N <= lim; N += 4)
2405 : {
2406 7666708 : long s = 0, limt = usqrt(N>>2), flsq = 0, ind, t, L, S;
2407 : GEN DN, DN2;
2408 7638765 : if (N + 2 >= lg(VDIV))
2409 : { /* use local cache */
2410 : GEN F;
2411 5849820 : if (N + 2 > cacheb)
2412 : {
2413 1565 : set_avma(av2); cachea = N;
2414 1565 : CACHE = update_factor_cache(N, lim+2, &cacheb);
2415 : }
2416 5849820 : F = gel(CACHE, ((N-cachea)>>1)+1); /* factoru(N) */
2417 5849820 : DN = divisorsu_fact(F);
2418 6027299 : F = gel(CACHE, ((N-cachea)>>1)+2); /* factoru(N+2) */
2419 6027299 : DN2 = divisorsu_fact(F);
2420 : }
2421 : else
2422 : { /* use global cache */
2423 1788945 : DN = gel(VDIV,N);
2424 1788945 : DN2 = gel(VDIV,N+2);
2425 : }
2426 7756850 : ind = N >> 1;
2427 850545024 : for (t = 1; t <= limt; t++)
2428 : {
2429 842788174 : ind -= (t<<2)-2; /* N/2 - 2t^2 */
2430 842788174 : if (ind) s += VHDH0[ind]; else flsq = 1;
2431 : }
2432 7756850 : lamsig(DN, &L,&S);
2433 7541418 : VHDH0[N >> 1] = 2*S - 3*L - 2*s + flsq;
2434 7541418 : s = 0; flsq = 0; limt = (usqrt(N+2) - 1) >> 1;
2435 7612679 : ind = (N+1) >> 1;
2436 852290214 : for (t = 1; t <= limt; t++)
2437 : {
2438 844677535 : ind -= t<<2; /* (N+1)/2 - 2t(t+1) */
2439 844677535 : if (ind) s += VHDH0[ind]; else flsq = 1;
2440 : }
2441 7612679 : lamsig(DN2, &L,&S);
2442 7635422 : VHDH0[(N+1) >> 1] = S - 3*(L >> 1) - s - flsq;
2443 : }
2444 89 : cache_set(cache_H, VHDH0); set_avma(av);
2445 : }
2446 :
2447 : /*************************************************************************/
2448 : /* Core functions using factorizations, divisors of class numbers caches */
2449 : /* TODO: myfactoru and factorization cache should be exported */
2450 : static GEN
2451 33835577 : myfactoru(long N)
2452 : {
2453 33835577 : GEN z = cache_get(cache_FACT, N);
2454 33835577 : return z? gcopy(z): factoru(N);
2455 : }
2456 : static GEN
2457 69527256 : mydivisorsu(long N)
2458 : {
2459 69527256 : GEN z = cache_get(cache_DIV, N);
2460 69527256 : return z? leafcopy(z): divisorsu(N);
2461 : }
2462 : /* write -n = Df^2, D < 0 fundamental discriminant. Return D, set f. */
2463 : static long
2464 177457990 : mycoredisc2neg(ulong n, long *pf)
2465 : {
2466 177457990 : ulong m, D = (ulong)cache_get(cache_D, n);
2467 177457990 : if (D) { *pf = usqrt(n/D); return -(long)D; }
2468 58 : m = mycore(n, pf);
2469 58 : if ((m&3) != 3) { m <<= 2; *pf >>= 1; }
2470 58 : return (long)-m;
2471 : }
2472 : /* write n = Df^2, D > 0 fundamental discriminant. Return D, set f. */
2473 : static long
2474 14 : mycoredisc2pos(ulong n, long *pf)
2475 : {
2476 14 : ulong m = mycore(n, pf);
2477 14 : if ((m&3) != 1) { m <<= 2; *pf >>= 1; }
2478 14 : return (long)m;
2479 : }
2480 :
2481 : /* D < 0 fundamental. Return 6*hclassno(-D); faster than quadclassunit up
2482 : * to 5*10^5 or so */
2483 : static ulong
2484 57 : hclassno6_count(long D)
2485 : {
2486 57 : ulong a, b, b2, h = 0, d = -D;
2487 57 : int f = 0;
2488 :
2489 57 : if (d > 500000) return 6 * quadclassnos(D);
2490 : /* this part would work with -d non fundamental */
2491 50 : b = d&1; b2 = (1+d)>>2;
2492 50 : if (!b)
2493 : {
2494 1520 : for (a=1; a*a<b2; a++)
2495 1515 : if (b2%a == 0) h++;
2496 5 : f = (a*a==b2); b=2; b2=(4+d)>>2;
2497 : }
2498 9660 : while (b2*3 < d)
2499 : {
2500 9610 : if (b2%b == 0) h++;
2501 1545888 : for (a=b+1; a*a < b2; a++)
2502 1536278 : if (b2%a == 0) h += 2;
2503 9610 : if (a*a == b2) h++;
2504 9610 : b += 2; b2 = (b*b+d)>>2;
2505 : }
2506 50 : if (b2*3 == d) return 6*h+2;
2507 50 : if (f) return 6*h+3;
2508 50 : return 6*h;
2509 : }
2510 : /* D0 < 0; 6 * hclassno(-D), using D = D0*F^2 */
2511 : static long
2512 84 : hclassno6u_2(long D0, long F)
2513 : {
2514 : long h;
2515 84 : if (F == 1) h = hclassno6_count(D0);
2516 : else
2517 : { /* second chance */
2518 28 : h = (ulong)cache_get(cache_H, -D0);
2519 28 : if (!h) h = hclassno6_count(D0);
2520 28 : h *= uhclassnoF_fact(myfactoru(F), D0);
2521 : }
2522 84 : return h;
2523 : }
2524 : /* D > 0; 6 * hclassno(D) (6*Hurwitz). Beware, cached value for D (=0,3 mod 4)
2525 : * is stored at D>>1 */
2526 : ulong
2527 2567014 : hclassno6u(ulong D)
2528 : {
2529 2567014 : ulong z = (ulong)cache_get(cache_H, D);
2530 : long D0, F;
2531 2567014 : if (z) return z;
2532 84 : D0 = mycoredisc2neg(D, &F);
2533 84 : return hclassno6u_2(D0,F);
2534 : }
2535 : /* same as hclassno6u without creating caches */
2536 : ulong
2537 86918 : hclassno6u_no_cache(ulong D)
2538 : {
2539 86918 : cache *S = &caches[cache_H];
2540 : long D0, F;
2541 86918 : if (S->cache)
2542 : {
2543 80134 : const ulong d = D>>1; /* compressed */
2544 80134 : if ((ulong)lg(S->cache) > d) return S->cache[d];
2545 : }
2546 86647 : S = &caches[cache_D];
2547 86647 : if (!S->cache || (ulong)lg(S->cache) <= D) return 0;
2548 0 : D0 = mycoredisc2neg(D, &F);
2549 0 : return hclassno6u_2(D0,F);
2550 : }
2551 : /* same, where the decomposition D = D0*F^2 is already known */
2552 : static ulong
2553 157401866 : hclassno6u_i(ulong D, long D0, long F)
2554 : {
2555 157401866 : ulong z = (ulong)cache_get(cache_H, D);
2556 157401866 : if (z) return z;
2557 0 : return hclassno6u_2(D0,F);
2558 : }
2559 :
2560 : /* D < -4 fundamental, 6 * h(D), ordinary class number */
2561 : static long
2562 10652124 : hclassno6u_fund(long D)
2563 : {
2564 10652124 : ulong z = (ulong)cache_get(cache_H, -D);
2565 10652124 : return z? z: 6 * quadclassnos(D);
2566 : }
2567 :
2568 : /*************************************************************************/
2569 : /* TRACE FORMULAS */
2570 : /* CHIP primitive, initialize for t_POLMOD output */
2571 : static GEN
2572 33243 : mfcharinit(GEN CHIP)
2573 : {
2574 33243 : long n, o, l, vt, N = mfcharmodulus(CHIP);
2575 : GEN c, v, V, G, Pn;
2576 33243 : if (N == 1) return mkvec2(mkvec(gen_1), pol_x(0));
2577 5607 : G = gel(CHIP,1);
2578 5607 : v = ncharvecexpo(G, znconrey_normalized(G, gel(CHIP,2)));
2579 5607 : l = lg(v); V = cgetg(l, t_VEC);
2580 5607 : o = mfcharorder(CHIP);
2581 5607 : Pn = mfcharpol(CHIP); vt = varn(Pn);
2582 5607 : if (o <= 2)
2583 : {
2584 59248 : for (n = 1; n < l; n++)
2585 : {
2586 54719 : if (v[n] < 0) c = gen_0; else c = v[n]? gen_m1: gen_1;
2587 54719 : gel(V,n) = c;
2588 : }
2589 : }
2590 : else
2591 : {
2592 17591 : for (n = 1; n < l; n++)
2593 : {
2594 16513 : if (v[n] < 0) c = gen_0;
2595 : else
2596 : {
2597 9394 : c = Qab_zeta(v[n], o, vt);
2598 9394 : if (typ(c) == t_POL && lg(c) >= lg(Pn)) c = RgX_rem(c, Pn);
2599 : }
2600 16513 : gel(V,n) = c;
2601 : }
2602 : }
2603 5607 : return mkvec2(V, Pn);
2604 : }
2605 : static GEN
2606 416304 : vchip_lift(GEN VCHI, long x, GEN C)
2607 : {
2608 416304 : GEN V = gel(VCHI,1);
2609 416304 : long F = lg(V)-1;
2610 416304 : if (F == 1) return C;
2611 21056 : x %= F;
2612 21056 : if (!x) return C;
2613 21056 : if (x <= 0) x += F;
2614 21056 : return gmul(C, gel(V, x));
2615 : }
2616 : static long
2617 280525828 : vchip_FC(GEN VCHI) { return lg(gel(VCHI,1))-1; }
2618 : static GEN
2619 6497695 : vchip_mod(GEN VCHI, GEN S)
2620 6497695 : { return (typ(S) == t_POL)? RgX_rem(S, gel(VCHI,2)): S; }
2621 : static GEN
2622 1954117 : vchip_polmod(GEN VCHI, GEN S)
2623 1954117 : { return (typ(S) == t_POL)? mkpolmod(S, gel(VCHI,2)): S; }
2624 :
2625 : /* contribution of scalar matrices in dimension formula */
2626 : static GEN
2627 364665 : A1(long N, long k) { return uutoQ(mypsiu(N)*(k-1), 12); }
2628 : static long
2629 7686 : ceilA1(long N, long k) { return ceildivuu(mypsiu(N) * (k-1), 12); }
2630 :
2631 : /* sturm bound, slightly larger than dimension */
2632 : long
2633 21910 : mfsturmNk(long N, long k) { return (mypsiu(N) * k) / 12; }
2634 : long
2635 3318 : mfsturmNgk(long N, GEN k)
2636 : {
2637 3318 : long n,d; Qtoss(k,&n,&d);
2638 3318 : return 1 + (mypsiu(N)*n)/(d == 1? 12: 24);
2639 : }
2640 : static long
2641 427 : mfsturmmf(GEN F) { return mfsturmNgk(mf_get_N(F), mf_get_gk(F)); }
2642 :
2643 : /* List of all solutions of x^2 + x + 1 = 0 modulo N, x modulo N */
2644 : static GEN
2645 581 : sqrtm3modN(long N)
2646 : {
2647 : pari_sp av;
2648 : GEN fa, P, E, B, mB, A, Q, T, R, v, gen_m3;
2649 581 : long l, i, n, ct, fl3 = 0, Ninit;
2650 581 : if (!odd(N) || (N%9) == 0) return cgetg(1,t_VECSMALL);
2651 553 : Ninit = N;
2652 553 : if ((N%3) == 0) { N /= 3; fl3 = 1; }
2653 553 : fa = myfactoru(N); P = gel(fa, 1); E = gel(fa, 2);
2654 553 : l = lg(P);
2655 749 : for (i = 1; i < l; i++)
2656 560 : if ((P[i]%3) == 2) return cgetg(1,t_VECSMALL);
2657 189 : A = cgetg(l, t_VECSMALL);
2658 189 : B = cgetg(l, t_VECSMALL);
2659 189 : mB= cgetg(l, t_VECSMALL);
2660 189 : Q = cgetg(l, t_VECSMALL); gen_m3 = utoineg(3);
2661 385 : for (i = 1; i < l; i++)
2662 : {
2663 196 : long p = P[i], e = E[i];
2664 196 : Q[i] = upowuu(p,e);
2665 196 : B[i] = itou( Zp_sqrt(gen_m3, utoipos(p), e) );
2666 196 : mB[i]= Q[i] - B[i];
2667 : }
2668 189 : ct = 1 << (l-1);
2669 189 : T = ZV_producttree(Q);
2670 189 : R = ZV_chinesetree(Q,T);
2671 189 : v = cgetg(ct+1, t_VECSMALL);
2672 189 : av = avma;
2673 581 : for (n = 1; n <= ct; n++)
2674 : {
2675 392 : long m = n-1, r;
2676 812 : for (i = 1; i < l; i++)
2677 : {
2678 420 : A[i] = (m&1L)? mB[i]: B[i];
2679 420 : m >>= 1;
2680 : }
2681 392 : r = itou( ZV_chinese_tree(A, Q, T, R) );
2682 462 : if (fl3) while (r%3) r += N;
2683 392 : set_avma(av); v[n] = odd(r) ? (r-1) >> 1 : (r+Ninit-1) >> 1;
2684 : }
2685 189 : return v;
2686 : }
2687 :
2688 : /* number of elliptic points of order 3 in X0(N) */
2689 : static long
2690 10220 : nu3(long N)
2691 : {
2692 : long i, l;
2693 : GEN P;
2694 10220 : if (!odd(N) || (N%9) == 0) return 0;
2695 8995 : if ((N%3) == 0) N /= 3;
2696 8995 : P = gel(myfactoru(N), 1); l = lg(P);
2697 13195 : for (i = 1; i < l; i++) if ((P[i]%3) == 2) return 0;
2698 4018 : return 1L<<(l-1);
2699 : }
2700 : /* number of elliptic points of order 2 in X0(N) */
2701 : static long
2702 17598 : nu2(long N)
2703 : {
2704 : long i, l;
2705 : GEN P;
2706 17598 : if ((N&3L) == 0) return 0;
2707 17598 : if (!odd(N)) N >>= 1;
2708 17598 : P = gel(myfactoru(N), 1); l = lg(P);
2709 22015 : for (i = 1; i < l; i++) if ((P[i]&3L) == 3) return 0;
2710 3955 : return 1L<<(l-1);
2711 : }
2712 :
2713 : /* contribution of elliptic matrices of order 3 in dimension formula
2714 : * Only depends on CHIP the primitive char attached to CHI */
2715 : static GEN
2716 43911 : A21(long N, long k, GEN CHI)
2717 : {
2718 : GEN res, G, chi, o;
2719 : long a21, i, limx, S;
2720 43911 : if ((N&1L) == 0) return gen_0;
2721 21287 : a21 = k%3 - 1;
2722 21287 : if (!a21) return gen_0;
2723 20517 : if (N <= 3) return sstoQ(a21, 3);
2724 10801 : if (!CHI) return sstoQ(nu3(N) * a21, 3);
2725 581 : res = sqrtm3modN(N); limx = (N - 1) >> 1;
2726 581 : G = gel(CHI,1); chi = gel(CHI,2);
2727 581 : o = gmfcharorder(CHI);
2728 973 : for (S = 0, i = 1; i < lg(res); i++)
2729 : { /* (x,N) = 1; S += chi(x) + chi(x^2) */
2730 392 : long x = res[i];
2731 392 : if (x <= limx)
2732 : { /* CHI(x)=e(c/o), 3rd-root of 1 */
2733 196 : GEN c = znchareval(G, chi, utoi(x), o);
2734 196 : if (!signe(c)) S += 2; else S--;
2735 : }
2736 : }
2737 581 : return sstoQ(a21 * S, 3);
2738 : }
2739 :
2740 : /* List of all square roots of -1 modulo N */
2741 : static GEN
2742 595 : sqrtm1modN(long N)
2743 : {
2744 : pari_sp av;
2745 : GEN fa, P, E, B, mB, A, Q, T, R, v;
2746 595 : long l, i, n, ct, fleven = 0;
2747 595 : if ((N&3L) == 0) return cgetg(1,t_VECSMALL);
2748 595 : if ((N&1L) == 0) { N >>= 1; fleven = 1; }
2749 595 : fa = myfactoru(N); P = gel(fa,1); E = gel(fa,2);
2750 595 : l = lg(P);
2751 945 : for (i = 1; i < l; i++)
2752 665 : if ((P[i]&3L) == 3) return cgetg(1,t_VECSMALL);
2753 280 : A = cgetg(l, t_VECSMALL);
2754 280 : B = cgetg(l, t_VECSMALL);
2755 280 : mB= cgetg(l, t_VECSMALL);
2756 280 : Q = cgetg(l, t_VECSMALL);
2757 574 : for (i = 1; i < l; i++)
2758 : {
2759 294 : long p = P[i], e = E[i];
2760 294 : Q[i] = upowuu(p,e);
2761 294 : B[i] = itou( Zp_sqrt(gen_m1, utoipos(p), e) );
2762 294 : mB[i]= Q[i] - B[i];
2763 : }
2764 280 : ct = 1 << (l-1);
2765 280 : T = ZV_producttree(Q);
2766 280 : R = ZV_chinesetree(Q,T);
2767 280 : v = cgetg(ct+1, t_VECSMALL);
2768 280 : av = avma;
2769 868 : for (n = 1; n <= ct; n++)
2770 : {
2771 588 : long m = n-1, r;
2772 1232 : for (i = 1; i < l; i++)
2773 : {
2774 644 : A[i] = (m&1L)? mB[i]: B[i];
2775 644 : m >>= 1;
2776 : }
2777 588 : r = itou( ZV_chinese_tree(A, Q, T, R) );
2778 588 : if (fleven && !odd(r)) r += N;
2779 588 : set_avma(av); v[n] = r;
2780 : }
2781 280 : return v;
2782 : }
2783 :
2784 : /* contribution of elliptic matrices of order 4 in dimension formula.
2785 : * Only depends on CHIP the primitive char attached to CHI */
2786 : static GEN
2787 43911 : A22(long N, long k, GEN CHI)
2788 : {
2789 : GEN G, chi, o, res;
2790 : long S, a22, i, limx, o2;
2791 43911 : if ((N&3L) == 0) return gen_0;
2792 30296 : a22 = (k & 3L) - 1; /* (k % 4) - 1 */
2793 30296 : if (!a22) return gen_0;
2794 30226 : if (N <= 2) return sstoQ(a22, 4);
2795 18403 : if (!CHI) return sstoQ(nu2(N)*a22, 4);
2796 805 : if (mfcharparity(CHI) == -1) return gen_0;
2797 595 : res = sqrtm1modN(N); limx = (N - 1) >> 1;
2798 595 : G = gel(CHI,1); chi = gel(CHI,2);
2799 595 : o = gmfcharorder(CHI);
2800 595 : o2 = itou(o)>>1;
2801 1183 : for (S = 0, i = 1; i < lg(res); i++)
2802 : { /* (x,N) = 1, S += real(chi(x)) */
2803 588 : long x = res[i];
2804 588 : if (x <= limx)
2805 : { /* CHI(x)=e(c/o), 4th-root of 1 */
2806 294 : long c = itou( znchareval(G, chi, utoi(x), o) );
2807 294 : if (!c) S++; else if (c == o2) S--;
2808 : }
2809 : }
2810 595 : return sstoQ(a22 * S, 2);
2811 : }
2812 :
2813 : /* sumdiv(N,d,eulerphi(gcd(d,N/d))) */
2814 : static long
2815 39032 : nuinf(long N)
2816 : {
2817 39032 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
2818 39032 : long i, t = 1, l = lg(P);
2819 82845 : for (i=1; i<l; i++)
2820 : {
2821 43813 : long p = P[i], e = E[i];
2822 43813 : if (odd(e))
2823 35021 : t *= upowuu(p,e>>1) << 1;
2824 : else
2825 8792 : t *= upowuu(p,(e>>1)-1) * (p+1);
2826 : }
2827 39032 : return t;
2828 : }
2829 :
2830 : /* contribution of hyperbolic matrices in dimension formula */
2831 : static GEN
2832 44359 : A3(long N, long FC)
2833 : {
2834 : long i, S, NF, l;
2835 : GEN D;
2836 44359 : if (FC == 1) return uutoQ(nuinf(N),2);
2837 5327 : D = mydivisorsu(N); l = lg(D);
2838 5327 : S = 0; NF = N/FC;
2839 41720 : for (i = 1; i < l; i++)
2840 : {
2841 36393 : long g = ugcd(D[i], D[l-i]);
2842 36393 : if (NF%g == 0) S += myeulerphiu(g);
2843 : }
2844 5327 : return uutoQ(S, 2);
2845 : }
2846 :
2847 : /* special contribution in weight 2 in dimension formula */
2848 : static long
2849 43442 : A4(long k, long FC)
2850 43442 : { return (k==2 && FC==1)? 1: 0; }
2851 : /* gcd(x,N) */
2852 : static long
2853 284283559 : myugcd(GEN GCD, ulong x)
2854 : {
2855 284283559 : ulong N = lg(GCD)-1;
2856 284283559 : if (x >= N) x %= N;
2857 284283559 : return GCD[x+1];
2858 : }
2859 : /* 1_{gcd(x,N) = 1} * chi(x), return NULL if 0 */
2860 : static GEN
2861 404190216 : mychicgcd(GEN GCD, GEN VCHI, long x)
2862 : {
2863 404190216 : long N = lg(GCD)-1;
2864 404190216 : if (N == 1) return gen_1;
2865 329043657 : x = umodsu(x, N);
2866 329043657 : if (GCD[x+1] != 1) return NULL;
2867 272843310 : x %= vchip_FC(VCHI); if (!x) return gen_1;
2868 4678940 : return gel(gel(VCHI,1), x);
2869 : }
2870 :
2871 : /* contribution of scalar matrices to trace formula */
2872 : static GEN
2873 6450498 : TA1(long N, long k, GEN VCHI, GEN GCD, long n)
2874 : {
2875 : GEN S;
2876 : ulong m;
2877 6450498 : if (!uissquareall(n, &m)) return gen_0;
2878 391475 : if (m == 1) return A1(N,k); /* common */
2879 350924 : S = mychicgcd(GCD, VCHI, m);
2880 350924 : return S? gmul(gmul(powuu(m, k-2), A1(N,k)), S): gen_0;
2881 : }
2882 :
2883 : /* All square roots modulo 4N, x modulo 2N, precomputed to accelerate TA2 */
2884 : static GEN
2885 128303 : mksqr(long N)
2886 : {
2887 128303 : pari_sp av = avma;
2888 128303 : long x, N2 = N << 1, N4 = N << 2;
2889 128303 : GEN v = const_vec(N2, cgetg(1, t_VECSMALL));
2890 128303 : gel(v, N2) = mkvecsmall(0); /* x = 0 */
2891 3507175 : for (x = 1; x <= N; x++)
2892 : {
2893 3378872 : long r = (((x*x - 1)%N4) >> 1) + 1;
2894 3378872 : gel(v,r) = vecsmall_append(gel(v,r), x);
2895 : }
2896 128303 : return gc_GEN(av, v);
2897 : }
2898 :
2899 : static GEN
2900 128303 : mkgcd(long N)
2901 : {
2902 : GEN GCD, d;
2903 : long i, N2;
2904 128303 : if (N == 1) return mkvecsmall(N);
2905 105462 : GCD = cgetg(N + 1, t_VECSMALL);
2906 105462 : d = GCD+1; /* GCD[i+1] = d[i] = gcd(i,N) = gcd(N-i,N), i = 0..N-1 */
2907 105462 : d[0] = N; d[1] = d[N-1] = 1; N2 = N>>1;
2908 1657026 : for (i = 2; i <= N2; i++) d[i] = d[N-i] = ugcd(N, i);
2909 105462 : return GCD;
2910 : }
2911 :
2912 : /* Table of \sum_{x^2-tx+n=0 mod Ng}chi(x) for all g dividing gcd(N,F),
2913 : * F^2 largest such that (t^2-4n)/F^2=0 or 1 mod 4; t >= 0 */
2914 : static GEN
2915 15231080 : mutglistall(long t, long N, long NF, GEN VCHI, long n, GEN MUP, GEN L, GEN GCD)
2916 : {
2917 15231080 : long i, lx = lg(L);
2918 15231080 : GEN DNF = mydivisorsu(NF), v = zerovec(NF);
2919 15231080 : long j, g, lDNF = lg(DNF);
2920 42489640 : for (i = 1; i < lx; i++)
2921 : {
2922 27258560 : long x = (L[i] + t) >> 1, y, lD;
2923 27258560 : GEN D, c = mychicgcd(GCD, VCHI, x);
2924 27258560 : if (L[i] && L[i] != N)
2925 : {
2926 18108670 : GEN c2 = mychicgcd(GCD, VCHI, t - x);
2927 18108670 : if (c2) c = c? gadd(c, c2): c2;
2928 : }
2929 27258560 : if (!c) continue;
2930 22109059 : y = (x*(x - t) + n) / N; /* exact division */
2931 22109059 : D = mydivisorsu(ugcd(labs(y), NF)); lD = lg(D);
2932 59563555 : for (j=1; j < lD; j++) { g = D[j]; gel(v,g) = gadd(gel(v,g), c); }
2933 : }
2934 : /* j = 1 corresponds to g = 1, and MUP[1] = 1 */
2935 35176993 : for (j=2; j < lDNF; j++) { g = DNF[j]; gel(v,g) = gmulsg(MUP[g], gel(v,g)); }
2936 15231080 : return v;
2937 : }
2938 :
2939 : /* special case (N,F) = 1: easier */
2940 : static GEN
2941 162226812 : mutg1(long t, long N, GEN VCHI, GEN L, GEN GCD)
2942 : {
2943 162226812 : GEN S = NULL;
2944 162226812 : long i, lx = lg(L);
2945 340338661 : for (i = 1; i < lx; i++)
2946 : {
2947 178111849 : long x = (L[i] + t) >> 1;
2948 178111849 : GEN c = mychicgcd(GCD, VCHI, x);
2949 178111849 : if (c) S = S? gadd(S, c): c;
2950 178111849 : if (L[i] && L[i] != N)
2951 : {
2952 98297640 : c = mychicgcd(GCD, VCHI, t - x);
2953 98297640 : if (c) S = S? gadd(S, c): c;
2954 : }
2955 178111849 : if (S && !signe(S)) S = NULL; /* strive hard to add gen_0 */
2956 : }
2957 162226812 : return S; /* single value */
2958 : }
2959 :
2960 : /* n > 2, return P_n = \sum_{0<=j<=n/2} (-1)^j binomial(n-j,j) X^j
2961 : * (2x)^n P_n (1 / (4x^2)) = polchebyshev(n, 2) */
2962 : GEN
2963 402687 : mfrhopol(long n)
2964 : {
2965 : #ifdef LONG_IS_64BIT
2966 345204 : const long M = 2642249;
2967 : #else
2968 57483 : const long M = 1629;
2969 : #endif
2970 402687 : long j, d = n >> 1; /* >= 1 */
2971 402687 : GEN P = cgetg(d + 3, t_POL);
2972 :
2973 402687 : if (n > M) pari_err_IMPL("mfrhopol for large weight"); /* avoid overflow */
2974 402687 : P[1] = evalvarn(0)|evalsigne(1);
2975 402687 : gel(P,2) = gen_1;
2976 402687 : gel(P,3) = utoineg(n-1); /* j = 1 */
2977 402687 : if (d > 1) gel(P,4) = utoipos(((n-3)*(n-2)) >> 1); /* j = 2 */
2978 402687 : if (d > 2) gel(P,5) = utoineg(((n-5)*(n-4)*(n-3)) / 6); /* j = 3 */
2979 1608468 : for (j = 4; j <= d; j++)
2980 1205781 : gel(P,j+2) = diviuexact(mulis(gel(P,j+1), -(n-2*j+1)*(n-2*j+2)), (n-j+1)*j);
2981 402687 : return P;
2982 : }
2983 :
2984 : /* polrecip(Q)(x), assume Q(0) = 1 */
2985 : GEN
2986 4054280 : mfrhopol_u_eval(GEN Q, ulong x)
2987 : {
2988 4054280 : GEN T = addiu(gel(Q,3), x);
2989 4054279 : long l = lg(Q), j;
2990 40990203 : for (j = 4; j < l; j++) T = addii(gel(Q,j), mului(x, T));
2991 4054282 : return T;
2992 : }
2993 : GEN
2994 56617 : mfrhopol_eval(GEN Q, GEN x)
2995 : {
2996 : long l, j;
2997 : GEN T;
2998 56617 : if (lgefint(x) == 3) return mfrhopol_u_eval(Q, x[2]);
2999 0 : l = lg(Q); T = addii(gel(Q,3), x);
3000 0 : for (j = 4; j < l; j++) T = addii(gel(Q,j), mulii(x, T));
3001 0 : return T;
3002 : }
3003 : /* t >= 0. If nu odd, let [N, T] = [(nu - 1)/2, t]; else let [N, T] = [nu/2, 1].
3004 : * We have t2 = t^2 and Q(X) = sum_{0<=j<=N} (-1)^j binomial(nu-j,j) n^j X^j
3005 : * U_nu(z) = polchebyshev(nu, 2, z)
3006 : * = sum_{0<=j<=N} (-1)^j binomial(nu-j,j) (2z)^(nu-2*j))
3007 : * Return C n^(nu/2) U_nu(t / (2*sqrt(n)))
3008 : * = C sum_{0<=j<=N} (-1)^j binomial(nu-j,j) n^j t^(nu - 2j)
3009 : * = C T sum_{0<=j<=N} (-1)^j binomial(nu-j,j) n^j (t^2)^(N - j)
3010 : * = C T polrecip(Q)(t^2); note that Q(0) = 1 */
3011 : static GEN
3012 169019315 : mfrhopow(GEN C, GEN Q, long nu, long t, long t2, long n)
3013 : {
3014 : GEN T;
3015 169019315 : switch (nu)
3016 : {
3017 162107834 : case 0: return C;
3018 1125446 : case 1: return gmulsg(t, C);
3019 1660813 : case 2: return gmulsg(t2 - n, C);
3020 51275 : case 3: return gmul(mulss(t, t2 - 2*n), C);
3021 4073947 : default:
3022 4073947 : if (!t) return gmul(gel(Q, lg(Q) - 1), C);
3023 3997663 : T = mfrhopol_u_eval(Q, t2); if (odd(nu)) T = mului(t, T);
3024 3997663 : return gmul(T, C);
3025 : }
3026 : }
3027 :
3028 : static GEN
3029 320848534 : TA2_t(long t, long N, long N4, long n, long n4, long nu, GEN Q,
3030 : GEN VCHI, GEN SQRTS, GEN MUP, GEN GCD)
3031 : {
3032 320848534 : long F, NF, D0, t2 = t*t, D = n4 - t2; /* > 0 */
3033 320848534 : GEN sh, L = gel(SQRTS, (umodsu(-D - 1, N4) >> 1) + 1);
3034 :
3035 320848534 : if (lg(L) == 1) return NULL;
3036 177457892 : D0 = mycoredisc2neg(D, &F);
3037 177457892 : NF = myugcd(GCD, F);
3038 177457892 : if (NF == 1)
3039 : { /* (N,F) = 1 => single value in mutglistall */
3040 162226812 : GEN mut = mutg1(t, N, VCHI, L, GCD);
3041 162226812 : if (!mut) return NULL;
3042 157401866 : sh = gmulgu(mut, hclassno6u_i(D,D0,F));
3043 : }
3044 : else
3045 : {
3046 15231080 : GEN v = mutglistall(t, N, NF, VCHI, n, MUP, L, GCD);
3047 15231080 : GEN DF = mydivisorsu(F);
3048 15231080 : long i, lDF = lg(DF);
3049 15231080 : sh = gen_0;
3050 61389665 : for (i = 1; i < lDF; i++)
3051 : {
3052 46158585 : long Ff, f = DF[i], g = myugcd(GCD, f);
3053 46158585 : GEN mut = gel(v, g);
3054 46158585 : if (gequal0(mut)) continue;
3055 31213441 : Ff = DF[lDF-i]; /* F/f */
3056 31213441 : if (Ff > 1)
3057 : {
3058 22367354 : GEN P = gel(myfactoru(Ff), 1);
3059 22367354 : long j, lP = lg(P);
3060 49329992 : for (j = 1; j < lP; j++) { long p = P[j]; Ff -= kross(D0, p)*Ff/p; }
3061 22367354 : mut = gmulsg(Ff, mut);
3062 : }
3063 31213441 : sh = gadd(sh, mut);
3064 : }
3065 15231080 : if (gequal0(sh)) return NULL;
3066 11617449 : if (D0 == -3) sh = gmul2n(sh, 1);
3067 11124029 : else if (D0 == -4) sh = gmulgu(sh, 3);
3068 10652124 : else sh = gmulgu(sh, hclassno6u_fund(D0));
3069 : }
3070 169019315 : return mfrhopow(sh, Q, nu, t, t2, n);
3071 : }
3072 :
3073 : /* contribution of elliptic matrices to trace formula */
3074 : static GEN
3075 6450498 : TA2(long N, long k, GEN VCHI, long n, GEN SQRTS, GEN MUP, GEN GCD)
3076 : {
3077 6450498 : long N4 = N << 2, n4 = n << 2, nu = k - 2;
3078 6450498 : long st = (!odd(N) && odd(n)) ? 2 : 1;
3079 6450498 : long t, limt = usqrt(n4 - 1);
3080 6450498 : GEN s, S = gen_0, Q = nu > 3 ? ZX_z_unscale(mfrhopol(nu), n) : NULL;
3081 :
3082 : /* actually compute 6*S to ensure integrality */
3083 321040747 : for (t = st; t <= limt; t += st) /* t^2 < 4n */
3084 : {
3085 314590249 : pari_sp av = avma;
3086 314590249 : s = TA2_t(t, N, N4, n, n4, nu, Q, VCHI, SQRTS, MUP, GCD);
3087 314590249 : if (s) S = gc_upto(av, gadd(S, s)); else set_avma(av);
3088 : }
3089 6450498 : if (!odd(k))
3090 : {
3091 6258285 : s = TA2_t(0, N, N4, n, n4, nu, Q, VCHI, SQRTS, MUP, GCD);
3092 : /* s/2 is the only term involving a denominator (= 2) */
3093 6258285 : if (s) S = gadd(S, gmul2n(s, -1));
3094 : }
3095 6450498 : return gdivgu(S, 6);
3096 : }
3097 :
3098 : /* compute global auxiliary data for TA3 */
3099 : static GEN
3100 128303 : mkbez(long N, long FC)
3101 : {
3102 128303 : long ct, i, NF = N/FC;
3103 128303 : GEN w, D = mydivisorsu(N);
3104 128303 : long l = lg(D);
3105 :
3106 128303 : w = cgetg(l, t_VEC);
3107 371336 : for (i = ct = 1; i < l; i++)
3108 : {
3109 348495 : long u, v, h, c = D[i], Nc = D[l-i];
3110 348495 : if (c > Nc) break;
3111 243033 : h = cbezout(c, Nc, &u, &v);
3112 243033 : if (h == 1) /* shortcut */
3113 175539 : gel(w, ct++) = mkvecsmall4(1,u*c,1,i);
3114 67494 : else if (!(NF%h))
3115 57582 : gel(w, ct++) = mkvecsmall4(h,u*(c/h),myeulerphiu(h),i);
3116 : }
3117 128303 : setlg(w,ct); stackdummy((pari_sp)(w+ct),(pari_sp)(w+l));
3118 128303 : return w;
3119 : }
3120 :
3121 : /* contribution of hyperbolic matrices to trace formula, d * nd = n,
3122 : * DN = divisorsu(N) */
3123 : static GEN
3124 33430095 : auxsum(GEN VCHI, GEN GCD, long d, long nd, GEN DN, GEN BEZ)
3125 : {
3126 33430095 : GEN S = gen_0;
3127 33430095 : long ct, g = nd - d, lDN = lg(DN), lBEZ = lg(BEZ);
3128 85746801 : for (ct = 1; ct < lBEZ; ct++)
3129 : {
3130 52316706 : GEN y, B = gel(BEZ, ct);
3131 52316706 : long ic, c, Nc, uch, h = B[1];
3132 52316706 : if (g%h) continue;
3133 51093820 : uch = B[2];
3134 51093820 : ic = B[4];
3135 51093820 : c = DN[ic];
3136 51093820 : Nc= DN[lDN - ic]; /* Nc = N/c */
3137 51093820 : if (ugcd(Nc, nd) == 1)
3138 43639142 : y = mychicgcd(GCD, VCHI, d + uch*g); /* 0 if (c,d) > 1 */
3139 : else
3140 7454678 : y = NULL;
3141 51093820 : if (c != Nc && ugcd(Nc, d) == 1)
3142 : {
3143 38423431 : GEN y2 = mychicgcd(GCD, VCHI, nd - uch*g); /* 0 if (c,nd) > 1 */
3144 38423431 : if (y2) y = y? gadd(y, y2): y2;
3145 : }
3146 51093820 : if (y) S = gadd(S, gmulsg(B[3], y));
3147 : }
3148 33430095 : return S;
3149 : }
3150 :
3151 : static GEN
3152 6450498 : TA3(long N, long k, GEN VCHI, GEN GCD, GEN Dn, GEN BEZ)
3153 : {
3154 6450498 : GEN S = gen_0, DN = mydivisorsu(N);
3155 6450498 : long i, l = lg(Dn);
3156 39880593 : for (i = 1; i < l; i++)
3157 : {
3158 39840042 : long d = Dn[i], nd = Dn[l-i]; /* = n/d */
3159 : GEN t, u;
3160 39840042 : if (d > nd) break;
3161 33430095 : t = auxsum(VCHI, GCD, d, nd, DN, BEZ);
3162 33430095 : if (isintzero(t)) continue;
3163 32302059 : u = powuu(d,k-1); if (d == nd) u = gmul2n(u,-1);
3164 32302059 : S = gadd(S, gmul(u,t));
3165 : }
3166 6450498 : return S;
3167 : }
3168 :
3169 : /* special contribution in weight 2 in trace formula */
3170 : static long
3171 6450498 : TA4(long k, GEN VCHIP, GEN Dn, GEN GCD)
3172 : {
3173 : long i, l, S;
3174 6450498 : if (k != 2 || vchip_FC(VCHIP) != 1) return 0;
3175 5687416 : l = lg(Dn); S = 0;
3176 66354498 : for (i = 1; i < l; i++)
3177 : {
3178 60667082 : long d = Dn[i]; /* gcd(N,n/d) == 1? */
3179 60667082 : if (myugcd(GCD, Dn[l-i]) == 1) S += d;
3180 : }
3181 5687416 : return S;
3182 : }
3183 :
3184 : /* precomputation of products occurring im mutg, again to accelerate TA2 */
3185 : static GEN
3186 128303 : mkmup(long N)
3187 : {
3188 128303 : GEN fa = myfactoru(N), P = gel(fa,1), D = divisorsu_fact(fa);
3189 128303 : long i, lP = lg(P), lD = lg(D);
3190 128303 : GEN MUP = zero_zv(N);
3191 128303 : MUP[1] = 1;
3192 447944 : for (i = 2; i < lD; i++)
3193 : {
3194 319641 : long j, g = D[i], Ng = D[lD-i]; /* N/g */
3195 874699 : for (j = 1; j < lP; j++) { long p = P[j]; if (Ng%p) g += g/p; }
3196 319641 : MUP[D[i]] = g;
3197 : }
3198 128303 : return MUP;
3199 : }
3200 :
3201 : /* quadratic nonresidues mod p; p odd prime, p^2 fits in a long */
3202 : static GEN
3203 2814 : non_residues(long p)
3204 : {
3205 2814 : long i, j, p2 = p >> 1;
3206 2814 : GEN v = cgetg(p2+1, t_VECSMALL), w = const_vecsmall(p-1, 1);
3207 4571 : for (i = 2; i <= p2; i++) w[(i*i) % p] = 0; /* no need to check 1 */
3208 9142 : for (i = 2, j = 1; i < p; i++) if (w[i]) v[j++] = i;
3209 2814 : return v;
3210 : }
3211 :
3212 : /* CHIP primitive. Return t_VECSMALL v of length q such that
3213 : * Tr^new_{N,CHIP}(n) = 0 whenever v[(n%q) + 1] is nonzero */
3214 : static GEN
3215 33341 : mfnewzerodata(long N, GEN CHIP)
3216 : {
3217 33341 : GEN V, M, L, faN = myfactoru(N), PN = gel(faN,1), EN = gel(faN,2);
3218 33341 : GEN G = gel(CHIP,1), chi = gel(CHIP,2);
3219 33341 : GEN fa = znstar_get_faN(G), P = ZV_to_zv(gel(fa,1)), E = gel(fa,2);
3220 33341 : long i, mod, j = 1, l = lg(PN);
3221 :
3222 33341 : M = cgetg(l, t_VECSMALL); M[1] = 0;
3223 33341 : V = cgetg(l, t_VEC);
3224 : /* Tr^new(n) = 0 if (n mod M[i]) in V[i] */
3225 33341 : if ((N & 3) == 0)
3226 : {
3227 12929 : long e = EN[1];
3228 12929 : long c = (lg(P) > 1 && P[1] == 2)? E[1]: 0; /* c = v_2(FC) */
3229 : /* e >= 2 */
3230 12929 : if (c == e-1) return NULL; /* Tr^new = 0 */
3231 12824 : if (c == e)
3232 : {
3233 3717 : if (e == 2)
3234 : { /* sc: -4 */
3235 1785 : gel(V,1) = mkvecsmall(3);
3236 1785 : M[1] = 4;
3237 : }
3238 1932 : else if (e == 3)
3239 : { /* sc: -8 (CHI_2(-1)=-1<=>chi[1]=1) and 8 (CHI_2(-1)=1 <=> chi[1]=0) */
3240 1932 : long t = signe(gel(chi,1))? 7: 3;
3241 1932 : gel(V,1) = mkvecsmall2(5, t);
3242 1932 : M[1] = 8;
3243 : }
3244 : }
3245 9107 : else if (e == 5 && c == 3)
3246 154 : { /* sc: -8 (CHI_2(-1)=-1<=>chi[1]=1) and 8 (CHI_2(-1)=1 <=> chi[1]=0) */
3247 154 : long t = signe(gel(chi,1))? 7: 3;
3248 154 : gel(V,1) = mkvecsmalln(6, 2L,4L,5L,6L,8L,t);
3249 154 : M[1] = 8;
3250 : }
3251 8953 : else if ((e == 4 && c == 2) || (e == 5 && c <= 2) || (e == 6 && c <= 2)
3252 7378 : || (e >= 7 && c == e - 3))
3253 : { /* sc: 4 */
3254 1575 : gel(V,1) = mkvecsmall3(0,2,3);
3255 1575 : M[1] = 4;
3256 : }
3257 7378 : else if ((e <= 4 && c == 0) || (e >= 5 && c == e - 2))
3258 : { /* sc: 2 */
3259 7021 : gel(V,1) = mkvecsmall(0);
3260 7021 : M[1] = 2;
3261 : }
3262 357 : else if ((e == 6 && c == 3) || (e >= 7 && c <= e - 4))
3263 : { /* sc: -2 */
3264 357 : gel(V,1) = mkvecsmalln(7, 0L,2L,3L,4L,5L,6L,7L);
3265 357 : M[1] = 8;
3266 : }
3267 : }
3268 33236 : j = M[1]? 2: 1;
3269 70798 : for (i = odd(N)? 1: 2; i < l; i++) /* skip p=2, done above */
3270 : {
3271 37562 : long p = PN[i], e = EN[i];
3272 37562 : long z = zv_search(P, p), c = z? E[z]: 0; /* c = v_p(FC) */
3273 37562 : if ((e <= 2 && c == 1 && itos(gel(chi,z)) == (p>>1)) /* ord(CHI_p)=2 */
3274 35371 : || (e >= 3 && c <= e - 2))
3275 2814 : { /* sc: -p */
3276 2814 : GEN v = non_residues(p);
3277 2814 : if (e != 1) v = vecsmall_prepend(v, 0);
3278 2814 : gel(V,j) = v;
3279 2814 : M[j] = p; j++;
3280 : }
3281 34748 : else if (e >= 2 && c < e)
3282 : { /* sc: p */
3283 2660 : gel(V,j) = mkvecsmall(0);
3284 2660 : M[j] = p; j++;
3285 : }
3286 : }
3287 33236 : if (j == 1) return cgetg(1, t_VECSMALL);
3288 15379 : setlg(V,j); setlg(M,j); mod = zv_prod(M);
3289 15379 : L = zero_zv(mod);
3290 33677 : for (i = 1; i < j; i++)
3291 : {
3292 18298 : GEN v = gel(V,i);
3293 18298 : long s, m = M[i], lv = lg(v);
3294 47621 : for (s = 1; s < lv; s++)
3295 : {
3296 29323 : long a = v[s] + 1;
3297 56392 : do { L[a] = 1; a += m; } while (a <= mod);
3298 : }
3299 : }
3300 15379 : return L;
3301 : }
3302 : /* v=mfnewzerodata(N,CHI); returns TRUE if newtrace(n) must be zero,
3303 : * (but newtrace(n) may still be zero if we return FALSE) */
3304 : static long
3305 2628827 : mfnewchkzero(GEN v, long n) { long q = lg(v)-1; return q && v[(n%q) + 1]; }
3306 :
3307 : /* if (!VCHIP): from mftraceform_cusp;
3308 : * else from initnewtrace and CHI is known to be primitive */
3309 : static GEN
3310 128303 : inittrace(long N, GEN CHI, GEN VCHIP)
3311 : {
3312 : long FC;
3313 128303 : if (VCHIP)
3314 128296 : FC = mfcharmodulus(CHI);
3315 : else
3316 7 : VCHIP = mfcharinit(mfchartoprimitive(CHI, &FC));
3317 128303 : return mkvecn(5, mksqr(N), mkmup(N), mkgcd(N), VCHIP, mkbez(N, FC));
3318 : }
3319 :
3320 : /* p > 2 prime; return a sorted t_VECSMALL of primes s.t Tr^new(p) = 0 for all
3321 : * weights > 2 */
3322 : static GEN
3323 33236 : inittrconj(long N, long FC)
3324 : {
3325 : GEN fa, P, E, v;
3326 : long i, k, l;
3327 :
3328 33236 : if (FC != 1) return cgetg(1,t_VECSMALL);
3329 :
3330 27629 : fa = myfactoru(N >> vals(N));
3331 27629 : P = gel(fa,1); l = lg(P);
3332 27629 : E = gel(fa,2);
3333 27629 : v = cgetg(l, t_VECSMALL);
3334 60102 : for (i = k = 1; i < l; i++)
3335 : {
3336 32473 : long j, p = P[i]; /* > 2 */
3337 78302 : for (j = 1; j < l; j++)
3338 45829 : if (j != i && E[j] == 1 && kross(-p, P[j]) == 1) v[k++] = p;
3339 : }
3340 27629 : setlg(v,k); return v;
3341 : }
3342 :
3343 : /* assume CHIP primitive, f(CHIP) | N; NZ = mfnewzerodata(N,CHIP) */
3344 : static GEN
3345 33236 : initnewtrace_i(long N, GEN CHIP, GEN NZ)
3346 : {
3347 33236 : GEN T = const_vec(N, cgetg(1,t_VEC)), D, VCHIP;
3348 33236 : long FC = mfcharmodulus(CHIP), N1, N2, i, l;
3349 :
3350 33236 : if (!NZ) NZ = mkvecsmall(1); /*Tr^new = 0; initialize data nevertheless*/
3351 33236 : VCHIP = mfcharinit(CHIP);
3352 33236 : N1 = N/FC; newd_params(N1, &N2);
3353 33236 : D = mydivisorsu(N1/N2); l = lg(D);
3354 33236 : N2 *= FC;
3355 161532 : for (i = 1; i < l; i++)
3356 : {
3357 128296 : long M = D[i]*N2;
3358 128296 : gel(T,M) = inittrace(M, CHIP, VCHIP);
3359 : }
3360 33236 : gel(T,N) = shallowconcat(gel(T,N), mkvec2(NZ, inittrconj(N,FC)));
3361 33236 : return T;
3362 : }
3363 : /* don't initialize if Tr^new = 0, return NULL */
3364 : static GEN
3365 33341 : initnewtrace(long N, GEN CHI)
3366 : {
3367 33341 : GEN CHIP = mfchartoprimitive(CHI, NULL), NZ = mfnewzerodata(N,CHIP);
3368 33341 : return NZ? initnewtrace_i(N, CHIP, NZ): NULL;
3369 : }
3370 :
3371 : /* (-1)^k */
3372 : static long
3373 8246 : m1pk(long k) { return odd(k)? -1 : 1; }
3374 : static long
3375 7889 : badchar(long N, long k, GEN CHI)
3376 7889 : { return mfcharparity(CHI) != m1pk(k) || (CHI && N % mfcharconductor(CHI)); }
3377 :
3378 :
3379 : static long
3380 43519 : mfcuspdim_i(long N, long k, GEN CHI, GEN vSP)
3381 : {
3382 43519 : pari_sp av = avma;
3383 : long FC;
3384 : GEN s;
3385 43519 : if (k <= 0) return 0;
3386 43519 : if (k == 1) return CHI? mf1cuspdim(N, CHI, vSP): 0;
3387 43260 : FC = CHI? mfcharconductor(CHI): 1;
3388 43260 : if (FC == 1) CHI = NULL;
3389 43260 : s = gsub(A1(N, k), gadd(A21(N, k, CHI), A22(N, k, CHI)));
3390 43260 : s = gadd(s, gsubsg(A4(k, FC), A3(N, FC)));
3391 43260 : return gc_long(av, itos(s));
3392 : }
3393 : /* dimension of space of cusp forms S_k(\G_0(N),CHI)
3394 : * Only depends on CHIP the primitive char attached to CHI */
3395 : long
3396 3423 : mfcuspdim(long N, long k, GEN CHI) { return mfcuspdim_i(N, k, CHI, NULL); }
3397 :
3398 : /* dimension of whole space M_k(\G_0(N),CHI)
3399 : * Only depends on CHIP the primitive char attached to CHI; assumes !badchar */
3400 : long
3401 868 : mffulldim(long N, long k, GEN CHI)
3402 : {
3403 868 : pari_sp av = avma;
3404 868 : long FC = CHI? mfcharconductor(CHI): 1;
3405 : GEN s;
3406 868 : if (k <= 0) return (k == 0 && FC == 1)? 1: 0;
3407 868 : if (k == 1) return gc_long(av, itos(A3(N, FC)) + mf1cuspdim(N, CHI, NULL));
3408 651 : if (FC == 1) CHI = NULL;
3409 651 : s = gsub(A1(N, k), gadd(A21(N, k, CHI), A22(N, k, CHI)));
3410 651 : s = gadd(s, A3(N, FC));
3411 651 : return gc_long(av, itos(s));
3412 : }
3413 :
3414 : /* Dimension of the space of Eisenstein series */
3415 : long
3416 231 : mfeisensteindim(long N, long k, GEN CHI)
3417 : {
3418 231 : pari_sp av = avma;
3419 231 : long s, FC = CHI? mfcharconductor(CHI): 1;
3420 231 : if (k <= 0) return (k == 0 && FC == 1)? 1: 0;
3421 231 : s = itos(gmul2n(A3(N, FC), 1));
3422 231 : if (k > 1) s -= A4(k, FC); else s >>= 1;
3423 231 : return gc_long(av,s);
3424 : }
3425 :
3426 : enum { _SQRTS = 1, _MUP, _GCD, _VCHIP, _BEZ, _NEWLZ, _TRCONJ };
3427 : /* Trace of T(n) on space of cuspforms; only depends on CHIP the primitive char
3428 : * attached to CHI */
3429 : static GEN
3430 6450498 : mfcusptrace_i(long N, long k, long n, GEN Dn, GEN S)
3431 : {
3432 6450498 : pari_sp av = avma;
3433 : GEN a, b, VCHIP, GCD;
3434 : long t;
3435 6450498 : if (!n) return gen_0;
3436 6450498 : VCHIP = gel(S,_VCHIP);
3437 6450498 : GCD = gel(S,_GCD);
3438 6450498 : t = TA4(k, VCHIP, Dn, GCD);
3439 6450498 : a = TA1(N, k, VCHIP, GCD, n); if (t) a = gaddgs(a,t);
3440 6450498 : b = TA2(N, k, VCHIP, n, gel(S,_SQRTS), gel(S,_MUP), GCD);
3441 6450498 : b = gadd(b, TA3(N, k, VCHIP, GCD, Dn, gel(S,_BEZ)));
3442 6450498 : b = gsub(a,b);
3443 6450498 : if (typ(b) != t_POL) return gc_upto(av, b);
3444 50834 : return gc_GEN(av, vchip_polmod(VCHIP, b));
3445 : }
3446 :
3447 : static GEN
3448 7714544 : mfcusptracecache(long N, long k, long n, GEN Dn, GEN S, cachenew_t *cache)
3449 : {
3450 7714544 : GEN C = NULL, T = gel(cache->vfull,N);
3451 7714544 : long lcache = lg(T);
3452 7714544 : if (n < lcache) C = gel(T, n);
3453 7714544 : if (C) cache->cuspHIT++; else C = mfcusptrace_i(N, k, n, Dn, S);
3454 7714544 : cache->cuspTOTAL++;
3455 7714544 : if (n < lcache) gel(T,n) = C;
3456 7714544 : return C;
3457 : }
3458 :
3459 : /* return the divisors of n, known to be among the elements of D */
3460 : static GEN
3461 324443 : div_restrict(GEN D, ulong n)
3462 : {
3463 : long i, j, l;
3464 324443 : GEN v, VDIV = caches[cache_DIV].cache;
3465 324443 : if (lg(VDIV) > n) return gel(VDIV,n);
3466 0 : l = lg(D);
3467 0 : v = cgetg(l, t_VECSMALL);
3468 0 : for (i = j = 1; i < l; i++)
3469 : {
3470 0 : ulong d = D[i];
3471 0 : if (n % d == 0) v[j++] = d;
3472 : }
3473 0 : setlg(v,j); return v;
3474 : }
3475 :
3476 : /* for some prime divisors of N, Tr^new(p) = 0 */
3477 : static int
3478 232579 : trconj(GEN T, long N, long n)
3479 232579 : { return (lg(T) > 1 && N % n == 0 && zv_search(T, n)); }
3480 :
3481 : /* n > 0; trace formula on new space */
3482 : static GEN
3483 2628827 : mfnewtrace_i(long N, long k, long n, cachenew_t *cache)
3484 : {
3485 2628827 : GEN VCHIP, s, Dn, DN1, SN, S = cache->DATA;
3486 : long FC, N1, N2, N1N2, g, i, j, lDN1;
3487 :
3488 2628827 : if (!S) return gen_0;
3489 2628827 : SN = gel(S,N);
3490 2628827 : if (mfnewchkzero(gel(SN,_NEWLZ), n)) return gen_0;
3491 1903311 : if (k > 2 && trconj(gel(SN,_TRCONJ), N, n)) return gen_0;
3492 1903283 : VCHIP = gel(SN, _VCHIP); FC = vchip_FC(VCHIP);
3493 1903283 : N1 = N/FC; newt_params(N1, n, FC, &g, &N2);
3494 1903283 : N1N2 = N1/N2;
3495 1903283 : DN1 = mydivisorsu(N1N2); lDN1 = lg(DN1);
3496 1903283 : N2 *= FC;
3497 1903283 : Dn = mydivisorsu(n); /* this one is probably out of cache */
3498 1903283 : s = gmulsg(mubeta2(N1N2,n), mfcusptracecache(N2, k, n, Dn, gel(S,N2), cache));
3499 7390101 : for (i = 2; i < lDN1; i++)
3500 : { /* skip M1 = 1, done above */
3501 5486818 : long M1 = DN1[i], N1M1 = DN1[lDN1-i];
3502 5486818 : GEN Dg = mydivisorsu(ugcd(M1, g));
3503 5486818 : M1 *= N2;
3504 5486818 : s = gadd(s, gmulsg(mubeta2(N1M1,n),
3505 5486818 : mfcusptracecache(M1, k, n, Dn, gel(S,M1), cache)));
3506 5811261 : for (j = 2; j < lg(Dg); j++) /* skip d = 1, done above */
3507 : {
3508 324443 : long d = Dg[j], ndd = n/(d*d), M = M1/d;
3509 324443 : GEN z = mulsi(mubeta2(N1M1,ndd), powuu(d,k-1)), C = vchip_lift(VCHIP,d,z);
3510 324443 : GEN Dndd = div_restrict(Dn, ndd);
3511 324443 : s = gadd(s, gmul(C, mfcusptracecache(M, k, ndd, Dndd, gel(S,M), cache)));
3512 : }
3513 5486818 : s = vchip_mod(VCHIP, s);
3514 : }
3515 1903283 : return vchip_polmod(VCHIP, s);
3516 : }
3517 :
3518 : static GEN
3519 12355 : get_DIH(long N)
3520 : {
3521 12355 : GEN x = cache_get(cache_DIH, N);
3522 12355 : return x? gcopy(x): mfdihedral(N);
3523 : }
3524 : static GEN
3525 2373 : get_vDIH(long N, GEN D)
3526 : {
3527 2373 : GEN x = const_vec(N, NULL);
3528 : long i, l;
3529 2373 : if (!D) D = mydivisorsu(N);
3530 2373 : l = lg(D);
3531 14504 : for (i = 1; i < l; i++) { long d = D[i]; gel(x, d) = get_DIH(d); }
3532 2373 : return x;
3533 : }
3534 :
3535 : /* divisors of N which are multiple of F */
3536 : static GEN
3537 322 : divisorsNF(long N, long F)
3538 : {
3539 322 : GEN D = mydivisorsu(N / F);
3540 322 : long l = lg(D), i;
3541 833 : for (i = 1; i < l; i++) D[i] = N / D[i];
3542 322 : return D;
3543 : }
3544 : /* mfcuspdim(N,k,CHI) - mfnewdim(N,k,CHI); CHIP primitive (for efficiency) */
3545 : static long
3546 8442 : mfolddim_i(long N, long k, GEN CHIP, GEN vSP)
3547 : {
3548 8442 : long S, i, l, F = mfcharmodulus(CHIP), N1 = N / F, N2;
3549 : GEN D;
3550 8442 : newd_params(N1, &N2); /* will ensure mubeta != 0 */
3551 8442 : D = mydivisorsu(N1/N2); l = lg(D); S = 0;
3552 8442 : if (k == 1 && !vSP) vSP = get_vDIH(N, divisorsNF(N, F));
3553 32627 : for (i = 2; i < l; i++)
3554 : {
3555 24185 : long d = mfcuspdim_i(N / D[i], k, CHIP, vSP);
3556 24185 : if (d) S -= mubeta(D[i]) * d;
3557 : }
3558 8442 : return S;
3559 : }
3560 : long
3561 224 : mfolddim(long N, long k, GEN CHI)
3562 : {
3563 224 : pari_sp av = avma;
3564 224 : GEN CHIP = mfchartoprimitive(CHI, NULL);
3565 224 : return gc_long(av, mfolddim_i(N, k, CHIP, NULL));
3566 : }
3567 : /* Only depends on CHIP the primitive char attached to CHI; assumes !badchar */
3568 : long
3569 15911 : mfnewdim(long N, long k, GEN CHI)
3570 : {
3571 : pari_sp av;
3572 : long S, F;
3573 15911 : GEN vSP, CHIP = mfchartoprimitive(CHI, &F);
3574 15911 : vSP = (k == 1)? get_vDIH(N, divisorsNF(N, F)): NULL;
3575 15911 : S = mfcuspdim_i(N, k, CHIP, vSP); if (!S) return 0;
3576 7945 : av = avma; return gc_long(av, S - mfolddim_i(N, k, CHIP, vSP));
3577 : }
3578 :
3579 : /* trace form, given as closure */
3580 : static GEN
3581 980 : mftraceform_new(long N, long k, GEN CHI)
3582 : {
3583 : GEN T;
3584 980 : if (k == 1) return initwt1newtrace(mfinit_Nkchi(N, 1, CHI, mf_CUSP, 0));
3585 959 : T = initnewtrace(N,CHI); if (!T) return mftrivial();
3586 959 : return tag(t_MF_NEWTRACE, mkNK(N,k,CHI), T);
3587 : }
3588 : static GEN
3589 14 : mftraceform_cusp(long N, long k, GEN CHI)
3590 : {
3591 14 : if (k == 1) return initwt1trace(mfinit_Nkchi(N, 1, CHI, mf_CUSP, 0));
3592 7 : return tag(t_MF_TRACE, mkNK(N,k,CHI), inittrace(N,CHI,NULL));
3593 : }
3594 : static GEN
3595 98 : mftraceform_i(GEN NK, long space)
3596 : {
3597 : GEN CHI;
3598 : long N, k;
3599 98 : checkNK(NK, &N, &k, &CHI, 0);
3600 98 : if (!mfdim_Nkchi(N, k, CHI, space)) return mftrivial();
3601 77 : switch(space)
3602 : {
3603 56 : case mf_NEW: return mftraceform_new(N, k, CHI);
3604 14 : case mf_CUSP:return mftraceform_cusp(N, k, CHI);
3605 : }
3606 7 : pari_err_DOMAIN("mftraceform", "space", "=", utoi(space), NK);
3607 : return NULL;/*LCOV_EXCL_LINE*/
3608 : }
3609 : GEN
3610 98 : mftraceform(GEN NK, long space)
3611 98 : { pari_sp av = avma; return gc_GEN(av, mftraceform_i(NK,space)); }
3612 :
3613 : static GEN
3614 18186 : hecke_data(long N, long n)
3615 18186 : { return mkvecsmall3(n, u_ppo(n, N), N); }
3616 : /* 1/2-integral weight */
3617 : static GEN
3618 84 : heckef2_data(long N, long n)
3619 : {
3620 : ulong f, fN, fN2;
3621 84 : if (!uissquareall(n, &f)) return NULL;
3622 77 : fN = u_ppo(f, N); fN2 = fN*fN;
3623 77 : return mkvec2(myfactoru(fN), mkvecsmall4(n, N, fN2, n/fN2));
3624 : }
3625 : /* N = mf_get_N(F) or a multiple */
3626 : static GEN
3627 25368 : mfhecke_i(long n, long N, GEN F)
3628 : {
3629 25368 : if (n == 1) return F;
3630 17815 : return tag2(t_MF_HECKE, mf_get_NK(F), hecke_data(N,n), F);
3631 : }
3632 :
3633 : GEN
3634 105 : mfhecke(GEN mf, GEN F, long n)
3635 : {
3636 105 : pari_sp av = avma;
3637 : GEN NK, CHI, gk, DATA;
3638 : long N, nk, dk;
3639 105 : mf = checkMF(mf);
3640 105 : if (!checkmf_i(F)) pari_err_TYPE("mfhecke",F);
3641 105 : if (n <= 0) pari_err_TYPE("mfhecke [n <= 0]", stoi(n));
3642 105 : if (n == 1) return gcopy(F);
3643 105 : gk = mf_get_gk(F);
3644 105 : Qtoss(gk,&nk,&dk);
3645 105 : CHI = mf_get_CHI(F);
3646 105 : N = MF_get_N(mf);
3647 105 : if (dk == 2)
3648 : {
3649 77 : DATA = heckef2_data(N,n);
3650 77 : if (!DATA) return mftrivial();
3651 : }
3652 : else
3653 28 : DATA = hecke_data(N,n);
3654 98 : NK = mkgNK(lcmii(stoi(N), mf_get_gN(F)), gk, CHI, mf_get_field(F));
3655 98 : return gc_GEN(av, tag2(t_MF_HECKE, NK, DATA, F));
3656 : }
3657 :
3658 : /* form F given by closure, compute B(d)(F) as closure (q -> q^d) */
3659 : static GEN
3660 36456 : mfbd_i(GEN F, long d)
3661 : {
3662 : GEN D, NK, gk, CHI;
3663 36456 : if (d == 1) return F;
3664 13587 : if (d <= 0) pari_err_TYPE("mfbd [d <= 0]", stoi(d));
3665 13587 : if (mf_get_type(F) != t_MF_BD) D = utoi(d);
3666 7 : else { D = mului(d, gel(F,3)); F = gel(F,2); }
3667 13587 : gk = mf_get_gk(F); CHI = mf_get_CHI(F);
3668 13587 : if (typ(gk) != t_INT) CHI = mfcharmul(CHI, get_mfchar(utoi(d << 2)));
3669 13587 : NK = mkgNK(muliu(mf_get_gN(F), d), gk, CHI, mf_get_field(F));
3670 13587 : return tag2(t_MF_BD, NK, F, D);
3671 : }
3672 : GEN
3673 266 : mfbd(GEN F, long d)
3674 : {
3675 266 : pari_sp av = avma;
3676 266 : if (!checkmf_i(F)) pari_err_TYPE("mfbd",F);
3677 266 : return gc_GEN(av, mfbd_i(F, d));
3678 : }
3679 :
3680 : /* A[i+1] = a(t*i^2) */
3681 : static GEN
3682 105 : RgV_shimura(GEN A, long n, long t, long N, long r, GEN CHI)
3683 : {
3684 105 : GEN R, a0, Pn = mfcharpol(CHI);
3685 105 : long m, st, ord = mfcharorder(CHI), vt = varn(Pn), Nt = t == 1? N: ulcm(N,t);
3686 :
3687 105 : R = cgetg(n + 2, t_VEC);
3688 105 : st = odd(r)? -t: t;
3689 105 : a0 = gel(A, 1);
3690 105 : if (!gequal0(a0))
3691 : {
3692 14 : long o = mfcharorder(CHI);
3693 14 : if (st != 1 && odd(o)) o <<= 1;
3694 14 : a0 = gmul(a0, charLFwtk(Nt, r, CHI, o, st));
3695 : }
3696 105 : gel(R, 1) = a0;
3697 637 : for (m = 1; m <= n; m++)
3698 : {
3699 532 : GEN Dm = mydivisorsu(u_ppo(m, Nt)), S = gel(A, m*m + 1);
3700 532 : long i, l = lg(Dm);
3701 805 : for (i = 2; i < l; i++)
3702 : { /* (e,Nt) = 1; skip i = 1: e = 1, done above */
3703 273 : long e = Dm[i], me = m / e, a = mfcharevalord(CHI, e, ord);
3704 273 : GEN c, C = powuu(e, r - 1);
3705 273 : if (kross(st, e) == -1) C = negi(C);
3706 273 : c = Qab_Czeta(a, ord, C, vt);
3707 273 : S = gadd(S, gmul(c, gel(A, me*me + 1)));
3708 : }
3709 532 : gel(R, m+1) = S;
3710 : }
3711 105 : return degpol(Pn) > 1? gmodulo(R, Pn): R;
3712 : }
3713 :
3714 : static long
3715 28 : mfisinkohnen(GEN mf, GEN F)
3716 : {
3717 28 : GEN v, gk = MF_get_gk(mf), CHI = MF_get_CHI(mf);
3718 28 : long i, eps, N4 = MF_get_N(mf) >> 2, sb = mfsturmNgk(N4 << 4, gk) + 1;
3719 28 : eps = N4 % mfcharconductor(CHI)? -1 : 1;
3720 28 : if (odd(MF_get_r(mf))) eps = -eps;
3721 28 : v = mfcoefs(F, sb, 1);
3722 686 : for (i = 2; i <= sb; i+=4) if (!gequal0(gel(v,i+1))) return 0;
3723 245 : for (i = 2+eps; i <= sb; i+=4) if (!gequal0(gel(v,i+1))) return 0;
3724 14 : return 1;
3725 : }
3726 :
3727 : static long
3728 42 : mfshimura_space_cusp(GEN mf)
3729 : {
3730 : long N4;
3731 42 : if (MF_get_r(mf) == 1 && (N4 = MF_get_N(mf) >> 2) >= 4)
3732 : {
3733 21 : GEN E = gel(myfactoru(N4), 2);
3734 21 : long ma = vecsmall_max(E);
3735 21 : if (ma > 2 || (ma == 2 && !mfcharistrivial(MF_get_CHI(mf)))) return 0;
3736 : }
3737 28 : return 1;
3738 : }
3739 :
3740 : /* D is either a discriminant (not necessarily fundamental) with
3741 : sign(D)=(-1)^{k-1/2}*eps, or a positive squarefree integer t, which is then
3742 : transformed into a fundamental discriminant of the correct sign. */
3743 : GEN
3744 49 : mfshimura(GEN mf, GEN F, long t)
3745 : {
3746 49 : pari_sp av = avma;
3747 : GEN G, res, mf2, CHI;
3748 49 : long sb, M, r, N, space = mf_FULL;
3749 :
3750 49 : if (!checkmf_i(F)) pari_err_TYPE("mfshimura",F);
3751 49 : mf = checkMF(mf);
3752 49 : r = MF_get_r(mf);
3753 49 : if (r <= 0) pari_err_DOMAIN("mfshimura", "weight", "<=", ghalf, mf_get_gk(F));
3754 49 : if (t <= 0 || !uissquarefree(t)) pari_err_TYPE("mfshimura [t]", stoi(t));
3755 42 : N = MF_get_N(mf); M = N >> 1;
3756 42 : if (mfiscuspidal(mf,F))
3757 : {
3758 28 : if (mfshimura_space_cusp(mf)) space = mf_CUSP;
3759 28 : if (mfisinkohnen(mf,F)) M = N >> 2;
3760 : }
3761 42 : CHI = MF_get_CHI(mf);
3762 42 : mf2 = mfinit_Nkchi(M, r << 1, mfcharpow(CHI, gen_2), space, 0);
3763 42 : sb = mfsturm(mf2);
3764 42 : G = RgV_shimura(mfcoefs_i(F, sb*sb, t), sb, t, N, r, CHI);
3765 42 : res = mftobasis_i(mf2, G);
3766 : /* not mflinear(mf2,): we want lowest possible level */
3767 42 : G = mflinear(MF_get_basis(mf2), res);
3768 42 : return gc_GEN(av, mkvec3(mf2, G, res));
3769 : }
3770 :
3771 : /* W ZabM (ZM if n = 1), a t_INT or NULL, b t_INT, ZXQ mod P or NULL.
3772 : * Write a/b = A/d with d t_INT and A Zab return [W,d,A,P] */
3773 : static GEN
3774 7791 : mkMinv(GEN W, GEN a, GEN b, GEN P)
3775 : {
3776 7791 : GEN A = (b && typ(b) == t_POL)? Q_remove_denom(QXQ_inv(b,P), &b): NULL;
3777 7791 : if (a && b)
3778 : {
3779 1351 : a = Qdivii(a,b);
3780 1351 : if (typ(a) == t_INT) b = gen_1; else { b = gel(a,2); a = gel(a,1); }
3781 1351 : if (is_pm1(a)) a = NULL;
3782 : }
3783 7791 : if (a) A = A? ZX_Z_mul(A,a): a; else if (!A) A = gen_1;
3784 7791 : if (!b) b = gen_1;
3785 7791 : if (!P) P = gen_0;
3786 7791 : return mkvec4(W,b,A,P);
3787 : }
3788 : /* M square invertible QabM, return [M',d], M*M' = d*Id */
3789 : static GEN
3790 609 : QabM_Minv(GEN M, GEN P, long n)
3791 : {
3792 : GEN dW, W, dM;
3793 609 : M = Q_remove_denom(M, &dM);
3794 609 : W = P? ZabM_inv(liftpol_shallow(M), P, n, &dW): ZM_inv(M, &dW);
3795 609 : return mkMinv(W, dM, dW, P);
3796 : }
3797 : /* Simplified form of mfclean, after a QabM_indexrank: M a ZabM with full
3798 : * column rank and z = indexrank(M) is known */
3799 : static GEN
3800 861 : mfclean2(GEN M, GEN z, GEN P, long n)
3801 : {
3802 861 : GEN d, Minv, y = gel(z,1), W = rowpermute(M, y);
3803 861 : W = P? ZabM_inv(liftpol_shallow(W), P, n, &d): ZM_inv(W, &d);
3804 861 : M = rowslice(M, 1, y[lg(y)-1]);
3805 861 : Minv = mkMinv(W, NULL, d, P);
3806 861 : return mkvec3(y, Minv, M);
3807 : }
3808 : /* M QabM, lg(M)>1 and [y,z] its rank profile. Let Minv be the inverse of the
3809 : * invertible square matrix in mkMinv format. Return [y,Minv, M[..y[#y],]]
3810 : * P cyclotomic polynomial of order n > 2 or NULL */
3811 : static GEN
3812 5026 : mfclean(GEN M, GEN P, long n, int ratlift)
3813 : {
3814 5026 : GEN W, v, y, z, d, Minv, dM, MdM = Q_remove_denom(M, &dM);
3815 5026 : if (n <= 2)
3816 3920 : W = ZM_pseudoinv(MdM, &v, &d);
3817 : else
3818 1106 : W = ZabM_pseudoinv_i(liftpol_shallow(MdM), P, n, &v, &d, ratlift);
3819 5026 : y = gel(v,1);
3820 5026 : z = gel(v,2);
3821 5026 : if (lg(z) != lg(MdM)) M = vecpermute(M,z);
3822 5026 : M = rowslice(M, 1, y[lg(y)-1]);
3823 5026 : Minv = mkMinv(W, dM, d, P);
3824 5026 : return mkvec3(y, Minv, M);
3825 : }
3826 : /* call mfclean using only CHI */
3827 : static GEN
3828 4074 : mfcleanCHI(GEN M, GEN CHI, int ratlift)
3829 : {
3830 4074 : long n = mfcharorder(CHI);
3831 4074 : GEN P = (n <= 2)? NULL: mfcharpol(CHI);
3832 4074 : return mfclean(M, P, n, ratlift);
3833 : }
3834 :
3835 : /* DATA component of a t_MF_NEWTRACE. Was it stripped to save memory ? */
3836 : static int
3837 34293 : newtrace_stripped(GEN DATA)
3838 34293 : { return DATA && (lg(DATA) == 5 && typ(gel(DATA,3)) == t_INT); }
3839 : /* f a t_MF_NEWTRACE */
3840 : static GEN
3841 34293 : newtrace_DATA(long N, GEN f)
3842 : {
3843 34293 : GEN DATA = gel(f,2);
3844 34293 : return newtrace_stripped(DATA)? initnewtrace(N, DATA): DATA;
3845 : }
3846 : /* reset cachenew for new level incorporating new DATA, tf a t_MF_NEWTRACE
3847 : * (+ possibly initialize 'full' for new allowed levels) */
3848 : static void
3849 34293 : reset_cachenew(cachenew_t *cache, long N, GEN tf)
3850 : {
3851 : long i, n, l;
3852 34293 : GEN v, DATA = newtrace_DATA(N,tf);
3853 34293 : cache->DATA = DATA;
3854 34293 : if (!DATA) return;
3855 34188 : n = cache->n;
3856 34188 : v = cache->vfull; l = N+1; /* = lg(DATA) */
3857 2213008 : for (i = 1; i < l; i++)
3858 2178820 : if (typ(gel(v,i)) == t_INT && lg(gel(DATA,i)) != 1)
3859 54236 : gel(v,i) = const_vec(n, NULL);
3860 34188 : cache->VCHIP = gel(gel(DATA,N),_VCHIP);
3861 : }
3862 : /* initialize a cache of newtrace / cusptrace up to index n and level | N;
3863 : * DATA may be NULL (<=> Tr^new = 0). tf a t_MF_NEWTRACE */
3864 : static void
3865 13566 : init_cachenew(cachenew_t *cache, long n, long N, GEN tf)
3866 : {
3867 13566 : long i, l = N+1; /* = lg(tf.DATA) when DATA != NULL */
3868 : GEN v;
3869 13566 : cache->n = n;
3870 13566 : cache->vnew = v = cgetg(l, t_VEC);
3871 952987 : for (i = 1; i < l; i++) gel(v,i) = (N % i)? gen_0: const_vec(n, NULL);
3872 13566 : cache->newHIT = cache->newTOTAL = cache->cuspHIT = cache->cuspTOTAL = 0;
3873 13566 : cache->vfull = v = zerovec(N);
3874 13566 : reset_cachenew(cache, N, tf);
3875 13566 : }
3876 : static void
3877 17626 : dbg_cachenew(cachenew_t *C)
3878 : {
3879 17626 : if (DEBUGLEVEL >= 2 && C)
3880 0 : err_printf("newtrace cache hits: new = %ld/%ld, cusp = %ld/%ld\n",
3881 : C->newHIT, C->newTOTAL, C->cuspHIT, C->cuspTOTAL);
3882 17626 : }
3883 :
3884 : /* newtrace_{N,k}(d*i), i = n0, ..., n */
3885 : static GEN
3886 184261 : colnewtrace(long n0, long n, long d, long N, long k, cachenew_t *cache)
3887 : {
3888 184261 : GEN v = cgetg(n-n0+2, t_COL);
3889 : long i;
3890 4767189 : for (i = n0; i <= n; i++) gel(v, i-n0+1) = mfnewtracecache(N, k, i*d, cache);
3891 184261 : return v;
3892 : }
3893 : /* T_n(l*m0, l*(m0+1), ..., l*m) F, F = t_MF_NEWTRACE [N,k],DATA, cache
3894 : * contains DATA != NULL as well as cached values of F */
3895 : static GEN
3896 90482 : heckenewtrace(long m0, long m, long l, long N, long NBIG, long k, long n, cachenew_t *cache)
3897 : {
3898 90482 : long lD, a, k1, nl = n*l;
3899 90482 : GEN D, V, v = colnewtrace(m0, m, nl, N, k, cache); /* d=1 */
3900 : GEN VCHIP;
3901 90482 : if (n == 1) return v;
3902 62412 : VCHIP = cache->VCHIP;
3903 62412 : D = mydivisorsu(u_ppo(n, NBIG)); lD = lg(D);
3904 62412 : k1 = k - 1;
3905 154273 : for (a = 2; a < lD; a++)
3906 : { /* d > 1, (d,NBIG) = 1 */
3907 91861 : long i, j, d = D[a], c = ugcd(l, d), dl = d/c, m0d = ceildivuu(m0, dl);
3908 91861 : GEN C = vchip_lift(VCHIP, d, powuu(d, k1));
3909 : /* m0=0: i = 1 => skip F(0) = 0 */
3910 91861 : if (!m0) { i = 1; j = dl; } else { i = 0; j = m0d*dl; }
3911 91861 : V = colnewtrace(m0d, m/dl, nl/(d*c), N, k, cache);
3912 : /* C = chi(d) d^(k-1) */
3913 1102738 : for (; j <= m; i++, j += dl)
3914 1010877 : gel(v,j-m0+1) = gadd(gel(v,j-m0+1), vchip_mod(VCHIP, gmul(C,gel(V,i+1))));
3915 : }
3916 62412 : return v;
3917 : }
3918 :
3919 : /* Given v = an[i], return an[d*i], i=0..n */
3920 : static GEN
3921 2618 : anextract(GEN v, long n, long d)
3922 : {
3923 2618 : long i, id, l = n + 2;
3924 2618 : GEN w = cgetg(l, t_VEC);
3925 2618 : if (d == 1)
3926 7245 : for (i = 1; i < l; i++) gel(w, i) = gel(v, i);
3927 : else
3928 22036 : for (i = id = 1; i < l; i++, id += d) gel(w, i) = gel(v, id);
3929 2618 : return w;
3930 : }
3931 : /* T_n(F)(0, l, ..., l*m) */
3932 : static GEN
3933 2709 : hecke_i(long m, long l, GEN V, GEN F, GEN DATA)
3934 : {
3935 : long k, n, nNBIG, NBIG, lD, M, a, t, nl;
3936 : GEN D, v, CHI;
3937 2709 : if (typ(DATA) == t_VEC)
3938 : { /* 1/2-integral k */
3939 98 : if (!V) { GEN S = gel(DATA,2); V = mfcoefs_i(F, m*l*S[3], S[4]); }
3940 98 : return RgV_heckef2(m, l, V, F, DATA);
3941 : }
3942 2611 : k = mf_get_k(F);
3943 2611 : n = DATA[1]; nl = n*l;
3944 2611 : nNBIG = DATA[2];
3945 2611 : NBIG = DATA[3];
3946 2611 : if (nNBIG == 1) return V? V: mfcoefs_i(F,m,nl);
3947 1855 : if (!V && mf_get_type(F) == t_MF_NEWTRACE)
3948 : { /* inline F to allow cache, T_n at level NBIG acting on Tr^new(N,k,CHI) */
3949 : cachenew_t cache;
3950 546 : long N = mf_get_N(F);
3951 546 : init_cachenew(&cache, m*nl, N, F);
3952 546 : v = heckenewtrace(0, m, l, N, NBIG, k, n, &cache);
3953 546 : dbg_cachenew(&cache);
3954 546 : settyp(v, t_VEC); return v;
3955 : }
3956 1309 : CHI = mf_get_CHI(F);
3957 1309 : D = mydivisorsu(nNBIG); lD = lg(D);
3958 1309 : M = m + 1;
3959 1309 : t = nNBIG * ugcd(nNBIG, l);
3960 1309 : if (!V) V = mfcoefs_i(F, m * t, nl / t); /* usually nl = t */
3961 1309 : v = anextract(V, m, t); /* mfcoefs(F, m, nl); d = 1 */
3962 2618 : for (a = 2; a < lD; a++)
3963 : { /* d > 1, (d, NBIG) = 1 */
3964 1309 : long d = D[a], c = ugcd(l, d), dl = d/c, i, idl;
3965 1309 : GEN C = gmul(mfchareval(CHI, d), powuu(d, k-1));
3966 1309 : GEN w = anextract(V, m/dl, t/(d*c)); /* mfcoefs(F, m/dl, nl/(d*c)) */
3967 7245 : for (i = idl = 1; idl <= M; i++, idl += dl)
3968 5936 : gel(v,idl) = gadd(gel(v,idl), gmul(C, gel(w,i)));
3969 : }
3970 1309 : return v;
3971 : }
3972 :
3973 : static GEN
3974 12439 : mkmf(GEN x1, GEN x2, GEN x3, GEN x4, GEN x5)
3975 : {
3976 12439 : GEN MF = obj_init(5, MF_SPLITN);
3977 12439 : gel(MF,1) = x1;
3978 12439 : gel(MF,2) = x2;
3979 12439 : gel(MF,3) = x3;
3980 12439 : gel(MF,4) = x4;
3981 12439 : gel(MF,5) = x5; return MF;
3982 : }
3983 :
3984 : /* return an integer b such that p | b => T_p^k Tr^new = 0, for all k > 0 */
3985 : static long
3986 7686 : get_badj(long N, long FC)
3987 : {
3988 7686 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
3989 7686 : long i, b = 1, l = lg(P);
3990 20391 : for (i = 1; i < l; i++)
3991 12705 : if (E[i] > 1 && u_lval(FC, P[i]) < E[i]) b *= P[i];
3992 7686 : return b;
3993 : }
3994 : /* in place, assume perm strictly increasing */
3995 : static void
3996 1358 : vecpermute_inplace(GEN v, GEN perm)
3997 : {
3998 1358 : long i, l = lg(perm);
3999 11690 : for (i = 1; i < l; i++) gel(v,i) = gel(v,perm[i]);
4000 1358 : }
4001 :
4002 : /* Find basis of newspace using closures; assume k >= 2 and !badchar.
4003 : * Return NULL if space is empty, else
4004 : * [mf1, list of closures T(j)traceform, list of corresponding j, matrix] */
4005 : static GEN
4006 15666 : mfnewinit(long N, long k, GEN CHI, cachenew_t *cache, long init)
4007 : {
4008 : GEN S, vj, M, CHIP, mf1, listj, P, tf;
4009 : long j, ct, ctlj, dim, jin, SB, sb, two, ord, FC, badj;
4010 :
4011 15666 : dim = mfnewdim(N, k, CHI);
4012 15666 : if (!dim && !init) return NULL;
4013 7686 : sb = mfsturmNk(N, k);
4014 7686 : CHIP = mfchartoprimitive(CHI, &FC);
4015 : /* remove newtrace data from S to save space in output: negligible slowdown */
4016 7686 : tf = tag(t_MF_NEWTRACE, mkNK(N,k,CHIP), CHIP);
4017 7686 : badj = get_badj(N, FC);
4018 : /* try sbsmall first: Sturm bound not sharp for new space */
4019 7686 : SB = ceilA1(N, k);
4020 7686 : listj = cgetg(2*sb + 3, t_VECSMALL);
4021 374899 : for (j = ctlj = 1; ctlj < 2*sb + 3; j++)
4022 367213 : if (ugcd(j, badj) == 1) listj[ctlj++] = j;
4023 7686 : if (init)
4024 : {
4025 4172 : init_cachenew(cache, (SB+1)*listj[dim+1], N, tf);
4026 4172 : if (init == -1 || !dim) return NULL; /* old space or dim = 0 */
4027 : }
4028 : else
4029 3514 : reset_cachenew(cache, N, tf);
4030 : /* cache.DATA is not NULL */
4031 7217 : ord = mfcharorder(CHIP);
4032 7217 : P = ord <= 2? NULL: mfcharpol(CHIP);
4033 7217 : vj = cgetg(dim+1, t_VECSMALL);
4034 7217 : M = cgetg(dim+1, t_MAT);
4035 7224 : for (two = 1, ct = 0, jin = 1; two <= 2; two++)
4036 : {
4037 7224 : long a, jlim = jin + sb;
4038 22533 : for (a = jin; a <= jlim; a++)
4039 : {
4040 : GEN z, vecz;
4041 22526 : ct++; vj[ct] = listj[a];
4042 22526 : gel(M, ct) = heckenewtrace(0, SB, 1, N, N, k, vj[ct], cache);
4043 22526 : if (ct < dim) continue;
4044 :
4045 7896 : z = QabM_indexrank(M, P, ord);
4046 7896 : vecz = gel(z, 2); ct = lg(vecz) - 1;
4047 7896 : if (ct == dim) { M = mkvec3(z, gen_0, M); break; } /*maximal rank, done*/
4048 679 : vecpermute_inplace(M, vecz);
4049 679 : vecpermute_inplace(vj, vecz);
4050 : }
4051 7224 : if (a <= jlim) break;
4052 : /* sbsmall was not sufficient, use Sturm bound: must extend M */
4053 70 : for (j = 1; j <= ct; j++)
4054 : {
4055 63 : GEN t = heckenewtrace(SB + 1, sb, 1, N, N, k, vj[j], cache);
4056 63 : gel(M,j) = shallowconcat(gel(M, j), t);
4057 : }
4058 7 : jin = jlim + 1; SB = sb;
4059 : }
4060 7217 : S = cgetg(dim + 1, t_VEC);
4061 29022 : for (j = 1; j <= dim; j++) gel(S, j) = mfhecke_i(vj[j], N, tf);
4062 7217 : dbg_cachenew(cache);
4063 7217 : mf1 = mkvec4(utoipos(N), utoipos(k), CHI, utoi(mf_NEW));
4064 7217 : return mkmf(mf1, cgetg(1,t_VEC), S, vj, M);
4065 : }
4066 : /* k > 1 integral, mf space is mf_CUSP or mf_FULL */
4067 : static GEN
4068 49 : mfinittonew(GEN mf)
4069 : {
4070 49 : GEN CHI = MF_get_CHI(mf), S = MF_get_S(mf), vMjd = MFcusp_get_vMjd(mf);
4071 49 : GEN M = MF_get_M(mf), vj, mf1;
4072 49 : long i, j, l, l0 = lg(S), N0 = MF_get_N(mf);
4073 252 : for (i = l0-1; i > 0; i--)
4074 : {
4075 238 : long N = gel(vMjd,i)[1];
4076 238 : if (N != N0) break;
4077 : }
4078 49 : if (i == l0-1) return NULL;
4079 42 : S = vecslice(S, i+1, l0-1); /* forms of conductor N0 */
4080 42 : l = lg(S); vj = cgetg(l, t_VECSMALL);
4081 245 : for (j = 1; j < l; j++) vj[j] = gel(vMjd,j+i)[2];
4082 42 : M = vecslice(M, lg(M)-lg(S)+1, lg(M)-1); /* their coefficients */
4083 42 : M = mfcleanCHI(M, CHI, 0);
4084 42 : mf1 = mkvec4(utoipos(N0), MF_get_gk(mf), CHI, utoi(mf_NEW));
4085 42 : return mkmf(mf1, cgetg(1,t_VEC), S, vj, M);
4086 : }
4087 :
4088 : /* Bd(f)[m0..m], v = f[ceil(m0/d)..floor(m/d)], m0d = ceil(m0/d) */
4089 : static GEN
4090 83601 : RgC_Bd_expand(long m0, long m, GEN v, long d, long m0d)
4091 : {
4092 : long i, j;
4093 : GEN w;
4094 83601 : if (d == 1) return v;
4095 23758 : w = zerocol(m-m0+1);
4096 23758 : if (!m0) { i = 1; j = d; } else { i = 0; j = m0d*d; }
4097 470575 : for (; j <= m; i++, j += d) gel(w,j-m0+1) = gel(v,i+1);
4098 23758 : return w;
4099 : }
4100 : /* S a nonempty vector of t_MF_BD(t_MF_HECKE(t_MF_NEWTRACE)); M the matrix
4101 : * of their coefficients r*0, r*1, ..., r*m0 (~ mfvectomat) or NULL (empty),
4102 : * extend it to coeffs up to m > m0. The forms B_d(T_j(tf_N))in S should be
4103 : * sorted by level N, then j, then increasing d. No reordering here. */
4104 : static GEN
4105 9212 : bhnmat_extend(GEN M, long m, long r, GEN S, cachenew_t *cache)
4106 : {
4107 9212 : long i, mr, m0, m0r, Nold = 0, jold = 0, l = lg(S);
4108 9212 : GEN MAT = cgetg(l, t_MAT), v = NULL;
4109 9212 : if (M) { m0 = nbrows(M); m0r = m0 * r; } else m0 = m0r = 0;
4110 9212 : mr = m*r;
4111 92813 : for (i = 1; i < l; i++)
4112 : {
4113 : long d, j, md, N;
4114 83601 : GEN c, f = bhn_parse(gel(S,i), &d,&j); /* t_MF_NEWTRACE */
4115 83601 : N = mf_get_N(f);
4116 83601 : md = ceildivuu(m0r,d);
4117 83601 : if (N != Nold) { reset_cachenew(cache, N, f); Nold = N; jold = 0; }
4118 83601 : if (!cache->DATA) { gel(MAT,i) = zerocol(m+1); continue; }
4119 83601 : if (j != jold || md)
4120 67347 : { v = heckenewtrace(md, mr/d, 1, N, N, mf_get_k(f), j,cache); jold=j; }
4121 83601 : c = RgC_Bd_expand(m0r, mr, v, d, md);
4122 83601 : if (r > 1) c = c_deflate(m-m0, r, c);
4123 83601 : if (M) c = shallowconcat(gel(M,i), c);
4124 83601 : gel(MAT,i) = c;
4125 : }
4126 9212 : return MAT;
4127 : }
4128 :
4129 : /* k > 1 */
4130 : static GEN
4131 3241 : mfinitcusp(long N, long k, GEN CHI, cachenew_t *cache, long space)
4132 : {
4133 : long L, l, lDN1, FC, N1, d1, i, init;
4134 3241 : GEN vS, vMjd, DN1, vmf, CHIP = mfchartoprimitive(CHI, &FC);
4135 :
4136 3241 : d1 = (space == mf_OLD)? mfolddim_i(N, k, CHIP, NULL): mfcuspdim(N, k, CHIP);
4137 3241 : if (!d1) return NULL;
4138 2933 : N1 = N/FC; DN1 = mydivisorsu(N1); lDN1 = lg(DN1);
4139 2933 : init = (space == mf_OLD)? -1: 1;
4140 2933 : vmf = cgetg(lDN1, t_VEC);
4141 17360 : for (i = lDN1 - 1, l = 1; i; i--)
4142 : { /* by decreasing level to allow cache */
4143 14427 : GEN mf = mfnewinit(FC*DN1[i], k, CHIP, cache, init);
4144 14427 : if (mf) gel(vmf, l++) = mf;
4145 14427 : init = 0;
4146 : }
4147 2933 : setlg(vmf,l); vmf = vecreverse(vmf); /* reorder by increasing level */
4148 :
4149 2933 : L = mfsturmNk(N, k)+1;
4150 2933 : vS = vectrunc_init(L);
4151 2933 : vMjd = vectrunc_init(L);
4152 9282 : for (i = 1; i < l; i++)
4153 : {
4154 6349 : GEN DNM, mf = gel(vmf,i), S = MF_get_S(mf), vj = MFnew_get_vj(mf);
4155 6349 : long a, lDNM, lS = lg(S), M = MF_get_N(mf);
4156 6349 : DNM = mydivisorsu(N / M); lDNM = lg(DNM);
4157 25998 : for (a = 1; a < lS; a++)
4158 : {
4159 19649 : GEN tf = gel(S,a);
4160 19649 : long b, j = vj[a];
4161 48727 : for (b = 1; b < lDNM; b++)
4162 : {
4163 29078 : long d = DNM[b];
4164 29078 : vectrunc_append(vS, mfbd_i(tf, d));
4165 29078 : vectrunc_append(vMjd, mkvecsmall3(M, j, d));
4166 : }
4167 : }
4168 : }
4169 2933 : return mkmf(NULL, cgetg(1, t_VEC), vS, vMjd, NULL);
4170 : }
4171 :
4172 : long
4173 4585 : mfsturm_mf(GEN mf)
4174 : {
4175 4585 : GEN Mindex = MF_get_Mindex(mf);
4176 4585 : long n = lg(Mindex)-1;
4177 4585 : return n? Mindex[n]-1: 0;
4178 : }
4179 :
4180 : long
4181 826 : mfsturm(GEN T)
4182 : {
4183 : long N, nk, dk;
4184 826 : GEN CHI, mf = checkMF_i(T);
4185 826 : if (mf) return mfsturm_mf(mf);
4186 7 : checkNK2(T, &N, &nk, &dk, &CHI, 0);
4187 7 : return dk == 1 ? mfsturmNk(N, nk) : mfsturmNk(N, (nk + 1) >> 1);
4188 : }
4189 : long
4190 196 : mfisequal(GEN F, GEN G, long lim)
4191 : {
4192 196 : pari_sp av = avma;
4193 : long b;
4194 196 : if (!checkmf_i(F)) pari_err_TYPE("mfisequal",F);
4195 196 : if (!checkmf_i(G)) pari_err_TYPE("mfisequal",G);
4196 196 : b = lim? lim: maxss(mfsturmmf(F), mfsturmmf(G));
4197 196 : return gc_long(av, gequal(mfcoefs_i(F, b, 1), mfcoefs_i(G, b, 1)));
4198 : }
4199 :
4200 : GEN
4201 35 : mffields(GEN mf)
4202 : {
4203 35 : if (checkmf_i(mf)) return gcopy(mf_get_field(mf));
4204 35 : mf = checkMF(mf); return gcopy(MF_get_fields(mf));
4205 : }
4206 :
4207 : GEN
4208 364 : mfeigenbasis(GEN mf)
4209 : {
4210 364 : pari_sp ltop = avma;
4211 : GEN F, S, v, vP;
4212 : long i, l, k, dS;
4213 :
4214 364 : mf = checkMF(mf);
4215 364 : k = MF_get_k(mf);
4216 364 : S = MF_get_S(mf); dS = lg(S)-1;
4217 364 : if (!dS) return cgetg(1, t_VEC);
4218 357 : F = MF_get_newforms(mf);
4219 357 : vP = MF_get_fields(mf);
4220 357 : if (k == 1)
4221 : {
4222 210 : if (MF_get_space(mf) == mf_FULL)
4223 : {
4224 14 : long dE = lg(MF_get_E(mf)) - 1;
4225 14 : if (dE) F = rowslice(F, dE+1, dE+dS);
4226 : }
4227 210 : v = vecmflineardiv_linear(S, F);
4228 210 : l = lg(v);
4229 : }
4230 : else
4231 : {
4232 147 : GEN (*L)(GEN, GEN) = (MF_get_space(mf) == mf_FULL)? mflinear: mflinear_bhn;
4233 147 : l = lg(F); v = cgetg(l, t_VEC);
4234 511 : for (i = 1; i < l; i++) gel(v,i) = L(mf, gel(F,i));
4235 : }
4236 945 : for (i = 1; i < l; i++) mf_setfield(gel(v,i), gel(vP,i));
4237 357 : return gc_GEN(ltop, v);
4238 : }
4239 :
4240 : /* Minv = [M, d, A], v a t_COL; A a Zab, d a t_INT; return (A/d) * M*v */
4241 : static GEN
4242 7924 : Minv_RgC_mul(GEN Minv, GEN v)
4243 : {
4244 7924 : GEN M = gel(Minv,1), d = gel(Minv,2), A = gel(Minv,3);
4245 7924 : v = RgM_RgC_mul(M, v);
4246 7924 : if (!equali1(A))
4247 : {
4248 2072 : if (typ(A) == t_POL && degpol(A) > 0) A = mkpolmod(A, gel(Minv,4));
4249 2072 : v = RgC_Rg_mul(v, A);
4250 : }
4251 7924 : if (!equali1(d)) v = RgC_Rg_div(v, d);
4252 7924 : return v;
4253 : }
4254 : static GEN
4255 1309 : Minv_RgM_mul(GEN Minv, GEN B)
4256 : {
4257 1309 : long j, l = lg(B);
4258 1309 : GEN M = cgetg(l, t_MAT);
4259 6090 : for (j = 1; j < l; j++) gel(M,j) = Minv_RgC_mul(Minv, gel(B,j));
4260 1309 : return M;
4261 : }
4262 : /* B * Minv; allow B = NULL for Id */
4263 : static GEN
4264 2436 : RgM_Minv_mul(GEN B, GEN Minv)
4265 : {
4266 2436 : GEN M = gel(Minv,1), d = gel(Minv,2), A = gel(Minv,3);
4267 2436 : if (B) M = RgM_mul(B, M);
4268 2436 : if (!equali1(A))
4269 : {
4270 980 : if (typ(A) == t_POL) A = mkpolmod(A, gel(Minv,4));
4271 980 : M = RgM_Rg_mul(M, A);
4272 : }
4273 2436 : if (!equali1(d)) M = RgM_Rg_div(M,d);
4274 2436 : return M;
4275 : }
4276 :
4277 : /* perm vector of strictly increasing indices, v a vector or arbitrary length;
4278 : * the last r entries of perm fall beyond v.
4279 : * Return v o perm[1..(-r)], discarding the last r entries of v */
4280 : static GEN
4281 1603 : vecpermute_partial(GEN v, GEN perm, long *r)
4282 : {
4283 1603 : long i, n = lg(v)-1, l = lg(perm);
4284 : GEN w;
4285 1603 : if (perm[l-1] <= n) { *r = 0; return vecpermute(v,perm); }
4286 63 : for (i = 1; i < l; i++)
4287 63 : if (perm[i] > n) break;
4288 21 : *r = l - i; l = i;
4289 21 : w = cgetg(l, typ(v));
4290 63 : for (i = 1; i < l; i++) gel(w,i) = gel(v,perm[i]);
4291 21 : return w;
4292 : }
4293 :
4294 : /* given form F, find coeffs of F on mfbasis(mf). If power series, not
4295 : * guaranteed correct if precision less than Sturm bound */
4296 : static GEN
4297 1435 : mftobasis_i(GEN mf, GEN F)
4298 : {
4299 : GEN v, Mindex, Minv;
4300 1435 : if (!MF_get_dim(mf)) return cgetg(1, t_COL);
4301 1435 : Mindex = MF_get_Mindex(mf);
4302 1435 : Minv = MF_get_Minv(mf);
4303 1435 : if (checkmf_i(F))
4304 : {
4305 287 : long n = Mindex[lg(Mindex)-1];
4306 287 : v = vecpermute(mfcoefs_i(F, n, 1), Mindex);
4307 287 : return Minv_RgC_mul(Minv, v);
4308 : }
4309 : else
4310 : {
4311 1148 : GEN A = gel(Minv,1), d = gel(Minv,2);
4312 : long r;
4313 1148 : v = F;
4314 1148 : switch(typ(F))
4315 : {
4316 0 : case t_SER: v = sertocol(v);
4317 1148 : case t_VEC: case t_COL: break;
4318 0 : default: pari_err_TYPE("mftobasis", F);
4319 : }
4320 1148 : if (lg(v) == 1) pari_err_TYPE("mftobasis",v);
4321 1148 : v = vecpermute_partial(v, Mindex, &r);
4322 1148 : if (!r) return Minv_RgC_mul(Minv, v); /* single solution */
4323 : /* affine space of dimension r */
4324 21 : v = RgM_RgC_mul(vecslice(A, 1, lg(v)-1), v);
4325 21 : if (!equali1(d)) v = RgC_Rg_div(v,d);
4326 21 : return mkvec2(v, vecslice(A, lg(A)-r, lg(A)-1));
4327 : }
4328 : }
4329 :
4330 : static GEN
4331 910 : const_mat(long n, GEN x)
4332 : {
4333 910 : long j, l = n+1;
4334 910 : GEN A = cgetg(l,t_MAT);
4335 6902 : for (j = 1; j < l; j++) gel(A,j) = const_col(n, x);
4336 910 : return A;
4337 : }
4338 :
4339 : /* L is the mftobasis of a form on CUSP space. We allow mf_FULL or mf_CUSP */
4340 : static GEN
4341 455 : mftonew_i(GEN mf, GEN L, long *plevel)
4342 : {
4343 : GEN S, listMjd, CHI, res, Aclos, Acoef, D, perm;
4344 455 : long N1, LC, lD, i, l, t, level, N = MF_get_N(mf);
4345 :
4346 455 : if (MF_get_k(mf) == 1) pari_err_IMPL("mftonew in weight 1");
4347 455 : listMjd = MFcusp_get_vMjd(mf);
4348 455 : CHI = MF_get_CHI(mf); LC = mfcharconductor(CHI);
4349 455 : S = MF_get_S(mf);
4350 :
4351 455 : N1 = N/LC;
4352 455 : D = mydivisorsu(N1); lD = lg(D);
4353 455 : perm = cgetg(N1+1, t_VECSMALL);
4354 3451 : for (i = 1; i < lD; i++) perm[D[i]] = i;
4355 455 : Aclos = const_mat(lD-1, cgetg(1,t_VEC));
4356 455 : Acoef = const_mat(lD-1, cgetg(1,t_VEC));
4357 455 : l = lg(listMjd);
4358 4669 : for (i = 1; i < l; i++)
4359 : {
4360 : long M, d;
4361 : GEN v;
4362 4214 : if (gequal0(gel(L,i))) continue;
4363 469 : v = gel(listMjd, i);
4364 469 : M = perm[ v[1]/LC ];
4365 469 : d = perm[ v[3] ];
4366 469 : gcoeff(Aclos,M,d) = vec_append(gcoeff(Aclos,M,d), gel(S,i));
4367 469 : gcoeff(Acoef,M,d) = shallowconcat(gcoeff(Acoef,M,d), gel(L,i));
4368 : }
4369 455 : res = cgetg(l, t_VEC); level = 1;
4370 3451 : for (i = t = 1; i < lD; i++)
4371 : {
4372 2996 : long j, M = D[i]*LC;
4373 2996 : GEN gM = utoipos(M);
4374 26530 : for (j = 1; j < lD; j++)
4375 : {
4376 23534 : GEN vf = gcoeff(Aclos,i,j), C, NK;
4377 : long d;
4378 23534 : if (lg(vf) == 1) continue;
4379 427 : d = D[j];
4380 427 : C = gcoeff(Acoef,i,j);
4381 427 : NK = mf_get_NK(gel(vf, 1));
4382 427 : if (d > 1)
4383 : { /* remove mfbd(, d) wrappers */
4384 175 : long h, lf = lg(vf);
4385 357 : for (h = 1; h < lf; h++)
4386 : {
4387 182 : GEN fd = gel(vf, h);
4388 182 : if (mf_get_type(fd) != t_MF_BD || !equaliu(gel(fd,3), d))
4389 0 : pari_err_BUG("mftonew [inconsistent multiplier]");
4390 182 : gel(vf, h) = gel(fd, 2);
4391 : }
4392 : }
4393 427 : level = ulcm(level, M*d);
4394 427 : gel(res,t++) = mkvec3(gM, utoipos(d), mflinear_i(NK,vf,C));
4395 : }
4396 : }
4397 455 : if (plevel) *plevel = level;
4398 455 : setlg(res, t); return res;
4399 : }
4400 : GEN
4401 217 : mftonew(GEN mf, GEN F)
4402 : {
4403 217 : pari_sp av = avma;
4404 : GEN ES;
4405 : long s;
4406 217 : mf = checkMF(mf);
4407 217 : s = MF_get_space(mf);
4408 217 : if (s != mf_FULL && s != mf_CUSP)
4409 7 : pari_err_TYPE("mftonew [not a full or cuspidal space]", mf);
4410 210 : ES = mftobasisES(mf,F);
4411 203 : if (!gequal0(gel(ES,1)))
4412 0 : pari_err_TYPE("mftonew [not a cuspidal form]", F);
4413 203 : F = gel(ES,2);
4414 203 : return gc_GEN(av, mftonew_i(mf,F, NULL));
4415 : }
4416 :
4417 : static GEN mfeisenstein_i(long k, GEN CHI1, GEN CHI2);
4418 :
4419 : /* mfinit(F * Theta) */
4420 : static GEN
4421 98 : mf2init(GEN mf)
4422 : {
4423 98 : GEN CHI = MF_get_CHI(mf), gk = gadd(MF_get_gk(mf), ghalf);
4424 98 : long N = MF_get_N(mf);
4425 98 : return mfinit_Nkchi(N, itou(gk), mfchiadjust(CHI, gk, N), mf_FULL, 0);
4426 : }
4427 :
4428 : static long
4429 623 : mfvec_first_cusp(GEN v)
4430 : {
4431 623 : long i, l = lg(v);
4432 1519 : for (i = 1; i < l; i++)
4433 : {
4434 1414 : GEN F = gel(v,i);
4435 1414 : long t = mf_get_type(F);
4436 1414 : if (t == t_MF_BD) { F = gel(F,2); t = mf_get_type(F); }
4437 1414 : if (t == t_MF_HECKE) { F = gel(F,3); t = mf_get_type(F); }
4438 1414 : if (t == t_MF_NEWTRACE) break;
4439 : }
4440 623 : return i;
4441 : }
4442 : /* vF a vector of mf F of type DIV(LINEAR(BAS,L), f) in (lcm) level N,
4443 : * F[2]=LINEAR(BAS,L), F[2][2]=BAS=fixed basis (Eisenstein or bhn type),
4444 : * F[2][3]=L, F[3]=f; mfvectomat(vF, n) */
4445 : static GEN
4446 630 : mflineardivtomat(long N, GEN vF, long n)
4447 : {
4448 630 : GEN F, M, f, fc, ME, dB, B, a0, V = NULL;
4449 630 : long lM, lF = lg(vF), j;
4450 :
4451 630 : if (lF == 1) return cgetg(1,t_MAT);
4452 623 : F = gel(vF,1);
4453 623 : if (lg(F) == 5)
4454 : { /* chicompat */
4455 273 : V = gmael(F,4,4);
4456 273 : if (typ(V) == t_INT) V = NULL;
4457 : }
4458 623 : M = gmael(F,2,2); /* BAS */
4459 623 : lM = lg(M);
4460 623 : j = mfvec_first_cusp(M);
4461 623 : if (j == 1) ME = NULL;
4462 : else
4463 : { /* BAS starts by Eisenstein */
4464 161 : ME = mfvectomat(vecslice(M,1,j-1), n, 1);
4465 161 : M = vecslice(M, j,lM-1);
4466 : }
4467 623 : M = bhnmat_extend_nocache(NULL, N, n, 1, M);
4468 623 : if (ME) M = shallowconcat(ME,M);
4469 : /* M = mfcoefs of BAS */
4470 623 : B = cgetg(lF, t_MAT);
4471 623 : dB= cgetg(lF, t_VEC);
4472 2947 : for (j = 1; j < lF; j++)
4473 : {
4474 2324 : GEN g = gel(vF, j); /* t_MF_DIV */
4475 2324 : gel(B,j) = RgM_RgC_mul(M, gmael(g,2,3));
4476 2324 : gel(dB,j)= gmael(g,2,4);
4477 : }
4478 623 : f = mfcoefsser(gel(F,3),n);
4479 623 : a0 = polcoef_i(f, 0, -1);
4480 623 : if (gequal0(a0) || gequal1(a0))
4481 322 : a0 = NULL;
4482 : else
4483 301 : f = gdiv(ser_unscale(f, a0), a0);
4484 623 : fc = ginv(f);
4485 2947 : for (j = 1; j < lF; j++)
4486 : {
4487 2324 : pari_sp av = avma;
4488 2324 : GEN LISer = RgV_to_ser_full(gel(B,j)), f;
4489 2324 : if (a0) LISer = gdiv(ser_unscale(LISer, a0), a0);
4490 2324 : f = gmul(LISer, fc);
4491 2324 : if (a0) f = ser_unscale(f, ginv(a0));
4492 2324 : f = sertocol(f); setlg(f, n+2);
4493 2324 : if (!gequal1(gel(dB,j))) f = RgC_Rg_div(f, gel(dB,j));
4494 2324 : gel(B,j) = gc_upto(av,f);
4495 : }
4496 623 : if (V) B = gmodulo(QabM_tracerel(V, 0, B), gel(V,1));
4497 623 : return B;
4498 : }
4499 :
4500 : static GEN
4501 350 : mfheckemat_mfcoefs(GEN mf, GEN B, GEN DATA)
4502 : {
4503 350 : GEN Mindex = MF_get_Mindex(mf), Minv = MF_get_Minv(mf);
4504 350 : long j, l = lg(B), sb = mfsturm_mf(mf);
4505 350 : GEN b = MF_get_basis(mf), Q = cgetg(l, t_VEC);
4506 1827 : for (j = 1; j < l; j++)
4507 : {
4508 1477 : GEN v = hecke_i(sb, 1, gel(B,j), gel(b,j), DATA); /* Tn b[j] */
4509 1477 : settyp(v,t_COL); gel(Q,j) = vecpermute(v, Mindex);
4510 : }
4511 350 : return Minv_RgM_mul(Minv,Q);
4512 : }
4513 : /* T_p^2, p prime, 1/2-integral weight; B = mfcoefs(mf,sb*p^2,1) or (mf,sb,p^2)
4514 : * if p|N */
4515 : static GEN
4516 7 : mfheckemat_mfcoefs_p2(GEN mf, long p, GEN B)
4517 : {
4518 7 : pari_sp av = avma;
4519 7 : GEN DATA = heckef2_data(MF_get_N(mf), p*p);
4520 7 : return gc_upto(av, mfheckemat_mfcoefs(mf, B, DATA));
4521 : }
4522 : /* convert Mindex from row-index to mfcoef indexation: a(n) is stored in
4523 : * mfcoefs()[n+1], so subtract 1 from all indices */
4524 : static GEN
4525 49 : Mindex_as_coef(GEN mf)
4526 : {
4527 49 : GEN v, Mindex = MF_get_Mindex(mf);
4528 49 : long i, l = lg(Mindex);
4529 49 : v = cgetg(l, t_VECSMALL);
4530 210 : for (i = 1; i < l; i++) v[i] = Mindex[i]-1;
4531 49 : return v;
4532 : }
4533 : /* T_p, p prime; B = mfcoefs(mf,sb*p,1) or (mf,sb,p) if p|N; integral weight */
4534 : static GEN
4535 35 : mfheckemat_mfcoefs_p(GEN mf, long p, GEN B)
4536 : {
4537 35 : pari_sp av = avma;
4538 35 : GEN vm, Q, C, Minv = MF_get_Minv(mf);
4539 35 : long lm, k, i, j, l = lg(B), N = MF_get_N(mf);
4540 :
4541 35 : if (N % p == 0) return Minv_RgM_mul(Minv, rowpermute(B, MF_get_Mindex(mf)));
4542 21 : k = MF_get_k(mf);
4543 21 : C = gmul(mfchareval(MF_get_CHI(mf), p), powuu(p, k-1));
4544 21 : vm = Mindex_as_coef(mf); lm = lg(vm);
4545 21 : Q = cgetg(l, t_MAT);
4546 147 : for (j = 1; j < l; j++) gel(Q,j) = cgetg(lm, t_COL);
4547 147 : for (i = 1; i < lm; i++)
4548 : {
4549 126 : long m = vm[i], mp = m*p;
4550 126 : GEN Cm = (m % p) == 0? C : NULL;
4551 1260 : for (j = 1; j < l; j++)
4552 : {
4553 1134 : GEN S = gel(B,j), s = gel(S, mp + 1);
4554 1134 : if (Cm) s = gadd(s, gmul(C, gel(S, m/p + 1)));
4555 1134 : gcoeff(Q, i, j) = s;
4556 : }
4557 : }
4558 21 : return gc_upto(av, Minv_RgM_mul(Minv,Q));
4559 : }
4560 : /* Matrix of T(p), p prime, dim(mf) > 0 and integral weight */
4561 : static GEN
4562 343 : mfheckemat_p(GEN mf, long p)
4563 : {
4564 343 : pari_sp av = avma;
4565 343 : long N = MF_get_N(mf), sb = mfsturm_mf(mf);
4566 343 : GEN B = (N % p)? mfcoefs_mf(mf, sb * p, 1): mfcoefs_mf(mf, sb, p);
4567 343 : return gc_upto(av, mfheckemat_mfcoefs(mf, B, hecke_data(N,p)));
4568 : }
4569 :
4570 : /* mf_NEW != (0), weight > 1, p prime. Use
4571 : * T(p) T(j) = T(j*p) + p^{k-1} \chi(p) 1_{p | j, p \nmid N} T(j/p) */
4572 : static GEN
4573 924 : mfnewmathecke_p(GEN mf, long p)
4574 : {
4575 924 : pari_sp av = avma;
4576 924 : GEN tf, vj = MFnew_get_vj(mf), CHI = MF_get_CHI(mf);
4577 924 : GEN Mindex = MF_get_Mindex(mf), Minv = MF_get_Minv(mf);
4578 924 : long N = MF_get_N(mf), k = MF_get_k(mf);
4579 924 : long i, j, lvj = lg(vj), lim = vj[lvj-1] * p;
4580 924 : GEN M, perm, V, need = zero_zv(lim);
4581 924 : GEN C = (N % p)? gmul(mfchareval(CHI,p), powuu(p,k-1)): NULL;
4582 924 : tf = mftraceform_new(N, k, CHI);
4583 4004 : for (i = 1; i < lvj; i++)
4584 : {
4585 3080 : j = vj[i]; need[j*p] = 1;
4586 3080 : if (N % p && j % p == 0) need[j/p] = 1;
4587 : }
4588 924 : perm = zero_zv(lim);
4589 924 : V = cgetg(lim+1, t_VEC);
4590 12754 : for (i = j = 1; i <= lim; i++)
4591 11830 : if (need[i]) { gel(V,j) = mfhecke_i(i, N, tf); perm[i] = j; j++; }
4592 924 : setlg(V, j);
4593 924 : V = bhnmat_extend_nocache(NULL, N, mfsturm_mf(mf), 1, V);
4594 924 : V = rowpermute(V, Mindex); /* V[perm[i]] = coeffs(T_i newtrace) */
4595 924 : M = cgetg(lvj, t_MAT);
4596 4004 : for (i = 1; i < lvj; i++)
4597 : {
4598 : GEN t;
4599 3080 : j = vj[i]; t = gel(V, perm[j*p]);
4600 3080 : if (C && j % p == 0) t = RgC_add(t, RgC_Rg_mul(gel(V, perm[j/p]),C));
4601 3080 : gel(M,i) = t;
4602 : }
4603 924 : return gc_upto(av, Minv_RgM_mul(Minv, M));
4604 : }
4605 :
4606 : GEN
4607 77 : mfheckemat(GEN mf, GEN vn)
4608 : {
4609 77 : pari_sp av = avma;
4610 77 : long lv, lvP, i, N, dim, nk, dk, p, sb, flint = (typ(vn)==t_INT);
4611 : GEN CHI, res, vT, FA, B, vP;
4612 :
4613 77 : mf = checkMF(mf);
4614 77 : if (typ(vn) != t_VECSMALL) vn = gtovecsmall(vn);
4615 77 : N = MF_get_N(mf); CHI = MF_get_CHI(mf); Qtoss(MF_get_gk(mf), &nk, &dk);
4616 77 : dim = MF_get_dim(mf);
4617 77 : lv = lg(vn);
4618 77 : res = cgetg(lv, t_VEC);
4619 77 : FA = cgetg(lv, t_VEC);
4620 77 : vP = cgetg(lv, t_VEC);
4621 77 : vT = const_vec(vecsmall_max(vn), NULL);
4622 182 : for (i = 1; i < lv; i++)
4623 : {
4624 105 : ulong n = (ulong)labs(vn[i]);
4625 : GEN fa;
4626 105 : if (!n) pari_err_TYPE("mfheckemat", vn);
4627 105 : if (dk == 1 || uissquareall(n, &n)) fa = myfactoru(n);
4628 0 : else { n = 0; fa = myfactoru(1); } /* dummy: T_{vn[i]} = 0 */
4629 105 : vn[i] = n;
4630 105 : gel(FA,i) = fa;
4631 105 : gel(vP,i) = gel(fa,1);
4632 : }
4633 77 : vP = shallowconcat1(vP); vecsmall_sort(vP);
4634 77 : vP = vecsmall_uniq_sorted(vP); /* all primes occurring in vn */
4635 77 : lvP = lg(vP); if (lvP == 1) goto END;
4636 56 : p = vP[lvP-1];
4637 56 : sb = mfsturm_mf(mf);
4638 56 : if (dk == 1 && nk != 1 && MF_get_space(mf) == mf_NEW)
4639 21 : B = NULL; /* special purpose mfnewmathecke_p is faster */
4640 35 : else if (lvP == 2 && N % p == 0)
4641 21 : B = mfcoefs_mf(mf, sb, dk==2? p*p: p); /* single prime | N, can optimize */
4642 : else
4643 14 : B = mfcoefs_mf(mf, sb * (dk==2? p*p: p), 1); /* general initialization */
4644 126 : for (i = 1; i < lvP; i++)
4645 : {
4646 70 : long j, l, q, e = 1;
4647 : GEN C, Tp, u1, u0;
4648 70 : p = vP[i];
4649 189 : for (j = 1; j < lv; j++) e = maxss(e, z_lval(vn[j], p));
4650 70 : if (!B)
4651 28 : Tp = mfnewmathecke_p(mf, p);
4652 42 : else if (dk == 2)
4653 7 : Tp = mfheckemat_mfcoefs_p2(mf,p, (lvP==2||N%p)? B: matdeflate(sb,p*p,B));
4654 : else
4655 35 : Tp = mfheckemat_mfcoefs_p(mf, p, (lvP==2||N%p)? B: matdeflate(sb,p,B));
4656 70 : gel(vT, p) = Tp;
4657 70 : if (e == 1) continue;
4658 14 : u0 = gen_1;
4659 14 : if (dk == 2)
4660 : {
4661 0 : C = N % p? gmul(mfchareval(CHI,p*p), powuu(p, nk-2)): NULL;
4662 0 : if (e == 2) u0 = uutoQ(p+1,p); /* special case T_{p^4} */
4663 : }
4664 : else
4665 14 : C = N % p? gmul(mfchareval(CHI,p), powuu(p, nk-1)): NULL;
4666 28 : for (u1=Tp, q=p, l=2; l <= e; l++)
4667 : { /* u0 = T_{p^{l-2}}, u1 = T_{p^{l-1}} for l > 2 */
4668 14 : GEN v = gmul(Tp, u1);
4669 14 : if (C) v = gsub(v, gmul(C, u0));
4670 : /* q = p^l, vT[q] = T_q for k integer else T_{q^2} */
4671 14 : q *= p; u0 = u1; gel(vT, q) = u1 = v;
4672 : }
4673 : }
4674 56 : END:
4675 : /* vT[p^e] = T_{p^e} for all p^e occurring below */
4676 182 : for (i = 1; i < lv; i++)
4677 : {
4678 105 : long n = vn[i], j, lP;
4679 : GEN fa, P, E, M;
4680 105 : if (n == 0) { gel(res,i) = zeromat(dim,dim); continue; }
4681 105 : if (n == 1) { gel(res,i) = matid(dim); continue; }
4682 77 : fa = gel(FA,i);
4683 77 : P = gel(fa,1); lP = lg(P);
4684 77 : E = gel(fa,2); M = gel(vT, upowuu(P[1], E[1]));
4685 84 : for (j = 2; j < lP; j++) M = RgM_mul(M, gel(vT, upowuu(P[j], E[j])));
4686 77 : gel(res,i) = M;
4687 : }
4688 77 : if (flint) res = gel(res,1);
4689 77 : return gc_GEN(av, res);
4690 : }
4691 :
4692 : /* f = \sum_i v[i] T_listj[i] (Trace Form) attached to v; replace by f/a_1(f) */
4693 : static GEN
4694 1540 : mf_normalize(GEN mf, GEN v)
4695 : {
4696 1540 : GEN c, dc = NULL, M = MF_get_M(mf), Mindex = MF_get_Mindex(mf);
4697 1540 : v = Q_primpart(v);
4698 1540 : c = RgMrow_RgC_mul(M, v, 2); /* a_1(f) */
4699 1540 : if (gequal1(c)) return v;
4700 945 : if (typ(c) == t_POL) c = gmodulo(c, mfcharpol(MF_get_CHI(mf)));
4701 945 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1 && degpol(gel(c,1)) >= 40
4702 7 : && Mindex[1] == 2
4703 7 : && mfcharorder(MF_get_CHI(mf)) <= 2)
4704 7 : { /* normalize using expansion at infinity (small coefficients) */
4705 7 : GEN w, P = gel(c,1), a1 = gel(c,2);
4706 7 : long i, l = lg(Mindex);
4707 7 : w = cgetg(l, t_COL);
4708 7 : gel(w,1) = gen_1;
4709 280 : for (i = 2; i < l; i++)
4710 : {
4711 273 : c = liftpol_shallow(RgMrow_RgC_mul(M, v, Mindex[i]));
4712 273 : gel(w,i) = QXQ_div(c, a1, P);
4713 : }
4714 : /* w = expansion at oo of normalized form */
4715 7 : v = Minv_RgC_mul(MF_get_Minv(mf), Q_remove_denom(w, &dc));
4716 7 : v = gmodulo(v, P); /* back to mfbasis coefficients */
4717 : }
4718 : else
4719 : {
4720 938 : c = ginv(c);
4721 938 : if (typ(c) == t_POLMOD) c = Q_remove_denom(c, &dc);
4722 938 : v = RgC_Rg_mul(v, c);
4723 : }
4724 945 : if (dc) v = RgC_Rg_div(v, dc);
4725 945 : return v;
4726 : }
4727 : static void
4728 455 : pol_red(GEN NF, GEN *pP, GEN *pa, long flag)
4729 : {
4730 455 : GEN dP, a, P = *pP;
4731 455 : long d = degpol(P);
4732 :
4733 455 : *pa = a = pol_x(varn(P));
4734 455 : if (d * (NF ? nf_get_degree(NF): 1) > 30) return;
4735 :
4736 448 : dP = RgX_disc(P);
4737 448 : if (typ(dP) != t_INT)
4738 112 : { dP = gnorm(dP); if (typ(dP) != t_INT) pari_err_BUG("mfnewsplit"); }
4739 448 : if (d == 2 || expi(dP) < 62)
4740 : {
4741 413 : if (expi(dP) < 31)
4742 406 : P = NF? rnfpolredabs(NF, P,flag): polredabs0(P,flag);
4743 : else
4744 7 : P = NF? rnfpolredbest(NF,P,flag): polredbest(P,flag);
4745 413 : if (flag)
4746 : {
4747 385 : a = gel(P,2); if (typ(a) == t_POLMOD) a = gel(a,2);
4748 385 : P = gel(P,1);
4749 : }
4750 : }
4751 448 : *pP = P;
4752 448 : *pa = a;
4753 : }
4754 :
4755 : /* Diagonalize and normalize. See mfsplit for meaning of flag. */
4756 : static GEN
4757 1092 : mfspclean(GEN mf, GEN mf0, GEN NF, long ord, GEN simplesp, long flag)
4758 : {
4759 1092 : const long vz = 1;
4760 1092 : long i, l = lg(simplesp), dim = MF_get_dim(mf);
4761 1092 : GEN res = cgetg(l, t_MAT), pols = cgetg(l, t_VEC);
4762 1092 : GEN zeros = (mf == mf0)? NULL: zerocol(dim - MF_get_dim(mf0));
4763 2660 : for (i = 1; i < l; i++)
4764 : {
4765 1568 : GEN ATP = gel(simplesp, i), A = gel(ATP,1), P = gel(ATP,3);
4766 1568 : long d = degpol(P);
4767 1568 : GEN a, v = (flag && d > flag)? NULL: gel(A,1);
4768 1568 : if (d == 1) P = pol_x(vz);
4769 : else
4770 : {
4771 455 : pol_red(NF, &P, &a, !!v);
4772 455 : if (v)
4773 : { /* Mod(a,P) root of charpoly(T), K*gpowers(a) = eigenvector of T */
4774 427 : GEN K, den, M = cgetg(d+1, t_MAT), T = gel(ATP,2);
4775 : long j;
4776 427 : T = shallowtrans(T);
4777 427 : gel(M,1) = vec_ei(d,1); /* basis of cyclic vectors */
4778 1372 : for (j = 2; j <= d; j++) gel(M,j) = RgM_RgC_mul(T, gel(M,j-1));
4779 427 : M = Q_primpart(M);
4780 147 : K = NF? ZabM_inv(liftpol_shallow(M), nf_get_pol(NF), ord, &den)
4781 427 : : ZM_inv(M,&den);
4782 427 : K = shallowtrans(K);
4783 427 : v = gequalX(a)? pol_x_powers(d, vz): RgXQ_powers(a, d-1, P);
4784 427 : v = gmodulo(RgM_RgC_mul(A, RgM_RgC_mul(K,v)), P);
4785 : }
4786 : }
4787 1568 : if (v)
4788 : {
4789 1540 : v = mf_normalize(mf0, v); if (zeros) v = shallowconcat(zeros,v);
4790 1540 : gel(res,i) = v; if (flag) setlg(res,i+1);
4791 : }
4792 : else
4793 28 : gel(res,i) = zerocol(dim);
4794 1568 : gel(pols,i) = P;
4795 : }
4796 1092 : return mkvec2(res, pols);
4797 : }
4798 :
4799 : /* return v = v_{X-r}(P), and set Z = P / (X-r)^v */
4800 : static long
4801 70 : RgX_valrem_root(GEN P, GEN r, GEN *Z)
4802 : {
4803 : long v;
4804 140 : for (v = 0; degpol(P); v++)
4805 : {
4806 140 : GEN t, Q = RgX_div_by_X_x(P, r, &t);
4807 140 : if (!gequal0(t)) break;
4808 70 : P = Q;
4809 : }
4810 70 : *Z = P; return v;
4811 : }
4812 : static GEN
4813 1533 : mynffactor(GEN NF, GEN P, long dimlim)
4814 : {
4815 : long i, l, v;
4816 : GEN R, E;
4817 1533 : if (dimlim != 1)
4818 : {
4819 966 : R = NF? nffactor(NF, P): QX_factor(P);
4820 966 : if (!dimlim) return R;
4821 21 : E = gel(R,2);
4822 21 : R = gel(R,1); l = lg(R);
4823 98 : for (i = 1; i < l; i++)
4824 91 : if (degpol(gel(R,i)) > dimlim) break;
4825 21 : if (i == 1) return NULL;
4826 21 : setlg(E,i);
4827 21 : setlg(R,i); return mkmat2(R, E);
4828 : }
4829 : /* dimlim = 1 */
4830 567 : R = nfroots(NF, P); l = lg(R);
4831 567 : if (l == 1) return NULL;
4832 504 : v = varn(P);
4833 504 : settyp(R, t_COL);
4834 504 : if (degpol(P) == l-1)
4835 448 : E = const_col(l-1, gen_1);
4836 : else
4837 : {
4838 56 : E = cgetg(l, t_COL);
4839 126 : for (i = 1; i < l; i++) gel(E,i) = utoi(RgX_valrem_root(P, gel(R,i), &P));
4840 : }
4841 504 : R = deg1_from_roots(R, v);
4842 504 : return mkmat2(R, E);
4843 : }
4844 :
4845 : /* Let K be a number field attached to NF (Q if NF = NULL). A K-vector
4846 : * space of dimension d > 0 is given by a t_MAT A (n x d, full column rank)
4847 : * giving a K-basis, X a section (d x n: left pseudo-inverse of A). Return a
4848 : * pair (T, fa), where T is an element of the Hecke algebra (a sum of Tp taken
4849 : * from vector vTp) acting on A (a d x d t_MAT) and fa is the factorization of
4850 : * its characteristic polynomial, limited to factors of degree <= dimlim if
4851 : * dimlim != 0 (return NULL if there are no factors of degree <= dimlim) */
4852 : static GEN
4853 1358 : findbestsplit(GEN NF, GEN vTp, GEN A, GEN X, long dimlim, long vz)
4854 : {
4855 1358 : GEN T = NULL, Tkeep = NULL, fakeep = NULL;
4856 1358 : long lmax = 0, i, lT = lg(vTp);
4857 1785 : for (i = 1; i < lT; i++)
4858 : {
4859 1785 : GEN D, P, E, fa, TpA = gel(vTp,i);
4860 : long l;
4861 2828 : if (typ(TpA) == t_INT) break;
4862 1533 : if (lg(TpA) > lg(A)) TpA = RgM_mul(X, RgM_mul(TpA, A)); /* Tp | A */
4863 1533 : T = T ? RgM_add(T, TpA) : TpA;
4864 1533 : if (!NF) { P = QM_charpoly_ZX(T); setvarn(P, vz); }
4865 : else
4866 : {
4867 294 : P = charpoly(Q_remove_denom(T, &D), vz);
4868 294 : if (D) P = gdiv(RgX_unscale(P, D), powiu(D, degpol(P)));
4869 : }
4870 1533 : fa = mynffactor(NF, P, dimlim);
4871 1533 : if (!fa) return NULL;
4872 1470 : E = gel(fa, 2);
4873 : /* characteristic polynomial is separable ? */
4874 1470 : if (isint1(vecmax(E))) { Tkeep = T; fakeep = fa; break; }
4875 427 : l = lg(E);
4876 : /* characteristic polynomial has more factors than before ? */
4877 427 : if (l > lmax) { lmax = l; Tkeep = T; fakeep = fa; }
4878 : }
4879 1295 : return mkvec2(Tkeep, fakeep);
4880 : }
4881 :
4882 : static GEN
4883 294 : nfcontent(GEN nf, GEN v)
4884 : {
4885 294 : long i, l = lg(v);
4886 294 : GEN c = gel(v,1);
4887 1512 : for (i = 2; i < l; i++) c = idealadd(nf, c, gel(v,i));
4888 294 : if (typ(c) == t_MAT && gequal1(gcoeff(c,1,1))) c = gen_1;
4889 294 : return c;
4890 : }
4891 : static GEN
4892 455 : nf_primpart(GEN nf, GEN x)
4893 : {
4894 455 : switch(typ(x))
4895 : {
4896 294 : case t_COL:
4897 : {
4898 294 : GEN A = matalgtobasis(nf, x), c = nfcontent(nf, A);
4899 294 : if (typ(c) == t_INT) return x;
4900 35 : c = idealred_elt(nf,c);
4901 35 : A = Q_primpart( nfC_nf_mul(nf, A, Q_primpart(nfinv(nf,c))) );
4902 35 : A = liftpol_shallow( matbasistoalg(nf, A) );
4903 35 : if (gexpo(A) > gexpo(x)) A = x;
4904 35 : return A;
4905 : }
4906 455 : case t_MAT: pari_APPLY_same(nf_primpart(nf, gel(x,i)));
4907 0 : default:
4908 0 : pari_err_TYPE("nf_primpart", x);
4909 : return NULL; /*LCOV_EXCL_LINE*/
4910 : }
4911 : }
4912 :
4913 : /* rotate entries of v to accomodate new entry 'x' (push out oldest entry) */
4914 : static void
4915 1239 : vecpush(GEN v, GEN x)
4916 : {
4917 : long i;
4918 6195 : for (i = lg(v)-1; i > 1; i--) gel(v,i) = gel(v,i-1);
4919 1239 : gel(v,1) = x;
4920 1239 : }
4921 :
4922 : /* sort t_VEC of vector spaces by increasing dimension */
4923 : static GEN
4924 1092 : sort_by_dim(GEN v)
4925 : {
4926 1092 : long i, l = lg(v);
4927 1092 : GEN D = cgetg(l, t_VECSMALL);
4928 2660 : for (i = 1; i < l; i++) D[i] = lg(gmael(v,i,2));
4929 1092 : return vecpermute(v , vecsmall_indexsort(D));
4930 : }
4931 : static GEN
4932 1092 : split_starting_space(GEN mf)
4933 : {
4934 1092 : long d = MF_get_dim(mf), d2;
4935 1092 : GEN id = matid(d);
4936 1092 : switch(MF_get_space(mf))
4937 : {
4938 1085 : case mf_NEW:
4939 1085 : case mf_CUSP: return mkvec2(id, id);
4940 : }
4941 7 : d2 = lg(MF_get_S(mf))-1;
4942 7 : return mkvec2(vecslice(id, d-d2+1,d),
4943 : shallowconcat(zeromat(d2,d-d2),matid(d2)));
4944 : }
4945 : /* If dimlim > 0, keep only the dimension <= dimlim eigenspaces.
4946 : * See mfsplit for the meaning of flag. */
4947 : static GEN
4948 1491 : split_ii(GEN mf, long dimlim, long flag, GEN vSP, long *pnewd)
4949 : {
4950 : forprime_t iter;
4951 1491 : GEN CHI = MF_get_CHI(mf), empty = cgetg(1, t_VEC), mf0 = mf;
4952 : GEN NF, POLCYC, todosp, Tpbigvec, simplesp;
4953 1491 : long N = MF_get_N(mf), k = MF_get_k(mf);
4954 1491 : long ord, FC, NEWT, dimsimple = 0, newd = -1;
4955 1491 : const long NBH = 5, vz = 1;
4956 : ulong p;
4957 :
4958 1491 : switch(MF_get_space(mf))
4959 : {
4960 1197 : case mf_NEW: break;
4961 287 : case mf_CUSP:
4962 : case mf_FULL:
4963 : {
4964 : GEN CHIP;
4965 287 : if (k > 1) { mf0 = mfinittonew(mf); break; }
4966 259 : CHIP = mfchartoprimitive(CHI, NULL);
4967 259 : newd = lg(MF_get_S(mf))-1 - mfolddim_i(N, k, CHIP, vSP);
4968 259 : break;
4969 : }
4970 7 : default: pari_err_TYPE("mfsplit [space does not contain newspace]", mf);
4971 : return NULL; /*LCOV_EXCL_LINE*/
4972 : }
4973 1484 : if (newd < 0) newd = mf0? MF_get_dim(mf0): 0;
4974 1484 : *pnewd = newd;
4975 1484 : if (!newd) return mkvec2(cgetg(1, t_MAT), empty);
4976 :
4977 1092 : NEWT = (k > 1 && MF_get_space(mf0) == mf_NEW);
4978 1092 : todosp = mkvec( split_starting_space(mf0) );
4979 1092 : simplesp = empty;
4980 1092 : FC = mfcharconductor(CHI);
4981 1092 : ord = mfcharorder(CHI);
4982 1092 : if (ord <= 2) NF = POLCYC = NULL;
4983 : else
4984 : {
4985 210 : POLCYC = mfcharpol(CHI);
4986 210 : NF = nfinit(POLCYC,DEFAULTPREC);
4987 : }
4988 1092 : Tpbigvec = zerovec(NBH);
4989 1092 : u_forprime_init(&iter, 2, ULONG_MAX);
4990 1526 : while (dimsimple < newd && (p = u_forprime_next(&iter)))
4991 : {
4992 : GEN nextsp;
4993 : long ind;
4994 1526 : if (N % (p*p) == 0 && N/p % FC == 0) continue; /* T_p = 0 in this case */
4995 1239 : vecpush(Tpbigvec, NEWT? mfnewmathecke_p(mf0,p): mfheckemat_p(mf0,p));
4996 1239 : nextsp = empty;
4997 1638 : for (ind = 1; ind < lg(todosp); ind++)
4998 : {
4999 1358 : GEN tmp = gel(todosp, ind), fa, P, E, D, Tp, DTp;
5000 1358 : GEN A = gel(tmp, 1);
5001 1358 : GEN X = gel(tmp, 2);
5002 : long lP, i;
5003 1358 : tmp = findbestsplit(NF, Tpbigvec, A, X, dimlim, vz);
5004 1477 : if (!tmp) continue; /* nothing there */
5005 1295 : Tp = gel(tmp, 1);
5006 1295 : fa = gel(tmp, 2);
5007 1295 : P = gel(fa, 1);
5008 1295 : E = gel(fa, 2); lP = lg(P);
5009 : /* lP > 1 */
5010 1295 : if (DEBUGLEVEL) err_printf("Exponents = %Ps\n", E);
5011 1295 : if (lP == 2)
5012 : {
5013 868 : GEN P1 = gel(P,1);
5014 868 : long e1 = itos(gel(E,1)), d1 = degpol(P1);
5015 868 : if (e1 * d1 == lg(Tp)-1)
5016 : {
5017 819 : if (e1 > 1) nextsp = vec_append(nextsp, mkvec2(A,X));
5018 : else
5019 : { /* simple module */
5020 721 : simplesp = vec_append(simplesp, mkvec3(A,Tp,P1));
5021 980 : if ((dimsimple += d1) == newd) goto END;
5022 : }
5023 119 : continue;
5024 : }
5025 : }
5026 : /* Found splitting */
5027 476 : DTp = Q_remove_denom(Tp, &D);
5028 1295 : for (i = 1; i < lP; i++)
5029 : {
5030 1078 : GEN Ai, Xi, dXi, AAi, v, y, Pi = gel(P,i);
5031 1078 : Ai = RgX_RgM_eval(D? RgX_rescale(Pi,D): Pi, DTp);
5032 1078 : Ai = QabM_ker(Ai, POLCYC, ord);
5033 1078 : if (NF) Ai = nf_primpart(NF, Ai);
5034 :
5035 1078 : AAi = RgM_mul(A, Ai);
5036 : /* gives section, works on nonsquare matrices */
5037 1078 : Xi = QabM_pseudoinv(Ai, POLCYC, ord, &v, &dXi);
5038 1078 : Xi = RgM_Rg_div(Xi, dXi);
5039 1078 : y = gel(v,1);
5040 1078 : if (isint1(gel(E,i)))
5041 : {
5042 847 : GEN Tpi = RgM_mul(Xi, RgM_mul(rowpermute(Tp,y), Ai));
5043 847 : simplesp = vec_append(simplesp, mkvec3(AAi, Tpi, Pi));
5044 847 : if ((dimsimple += degpol(Pi)) == newd) goto END;
5045 : }
5046 : else
5047 : {
5048 231 : Xi = RgM_mul(Xi, rowpermute(X,y));
5049 231 : nextsp = vec_append(nextsp, mkvec2(AAi, Xi));
5050 : }
5051 : }
5052 : }
5053 280 : todosp = nextsp; if (lg(todosp) == 1) break;
5054 : }
5055 0 : END:
5056 1092 : if (DEBUGLEVEL) err_printf("end split, need to clean\n");
5057 1092 : return mfspclean(mf, mf0, NF, ord, sort_by_dim(simplesp), flag);
5058 : }
5059 : static GEN
5060 42 : dim_filter(GEN v, long dim)
5061 : {
5062 42 : GEN P = gel(v,2);
5063 42 : long j, l = lg(P);
5064 175 : for (j = 1; j < l; j++)
5065 161 : if (degpol(gel(P,j)) > dim)
5066 : {
5067 28 : v = mkvec2(vecslice(gel(v,1),1,j-1), vecslice(P,1,j-1));
5068 28 : break;
5069 : }
5070 42 : return v;
5071 : }
5072 : static long
5073 287 : dim_sum(GEN v)
5074 : {
5075 287 : GEN P = gel(v,2);
5076 287 : long j, l = lg(P), d = 0;
5077 707 : for (j = 1; j < l; j++) d += degpol(gel(P,j));
5078 287 : return d;
5079 : }
5080 : static GEN
5081 1169 : split_i(GEN mf, long dimlim, long flag)
5082 1169 : { long junk; return split_ii(mf, dimlim, flag, NULL, &junk); }
5083 : /* mf is either already split or output by mfinit. Splitting is done only for
5084 : * newspace except in weight 1. If flag = 0 (default) split completely.
5085 : * If flag = d > 0, only give the Galois polynomials in degree > d
5086 : * Flag is ignored if dimlim = 1. */
5087 : GEN
5088 112 : mfsplit(GEN mf0, long dimlim, long flag)
5089 : {
5090 112 : pari_sp av = avma;
5091 112 : GEN v, mf = checkMF_i(mf0);
5092 112 : if (!mf) pari_err_TYPE("mfsplit", mf0);
5093 112 : if ((v = obj_check(mf, MF_SPLIT)))
5094 42 : { if (dimlim) v = dim_filter(v, dimlim); }
5095 70 : else if (dimlim && (v = obj_check(mf, MF_SPLITN)))
5096 21 : { v = (itos(gel(v,1)) >= dimlim)? dim_filter(gel(v,2), dimlim): NULL; }
5097 112 : if (!v)
5098 : {
5099 : long newd;
5100 70 : v = split_ii(mf, dimlim, flag, NULL, &newd);
5101 70 : if (lg(v) == 1) obj_insert(mf, MF_SPLITN, mkvec2(utoi(dimlim), v));
5102 70 : else if (!flag)
5103 : {
5104 49 : if (dim_sum(v) == newd) obj_insert(mf, MF_SPLIT,v);
5105 21 : else obj_insert(mf, MF_SPLITN, mkvec2(utoi(dimlim), v));
5106 : }
5107 : }
5108 112 : return gc_GEN(av, v);
5109 : }
5110 : static GEN
5111 252 : split(GEN mf) { return split_i(mf,0,0); }
5112 : GEN
5113 819 : MF_get_newforms(GEN mf) { return gel(obj_checkbuild(mf,MF_SPLIT,&split),1); }
5114 : GEN
5115 616 : MF_get_fields(GEN mf) { return gel(obj_checkbuild(mf,MF_SPLIT,&split),2); }
5116 :
5117 : /*************************************************************************/
5118 : /* Modular forms of Weight 1 */
5119 : /*************************************************************************/
5120 : /* S_1(G_0(N)), small N. Return 1 if definitely empty; return 0 if maybe
5121 : * nonempty */
5122 : static int
5123 16632 : wt1empty(long N)
5124 : {
5125 16632 : if (N <= 100) switch (N)
5126 : { /* nonempty [32/100] */
5127 5453 : case 23: case 31: case 39: case 44: case 46:
5128 : case 47: case 52: case 55: case 56: case 57:
5129 : case 59: case 62: case 63: case 68: case 69:
5130 : case 71: case 72: case 76: case 77: case 78:
5131 : case 79: case 80: case 83: case 84: case 87:
5132 : case 88: case 92: case 93: case 94: case 95:
5133 5453 : case 99: case 100: return 0;
5134 3549 : default: return 1;
5135 : }
5136 7630 : if (N <= 600) switch(N)
5137 : { /* empty [111/500] */
5138 336 : case 101: case 102: case 105: case 106: case 109:
5139 : case 113: case 121: case 122: case 123: case 125:
5140 : case 130: case 134: case 137: case 146: case 149:
5141 : case 150: case 153: case 157: case 162: case 163:
5142 : case 169: case 170: case 173: case 178: case 181:
5143 : case 182: case 185: case 187: case 193: case 194:
5144 : case 197: case 202: case 205: case 210: case 218:
5145 : case 221: case 226: case 233: case 241: case 242:
5146 : case 245: case 246: case 250: case 257: case 265:
5147 : case 267: case 269: case 274: case 277: case 281:
5148 : case 289: case 293: case 298: case 305: case 306:
5149 : case 313: case 314: case 317: case 326: case 337:
5150 : case 338: case 346: case 349: case 353: case 361:
5151 : case 362: case 365: case 369: case 370: case 373:
5152 : case 374: case 377: case 386: case 389: case 394:
5153 : case 397: case 401: case 409: case 410: case 421:
5154 : case 425: case 427: case 433: case 442: case 449:
5155 : case 457: case 461: case 466: case 481: case 482:
5156 : case 485: case 490: case 493: case 509: case 514:
5157 : case 521: case 530: case 533: case 534: case 538:
5158 : case 541: case 545: case 554: case 557: case 562:
5159 : case 565: case 569: case 577: case 578: case 586:
5160 336 : case 593: return 1;
5161 6979 : default: return 0;
5162 : }
5163 315 : return 0;
5164 : }
5165 :
5166 : static GEN
5167 28 : initwt1trace(GEN mf)
5168 : {
5169 28 : GEN S = MF_get_S(mf), v, H;
5170 : long l, i;
5171 28 : if (lg(S) == 1) return mftrivial();
5172 28 : H = mfheckemat(mf, Mindex_as_coef(mf));
5173 28 : l = lg(H); v = cgetg(l, t_VEC);
5174 63 : for (i = 1; i < l; i++) gel(v,i) = gtrace(gel(H,i));
5175 28 : v = Minv_RgC_mul(MF_get_Minv(mf), v);
5176 28 : return mflineardiv_linear(S, v, 1);
5177 : }
5178 : static GEN
5179 21 : initwt1newtrace(GEN mf)
5180 : {
5181 21 : GEN v, D, S, Mindex, CHI = MF_get_CHI(mf);
5182 21 : long FC, lD, i, sb, N1, N2, lM, N = MF_get_N(mf);
5183 21 : CHI = mfchartoprimitive(CHI, &FC);
5184 21 : if (N % FC || mfcharparity(CHI) == 1) return mftrivial();
5185 21 : D = mydivisorsu(N/FC); lD = lg(D);
5186 21 : S = MF_get_S(mf);
5187 21 : if (lg(S) == 1) return mftrivial();
5188 21 : N2 = newd_params2(N);
5189 21 : N1 = N / N2;
5190 21 : Mindex = MF_get_Mindex(mf);
5191 21 : lM = lg(Mindex);
5192 21 : sb = Mindex[lM-1];
5193 21 : v = zerovec(sb+1);
5194 42 : for (i = 1; i < lD; i++)
5195 : {
5196 21 : long M = FC*D[i], j;
5197 21 : GEN tf = initwt1trace(M == N? mf: mfinit_Nkchi(M, 1, CHI, mf_CUSP, 0));
5198 : GEN listd, w;
5199 21 : if (mf_get_type(tf) == t_MF_CONST) continue;
5200 21 : w = mfcoefs_i(tf, sb, 1);
5201 21 : if (M == N) { v = gadd(v, w); continue; }
5202 0 : listd = mydivisorsu(u_ppo(ugcd(N/M, N1), FC));
5203 0 : for (j = 1; j < lg(listd); j++)
5204 : {
5205 0 : long d = listd[j], d2 = d*d; /* coprime to FC */
5206 0 : GEN dk = mfchareval(CHI, d);
5207 0 : long NMd = N/(M*d), m;
5208 0 : for (m = 1; m <= sb/d2; m++)
5209 : {
5210 0 : long be = mubeta2(NMd, m);
5211 0 : if (be)
5212 : {
5213 0 : GEN c = gmul(dk, gmulsg(be, gel(w, m+1)));
5214 0 : long n = m*d2;
5215 0 : gel(v, n+1) = gadd(gel(v, n+1), c);
5216 : }
5217 : }
5218 : }
5219 : }
5220 21 : if (gequal0(gel(v,2))) return mftrivial();
5221 21 : v = vecpermute(v,Mindex);
5222 21 : v = Minv_RgC_mul(MF_get_Minv(mf), v);
5223 21 : return mflineardiv_linear(S, v, 1);
5224 : }
5225 :
5226 : /* i*p + 1, i*p < lim corresponding to a_p(f_j), a_{2p}(f_j)... */
5227 : static GEN
5228 1834 : pindices(long p, long lim)
5229 : {
5230 1834 : GEN v = cgetg(lim, t_VECSMALL);
5231 : long i, ip;
5232 22190 : for (i = 1, ip = p + 1; ip < lim; i++, ip += p) v[i] = ip;
5233 1834 : setlg(v, i); return v;
5234 : }
5235 :
5236 : /* assume !wt1empty(N), in particular N>25 */
5237 : /* Returns [[lim,p], mf (weight 2), p*lim x dim matrix] */
5238 : static GEN
5239 1834 : mf1_pre(long N)
5240 : {
5241 : pari_timer tt;
5242 : GEN mf, v, L, I, M, Minv, den;
5243 : long B, lim, LIM, p;
5244 :
5245 1834 : if (DEBUGLEVEL) timer_start(&tt);
5246 1834 : mf = mfinit_Nkchi(N, 2, mfchartrivial(), mf_CUSP, 0);
5247 1834 : if (DEBUGLEVEL)
5248 0 : timer_printf(&tt, "mf1basis [pre]: S_2(%ld), dim = %ld",
5249 : N, MF_get_dim(mf));
5250 1834 : M = MF_get_M(mf); Minv = MF_get_Minv(mf); den = gel(Minv,2);
5251 1834 : B = mfsturm_mf(mf);
5252 1834 : if (uisprime(N))
5253 : {
5254 392 : lim = 2 * MF_get_dim(mf); /* ensure mfstabiter's first kernel ~ square */
5255 392 : p = 2;
5256 : }
5257 : else
5258 : {
5259 : forprime_t S;
5260 1442 : u_forprime_init(&S, 2, N);
5261 2576 : while ((p = u_forprime_next(&S)))
5262 2576 : if (N % p) break;
5263 1442 : lim = B + 1;
5264 : }
5265 1834 : LIM = (N & (N - 1))? 2 * lim: 3 * lim; /* N power of 2 ? */
5266 1834 : L = mkvecsmall4(lim, LIM, mfsturmNk(N,1), p);
5267 1834 : M = bhnmat_extend_nocache(M, N, LIM-1, 1, MF_get_S(mf));
5268 1834 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis [pre]: bnfmat_extend");
5269 1834 : v = pindices(p, LIM);
5270 1834 : if (!LIM) return mkvec4(L, mf, M, v);
5271 1834 : I = RgM_Rg_div(ZM_mul(rowslice(M, B+2, LIM), gel(Minv,1)), den);
5272 1834 : I = Q_remove_denom(I, &den);
5273 1834 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis [prec]: Iden");
5274 1834 : return mkvec5(L, mf, M, v, mkvec2(I, den));
5275 : }
5276 :
5277 : /* lg(A) > 1, E a t_POL */
5278 : static GEN
5279 686 : mfmatsermul(GEN A, GEN E)
5280 : {
5281 686 : long j, l = lg(A), r = nbrows(A);
5282 686 : GEN M = cgetg(l, t_MAT);
5283 686 : E = RgXn_red_shallow(E, r+1);
5284 5866 : for (j = 1; j < l; j++)
5285 : {
5286 5180 : GEN c = RgV_to_RgX(gel(A,j), 0);
5287 5180 : gel(M, j) = RgX_to_RgC(RgXn_mul(c, E, r+1), r);
5288 : }
5289 686 : return M;
5290 : }
5291 : /* lg(Ap) > 1, Ep an Flxn */
5292 : static GEN
5293 1141 : mfmatsermul_Fl(GEN Ap, GEN Ep, ulong p)
5294 : {
5295 1141 : long j, l = lg(Ap), r = nbrows(Ap);
5296 1141 : GEN M = cgetg(l, t_MAT);
5297 42630 : for (j = 1; j < l; j++)
5298 : {
5299 41489 : GEN c = Flv_to_Flx(gel(Ap,j), 0);
5300 41489 : gel(M,j) = Flx_to_Flv(Flxn_mul(c, Ep, r+1, p), r);
5301 : }
5302 1141 : return M;
5303 : }
5304 :
5305 : /* CHI mod F | N, return mfchar of modulus N.
5306 : * FIXME: wasteful, G should be precomputed */
5307 : static GEN
5308 13048 : mfcharinduce(GEN CHI, long N)
5309 : {
5310 : GEN G, chi;
5311 13048 : if (mfcharmodulus(CHI) == N) return CHI;
5312 1463 : G = znstar0(utoipos(N), 1);
5313 1463 : chi = zncharinduce(gel(CHI,1), gel(CHI,2), G);
5314 1463 : CHI = leafcopy(CHI);
5315 1463 : gel(CHI,1) = G;
5316 1463 : gel(CHI,2) = chi; return CHI;
5317 : }
5318 :
5319 : static GEN
5320 3983 : gmfcharno(GEN CHI)
5321 : {
5322 3983 : GEN G = gel(CHI,1), chi = gel(CHI,2);
5323 3983 : return mkintmod(znconreyexp(G, chi), znstar_get_N(G));
5324 : }
5325 : static long
5326 13699 : mfcharno(GEN CHI)
5327 : {
5328 13699 : GEN n = znconreyexp(gel(CHI,1), gel(CHI,2));
5329 13699 : return itou(n);
5330 : }
5331 :
5332 : /* return k such that minimal mfcharacter in Galois orbit of CHI is CHI^k */
5333 : static long
5334 12138 : mfconreyminimize(GEN CHI)
5335 : {
5336 12138 : GEN G = gel(CHI,1), cyc, chi;
5337 12138 : cyc = ZV_to_zv(znstar_get_cyc(G));
5338 12138 : chi = ZV_to_zv(znconreychar(G, gel(CHI,2)));
5339 12138 : return zv_cyc_minimize(cyc, chi, coprimes_zv(mfcharorder(CHI)));
5340 : }
5341 :
5342 : /* find scalar c such that first nonzero entry of c*v is 1; return c*v */
5343 : static GEN
5344 2065 : RgV_normalize(GEN v, GEN *pc)
5345 : {
5346 2065 : long i, l = lg(v);
5347 5313 : for (i = 1; i < l; i++)
5348 : {
5349 5313 : GEN c = gel(v,i);
5350 5313 : if (!gequal0(c))
5351 : {
5352 2065 : if (gequal1(c)) break;
5353 679 : *pc = ginv(c); return RgV_Rg_mul(v, *pc);
5354 : }
5355 : }
5356 1386 : *pc = gen_1; return v;
5357 : }
5358 : /* pS != NULL; dim > 0 */
5359 : static GEN
5360 784 : mftreatdihedral(long N, GEN DIH, GEN POLCYC, long ordchi, GEN *pS)
5361 : {
5362 784 : long l = lg(DIH), lim = mfsturmNk(N, 1), i;
5363 784 : GEN Minv, C = cgetg(l, t_VEC), M = cgetg(l, t_MAT);
5364 2436 : for (i = 1; i < l; i++)
5365 : {
5366 1652 : GEN c, v = mfcoefs_i(gel(DIH,i), lim, 1);
5367 1652 : gel(M,i) = RgV_normalize(v, &c);
5368 1652 : gel(C,i) = Rg_col_ei(c, l-1, i);
5369 : }
5370 784 : Minv = gel(mfclean(M,POLCYC,ordchi,0),2);
5371 784 : M = RgM_Minv_mul(M, Minv);
5372 784 : C = RgM_Minv_mul(C, Minv);
5373 784 : *pS = vecmflinear(DIH, C); return M;
5374 : }
5375 :
5376 : /* same mode a maximal ideal above q */
5377 : static GEN
5378 2408 : Tpmod(GEN Ap, GEN A, ulong chip, long p, ulong q)
5379 : {
5380 2408 : GEN B = leafcopy(Ap);
5381 2408 : long i, ip, l = lg(B);
5382 86345 : for (i = 1, ip = p; ip < l; i++, ip += p)
5383 83937 : B[ip] = Fl_add(B[ip], Fl_mul(A[i], chip, q), q);
5384 2408 : return B;
5385 : }
5386 : /* Tp(f_1), ..., Tp(f_d) mod q */
5387 : static GEN
5388 301 : matTpmod(GEN xp, GEN x, ulong chip, long p, ulong q)
5389 2709 : { pari_APPLY_same(Tpmod(gel(xp,i), gel(x,i), chip, p, q)); }
5390 :
5391 : /* Ap[i] = a_{p*i}(F), A[i] = a_i(F), i = 1..lim
5392 : * Tp(f)[n] = a_{p*n}(f) + chi(p) a_{n/p}(f) * 1_{p | n} */
5393 : static GEN
5394 469 : Tp(GEN Ap, GEN A, GEN chip, long p)
5395 : {
5396 469 : GEN B = leafcopy(Ap);
5397 469 : long i, ip, l = lg(B);
5398 12915 : for (i = 1, ip = p; ip < l; i++, ip += p)
5399 12446 : gel(B,ip) = gadd(gel(B,ip), gmul(gel(A,i), chip));
5400 469 : return B;
5401 : }
5402 : /* Tp(f_1), ..., Tp(f_d) */
5403 : static GEN
5404 56 : matTp(GEN xp, GEN x, GEN chip, long p)
5405 525 : { pari_APPLY_same(Tp(gel(xp,i), gel(x,i), chip, p)); }
5406 :
5407 : static GEN
5408 378 : _RgXQM_mul(GEN x, GEN y, GEN T)
5409 378 : { return T? RgXQM_mul(x, y, T): RgM_mul(x, y); }
5410 : /* largest T-stable Q(CHI)-subspace of Q(CHI)-vector space spanned by columns
5411 : * of A */
5412 : static GEN
5413 28 : mfstabiter(GEN *pC, GEN A0, GEN chip, GEN TMP, GEN P, long ordchi)
5414 : {
5415 28 : GEN A, Ap, vp = gel(TMP,4), C = NULL;
5416 28 : long i, lA, lim1 = gel(TMP,1)[3], p = gel(TMP,1)[4];
5417 : pari_timer tt;
5418 :
5419 28 : Ap = rowpermute(A0, vp);
5420 28 : A = rowslice(A0, 2, nbrows(Ap)+1); /* remove a0 */
5421 : for(;;)
5422 28 : {
5423 56 : GEN R = shallowconcat(matTp(Ap, A, chip, p), A);
5424 56 : GEN B = QabM_ker(R, P, ordchi);
5425 56 : long lB = lg(B);
5426 56 : if (DEBUGLEVEL)
5427 0 : timer_printf(&tt, "mf1basis: Hecke intersection (dim %ld)", lB-1);
5428 56 : if (lB == 1) return NULL;
5429 56 : lA = lg(A); if (lB == lA) break;
5430 28 : B = rowslice(B, 1, lA-1);
5431 28 : Ap = _RgXQM_mul(Ap, B, P);
5432 28 : A = _RgXQM_mul(A, B, P);
5433 28 : C = C? _RgXQM_mul(C, B, P): B;
5434 : }
5435 28 : if (nbrows(A) < lim1)
5436 : {
5437 14 : A0 = rowslice(A0, 2, lim1);
5438 14 : A = C? _RgXQM_mul(A0, C, P): A0;
5439 : }
5440 : else /* all needed coefs computed */
5441 14 : A = rowslice(A, 1, lim1-1);
5442 28 : if (*pC) C = C? _RgXQM_mul(*pC, C, P): *pC;
5443 : /* put back a0 */
5444 119 : for (i = 1; i < lA; i++) gel(A,i) = vec_prepend(gel(A,i), gen_0);
5445 28 : *pC = C; return A;
5446 : }
5447 :
5448 : static long
5449 252 : mfstabitermod(GEN A, GEN vp, ulong chip, long p, ulong q)
5450 : {
5451 252 : GEN Ap, C = NULL;
5452 252 : Ap = rowpermute(A, vp);
5453 252 : A = rowslice(A, 2, nbrows(Ap)+1);
5454 : while (1)
5455 49 : {
5456 301 : GEN Rp = shallowconcat(matTpmod(Ap, A, chip, p, q), A);
5457 301 : GEN B = Flm_ker(Rp, q);
5458 301 : long lA = lg(A), lB = lg(B);
5459 301 : if (lB == 1) return 0;
5460 266 : if (lB == lA) return lA-1;
5461 49 : B = rowslice(B, 1, lA-1);
5462 49 : Ap = Flm_mul(Ap, B, q);
5463 49 : A = Flm_mul(A, B, q);
5464 49 : C = C? Flm_mul(C, B, q): B;
5465 : }
5466 : }
5467 :
5468 : static GEN
5469 595 : mfcharinv_i(GEN CHI)
5470 : {
5471 595 : GEN G = gel(CHI,1);
5472 595 : CHI = leafcopy(CHI); gel(CHI,2) = zncharconj(G, gel(CHI,2)); return CHI;
5473 : }
5474 :
5475 : /* upper bound dim S_1(Gamma_0(N),chi) performing the linear algebra mod p */
5476 : static long
5477 595 : mf1dimmod(GEN E1, GEN E, GEN chip, long ordchi, long dih, GEN TMP)
5478 : {
5479 595 : GEN E1i, A, vp, mf, C = NULL;
5480 595 : ulong q, r = QabM_init(ordchi, &q);
5481 : long lim, LIM, p;
5482 :
5483 595 : LIM = gel(TMP,1)[2]; lim = gel(TMP,1)[1];
5484 595 : mf= gel(TMP,2);
5485 595 : A = gel(TMP,3);
5486 595 : A = QabM_to_Flm(A, r, q);
5487 595 : E1 = QabX_to_Flx(E1, r, q);
5488 595 : E1i = Flxn_inv(E1, nbrows(A), q);
5489 595 : if (E)
5490 : {
5491 574 : GEN Iden = gel(TMP,5), I = gel(Iden,1), den = gel(Iden,2);
5492 574 : GEN Mindex = MF_get_Mindex(mf), F = rowslice(A, 1, LIM);
5493 574 : GEN E1ip = Flxn_red(E1i, LIM);
5494 574 : ulong d = den? umodiu(den, q): 1;
5495 574 : long i, nE = lg(E) - 1;
5496 : pari_sp av;
5497 :
5498 574 : I = ZM_to_Flm(I, q);
5499 574 : if (d != 1) I = Flm_Fl_mul(I, Fl_inv(d, q), q);
5500 574 : av = avma;
5501 1120 : for (i = 1; i <= nE; i++)
5502 : {
5503 889 : GEN e = Flxn_mul(E1ip, QabX_to_Flx(gel(E,i), r, q), LIM, q);
5504 889 : GEN B = mfmatsermul_Fl(F, e, q), z;
5505 889 : GEN B2 = Flm_mul(I, rowpermute(B, Mindex), q);
5506 889 : B = rowslice(B, lim+1,LIM);
5507 889 : z = Flm_ker(Flm_sub(B2, B, q), q);
5508 889 : if (lg(z)-1 == dih) return dih;
5509 546 : C = C? Flm_mul(C, z, q): z;
5510 546 : F = Flm_mul(F, z, q);
5511 546 : (void)gc_all(av, 2, &F,&C);
5512 : }
5513 231 : A = F;
5514 : }
5515 : /* use Schaeffer */
5516 252 : p = gel(TMP,1)[4]; vp = gel(TMP,4);
5517 252 : A = mfmatsermul_Fl(A, E1i, q);
5518 252 : return mfstabitermod(A, vp, Qab_to_Fl(chip, r, q), p, q);
5519 : }
5520 :
5521 : static GEN
5522 224 : mf1intermat(GEN A, GEN Mindex, GEN e, GEN Iden, long lim, GEN POLCYC)
5523 : {
5524 224 : long j, l = lg(A), LIM = nbrows(A);
5525 224 : GEN I = gel(Iden,1), den = gel(Iden,2), B = cgetg(l, t_MAT);
5526 :
5527 5257 : for (j = 1; j < l; j++)
5528 : {
5529 5033 : pari_sp av = avma;
5530 5033 : GEN c = RgV_to_RgX(gel(A,j), 0), c1, c2;
5531 5033 : c = RgX_to_RgC(RgXn_mul(c, e, LIM), LIM);
5532 5033 : if (POLCYC) c = liftpol_shallow(c);
5533 5033 : c1 = vecslice(c, lim+1, LIM);
5534 5033 : if (den) c1 = RgC_Rg_mul(c1, den);
5535 5033 : c2 = RgM_RgC_mul(I, vecpermute(c, Mindex));
5536 5033 : gel(B, j) = gc_upto(av, RgC_sub(c2, c1));
5537 : }
5538 224 : return B;
5539 : }
5540 : /* Compute the full S_1(\G_0(N),\chi); return NULL if space is empty; else
5541 : * if pS is NULL, return stoi(dim), where dim is the dimension; else *pS is
5542 : * set to a vector of forms making up a basis, and return the matrix of their
5543 : * Fourier expansions. pdih gives the dimension of the subspace generated by
5544 : * dihedral forms; TMP is from mf1_pre or NULL. */
5545 : static GEN
5546 11284 : mf1basis(long N, GEN CHI, GEN TMP, GEN vSP, GEN *pS, long *pdih)
5547 : {
5548 11284 : GEN E = NULL, EB, E1, E1i, dE1i, mf, A, C, POLCYC, DIH, Minv, chip;
5549 11284 : long nE = 0, p, LIM, lim, lim1, i, lA, dimp, ordchi, dih;
5550 : pari_timer tt;
5551 : pari_sp av;
5552 :
5553 11284 : if (pdih) *pdih = 0;
5554 11284 : if (pS) *pS = NULL;
5555 11284 : if (wt1empty(N) || mfcharparity(CHI) != -1) return NULL;
5556 10990 : ordchi = mfcharorder(CHI);
5557 10990 : if (uisprime(N) && ordchi > 4) return NULL;
5558 10962 : if (pS)
5559 : {
5560 3857 : DIH = mfdihedralcusp(N, CHI, vSP);
5561 3857 : dih = lg(DIH) - 1;
5562 : }
5563 : else
5564 : {
5565 7105 : DIH = NULL;
5566 7105 : dih = mfdihedralcuspdim(N, CHI, vSP);
5567 : }
5568 10962 : POLCYC = (ordchi <= 2)? NULL: mfcharpol(CHI);
5569 10962 : if (pdih) *pdih = dih;
5570 10962 : if (N <= 600) switch(N)
5571 : {
5572 : long m;
5573 126 : case 219: case 273: case 283: case 331: case 333: case 344: case 416:
5574 : case 438: case 468: case 491: case 504: case 546: case 553: case 563:
5575 : case 566: case 581: case 592:
5576 126 : break; /* one chi with both exotic and dihedral forms */
5577 9499 : default: /* only dihedral forms */
5578 9499 : if (!dih) return NULL;
5579 : /* fall through */
5580 : case 124: case 133: case 148: case 171: case 201: case 209: case 224:
5581 : case 229: case 248: case 261: case 266: case 288: case 296: case 301:
5582 : case 309: case 325: case 342: case 371: case 372: case 380: case 399:
5583 : case 402: case 403: case 404: case 408: case 418: case 432: case 444:
5584 : case 448: case 451: case 453: case 458: case 496: case 497: case 513:
5585 : case 522: case 527: case 532: case 576: case 579:
5586 : /* no chi with both exotic and dihedral; one chi with exotic forms */
5587 3248 : if (dih)
5588 : {
5589 2338 : if (!pS) return utoipos(dih);
5590 728 : return mftreatdihedral(N, DIH, POLCYC, ordchi, pS) ;
5591 : }
5592 910 : m = mfcharno(mfcharinduce(CHI,N));
5593 910 : if (N == 124 && (m != 67 && m != 87)) return NULL;
5594 784 : if (N == 133 && (m != 83 && m !=125)) return NULL;
5595 490 : if (N == 148 && (m !=105 && m !=117)) return NULL;
5596 364 : if (N == 171 && (m != 94 && m !=151)) return NULL;
5597 364 : if (N == 201 && (m != 29 && m !=104)) return NULL;
5598 364 : if (N == 209 && (m != 87 && m !=197)) return NULL;
5599 364 : if (N == 224 && (m != 95 && m !=191)) return NULL;
5600 364 : if (N == 229 && (m !=107 && m !=122)) return NULL;
5601 364 : if (N == 248 && (m != 87 && m !=191)) return NULL;
5602 273 : if (N == 261 && (m != 46 && m !=244)) return NULL;
5603 273 : if (N == 266 && (m != 83 && m !=125)) return NULL;
5604 273 : if (N == 288 && (m != 31 && m !=223)) return NULL;
5605 273 : if (N == 296 && (m !=105 && m !=265)) return NULL;
5606 : }
5607 595 : if (DEBUGLEVEL)
5608 0 : err_printf("mf1basis: start character %Ps, conductor = %ld, order = %ld\n",
5609 : gmfcharno(CHI), mfcharconductor(CHI), ordchi);
5610 595 : if (!TMP) TMP = mf1_pre(N);
5611 595 : lim = gel(TMP,1)[1]; LIM = gel(TMP,1)[2]; lim1 = gel(TMP,1)[3];
5612 595 : p = gel(TMP,1)[4];
5613 595 : mf = gel(TMP,2);
5614 595 : A = gel(TMP,3);
5615 595 : EB = mfeisensteinbasis(N, 1, mfcharinv_i(CHI));
5616 595 : nE = lg(EB) - 1;
5617 595 : E1 = RgV_to_RgX(mftocol(gel(EB,1), LIM-1, 1), 0); /* + O(x^LIM) */
5618 595 : if (--nE)
5619 574 : E = RgM_to_RgXV(mfvectomat(vecslice(EB, 2, nE+1), LIM-1, 1), 0);
5620 595 : chip = mfchareval(CHI, p); /* != 0 */
5621 595 : if (DEBUGLEVEL) timer_start(&tt);
5622 595 : av = avma; dimp = mf1dimmod(E1, E, chip, ordchi, dih, TMP);
5623 595 : set_avma(av);
5624 595 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: dim mod p is %ld", dimp);
5625 595 : if (!dimp) return NULL;
5626 280 : if (!pS) return utoi(dimp);
5627 224 : if (dimp == dih) return mftreatdihedral(N, DIH, POLCYC, ordchi, pS);
5628 168 : E1i = RgXn_inv(E1, LIM); /* E[1] does not vanish at oo */
5629 168 : if (POLCYC) E1i = liftpol_shallow(E1i);
5630 168 : E1i = Q_remove_denom(E1i, &dE1i);
5631 168 : if (DEBUGLEVEL)
5632 : {
5633 0 : GEN a0 = gel(E1,2);
5634 0 : if (typ(a0) == t_POLMOD) a0 = gnorm(a0);
5635 0 : a0 = Q_abs_shallow(a0);
5636 0 : timer_printf(&tt, "mf1basis: invert E; norm(a0(E)) = %Ps", a0);
5637 : }
5638 168 : C = NULL;
5639 168 : if (nE)
5640 : { /* mf attached to S2(N), fi = mfbasis(mf)
5641 : * M = coefs(f1,...,fd) up to LIM
5642 : * F = coefs(F1,...,FD) = M * C, for some matrix C over Q(chi),
5643 : * initially 1, eventually giving \cap_E S2 / E; D <= d.
5644 : * B = coefs(E/E1 F1, .., E/E1 FD); we want X in Q(CHI)^d and
5645 : * Y in Q(CHI)^D such that
5646 : * B * X = M * Y, i.e. Minv * rowpermute(B, Mindex * X) = Y
5647 : *(B - I * rowpermute(B, Mindex)) * X = 0.
5648 : * where I = M * Minv. Rows of (B - I * ...) are 0 up to lim so
5649 : * are not included */
5650 154 : GEN Mindex = MF_get_Mindex(mf), Iden = gel(TMP,5);
5651 : pari_timer TT;
5652 154 : pari_sp av = avma;
5653 154 : if (DEBUGLEVEL) timer_start(&TT);
5654 238 : for (i = 1; i <= nE; i++)
5655 : {
5656 224 : pari_sp av2 = avma;
5657 : GEN e, z, B;
5658 :
5659 224 : e = Q_primpart(RgXn_mul(E1i, gel(E,i), LIM));
5660 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: E[%ld] / E[1]", i+1);
5661 : /* the first time A is over Z and it is more efficient to lift than
5662 : * to let RgXn_mul use Kronecker's trick */
5663 224 : if (POLCYC && i == 1) e = liftpol_shallow(e);
5664 224 : B = mf1intermat(A, Mindex, e, Iden, lim, i == 1? NULL: POLCYC);
5665 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: ... intermat");
5666 224 : z = gc_upto(av2, QabM_ker(B, POLCYC, ordchi));
5667 224 : if (DEBUGLEVEL)
5668 0 : timer_printf(&TT, "mf1basis: ... kernel (dim %ld)",lg(z)-1);
5669 224 : if (lg(z) == 1) return NULL;
5670 224 : if (lg(z) == lg(A)) { set_avma(av2); continue; } /* no progress */
5671 224 : C = C? _RgXQM_mul(C, z, POLCYC): z;
5672 224 : A = _RgXQM_mul(A, z, POLCYC);
5673 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: ... updates");
5674 224 : if (lg(z)-1 == dimp) break;
5675 84 : if (gc_needed(av, 1))
5676 : {
5677 0 : if (DEBUGMEM > 1) pari_warn(warnmem,"mf1basis i = %ld", i);
5678 0 : (void)gc_all(av, 2, &A, &C);
5679 : }
5680 : }
5681 154 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: intersection [total]");
5682 : }
5683 168 : lA = lg(A);
5684 168 : if (lA-1 == dimp)
5685 : {
5686 140 : A = mfmatsermul(rowslice(A, 1, lim1), E1i);
5687 140 : if (POLCYC) A = RgXQM_red(A, POLCYC);
5688 140 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: matsermul [1]");
5689 : }
5690 : else
5691 : {
5692 28 : A = mfmatsermul(A, E1i);
5693 28 : if (POLCYC) A = RgXQM_red(A, POLCYC);
5694 28 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: matsermul [2]");
5695 28 : A = mfstabiter(&C, A, chip, TMP, POLCYC, ordchi);
5696 28 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: Hecke stability");
5697 28 : if (!A) return NULL;
5698 : }
5699 168 : if (dE1i) C = RgM_Rg_mul(C, dE1i);
5700 168 : if (POLCYC)
5701 : {
5702 147 : A = QXQM_to_mod_shallow(A, POLCYC);
5703 147 : C = QXQM_to_mod_shallow(C, POLCYC);
5704 : }
5705 168 : lA = lg(A);
5706 581 : for (i = 1; i < lA; i++)
5707 : {
5708 413 : GEN c, v = gel(A,i);
5709 413 : gel(A,i) = RgV_normalize(v, &c);
5710 413 : gel(C,i) = RgC_Rg_mul(gel(C,i), c);
5711 : }
5712 168 : Minv = gel(mfclean(A, POLCYC, ordchi, 0), 2);
5713 168 : A = RgM_Minv_mul(A, Minv);
5714 168 : C = RgM_Minv_mul(C, Minv);
5715 168 : *pS = vecmflineardiv0(MF_get_S(mf), C, gel(EB,1));
5716 168 : return A;
5717 : }
5718 :
5719 : static void
5720 413 : MF_set_space(GEN mf, long x) { gmael(mf,1,4) = utoi(x); }
5721 : static GEN
5722 252 : mf1_cusptonew(GEN mf, GEN vSP)
5723 : {
5724 252 : const long vy = 1;
5725 : long i, lP, dSnew, ct;
5726 252 : GEN vP, F, S, Snew, vF, v = split_ii(mf, 0, 0, vSP, &i);
5727 :
5728 252 : F = gel(v,1);
5729 252 : vP= gel(v,2); lP = lg(vP);
5730 252 : if (lP == 1) { obj_insert(mf, MF_SPLIT, v); return NULL; }
5731 238 : MF_set_space(mf, mf_NEW);
5732 238 : S = MF_get_S(mf);
5733 238 : dSnew = dim_sum(v);
5734 238 : Snew = cgetg(dSnew + 1, t_VEC); ct = 0;
5735 238 : vF = cgetg(lP, t_MAT);
5736 546 : for (i = 1; i < lP; i++)
5737 : {
5738 308 : GEN V, P = gel(vP,i), f = liftpol_shallow(gel(F,i));
5739 308 : long j, d = degpol(P);
5740 308 : gel(vF,i) = V = zerocol(dSnew);
5741 308 : if (d == 1)
5742 : {
5743 140 : gel(Snew, ct+1) = mflineardiv_linear(S, f, 0);
5744 140 : gel(V, ct+1) = gen_1;
5745 : }
5746 : else
5747 : {
5748 168 : f = RgXV_to_RgM(f,d);
5749 511 : for (j = 1; j <= d; j++)
5750 : {
5751 343 : gel(Snew, ct+j) = mflineardiv_linear(S, row(f,j), 0);
5752 343 : gel(V, ct+j) = mkpolmod(pol_xn(j-1,vy), P);
5753 : }
5754 : }
5755 308 : ct += d;
5756 : }
5757 238 : obj_insert(mf, MF_SPLIT, mkvec2(vF, vP));
5758 238 : gel(mf,3) = Snew; return mf;
5759 : }
5760 : static GEN
5761 3969 : mf1init(long N, GEN CHI, GEN TMP, GEN vSP, long space, long flraw)
5762 : {
5763 3969 : GEN mf, mf1, S, M = mf1basis(N, CHI, TMP, vSP, &S, NULL);
5764 3969 : if (!M) return NULL;
5765 952 : mf1 = mkvec4(stoi(N), gen_1, CHI, utoi(mf_CUSP));
5766 952 : mf = mkmf(mf1, cgetg(1,t_VEC), S, gen_0, NULL);
5767 952 : if (space == mf_NEW)
5768 : {
5769 252 : gel(mf,5) = mfcleanCHI(M,CHI, 0);
5770 252 : mf = mf1_cusptonew(mf, vSP); if (!mf) return NULL;
5771 238 : if (!flraw) M = mfcoefs_mf(mf, mfsturmNk(N,1)+1, 1);
5772 : }
5773 938 : gel(mf,5) = flraw? zerovec(3): mfcleanCHI(M, CHI, 0);
5774 938 : return mf;
5775 : }
5776 :
5777 : static GEN
5778 1029 : mfEMPTY(GEN mf1)
5779 : {
5780 1029 : GEN Minv = mkMinv(cgetg(1,t_MAT), NULL,NULL,NULL);
5781 1029 : GEN M = mkvec3(cgetg(1,t_VECSMALL), Minv, cgetg(1,t_MAT));
5782 1029 : return mkmf(mf1, cgetg(1,t_VEC), cgetg(1,t_VEC), cgetg(1,t_VEC), M);
5783 : }
5784 : static GEN
5785 616 : mfEMPTYall(long N, GEN gk, GEN vCHI, long space)
5786 : {
5787 : long i, l;
5788 : GEN v, gN, gs;
5789 616 : if (!vCHI) return cgetg(1, t_VEC);
5790 14 : gN = utoipos(N); gs = utoi(space);
5791 14 : l = lg(vCHI); v = cgetg(l, t_VEC);
5792 42 : for (i = 1; i < l; i++) gel(v,i) = mfEMPTY(mkvec4(gN,gk,gel(vCHI,i),gs));
5793 14 : return v;
5794 : }
5795 :
5796 : static GEN
5797 3983 : fmt_dim(GEN CHI, long d, long dih)
5798 3983 : { return mkvec4(gmfcharorder(CHI), gmfcharno(CHI), utoi(d), stoi(dih)); }
5799 : /* merge two vector of fmt_dim's for the same vector of characters. If CHI
5800 : * is not NULL, remove dim-0 spaces and add character from CHI */
5801 : static GEN
5802 7 : merge_dims(GEN V, GEN W, GEN CHI)
5803 : {
5804 7 : long i, j, id, l = lg(V);
5805 7 : GEN A = cgetg(l, t_VEC);
5806 7 : if (l == 1) return A;
5807 7 : id = CHI? 1: 3;
5808 21 : for (i = j = 1; i < l; i++)
5809 : {
5810 14 : GEN v = gel(V,i), w = gel(W,i);
5811 14 : long dv = itou(gel(v,id)), dvh = itou(gel(v,id+1)), d;
5812 14 : long dw = itou(gel(w,id)), dwh = itou(gel(w,id+1));
5813 14 : d = dv + dw;
5814 14 : if (d || CHI)
5815 14 : gel(A,j++) = CHI? fmt_dim(gel(CHI,i),d, dvh+dwh)
5816 14 : : mkvec2s(d,dvh+dwh);
5817 : }
5818 7 : setlg(A, j); return A;
5819 : }
5820 : static GEN
5821 3010 : mfdim0all(GEN w)
5822 : {
5823 3038 : if (w) retconst_vec(lg(w)-1, zerovec(2));
5824 3003 : return cgetg(1,t_VEC);
5825 : }
5826 : static long
5827 7315 : mf1cuspdim_i(long N, GEN CHI, GEN TMP, GEN vSP, long *dih)
5828 : {
5829 7315 : pari_sp av = avma;
5830 7315 : GEN b = mf1basis(N, CHI, TMP, vSP, NULL, dih);
5831 7315 : return gc_long(av, b? itou(b): 0);
5832 : }
5833 :
5834 : static long
5835 476 : mf1cuspdim(long N, GEN CHI, GEN vSP)
5836 : {
5837 476 : if (!vSP) vSP = get_vDIH(N, divisorsNF(N, mfcharconductor(CHI)));
5838 476 : return mf1cuspdim_i(N, CHI, NULL, vSP, NULL);
5839 : }
5840 : static GEN
5841 4144 : mf1cuspdimall(long N, GEN vCHI)
5842 : {
5843 : GEN z, TMP, w, vSP;
5844 : long i, j, l;
5845 4144 : if (wt1empty(N)) return mfdim0all(vCHI);
5846 1141 : w = mf1chars(N,vCHI);
5847 1141 : l = lg(w); if (l == 1) return cgetg(1,t_VEC);
5848 1141 : z = cgetg(l, t_VEC);
5849 1141 : TMP = mf1_pre(N); vSP = get_vDIH(N, NULL);
5850 7861 : for (i = j = 1; i < l; i++)
5851 : {
5852 6720 : GEN CHI = gel(w,i);
5853 6720 : long dih, d = mf1cuspdim_i(N, CHI, TMP, vSP, &dih);
5854 6720 : if (vCHI)
5855 42 : gel(z,j++) = mkvec2s(d, dih);
5856 6678 : else if (d)
5857 1428 : gel(z,j++) = fmt_dim(CHI, d, dih);
5858 : }
5859 1141 : setlg(z,j); return z;
5860 : }
5861 :
5862 : /* dimension of S_1(Gamma_1(N)) */
5863 : static long
5864 4123 : mf1cuspdimsum(long N)
5865 : {
5866 4123 : pari_sp av = avma;
5867 4123 : GEN v = mf1cuspdimall(N, NULL);
5868 4123 : long i, ct = 0, l = lg(v);
5869 5544 : for (i = 1; i < l; i++)
5870 : {
5871 1421 : GEN w = gel(v,i); /* [ord(CHI),*,dim,*] */
5872 1421 : ct += itou(gel(w,3))*myeulerphiu(itou(gel(w,1)));
5873 : }
5874 4123 : return gc_long(av,ct);
5875 : }
5876 :
5877 : static GEN
5878 56 : mf1newdimall(long N, GEN vCHI)
5879 : {
5880 : GEN z, w, vTMP, vSP, fa, P, E;
5881 : long i, c, l, lw, P1;
5882 56 : if (wt1empty(N)) return mfdim0all(vCHI);
5883 56 : w = mf1chars(N,vCHI);
5884 56 : lw = lg(w); if (lw == 1) return cgetg(1,t_VEC);
5885 56 : vTMP = const_vec(N, NULL);
5886 56 : vSP = get_vDIH(N, NULL);
5887 56 : gel(vTMP,N) = mf1_pre(N);
5888 : /* if p || N and p \nmid F(CHI), S_1^new(G0(N),chi) = 0 */
5889 56 : fa = znstar_get_faN(gmael(w,1,1));
5890 56 : P = gel(fa,1); l = lg(P);
5891 56 : E = gel(fa,2);
5892 154 : for (i = P1 = 1; i < l; i++)
5893 98 : if (E[i] == 1) P1 *= itou(gel(P,i));
5894 : /* P1 = \prod_{v_p(N) = 1} p */
5895 56 : z = cgetg(lw, t_VEC);
5896 182 : for (i = c = 1; i < lw; i++)
5897 : {
5898 : long S, j, l, F, dihnew;
5899 126 : GEN D, CHI = gel(w,i), CHIP = mfchartoprimitive(CHI,&F);
5900 :
5901 126 : S = F % P1? 0: mf1cuspdim_i(N, CHI, gel(vTMP,N), vSP, &dihnew);
5902 126 : if (!S)
5903 : {
5904 56 : if (vCHI) gel(z, c++) = zerovec(2);
5905 56 : continue;
5906 : }
5907 70 : D = mydivisorsu(N/F); l = lg(D);
5908 77 : for (j = l-2; j > 0; j--) /* skip last M = N */
5909 : {
5910 7 : long M = D[j]*F, m, s, dih;
5911 7 : GEN TMP = gel(vTMP,M);
5912 7 : if (wt1empty(M) || !(m = mubeta(D[l-j]))) continue; /*m = mubeta(N/M)*/
5913 7 : if (!TMP) gel(vTMP,M) = TMP = mf1_pre(M);
5914 7 : s = mf1cuspdim_i(M, CHIP, TMP, vSP, &dih);
5915 7 : if (s) { S += m * s; dihnew += m * dih; }
5916 : }
5917 70 : if (vCHI)
5918 63 : gel(z,c++) = mkvec2s(S, dihnew);
5919 7 : else if (S)
5920 7 : gel(z, c++) = fmt_dim(CHI, S, dihnew);
5921 : }
5922 56 : setlg(z,c); return z;
5923 : }
5924 :
5925 : static GEN
5926 28 : mf1olddimall(long N, GEN vCHI)
5927 : {
5928 : long i, j, l;
5929 : GEN z, w;
5930 28 : if (wt1empty(N)) return mfdim0all(vCHI);
5931 28 : w = mf1chars(N,vCHI);
5932 28 : l = lg(w); z = cgetg(l, t_VEC);
5933 84 : for (i = j = 1; i < l; i++)
5934 : {
5935 56 : GEN CHI = gel(w,i);
5936 56 : long d = mfolddim(N, 1, CHI);
5937 56 : if (vCHI)
5938 28 : gel(z,j++) = mkvec2s(d,d?-1:0);
5939 28 : else if (d)
5940 7 : gel(z, j++) = fmt_dim(CHI, d, -1);
5941 : }
5942 28 : setlg(z,j); return z;
5943 : }
5944 :
5945 : static long
5946 469 : mf1olddimsum(long N)
5947 : {
5948 : GEN D;
5949 469 : long N2, i, l, S = 0;
5950 469 : newd_params(N, &N2); /* will ensure mubeta != 0 */
5951 469 : D = mydivisorsu(N/N2); l = lg(D);
5952 2485 : for (i = 2; i < l; i++)
5953 : {
5954 2016 : long M = D[l-i]*N2, d = mf1cuspdimsum(M);
5955 2016 : if (d) S -= mubeta(D[i]) * d;
5956 : }
5957 469 : return S;
5958 : }
5959 : static long
5960 1050 : mf1newdimsum(long N)
5961 : {
5962 1050 : long S = mf1cuspdimsum(N);
5963 1050 : return S? S - mf1olddimsum(N): 0;
5964 : }
5965 :
5966 : /* return the automorphism of a degree-2 nf */
5967 : static GEN
5968 5768 : nf2_get_conj(GEN nf)
5969 : {
5970 5768 : GEN pol = nf_get_pol(nf);
5971 5768 : return deg1pol_shallow(gen_m1, negi(gel(pol,3)), varn(pol));
5972 : }
5973 : static int
5974 42 : foo_stable(GEN foo)
5975 42 : { return lg(foo) != 3 || equalii(gel(foo,1), gel(foo,2)); }
5976 :
5977 : static long
5978 224 : mfisdihedral(GEN vF, GEN DIH)
5979 : {
5980 224 : GEN vG = gel(DIH,1), M = gel(DIH,2), v, G, bnr, w, gen, D, f, nf, tau;
5981 224 : GEN bnr0 = NULL, f0, f0b, xin, foo;
5982 : long i, l, e, j, L, n;
5983 224 : if (lg(M) == 1) return 0;
5984 42 : v = RgM_RgC_invimage(M, vF);
5985 42 : if (!v) return 0;
5986 42 : l = lg(v);
5987 42 : for (i = 1; i < l; i++)
5988 42 : if (!gequal0(gel(v,i))) break;
5989 42 : if (i == l) return 0;
5990 42 : G = gel(vG,i);
5991 42 : bnr = gel(G,2); D = cyc_get_expo(bnr_get_cyc(bnr));
5992 42 : w = gel(G,3);
5993 42 : f = bnr_get_mod(bnr);
5994 42 : nf = bnr_get_nf(bnr);
5995 42 : tau = nf2_get_conj(nf);
5996 42 : f0 = gel(f,1); foo = gel(f,2);
5997 42 : f0b = galoisapply(nf, tau, f0);
5998 42 : xin = zv_to_ZV(gel(w,2)); /* xi(bnr.gen[i]) = e(xin[i] / D) */
5999 42 : if (!foo_stable(foo)) { foo = mkvec2(gen_1, gen_1); bnr0 = bnr; }
6000 42 : if (!gequal(f0, f0b))
6001 : {
6002 21 : f0 = idealmul(nf, f0, idealdivexact(nf, f0b, idealadd(nf, f0, f0b)));
6003 21 : bnr0 = bnr;
6004 : }
6005 42 : if (bnr0)
6006 : { /* conductor not ambiguous */
6007 : GEN S;
6008 28 : bnr = Buchray(bnr_get_bnf(bnr), mkvec2(f0, foo), nf_INIT | nf_GEN);
6009 28 : S = bnrsurjection(bnr, bnr0);
6010 28 : xin = FpV_red(RgV_RgM_mul(xin, gel(S,1)), D);
6011 : /* still xi(gen[i]) = e(xin[i] / D), for the new generators; D stays
6012 : * the same, not exponent(bnr.cyc) ! */
6013 : }
6014 42 : gen = bnr_get_gen(bnr); L = lg(gen);
6015 77 : for (j = 1, e = itou(D); j < L; j++)
6016 : {
6017 63 : GEN Ng = idealnorm(nf, gel(gen,j));
6018 63 : GEN a = shifti(gel(xin,j), 1); /* xi(g_j^2) = e(a/D) */
6019 63 : GEN b = FpV_dotproduct(xin, isprincipalray(bnr,Ng), D);
6020 63 : GEN m = Fp_sub(a, b, D); /* xi(g_j/g_j^\tau) = e(m/D) */
6021 63 : e = ugcd(e, itou(m)); if (e == 1) break;
6022 : }
6023 42 : n = itou(D) / e;
6024 42 : return n == 1? 4: 2*n;
6025 : }
6026 :
6027 : static ulong
6028 119 : myradicalu(ulong n) { return zv_prod(gel(myfactoru(n),1)); }
6029 :
6030 : /* list of fundamental discriminants unramified outside N, with sign s
6031 : * [s = 0 => no sign condition] */
6032 : static GEN
6033 119 : mfunram(long N, long s)
6034 : {
6035 119 : long cN = myradicalu(N >> vals(N)), p = 1, m = 1, l, c, i;
6036 119 : GEN D = mydivisorsu(cN), res;
6037 119 : l = lg(D);
6038 119 : if (s == 1) m = 0; else if (s == -1) p = 0;
6039 119 : res = cgetg(6*l - 5, t_VECSMALL);
6040 119 : c = 1;
6041 119 : if (!odd(N))
6042 : { /* d = 1 */
6043 56 : if (p) res[c++] = 8;
6044 56 : if (m) { res[c++] =-8; res[c++] =-4; }
6045 : }
6046 364 : for (i = 2; i < l; i++)
6047 : { /* skip d = 1, done above */
6048 245 : long d = D[i], d4 = d & 3L; /* d odd, squarefree, d4 = 1 or 3 */
6049 245 : if (d4 == 1) { if (p) res[c++] = d; }
6050 182 : else { if (m) res[c++] =-d; }
6051 245 : if (!odd(N))
6052 : {
6053 56 : if (p) { res[c++] = 8*d; if (d4 == 3) res[c++] = 4*d; }
6054 56 : if (m) { res[c++] =-8*d; if (d4 == 1) res[c++] =-4*d; }
6055 : }
6056 : }
6057 119 : setlg(res, c); return res;
6058 : }
6059 :
6060 : /* Return 1 if F is definitely not S4 type; return 0 on failure. */
6061 : static long
6062 105 : mfisnotS4(long N, GEN w)
6063 : {
6064 105 : GEN D = mfunram(N, 0);
6065 105 : long i, lD = lg(D), lw = lg(w);
6066 616 : for (i = 1; i < lD; i++)
6067 : {
6068 511 : long p, d = D[i], ok = 0;
6069 1442 : for (p = 2; p < lw; p++)
6070 1442 : if (w[p] && kross(d,p) == -1) { ok = 1; break; }
6071 511 : if (!ok) return 0;
6072 : }
6073 105 : return 1;
6074 : }
6075 :
6076 : /* Return 1 if Q(sqrt(5)) \not\subset Q(F), i.e. F is definitely not A5 type;
6077 : * return 0 on failure. */
6078 : static long
6079 105 : mfisnotA5(GEN F)
6080 : {
6081 105 : GEN CHI = mf_get_CHI(F), P = mfcharpol(CHI), T, Q;
6082 :
6083 105 : if (mfcharorder(CHI) % 5 == 0) return 0;
6084 105 : T = mf_get_field(F); if (degpol(T) == 1) return 1;
6085 105 : if (degpol(P) > 1) T = rnfequation(P,T);
6086 105 : Q = gsubgs(pol_xn(2,varn(T)), 5);
6087 105 : return (typ(nfisincl(Q, T)) == t_INT);
6088 : }
6089 :
6090 : /* v[p+1]^2 / chi(p) - 2 = z + 1/z with z primitive root of unity of order n,
6091 : * return n */
6092 : static long
6093 6741 : mffindrootof1(GEN v, long p, GEN CHI)
6094 : {
6095 6741 : GEN ap = gel(v,p+1), u0, u1, u1k, u2;
6096 6741 : long c = 1;
6097 6741 : if (gequal0(ap)) return 2;
6098 5033 : u0 = gen_2; u1k = u1 = gsubgs(gdiv(gsqr(ap), mfchareval(CHI, p)), 2);
6099 14812 : while (!gequalsg(2, liftpol_shallow(u1))) /* u1 = z^c + z^-c */
6100 : {
6101 9779 : u2 = gsub(gmul(u1k, u1), u0);
6102 9779 : u0 = u1; u1 = u2; c++;
6103 : }
6104 5033 : return c;
6105 : }
6106 :
6107 : /* we known that F is not dihedral */
6108 : static long
6109 182 : mfgaloistype_i(long N, GEN CHI, GEN F, GEN v)
6110 : {
6111 : forprime_t iter;
6112 182 : long lim = lg(v)-2;
6113 182 : GEN w = zero_zv(lim);
6114 : pari_sp av;
6115 : ulong p;
6116 182 : u_forprime_init(&iter, 2, lim);
6117 182 : av = avma;
6118 5292 : while((p = u_forprime_next(&iter))) if (N%p) switch(mffindrootof1(v, p, CHI))
6119 : {
6120 1400 : case 1: case 2: continue;
6121 3451 : case 3: w[p] = 1; break;
6122 70 : case 4: return -24; /* S4 */
6123 0 : case 5: return -60; /* A5 */
6124 7 : default: pari_err_DOMAIN("mfgaloistype", "form", "not a",
6125 : strtoGENstr("cuspidal eigenform"), F);
6126 0 : set_avma(av);
6127 : }
6128 105 : if (mfisnotS4(N,w) && mfisnotA5(F)) return -12; /* A4 */
6129 0 : return 0; /* FAILURE */
6130 : }
6131 :
6132 : static GEN
6133 224 : mfgaloistype0(long N, GEN CHI, GEN F, GEN DIH, long lim)
6134 : {
6135 224 : pari_sp av = avma;
6136 224 : GEN vF = mftocol(F, lim, 1);
6137 224 : long t = mfisdihedral(vF, DIH), bound;
6138 224 : if (t) return gc_stoi(av,t);
6139 182 : bound = maxss(200, 5*expu(N)*expu(N));
6140 : for(;;)
6141 : {
6142 182 : t = mfgaloistype_i(N, CHI, F, vF);
6143 175 : set_avma(av); if (t) return stoi(t);
6144 0 : if (lim > bound) return gen_0;
6145 0 : lim += lim >> 1;
6146 0 : vF = mfcoefs_i(F,lim,1);
6147 : }
6148 : }
6149 :
6150 : /* If f is NULL, give all the galoistypes, otherwise just for f */
6151 : /* Return 0 to indicate failure; in this case the type is either -12 or -60,
6152 : * most likely -12. FIXME using the Galois representation. */
6153 : GEN
6154 231 : mfgaloistype(GEN NK, GEN f)
6155 : {
6156 231 : pari_sp av = avma;
6157 231 : GEN CHI, T, F, DIH, SP, mf = checkMF_i(NK);
6158 : long N, k, lL, i, lim, SB;
6159 :
6160 231 : if (f && !checkmf_i(f)) pari_err_TYPE("mfgaloistype", f);
6161 224 : if (mf)
6162 : {
6163 189 : N = MF_get_N(mf);
6164 189 : k = MF_get_k(mf);
6165 189 : CHI = MF_get_CHI(mf);
6166 : }
6167 : else
6168 : {
6169 35 : checkNK(NK, &N, &k, &CHI, 0);
6170 35 : mf = f? NULL: mfinit_i(NK, mf_NEW);
6171 : }
6172 224 : if (k != 1) pari_err_DOMAIN("mfgaloistype", "k", "!=", gen_1, stoi(k));
6173 224 : SB = mf? mfsturm_mf(mf): mfsturmNk(N,1);
6174 224 : SP = get_DIH(N);
6175 224 : DIH = mfdihedralnew(N, CHI, SP);
6176 224 : lim = lg(DIH) == 1? 200: SB;
6177 224 : DIH = mkvec2(DIH, mfvectomat(DIH,SB,1));
6178 224 : if (f) return gc_INT(av, mfgaloistype0(N,CHI, f, DIH, lim));
6179 126 : F = mfeigenbasis(mf); lL = lg(F);
6180 126 : T = cgetg(lL, t_VEC);
6181 252 : for (i=1; i < lL; i++) gel(T,i) = mfgaloistype0(N, CHI, gel(F,i), DIH, lim);
6182 126 : return gc_upto(av, T);
6183 : }
6184 :
6185 : /******************************************************************/
6186 : /* Find all dihedral forms. */
6187 : /******************************************************************/
6188 : /* lim >= 2 */
6189 : static void
6190 14 : consttabdihedral(long lim) { cache_set(cache_DIH, mfdihedralall(lim)); }
6191 :
6192 : /* a ideal coprime to bnr modulus */
6193 : static long
6194 107611 : mfdiheval(GEN bnr, GEN w, GEN a)
6195 : {
6196 107611 : GEN L, cycn = gel(w,1), chin = gel(w,2);
6197 107611 : long ordmax = cycn[1];
6198 107611 : L = ZV_to_Flv(isprincipalray(bnr,a), ordmax);
6199 107611 : return Flv_dotproduct(chin, L, ordmax);
6200 : }
6201 :
6202 : /* x(t^k) mod T = polcyclo(m), 0 <= k < m */
6203 : static GEN
6204 30331 : Galois(GEN x, long k, GEN T, long m)
6205 : {
6206 : GEN B;
6207 : long i, ik, d;
6208 30331 : if (typ(x) != t_POL) return x;
6209 7455 : if (varn(x) != varn(T)) pari_APPLY_pol_normalized(Galois(gel(x,i), k, T, m));
6210 7420 : if ((d = degpol(x)) <= 0) return x;
6211 7063 : B = cgetg(m + 2, t_POL); B[1] = x[1]; gel(B,2) = gel(x,2);
6212 61565 : for (i = 1; i < m; i++) gel(B, i+2) = gen_0;
6213 23940 : for (i = 1, ik = k; i <= d; i++, ik = Fl_add(ik, k, m))
6214 16877 : gel(B, ik + 2) = gel(x, i+2);
6215 7063 : return QX_ZX_rem(normalizepol(B), T);
6216 : }
6217 : static GEN
6218 1022 : vecGalois(GEN x, long k, GEN T, long m)
6219 31332 : { pari_APPLY_same(Galois(gel(x,i), k, T, m)); }
6220 :
6221 : static GEN
6222 234178 : fix_pol(GEN S, GEN Pn, int *trace)
6223 : {
6224 234178 : if (typ(S) != t_POL) return S;
6225 118069 : S = RgX_rem(S, Pn);
6226 118069 : if (typ(S) == t_POL)
6227 : {
6228 118069 : switch(lg(S))
6229 : {
6230 45108 : case 2: return gen_0;
6231 20517 : case 3: return gel(S,2);
6232 : }
6233 52444 : *trace = 1;
6234 : }
6235 52444 : return S;
6236 : }
6237 :
6238 : static GEN
6239 13573 : dihan(GEN bnr, GEN w, GEN k0j, long m, ulong lim)
6240 : {
6241 13573 : GEN nf = bnr_get_nf(bnr), f = bid_get_ideal(bnr_get_bid(bnr));
6242 13573 : GEN v = zerovec(lim+1), cycn = gel(w,1), Tinit = gel(w,3);
6243 13573 : GEN Pn = gel(Tinit,lg(Tinit)==4? 2: 1);
6244 13573 : long j, ordmax = cycn[1];
6245 13573 : long D = itos(nf_get_disc(nf)), vt = varn(Pn);
6246 13573 : int trace = 0;
6247 : ulong p, n;
6248 : forprime_t T;
6249 :
6250 13573 : if (!lim) return v;
6251 13363 : gel(v,2) = gen_1;
6252 13363 : u_forprime_init(&T, 2, lim);
6253 : /* fill in prime powers first */
6254 116207 : while ((p = u_forprime_next(&T)))
6255 : {
6256 : GEN vP, vchiP, S;
6257 : long k, lP;
6258 : ulong q, qk;
6259 102844 : if (kross(D,p) >= 0) q = p;
6260 45192 : else if (!(q = umuluu_le(p,p,lim))) continue;
6261 : /* q = Norm P */
6262 65856 : vP = idealprimedec(nf, utoipos(p));
6263 65856 : lP = lg(vP);
6264 65856 : vchiP = cgetg(lP, t_VECSMALL);
6265 179081 : for (j = k = 1; j < lP; j++)
6266 : {
6267 113225 : GEN P = gel(vP,j);
6268 113225 : if (!idealval(nf, f, P)) vchiP[k++] = mfdiheval(bnr,w,P);
6269 : }
6270 65856 : if (k == 1) continue;
6271 62188 : setlg(vchiP, k); lP = k;
6272 62188 : if (lP == 2)
6273 : { /* one prime above p not dividing f */
6274 16765 : long s, s0 = vchiP[1];
6275 27069 : for (qk=q, s = s0;; s = Fl_add(s,s0,ordmax))
6276 : {
6277 27069 : S = Qab_zeta(s, ordmax, vt);
6278 27069 : gel(v, qk+1) = fix_pol(S, Pn, &trace);
6279 27069 : if (!(qk = umuluu_le(qk,q,lim))) break;
6280 : }
6281 : }
6282 : else /* two primes above p not dividing f */
6283 : {
6284 45423 : long s, s0 = vchiP[1], s1 = vchiP[2];
6285 45423 : for (qk=q, k = 1;; k++)
6286 18424 : { /* sum over a,b s.t. Norm( P1^a P2^b ) = q^k, i.e. a+b = k */
6287 : long a;
6288 63847 : GEN S = gen_0;
6289 220752 : for (a = 0; a <= k; a++)
6290 : {
6291 156905 : s = Fl_add(Fl_mul(a, s0, ordmax), Fl_mul(k-a, s1, ordmax), ordmax);
6292 156905 : S = gadd(S, Qab_zeta(s, ordmax, vt));
6293 : }
6294 63847 : gel(v, qk+1) = fix_pol(S, Pn, &trace);
6295 63847 : if (!(qk = umuluu_le(qk,q,lim))) break;
6296 : }
6297 : }
6298 : }
6299 : /* complete with nonprime powers */
6300 308098 : for (n = 2; n <= lim; n++)
6301 : {
6302 294735 : GEN S, fa = myfactoru(n), P = gel(fa, 1), E = gel(fa, 2);
6303 : long q;
6304 294735 : if (lg(P) == 2) continue;
6305 : /* not a prime power */
6306 143262 : q = upowuu(P[1],E[1]);
6307 143262 : S = gmul(gel(v, q + 1), gel(v, n/q + 1));
6308 143262 : gel(v, n+1) = fix_pol(S, Pn, &trace);
6309 : }
6310 13363 : if (trace)
6311 : {
6312 7154 : long k0 = k0j[1], jdeg = k0j[2];
6313 7154 : v = QabV_tracerel(Tinit, jdeg, v); /* Apply Galois Mod(k0, ordw) */
6314 7154 : if (k0 > 1) v = vecGalois(v, k0, gel(Tinit,1), m);
6315 : }
6316 13363 : return v;
6317 : }
6318 :
6319 : /* as cyc_normalize for t_VECSMALL cyc */
6320 : static GEN
6321 26810 : cyc_normalize_zv(GEN cyc)
6322 : {
6323 26810 : long i, o = cyc[1], l = lg(cyc); /* > 1 */
6324 26810 : GEN D = cgetg(l, t_VECSMALL);
6325 31185 : D[1] = o; for (i = 2; i < l; i++) D[i] = o / cyc[i];
6326 26810 : return D;
6327 : }
6328 : /* as char_normalize for t_VECSMALLs */
6329 : static GEN
6330 118517 : char_normalize_zv(GEN chi, GEN ncyc)
6331 : {
6332 118517 : long i, l = lg(chi);
6333 118517 : GEN c = cgetg(l, t_VECSMALL);
6334 118517 : if (l > 1) {
6335 118517 : c[1] = chi[1];
6336 160454 : for (i = 2; i < l; i++) c[i] = chi[i] * ncyc[i];
6337 : }
6338 118517 : return c;
6339 : }
6340 :
6341 : static GEN
6342 9331 : dihan_bnf(long D)
6343 : {
6344 9331 : GEN c = getrand(), bnf;
6345 9331 : setrand(gen_1);
6346 9331 : bnf = Buchall(quadpoly_i(stoi(D)), nf_FORCE, LOWDEFAULTPREC);
6347 9331 : setrand(c);
6348 9331 : return bnf;
6349 : }
6350 : static GEN
6351 37758 : dihan_bnr(GEN bnf, GEN A)
6352 : {
6353 37758 : GEN c = getrand(), bnr;
6354 37758 : setrand(gen_1);
6355 37758 : bnr = Buchray(bnf, A, nf_INIT|nf_GEN);
6356 37758 : setrand(c);
6357 37758 : return bnr;
6358 : }
6359 : /* Hecke xi * (D/.) = Dirichlet chi, return v in Q^r st chi(g_i) = e(v[i]).
6360 : * cycn = cyc_normalize_zv(bnr.cyc), chin = char_normalize_zv(chi,cyc) */
6361 : static GEN
6362 34489 : bnrchartwist2conrey(GEN chin, GEN cycn, GEN bnrconreyN, GEN kroconreyN)
6363 : {
6364 34489 : long l = lg(bnrconreyN), c1 = cycn[1], i;
6365 34489 : GEN v = cgetg(l, t_COL);
6366 125363 : for (i = 1; i < l; i++)
6367 : {
6368 90874 : GEN d = sstoQ(zv_dotproduct(chin, gel(bnrconreyN,i)), c1);
6369 90874 : if (kroconreyN[i] < 0) d = gadd(d, ghalf);
6370 90874 : gel(v,i) = d;
6371 : }
6372 34489 : return v;
6373 : }
6374 :
6375 : /* chi(g_i) = e(v[i]) denormalize wrt Conrey generators orders */
6376 : static GEN
6377 34489 : conreydenormalize(GEN znN, GEN v)
6378 : {
6379 34489 : GEN gcyc = znstar_get_conreycyc(znN), w;
6380 34489 : long l = lg(v), i;
6381 34489 : w = cgetg(l, t_COL);
6382 125363 : for (i = 1; i < l; i++)
6383 90874 : gel(w,i) = modii(gmul(gel(v,i), gel(gcyc,i)), gel(gcyc,i));
6384 34489 : return w;
6385 : }
6386 :
6387 : static long
6388 84028 : Miyake(GEN vchi, GEN gb, GEN cycn)
6389 : {
6390 84028 : long i, e = cycn[1], lb = lg(gb);
6391 84028 : GEN v = char_normalize_zv(vchi, cycn);
6392 124992 : for (i = 1; i < lb; i++)
6393 100268 : if ((zv_dotproduct(v, gel(gb,i)) - v[i]) % e) return 1;
6394 24724 : return 0;
6395 : }
6396 :
6397 : /* list of Hecke characters not induced by a Dirichlet character up to Galois
6398 : * conjugation, whose conductor is bnr.cond; cycn = cyc_normalize(bnr.cyc)*/
6399 : static GEN
6400 26810 : mklvchi(GEN bnr, GEN cycn, GEN gb)
6401 : {
6402 26810 : GEN cyc = bnr_get_cyc(bnr), cycsmall = ZV_to_zv(cyc);
6403 26810 : GEN vchi = cyc2elts(cycsmall);
6404 26810 : long ordmax = cycsmall[1], c, i, l;
6405 26810 : l = lg(vchi);
6406 304024 : for (i = c = 1; i < l; i++)
6407 : {
6408 277214 : GEN chi = gel(vchi,i);
6409 277214 : if (!gb || Miyake(chi, gb, cycn)) gel(vchi, c++) = Flv_to_ZV(chi);
6410 : }
6411 26810 : setlg(vchi, c); l = c;
6412 279300 : for (i = 1; i < l; i++)
6413 : {
6414 252490 : GEN chi = gel(vchi,i);
6415 : long n;
6416 252490 : if (!chi) continue;
6417 1055754 : for (n = 2; n < ordmax; n++)
6418 966476 : if (ugcd(n, ordmax) == 1)
6419 : {
6420 397670 : GEN tmp = ZV_ZV_mod(gmulsg(n, chi), cyc);
6421 : long j;
6422 7623539 : for (j = i+1; j < l; j++)
6423 7225869 : if (gel(vchi,j) && gequal(gel(vchi,j), tmp)) gel(vchi,j) = NULL;
6424 : }
6425 : }
6426 279300 : for (i = c = 1; i < l; i++)
6427 : {
6428 252490 : GEN chi = gel(vchi,i);
6429 252490 : if (chi && bnrisconductor(bnr, chi)) gel(vchi, c++) = chi;
6430 : }
6431 26810 : setlg(vchi, c); return vchi;
6432 : }
6433 :
6434 : static GEN
6435 7805 : get_gb(GEN bnr, GEN con)
6436 : {
6437 7805 : GEN gb, g = bnr_get_gen(bnr), nf = bnr_get_nf(bnr);
6438 7805 : long i, l = lg(g);
6439 7805 : gb = cgetg(l, t_VEC);
6440 18326 : for (i = 1; i < l; i++)
6441 10521 : gel(gb,i) = ZV_to_zv(isprincipalray(bnr, galoisapply(nf, con, gel(g,i))));
6442 7805 : return gb;
6443 : }
6444 : static GEN
6445 15862 : get_bnrconreyN(GEN bnr, GEN znN)
6446 : {
6447 15862 : GEN z, g = znstar_get_conreygen(znN);
6448 15862 : long i, l = lg(g);
6449 15862 : z = cgetg(l, t_VEC);
6450 57134 : for (i = 1; i < l; i++) gel(z,i) = ZV_to_zv(isprincipalray(bnr,gel(g,i)));
6451 15862 : return z;
6452 : }
6453 : /* con = NULL if D > 0 or if D < 0 and id != idcon. */
6454 : static GEN
6455 33698 : mfdihedralcommon(GEN bnf, GEN id, GEN znN, GEN kroconreyN, long vt,
6456 : long N, long D, GEN con)
6457 : {
6458 33698 : GEN bnr = dihan_bnr(bnf, id), cyc = ZV_to_zv( bnr_get_cyc(bnr) );
6459 : GEN bnrconreyN, cycn, cycN, Lvchi, res, P, vT;
6460 : long j, ordmax, l, lc, deghecke;
6461 :
6462 33698 : lc = lg(cyc); if (lc == 1) return NULL;
6463 26810 : cycn = cyc_normalize_zv(cyc);
6464 26810 : Lvchi = mklvchi(bnr, cycn, con? get_gb(bnr, con): NULL);
6465 26810 : l = lg(Lvchi);
6466 26810 : if (l == 1) return NULL;
6467 :
6468 15862 : bnrconreyN = get_bnrconreyN(bnr, znN);
6469 15862 : cycN = ZV_to_zv(znstar_get_cyc(znN));
6470 15862 : ordmax = cyc[1];
6471 15862 : vT = const_vec(odd(ordmax)? ordmax << 1: ordmax, NULL);
6472 15862 : P = polcyclo(ordmax, vt);
6473 15862 : gel(vT,ordmax) = Qab_trace_init(ordmax, ordmax, P, P);
6474 15862 : deghecke = myeulerphiu(ordmax);
6475 15862 : res = cgetg(l, t_VEC);
6476 50351 : for (j = 1; j < l; j++)
6477 : {
6478 34489 : GEN T, v, vchi = ZV_to_zv(gel(Lvchi,j));
6479 34489 : GEN chi, chin = char_normalize_zv(vchi, cycn);
6480 : long o, vnum, k0, degrel;
6481 34489 : v = bnrchartwist2conrey(chin, cycn, bnrconreyN, kroconreyN);
6482 34489 : o = itou(Q_denom(v));
6483 34489 : T = gel(vT, o);
6484 34489 : if (!T) gel(vT,o) = T = Qab_trace_init(ordmax, o, P, polcyclo(o,vt));
6485 34489 : chi = conreydenormalize(znN, v);
6486 34489 : vnum = itou(znconreyexp(znN, chi));
6487 34489 : chi = ZV_to_zv(znconreychar(znN,chi));
6488 34489 : degrel = deghecke / degpol(gel(T,1));
6489 34489 : k0 = zv_cyc_minimize(cycN, chi, coprimes_zv(o));
6490 34489 : vnum = Fl_powu(vnum, k0, N);
6491 : /* encodes degrel forms: jdeg = 0..degrel-1 */
6492 34489 : gel(res,j) = mkvec3(mkvecsmalln(5, N, k0 % o, vnum, D, degrel),
6493 : id, mkvec3(cycn,chin,T));
6494 : }
6495 15862 : return res;
6496 : }
6497 :
6498 : static long
6499 49364 : is_cond(long D, long n)
6500 : {
6501 49364 : if (D > 0) return n != 4 || (D&7L) == 1;
6502 30114 : return n != 2 && n != 3 && (n != 4 || (D&7L)!=1);
6503 : }
6504 : /* Append to v all dihedral weight 1 forms coming from D, if fundamental.
6505 : * level in [l1, l2] */
6506 : static void
6507 18718 : append_dihedral(GEN v, long D, long l1, long l2, long vt)
6508 : {
6509 18718 : long Da = labs(D), no, i, numi, ct, min, max;
6510 : GEN bnf, con, vI, resall, arch1, arch2;
6511 : pari_sp av;
6512 :
6513 : /* min <= Nf <= max */
6514 18718 : max = l2 / Da;
6515 18718 : if (l1 == l2)
6516 : { /* assume Da | l2 */
6517 140 : min = max;
6518 140 : if (D > 0 && min < 3) return;
6519 : }
6520 : else /* assume l1 < l2 */
6521 18578 : min = (l1 + Da-1)/Da;
6522 18718 : if (!sisfundamental(D)) return;
6523 :
6524 5726 : av = avma;
6525 5726 : bnf = dihan_bnf(D);
6526 5726 : con = nf2_get_conj(bnf_get_nf(bnf));
6527 5726 : vI = ideallist(bnf, max);
6528 55090 : numi = 0; for (i = min; i <= max; i++) numi += lg(gel(vI, i)) - 1;
6529 5726 : if (D > 0)
6530 : {
6531 1428 : numi <<= 1;
6532 1428 : arch1 = mkvec2(gen_1,gen_0);
6533 1428 : arch2 = mkvec2(gen_0,gen_1);
6534 : }
6535 : else
6536 4298 : arch1 = arch2 = NULL;
6537 5726 : resall = cgetg(numi+1, t_VEC); ct = 1;
6538 55090 : for (no = min; no <= max; no++) if (is_cond(D, no))
6539 : {
6540 44646 : long N = Da*no, lc, lI;
6541 44646 : GEN I = gel(vI, no), znN = znstar0(utoipos(N), 1), conreyN, kroconreyN;
6542 :
6543 44646 : conreyN = znstar_get_conreygen(znN); lc = lg(conreyN);
6544 44646 : kroconreyN = cgetg(lc, t_VECSMALL);
6545 166054 : for (i = 1; i < lc; i++) kroconreyN[i] = krosi(D, gel(conreyN, i));
6546 44646 : lI = lg(I);
6547 87822 : for (i = 1; i < lI; i++)
6548 : {
6549 43176 : GEN id = gel(I, i), idcon, z;
6550 : long j;
6551 43176 : if (typ(id) == t_INT) continue;
6552 28182 : idcon = galoisapply(bnf, con, id);
6553 51408 : for (j = i; j < lI; j++)
6554 51408 : if (gequal(idcon, gel(I, j))) { gel(I, j) = gen_0; break; }
6555 28182 : if (D < 0)
6556 : {
6557 17479 : GEN conk = i == j ? con : NULL;
6558 17479 : z = mfdihedralcommon(bnf, id, znN, kroconreyN, vt, N, D, conk);
6559 17479 : if (z) gel(resall, ct++) = z;
6560 : }
6561 : else
6562 : {
6563 : GEN ide;
6564 10703 : ide = mkvec2(id, arch1);
6565 10703 : z = mfdihedralcommon(bnf, ide, znN, kroconreyN, vt, N, D, NULL);
6566 10703 : if (z) gel(resall, ct++) = z;
6567 10703 : if (gequal(idcon,id)) continue;
6568 5516 : ide = mkvec2(id, arch2);
6569 5516 : z = mfdihedralcommon(bnf, ide, znN, kroconreyN, vt, N, D, NULL);
6570 5516 : if (z) gel(resall, ct++) = z;
6571 : }
6572 : }
6573 : }
6574 5726 : if (ct == 1) set_avma(av);
6575 : else
6576 : {
6577 4816 : setlg(resall, ct);
6578 4816 : vectrunc_append(v, gc_GEN(av, shallowconcat1(resall)));
6579 : }
6580 : }
6581 :
6582 : static long
6583 42042 : di_N(GEN a) { return gel(a,1)[1]; }
6584 : static GEN
6585 14 : mfdihedral(long N)
6586 : {
6587 14 : GEN D = mydivisorsu(N), res = vectrunc_init(2*N);
6588 14 : long j, l = lg(D), vt = fetch_user_var("t");
6589 105 : for (j = 2; j < l; j++)
6590 : { /* skip d = 1 */
6591 91 : long d = D[j];
6592 91 : if (d == 2) continue;
6593 84 : append_dihedral(res, -d, N,N, vt);
6594 84 : if (d >= 5 && D[l-j] >= 3) append_dihedral(res, d, N,N, vt);/* Nf >= 3 */
6595 : }
6596 14 : if (lg(res) > 1) res = shallowconcat1(res);
6597 14 : return res;
6598 : }
6599 : /* All primitive dihedral weight 1 forms of leven in [1, N], N > 1 */
6600 : static GEN
6601 14 : mfdihedralall(long N)
6602 : {
6603 14 : GEN res = vectrunc_init(2*N), z;
6604 14 : long D, ct, i, vt = fetch_user_var("t");
6605 :
6606 13986 : for (D = -3; D >= -N; D--) append_dihedral(res, D, 1,N, vt);
6607 : /* Nf >= 3 (GTM 193, prop 3.3.18) */
6608 4620 : for (D = N / 3; D >= 5; D--) append_dihedral(res, D, 1,N, vt);
6609 14 : ct = lg(res);
6610 14 : if (ct > 1)
6611 : { /* sort wrt N */
6612 14 : res = shallowconcat1(res);
6613 14 : res = vecpermute(res, indexvecsort(res, mkvecsmall(1)));
6614 14 : ct = lg(res);
6615 : }
6616 14 : z = const_vec(N, cgetg(1,t_VEC));
6617 7658 : for (i = 1; i < ct;)
6618 : { /* regroup result sharing the same N */
6619 7644 : long n = di_N(gel(res,i)), j = i+1, k;
6620 : GEN v;
6621 34412 : while (j < ct && di_N(gel(res,j)) == n) j++;
6622 7644 : gel(z, n) = v = cgetg(j-i+1, t_VEC);
6623 42056 : for (k = 1; i < j; k++,i++) gel(v,k) = gel(res,i);
6624 : }
6625 14 : return z;
6626 : }
6627 :
6628 : /* return [vF, index], where vecpermute(vF,index) generates dihedral forms
6629 : * for character CHI */
6630 : static GEN
6631 24969 : mfdihedralnew_i(long N, GEN CHI, GEN SP)
6632 : {
6633 : GEN bnf, Tinit, Pm, vf, M, V, NK;
6634 : long Dold, d, ordw, i, SB, c, l, k0, k1, chino, chinoorig, lv;
6635 :
6636 24969 : lv = lg(SP); if (lv == 1) return NULL;
6637 12138 : CHI = mfcharinduce(CHI,N);
6638 12138 : ordw = mfcharorder(CHI);
6639 12138 : chinoorig = mfcharno(CHI);
6640 12138 : k0 = mfconreyminimize(CHI);
6641 12138 : chino = Fl_powu(chinoorig, k0, N);
6642 12138 : k1 = Fl_inv(k0 % ordw, ordw);
6643 12138 : V = cgetg(lv, t_VEC);
6644 12138 : d = 0;
6645 39039 : for (i = l = 1; i < lv; i++)
6646 : {
6647 26901 : GEN sp = gel(SP,i), T = gel(sp,1);
6648 26901 : if (T[3] != chino) continue;
6649 4060 : d += T[5];
6650 4060 : if (k1 != 1)
6651 : {
6652 77 : GEN t = leafcopy(T);
6653 77 : t[3] = chinoorig;
6654 77 : t[2] = (t[2]*k1) % ordw;
6655 77 : sp = mkvec4(t, gel(sp,2), gel(sp,3), gel(sp,4));
6656 : }
6657 4060 : gel(V, l++) = sp;
6658 : }
6659 12138 : setlg(V, l); /* dihedral forms of level N and character CHI */
6660 12138 : if (l == 1) return NULL;
6661 :
6662 2555 : SB = mfsturmNk(N,1) + 1;
6663 2555 : M = cgetg(d+1, t_MAT);
6664 2555 : vf = cgetg(d+1, t_VEC);
6665 2555 : NK = mkNK(N, 1, CHI);
6666 2555 : bnf = NULL; Dold = 0;
6667 6615 : for (i = c = 1; i < l; i++)
6668 : { /* T = [N, k0, conreyno, D, degrel] */
6669 4060 : GEN bnr, Vi = gel(V,i), T = gel(Vi,1), id = gel(Vi,2), w = gel(Vi,3);
6670 4060 : long jdeg, k0i = T[2], D = T[4], degrel = T[5];
6671 :
6672 4060 : if (D != Dold) { Dold = D; bnf = dihan_bnf(D); }
6673 4060 : bnr = dihan_bnr(bnf, id);
6674 12054 : for (jdeg = 0; jdeg < degrel; jdeg++,c++)
6675 : {
6676 7994 : GEN k0j = mkvecsmall2(k0i, jdeg), an = dihan(bnr, w, k0j, ordw, SB);
6677 7994 : settyp(an, t_COL); gel(M,c) = an;
6678 7994 : gel(vf,c) = tag3(t_MF_DIHEDRAL, NK, bnr, w, k0j);
6679 : }
6680 : }
6681 2555 : Tinit = gmael3(V,1,3,3); Pm = gel(Tinit,1);
6682 2555 : V = QabM_indexrank(M, degpol(Pm)==1? NULL: Pm, ordw);
6683 2555 : return mkvec2(vf,gel(V,2));
6684 : }
6685 : static long
6686 16149 : mfdihedralnewdim(long N, GEN CHI, GEN SP)
6687 : {
6688 16149 : pari_sp av = avma;
6689 16149 : GEN S = mfdihedralnew_i(N, CHI, SP);
6690 16149 : return gc_long(av, S? lg(gel(S,2))-1: 0);
6691 : }
6692 : static GEN
6693 8820 : mfdihedralnew(long N, GEN CHI, GEN SP)
6694 : {
6695 8820 : pari_sp av = avma;
6696 8820 : GEN S = mfdihedralnew_i(N, CHI, SP);
6697 8820 : if (!S) retgc_const(av, cgetg(1, t_VEC));
6698 917 : return vecpermute(gel(S,1), gel(S,2));
6699 : }
6700 :
6701 : static long
6702 7105 : mfdihedralcuspdim(long N, GEN CHI, GEN vSP)
6703 : {
6704 7105 : pari_sp av = avma;
6705 : GEN D, CHIP;
6706 : long F, i, lD, dim;
6707 :
6708 7105 : CHIP = mfchartoprimitive(CHI, &F);
6709 7105 : D = mydivisorsu(N/F); lD = lg(D);
6710 7105 : dim = mfdihedralnewdim(N, CHI, gel(vSP,N)); /* d = 1 */
6711 16149 : for (i = 2; i < lD; i++)
6712 : {
6713 9044 : long d = D[i], a = mfdihedralnewdim(N/d, CHIP, gel(vSP, N/d));
6714 9044 : if (a) dim += a * mynumdivu(d);
6715 : }
6716 7105 : return gc_long(av,dim);
6717 : }
6718 :
6719 : static GEN
6720 7378 : mfbdall(GEN E, long N)
6721 : {
6722 7378 : GEN v, D = mydivisorsu(N);
6723 7378 : long i, j, nD = lg(D) - 1, nE = lg(E) - 1;
6724 7378 : v = cgetg(nD*nE + 1, t_VEC);
6725 10486 : for (j = 1; j <= nE; j++)
6726 : {
6727 3108 : GEN Ej = gel(E, j);
6728 9499 : for (i = 0; i < nD; i++) gel(v, i*nE + j) = mfbd_i(Ej, D[i+1]);
6729 : }
6730 7378 : return v;
6731 : }
6732 : static GEN
6733 3857 : mfdihedralcusp(long N, GEN CHI, GEN vSP)
6734 : {
6735 3857 : pari_sp av = avma;
6736 : GEN D, CHIP, z;
6737 : long F, i, lD;
6738 :
6739 3857 : CHIP = mfchartoprimitive(CHI, &F);
6740 3857 : D = mydivisorsu(N/F); lD = lg(D);
6741 3857 : z = cgetg(lD, t_VEC);
6742 3857 : gel(z,1) = mfdihedralnew(N, CHI, gel(vSP,N));
6743 8596 : for (i = 2; i < lD; i++) /* skip 1 */
6744 : {
6745 4739 : GEN LF = mfdihedralnew(N / D[i], CHIP, gel(vSP, N / D[i]));
6746 4739 : gel(z,i) = mfbdall(LF, D[i]);
6747 : }
6748 3857 : return gc_GEN(av, shallowconcat1(z));
6749 : }
6750 :
6751 : /* used to decide between ratlift and comatrix for ZM_inv; ratlift is better
6752 : * when N has many divisors */
6753 : static int
6754 2590 : abundant(ulong N) { return mynumdivu(N) >= 8; }
6755 :
6756 : /* CHI an mfchar */
6757 : static int
6758 371 : cmp_ord(void *E, GEN a, GEN b)
6759 : {
6760 371 : GEN chia = MF_get_CHI(a), chib = MF_get_CHI(b);
6761 371 : (void)E; return cmpii(gmfcharorder(chia), gmfcharorder(chib));
6762 : }
6763 : /* mfinit structure.
6764 : -- mf[1] contains [N,k,CHI,space],
6765 : -- mf[2] contains vector of closures of Eisenstein series, empty if not
6766 : full space.
6767 : -- mf[3] contains vector of closures, so #mf[3] = dimension of cusp/new space.
6768 : -- mf[4] contains the corresponding indices: either j for T(j)tf if newspace,
6769 : or [M,j,d] for B(d)T(j)tf_M if cuspspace or oldspace.
6770 : -- mf[5] contains the matrix M of first coefficients of basis, never cleaned.
6771 : * NK is either [N,k] or [N,k,CHI].
6772 : * mfinit does not do the splitting, only the basis generation. */
6773 :
6774 : /* Set flraw to 1 if do not need mf[5]: no mftobasis etc..., only the
6775 : expansions of the basis elements are needed. */
6776 :
6777 : static GEN
6778 5047 : mfinit_Nkchi(long N, long k, GEN CHI, long space, long flraw)
6779 : {
6780 5047 : GEN M = NULL, mf = NULL, mf1 = mkvec4(utoi(N), stoi(k), CHI, utoi(space));
6781 5047 : long sb = mfsturmNk(N, k);
6782 5047 : if (k < 0 || badchar(N, k, CHI)) return mfEMPTY(mf1);
6783 5012 : if (k == 0 || space == mf_EISEN) /*nothing*/;
6784 4851 : else if (k == 1)
6785 : {
6786 364 : switch (space)
6787 : {
6788 350 : case mf_NEW:
6789 : case mf_FULL:
6790 350 : case mf_CUSP: mf = mf1init(N, CHI, NULL, get_vDIH(N,NULL), space, flraw);
6791 350 : break;
6792 7 : case mf_OLD: pari_err_IMPL("mfinit in weight 1 for old space");
6793 7 : default: pari_err_FLAG("mfinit");
6794 : }
6795 : }
6796 : else /* k >= 2 */
6797 : {
6798 4487 : long ord = mfcharorder(CHI);
6799 4487 : GEN z = NULL, P = (ord <= 2)? NULL: mfcharpol(CHI);
6800 : cachenew_t cache;
6801 4487 : switch(space)
6802 : {
6803 1239 : case mf_NEW:
6804 1239 : mf = mfnewinit(N, k, CHI, &cache, 1);
6805 1239 : if (mf && !flraw) { M = MF_get_M(mf); z = MF_get_Mindex(mf); }
6806 1239 : break;
6807 3241 : case mf_OLD:
6808 : case mf_CUSP:
6809 : case mf_FULL:
6810 3241 : if (!(mf = mfinitcusp(N, k, CHI, &cache, space))) break;
6811 2933 : if (!flraw)
6812 : {
6813 2282 : M = bhnmat_extend(M, sb+1, 1, MF_get_S(mf), &cache);
6814 2282 : if (space != mf_FULL) gel(mf,5) = mfcleanCHI(M, CHI, abundant(N));
6815 : }
6816 2933 : dbg_cachenew(&cache); break;
6817 7 : default: pari_err_FLAG("mfinit");
6818 : }
6819 4480 : if (z) gel(mf,5) = mfclean2(M, z, P, ord);
6820 : }
6821 4991 : if (!mf) mf = mfEMPTY(mf1);
6822 : else
6823 : {
6824 4025 : gel(mf,1) = mf1;
6825 4025 : if (flraw) gel(mf,5) = zerovec(3);
6826 : }
6827 4991 : if (!space_is_cusp(space))
6828 : {
6829 854 : GEN E = mfeisensteinbasis(N, k, CHI);
6830 854 : gel(mf,2) = E;
6831 854 : if (!flraw)
6832 : {
6833 532 : if (M)
6834 224 : M = shallowconcat(mfvectomat(E, sb+1, 1), M);
6835 : else
6836 308 : M = mfcoefs_mf(mf, sb+1, 1);
6837 532 : gel(mf,5) = mfcleanCHI(M, CHI, abundant(N));
6838 : }
6839 : }
6840 4991 : return mf;
6841 : }
6842 :
6843 : /* mfinit for k = nk/dk */
6844 : static GEN
6845 2751 : mfinit_Nndkchi(long N, long nk, long dk, GEN CHI, long space, long flraw)
6846 266 : { return (dk == 2)? mf2init_Nkchi(N, nk >> 1, CHI, space, flraw)
6847 3017 : : mfinit_Nkchi(N, nk, CHI, space, flraw); }
6848 : static GEN
6849 3416 : mfinit_i(GEN NK, long space)
6850 : {
6851 : GEN CHI, mf;
6852 : long N, k, dk, joker;
6853 3416 : if (checkmf_i(NK))
6854 : {
6855 161 : N = mf_get_N(NK);
6856 161 : Qtoss(mf_get_gk(NK), &k, &dk);
6857 161 : CHI = mf_get_CHI(NK);
6858 : }
6859 3255 : else if ((mf = checkMF_i(NK)))
6860 : {
6861 21 : long s = MF_get_space(mf);
6862 21 : if (s == space) return mf;
6863 21 : Qtoss(MF_get_gk(mf), &k, &dk);
6864 21 : if (dk == 1 && k > 1 && space == mf_NEW && (s == mf_CUSP || s == mf_FULL))
6865 21 : return mfinittonew(mf);
6866 0 : N = MF_get_N(mf);
6867 0 : CHI = MF_get_CHI(mf);
6868 : }
6869 : else
6870 3234 : checkNK2(NK, &N, &k, &dk, &CHI, 1);
6871 3374 : joker = !CHI || typ(CHI) == t_COL;
6872 3374 : if (joker)
6873 : {
6874 1162 : GEN mf, vCHI = CHI;
6875 : long i, j, l;
6876 1162 : if (CHI && lg(CHI) == 1) return cgetg(1,t_VEC);
6877 1155 : if (k < 0) return mfEMPTYall(N, uutoQ(k,dk), CHI, space);
6878 1141 : if (k == 1 && dk == 1 && space != mf_EISEN)
6879 504 : {
6880 : GEN TMP, vSP, gN, gs;
6881 : pari_timer tt;
6882 1106 : if (space != mf_CUSP && space != mf_NEW)
6883 0 : pari_err_IMPL("mfinit([N,1,wildcard], space != cusp or new space)");
6884 1106 : if (wt1empty(N)) return mfEMPTYall(N, gen_1, CHI, space);
6885 504 : vCHI = mf1chars(N,vCHI);
6886 504 : l = lg(vCHI); mf = cgetg(l, t_VEC); if (l == 1) return mf;
6887 504 : TMP = mf1_pre(N); vSP = get_vDIH(N, NULL);
6888 504 : gN = utoipos(N); gs = utoi(space);
6889 504 : if (DEBUGLEVEL) timer_start(&tt);
6890 4123 : for (i = j = 1; i < l; i++)
6891 : {
6892 3619 : pari_sp av = avma;
6893 3619 : GEN c = gel(vCHI,i), z = mf1init(N, c, TMP, vSP, space, 0);
6894 3619 : if (z) z = gc_GEN(av, z);
6895 : else
6896 : {
6897 2905 : set_avma(av);
6898 2905 : if (CHI) z = mfEMPTY(mkvec4(gN,gen_1,c,gs));
6899 : }
6900 3619 : if (z) gel(mf, j++) = z;
6901 3619 : if (DEBUGLEVEL)
6902 0 : timer_printf(&tt, "mf1basis: character %ld / %ld (order = %ld)",
6903 : i, l-1, mfcharorder(c));
6904 : }
6905 : }
6906 : else
6907 : {
6908 35 : vCHI = mfchars(N,k,dk,vCHI);
6909 35 : l = lg(vCHI); mf = cgetg(l, t_VEC);
6910 119 : for (i = j = 1; i < l; i++)
6911 : {
6912 84 : pari_sp av = avma;
6913 84 : GEN v = mfinit_Nndkchi(N, k, dk, gel(vCHI,i), space, 0);
6914 84 : if (MF_get_dim(v) || CHI) gel(mf, j++) = v; else set_avma(av);
6915 : }
6916 : }
6917 539 : setlg(mf,j);
6918 539 : if (!CHI) gen_sort_inplace(mf, NULL, &cmp_ord, NULL);
6919 539 : return mf;
6920 : }
6921 2212 : return mfinit_Nndkchi(N, k, dk, CHI, space, 0);
6922 : }
6923 : GEN
6924 2436 : mfinit(GEN NK, long space)
6925 : {
6926 2436 : pari_sp av = avma;
6927 2436 : return gc_GEN(av, mfinit_i(NK, space));
6928 : }
6929 :
6930 : /* UTILITY FUNCTIONS */
6931 : static void
6932 364 : cusp_canon(GEN cusp, long N, long *pA, long *pC)
6933 : {
6934 364 : pari_sp av = avma;
6935 : long A, C, tc, cg;
6936 364 : if (N <= 0) pari_err_DOMAIN("mfcuspwidth","N","<=",gen_0,stoi(N));
6937 357 : if (!cusp || (tc = typ(cusp)) == t_INFINITY) { *pA = 1; *pC = N; return; }
6938 350 : if (tc != t_INT && tc != t_FRAC) pari_err_TYPE("checkcusp", cusp);
6939 350 : Qtoss(cusp, &A,&C);
6940 350 : if (N % C)
6941 : {
6942 : ulong uC;
6943 14 : long u = Fl_invgen((C-1)%N + 1, N, &uC);
6944 14 : A = Fl_mul(A, u, N);
6945 14 : C = (long)uC;
6946 : }
6947 350 : cg = ugcd(C, N/C);
6948 420 : while (ugcd(A, N) > 1) A += cg;
6949 350 : *pA = A % N; *pC = C; set_avma(av);
6950 : }
6951 : static long
6952 1001 : mfcuspcanon_width(long N, long C)
6953 1001 : { return (!C || C == N)? 1 : N / ugcd(N, Fl_sqr(umodsu(C,N),N)); }
6954 : /* v = [a,c] a ZC, width of cusp (a:c) */
6955 : static long
6956 9975 : mfZC_width(long N, GEN v)
6957 : {
6958 9975 : ulong C = umodiu(gel(v,2), N);
6959 9975 : return (C == 0)? 1: N / ugcd(N, Fl_sqr(C,N));
6960 : }
6961 : long
6962 161 : mfcuspwidth(GEN gN, GEN cusp)
6963 : {
6964 161 : long N = 0, A, C;
6965 : GEN mf;
6966 161 : if (typ(gN) == t_INT) N = itos(gN);
6967 42 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
6968 0 : else pari_err_TYPE("mfcuspwidth", gN);
6969 161 : cusp_canon(cusp, N, &A, &C);
6970 154 : return mfcuspcanon_width(N, C);
6971 : }
6972 :
6973 : /* Q a t_INT */
6974 : static GEN
6975 14 : findq(GEN al, GEN Q)
6976 : {
6977 : long n;
6978 14 : if (typ(al) == t_FRAC && cmpii(gel(al,2), Q) <= 0)
6979 0 : return mkvec(mkvec2(gel(al,1), gel(al,2)));
6980 14 : n = 1 + (long)ceil(2.0781*gtodouble(glog(Q, LOWDEFAULTPREC)));
6981 14 : return contfracpnqn(gboundcf(al,n), n);
6982 : }
6983 : static GEN
6984 91 : findqga(long N, GEN z)
6985 : {
6986 91 : GEN Q, LDC, CK = NULL, DK = NULL, ma, x, y = imag_i(z);
6987 : long j, l;
6988 91 : if (gcmpgs(gmulsg(2*N, y), 1) >= 0) return NULL;
6989 14 : x = real_i(z);
6990 14 : Q = ground(ginv(gsqrt(gmulsg(N, y), LOWDEFAULTPREC)));
6991 14 : LDC = findq(gmulsg(-N,x), Q);
6992 14 : ma = gen_1; l = lg(LDC);
6993 35 : for (j = 1; j < l; j++)
6994 : {
6995 21 : GEN D, DC = gel(LDC,j), C1 = gel(DC,2);
6996 21 : if (cmpii(C1,Q) > 0) break;
6997 21 : D = gel(DC,1);
6998 21 : if (ugcdiu(D,N) == 1)
6999 : {
7000 7 : GEN C = mului(N, C1), den;
7001 7 : den = gadd(gsqr(gmul(C,y)), gsqr(gadd(D, gmul(C,x))));
7002 7 : if (gcmp(den, ma) < 0) { ma = den; CK = C; DK = D; }
7003 : }
7004 : }
7005 14 : return DK? mkvec2(CK, DK): NULL;
7006 : }
7007 :
7008 : static long
7009 168 : valNC2(GEN P, GEN E, long e)
7010 : {
7011 168 : long i, d = 1, l = lg(P);
7012 504 : for (i = 1; i < l; i++)
7013 : {
7014 336 : long v = u_lval(e, P[i]) << 1;
7015 336 : if (v == E[i] + 1) v--;
7016 336 : d *= upowuu(P[i], v);
7017 : }
7018 168 : return d;
7019 : }
7020 :
7021 : static GEN
7022 49 : findqganew(long N, GEN z)
7023 : {
7024 49 : GEN MI, DI, x = real_i(z), y = imag_i(z), Ck = gen_0, Dk = gen_1, fa, P, E;
7025 : long i;
7026 49 : MI = uutoQ(1,N);
7027 49 : DI = mydivisorsu(mysqrtu(N));
7028 49 : fa = myfactoru(N); P = gel(fa,1); E = gel(fa,2);
7029 217 : for (i = 1; i < lg(DI); i++)
7030 : {
7031 168 : long e = DI[i], g;
7032 : GEN U, C, D, m;
7033 168 : (void)cxredsl2(gmulsg(e, z), &U);
7034 168 : C = gcoeff(U,2,1); if (!signe(C)) continue;
7035 168 : D = gcoeff(U,2,2);
7036 168 : g = ugcdiu(D,e);
7037 168 : if (g > 1) { C = muliu(C,e/g); D = diviuexact(D,g); } else C = muliu(C,e);
7038 168 : m = gadd(gsqr(gadd(gmul(C, x), D)), gsqr(gmul(C, y)));
7039 168 : m = gdivgu(m, valNC2(P, E, e));
7040 168 : if (gcmp(m, MI) < 0) { MI = m; Ck = C; Dk = D; }
7041 : }
7042 49 : return signe(Ck)? mkvec2(Ck, Dk): NULL;
7043 : }
7044 :
7045 : /* Return z' and U = [a,b;c,d] \in SL_2(Z), z' = U*z,
7046 : * Im(z')/width(U.oo) > sqrt(3)/(2N). Set *pczd = c*z+d */
7047 : static GEN
7048 182 : cxredga0N(long N, GEN z, GEN *pU, GEN *pczd, long flag)
7049 : {
7050 182 : GEN v = NULL, A, B, C, D;
7051 : long e;
7052 182 : if (N == 1) return cxredsl2_i(z, pU, pczd);
7053 140 : e = gexpo(gel(z,2));
7054 140 : if (e < 0) z = gprec_wensure(z, precision(z) + nbits2extraprec(-e));
7055 140 : v = flag? findqganew(N,z): findqga(N,z);
7056 140 : if (!v) { *pU = matid(2); *pczd = gen_1; return z; }
7057 56 : C = gel(v,1);
7058 56 : D = gel(v,2);
7059 56 : if (!is_pm1(bezout(C,D, &B,&A))) pari_err_BUG("cxredga0N [gcd > 1]");
7060 56 : B = negi(B);
7061 56 : *pU = mkmat2(mkcol2(A,C), mkcol2(B,D));
7062 56 : *pczd = gadd(gmul(C,z), D);
7063 56 : return gdiv(gadd(gmul(A,z), B), *pczd);
7064 : }
7065 :
7066 : static GEN
7067 161 : lfunthetaall(GEN b, GEN vL, GEN t, long bitprec)
7068 : {
7069 161 : long i, l = lg(vL);
7070 161 : GEN v = cgetg(l, t_VEC);
7071 350 : for (i = 1; i < l; i++)
7072 : {
7073 189 : GEN T, L = gel(vL,i), a0 = gel(L,1), ldata = gel(L,2);
7074 189 : GEN van = gel(ldata_get_an(ldata),2);
7075 189 : if (lg(van) == 1)
7076 : {
7077 0 : T = gmul(b, a0);
7078 0 : if (isexactzero(T)) { GEN z = real_0_bit(-bitprec); T = mkcomplex(z,z); }
7079 : }
7080 : else
7081 : {
7082 189 : T = gmul2n(lfuntheta(ldata, t, 0, bitprec), -1);
7083 189 : T = gmul(b, gadd(a0, T));
7084 : }
7085 189 : gel(v,i) = T;
7086 : }
7087 161 : return l == 2? gel(v,1): v;
7088 : }
7089 :
7090 : /* P in ZX, irreducible */
7091 : static GEN
7092 182 : ZX_roots(GEN P, long prec)
7093 : {
7094 182 : long d = degpol(P);
7095 182 : if (d == 1) return mkvec(gen_0);
7096 182 : if (d == 2 && isint1(gel(P,2)) && isintzero(gel(P,3)) && isint1(gel(P,4)))
7097 7 : return mkvec2(powIs(3), gen_I()); /* order as polroots */
7098 294 : return (ZX_sturm_irred(P) == d)? ZX_realroots_irred(P, prec)
7099 294 : : QX_complex_roots(P, prec);
7100 : }
7101 : /* initializations for RgX_RgV_eval / RgC_embed */
7102 : static GEN
7103 217 : rootspowers(GEN v)
7104 : {
7105 217 : long i, l = lg(v);
7106 217 : GEN w = cgetg(l, t_VEC);
7107 868 : for (i = 1; i < l; i++) gel(w,i) = gpowers(gel(v,i), l-2);
7108 217 : return w;
7109 : }
7110 : /* mf embeddings attached to Q(chi)/(T), chi attached to cyclotomic P */
7111 : static GEN
7112 938 : getembed(GEN P, GEN T, GEN zcyclo, long prec)
7113 : {
7114 : long i, l;
7115 : GEN v;
7116 938 : if (degpol(P) == 1) P = NULL; /* mfcharpol for quadratic char */
7117 938 : if (degpol(T) == 1) T = NULL; /* dim 1 orbit */
7118 938 : if (T && P)
7119 35 : { /* K(y) / (T(y)), K = Q(t)/(P) cyclotomic */
7120 35 : GEN vr = RgX_is_ZX(T)? ZX_roots(T,prec): roots(RgX_embed1(T,zcyclo), prec);
7121 35 : v = rootspowers(vr); l = lg(v);
7122 105 : for (i = 1; i < l; i++) gel(v,i) = mkcol3(P,zcyclo,gel(v,i));
7123 : }
7124 903 : else if (T)
7125 : { /* Q(y) / (T(y)), T noncyclotomic */
7126 182 : GEN vr = ZX_roots(T, prec);
7127 182 : v = rootspowers(vr); l = lg(v);
7128 763 : for (i = 1; i < l; i++) gel(v,i) = mkcol2(T, gel(v,i));
7129 : }
7130 : else /* cyclotomic or rational */
7131 721 : v = mkvec(P? mkvec2(P, zcyclo): cgetg(1,t_VEC));
7132 938 : return v;
7133 : }
7134 : static GEN
7135 791 : grootsof1_CHI(GEN CHI, long prec)
7136 791 : { return grootsof1(mfcharorder(CHI), prec); }
7137 : /* return the [Q(F):Q(chi)] embeddings of F */
7138 : static GEN
7139 623 : mfgetembed(GEN F, long prec)
7140 : {
7141 623 : GEN T = mf_get_field(F), CHI = mf_get_CHI(F), P = mfcharpol(CHI);
7142 623 : return getembed(P, T, grootsof1_CHI(CHI, prec), prec);
7143 : }
7144 : static GEN
7145 7 : mfchiembed(GEN mf, long prec)
7146 : {
7147 7 : GEN CHI = MF_get_CHI(mf), P = mfcharpol(CHI);
7148 7 : return getembed(P, pol_x(0), grootsof1_CHI(CHI, prec), prec);
7149 : }
7150 : /* mfgetembed for the successive eigenforms in MF_get_newforms */
7151 : static GEN
7152 161 : mfeigenembed(GEN mf, long prec)
7153 : {
7154 161 : GEN vP = MF_get_fields(mf), vF = MF_get_newforms(mf);
7155 161 : GEN zcyclo, vE, CHI = MF_get_CHI(mf), P = mfcharpol(CHI);
7156 161 : long i, l = lg(vP);
7157 161 : vF = Q_remove_denom(liftpol_shallow(vF), NULL);
7158 161 : prec += nbits2extraprec(gexpo(vF));
7159 161 : zcyclo = grootsof1_CHI(CHI, prec);
7160 161 : vE = cgetg(l, t_VEC);
7161 469 : for (i = 1; i < l; i++) gel(vE,i) = getembed(P, gel(vP,i), zcyclo, prec);
7162 161 : return vE;
7163 : }
7164 :
7165 : static int
7166 28 : checkPv(GEN P, GEN v)
7167 28 : { return typ(P) == t_POL && is_vec_t(typ(v)) && lg(v)-1 >= degpol(P); }
7168 : static int
7169 28 : checkemb_i(GEN E)
7170 : {
7171 28 : long t = typ(E), l = lg(E);
7172 28 : if (t == t_VEC) return l == 1 || (l == 3 && checkPv(gel(E,1), gel(E,2)));
7173 21 : if (t != t_COL) return 0;
7174 21 : if (l == 3) return checkPv(gel(E,1), gel(E,2));
7175 21 : return l == 4 && is_vec_t(typ(gel(E,2))) && checkPv(gel(E,1), gel(E,3));
7176 : }
7177 : static GEN
7178 28 : anyembed(GEN v, GEN E)
7179 : {
7180 28 : switch(typ(v))
7181 : {
7182 21 : case t_VEC: case t_COL: return mfvecembed(E, v);
7183 7 : case t_MAT: return mfmatembed(E, v);
7184 : }
7185 0 : return mfembed(E, v);
7186 : }
7187 : GEN
7188 49 : mfembed0(GEN E, GEN v, long prec)
7189 : {
7190 49 : pari_sp av = avma;
7191 49 : GEN mf, vE = NULL;
7192 49 : if (checkmf_i(E)) vE = mfgetembed(E, prec);
7193 35 : else if ((mf = checkMF_i(E))) vE = mfchiembed(mf, prec);
7194 49 : if (vE)
7195 : {
7196 21 : long i, l = lg(vE);
7197 : GEN w;
7198 21 : if (!v) return gc_GEN(av, l == 2? gel(vE,1): vE);
7199 0 : w = cgetg(l, t_VEC);
7200 0 : for (i = 1; i < l; i++) gel(w,i) = anyembed(v, gel(vE,i));
7201 0 : return gc_GEN(av, l == 2? gel(w,1): w);
7202 : }
7203 28 : if (!checkemb_i(E) || !v) pari_err_TYPE("mfembed", E);
7204 28 : return gc_GEN(av, anyembed(v,E));
7205 : }
7206 :
7207 : /* dummy lfun create for theta evaluation */
7208 : static GEN
7209 980 : mfthetaancreate(GEN van, GEN N, GEN k)
7210 : {
7211 980 : GEN L = zerovec(6);
7212 980 : gel(L,1) = lfuntag(t_LFUN_GENERIC, van);
7213 980 : gel(L,3) = mkvec2(gen_0, gen_1);
7214 980 : gel(L,4) = k;
7215 980 : gel(L,5) = N; return L;
7216 : }
7217 : /* destroy van and prepare to evaluate theta(sigma(van)), for all sigma in
7218 : * embeddings vector vE */
7219 : static GEN
7220 357 : van_embedall(GEN van, GEN vE, GEN gN, GEN gk)
7221 : {
7222 357 : GEN a0 = gel(van,1), vL;
7223 357 : long i, lE = lg(vE), l = lg(van);
7224 357 : van++; van[0] = evaltyp(t_VEC) | _evallg(l-1); /* remove a0 */
7225 357 : vL = cgetg(lE, t_VEC);
7226 945 : for (i = 1; i < lE; i++)
7227 : {
7228 588 : GEN E = gel(vE,i), v = mfvecembed(E, van);
7229 588 : gel(vL,i) = mkvec2(mfembed(E,a0), mfthetaancreate(v, gN, gk));
7230 : }
7231 357 : return vL;
7232 : }
7233 :
7234 : static int
7235 1134 : cusp_AC(GEN cusp, long *A, long *C)
7236 : {
7237 1134 : switch(typ(cusp))
7238 : {
7239 140 : case t_INFINITY: *A = 1; *C = 0; break;
7240 301 : case t_INT: *A = itos(cusp); *C = 1; break;
7241 448 : case t_FRAC: *A = itos(gel(cusp, 1)); *C = itos(gel(cusp, 2)); break;
7242 245 : case t_REAL: case t_COMPLEX:
7243 245 : *A = 0; *C = 0;
7244 245 : if (gsigne(imag_i(cusp)) <= 0)
7245 7 : pari_err_DOMAIN("mfeval","imag(tau)","<=",gen_0,cusp);
7246 238 : return 0;
7247 0 : default: pari_err_TYPE("cusp_AC", cusp);
7248 : }
7249 889 : return 1;
7250 : }
7251 : static GEN
7252 518 : cusp2mat(long A, long C)
7253 : { long B, D;
7254 518 : cbezout(A, C, &D, &B);
7255 518 : return mkmat22s(A, -B, C, D);
7256 : }
7257 : static GEN
7258 21 : mkS(void) { return mkmat22s(0,-1,1,0); }
7259 :
7260 : /* if t is a cusp, return F(t), else NULL */
7261 : static GEN
7262 364 : evalcusp(GEN mf, GEN F, GEN t, long prec)
7263 : {
7264 : long A, C;
7265 : GEN R;
7266 364 : if (!cusp_AC(t, &A,&C)) return NULL;
7267 196 : if (C % mf_get_N(F) == 0) return gel(mfcoefs_i(F, 0, 1), 1);
7268 175 : R = mfgaexpansion(mf, F, cusp2mat(A,C), 0, prec);
7269 175 : return gequal0(gel(R,1))? gmael(R,3,1): gen_0;
7270 : }
7271 : /* Evaluate an mf closure numerically, i.e., in the usual sense, either for a
7272 : * single tau or a vector of tau; for each, return a vector of results
7273 : * corresponding to all complex embeddings of F. If flag is nonzero, allow
7274 : * replacing F by F | gamma to increase imag(gamma^(-1).tau) [ expensive if
7275 : * MF_EISENSPACE not present ] */
7276 : static GEN
7277 168 : mfeval_i(GEN mf, GEN F, GEN vtau, long flag, long bitprec)
7278 : {
7279 : GEN L0, vL, vb, sqN, vczd, vTAU, vs, van, vE;
7280 168 : long N = MF_get_N(mf), N0, ta, lv, i, prec = nbits2prec(bitprec);
7281 168 : GEN gN = utoipos(N), gk = mf_get_gk(F), gk1 = gsubgs(gk,1), vgk;
7282 168 : long flscal = 0;
7283 :
7284 : /* gen_0 is ignored, second component assumes Ramanujan-Petersson in
7285 : * 1/2-integer weight */
7286 168 : vgk = mkvec2(gen_0, mfiscuspidal(mf,F)? gmul2n(gk1,-1): gk1);
7287 168 : ta = typ(vtau);
7288 168 : if (!is_vec_t(ta)) { flscal = 1; vtau = mkvec(vtau); ta = t_VEC; }
7289 168 : lv = lg(vtau);
7290 168 : sqN = sqrtr_abs(utor(N, prec));
7291 168 : vs = const_vec(lv-1, NULL);
7292 168 : vb = const_vec(lv-1, NULL);
7293 168 : vL = cgetg(lv, t_VEC);
7294 168 : vTAU = cgetg(lv, t_VEC);
7295 168 : vczd = cgetg(lv, t_VEC);
7296 168 : L0 = mfthetaancreate(NULL, gN, vgk); /* only for thetacost */
7297 168 : vE = mfgetembed(F, prec);
7298 168 : N0 = 0;
7299 357 : for (i = 1; i < lv; i++)
7300 : {
7301 196 : GEN z = gel(vtau,i), tau, U;
7302 : long w, n;
7303 :
7304 196 : gel(vs,i) = evalcusp(mf, F, z, prec);
7305 189 : if (gel(vs,i)) continue;
7306 161 : tau = cxredga0N(N, z, &U, &gel(vczd,i), flag);
7307 161 : if (!flag) w = 0; else { w = mfZC_width(N, gel(U,1)); tau = gdivgu(tau,w); }
7308 161 : gel(vTAU,i) = mulcxmI(gmul(tau, sqN));
7309 161 : n = lfunthetacost(L0, real_i(gel(vTAU,i)), 0, bitprec);
7310 161 : if (N0 < n) N0 = n;
7311 161 : if (flag)
7312 : {
7313 49 : GEN A, al, v = mfslashexpansion(mf, F, ZM_inv(U,NULL), n, 0, &A, prec);
7314 49 : gel(vL,i) = van_embedall(v, vE, gN, vgk);
7315 49 : al = gel(A,1);
7316 49 : if (!gequal0(al))
7317 7 : gel(vb,i) = gexp(gmul(gmul(gmulsg(w,al),PiI2(prec)), tau), prec);
7318 : }
7319 : }
7320 161 : if (!flag)
7321 : {
7322 112 : van = mfcoefs_i(F, N0, 1);
7323 112 : vL = const_vec(lv-1, van_embedall(van, vE, gN, vgk));
7324 : }
7325 350 : for (i = 1; i < lv; i++)
7326 : {
7327 : GEN T;
7328 189 : if (gel(vs,i)) continue;
7329 161 : T = gpow(gel(vczd,i), gneg(gk), prec);
7330 161 : if (flag && gel(vb,i)) T = gmul(T, gel(vb,i));
7331 161 : gel(vs,i) = lfunthetaall(T, gel(vL,i), gel(vTAU,i), bitprec);
7332 : }
7333 161 : return flscal? gel(vs,1): vs;
7334 : }
7335 :
7336 : static long
7337 1372 : mfistrivial(GEN F)
7338 : {
7339 1372 : switch(mf_get_type(F))
7340 : {
7341 7 : case t_MF_CONST: return lg(gel(F,2)) == 1;
7342 287 : case t_MF_LINEAR: case t_MF_LINEAR_BHN: return gequal0(gel(F,3));
7343 1078 : default: return 0;
7344 : }
7345 : }
7346 :
7347 : static long
7348 1190 : mf_same_k(GEN mf, GEN f) { return gequal(MF_get_gk(mf), mf_get_gk(f)); }
7349 : static long
7350 1148 : mf_same_CHI(GEN mf, GEN f)
7351 : {
7352 1148 : GEN F1, F2, chi1, chi2, CHI1 = MF_get_CHI(mf), CHI2 = mf_get_CHI(f);
7353 : /* are the primitive chars attached to CHI1 and CHI2 equal ? */
7354 1148 : F1 = znconreyconductor(gel(CHI1,1), gel(CHI1,2), &chi1);
7355 1148 : if (typ(F1) == t_VEC) F1 = gel(F1,1);
7356 1148 : F2 = znconreyconductor(gel(CHI2,1), gel(CHI2,2), &chi2);
7357 1148 : if (typ(F2) == t_VEC) F2 = gel(F2,1);
7358 1148 : return equalii(F1,F2) && ZV_equal(chi1,chi2);
7359 : }
7360 : /* check k and CHI rigorously, but not coefficients nor N */
7361 : static long
7362 259 : mfisinspace_i(GEN mf, GEN F)
7363 : {
7364 259 : return mfistrivial(F) || (mf_same_k(mf,F) && mf_same_CHI(mf,F));
7365 : }
7366 : static void
7367 7 : err_space(GEN F)
7368 7 : { pari_err_DOMAIN("mftobasis", "form", "does not belong to",
7369 0 : strtoGENstr("space"), F); }
7370 :
7371 : static long
7372 154 : mfcheapeisen(GEN mf)
7373 : {
7374 154 : long k, L, N = MF_get_N(mf);
7375 : GEN P;
7376 154 : if (N <= 70) return 1;
7377 84 : k = itos(gceil(MF_get_gk(mf)));
7378 84 : if (odd(k)) k--;
7379 84 : switch (k)
7380 : {
7381 0 : case 2: L = 190; break;
7382 14 : case 4: L = 162; break;
7383 70 : case 6:
7384 70 : case 8: L = 88; break;
7385 0 : case 10: L = 78; break;
7386 0 : default: L = 66; break;
7387 : }
7388 84 : P = gel(myfactoru(N), 1);
7389 84 : return P[lg(P)-1] <= L;
7390 : }
7391 :
7392 : static GEN
7393 189 : myimag_i(GEN x)
7394 : {
7395 189 : long tc = typ(x);
7396 189 : if (tc == t_INFINITY || tc == t_INT || tc == t_FRAC) return gen_1;
7397 196 : if (tc == t_VEC) pari_APPLY_same(myimag_i(gel(x,i)));
7398 154 : return imag_i(x);
7399 : }
7400 :
7401 : static GEN
7402 154 : mintau(GEN vtau)
7403 : {
7404 154 : if (!is_vec_t(typ(vtau))) return myimag_i(vtau);
7405 7 : return (lg(vtau) == 1)? gen_1: vecmin(myimag_i(vtau));
7406 : }
7407 :
7408 : /* initialization for mfgaexpansion: what does not depend on cusp */
7409 : static GEN
7410 1218 : mf_eisendec(GEN mf, GEN F, long prec)
7411 : {
7412 1218 : GEN B = liftpol_shallow(mfeisensteindec(mf, F)), v = variables_vecsmall(B);
7413 1218 : GEN Mvecj = obj_check(mf, MF_EISENSPACE);
7414 1218 : long l = lg(v), i, ord;
7415 1218 : if (lg(Mvecj) < 5) Mvecj = gel(Mvecj,1);
7416 1218 : ord = itou(gel(Mvecj,4));
7417 1274 : for (i = 1; i < l; i++)
7418 924 : if (v[i] != 1)
7419 : {
7420 : GEN d;
7421 : long e;
7422 868 : B = Q_remove_denom(B, &d);
7423 868 : e = gexpo(B);
7424 868 : if (e > 0) prec += nbits2prec(e);
7425 868 : B = gsubst(B, v[i], rootsof1u_cx(ord, prec));
7426 868 : if (d) B = gdiv(B, d);
7427 868 : break;
7428 : }
7429 1218 : return B;
7430 : }
7431 :
7432 : GEN
7433 168 : mfeval(GEN mf0, GEN F, GEN vtau, long bitprec)
7434 : {
7435 168 : pari_sp av = avma;
7436 168 : long flnew = 1;
7437 168 : GEN mf = checkMF_i(mf0);
7438 168 : if (!mf) pari_err_TYPE("mfeval", mf0);
7439 168 : if (!checkmf_i(F)) pari_err_TYPE("mfeval", F);
7440 168 : if (!mfisinspace_i(mf, F)) err_space(F);
7441 168 : if (!obj_check(mf, MF_EISENSPACE)) flnew = mfcheapeisen(mf);
7442 168 : if (flnew && gcmpgs(gmulsg(2*MF_get_N(mf), mintau(vtau)), 1) >= 0) flnew = 0;
7443 168 : return gc_GEN(av, mfeval_i(mf, F, vtau, flnew, bitprec));
7444 : }
7445 :
7446 : static long
7447 189 : val(GEN v, long bit)
7448 : {
7449 189 : long c, l = lg(v);
7450 392 : for (c = 1; c < l; c++)
7451 378 : if (gexpo(gel(v,c)) > -bit) return c-1;
7452 14 : return -1;
7453 : }
7454 : GEN
7455 203 : mfcuspval(GEN mf, GEN F, GEN cusp, long bitprec)
7456 : {
7457 203 : pari_sp av = avma;
7458 203 : long lvE, w, N, sb, n, A, C, prec = nbits2prec(bitprec);
7459 : GEN ga, gk, vE;
7460 203 : mf = checkMF(mf);
7461 203 : if (!checkmf_i(F)) pari_err_TYPE("mfcuspval",F);
7462 203 : N = MF_get_N(mf);
7463 203 : cusp_canon(cusp, N, &A, &C);
7464 203 : gk = mf_get_gk(F);
7465 203 : if (typ(gk) != t_INT)
7466 : {
7467 42 : GEN FT = mfmultheta(F), mf2 = obj_checkbuild(mf, MF_MF2INIT, &mf2init);
7468 42 : GEN r = mfcuspval(mf2, FT, cusp, bitprec);
7469 42 : if ((C & 3L) == 2)
7470 : {
7471 14 : GEN z = uutoQ(1,4);
7472 14 : r = gsub(r, typ(r) == t_VEC? const_vec(lg(r)-1, z): z);
7473 : }
7474 42 : return gc_upto(av, r);
7475 : }
7476 161 : vE = mfgetembed(F, prec);
7477 161 : lvE = lg(vE);
7478 161 : w = mfcuspcanon_width(N, C);
7479 161 : sb = w * mfsturmNk(N, itos(gk));
7480 161 : ga = cusp2mat(A,C);
7481 168 : for (n = 8;; n = minss(sb, n << 1))
7482 7 : {
7483 168 : GEN R = mfgaexpansion(mf, F, ga, n, prec), res = liftpol_shallow(gel(R,3));
7484 168 : GEN v = cgetg(lvE-1, t_VECSMALL);
7485 168 : long j, ok = 1;
7486 168 : res = RgC_embedall(res, vE);
7487 357 : for (j = 1; j < lvE; j++)
7488 : {
7489 189 : v[j] = val(gel(res,j), bitprec/2);
7490 189 : if (v[j] < 0) ok = 0;
7491 : }
7492 168 : if (ok)
7493 : {
7494 154 : res = cgetg(lvE, t_VEC);
7495 329 : for (j = 1; j < lvE; j++) gel(res,j) = gadd(gel(R,1), uutoQ(v[j], w));
7496 154 : return gc_GEN(av, lvE==2? gel(res,1): res);
7497 : }
7498 14 : if (n == sb) return lvE==2? mkoo(): const_vec(lvE-1, mkoo()); /* 0 */
7499 : }
7500 : }
7501 :
7502 : long
7503 231 : mfiscuspidal(GEN mf, GEN F)
7504 : {
7505 231 : pari_sp av = avma;
7506 : GEN mf2;
7507 231 : if (space_is_cusp(MF_get_space(mf))) return 1;
7508 105 : if (typ(mf_get_gk(F)) == t_INT)
7509 : {
7510 63 : GEN v = mftobasis(mf,F,0), vE = vecslice(v, 1, lg(MF_get_E(mf))-1);
7511 63 : return gc_long(av, gequal0(vE));
7512 : }
7513 42 : if (!gequal0(mfak_i(F, 0))) return 0;
7514 21 : mf2 = obj_checkbuild(mf, MF_MF2INIT, &mf2init);
7515 21 : return mfiscuspidal(mf2, mfmultheta(F));
7516 : }
7517 :
7518 : /* F = vector of newforms in mftobasis format */
7519 : static GEN
7520 119 : mffrickeeigen_i(GEN mf, GEN F, GEN vE, long prec)
7521 : {
7522 119 : GEN M, Z, L0, gN = MF_get_gN(mf), gk = MF_get_gk(mf);
7523 119 : long N0, i, lM, bit = prec2nbits(prec), k = itou(gk);
7524 119 : long LIM = 5; /* Sturm bound is enough */
7525 :
7526 119 : L0 = mfthetaancreate(NULL, gN, gk); /* only for thetacost */
7527 119 : START:
7528 119 : N0 = lfunthetacost(L0, gen_1, LIM, bit);
7529 119 : M = mfcoefs_mf(mf, N0, 1);
7530 119 : lM = lg(F);
7531 119 : Z = cgetg(lM, t_VEC);
7532 315 : for (i = 1; i < lM; i++)
7533 : { /* expansion of D * F[i] */
7534 196 : GEN D, z, van = RgM_RgC_mul(M, Q_remove_denom(gel(F,i), &D));
7535 196 : GEN L = van_embedall(van, gel(vE,i), gN, gk);
7536 196 : long l = lg(L), j, bit_add = D? expi(D): 0;
7537 196 : gel(Z,i) = z = cgetg(l, t_VEC);
7538 595 : for (j = 1; j < l; j++)
7539 : {
7540 : GEN v, C, C0;
7541 : long m, e;
7542 546 : for (m = 0; m <= LIM; m++)
7543 : {
7544 546 : v = lfuntheta(gmael(L,j,2), gen_1, m, bit);
7545 546 : if (gexpo(v) > bit_add - bit/2) break;
7546 : }
7547 399 : if (m > LIM) { LIM <<= 1; goto START; }
7548 399 : C = mulcxpowIs(gdiv(v,conj_i(v)), 2*m - k);
7549 399 : C0 = grndtoi(C, &e); if (e < 5-prec2nbits(precision(C))) C = C0;
7550 399 : gel(z,j) = C;
7551 : }
7552 : }
7553 119 : return Z;
7554 : }
7555 : static GEN
7556 84 : mffrickeeigen(GEN mf, GEN vE, long prec)
7557 : {
7558 84 : GEN D = obj_check(mf, MF_FRICKE);
7559 84 : if (D) { long p = gprecision(D); if (!p || p >= prec) return D; }
7560 77 : D = mffrickeeigen_i(mf, MF_get_newforms(mf), vE, prec);
7561 77 : return obj_insert(mf, MF_FRICKE, D);
7562 : }
7563 :
7564 : /* integral weight, new space for primitive quadratic character CHIP;
7565 : * MF = vector of embedded eigenforms coefs on mfbasis, by orbit.
7566 : * Assume N > Q > 1 and (Q,f(CHIP)) = 1 */
7567 : static GEN
7568 56 : mfatkineigenquad(GEN mf, GEN CHIP, long Q, GEN MF, long bitprec)
7569 : {
7570 : GEN L0, la2, S, F, vP, tau, wtau, Z, va, vb, den, coe, sqrtQ, sqrtN;
7571 56 : GEN M, gN, gk = MF_get_gk(mf);
7572 56 : long N0, x, yq, i, j, lF, dim, muQ, prec = nbits2prec(bitprec);
7573 56 : long N = MF_get_N(mf), k = itos(gk), NQ = N / Q;
7574 :
7575 : /* Q coprime to FC */
7576 56 : F = MF_get_newforms(mf);
7577 56 : vP = MF_get_fields(mf);
7578 56 : lF = lg(F);
7579 56 : Z = cgetg(lF, t_VEC);
7580 56 : S = MF_get_S(mf); dim = lg(S) - 1;
7581 56 : muQ = mymoebiusu(Q);
7582 56 : if (muQ)
7583 : {
7584 42 : GEN SQ = cgetg(dim+1,t_VEC), Qk = gpow(stoi(Q), sstoQ(k-2, 2), prec);
7585 42 : long i, bit2 = bitprec >> 1;
7586 154 : for (j = 1; j <= dim; j++) gel(SQ,j) = mfak_i(gel(S,j), Q);
7587 84 : for (i = 1; i < lF; i++)
7588 : {
7589 42 : GEN S = RgV_dotproduct(gel(F,i), SQ), T = gel(vP,i);
7590 : long e;
7591 42 : if (degpol(T) > 1 && typ(S) != t_POLMOD) S = gmodulo(S, T);
7592 42 : S = grndtoi(gdiv(conjvec(S, prec), Qk), &e);
7593 42 : if (e > -bit2) pari_err_PREC("mfatkineigenquad");
7594 42 : if (muQ == -1) S = gneg(S);
7595 42 : gel(Z,i) = S;
7596 : }
7597 42 : return Z;
7598 : }
7599 14 : la2 = mfchareval(CHIP, Q); /* 1 or -1 */
7600 14 : (void)cbezout(Q, NQ, &x, &yq);
7601 14 : sqrtQ = sqrtr_abs(utor(Q,prec));
7602 14 : tau = mkcomplex(gadd(sstoQ(-1, NQ), uutoQ(1, 1000)),
7603 : divru(sqrtQ, N));
7604 14 : den = gaddgs(gmulsg(NQ, tau), 1);
7605 14 : wtau = gdiv(gsub(gmulsg(x, tau), sstoQ(yq, Q)), den);
7606 14 : coe = gpowgs(gmul(sqrtQ, den), k);
7607 :
7608 14 : sqrtN = sqrtr_abs(utor(N,prec));
7609 14 : tau = mulcxmI(gmul(tau, sqrtN));
7610 14 : wtau = mulcxmI(gmul(wtau, sqrtN));
7611 14 : gN = utoipos(N);
7612 14 : L0 = mfthetaancreate(NULL, gN, gk); /* only for thetacost */
7613 14 : N0 = maxss(lfunthetacost(L0,real_i(tau), 0,bitprec),
7614 : lfunthetacost(L0,real_i(wtau),0,bitprec));
7615 14 : M = mfcoefs_mf(mf, N0, 1);
7616 14 : va = cgetg(dim+1, t_VEC);
7617 14 : vb = cgetg(dim+1, t_VEC);
7618 105 : for (j = 1; j <= dim; j++)
7619 : {
7620 91 : GEN L, v = vecslice(gel(M,j), 2, N0+1); /* remove a0 */
7621 91 : settyp(v, t_VEC); L = mfthetaancreate(v, gN, gk);
7622 91 : gel(va,j) = lfuntheta(L, tau,0,bitprec);
7623 91 : gel(vb,j) = lfuntheta(L,wtau,0,bitprec);
7624 : }
7625 84 : for (i = 1; i < lF; i++)
7626 : {
7627 70 : GEN z, FE = gel(MF,i);
7628 70 : long l = lg(FE);
7629 70 : z = cgetg(l, t_VEC);
7630 70 : for (j = 1; j < l; j++)
7631 : {
7632 70 : GEN f = gel(FE,j), a = RgV_dotproduct(va,f), b = RgV_dotproduct(vb,f);
7633 70 : GEN la = ground( gdiv(b, gmul(a,coe)) );
7634 70 : if (!gequal(gsqr(la), la2)) pari_err_PREC("mfatkineigenquad");
7635 70 : if (typ(la) == t_INT)
7636 : {
7637 70 : if (j != 1) pari_err_BUG("mfatkineigenquad");
7638 70 : z = const_vec(l-1, la); break;
7639 : }
7640 0 : gel(z,j) = la;
7641 : }
7642 70 : gel(Z,i) = z;
7643 : }
7644 14 : return Z;
7645 : }
7646 :
7647 : static GEN
7648 84 : myusqrt(ulong a, long prec)
7649 : {
7650 84 : if (a == 1UL) return gen_1;
7651 70 : if (uissquareall(a, &a)) return utoipos(a);
7652 49 : return sqrtr_abs(utor(a, prec));
7653 : }
7654 : /* Assume mf is a nontrivial new space, rational primitive character CHIP
7655 : * and (Q,FC) = 1 */
7656 : static GEN
7657 112 : mfatkinmatnewquad(GEN mf, GEN CHIP, long Q, long flag, long PREC)
7658 : {
7659 112 : GEN cM, M, D, MF, den, vE, F = MF_get_newforms(mf);
7660 112 : long i, c, e, prec, bitprec, lF = lg(F), N = MF_get_N(mf), k = MF_get_k(mf);
7661 :
7662 112 : if (Q == 1) return mkvec4(gen_0, matid(MF_get_dim(mf)), gen_1, mf);
7663 112 : den = gel(MF_get_Minv(mf), 2);
7664 112 : bitprec = expi(den) + 64;
7665 112 : if (!flag) bitprec = maxss(bitprec, prec2nbits(PREC));
7666 :
7667 35 : START:
7668 112 : prec = nbits2prec(bitprec);
7669 112 : vE = mfeigenembed(mf, prec);
7670 112 : M = cgetg(lF, t_VEC);
7671 294 : for (i = 1; i < lF; i++) gel(M,i) = RgC_embedall(gel(F,i), gel(vE,i));
7672 112 : if (Q != N)
7673 : {
7674 56 : D = mfatkineigenquad(mf, CHIP, Q, M, bitprec);
7675 56 : c = odd(k)? Q: 1;
7676 : }
7677 : else
7678 : {
7679 56 : D = mffrickeeigen(mf, vE, prec);
7680 56 : c = mfcharmodulus(CHIP); if (odd(k)) c = -Q/c;
7681 : }
7682 112 : D = shallowconcat1(D);
7683 112 : if (vec_isconst(D)) { MF = diagonal_shallow(D); flag = 0; }
7684 : else
7685 : {
7686 63 : M = shallowconcat1(M);
7687 63 : MF = RgM_mul(matmuldiagonal(M,D), ginv(M));
7688 : }
7689 112 : if (!flag) return mkvec4(gen_0, MF, gen_1, mf);
7690 :
7691 21 : if (c > 0)
7692 21 : cM = myusqrt(c, PREC);
7693 : else
7694 : {
7695 0 : MF = imag_i(MF); c = -c;
7696 0 : cM = mkcomplex(gen_0, myusqrt(c,PREC));
7697 : }
7698 21 : if (c != 1) MF = RgM_Rg_mul(MF, myusqrt(c,prec));
7699 21 : MF = grndtoi(RgM_Rg_mul(MF,den), &e);
7700 21 : if (e > -32) { bitprec <<= 1; goto START; }
7701 21 : MF = RgM_Rg_div(MF, den);
7702 21 : if (is_rational_t(typ(cM)) && !isint1(cM))
7703 0 : { MF = RgM_Rg_div(MF, cM); cM = gen_1; }
7704 21 : return mkvec4(gen_0, MF, cM, mf);
7705 : }
7706 :
7707 : /* let CHI mod N, Q || N, return \bar{CHI_Q} * CHI_{N/Q} */
7708 : static GEN
7709 112 : mfcharAL(GEN CHI, long Q)
7710 : {
7711 112 : GEN G = gel(CHI,1), c = gel(CHI,2), cycc, d, P, E, F;
7712 112 : long l = lg(c), N = mfcharmodulus(CHI), i;
7713 112 : if (N == Q) return mfcharconj(CHI);
7714 56 : if (N == 1) return CHI;
7715 42 : CHI = leafcopy(CHI);
7716 42 : gel(CHI,2) = d = leafcopy(c);
7717 42 : F = znstar_get_faN(G);
7718 42 : P = gel(F,1);
7719 42 : E = gel(F,2);
7720 42 : cycc = znstar_get_conreycyc(G);
7721 42 : if (!odd(Q) && equaliu(gel(P,1), 2) && E[1] >= 3)
7722 14 : gel(d,2) = Fp_neg(gel(d,2), gel(cycc,2));
7723 56 : else for (i = 1; i < l; i++)
7724 28 : if (!umodui(Q, gel(P,i))) gel(d,i) = Fp_neg(gel(d,i), gel(cycc,i));
7725 42 : return CHI;
7726 : }
7727 : static long
7728 245 : atkin_get_NQ(long N, long Q, const char *f)
7729 : {
7730 245 : long NQ = N / Q;
7731 245 : if (N % Q) pari_err_DOMAIN(f,"N % Q","!=",gen_0,utoi(Q));
7732 245 : if (ugcd(NQ, Q) > 1) pari_err_DOMAIN(f,"gcd(Q,N/Q)","!=",gen_1,utoi(Q));
7733 245 : return NQ;
7734 : }
7735 :
7736 : /* transform mf to new_NEW if possible */
7737 : static GEN
7738 1589 : MF_set_new(GEN mf)
7739 : {
7740 1589 : GEN vMjd, vj, gk = MF_get_gk(mf);
7741 : long l, j;
7742 1589 : if (MF_get_space(mf) != mf_CUSP
7743 1589 : || typ(gk) != t_INT || itou(gk) == 1) return mf;
7744 182 : vMjd = MFcusp_get_vMjd(mf); l = lg(vMjd);
7745 182 : if (l > 1 && gel(vMjd,1)[1] != MF_get_N(mf)) return mf; /* oldspace != 0 */
7746 175 : mf = shallowcopy(mf);
7747 175 : gel(mf,1) = shallowcopy(gel(mf,1));
7748 175 : MF_set_space(mf, mf_NEW);
7749 175 : vj = cgetg(l, t_VECSMALL);
7750 938 : for (j = 1; j < l; j++) vj[j] = gel(vMjd, j)[2];
7751 175 : gel(mf,4) = vj; return mf;
7752 : }
7753 :
7754 : /* if flag = 1, rationalize, else don't */
7755 : static GEN
7756 224 : mfatkininit_i(GEN mf, long Q, long flag, long prec)
7757 : {
7758 : GEN M, B, C, CHI, CHIAL, G, chi, P, z, g, mfB, s, Mindex, Minv;
7759 224 : long j, l, lim, ord, FC, NQ, cQ, nk, dk, N = MF_get_N(mf);
7760 :
7761 224 : B = MF_get_basis(mf); l = lg(B);
7762 224 : M = cgetg(l, t_MAT); if (l == 1) return mkvec4(gen_0,M,gen_1,mf);
7763 224 : Qtoss(MF_get_gk(mf), &nk,&dk);
7764 224 : Q = labs(Q);
7765 224 : NQ = atkin_get_NQ(N, Q, "mfatkininit");
7766 224 : CHI = MF_get_CHI(mf);
7767 224 : CHI = mfchartoprimitive(CHI, &FC);
7768 224 : ord = mfcharorder(CHI);
7769 224 : mf = MF_set_new(mf);
7770 224 : if (MF_get_space(mf) == mf_NEW && ord <= 2 && NQ % FC == 0 && dk == 1)
7771 112 : return mfatkinmatnewquad(mf, CHI, Q, flag, prec);
7772 : /* now flag != 0 */
7773 112 : G = gel(CHI,1);
7774 112 : chi = gel(CHI,2);
7775 112 : if (Q == N) { g = mkmat22s(0, -1, N, 0); cQ = NQ; } /* Fricke */
7776 : else
7777 : {
7778 28 : GEN F, gQP = utoi(ugcd(Q, FC));
7779 : long t, v;
7780 28 : chi = znchardecompose(G, chi, gQP);
7781 28 : F = znconreyconductor(G, chi, &chi);
7782 28 : G = znstar0(F,1);
7783 28 : (void)cbezout(Q, NQ, &t, &v);
7784 28 : g = mkmat22s(Q*t, 1, -N*v, Q);
7785 28 : cQ = -NQ*v;
7786 : }
7787 112 : C = s = gen_1;
7788 : /* N.B. G,chi are G_Q,chi_Q [primitive] at this point */
7789 112 : if (lg(chi) != 1) C = ginv( znchargauss(G, chi, gen_1, prec2nbits(prec)) );
7790 112 : if (dk == 1)
7791 91 : { if (odd(nk)) s = myusqrt(Q,prec); }
7792 : else
7793 : {
7794 21 : long r = nk >> 1; /* k-1/2 */
7795 21 : s = gpow(utoipos(Q), mkfracss(odd(r)? 1: 3, 4), prec);
7796 21 : if (odd(cQ))
7797 : {
7798 21 : long t = r + ((cQ-1) >> 1);
7799 21 : s = mkcomplex(s, odd(t)? gneg(s): s);
7800 : }
7801 : }
7802 112 : if (!isint1(s)) C = gmul(C, s);
7803 112 : CHIAL = mfcharAL(CHI, Q);
7804 112 : if (dk == 2)
7805 : {
7806 21 : ulong q = odd(Q)? Q << 2: Q, Nq = ulcm(q, mfcharmodulus(CHIAL));
7807 21 : CHIAL = induceN(Nq, CHIAL);
7808 21 : CHIAL = mfcharmul(CHIAL, induce(gel(CHIAL,1), utoipos(q)));
7809 : }
7810 112 : CHIAL = mfchartoprimitive(CHIAL,NULL);
7811 112 : mfB = gequal(CHIAL,CHI)? mf: mfinit_Nndkchi(N,nk,dk,CHIAL,MF_get_space(mf),0);
7812 112 : Mindex = MF_get_Mindex(mfB);
7813 112 : Minv = MF_get_Minv(mfB);
7814 112 : P = z = NULL;
7815 112 : if (ord > 2) { P = mfcharpol(CHI); z = rootsof1u_cx(ord, prec); }
7816 112 : lim = maxss(mfsturm(mfB), mfsturm(mf)) + 1;
7817 567 : for (j = 1; j < l; j++)
7818 : {
7819 455 : GEN v = mfslashexpansion(mf, gel(B,j), g, lim, 0, NULL, prec+EXTRAPREC64);
7820 : long junk;
7821 455 : if (!isint1(C)) v = RgV_Rg_mul(v, C);
7822 455 : v = bestapprnf(v, P, z, prec);
7823 455 : v = vecpermute_partial(v, Mindex, &junk);
7824 455 : v = Minv_RgC_mul(Minv, v); /* cf mftobasis_i */
7825 455 : gel(M, j) = v;
7826 : }
7827 112 : if (is_rational_t(typ(C)) && !gequal1(C)) { M = gdiv(M, C); C = gen_1; }
7828 112 : if (mfB == mf) mfB = gen_0;
7829 112 : return mkvec4(mfB, M, C, mf);
7830 : }
7831 : GEN
7832 98 : mfatkininit(GEN mf, long Q, long prec)
7833 : {
7834 98 : pari_sp av = avma;
7835 98 : mf = checkMF(mf); return gc_GEN(av, mfatkininit_i(mf, Q, 1, prec));
7836 : }
7837 : static void
7838 63 : checkmfa(GEN z)
7839 : {
7840 63 : if (typ(z) != t_VEC || lg(z) != 5 || typ(gel(z,2)) != t_MAT
7841 63 : || !checkMF_i(gel(z,4))
7842 63 : || (!isintzero(gel(z,1)) && !checkMF_i(gel(z,1))))
7843 0 : pari_err_TYPE("mfatkin [please apply mfatkininit()]",z);
7844 63 : }
7845 :
7846 : /* Apply atkin Q to closure F */
7847 : GEN
7848 63 : mfatkin(GEN mfa, GEN F)
7849 : {
7850 63 : pari_sp av = avma;
7851 : GEN z, mfB, MQ, mf;
7852 63 : checkmfa(mfa);
7853 63 : mfB= gel(mfa,1);
7854 63 : MQ = gel(mfa,2);
7855 63 : mf = gel(mfa,4);
7856 63 : if (typ(mfB) == t_INT) mfB = mf;
7857 63 : z = RgM_RgC_mul(MQ, mftobasis_i(mf,F));
7858 63 : return gc_upto(av, mflinear(mfB, z));
7859 : }
7860 :
7861 : GEN
7862 49 : mfatkineigenvalues(GEN mf, long Q, long prec)
7863 : {
7864 49 : pari_sp av = avma;
7865 : GEN vF, L, CHI, M, mfatk, C, MQ, vE, mfB;
7866 : long N, NQ, l, i;
7867 :
7868 49 : mf = checkMF(mf); N = MF_get_N(mf);
7869 49 : vF = MF_get_newforms(mf); l = lg(vF);
7870 : /* N.B. k is integral */
7871 49 : if (l == 1) retgc_const(av, cgetg(1, t_VEC));
7872 49 : L = cgetg(l, t_VEC);
7873 49 : if (Q == 1)
7874 : {
7875 7 : GEN vP = MF_get_fields(mf);
7876 21 : for (i = 1; i < l; i++) gel(L,i) = const_vec(degpol(gel(vP,i)), gen_1);
7877 7 : return L;
7878 : }
7879 42 : vE = mfeigenembed(mf,prec);
7880 42 : if (Q == N) return gc_upto(av, mffrickeeigen(mf, vE, prec));
7881 21 : Q = labs(Q);
7882 21 : NQ = atkin_get_NQ(N, Q, "mfatkineigenvalues"); /* != 1 */
7883 21 : mfatk = mfatkininit(mf, Q, prec);
7884 21 : mfB= gel(mfatk,1); if (typ(mfB) != t_VEC) mfB = mf;
7885 21 : MQ = gel(mfatk,2);
7886 21 : C = gel(mfatk,3);
7887 21 : M = row(mfcoefs_mf(mfB,1,1), 2); /* vec of a_1(b_i) for mfbasis functions */
7888 56 : for (i = 1; i < l; i++)
7889 : {
7890 35 : GEN c = RgV_dotproduct(RgM_RgC_mul(MQ,gel(vF,i)), M); /* C * eigen_i */
7891 35 : gel(L,i) = Rg_embedall_i(c, gel(vE,i));
7892 : }
7893 21 : if (!gequal1(C)) L = gdiv(L, C);
7894 21 : CHI = MF_get_CHI(mf);
7895 21 : if (mfcharorder(CHI) <= 2 && NQ % mfcharconductor(CHI) == 0) L = ground(L);
7896 21 : return gc_GEN(av, L);
7897 : }
7898 :
7899 : /* expand B_d V, keeping same length */
7900 : static GEN
7901 14168 : bdexpand(GEN V, long d)
7902 : {
7903 : GEN W;
7904 : long N, n;
7905 14168 : if (d == 1) return V;
7906 2730 : N = lg(V)-1; W = zerovec(N);
7907 47768 : for (n = 0; n <= (N-1)/d; n++) gel(W, n*d+1) = gel(V, n+1);
7908 2730 : return W;
7909 : }
7910 : /* expand B_d V, increasing length up to lim */
7911 : static GEN
7912 343 : bdexpandall(GEN V, long d, long lim)
7913 : {
7914 : GEN W;
7915 : long N, n;
7916 343 : if (d == 1) return V;
7917 49 : N = lg(V)-1; W = zerovec(lim);
7918 301 : for (n = 0; n <= N-1 && n*d <= lim; n++) gel(W, n*d+1) = gel(V, n+1);
7919 49 : return W;
7920 : }
7921 :
7922 : static void
7923 15491 : parse_vecj(GEN T, GEN *E1, GEN *E2)
7924 : {
7925 15491 : if (lg(T)==3) { *E1 = gel(T,1); *E2 = gel(T,2); }
7926 5600 : else { *E1 = T; *E2 = NULL; }
7927 15491 : }
7928 :
7929 : /* g in M_2(Z) ? */
7930 : static int
7931 3486 : check_M2Z(GEN g)
7932 3486 : { return typ(g) == t_MAT && lg(g) == 3 && lgcols(g) == 3 && RgM_is_ZM(g); }
7933 : /* g in SL_2(Z) ? */
7934 : static int
7935 2058 : check_SL2Z(GEN g) { return check_M2Z(g) && equali1(ZM_det(g)); }
7936 :
7937 : static GEN
7938 9513 : mfcharcxeval(GEN CHI, long n, long prec)
7939 : {
7940 9513 : ulong ord, N = mfcharmodulus(CHI);
7941 : GEN ordg;
7942 9513 : if (N == 1) return gen_1;
7943 3696 : if (ugcd(N, labs(n)) > 1) return gen_0;
7944 3696 : ordg = gmfcharorder(CHI);
7945 3696 : ord = itou(ordg);
7946 3696 : return rootsof1q_cx(znchareval_i(CHI,n,ordg), ord, prec);
7947 : }
7948 :
7949 : static GEN
7950 11039 : RgV_shift(GEN V, GEN gn)
7951 : {
7952 : long i, n, l;
7953 : GEN W;
7954 11039 : if (typ(gn) != t_INT) pari_err_BUG("RgV_shift [n not integral]");
7955 11039 : n = itos(gn);
7956 11039 : if (n < 0) pari_err_BUG("RgV_shift [n negative]");
7957 11039 : if (!n) return V;
7958 112 : W = cgetg_copy(V, &l); if (n > l-1) n = l-1;
7959 308 : for (i=1; i <= n; i++) gel(W,i) = gen_0;
7960 4900 : for ( ; i < l; i++) gel(W,i) = gel(V, i-n);
7961 112 : return W;
7962 : }
7963 : static GEN
7964 19236 : hash_eisengacx(hashtable *H, void *E, long w, GEN ga, long n, long prec)
7965 : {
7966 19236 : ulong h = H->hash(E);
7967 19236 : hashentry *e = hash_search2(H, E, h);
7968 : GEN v;
7969 19236 : if (e) v = (GEN)e->val;
7970 : else
7971 : {
7972 12971 : v = mfeisensteingacx((GEN)E, w, ga, n, prec);
7973 12971 : hash_insert2(H, E, (void*)v, h);
7974 : }
7975 19236 : return v;
7976 : }
7977 : static GEN
7978 11039 : vecj_expand(GEN B, hashtable *H, long w, GEN ga, long n, long prec)
7979 : {
7980 : GEN E1, E2, v;
7981 11039 : parse_vecj(B, &E1, &E2);
7982 11039 : v = hash_eisengacx(H, (void*)E1, w, ga, n, prec);
7983 11039 : if (E2)
7984 : {
7985 8141 : GEN u = hash_eisengacx(H, (void*)E2, w, ga, n, prec);
7986 8141 : GEN a = gadd(gel(v,1), gel(u,1));
7987 8141 : GEN b = RgV_mul_RgXn(gel(v,2), gel(u,2));
7988 8141 : v = mkvec2(a,b);
7989 : }
7990 11039 : return v;
7991 : }
7992 : static GEN
7993 1288 : shift_M(GEN M, GEN Valpha, long w)
7994 : {
7995 1288 : long i, l = lg(Valpha);
7996 1288 : GEN almin = vecmin(Valpha);
7997 12327 : for (i = 1; i < l; i++)
7998 : {
7999 11039 : GEN alpha = gel(Valpha, i), gsh = gmulsg(w, gsub(alpha,almin));
8000 11039 : gel(M,i) = RgV_shift(gel(M,i), gsh);
8001 : }
8002 1288 : return almin;
8003 : }
8004 : static GEN mfeisensteinspaceinit(GEN NK);
8005 : #if 0
8006 : /* ga in M_2^+(Z)), n >= 0 */
8007 : static GEN
8008 : mfgaexpansion_init(GEN mf, GEN ga, long n, long prec)
8009 : {
8010 : GEN M, Mvecj, vecj, almin, Valpha;
8011 : long i, w, l, N = MF_get_N(mf), c = itos(gcoeff(ga,2,1));
8012 : hashtable *H;
8013 :
8014 : if (c % N == 0)
8015 : { /* ga in G_0(N), trivial case; w = 1 */
8016 : GEN chid = mfcharcxeval(MF_get_CHI(mf), itos(gcoeff(ga,2,2)), prec);
8017 : return mkvec2(chid, utoi(n));
8018 : }
8019 :
8020 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
8021 : if (lg(Mvecj) < 5) pari_err_IMPL("mfgaexpansion_init in this case");
8022 : w = mfcuspcanon_width(N, c);
8023 : vecj = gel(Mvecj, 3);
8024 : l = lg(vecj);
8025 : M = cgetg(l, t_VEC);
8026 : Valpha = cgetg(l, t_VEC);
8027 : H = hash_create_GEN(l, 1);
8028 : for (i = 1; i < l; i++)
8029 : {
8030 : GEN v = vecj_expand(gel(vecj,i), H, w, ga, n, prec);
8031 : gel(Valpha,i) = gel(v,1);
8032 : gel(M,i) = gel(v,2);
8033 : }
8034 : almin = shift_M(M, Valpha, w);
8035 : return mkvec3(almin, utoi(w), M);
8036 : }
8037 : /* half-integer weight not supported; vF = [F,eisendec(F)].
8038 : * Minit = mfgaexpansion_init(mf, ga, n, prec) */
8039 : static GEN
8040 : mfgaexpansion_with_init(GEN Minit, GEN vF)
8041 : {
8042 : GEN v;
8043 : if (lg(Minit) == 3)
8044 : { /* ga in G_0(N) */
8045 : GEN chid = gel(Minit,1), gn = gel(Minit,2);
8046 : v = mfcoefs_i(gel(vF,1), itou(gn), 1);
8047 : v = mkvec3(gen_0, gen_1, RgV_Rg_mul(v,chid));
8048 : }
8049 : else
8050 : {
8051 : GEN V = RgM_RgC_mul(gel(Minit,3), gel(vF,2));
8052 : v = mkvec3(gel(Minit,1), gel(Minit,2), V);
8053 : }
8054 : return v;
8055 : }
8056 : #endif
8057 :
8058 : /* B = mfeisensteindec(F) already embedded, ga in M_2^+(Z)), n >= 0 */
8059 : static GEN
8060 1288 : mfgaexpansion_i(GEN mf, GEN B0, GEN ga, long n, long prec)
8061 : {
8062 1288 : GEN M, Mvecj, vecj, almin, Valpha, B, E = NULL;
8063 1288 : long i, j, w, nw, l, N = MF_get_N(mf), bit = prec2nbits(prec) / 2;
8064 : hashtable *H;
8065 :
8066 1288 : Mvecj = obj_check(mf, MF_EISENSPACE);
8067 1288 : if (lg(Mvecj) < 5) { E = gel(Mvecj, 2); Mvecj = gel(Mvecj, 1); }
8068 1288 : vecj = gel(Mvecj, 3);
8069 1288 : l = lg(vecj);
8070 1288 : B = cgetg(l, t_COL);
8071 1288 : M = cgetg(l, t_VEC);
8072 1288 : Valpha = cgetg(l, t_VEC);
8073 1288 : w = mfZC_width(N, gel(ga,1));
8074 1288 : nw = E ? n + w : n;
8075 1288 : H = hash_create_GEN(l, 1);
8076 15673 : for (i = j = 1; i < l; i++)
8077 : {
8078 : GEN v;
8079 14385 : if (gequal0(gel(B0,i))) continue;
8080 11039 : v = vecj_expand(gel(vecj,i), H, w, ga, nw, prec);
8081 11039 : gel(B,j) = gel(B0,i);
8082 11039 : gel(Valpha,j) = gel(v,1);
8083 11039 : gel(M,j) = gel(v,2); j++;
8084 : }
8085 1288 : setlg(Valpha, j);
8086 1288 : setlg(B, j);
8087 1288 : setlg(M, j); l = j;
8088 1288 : if (l == 1) return mkvec3(gen_0, utoi(w), zerovec(n+1));
8089 1288 : almin = shift_M(M, Valpha, w);
8090 1288 : B = RgM_RgC_mul(M, B); l = lg(B);
8091 158347 : for (i = 1; i < l; i++)
8092 157059 : if (gexpo(gel(B,i)) < -bit) gel(B,i) = gen_0;
8093 1288 : settyp(B, t_VEC);
8094 1288 : if (E)
8095 : {
8096 : GEN v, e;
8097 56 : long ell = 0, vB, ve;
8098 126 : for (i = 1; i < l; i++)
8099 126 : if (!gequal0(gel(B,i))) break;
8100 56 : vB = i-1;
8101 56 : v = hash_eisengacx(H, (void*)E, w, ga, n + vB, prec);
8102 56 : e = gel(v,2); l = lg(e);
8103 56 : for (i = 1; i < l; i++)
8104 56 : if (!gequal0(gel(e,i))) break;
8105 56 : ve = i-1;
8106 56 : almin = gsub(almin, gel(v,1));
8107 56 : if (gsigne(almin) < 0)
8108 : {
8109 0 : GEN gell = gceil(gmulsg(-w, almin));
8110 0 : ell = itos(gell);
8111 0 : almin = gadd(almin, gdivgu(gell, w));
8112 0 : if (nw < ell) pari_err_IMPL("alpha < 0 in mfgaexpansion");
8113 : }
8114 56 : if (ve) { ell += ve; e = vecslice(e, ve+1, l-1); }
8115 56 : B = vecslice(B, ell + 1, minss(n + ell + 1, lg(B)-1));
8116 56 : B = RgV_div_RgXn(B, e);
8117 : }
8118 1288 : return mkvec3(almin, utoi(w), B);
8119 : }
8120 :
8121 : /* Theta multiplier: assume 4 | C, (C,D)=1 */
8122 : static GEN
8123 343 : mfthetamultiplier(GEN C, GEN D)
8124 : {
8125 343 : long s = kronecker(C, D);
8126 343 : if (Mod4(D) == 1) return s > 0 ? gen_1: gen_m1;
8127 84 : return s > 0? powIs(3): gen_I();
8128 : }
8129 : /* theta | [*,*;C,D] defined over Q(i) [else over Q] */
8130 : static int
8131 56 : mfthetaI(long C, long D) { return odd(C) || (D & 3) == 3; }
8132 : /* (theta | M) [0..n], assume (C,D) = 1 */
8133 : static GEN
8134 343 : mfthetaexpansion(GEN M, long n)
8135 : {
8136 343 : GEN w, s, al, sla, E, V = zerovec(n+1), C = gcoeff(M,2,1), D = gcoeff(M,2,2);
8137 343 : long lim, la, f, C4 = Mod4(C);
8138 343 : switch (C4)
8139 : {
8140 70 : case 0: al = gen_0; w = gen_1;
8141 70 : s = mfthetamultiplier(C,D);
8142 70 : lim = usqrt(n); gel(V, 1) = s;
8143 70 : s = gmul2n(s, 1);
8144 756 : for (f = 1; f <= lim; f++) gel(V, f*f + 1) = s;
8145 70 : break;
8146 105 : case 2: al = uutoQ(1,4); w = gen_1;
8147 105 : E = subii(C, shifti(D,1)); /* (E, D) = 1 */
8148 105 : s = gmul2n(mfthetamultiplier(E, D), 1);
8149 105 : if ((!signe(E) && equalim1(D)) || (signe(E) > 0 && signe(C) < 0))
8150 14 : s = gneg(s);
8151 105 : lim = (usqrt(n << 2) - 1) >> 1;
8152 966 : for (f = 0; f <= lim; f++) gel(V, f*(f+1) + 1) = s;
8153 105 : break;
8154 168 : default: al = gen_0; w = utoipos(4);
8155 168 : la = (-Mod4(D)*C4) & 3L;
8156 168 : E = negi(addii(D, mului(la, C)));
8157 168 : s = mfthetamultiplier(E, C); /* (E,C) = 1 */
8158 168 : if (signe(C) < 0 && signe(E) >= 0) s = gneg(s);
8159 168 : s = gsub(s, mulcxI(s));
8160 168 : sla = gmul(s, powIs(-la));
8161 168 : lim = usqrt(n); gel(V, 1) = gmul2n(s, -1);
8162 1708 : for (f = 1; f <= lim; f++) gel(V, f*f + 1) = odd(f) ? sla : s;
8163 168 : break;
8164 : }
8165 343 : return mkvec3(al, w, V);
8166 : }
8167 :
8168 : /* F 1/2 integral weight */
8169 : static GEN
8170 343 : mf2gaexpansion(GEN mf2, GEN F, GEN ga, long n, long prec)
8171 : {
8172 343 : GEN FT = mfmultheta(F), mf = obj_checkbuild(mf2, MF_MF2INIT, &mf2init);
8173 343 : GEN res, V1, Tres, V2, al, V, gsh, C = gcoeff(ga,2,1);
8174 343 : long w2, N = MF_get_N(mf), w = mfcuspcanon_width(N, umodiu(C,N));
8175 343 : long ext = (Mod4(C) != 2)? 0: (w+3) >> 2;
8176 343 : long prec2 = prec + nbits2extraprec((long)M_PI/(2*M_LN2)*sqrt(n + ext));
8177 343 : res = mfgaexpansion(mf, FT, ga, n + ext, prec2);
8178 343 : Tres = mfthetaexpansion(ga, n + ext);
8179 343 : V1 = gel(res,3);
8180 343 : V2 = gel(Tres,3);
8181 343 : al = gsub(gel(res,1), gel(Tres,1));
8182 343 : w2 = itos(gel(Tres,2));
8183 343 : if (w != itos(gel(res,2)) || w % w2)
8184 0 : pari_err_BUG("mf2gaexpansion [incorrect w2 or w]");
8185 343 : if (w2 != w) V2 = bdexpand(V2, w/w2);
8186 343 : V = RgV_div_RgXn(V1, V2);
8187 343 : gsh = gfloor(gmulsg(w, al));
8188 343 : if (!gequal0(gsh))
8189 : {
8190 35 : al = gsub(al, gdivgu(gsh, w));
8191 35 : if (gsigne(gsh) > 0)
8192 : {
8193 0 : V = RgV_shift(V, gsh);
8194 0 : V = vecslice(V, 1, n + 1);
8195 : }
8196 : else
8197 : {
8198 35 : long sh = -itos(gsh), i;
8199 35 : if (sh > ext) pari_err_BUG("mf2gaexpansion [incorrect sh]");
8200 154 : for (i = 1; i <= sh; i++)
8201 119 : if (!gequal0(gel(V,i))) pari_err_BUG("mf2gaexpansion [sh too large]");
8202 35 : V = vecslice(V, sh+1, n + sh+1);
8203 : }
8204 : }
8205 343 : obj_free(mf); return mkvec3(al, stoi(w), gprec_wtrunc(V, prec));
8206 : }
8207 :
8208 : static GEN
8209 77 : mfgaexpansionatkin(GEN mf, GEN F, GEN C, GEN D, long Q, long n, long prec)
8210 : {
8211 77 : GEN mfa = mfatkininit_i(mf, Q, 0, prec), MQ = gel(mfa,2);
8212 77 : long i, FC, k = MF_get_k(mf);
8213 77 : GEN x, v, V, z, s, CHI = mfchartoprimitive(MF_get_CHI(mf), &FC);
8214 :
8215 : /* V = mfcoefs(F | w_Q, n), can't use mfatkin because MQ nonrational */
8216 77 : V = RgM_RgC_mul(mfcoefs_mf(mf,n,1), RgM_RgC_mul(MQ, mftobasis_i(mf,F)));
8217 77 : (void)bezout(utoipos(Q), C, &x, &v);
8218 77 : s = mfchareval(CHI, (umodiu(x, FC) * umodiu(D, FC)) % FC);
8219 77 : s = gdiv(s, gpow(utoipos(Q), uutoQ(k,2), prec));
8220 77 : V = RgV_Rg_mul(V, s);
8221 77 : z = rootsof1powinit(umodiu(D,Q)*umodiu(v,Q) % Q, Q, prec);
8222 11613 : for (i = 1; i <= n+1; i++) gel(V,i) = gmul(gel(V,i), rootsof1pow(z, i-1));
8223 77 : return mkvec3(gen_0, utoipos(Q), V);
8224 : }
8225 :
8226 : static long
8227 70 : inveis_extraprec(long N, GEN ga, GEN Mvecj, long n)
8228 : {
8229 70 : long e, w = mfZC_width(N, gel(ga,1));
8230 70 : GEN f, E = gel(Mvecj,2), v = mfeisensteingacx(E, w, ga, n, DEFAULTPREC);
8231 70 : v = gel(v,2);
8232 70 : f = RgV_to_RgX(v,0); n -= RgX_valrem(f, &f);
8233 70 : e = gexpo(RgXn_inv(f, n+1));
8234 70 : return (e > 0)? nbits2extraprec(e): 0;
8235 : }
8236 : /* allow F of the form [F, mf_eisendec(F)]~ */
8237 : static GEN
8238 2051 : mfgaexpansion(GEN mf, GEN F, GEN ga, long n, long prec)
8239 : {
8240 2051 : GEN v, EF = NULL, res, Mvecj, c, d;
8241 : long precnew, N;
8242 :
8243 2051 : if (n < 0) pari_err_DOMAIN("mfgaexpansion", "n", "<", gen_0, stoi(n));
8244 2051 : if (typ(F) == t_COL && lg(F) == 3) { EF = gel(F,2); F = gel(F,1); }
8245 2051 : if (!checkmf_i(F)) pari_err_TYPE("mfgaexpansion", F);
8246 2051 : if (!check_SL2Z(ga)) pari_err_TYPE("mfgaexpansion",ga);
8247 2051 : if (typ(mf_get_gk(F)) != t_INT) return mf2gaexpansion(mf, F, ga, n, prec);
8248 1708 : c = gcoeff(ga,2,1);
8249 1708 : d = gcoeff(ga,2,2);
8250 1708 : N = MF_get_N(mf);
8251 1708 : if (!umodiu(c, mf_get_N(F)))
8252 : { /* trivial case: ga in Gamma_0(N) */
8253 343 : long w = mfcuspcanon_width(N, umodiu(c,N));
8254 343 : GEN CHI = mf_get_CHI(F);
8255 343 : GEN chid = mfcharcxeval(CHI, umodiu(d,mfcharmodulus(CHI)), prec);
8256 343 : v = mfcoefs_i(F, n/w, 1); if (!isint1(chid)) v = RgV_Rg_mul(v,chid);
8257 343 : return mkvec3(gen_0, stoi(w), bdexpandall(v,w,n+1));
8258 : }
8259 1365 : mf = MF_set_new(mf);
8260 1365 : if (MF_get_space(mf) == mf_NEW)
8261 : {
8262 483 : long cN = umodiu(c,N), g = ugcd(cN,N), Q = N/g;
8263 483 : GEN CHI = MF_get_CHI(mf);
8264 483 : if (ugcd(cN, Q)==1 && mfcharorder(CHI) <= 2
8265 231 : && g % mfcharconductor(CHI) == 0
8266 119 : && degpol(mf_get_field(F)) == 1)
8267 77 : return mfgaexpansionatkin(mf, F, c, d, Q, n, prec);
8268 : }
8269 1288 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
8270 1288 : precnew = prec;
8271 1288 : if (lg(Mvecj) < 5) precnew += inveis_extraprec(N, ga, Mvecj, n);
8272 1288 : if (!EF) EF = mf_eisendec(mf, F, precnew);
8273 1288 : res = mfgaexpansion_i(mf, EF, ga, n, precnew);
8274 1288 : return precnew == prec ? res : gprec_wtrunc(res, prec);
8275 : }
8276 :
8277 : /* parity = -1 or +1 */
8278 : static GEN
8279 217 : findd(long N, long parity)
8280 : {
8281 217 : GEN L, D = mydivisorsu(N);
8282 217 : long i, j, l = lg(D);
8283 217 : L = cgetg(l, t_VEC);
8284 1218 : for (i = j = 1; i < l; i++)
8285 : {
8286 1001 : long d = D[i];
8287 1001 : if (parity == -1) d = -d;
8288 1001 : if (sisfundamental(d)) gel(L,j++) = stoi(d);
8289 : }
8290 217 : setlg(L,j); return L;
8291 : }
8292 : /* does ND contain a divisor of N ? */
8293 : static int
8294 413 : seenD(long N, GEN ND)
8295 : {
8296 413 : long j, l = lg(ND);
8297 427 : for (j = 1; j < l; j++)
8298 14 : if (N % ND[j] == 0) return 1;
8299 413 : return 0;
8300 : }
8301 : static GEN
8302 63 : search_levels(GEN vN, const char *f)
8303 : {
8304 63 : switch(typ(vN))
8305 : {
8306 28 : case t_INT: vN = mkvecsmall(itos(vN)); break;
8307 35 : case t_VEC: case t_COL: vN = ZV_to_zv(vN); break;
8308 0 : case t_VECSMALL: vN = leafcopy(vN); break;
8309 0 : default: pari_err_TYPE(f, vN);
8310 : }
8311 63 : vecsmall_sort(vN); return vN;
8312 : }
8313 : GEN
8314 28 : mfsearch(GEN NK, GEN V, long space)
8315 : {
8316 28 : pari_sp av = avma;
8317 : GEN F, gk, NbyD, vN;
8318 : long n, nk, dk, parity, nV, i, lvN;
8319 :
8320 28 : if (typ(NK) != t_VEC || lg(NK) != 3) pari_err_TYPE("mfsearch", NK);
8321 28 : gk = gel(NK,2);
8322 28 : if (typ(gmul2n(gk, 1)) != t_INT) pari_err_TYPE("mfsearch [k]", gk);
8323 28 : switch(typ(V))
8324 : {
8325 28 : case t_VEC: V = shallowtrans(V);
8326 28 : case t_COL: break;
8327 0 : default: pari_err_TYPE("mfsearch [V]", V);
8328 : }
8329 28 : vN = search_levels(gel(NK,1), "mfsearch [N]");
8330 28 : if (gequal0(V)) { set_avma(av); retmkvec(mftrivial()); }
8331 14 : lvN = lg(vN);
8332 :
8333 14 : Qtoss(gk, &nk,&dk);
8334 14 : parity = (dk == 1 && odd(nk)) ? -1 : 1;
8335 14 : nV = lg(V)-2;
8336 14 : F = cgetg(1, t_VEC);
8337 14 : NbyD = const_vec(vN[lvN-1], cgetg(1,t_VECSMALL));
8338 231 : for (n = 1; n < lvN; n++)
8339 : {
8340 217 : long N = vN[n];
8341 : GEN L;
8342 217 : if (N <= 0 || (dk == 2 && (N & 3))) continue;
8343 217 : L = findd(N, parity);
8344 630 : for (i = 1; i < lg(L); i++)
8345 : {
8346 413 : GEN mf, M, CO, gD = gel(L,i);
8347 413 : GEN *ND = (GEN*)NbyD + itou(gD); /* points to NbyD[|D|] */
8348 :
8349 413 : if (seenD(N, *ND)) continue;
8350 413 : mf = mfinit_Nndkchi(N, nk, dk, get_mfchar(gD), space, 1);
8351 413 : M = mfcoefs_mf(mf, nV, 1);
8352 413 : CO = inverseimage(M, V); if (lg(CO) == 1) continue;
8353 :
8354 42 : F = vec_append(F, mflinear(mf,CO));
8355 42 : *ND = vecsmall_append(*ND, N); /* add to NbyD[|D|] */
8356 : }
8357 : }
8358 14 : return gc_GEN(av, F);
8359 : }
8360 :
8361 : static GEN
8362 889 : search_from_split(GEN mf, GEN vap, GEN vlp)
8363 : {
8364 889 : pari_sp av = avma;
8365 889 : long lvlp = lg(vlp), j, jv, l1;
8366 889 : GEN v, NK, S1, S, M = NULL;
8367 :
8368 889 : S1 = gel(split_i(mf, 1, 0), 1); /* rational newforms */
8369 889 : l1 = lg(S1);
8370 889 : if (l1 == 1) return gc_NULL(av);
8371 455 : v = cgetg(l1, t_VEC);
8372 455 : S = MF_get_S(mf);
8373 455 : NK = mf_get_NK(gel(S,1));
8374 455 : if (lvlp > 1) M = rowpermute(mfcoefs_mf(mf, vlp[lvlp-1], 1), vlp);
8375 980 : for (j = jv = 1; j < l1; j++)
8376 : {
8377 525 : GEN vF = gel(S1,j);
8378 : long t;
8379 658 : for (t = lvlp-1; t > 0; t--)
8380 : { /* lhs = vlp[j]-th coefficient of eigenform */
8381 595 : GEN rhs = gel(vap,t), lhs = RgMrow_RgC_mul(M, vF, t);
8382 595 : if (!gequal(lhs, rhs)) break;
8383 : }
8384 525 : if (!t) gel(v,jv++) = mflinear_i(NK,S,vF);
8385 : }
8386 455 : if (jv == 1) return gc_NULL(av);
8387 63 : setlg(v,jv); return v;
8388 : }
8389 : GEN
8390 35 : mfeigensearch(GEN NK, GEN AP)
8391 : {
8392 35 : pari_sp av = avma;
8393 35 : GEN k, vN, vap, vlp, vres = cgetg(1, t_VEC), D;
8394 : long n, lvN, i, l, even;
8395 :
8396 35 : if (!AP) l = 1;
8397 : else
8398 : {
8399 28 : l = lg(AP);
8400 28 : if (typ(AP) != t_VEC) pari_err_TYPE("mfeigensearch",AP);
8401 : }
8402 35 : vap = cgetg(l, t_VEC);
8403 35 : vlp = cgetg(l, t_VECSMALL);
8404 35 : if (l > 1)
8405 : {
8406 28 : GEN perm = indexvecsort(AP, mkvecsmall(1));
8407 77 : for (i = 1; i < l; i++)
8408 : {
8409 49 : GEN v = gel(AP,perm[i]), gp, ap;
8410 49 : if (typ(v) != t_VEC || lg(v) != 3) pari_err_TYPE("mfeigensearch", AP);
8411 49 : gp = gel(v,1);
8412 49 : ap = gel(v,2);
8413 49 : if (typ(gp) != t_INT || (typ(ap) != t_INT && typ(ap) != t_INTMOD))
8414 0 : pari_err_TYPE("mfeigensearch", AP);
8415 49 : gel(vap,i) = ap;
8416 49 : vlp[i] = itos(gp)+1; if (vlp[i] < 0) pari_err_TYPE("mfeigensearch", AP);
8417 : }
8418 : }
8419 35 : l = lg(NK);
8420 35 : if (typ(NK) != t_VEC || l != 3) pari_err_TYPE("mfeigensearch",NK);
8421 35 : k = gel(NK,2);
8422 35 : vN = search_levels(gel(NK,1), "mfeigensearch [N]");
8423 35 : lvN = lg(vN);
8424 35 : vecsmall_sort(vlp);
8425 35 : even = !mpodd(k);
8426 980 : for (n = 1; n < lvN; n++)
8427 : {
8428 945 : pari_sp av2 = avma;
8429 : GEN mf, L;
8430 945 : long N = vN[n];
8431 945 : if (even) D = gen_1;
8432 : else
8433 : {
8434 112 : long r = (N&3L);
8435 112 : if (r == 1 || r == 2) continue;
8436 56 : D = stoi( corediscs(-N, NULL) ); /* < 0 */
8437 : }
8438 889 : mf = mfinit_i(mkvec3(utoipos(N), k, D), mf_NEW);
8439 889 : L = search_from_split(mf, vap, vlp);
8440 889 : if (L) vres = shallowconcat(vres, L); else set_avma(av2);
8441 : }
8442 35 : return gc_GEN(av, vres);
8443 : }
8444 :
8445 : /* tf_{N,k}(n) */
8446 : static GEN
8447 4582928 : mfnewtracecache(long N, long k, long n, cachenew_t *cache)
8448 : {
8449 4582928 : GEN C = NULL, S;
8450 : long lcache;
8451 4582928 : if (!n) return gen_0;
8452 4439876 : S = gel(cache->vnew,N);
8453 4439876 : lcache = lg(S);
8454 4439876 : if (n < lcache) C = gel(S, n);
8455 4439876 : if (C) cache->newHIT++;
8456 2628827 : else C = mfnewtrace_i(N,k,n,cache);
8457 4439876 : cache->newTOTAL++;
8458 4439876 : if (n < lcache) gel(S,n) = C;
8459 4439876 : return C;
8460 : }
8461 :
8462 : static long
8463 1393 : mfdim_Nkchi(long N, long k, GEN CHI, long space)
8464 : {
8465 1393 : if (k < 0 || badchar(N,k,CHI)) return 0;
8466 1092 : if (k == 0)
8467 35 : return mfcharistrivial(CHI) && !space_is_cusp(space)? 1: 0;
8468 1057 : switch(space)
8469 : {
8470 245 : case mf_NEW: return mfnewdim(N,k,CHI);
8471 196 : case mf_CUSP:return mfcuspdim(N,k,CHI);
8472 168 : case mf_OLD: return mfolddim(N,k,CHI);
8473 217 : case mf_FULL:return mffulldim(N,k,CHI);
8474 231 : case mf_EISEN: return mfeisensteindim(N,k,CHI);
8475 0 : default: pari_err_FLAG("mfdim");
8476 : }
8477 : return 0;/*LCOV_EXCL_LINE*/
8478 : }
8479 : static long
8480 2114 : mf1dimsum(long N, long space)
8481 : {
8482 2114 : switch(space)
8483 : {
8484 1050 : case mf_NEW: return mf1newdimsum(N);
8485 1057 : case mf_CUSP: return mf1cuspdimsum(N);
8486 7 : case mf_OLD: return mf1olddimsum(N);
8487 : }
8488 0 : pari_err_FLAG("mfdim");
8489 : return 0; /*LCOV_EXCL_LINE*/
8490 : }
8491 : /* mfdim for k = nk/dk */
8492 : static long
8493 44744 : mfdim_Nndkchi(long N, long nk, long dk, GEN CHI, long space)
8494 43463 : { return (dk == 2)? mf2dim_Nkchi(N, nk >> 1, CHI, space)
8495 88186 : : mfdim_Nkchi(N, nk, CHI, space); }
8496 : /* FIXME: use direct dim Gamma1(N) formula, don't compute individual spaces */
8497 : static long
8498 252 : mfkdimsum(long N, long k, long dk, long space)
8499 : {
8500 252 : GEN w = mfchars(N, k, dk, NULL);
8501 252 : long i, j, D = 0, l = lg(w);
8502 1239 : for (i = j = 1; i < l; i++)
8503 : {
8504 987 : GEN CHI = gel(w,i);
8505 987 : long d = mfdim_Nndkchi(N,k,dk,CHI,space);
8506 987 : if (d) D += d * myeulerphiu(mfcharorder(CHI));
8507 : }
8508 252 : return D;
8509 : }
8510 : static GEN
8511 105 : mf1dims(long N, GEN vCHI, long space)
8512 : {
8513 105 : GEN D = NULL;
8514 105 : switch(space)
8515 : {
8516 56 : case mf_NEW: D = mf1newdimall(N, vCHI); break;
8517 21 : case mf_CUSP:D = mf1cuspdimall(N, vCHI); break;
8518 28 : case mf_OLD: D = mf1olddimall(N, vCHI); break;
8519 0 : default: pari_err_FLAG("mfdim");
8520 : }
8521 105 : return D;
8522 : }
8523 : static GEN
8524 2961 : mfkdims(long N, long k, long dk, GEN vCHI, long space)
8525 : {
8526 2961 : GEN D, w = mfchars(N, k, dk, vCHI);
8527 2961 : long i, j, l = lg(w);
8528 2961 : D = cgetg(l, t_VEC);
8529 46592 : for (i = j = 1; i < l; i++)
8530 : {
8531 43631 : GEN CHI = gel(w,i);
8532 43631 : long d = mfdim_Nndkchi(N,k,dk,CHI,space);
8533 43631 : if (vCHI)
8534 574 : gel(D, j++) = mkvec2s(d, 0);
8535 43057 : else if (d)
8536 2520 : gel(D, j++) = fmt_dim(CHI, d, 0);
8537 : }
8538 2961 : setlg(D,j); return D;
8539 : }
8540 : GEN
8541 5719 : mfdim(GEN NK, long space)
8542 : {
8543 5719 : pari_sp av = avma;
8544 : long N, k, dk, joker;
8545 : GEN CHI, mf;
8546 5719 : if ((mf = checkMF_i(NK))) return utoi(MF_get_dim(mf));
8547 5586 : checkNK2(NK, &N, &k, &dk, &CHI, 2);
8548 5586 : if (!CHI) joker = 1;
8549 : else
8550 2611 : switch(typ(CHI))
8551 : {
8552 2373 : case t_INT: joker = 2; break;
8553 112 : case t_COL: joker = 3; break;
8554 126 : default: joker = 0; break;
8555 : }
8556 5586 : if (joker)
8557 : {
8558 : long d;
8559 : GEN D;
8560 5460 : if (k < 0) switch(joker)
8561 : {
8562 0 : case 1: return cgetg(1,t_VEC);
8563 7 : case 2: return gen_0;
8564 0 : case 3: return mfdim0all(CHI);
8565 : }
8566 5453 : if (k == 0)
8567 : {
8568 28 : if (space_is_cusp(space)) switch(joker)
8569 : {
8570 7 : case 1: return cgetg(1,t_VEC);
8571 0 : case 2: return gen_0;
8572 7 : case 3: return mfdim0all(CHI);
8573 : }
8574 14 : switch(joker)
8575 : {
8576 : long i, l;
8577 7 : case 1: retmkvec(fmt_dim(mfchartrivial(),0,0));
8578 0 : case 2: return gen_1;
8579 7 : case 3: l = lg(CHI); D = cgetg(l,t_VEC);
8580 35 : for (i = 1; i < l; i++)
8581 : {
8582 28 : long t = mfcharistrivial(gel(CHI,i));
8583 28 : gel(D,i) = mkvec2(t? gen_1: gen_0, gen_0);
8584 : }
8585 7 : return D;
8586 : }
8587 : }
8588 5425 : if (dk == 1 && k == 1 && space != mf_EISEN)
8589 105 : {
8590 2219 : long fix = 0, space0 = space;
8591 2219 : if (space == mf_FULL) space = mf_CUSP; /* remove Eisenstein part */
8592 2219 : if (joker == 2)
8593 : {
8594 2114 : d = mf1dimsum(N, space);
8595 2114 : if (space0 == mf_FULL) d += mfkdimsum(N,k,dk,mf_EISEN);/*add it back*/
8596 2114 : return gc_utoi(av, d);
8597 : }
8598 : /* must initialize explicitly: trivial spaces for E_k/S_k differ */
8599 105 : if (space0 == mf_FULL)
8600 : {
8601 7 : if (!CHI) fix = 1; /* must remove 0 spaces */
8602 7 : CHI = mfchars(N, k, dk, CHI);
8603 : }
8604 105 : D = mf1dims(N, CHI, space);
8605 105 : if (space0 == mf_FULL)
8606 : {
8607 7 : GEN D2 = mfkdims(N, k, dk, CHI, mf_EISEN);
8608 7 : D = merge_dims(D, D2, fix? CHI: NULL);
8609 : }
8610 : }
8611 : else
8612 : {
8613 3206 : if (joker==2) { d = mfkdimsum(N,k,dk,space); return gc_utoi(av,d); }
8614 2954 : D = mfkdims(N, k, dk, CHI, space);
8615 : }
8616 3059 : if (!CHI) return gc_upto(av, vecsort(D, mkvecsmall(1)));
8617 105 : return gc_GEN(av, D);
8618 : }
8619 126 : return utoi( mfdim_Nndkchi(N, k, dk, CHI, space) );
8620 : }
8621 :
8622 : GEN
8623 364 : mfbasis(GEN NK, long space)
8624 : {
8625 364 : pari_sp av = avma;
8626 : long N, k, dk;
8627 : GEN mf, CHI;
8628 364 : if ((mf = checkMF_i(NK))) return gconcat(gel(mf,2), gel(mf,3));
8629 14 : checkNK2(NK, &N, &k, &dk, &CHI, 0);
8630 14 : if (dk == 2) return gc_GEN(av, mf2basis(N, k>>1, CHI, NULL, space));
8631 14 : mf = mfinit_Nkchi(N, k, CHI, space, 1);
8632 14 : return gc_GEN(av, MF_get_basis(mf));
8633 : }
8634 :
8635 : static GEN
8636 49 : deg1ser_shallow(GEN a1, GEN a0, long v, long e)
8637 49 : { return RgX_to_ser(deg1pol_shallow(a1, a0, v), e+2); }
8638 : /* r / x + O(1) */
8639 : static GEN
8640 49 : simple_pole(GEN r)
8641 : {
8642 49 : GEN S = deg1ser_shallow(gen_0, r, 0, 1);
8643 49 : setvalser(S, -1); return S;
8644 : }
8645 :
8646 : /* F form, E embedding; mfa = mfatkininit or root number (eigenform case) */
8647 : static GEN
8648 175 : mflfuncreate(GEN mfa, GEN F, GEN E, GEN N, GEN gk)
8649 : {
8650 175 : GEN LF = cgetg(8,t_VEC), polar = cgetg(1,t_COL), eps;
8651 175 : long k = itou(gk);
8652 175 : gel(LF,1) = lfuntag(t_LFUN_MFCLOS, mkvec3(F,E,gen_1));
8653 175 : if (typ(mfa) != t_VEC)
8654 112 : eps = mfa; /* cuspidal eigenform: root number; no poles */
8655 : else
8656 : { /* mfatkininit */
8657 63 : GEN a0, b0, vF, vG, G = NULL;
8658 63 : GEN M = gel(mfa,2), C = gel(mfa,3), mf = gel(mfa,4);
8659 63 : M = gdiv(mfmatembed(E, M), C);
8660 63 : vF = mfvecembed(E, mftobasis_i(mf, F));
8661 63 : vG = RgM_RgC_mul(M, vF);
8662 63 : if (gequal(vF,vG)) eps = gen_1;
8663 49 : else if (gequal(vF,gneg(vG))) eps = gen_m1;
8664 : else
8665 : { /* not self-dual */
8666 42 : eps = NULL;
8667 42 : G = mfatkin(mfa, F);
8668 42 : gel(LF,2) = lfuntag(t_LFUN_MFCLOS, mkvec3(G,E,ginv(C)));
8669 42 : gel(LF,6) = powIs(k);
8670 : }
8671 : /* polar part */
8672 63 : a0 = mfembed(E, mfcoef(F,0));
8673 63 : b0 = eps? gmul(eps,a0): gdiv(mfembed(E, mfcoef(G,0)), C);
8674 63 : if (!gequal0(b0))
8675 : {
8676 28 : b0 = mulcxpowIs(gmul2n(b0,1), k);
8677 28 : polar = vec_append(polar, mkvec2(gk, simple_pole(b0)));
8678 : }
8679 63 : if (!gequal0(a0))
8680 : {
8681 21 : a0 = gneg(gmul2n(a0,1));
8682 21 : polar = vec_append(polar, mkvec2(gen_0, simple_pole(a0)));
8683 : }
8684 : }
8685 175 : if (eps) /* self-dual */
8686 : {
8687 133 : gel(LF,2) = mfcharorder(mf_get_CHI(F)) <= 2? gen_0: gen_1;
8688 133 : gel(LF,6) = mulcxpowIs(eps,k);
8689 : }
8690 175 : gel(LF,3) = mkvec2(gen_0, gen_1);
8691 175 : gel(LF,4) = gk;
8692 175 : gel(LF,5) = N;
8693 175 : if (lg(polar) == 1) setlg(LF,7); else gel(LF,7) = polar;
8694 175 : return LF;
8695 : }
8696 : static GEN
8697 147 : mflfuncreateall(long sd, GEN mfa, GEN F, GEN vE, GEN gN, GEN gk)
8698 : {
8699 147 : long i, l = lg(vE);
8700 147 : GEN L = cgetg(l, t_VEC);
8701 322 : for (i = 1; i < l; i++)
8702 175 : gel(L,i) = mflfuncreate(sd? gel(mfa,i): mfa, F, gel(vE,i), gN, gk);
8703 147 : return L;
8704 : }
8705 : GEN
8706 98 : lfunmf(GEN mf, GEN F, long bitprec)
8707 : {
8708 98 : pari_sp av = avma;
8709 98 : long i, l, prec = nbits2prec(bitprec);
8710 : GEN L, gk, gN;
8711 98 : mf = checkMF(mf);
8712 98 : gk = MF_get_gk(mf);
8713 98 : gN = MF_get_gN(mf);
8714 98 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
8715 98 : if (F)
8716 : {
8717 : GEN v;
8718 91 : long s = MF_get_space(mf);
8719 91 : if (!checkmf_i(F)) pari_err_TYPE("lfunmf", F);
8720 91 : if (!mfisinspace_i(mf, F)) err_space(F);
8721 91 : L = NULL;
8722 91 : if ((s == mf_NEW || s == mf_CUSP || s == mf_FULL)
8723 77 : && gequal(mfcoefs_i(F,1,1), mkvec2(gen_0,gen_1)))
8724 : { /* check if eigenform */
8725 49 : GEN vP, vF, b = mftobasis_i(mf, F);
8726 49 : long lF, d = degpol(mf_get_field(F));
8727 49 : v = mfsplit(mf, d, 0);
8728 49 : vF = gel(v,1);
8729 49 : vP = gel(v,2); lF = lg(vF);
8730 49 : for (i = 1; i < lF; i++)
8731 42 : if (degpol(gel(vP,i)) == d && gequal(gel(vF,i), b))
8732 : {
8733 42 : GEN vE = mfgetembed(F, prec);
8734 42 : GEN Z = mffrickeeigen_i(mf, mkvec(b), mkvec(vE), prec);
8735 42 : L = mflfuncreateall(1, gel(Z,1), F, vE, gN, gk);
8736 42 : break;
8737 : }
8738 : }
8739 91 : if (!L)
8740 : { /* not an eigenform: costly general case */
8741 49 : GEN mfa = mfatkininit_i(mf, itou(gN), 1, prec);
8742 49 : L = mflfuncreateall(0,mfa, F, mfgetembed(F,prec), gN, gk);
8743 : }
8744 91 : if (lg(L) == 2) L = gel(L,1);
8745 : }
8746 : else
8747 : {
8748 7 : GEN M = mfeigenbasis(mf), vE = mfeigenembed(mf, prec);
8749 7 : GEN v = mffrickeeigen(mf, vE, prec);
8750 7 : l = lg(vE); L = cgetg(l, t_VEC);
8751 63 : for (i = 1; i < l; i++)
8752 56 : gel(L,i) = mflfuncreateall(1,gel(v,i), gel(M,i), gel(vE,i), gN, gk);
8753 : }
8754 98 : return gc_GEN(av, L);
8755 : }
8756 :
8757 : GEN
8758 28 : mffromell(GEN E)
8759 : {
8760 28 : pari_sp av = avma;
8761 : GEN mf, F, z, v, S;
8762 : long N, i, l;
8763 :
8764 28 : checkell(E);
8765 28 : if (ell_get_type(E) != t_ELL_Q) pari_err_TYPE("mfffromell [E not over Q]", E);
8766 28 : N = itos(ellQ_get_N(E));
8767 28 : mf = mfinit_i(mkvec2(utoi(N), gen_2), mf_NEW);
8768 28 : v = split_i(mf, 1, 0);
8769 28 : S = gel(v,1); l = lg(S); /* rational newforms */
8770 28 : F = tag(t_MF_ELL, mkNK(N,2,mfchartrivial()), E);
8771 28 : z = mftobasis_i(mf, F);
8772 28 : for(i = 1; i < l; i++)
8773 28 : if (gequal(z, gel(S,i))) break;
8774 28 : if (i == l) pari_err_BUG("mffromell [E is not modular]");
8775 28 : return gc_GEN(av, mkvec3(mf, F, z));
8776 : }
8777 :
8778 : /* returns -1 if not, degree otherwise */
8779 : long
8780 140 : polishomogeneous(GEN P)
8781 : {
8782 : long i, D, l;
8783 140 : if (typ(P) != t_POL) return 0;
8784 77 : D = -1; l = lg(P);
8785 322 : for (i = 2; i < l; i++)
8786 : {
8787 245 : GEN c = gel(P,i);
8788 : long d;
8789 245 : if (gequal0(c)) continue;
8790 112 : d = polishomogeneous(c);
8791 112 : if (d < 0) return -1;
8792 112 : if (D < 0) D = d + i-2; else if (D != d + i-2) return -1;
8793 : }
8794 77 : return D;
8795 : }
8796 :
8797 : /* M a pp((Gram q)^(-1)) ZM; P a homogeneous t_POL, is P spherical ? */
8798 : static int
8799 28 : RgX_isspherical(GEN M, GEN P)
8800 : {
8801 28 : pari_sp av = avma;
8802 28 : GEN S, v = variables_vecsmall(P);
8803 28 : long i, j, l = lg(v);
8804 28 : if (l > lg(M)) pari_err(e_MISC, "too many variables in mffromqf");
8805 21 : S = gen_0;
8806 63 : for (j = 1; j < l; j++)
8807 : {
8808 42 : GEN Mj = gel(M, j), Pj = deriv(P, v[j]);
8809 105 : for (i = 1; i <= j; i++)
8810 : {
8811 63 : GEN c = gel(Mj, i);
8812 63 : if (!signe(c)) continue;
8813 42 : if (i != j) c = shifti(c, 1);
8814 42 : S = gadd(S, gmul(c, deriv(Pj, v[i])));
8815 : }
8816 : }
8817 21 : return gc_bool(av, gequal0(S));
8818 : }
8819 :
8820 : static GEN
8821 49 : c_QFsimple_i(long n, GEN Q, GEN P)
8822 : {
8823 49 : GEN V, v = qfrep0(Q, utoi(n), 1);
8824 49 : long i, l = lg(v);
8825 49 : V = cgetg(l+1, t_VEC);
8826 49 : if (!P || equali1(P))
8827 : {
8828 42 : gel(V,1) = gen_1;
8829 420 : for (i = 2; i <= l; i++) gel(V,i) = utoi(v[i-1] << 1);
8830 : }
8831 : else
8832 : {
8833 7 : gel(V,1) = gcopy(P);
8834 7 : for (i = 2; i <= l; i++) gel(V,i) = gmulgu(P, v[i-1] << 1);
8835 : }
8836 49 : return V;
8837 : }
8838 :
8839 : /* v a t_VECSMALL of variable numbers, lg(r) >= lg(v), r is a vector of
8840 : * scalars [not involving any variable in v] */
8841 : static GEN
8842 14 : gsubstvec_i(GEN e, GEN v, GEN r)
8843 : {
8844 14 : long i, l = lg(v);
8845 42 : for(i = 1; i < l; i++) e = gsubst(e, v[i], gel(r,i));
8846 14 : return e;
8847 : }
8848 : static GEN
8849 56 : c_QF_i(long n, GEN Q, GEN P)
8850 : {
8851 56 : pari_sp av = avma;
8852 : GEN V, v, va;
8853 : long i, l;
8854 56 : if (!P || typ(P) != t_POL) return gc_upto(av, c_QFsimple_i(n, Q, P));
8855 7 : v = gel(minim(Q, utoi(2*n), NULL), 3);
8856 7 : va = variables_vecsmall(P);
8857 7 : V = zerovec(n + 1); l = lg(v);
8858 21 : for (i = 1; i < l; i++)
8859 : {
8860 14 : pari_sp av = avma;
8861 14 : GEN X = gel(v,i);
8862 14 : long c = (itos(qfeval(Q, X)) >> 1) + 1;
8863 14 : gel(V, c) = gc_upto(av, gadd(gel(V, c), gsubstvec_i(P, va, X)));
8864 : }
8865 7 : return gmul2n(V, 1);
8866 : }
8867 :
8868 : GEN
8869 77 : mffromqf(GEN Q, GEN P)
8870 : {
8871 77 : pari_sp av = avma;
8872 : GEN G, Qi, F, D, N, mf, v, gk, chi;
8873 : long m, d, space;
8874 77 : if (typ(Q) != t_MAT) pari_err_TYPE("mffromqf", Q);
8875 77 : if (!RgM_is_ZM(Q) || !qfiseven(Q))
8876 0 : pari_err_TYPE("mffromqf [not integral or even]", Q);
8877 77 : m = lg(Q)-1;
8878 77 : Qi = ZM_inv(Q, &N);
8879 77 : if (!qfiseven(Qi)) N = shifti(N, 1);
8880 77 : d = 0;
8881 77 : if (!P || gequal1(P)) P = NULL;
8882 : else
8883 : {
8884 35 : P = simplify_shallow(P);
8885 35 : if (typ(P) == t_POL)
8886 : {
8887 28 : d = polishomogeneous(P);
8888 28 : if (d < 0) pari_err_TYPE("mffromqf [not homogeneous t_POL]", P);
8889 28 : if (!RgX_isspherical(Qi, P))
8890 7 : pari_err_TYPE("mffromqf [not a spherical t_POL]", P);
8891 : }
8892 : }
8893 63 : gk = uutoQ(m + 2*d, 2);
8894 63 : D = ZM_det(Q);
8895 63 : if (!odd(m)) { if ((m & 3) == 2) D = negi(D); } else D = shifti(D, 1);
8896 63 : space = d > 0 ? mf_CUSP : mf_FULL;
8897 63 : G = znstar0(N,1);
8898 63 : chi = mkvec2(G, znchar_quad(G,D));
8899 63 : mf = mfinit(mkvec3(N, gk, chi), space);
8900 63 : if (odd(d))
8901 : {
8902 7 : F = mftrivial();
8903 7 : v = zerocol(MF_get_dim(mf));
8904 : }
8905 : else
8906 : {
8907 56 : F = c_QF_i(mfsturm(mf), Q, P);
8908 56 : v = mftobasis_i(mf, F);
8909 56 : F = mflinear(mf, v);
8910 : }
8911 63 : return gc_GEN(av, mkvec3(mf, F, v));
8912 : }
8913 :
8914 : /***********************************************************************/
8915 : /* Eisenstein Series */
8916 : /***********************************************************************/
8917 : /* \sigma_{k-1}(\chi,n) */
8918 : static GEN
8919 24192 : sigchi(long k, GEN CHI, long n)
8920 : {
8921 24192 : pari_sp av = avma;
8922 24192 : GEN S = gen_1, D = mydivisorsu(u_ppo(n,mfcharmodulus(CHI)));
8923 24192 : long i, l = lg(D), ord = mfcharorder(CHI), vt = varn(mfcharpol(CHI));
8924 83671 : for (i = 2; i < l; i++) /* skip D[1] = 1 */
8925 : {
8926 59479 : long d = D[i], a = mfcharevalord(CHI, d, ord);
8927 59479 : S = gadd(S, Qab_Czeta(a, ord, powuu(d, k-1), vt));
8928 : }
8929 24192 : return gc_upto(av,S);
8930 : }
8931 :
8932 : /* write n = n0*n1*n2, (n0,N1*N2) = 1, n1 | N1^oo, n2 | N2^oo;
8933 : * return NULL if (n,N1,N2) > 1, else return factoru(n0) */
8934 : static GEN
8935 686175 : sigchi2_dec(long n, long N1, long N2, long *pn1, long *pn2)
8936 : {
8937 686175 : GEN P0, E0, P, E, fa = myfactoru(n);
8938 : long i, j, l;
8939 686175 : *pn1 = 1;
8940 686175 : *pn2 = 1;
8941 686175 : if (N1 == 1 && N2 == 1) return fa;
8942 669088 : P = gel(fa,1); l = lg(P);
8943 669088 : E = gel(fa,2);
8944 669088 : P0 = cgetg(l, t_VECSMALL);
8945 669088 : E0 = cgetg(l, t_VECSMALL);
8946 1553573 : for (i = j = 1; i < l; i++)
8947 : {
8948 989793 : long p = P[i], e = E[i];
8949 989793 : if (N1 % p == 0)
8950 : {
8951 142947 : if (N2 % p == 0) return NULL;
8952 37639 : *pn1 *= upowuu(p,e);
8953 : }
8954 846846 : else if (N2 % p == 0)
8955 129647 : *pn2 *= upowuu(p,e);
8956 717199 : else { P0[j] = p; E0[j] = e; j++; }
8957 : }
8958 563780 : setlg(P0, j);
8959 563780 : setlg(E0, j); return mkvec2(P0,E0);
8960 : }
8961 :
8962 : /* sigma_{k-1}(\chi_1,\chi_2,n), ord multiple of lcm(ord(CHI1),ord(CHI2)) */
8963 : static GEN
8964 608384 : sigchi2(long k, GEN CHI1, GEN CHI2, long n, long ord)
8965 : {
8966 608384 : pari_sp av = avma;
8967 : GEN S, D;
8968 608384 : long i, l, n1, n2, vt, N1 = mfcharmodulus(CHI1), N2 = mfcharmodulus(CHI2);
8969 608384 : D = sigchi2_dec(n, N1, N2, &n1, &n2); if (!D) return gc_const(av, gen_0);
8970 507759 : D = divisorsu_fact(D); l = lg(D);
8971 507759 : vt = varn(mfcharpol(CHI1));
8972 2191609 : for (i = 1, S = gen_0; i < l; i++)
8973 : { /* S += d^(k-1)*chi1(d)*chi2(n/d) */
8974 1683850 : long a, d = n2*D[i], nd = n1*D[l-i]; /* (d,N1)=1; (n/d,N2) = 1 */
8975 1683850 : a = mfcharevalord(CHI1, d, ord) + mfcharevalord(CHI2, nd, ord);
8976 1683850 : if (a >= ord) a -= ord;
8977 1683850 : S = gadd(S, Qab_Czeta(a, ord, powuu(d, k-1), vt));
8978 : }
8979 507759 : return gc_upto(av, S);
8980 : }
8981 :
8982 : /**************************************************************************/
8983 : /** Dirichlet characters with precomputed values **/
8984 : /**************************************************************************/
8985 : /* CHI mfchar */
8986 : static GEN
8987 33985 : mfcharcxinit(GEN CHI, long prec)
8988 : {
8989 33985 : GEN G = gel(CHI,1), chi = gel(CHI,2), z, V;
8990 33985 : GEN v = ncharvecexpo(G, znconrey_normalized(G,chi));
8991 33985 : long n, l = lg(v), o = mfcharorder(CHI);
8992 33985 : V = cgetg(l, t_VEC);
8993 33985 : z = grootsof1(o, prec); /* Mod(t, Phi_o(t)) -> e(1/o) */
8994 480851 : for (n = 1; n < l; n++) gel(V,n) = v[n] < 0? gen_0: gel(z, v[n]+1);
8995 33985 : return mkvecn(6, G, chi, gmfcharorder(CHI), v, V, mfcharpol(CHI));
8996 : }
8997 : /* v a "CHIvec" */
8998 : static long
8999 28601909 : CHIvec_N(GEN v) { return itou(znstar_get_N(gel(v,1))); }
9000 : static GEN
9001 25914 : CHIvec_CHI(GEN v)
9002 25914 : { return mkvec4(gel(v,1), gel(v,2), gel(v,3), gel(v,6)); }
9003 : /* character order */
9004 : static long
9005 66311 : CHIvec_ord(GEN v) { return itou(gel(v,3)); }
9006 : /* character exponents, i.e. t such that chi(n) = e(t) */
9007 : static GEN
9008 626913 : CHIvec_expo(GEN v) { return gel(v,4); }
9009 : /* character values chi(n) */
9010 : static GEN
9011 27670174 : CHIvec_val(GEN v) { return gel(v,5); }
9012 : /* CHI(n) */
9013 : static GEN
9014 27645779 : mychareval(GEN v, long n)
9015 : {
9016 27645779 : long N = CHIvec_N(v), ind = n%N;
9017 27645779 : if (ind <= 0) ind += N;
9018 27645779 : return gel(CHIvec_val(v), ind);
9019 : }
9020 : /* return c such that CHI(n) = e(c / ordz) or -1 if (n,N) > 1 */
9021 : static long
9022 626913 : mycharexpo(GEN v, long n)
9023 : {
9024 626913 : long N = CHIvec_N(v), ind = n%N;
9025 626913 : if (ind <= 0) ind += N;
9026 626913 : return CHIvec_expo(v)[ind];
9027 : }
9028 : /* faster than mfcharparity */
9029 : static long
9030 54754 : CHIvec_parity(GEN v) { return mycharexpo(v,-1) ? -1: 1; }
9031 : /**************************************************************************/
9032 :
9033 : static ulong
9034 77791 : sigchi2_Fl(long k, GEN CHI1vec, GEN CHI2vec, long n, GEN vz, ulong p)
9035 : {
9036 77791 : pari_sp av = avma;
9037 77791 : long ordz = lg(vz)-2, i, l, n1, n2;
9038 77791 : ulong S = 0;
9039 77791 : GEN D = sigchi2_dec(n, CHIvec_N(CHI1vec), CHIvec_N(CHI2vec), &n1, &n2);
9040 77791 : if (!D) return gc_ulong(av,S);
9041 73108 : D = divisorsu_fact(D);
9042 73108 : l = lg(D);
9043 276444 : for (i = 1; i < l; i++)
9044 : { /* S += d^(k-1)*chi1(d)*chi2(n/d) */
9045 203336 : long a, d = n2*D[i], nd = n1*D[l-i]; /* (d,N1)=1, (n/d,N2)=1 */
9046 203336 : a = mycharexpo(CHI2vec, nd) + mycharexpo(CHI1vec, d);
9047 203336 : if (a >= ordz) a -= ordz;
9048 203336 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, Fl_powu(d,k-1,p), p), p);
9049 : }
9050 73108 : return gc_ulong(av,S);
9051 : }
9052 :
9053 : /**********************************************************************/
9054 : /* Fourier expansions of Eisenstein series */
9055 : /**********************************************************************/
9056 : /* L(CHI_t,0) / 2, CHI_t(n) = CHI(n)(t/n) as a character modulo N*t,
9057 : * order(CHI) | ord != 0 */
9058 : static GEN
9059 2618 : charLFwt1(long N, GEN CHI, long ord, long t)
9060 : {
9061 : GEN S;
9062 : long r, vt;
9063 :
9064 2618 : if (N == 1 && t == 1) return mkfrac(gen_m1,stoi(4));
9065 2618 : S = gen_0; vt = varn(mfcharpol(CHI));
9066 295435 : for (r = 1; r < N; r++)
9067 : { /* S += r*chi(r) */
9068 : long a, c;
9069 292817 : if (ugcd(N,r) != 1) continue;
9070 233310 : a = mfcharevalord(CHI,r,ord);
9071 233310 : c = (t != 1 && kross(t, r) < 0)? -r: r;
9072 233310 : S = gadd(S, Qab_Czeta(a, ord, stoi(c), vt));
9073 : }
9074 2618 : return gdivgs(S, -2*N);
9075 : }
9076 : /* L(CHI,0) / 2, mod p */
9077 : static ulong
9078 2002 : charLFwt1_Fl(GEN CHIvec, GEN vz, ulong p)
9079 : {
9080 2002 : long r, m = CHIvec_N(CHIvec);
9081 : ulong S;
9082 2002 : if (m == 1) return Rg_to_Fl(mkfrac(gen_m1,stoi(4)), p);
9083 2002 : S = 0;
9084 95977 : for (r = 1; r < m; r++)
9085 : { /* S += r*chi(r) */
9086 93975 : long a = mycharexpo(CHIvec,r);
9087 93975 : if (a < 0) continue;
9088 91616 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, r, p), p);
9089 : }
9090 2002 : return Fl_div(Fl_neg(S,p), 2*m, p);
9091 : }
9092 : /* L(CHI_t,1-k) / 2, CHI_t(n) = CHI(n) * (t/n), order(CHI) | ord != 0;
9093 : * assume conductor of CHI_t divides N */
9094 : static GEN
9095 4543 : charLFwtk(long N, long k, GEN CHI, long ord, long t)
9096 : {
9097 : GEN S, P, dS;
9098 : long r, vt;
9099 :
9100 4543 : if (k == 1) return charLFwt1(N, CHI, ord, t);
9101 1925 : if (N == 1 && t == 1) return gdivgs(bernfrac(k),-2*k);
9102 1169 : vt = varn(mfcharpol(CHI));
9103 1169 : P = bern_init(N, k, &dS);
9104 1169 : dS = mul_denom(dS, stoi(-2*N*k));
9105 17598 : for (r = 1, S = gen_0; r < N; r++)
9106 : { /* S += P(r)*chi(r) */
9107 : long a;
9108 : GEN C;
9109 16429 : if (ugcd(r,N) != 1) continue;
9110 13832 : a = mfcharevalord(CHI,r,ord);
9111 13832 : C = ZX_Z_eval(P, utoi(r));
9112 13832 : if (t != 1 && kross(t, r) < 0) C = gneg(C);
9113 13832 : S = gadd(S, Qab_Czeta(a, ord, C, vt));
9114 : }
9115 1169 : return gdiv(S, dS);
9116 : }
9117 : /* L(CHI,1-k) / 2, mod p */
9118 : static ulong
9119 3227 : charLFwtk_Fl(long k, GEN CHIvec, GEN vz, ulong p)
9120 : {
9121 : GEN P, dS;
9122 : long r, m;
9123 : ulong S, d;
9124 3227 : if (k == 1) return charLFwt1_Fl(CHIvec, vz, p);
9125 1225 : m = CHIvec_N(CHIvec);
9126 1225 : if (m == 1) return Rg_to_Fl(gdivgs(bernfrac(k),-2*k), p);
9127 819 : P = ZX_to_Flx(bern_init(m, k, &dS), p);
9128 20167 : for (r = 1, S = 0; r < m; r++)
9129 : { /* S += P(r)*chi(r) */
9130 19348 : long a = mycharexpo(CHIvec,r);
9131 19348 : if (a < 0) continue;
9132 18088 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, Flx_eval(P,r,p), p), p);
9133 : }
9134 819 : d = (2 * k * m) % p; if (dS) d = Fl_mul(d, umodiu(dS, p), p);
9135 819 : return Fl_div(Fl_neg(S,p), d, p);
9136 : }
9137 :
9138 : static GEN
9139 8351 : mfeisenstein2_0(long k, GEN CHI1, GEN CHI2, long ord)
9140 : {
9141 8351 : long N1 = mfcharmodulus(CHI1), N2 = mfcharmodulus(CHI2);
9142 8351 : if (k == 1 && N1 == 1) return charLFwtk(N2, 1, CHI2, ord, 1);
9143 5740 : if (N2 == 1) return charLFwtk(N1, k, CHI1, ord, 1);
9144 4025 : return gen_0;
9145 : }
9146 : static ulong
9147 5054 : mfeisenstein2_0_Fl(long k, GEN CHI1vec, GEN CHI2vec, GEN vz, ulong p)
9148 : {
9149 5054 : if (k == 1 && CHIvec_N(CHI1vec) == 1)
9150 2002 : return charLFwtk_Fl(k, CHI2vec, vz, p);
9151 3052 : else if (CHIvec_N(CHI2vec) == 1)
9152 1225 : return charLFwtk_Fl(k, CHI1vec, vz, p);
9153 1827 : else return 0;
9154 : }
9155 : static GEN
9156 133 : NK_eisen2(long k, GEN CHI1, GEN CHI2, long ord)
9157 : {
9158 133 : long o, N = mfcharmodulus(CHI1)*mfcharmodulus(CHI2);
9159 133 : GEN CHI = mfcharmul(CHI1, CHI2);
9160 133 : o = mfcharorder(CHI);
9161 133 : if ((ord & 3) == 2) ord >>= 1;
9162 133 : if ((o & 3) == 2) o >>= 1;
9163 133 : if (ord != o) pari_err_IMPL("mfeisenstein for these characters");
9164 126 : return mkNK(N, k, CHI);
9165 : }
9166 : static GEN
9167 371 : mfeisenstein_i(long k, GEN CHI1, GEN CHI2)
9168 : {
9169 371 : long s = 1, ord, vt;
9170 : GEN E0, NK, vchi, T;
9171 371 : if (CHI2) { CHI2 = get_mfchar(CHI2); if (mfcharparity(CHI2) < 0) s = -s; }
9172 371 : if (CHI1) { CHI1 = get_mfchar(CHI1); if (mfcharparity(CHI1) < 0) s = -s; }
9173 357 : if (s != m1pk(k)) return mftrivial();
9174 336 : if (!CHI1) CHI1 = mfchartrivial();
9175 336 : if (!CHI2)
9176 : { /* E_k(chi1) */
9177 203 : vt = varn(mfcharpol(CHI1));
9178 203 : ord = mfcharorder(CHI1);
9179 203 : NK = mkNK(mfcharmodulus(CHI1), k, CHI1);
9180 203 : E0 = charLFwtk(mfcharmodulus(CHI1), k, CHI1, ord, 1);
9181 203 : vchi = mkvec3(E0, mkvec(mfcharpol(CHI1)), CHI1);
9182 203 : return tag(t_MF_EISEN, NK, vchi);
9183 : }
9184 : /* E_k(chi1,chi2) */
9185 133 : vt = varn(mfcharpol(CHI1));
9186 133 : ord = ulcm(mfcharorder(CHI1), mfcharorder(CHI2));
9187 133 : NK = NK_eisen2(k, CHI1, CHI2, ord);
9188 126 : E0 = mfeisenstein2_0(k, CHI1, CHI2, ord);
9189 126 : T = mkvec(polcyclo(ord, vt));
9190 126 : vchi = mkvec4(E0, T, CHI1, CHI2);
9191 126 : return tag2(t_MF_EISEN, NK, vchi, mkvecsmall2(ord,0));
9192 : }
9193 : GEN
9194 371 : mfeisenstein(long k, GEN CHI1, GEN CHI2)
9195 : {
9196 371 : pari_sp av = avma;
9197 371 : if (k < 1) pari_err_DOMAIN("mfeisenstein", "k", "<", gen_1, stoi(k));
9198 371 : return gc_GEN(av, mfeisenstein_i(k, CHI1, CHI2));
9199 : }
9200 :
9201 : static GEN
9202 2639 : mfeisenstein2all(long N0, GEN NK, long k, GEN CHI1, GEN CHI2, GEN T, long o)
9203 : {
9204 2639 : GEN E, E0 = mfeisenstein2_0(k, CHI1,CHI2, o), vchi = mkvec4(E0, T, CHI1,CHI2);
9205 2639 : long j, d = (lg(T)==4)? itou(gmael(T,3,1)): 1;
9206 2639 : E = cgetg(d+1, t_VEC);
9207 5397 : for (j=1; j<=d; j++) gel(E,j) = tag2(t_MF_EISEN, NK,vchi,mkvecsmall2(o,j-1));
9208 2639 : return mfbdall(E, N0 / mf_get_N(gel(E,1)));
9209 : }
9210 :
9211 : /* list of characters on G = (Z/NZ)^*, v[i] = NULL if (i,N) > 1, else
9212 : * the conductor of Conrey label i, [conductor, primitive char].
9213 : * Trivial chi (label 1) comes first */
9214 : static GEN
9215 1169 : zncharsG(GEN G)
9216 : {
9217 1169 : long i, l, N = itou(znstar_get_N(G));
9218 : GEN vCHI, V;
9219 1169 : if (N == 1) return mkvec2(gen_1,cgetg(1,t_COL));
9220 1169 : vCHI = const_vec(N,NULL);
9221 1169 : V = cyc2elts(znstar_get_conreycyc(G));
9222 1169 : l = lg(V);
9223 207739 : for (i = 1; i < l; i++)
9224 : {
9225 206570 : GEN chi0, chi = zc_to_ZC(gel(V,i)), n, F;
9226 206570 : F = znconreyconductor(G, chi, &chi0);
9227 206570 : if (typ(F) != t_INT) F = gel(F,1);
9228 206570 : n = znconreyexp(G, chi);
9229 206570 : gel(vCHI, itos(n)) = mkvec2(chi0, F);
9230 : }
9231 1169 : return vCHI;
9232 : }
9233 :
9234 : /* CHI primitive, f(CHI) | N. Return pairs (CHI1,CHI2) both primitive
9235 : * such that f(CHI1)*f(CHI2) | N and CHI1 * CHI2 = CHI;
9236 : * if k = 1, CHI1 is even; if k = 2, omit (1,1) if CHI = 1 */
9237 : static GEN
9238 1421 : mfeisensteinbasis_i(long N0, long k, GEN CHI)
9239 : {
9240 1421 : GEN G = gel(CHI,1), chi = gel(CHI,2), vT = const_vec(myeulerphiu(N0), NULL);
9241 1421 : GEN CHI0, GN, chiN, Lchi, LG, V, RES, NK, T, C = mfcharpol(CHI);
9242 1421 : long i, j, l, n, n1, N, ord = mfcharorder(CHI);
9243 1421 : long F = mfcharmodulus(CHI), vt = varn(mfcharpol(CHI));
9244 :
9245 1421 : CHI0 = (F == 1)? CHI: mfchartrivial();
9246 1421 : j = 1; RES = cgetg(N0+1, t_VEC);
9247 1421 : T = gel(vT,ord) = Qab_trace_init(ord, ord, C, C);
9248 1421 : if (F != 1 || k != 2)
9249 : { /* N1 = 1 */
9250 1267 : NK = mkNK(F, k, CHI);
9251 1267 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI0, CHI, T, ord);
9252 1267 : if (F != 1 && k != 1)
9253 329 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI, CHI0, T, ord);
9254 : }
9255 1421 : if (N0 == 1) { setlg(RES,j); return RES; }
9256 1330 : GN = G; chiN = chi;
9257 1330 : if (F == N0) N = N0;
9258 : else
9259 : {
9260 728 : GEN faN = myfactoru(N0), P = gel(faN,1), E = gel(faN,2);
9261 728 : long lP = lg(P);
9262 1876 : for (i = N = 1; i < lP; i++)
9263 : {
9264 1148 : long p = P[i];
9265 1148 : N *= upowuu(p, maxuu(E[i]/2, z_lval(F,p)));
9266 : }
9267 728 : if ((N & 3) == 2) N >>= 1;
9268 728 : if (N == 1) { setlg(RES,j); return RES; }
9269 567 : if (F != N)
9270 : {
9271 133 : GN = znstar0(utoipos(N),1);
9272 133 : chiN = zncharinduce(G, chi, GN);
9273 : }
9274 : }
9275 1169 : LG = const_vec(N, NULL); /* LG[d] = znstar(d,1) or NULL */
9276 1169 : gel(LG,1) = gel(CHI0,1);
9277 1169 : gel(LG,F) = G;
9278 1169 : gel(LG,N) = GN;
9279 1169 : Lchi = coprimes_zv(N);
9280 1169 : n = itou(znconreyexp(GN,chiN));
9281 1169 : V = zncharsG(GN); l = lg(V);
9282 263305 : for (n1 = 2; n1 < l; n1++) /* skip 1 (trivial char) */
9283 : {
9284 262136 : GEN v = gel(V,n1), w, chi1, chi2, G1, G2, CHI1, CHI2;
9285 : long N12, N1, N2, no, o12, t, m;
9286 262136 : if (!Lchi[n1] || n1 == n) continue; /* skip trivial chi2 */
9287 204197 : chi1 = gel(v,1); N1 = itou(gel(v,2)); /* conductor of chi1 */
9288 204197 : w = gel(V, Fl_div(n,n1,N));
9289 204197 : chi2 = gel(w,1); N2 = itou(gel(w,2)); /* conductor of chi2 */
9290 204197 : N12 = N1 * N2;
9291 204197 : if (N0 % N12) continue;
9292 :
9293 1771 : G1 = gel(LG,N1); if (!G1) gel(LG,N1) = G1 = znstar0(utoipos(N1), 1);
9294 1771 : if (k == 1 && zncharisodd(G1,chi1)) continue;
9295 1043 : G2 = gel(LG,N2); if (!G2) gel(LG,N2) = G2 = znstar0(utoipos(N2), 1);
9296 1043 : CHI1 = mfcharGL(G1, chi1);
9297 1043 : CHI2 = mfcharGL(G2, chi2);
9298 1043 : o12 = ulcm(mfcharorder(CHI1), mfcharorder(CHI2));
9299 : /* remove Galois orbit: same trace */
9300 1043 : no = Fl_powu(n1, ord, N);
9301 1414 : for (t = 1+ord, m = n1; t <= o12; t += ord)
9302 : { /* m <-> CHI1^t, if t in Gal(Q(chi1,chi2)/Q), omit (CHI1^t,CHI2^t) */
9303 371 : m = Fl_mul(m, no, N); if (!m) break;
9304 371 : if (ugcd(t, o12) == 1) Lchi[m] = 0;
9305 : }
9306 1043 : T = gel(vT,o12);
9307 1043 : if (!T) T = gel(vT,o12) = Qab_trace_init(o12, ord, polcyclo(o12,vt), C);
9308 1043 : NK = mkNK(N12, k, CHI);
9309 1043 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI1, CHI2, T, o12);
9310 : }
9311 1169 : setlg(RES,j); return RES;
9312 : }
9313 :
9314 : static GEN
9315 721 : mfbd_E2(GEN E2, long d, GEN CHI)
9316 : {
9317 721 : GEN E2d = mfbd_i(E2, d);
9318 721 : GEN F = mkvec2(E2, E2d), L = mkvec2(gen_1, utoineg(d));
9319 : /* cannot use mflinear_i: E2 and E2d do not have the same level */
9320 721 : return tag3(t_MF_LINEAR, mkNK(d,2,CHI), F, L, gen_1);
9321 : }
9322 : /* C-basis of E_k(Gamma_0(N),chi). If k = 1, the first basis element must not
9323 : * vanish at oo [used in mf1basis]. Here E_1(CHI), whose q^0 coefficient
9324 : * does not vanish (since L(CHI,0) does not) *if* CHI is not trivial; which
9325 : * must be the case in weight 1.
9326 : *
9327 : * (k>=3): In weight k >= 3, basis is B(d) E(CHI1,(CHI/CHI1)_prim), where
9328 : * CHI1 is primitive modulo N1, and if N2 is the conductor of CHI/CHI1
9329 : * then d*N1*N2 | N.
9330 : * (k=2): In weight k=2, same if CHI is nontrivial. If CHI is trivial, must
9331 : * not take CHI1 trivial, and must add E_2(tau)-dE_2(d tau)), where
9332 : * d|N, d > 1.
9333 : * (k=1): In weight k=1, same as k >= 3 except that we restrict to CHI1 even */
9334 : static GEN
9335 1449 : mfeisensteinbasis(long N, long k, GEN CHI)
9336 : {
9337 : long i, F;
9338 : GEN L;
9339 1449 : if (badchar(N, k, CHI)) return cgetg(1, t_VEC);
9340 1449 : if (k == 0) return mfcharistrivial(CHI)? mkvec(mf1()): cgetg(1, t_VEC);
9341 1421 : CHI = mfchartoprimitive(CHI, &F);
9342 1421 : L = mfeisensteinbasis_i(N, k, CHI);
9343 1421 : if (F == 1 && k == 2)
9344 : {
9345 154 : GEN v, E2 = mfeisenstein(2, NULL, NULL), D = mydivisorsu(N);
9346 154 : long nD = lg(D)-1;
9347 154 : v = cgetg(nD, t_VEC); L = vec_append(L,v);
9348 868 : for (i = 1; i < nD; i++) gel(v,i) = mfbd_E2(E2, D[i+1], CHI);
9349 : }
9350 1421 : return lg(L) == 1? L: shallowconcat1(L);
9351 : }
9352 :
9353 : static GEN
9354 77 : not_in_space(GEN F, long flag)
9355 : {
9356 77 : if (!flag) err_space(F);
9357 70 : return cgetg(1, t_COL);
9358 : }
9359 : /* when flag set, no error */
9360 : GEN
9361 1029 : mftobasis(GEN mf, GEN F, long flag)
9362 : {
9363 1029 : pari_sp av2, av = avma;
9364 : GEN G, v, y, gk;
9365 1029 : long N, B, ismf = checkmf_i(F);
9366 :
9367 1029 : mf = checkMF(mf);
9368 1029 : if (ismf)
9369 : {
9370 938 : if (mfistrivial(F)) return zerocol(MF_get_dim(mf));
9371 931 : if (!mf_same_k(mf, F) || !mf_same_CHI(mf, F)) return not_in_space(F, flag);
9372 : }
9373 980 : N = MF_get_N(mf);
9374 980 : gk = MF_get_gk(mf);
9375 980 : if (ismf)
9376 : {
9377 889 : long NF = mf_get_N(F);
9378 889 : B = maxuu(mfsturmNgk(NF,gk), mfsturmNgk(N,gk)) + 1;
9379 889 : v = mfcoefs_i(F,B,1);
9380 : }
9381 : else
9382 : {
9383 91 : B = mfsturmNgk(N, gk) + 1;
9384 91 : switch(typ(F))
9385 : { /* F(0),...,F(lg(v)-2) */
9386 63 : case t_SER: v = sertocol(F); settyp(v,t_VEC); break;
9387 14 : case t_VEC: v = F; break;
9388 7 : case t_COL: v = shallowtrans(F); break;
9389 7 : default: pari_err_TYPE("mftobasis",F);
9390 : v = NULL;/*LCOV_EXCL_LINE*/
9391 : }
9392 84 : if (flag) B = minss(B, lg(v)-2);
9393 : }
9394 973 : y = mftobasis_i(mf, v);
9395 973 : if (typ(y) == t_VEC)
9396 : {
9397 21 : if (flag) return gc_GEN(av, y);
9398 0 : pari_err(e_MISC, "not enough coefficients in mftobasis");
9399 : }
9400 952 : av2 = avma;
9401 952 : if (MF_get_space(mf) == mf_FULL || mfsturm(mf)+1 == B) return y;
9402 476 : G = mflinear(mf, y);
9403 476 : if (!gequal(v, mfcoefs_i(G, lg(v)-2,1))) y = NULL;
9404 476 : if (!y) { set_avma(av); return not_in_space(F, flag); }
9405 441 : set_avma(av2); return gc_upto(av, y);
9406 : }
9407 :
9408 : /* assume N > 0; first cusp is always 0 */
9409 : static GEN
9410 49 : mfcusps_i(long N)
9411 : {
9412 : long i, c, l;
9413 : GEN D, v;
9414 :
9415 49 : if (N == 1) return mkvec(gen_0);
9416 49 : D = mydivisorsu(N); l = lg(D); /* left on stack */
9417 49 : c = mfnumcuspsu_fact(myfactoru(N));
9418 49 : v = cgetg(c + 1, t_VEC);
9419 350 : for (i = c = 1; i < l; i++)
9420 : {
9421 301 : long C = D[i], NC = D[l-i], lima = ugcd(C, NC), A0, A;
9422 889 : for (A0 = 0; A0 < lima; A0++)
9423 588 : if (ugcd(A0, lima) == 1)
9424 : {
9425 539 : A = A0; while (ugcd(A,C) > 1) A += lima;
9426 392 : gel(v, c++) = uutoQ(A, C);
9427 : }
9428 : }
9429 49 : return v;
9430 : }
9431 : /* List of cusps of Gamma_0(N) */
9432 : GEN
9433 28 : mfcusps(GEN gN)
9434 : {
9435 : long N;
9436 : GEN mf;
9437 28 : if (typ(gN) == t_INT) N = itos(gN);
9438 14 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
9439 0 : else { pari_err_TYPE("mfcusps", gN); N = 0; }
9440 28 : if (N <= 0) pari_err_DOMAIN("mfcusps", "N", "<=", gen_0, stoi(N));
9441 28 : return mfcusps_i(N);
9442 : }
9443 :
9444 : long
9445 315 : mfcuspisregular(GEN NK, GEN cusp)
9446 : {
9447 : long v, N, dk, nk, t, o;
9448 : GEN mf, CHI, go, A, C, g, c, d;
9449 315 : if ((mf = checkMF_i(NK)))
9450 : {
9451 49 : GEN gk = MF_get_gk(mf);
9452 49 : N = MF_get_N(mf);
9453 49 : CHI = MF_get_CHI(mf);
9454 49 : Qtoss(gk, &nk, &dk);
9455 : }
9456 : else
9457 266 : checkNK2(NK, &N, &nk, &dk, &CHI, 0);
9458 315 : if (typ(cusp) == t_INFINITY) return 1;
9459 315 : if (typ(cusp) == t_FRAC) { A = gel(cusp,1); C = gel(cusp,2); }
9460 28 : else { A = cusp; C = gen_1; }
9461 315 : g = diviuexact(mului(N,C), ugcd(N, Fl_sqr(umodiu(C,N), N)));
9462 315 : c = mulii(negi(C),g);
9463 315 : d = addiu(mulii(A,g), 1);
9464 315 : if (!CHI) return 1;
9465 315 : go = gmfcharorder(CHI);
9466 315 : v = vali(go); if (v < 2) go = shifti(go, 2-v);
9467 315 : t = itou( znchareval(gel(CHI,1), gel(CHI,2), d, go) );
9468 315 : if (dk == 1) return t == 0;
9469 154 : o = itou(go);
9470 154 : if (kronecker(c,d) < 0) t = Fl_add(t, o/2, o);
9471 154 : if (Mod4(d) == 1) return t == 0;
9472 14 : t = Fl_sub(t, Fl_mul(o/4, nk, o), o);
9473 14 : return t == 0;
9474 : }
9475 :
9476 : /* Some useful closures */
9477 :
9478 : /* sum_{d|n} d^k */
9479 : static GEN
9480 48020 : mysumdivku(ulong n, ulong k)
9481 : {
9482 48020 : GEN fa = myfactoru(n);
9483 48020 : return k == 1? usumdiv_fact(fa): usumdivk_fact(fa,k);
9484 : }
9485 : static GEN
9486 882 : c_Ek(long n, long d, GEN F)
9487 : {
9488 882 : GEN E = cgetg(n + 2, t_VEC), C = gel(F,2);
9489 882 : long i, k = mf_get_k(F);
9490 882 : gel (E, 1) = gen_1;
9491 26264 : for (i = 1; i <= n; i++)
9492 : {
9493 25382 : pari_sp av = avma;
9494 25382 : gel(E, i+1) = gc_upto(av, gmul(C, mysumdivku(i*d, k-1)));
9495 : }
9496 882 : return E;
9497 : }
9498 :
9499 : GEN
9500 406 : mfEk(long k)
9501 : {
9502 406 : pari_sp av = avma;
9503 : GEN E0, NK;
9504 406 : if (k < 0 || odd(k)) pari_err_TYPE("mfEk [incorrect k]", stoi(k));
9505 406 : if (!k) return mf1();
9506 399 : E0 = gdivsg(-2*k, bernfrac(k));
9507 399 : NK = mkNK(1,k,mfchartrivial());
9508 399 : return gc_GEN(av, tag(t_MF_Ek, NK, E0));
9509 : }
9510 :
9511 : GEN
9512 56 : mfDelta(void)
9513 : {
9514 56 : pari_sp av = avma;
9515 56 : return gc_GEN(av, tag0(t_MF_DELTA, mkNK(1,12,mfchartrivial())));
9516 : }
9517 :
9518 : GEN
9519 798 : mfTheta(GEN psi)
9520 : {
9521 798 : pari_sp av = avma;
9522 : GEN N, gk, psi2;
9523 : long par;
9524 798 : if (!psi) { psi = mfchartrivial(); N = utoipos(4); par = 1; }
9525 : else
9526 : {
9527 : long FC;
9528 21 : psi = get_mfchar(psi);
9529 21 : FC = mfcharconductor(psi);
9530 21 : if (mfcharmodulus(psi) != FC)
9531 0 : pari_err_TYPE("mfTheta [nonprimitive character]", psi);
9532 21 : par = mfcharparity(psi);
9533 21 : N = shifti(sqru(FC),2);
9534 : }
9535 798 : if (par > 0) { gk = ghalf; psi2 = psi; }
9536 7 : else { gk = gsubsg(2, ghalf); psi2 = mfcharmul(psi, get_mfchar(stoi(-4))); }
9537 798 : return gc_GEN(av, tag(t_MF_THETA, mkgNK(N, gk, psi2, pol_x(1)), psi));
9538 : }
9539 :
9540 : /* Output 0 if not desired eta product: if flag=0 (default) require
9541 : * holomorphic at cusps. If flag set, accept meromorphic, but sill in some
9542 : * modular function space */
9543 : GEN
9544 210 : mffrometaquo(GEN eta, long flag)
9545 : {
9546 210 : pari_sp av = avma;
9547 : GEN NK, N, k, BR, P;
9548 210 : long v, cusp = 0;
9549 210 : if (!etaquotype(&eta, &N,&k,&P, &v, NULL, flag? NULL: &cusp) || cusp < 0)
9550 14 : return gc_const(av, gen_0);
9551 196 : if (lg(gel(eta,1)) == 1) { set_avma(av); return mf1(); }
9552 189 : BR = mkvec2(ZV_to_zv(gel(eta,1)), ZV_to_zv(gel(eta,2)));
9553 189 : if (v < 0) v = 0;
9554 189 : NK = mkgNK(N, k, get_mfchar(P), pol_x(1));
9555 189 : return gc_GEN(av, tag2(t_MF_ETAQUO, NK, BR, utoi(v)));
9556 : }
9557 :
9558 : /* Q^(-r) */
9559 : static GEN
9560 375 : RgXn_negpow(GEN Q, long r, long L)
9561 : {
9562 375 : if (r < 0) r = -r; else Q = RgXn_inv_i(Q, L);
9563 375 : if (r != 1) Q = RgXn_powu_i(Q, r, L);
9564 375 : return Q;
9565 : }
9566 : /* flag same as in mffrometaquo: if set, accept meromorphic. */
9567 : static GEN
9568 49 : mfisetaquo_i(GEN F, long flag)
9569 : {
9570 : GEN gk, P, E, M, S, G, CHI, v, w;
9571 : long b, l, L, N, vS, m, j;
9572 49 : const long bextra = 10;
9573 :
9574 49 : if (!checkmf_i(F)) pari_err_TYPE("mfisetaquo",F);
9575 49 : CHI = mf_get_CHI(F); if (mfcharorder(CHI) > 2) return NULL;
9576 49 : N = mf_get_N(F);
9577 49 : gk = mf_get_gk(F);
9578 49 : b = mfsturmNgk(N, gk);
9579 49 : L = maxss(N, b) + bextra;
9580 49 : S = mfcoefs_i(F, L, 1);
9581 49 : if (!RgV_is_ZV(S)) return NULL;
9582 889 : for (vS = 1; vS <= L+1; vS++)
9583 889 : if (signe(gel(S,vS))) break;
9584 49 : vS--;
9585 49 : if (vS >= bextra - 1) { L += vS; S = mfcoefs_i(F, L, 1); }
9586 49 : if (vS) { S = vecslice(S, vS+1, L+1); L -= vS; }
9587 49 : S = RgV_to_RgX(S, 0); l = lg(S)-2;
9588 49 : P = cgetg(l, t_COL);
9589 49 : E = cgetg(l, t_COL); w = v = gen_0; /* w = weight, v = valuation */
9590 1908 : for (m = j = 1; m+2 < lg(S); m++)
9591 : {
9592 1866 : GEN c = gel(S,m+2);
9593 : long r;
9594 1866 : if (is_bigint(c)) return NULL;
9595 1859 : r = -itos(c);
9596 1859 : if (r)
9597 : {
9598 375 : S = ZXn_mul(S, RgXn_negpow(eta_ZXn(m, L), r, L), L);
9599 375 : gel(P,j) = utoipos(m);
9600 375 : gel(E,j) = stoi(r);
9601 375 : v = addmuliu(v, gel(E,j), m);
9602 375 : w = addis(w, r);
9603 375 : j++;
9604 : }
9605 : }
9606 42 : if (!equalii(w, gmul2n(gk, 1)) || (!flag && !equalii(v, muluu(24,vS))))
9607 7 : return NULL;
9608 35 : setlg(P, j);
9609 35 : setlg(E, j); M = mkmat2(P, E); G = mffrometaquo(M, flag);
9610 35 : return (typ(G) != t_INT
9611 35 : && (mfsturmmf(G) <= b + bextra || mfisequal(F, G, b)))? M: NULL;
9612 : }
9613 : GEN
9614 49 : mfisetaquo(GEN F, long flag)
9615 : {
9616 49 : pari_sp av = avma;
9617 49 : GEN M = mfisetaquo_i(F, flag);
9618 49 : return M? gc_GEN(av, M): gc_const(av, gen_0);
9619 : }
9620 :
9621 : #if 0
9622 : /* number of primitive characters modulo N */
9623 : static ulong
9624 : numprimchars(ulong N)
9625 : {
9626 : GEN fa, P, E;
9627 : long i, l;
9628 : ulong n;
9629 : if ((N & 3) == 2) return 0;
9630 : fa = myfactoru(N);
9631 : P = gel(fa,1); l = lg(P);
9632 : E = gel(fa,2);
9633 : for (i = n = 1; i < l; i++)
9634 : {
9635 : ulong p = P[i], e = E[i];
9636 : if (e == 2) n *= p-2; else n *= (p-1)*(p-1)*upowuu(p,e-2);
9637 : }
9638 : return n;
9639 : }
9640 : #endif
9641 :
9642 : /* Space generated by products of two Eisenstein series */
9643 :
9644 : static int
9645 74431 : cmp_small_priority(void *E, GEN a, GEN b)
9646 : {
9647 74431 : GEN prio = (GEN)E;
9648 74431 : return cmpss(prio[(long)a], prio[(long)b]);
9649 : }
9650 : static long
9651 1302 : znstar_get_expo(GEN G) { return itou(cyc_get_expo(znstar_get_cyc(G))); }
9652 :
9653 : /* Return [vchi, bymod, vG]:
9654 : * vG[f] = znstar(f,1) for f a conductor of (at least) a char mod N; else NULL
9655 : * bymod[f] = vecsmall of conrey indexes of chars modulo f | N; else NULL
9656 : * vchi[n] = a list of CHIvec [G0,chi0,o,ncharvecexpo(G0,nchi0),...]:
9657 : * chi0 = primitive char attached to Conrey Mod(n,N)
9658 : * (resp. NULL if (n,N) > 1) */
9659 : static GEN
9660 651 : charsmodN(long N)
9661 : {
9662 651 : GEN D, G, prio, phio, dummy = cgetg(1,t_VEC);
9663 651 : GEN vP, vG = const_vec(N,NULL), vCHI = const_vec(N,NULL);
9664 651 : GEN bymod = const_vec(N,NULL);
9665 651 : long pn, i, l, vt = fetch_user_var("t");
9666 651 : D = mydivisorsu(N); l = lg(D);
9667 3941 : for (i = 1; i < l; i++)
9668 3290 : gel(bymod, D[i]) = vecsmalltrunc_init(myeulerphiu(D[i])+1);
9669 651 : gel(vG,N) = G = znstar0(utoipos(N),1);
9670 651 : pn = znstar_get_expo(G); /* exponent(Z/NZ)^* */
9671 651 : vP = const_vec(pn,NULL);
9672 27069 : for (i = 1; i <= N; i++)
9673 : {
9674 : GEN P, gF, G0, chi0, nchi0, chi, v, go;
9675 : long j, F, o;
9676 26418 : if (ugcd(i,N) != 1) continue;
9677 14147 : chi = znconreylog(G, utoipos(i));
9678 14147 : gF = znconreyconductor(G, chi, &chi0);
9679 14147 : F = (typ(gF) == t_INT)? itou(gF): itou(gel(gF,1));
9680 14147 : G0 = gel(vG, F); if (!G0) G0 = gel(vG,F) = znstar0(gF, 1);
9681 14147 : nchi0 = znconreylog_normalize(G0,chi0);
9682 14147 : go = gel(nchi0,1); o = itou(go); /* order(chi0) */
9683 14147 : v = ncharvecexpo(G0, nchi0);
9684 14147 : if (!equaliu(go, pn)) v = zv_z_mul(v, pn / o);
9685 14147 : P = gel(vP, o); if (!P) P = gel(vP,o) = polcyclo(o,vt);
9686 : /* mfcharcxinit with dummy complex powers */
9687 14147 : gel(vCHI,i) = mkvecn(6, G0, chi0, go, v, dummy, P);
9688 14147 : D = mydivisorsu(N / F); l = lg(D);
9689 40565 : for (j = 1; j < l; j++) vecsmalltrunc_append(gel(bymod, F*D[j]), i);
9690 : }
9691 651 : phio = zero_zv(pn); l = lg(vCHI); prio = cgetg(l, t_VEC);
9692 27069 : for (i = 1; i < l; i++)
9693 : {
9694 26418 : GEN CHI = gel(vCHI,i);
9695 : long o;
9696 26418 : if (!CHI) continue;
9697 14147 : o = CHIvec_ord(CHI);
9698 14147 : if (!phio[o]) phio[o] = myeulerphiu(o);
9699 14147 : prio[i] = phio[o];
9700 : }
9701 651 : l = lg(bymod);
9702 : /* sort characters by increasing value of phi(order) */
9703 27069 : for (i = 1; i < l; i++)
9704 : {
9705 26418 : GEN z = gel(bymod,i);
9706 26418 : if (z) gen_sort_inplace(z, (void*)prio, &cmp_small_priority, NULL);
9707 : }
9708 651 : return mkvec3(vCHI, bymod, vG);
9709 : }
9710 :
9711 : static GEN
9712 5586 : mfeisenstein2pure(long k, GEN CHI1, GEN CHI2, long ord, GEN P, long lim)
9713 : {
9714 5586 : GEN c, V = cgetg(lim+2, t_COL);
9715 : long n;
9716 5586 : c = mfeisenstein2_0(k, CHI1, CHI2, ord);
9717 5586 : if (P) c = grem(c, P);
9718 5586 : gel(V,1) = c;
9719 113400 : for (n=1; n <= lim; n++)
9720 : {
9721 107814 : c = sigchi2(k, CHI1, CHI2, n, ord);
9722 107814 : if (P) c = grem(c, P);
9723 107814 : gel(V,n+1) = c;
9724 : }
9725 5586 : return V;
9726 : }
9727 : static GEN
9728 5054 : mfeisenstein2pure_Fl(long k, GEN CHI1vec, GEN CHI2vec, GEN vz, ulong p, long lim)
9729 : {
9730 5054 : GEN V = cgetg(lim+2, t_VECSMALL);
9731 : long n;
9732 5054 : V[1] = mfeisenstein2_0_Fl(k, CHI1vec, CHI2vec, vz, p);
9733 82845 : for (n=1; n <= lim; n++) V[n+1] = sigchi2_Fl(k, CHI1vec, CHI2vec, n, vz, p);
9734 5054 : return V;
9735 : }
9736 :
9737 : static GEN
9738 252 : getcolswt2(GEN M, GEN D, ulong p)
9739 : {
9740 252 : GEN R, v = gel(M,1);
9741 252 : long i, l = lg(M) - 1;
9742 252 : R = cgetg(l, t_MAT); /* skip D[1] = 1 */
9743 1008 : for (i = 1; i < l; i++)
9744 : {
9745 756 : GEN w = Flv_Fl_mul(gel(M,i+1), D[i+1], p);
9746 756 : gel(R,i) = Flv_sub(v, w, p);
9747 : }
9748 252 : return R;
9749 : }
9750 : static GEN
9751 5845 : expandbd(GEN V, long d)
9752 : {
9753 : long L, n, nd;
9754 : GEN W;
9755 5845 : if (d == 1) return V;
9756 2114 : L = lg(V)-1; W = zerocol(L); /* nd = n/d */
9757 17955 : for (n = nd = 0; n < L; n += d, nd++) gel(W, n+1) = gel(V, nd+1);
9758 2114 : return W;
9759 : }
9760 : static GEN
9761 7714 : expandbd_Fl(GEN V, long d)
9762 : {
9763 : long L, n, nd;
9764 : GEN W;
9765 7714 : if (d == 1) return V;
9766 2660 : L = lg(V)-1; W = zero_Flv(L); /* nd = n/d */
9767 16429 : for (n = nd = 0; n < L; n += d, nd++) W[n+1] = V[nd+1];
9768 2660 : return W;
9769 : }
9770 : static void
9771 5054 : getcols_i(GEN *pM, GEN *pvj, GEN gk, GEN CHI1vec, GEN CHI2vec, long NN1, GEN vz,
9772 : ulong p, long lim)
9773 : {
9774 5054 : GEN CHI1 = CHIvec_CHI(CHI1vec), CHI2 = CHIvec_CHI(CHI2vec);
9775 5054 : long N2 = CHIvec_N(CHI2vec);
9776 5054 : GEN vj, M, D = mydivisorsu(NN1/N2);
9777 5054 : long i, l = lg(D), k = gk[2];
9778 5054 : GEN V = mfeisenstein2pure_Fl(k, CHI1vec, CHI2vec, vz, p, lim);
9779 5054 : M = cgetg(l, t_MAT);
9780 12768 : for (i = 1; i < l; i++) gel(M,i) = expandbd_Fl(V, D[i]);
9781 5054 : if (k == 2 && N2 == 1 && CHIvec_N(CHI1vec) == 1)
9782 : {
9783 252 : M = getcolswt2(M, D, p); l--;
9784 252 : D = vecslice(D, 2, l);
9785 : }
9786 5054 : *pM = M;
9787 5054 : *pvj = vj = cgetg(l, t_VEC);
9788 12516 : for (i = 1; i < l; i++) gel(vj,i) = mkvec4(gk, CHI1, CHI2, utoipos(D[i]));
9789 5054 : }
9790 :
9791 : /* find all CHI1, CHI2 mod N such that CHI1*CHI2 = CHI, f(CHI1)*f(CHI2) | N.
9792 : * set M = mfcoefs(B_e E(CHI1,CHI2), lim), vj = [e,i1,i2] */
9793 : static void
9794 2037 : getcols(GEN *pM, GEN *pv, long k, long nCHI, GEN allN, GEN vz, ulong p,
9795 : long lim)
9796 : {
9797 2037 : GEN vCHI = gel(allN,1), gk = utoi(k);
9798 2037 : GEN M = cgetg(1,t_MAT), v = cgetg(1,t_VEC);
9799 2037 : long i1, N = lg(vCHI)-1;
9800 93527 : for (i1 = 1; i1 <= N; i1++)
9801 : {
9802 91490 : GEN CHI1vec = gel(vCHI, i1), CHI2vec, M1, v1;
9803 : long NN1, i2;
9804 160972 : if (!CHI1vec) continue;
9805 73150 : if (k == 1 && CHIvec_parity(CHI1vec) == -1) continue;
9806 48391 : NN1 = N/CHIvec_N(CHI1vec); /* N/f(chi1) */;
9807 48391 : i2 = Fl_div(nCHI,i1, N);
9808 48391 : if (!i2) i2 = 1;
9809 48391 : CHI2vec = gel(vCHI,i2);
9810 48391 : if (NN1 % CHIvec_N(CHI2vec)) continue; /* f(chi1)f(chi2) | N ? */
9811 3668 : getcols_i(&M1, &v1, gk, CHI1vec, CHI2vec, NN1, vz, p, lim);
9812 3668 : M = shallowconcat(M, M1);
9813 3668 : v = shallowconcat(v, v1);
9814 : }
9815 2037 : *pM = M;
9816 2037 : *pv = v;
9817 2037 : }
9818 :
9819 : static void
9820 1239 : update_Mj(GEN *M, GEN *vecj, GEN *pz, ulong p)
9821 : {
9822 : GEN perm;
9823 1239 : *pz = Flm_indexrank(*M, p); perm = gel(*pz,2);
9824 1239 : *M = vecpermute(*M, perm);
9825 1239 : *vecj = vecpermute(*vecj, perm);
9826 1239 : }
9827 : static int
9828 441 : getcolsgen(long dim, GEN *pM, GEN *pvj, GEN *pz, long k, long ell, long nCHI,
9829 : GEN allN, GEN vz, ulong p, long lim)
9830 : {
9831 441 : GEN vCHI = gel(allN,1), bymod = gel(allN,2), gell = utoi(ell);
9832 441 : long i1, N = lg(vCHI)-1;
9833 441 : long L = lim+1;
9834 441 : if (lg(*pvj)-1 >= dim) update_Mj(pM, pvj, pz, p);
9835 441 : if (lg(*pvj)-1 == dim) return 1;
9836 1806 : for (i1 = 1; i1 <= N; i1++)
9837 : {
9838 1778 : GEN CHI1vec = gel(vCHI, i1), T;
9839 : long par1, j, l, N1, NN1;
9840 :
9841 1778 : if (!CHI1vec) continue;
9842 1750 : par1 = CHIvec_parity(CHI1vec);
9843 1750 : if (ell == 1 && par1 == -1) continue;
9844 1169 : if (odd(ell)) par1 = -par1;
9845 1169 : N1 = CHIvec_N(CHI1vec);
9846 1169 : NN1 = N/N1;
9847 1169 : T = gel(bymod, NN1); l = lg(T);
9848 4277 : for (j = 1; j < l; j++)
9849 : {
9850 3486 : long i2 = T[j], l1, l2, j1, s, nC;
9851 3486 : GEN M, M1, M2, vj, vj1, vj2, CHI2vec = gel(vCHI, i2);
9852 3486 : if (CHIvec_parity(CHI2vec) != par1) continue;
9853 1386 : nC = Fl_div(nCHI, Fl_mul(i1,i2,N), N);
9854 1386 : getcols(&M2, &vj2, k-ell, nC, allN, vz, p, lim);
9855 1386 : l2 = lg(M2); if (l2 == 1) continue;
9856 1386 : getcols_i(&M1, &vj1, gell, CHI1vec, CHI2vec, NN1, vz, p, lim);
9857 1386 : l1 = lg(M1);
9858 1386 : M1 = Flm_to_FlxV(M1, 0);
9859 1386 : M2 = Flm_to_FlxV(M2, 0);
9860 1386 : M = cgetg((l1-1)*(l2-1) + 1, t_MAT);
9861 1386 : vj = cgetg((l1-1)*(l2-1) + 1, t_VEC);
9862 3318 : for (j1 = s = 1; j1 < l1; j1++)
9863 : {
9864 1932 : GEN E = gel(M1,j1), v = gel(vj1,j1);
9865 : long j2;
9866 7805 : for (j2 = 1; j2 < l2; j2++, s++)
9867 : {
9868 5873 : GEN c = Flx_to_Flv(Flxn_mul(E, gel(M2,j2), L, p), L);
9869 5873 : gel(M,s) = c;
9870 5873 : gel(vj,s) = mkvec2(v, gel(vj2,j2));
9871 : }
9872 : }
9873 1386 : *pM = shallowconcat(*pM, M);
9874 1386 : *pvj = shallowconcat(*pvj, vj);
9875 1386 : if (lg(*pvj)-1 >= dim) update_Mj(pM, pvj, pz, p);
9876 1386 : if (lg(*pvj)-1 == dim) return 1;
9877 : }
9878 : }
9879 28 : if (ell == 1)
9880 : {
9881 21 : update_Mj(pM, pvj, pz, p);
9882 21 : return (lg(*pvj)-1 == dim);
9883 : }
9884 7 : return 0;
9885 : }
9886 :
9887 : static GEN
9888 1645 : mkF2bd(long d, long lim)
9889 : {
9890 1645 : GEN V = zerovec(lim + 1);
9891 : long n;
9892 1645 : gel(V, 1) = sstoQ(-1, 24);
9893 24248 : for (n = 1; n <= lim/d; n++) gel(V, n*d + 1) = mysumdivku(n, 1);
9894 1645 : return V;
9895 : }
9896 :
9897 : static GEN
9898 6202 : mkeisen(GEN E, long ord, GEN P, long lim)
9899 : {
9900 6202 : long k = itou(gel(E,1)), e = itou(gel(E,4));
9901 6202 : GEN CHI1 = gel(E,2), CHI2 = gel(E,3);
9902 6202 : if (k == 2 && mfcharistrivial(CHI1) && mfcharistrivial(CHI2))
9903 616 : return gsub(mkF2bd(1,lim), gmulgu(mkF2bd(e,lim), e));
9904 : else
9905 : {
9906 5586 : GEN V = mfeisenstein2pure(k, CHI1, CHI2, ord, P, lim);
9907 5586 : return expandbd(V, e);
9908 : }
9909 : }
9910 : static GEN
9911 609 : mkM(GEN vj, long pn, GEN P, long lim)
9912 : {
9913 609 : long j, l = lg(vj), L = lim+1;
9914 609 : GEN M = cgetg(l, t_MAT);
9915 5061 : for (j = 1; j < l; j++)
9916 : {
9917 : GEN E1, E2;
9918 4452 : parse_vecj(gel(vj,j), &E1,&E2);
9919 4452 : E1 = RgV_to_RgX(mkeisen(E1, pn, P, lim), 0);
9920 4452 : if (E2)
9921 : {
9922 1750 : E2 = RgV_to_RgX(mkeisen(E2, pn, P, lim), 0);
9923 1750 : E1 = RgXn_mul(E1, E2, L);
9924 : }
9925 4452 : E1 = RgX_to_RgC(E1, L);
9926 4452 : if (P && E2) E1 = RgXQV_red(E1, P);
9927 4452 : gel(M,j) = E1;
9928 : }
9929 609 : return M;
9930 : }
9931 :
9932 : /* assume N > 2 */
9933 : static GEN
9934 35 : mffindeisen1(long N)
9935 : {
9936 35 : GEN G = znstar0(utoipos(N), 1), L = chargalois(G, NULL), chi0 = NULL;
9937 35 : long j, m = N, l = lg(L);
9938 259 : for (j = 1; j < l; j++)
9939 : {
9940 245 : GEN chi = gel(L,j);
9941 245 : long r = myeulerphiu(itou(zncharorder(G,chi)));
9942 245 : if (r >= m) continue;
9943 182 : chi = znconreyfromchar(G, chi);
9944 182 : if (zncharisodd(G,chi)) { m = r; chi0 = chi; if (r == 1) break; }
9945 : }
9946 35 : if (!chi0) pari_err_BUG("mffindeisen1 [no Eisenstein series found]");
9947 35 : chi0 = znchartoprimitive(G,chi0);
9948 35 : return mfcharGL(gel(chi0,1), gel(chi0,2));
9949 : }
9950 :
9951 : static GEN
9952 651 : mfeisensteinspaceinit_i(long N, long k, GEN CHI)
9953 : {
9954 651 : GEN M, Minv, vj, vG, GN, allN, P, vz, z = NULL;
9955 651 : long nCHI, lim, ell, ord, dim = mffulldim(N, k, CHI);
9956 : ulong r, p;
9957 :
9958 651 : if (!dim) retmkvec3(cgetg(1,t_VECSMALL),
9959 : mkvec2(cgetg(1,t_MAT),gen_1),cgetg(1,t_VEC));
9960 651 : lim = mfsturmNk(N, k) + 1;
9961 651 : allN = charsmodN(N);
9962 651 : vG = gel(allN,3);
9963 651 : GN = gel(vG,N);
9964 651 : ord = znstar_get_expo(GN);
9965 651 : P = ord <= 2? NULL: polcyclo(ord, varn(mfcharpol(CHI)));
9966 651 : CHI = induce(GN, CHI); /* lift CHI mod N before mfcharno*/
9967 651 : nCHI = mfcharno(CHI);
9968 651 : r = QabM_init(ord, &p);
9969 651 : vz = Fl_powers(r, ord, p);
9970 651 : getcols(&M, &vj, k, nCHI, allN, vz, p, lim);
9971 679 : for (ell = k>>1; ell >= 1; ell--)
9972 441 : if (getcolsgen(dim, &M, &vj, &z, k, ell, nCHI, allN, vz, p, lim)) break;
9973 651 : if (!z) update_Mj(&M, &vj, &z, p);
9974 651 : if (lg(vj) - 1 < dim) return NULL;
9975 609 : M = mkM(vj, ord, P, lim);
9976 609 : Minv = QabM_Minv(rowpermute(M, gel(z,1)), P, ord);
9977 609 : return mkvec4(gel(z,1), Minv, vj, utoi(ord));
9978 : }
9979 : /* true mf */
9980 : static GEN
9981 609 : mfeisensteinspaceinit(GEN mf)
9982 : {
9983 609 : pari_sp av = avma;
9984 609 : GEN z, CHI = MF_get_CHI(mf);
9985 609 : long N = MF_get_N(mf), k = MF_get_k(mf);
9986 609 : if (!CHI) CHI = mfchartrivial();
9987 609 : z = mfeisensteinspaceinit_i(N, k, CHI);
9988 609 : if (!z)
9989 : {
9990 35 : GEN E, CHIN = mffindeisen1(N), CHI0 = mfchartrivial();
9991 35 : z = mfeisensteinspaceinit_i(N, k+1, mfcharmul(CHI, CHIN));
9992 35 : if (z) E = mkvec4(gen_1, CHI0, CHIN, gen_1);
9993 : else
9994 : {
9995 7 : z = mfeisensteinspaceinit_i(N, k+2, CHI);
9996 7 : E = mkvec4(gen_2, CHI0, CHI0, utoipos(N));
9997 : }
9998 35 : z = mkvec2(z, E);
9999 : }
10000 609 : return gc_GEN(av, z);
10001 : }
10002 :
10003 : /* decomposition of modular form on eisenspace */
10004 : static GEN
10005 1218 : mfeisensteindec(GEN mf, GEN F)
10006 : {
10007 1218 : pari_sp av = avma;
10008 : GEN M, Mindex, Mvecj, V, B, CHI;
10009 : long o, ord;
10010 :
10011 1218 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
10012 1218 : if (lg(Mvecj) < 5)
10013 : {
10014 56 : GEN E, e = gel(Mvecj,2), gkE = gel(e,1);
10015 56 : long dE = itou(gel(e,4));
10016 56 : Mvecj = gel(Mvecj,1);
10017 56 : E = mfeisenstein(itou(gkE), NULL, gel(e,3));
10018 56 : if (dE != 1) E = mfbd_E2(E, dE, gel(e,2)); /* here k = 2 */
10019 56 : F = mfmul(F, E);
10020 : }
10021 1218 : M = gel(Mvecj, 2);
10022 1218 : if (lg(M) == 1) return cgetg(1, t_VEC);
10023 1218 : Mindex = gel(Mvecj, 1);
10024 1218 : ord = itou(gel(Mvecj,4));
10025 1218 : V = mfcoefs(F, Mindex[lg(Mindex)-1]-1, 1); settyp(V, t_COL);
10026 1218 : CHI = mf_get_CHI(F);
10027 1218 : o = mfcharorder(CHI);
10028 1218 : if (o > 2 && o != ord)
10029 : { /* convert Mod(.,polcyclo(o)) to Mod(., polcyclo(N)) for o | N,
10030 : * o and N both != 2 (mod 4) */
10031 84 : GEN z, P = gel(M,4); /* polcyclo(ord) */
10032 84 : long vt = varn(P);
10033 84 : z = gmodulo(pol_xn(ord/o, vt), P);
10034 84 : if (ord % o) pari_err_TYPE("mfeisensteindec", V);
10035 84 : V = gsubst(liftpol_shallow(V), vt, z);
10036 : }
10037 1218 : B = Minv_RgC_mul(M, vecpermute(V, Mindex));
10038 1218 : return gc_upto(av, B);
10039 : }
10040 :
10041 : /*********************************************************************/
10042 : /* END EISENSPACE */
10043 : /*********************************************************************/
10044 :
10045 : static GEN
10046 70 : sertocol2(GEN S, long l)
10047 : {
10048 70 : GEN C = cgetg(l + 2, t_COL);
10049 : long i;
10050 420 : for (i = 0; i <= l; i++) gel(C, i+1) = polcoef_i(S, i, -1);
10051 70 : return C;
10052 : }
10053 :
10054 : /* Compute polynomial P0 such that F=E4^(k/4)P0(E6/E4^(3/2)). */
10055 : static GEN
10056 14 : mfcanfindp0(GEN F, long k)
10057 : {
10058 14 : pari_sp ltop = avma;
10059 : GEN E4, E6, V, V1, Q, W, res, M, B;
10060 : long l, j;
10061 14 : l = k/6 + 2;
10062 14 : V = mfcoefsser(F,l);
10063 14 : E4 = mfcoefsser(mfEk(4),l);
10064 14 : E6 = mfcoefsser(mfEk(6),l);
10065 14 : V1 = gdiv(V, gpow(E4, uutoQ(k,4), 0));
10066 14 : Q = gdiv(E6, gpow(E4, uutoQ(3,2), 0));
10067 14 : W = gpowers(Q, l - 1);
10068 14 : M = cgetg(l + 1, t_MAT);
10069 70 : for (j = 1; j <= l; j++) gel(M,j) = sertocol2(gel(W,j), l);
10070 14 : B = sertocol2(V1, l);
10071 14 : res = inverseimage(M, B);
10072 14 : if (lg(res) == 1) err_space(F);
10073 14 : return gc_GEN(ltop, gtopolyrev(res, 0));
10074 : }
10075 :
10076 : /* Compute the first n+1 Taylor coeffs at tau=I of a modular form
10077 : * on SL_2(Z). */
10078 : GEN
10079 14 : mftaylor(GEN F, long n, long flreal, long prec)
10080 : {
10081 14 : pari_sp ltop = avma;
10082 14 : GEN P0, Pm1 = gen_0, v;
10083 14 : GEN X2 = mkpoln(3, ghalf,gen_0,gneg(ghalf)); /* (x^2-1) / 2 */
10084 : long k, m;
10085 14 : if (!checkmf_i(F)) pari_err_TYPE("mftaylor",F);
10086 14 : k = mf_get_k(F);
10087 14 : if (mf_get_N(F) != 1 || k < 0) pari_err_IMPL("mftaylor for this form");
10088 14 : P0 = mfcanfindp0(F, k);
10089 14 : v = cgetg(n+2, t_VEC); gel(v, 1) = RgX_coeff(P0,0);
10090 154 : for (m = 0; m < n; m++)
10091 : {
10092 140 : GEN P1 = gdivgu(gmulsg(-(k + 2*m), RgX_shift(P0,1)), 12);
10093 140 : P1 = gadd(P1, gmul(X2, RgX_deriv(P0)));
10094 140 : if (m) P1 = gsub(P1, gdivgu(gmulsg(m*(m+k-1), Pm1), 144));
10095 140 : Pm1 = P0; P0 = P1;
10096 140 : gel(v, m+2) = RgX_coeff(P0, 0);
10097 : }
10098 14 : if (flreal)
10099 : {
10100 7 : GEN pi2 = Pi2n(1, prec), pim4 = gmulsg(-2, pi2), VPC;
10101 7 : GEN C = gmulsg(3, gdiv(gpowgs(ggamma(uutoQ(1,4), prec), 8), gpowgs(pi2, 6)));
10102 : /* E_4(i): */
10103 7 : GEN facn = gen_1;
10104 7 : VPC = gpowers(gmul(pim4, gsqrt(C, prec)), n);
10105 7 : C = gpow(C, uutoQ(k,4), prec);
10106 84 : for (m = 0; m <= n; m++)
10107 : {
10108 77 : gel(v, m+1) = gdiv(gmul(C, gmul(gel(v, m+1), gel(VPC, m+1))), facn);
10109 77 : facn = gmulgu(facn, m+1);
10110 : }
10111 : }
10112 14 : return gc_GEN(ltop, v);
10113 : }
10114 :
10115 : #if 0
10116 : /* To be used in mfeigensearch() */
10117 : GEN
10118 : mfreadratfile()
10119 : {
10120 : GEN eqn;
10121 : pariFILE *F = pari_fopengz("rateigen300.gp");
10122 : eqn = gp_readvec_stream(F->file);
10123 : pari_fclose(F);
10124 : return eqn;
10125 : }
10126 : #endif
10127 : /*****************************************************************/
10128 : /* EISENSTEIN CUSPS: COMPLEX DIRECTLY: one F_k */
10129 : /*****************************************************************/
10130 :
10131 : /* CHIvec = charinit(CHI); data = [N1g/g1,N2g/g2,g1/g,g2/g,C/g1,C/g2,
10132 : * (N1g/g1)^{-1},(N2g/g2)^{-1}] */
10133 :
10134 : /* nm = n/m;
10135 : * z1 = powers of \z_{C/g}^{(Ae/g)^{-1}},
10136 : * z2 = powers of \z_N^{A^{-1}(g1g2/C)}]
10137 : * N.B. : we compute value and conjugate at the end, so it is (Ae/g)^{-1}
10138 : * and not -(Ae/g)^{-1} */
10139 : static GEN
10140 9635178 : eiscnm(long nm, long m, GEN CHI1vec, GEN CHI2vec, GEN data, GEN z1)
10141 : {
10142 9635178 : long Cg1 = data[5], s10 = (nm*data[7]) % Cg1, r10 = (nm - data[1]*s10) / Cg1;
10143 9635178 : long Cg2 = data[6], s20 = (m *data[8]) % Cg2, r20 = (m - data[2]*s20) / Cg2;
10144 : long j1, r1, s1;
10145 9635178 : GEN T = gen_0;
10146 22465660 : for (j1 = 0, r1 = r10, s1 = s10; j1 < data[3]; j1++, r1 -= data[1], s1 += Cg1)
10147 : {
10148 12830482 : GEN c1 = mychareval(CHI1vec, r1);
10149 12830482 : if (!gequal0(c1))
10150 : {
10151 : long j2, r2, s2;
10152 9925790 : GEN S = gen_0;
10153 24733030 : for (j2 = 0, r2 = r20, s2 = s20; j2 < data[4]; j2++, r2 -= data[2], s2 += Cg2)
10154 : {
10155 14807240 : GEN c2 = mychareval(CHI2vec, r2);
10156 14807240 : if (!gequal0(c2)) S = gadd(S, gmul(c2, rootsof1pow(z1, s1*s2)));
10157 : }
10158 9925790 : T = gadd(T, gmul(c1, S));
10159 : }
10160 : }
10161 9635178 : return conj_i(T);
10162 : }
10163 :
10164 : static GEN
10165 855267 : fg1g2n(long n, long k, GEN CHI1vec, GEN CHI2vec, GEN data, GEN z1, GEN z2)
10166 : {
10167 855267 : pari_sp av = avma;
10168 855267 : GEN S = gen_0, D = mydivisorsu(n);
10169 855267 : long i, l = lg(D);
10170 5672856 : for (i = 1; i < l; i++)
10171 : {
10172 4817589 : long m = D[i], nm = D[l-i]; /* n/m */
10173 4817589 : GEN u = eiscnm( nm, m, CHI1vec, CHI2vec, data, z1);
10174 4817589 : GEN v = eiscnm(-nm, -m, CHI1vec, CHI2vec, data, z1);
10175 4817589 : GEN w = odd(k) ? gsub(u, v) : gadd(u, v);
10176 4817589 : S = gadd(S, gmul(powuu(m, k-1), w));
10177 : }
10178 855267 : return gc_upto(av, gmul(S, rootsof1pow(z2, n)));
10179 : }
10180 :
10181 : static GEN
10182 33985 : gausssumcx(GEN CHIvec, long prec)
10183 : {
10184 : GEN z, S, V;
10185 33985 : long m, N = CHIvec_N(CHIvec);
10186 33985 : if (N == 1) return gen_1;
10187 18277 : V = CHIvec_val(CHIvec);
10188 18277 : z = rootsof1u_cx(N, prec);
10189 18277 : S = gmul(z, gel(V, N));
10190 431158 : for (m = N-1; m >= 1; m--) S = gmul(z, gadd(gel(V, m), S));
10191 18277 : return S;
10192 : }
10193 :
10194 : /* Computation of Q_k(\z_N^s) as a polynomial in \z_N^s. FIXME: explicit
10195 : * formula ? */
10196 : static GEN
10197 6118 : mfqk(long k, long N)
10198 : {
10199 : GEN X, P, ZI, Q, Xm1, invden;
10200 : long i;
10201 6118 : ZI = gdivgu(RgX_shift_shallow(RgV_to_RgX(identity_ZV(N-1), 0), 1), N);
10202 6118 : if (k == 1) return ZI;
10203 4956 : P = gsubgs(pol_xn(N,0), 1);
10204 4956 : invden = RgXQ_powu(ZI, k, P);
10205 4956 : X = pol_x(0); Q = gneg(X); Xm1 = gsubgs(X, 1);
10206 21784 : for (i = 2; i < k; i++)
10207 16828 : Q = RgX_shift_shallow(ZX_add(gmul(Xm1, ZX_deriv(Q)), gmulsg(-i, Q)), 1);
10208 4956 : return RgXQ_mul(Q, invden, P);
10209 : }
10210 :
10211 : /* CHI mfchar; M is a multiple of the conductor of CHI, but is NOT
10212 : * necessarily its modulus */
10213 : static GEN
10214 7903 : mfskcx(long k, GEN CHI, long M, long prec)
10215 : {
10216 : GEN S, CHIvec, P;
10217 : long F, m, i, l;
10218 7903 : CHI = mfchartoprimitive(CHI, &F);
10219 7903 : CHIvec = mfcharcxinit(CHI, prec);
10220 7903 : if (F == 1) S = gdivgu(bernfrac(k), k);
10221 : else
10222 : {
10223 6118 : GEN Q = mfqk(k, F), V = CHIvec_val(CHIvec);
10224 6118 : S = gmul(gel(V, F), RgX_coeff(Q, 0));
10225 156611 : for (m = 1; m < F; m++) S = gadd(S, gmul(gel(V, m), RgX_coeff(Q, m)));
10226 6118 : S = conj_i(S);
10227 : }
10228 : /* prime divisors of M not dividing f(chi) */
10229 7903 : P = gel(myfactoru(u_ppo(M/F,F)), 1); l = lg(P);
10230 8057 : for (i = 1; i < l; i++)
10231 : {
10232 154 : long p = P[i];
10233 154 : S = gmul(S, gsubsg(1, gdiv(mychareval(CHIvec, p), powuu(p, k))));
10234 : }
10235 7903 : return gmul(gmul(gausssumcx(CHIvec, prec), S), powuu(M/F, k));
10236 : }
10237 :
10238 : static GEN
10239 13727 : f00_i(long k, GEN CHI1vec, GEN CHI2vec, GEN G2, GEN S, long prec)
10240 : {
10241 : GEN c, a;
10242 13727 : long N1 = CHIvec_N(CHI1vec), N2 = CHIvec_N(CHI2vec);
10243 13727 : if (S[2] != N1) return gen_0;
10244 7903 : c = mychareval(CHI1vec, S[3]);
10245 7903 : if (isintzero(c)) return gen_0;
10246 7903 : a = mfskcx(k, mfchardiv(CHIvec_CHI(CHI2vec), CHIvec_CHI(CHI1vec)), N1*N2, prec);
10247 7903 : a = gmul(a, conj_i(gmul(c,G2)));
10248 7903 : return gdiv(a, mulsi(-N2, powuu(S[1], k-1)));
10249 : }
10250 :
10251 : static GEN
10252 12446 : f00(long k, GEN CHI1vec,GEN CHI2vec, GEN G1,GEN G2, GEN data, long prec)
10253 : {
10254 : GEN T1, T2;
10255 12446 : T2 = f00_i(k, CHI1vec, CHI2vec, G2, data, prec);
10256 12446 : if (k > 1) return T2;
10257 1281 : T1 = f00_i(k, CHI2vec, CHI1vec, G1, data, prec);
10258 1281 : return gadd(T1, T2);
10259 : }
10260 :
10261 : /* ga in SL_2(Z), find beta [a,b;c,d] in Gamma_0(N) and mu in Z such that
10262 : * beta * ga * T^u = [A',B';C',D'] with C' | N and N | B', C' > 0 */
10263 : static void
10264 13041 : mfgatogap(GEN ga, long N, long *pA, long *pC, long *pD, long *pd, long *pmu)
10265 : {
10266 13041 : GEN A = gcoeff(ga,1,1), B = gcoeff(ga,1,2);
10267 13041 : GEN C = gcoeff(ga,2,1), D = gcoeff(ga,2,2), a, b, c, d;
10268 : long t, Ap, Cp, B1, D1, mu;
10269 13041 : Cp = itou(bezout(muliu(A,N), C, &c, &d)); /* divides N */
10270 13041 : t = 0;
10271 13041 : if (Cp > 1)
10272 : { /* (d, N/Cp) = 1, find t such that (d - t*(A*N/Cp), N) = 1 */
10273 2604 : long dN = umodiu(d,Cp), Q = (N/Cp * umodiu(A,Cp)) % Cp;
10274 2989 : while (ugcd(dN, Cp) > 1) { t++; dN = Fl_sub(dN, Q, Cp); }
10275 : }
10276 13041 : if (t)
10277 : {
10278 385 : c = addii(c, mului(t, diviuexact(C,Cp)));
10279 385 : d = subii(d, mului(t, muliu(A, N/Cp))); /* (d,N) = 1 */
10280 : }
10281 13041 : D1 = umodiu(mulii(d,D), N);
10282 13041 : (void)bezout(d, mulis(c,-N), &a, &b); /* = 1 */
10283 13041 : t = 0; Ap = umodiu(addii(mulii(a,A), mulii(b,C)), N); /* (Ap,Cp) = 1 */
10284 22267 : while (ugcd(Ap, N) > 1) { t++; Ap = Fl_add(Ap, Cp, N); }
10285 13041 : B1 = umodiu(a,N)*umodiu(B,N) + umodiu(b,N)*umodiu(D,N) + t*D1;
10286 13041 : B1 %= N;
10287 13041 : *pmu = mu = Fl_neg(Fl_div(B1, Ap, N), N);
10288 : /* A', D' and d only needed modulo N */
10289 13041 : *pd = umodiu(d, N);
10290 13041 : *pA = Ap;
10291 13041 : *pC = Cp; *pD = (D1 + Cp*mu) % N;
10292 13041 : }
10293 :
10294 : #if 0
10295 : /* CHI is a mfchar, return alpha(CHI) */
10296 : static long
10297 : mfalchi(GEN CHI, long AN, long cg)
10298 : {
10299 : GEN G = gel(CHI,1), chi = gel(CHI,2), go = gmfcharorder(CHI);
10300 : long o = itou(go), a = itos( znchareval(G, chi, stoi(1 + AN/cg), go) );
10301 : if (a < 0 || (cg * a) % o) pari_err_BUG("mfalchi");
10302 : return (cg * a) / o;
10303 : }
10304 : #endif
10305 : /* return A such that CHI1(t) * CHI2(t) = e(A) or NULL if (t,N1*N2) > 1 */
10306 : static GEN
10307 26082 : mfcharmuleval(GEN CHI1vec, GEN CHI2vec, long t)
10308 : {
10309 26082 : long a1 = mycharexpo(CHI1vec, t), o1 = CHIvec_ord(CHI1vec);
10310 26082 : long a2 = mycharexpo(CHI2vec, t), o2 = CHIvec_ord(CHI2vec);;
10311 26082 : if (a1 < 0 || a2 < 0) return NULL;
10312 26082 : return sstoQ(a1*o2 + a2*o1, o1*o2);
10313 : }
10314 : static GEN
10315 13041 : mfcharmulcxeval(GEN CHI1vec, GEN CHI2vec, long t, long prec)
10316 : {
10317 13041 : GEN A = mfcharmuleval(CHI1vec, CHI2vec, t);
10318 : long n, d;
10319 13041 : if (!A) return gen_0;
10320 13041 : Qtoss(A, &n,&d); return rootsof1q_cx(n, d, prec);
10321 : }
10322 : /* alpha(CHI1 * CHI2) */
10323 : static long
10324 13041 : mfalchi2(GEN CHI1vec, GEN CHI2vec, long AN, long cg)
10325 : {
10326 13041 : GEN A = mfcharmuleval(CHI1vec, CHI2vec, 1 + AN/cg);
10327 : long a;
10328 13041 : if (!A) pari_err_BUG("mfalchi2");
10329 13041 : A = gmulsg(cg, A);
10330 13041 : if (typ(A) != t_INT) pari_err_BUG("mfalchi2");
10331 13041 : a = itos(A) % cg; if (a < 0) a += cg;
10332 13041 : return a;
10333 : }
10334 :
10335 : /* return g = (a,b), set u >= 0 s.t. g = a * u (mod b) */
10336 : static long
10337 52164 : mybezout(long a, long b, long *pu)
10338 : {
10339 52164 : long junk, g = cbezout(a, b, pu, &junk);
10340 52164 : if (*pu < 0) *pu += b/g;
10341 52164 : return g;
10342 : }
10343 :
10344 : /* E = [k, CHI1,CHI2, e], CHI1 and CHI2 primitive mfchars such that,
10345 : * CHI1(-1)*CHI2(-1) = (-1)^k; expansion of (B_e (E_k(CHI1,CHI2))) | ga.
10346 : * w is the width for the space of the calling function. */
10347 : static GEN
10348 13041 : mfeisensteingacx(GEN E, long w, GEN ga, long lim, long prec)
10349 : {
10350 13041 : GEN CHI1vec, CHI2vec, CHI1 = gel(E,2), CHI2 = gel(E,3), v, S, ALPHA;
10351 : GEN G1, G2, z1, z2, data;
10352 13041 : long k = itou(gel(E,1)), e = itou(gel(E,4));
10353 13041 : long N1 = mfcharmodulus(CHI1);
10354 13041 : long N2 = mfcharmodulus(CHI2), N = e * N1 * N2;
10355 : long NsurC, cg, wN, A, C, Ai, d, mu, alchi, na, da;
10356 : long eg, g, gH, U, u0, u1, u2, Aig, H, m, n, t, Cg, NC1, NC2;
10357 :
10358 13041 : mfgatogap(ga, N, &A, &C, &Ai, &d, &mu);
10359 13041 : CHI1vec = mfcharcxinit(CHI1, prec);
10360 13041 : CHI2vec = mfcharcxinit(CHI2, prec);
10361 13041 : NsurC = N/C; cg = ugcd(C, NsurC); wN = NsurC / cg;
10362 13041 : if (w%wN) pari_err_BUG("mfeisensteingacx [wN does not divide w]");
10363 13041 : alchi = mfalchi2(CHI1vec, CHI2vec, A*N, cg);
10364 13041 : ALPHA = sstoQ(alchi, NsurC);
10365 :
10366 13041 : g = mybezout(A*e, C, &u0); Cg = C/g; eg = e/g;
10367 13041 : NC1 = mybezout(N1, Cg, &u1);
10368 13041 : NC2 = mybezout(N2, Cg, &u2);
10369 13041 : H = (NC1*NC2*g)/Cg;
10370 13041 : Aig = (Ai*H)%N; if (Aig < 0) Aig += N;
10371 13041 : z1 = rootsof1powinit(u0, Cg, prec);
10372 13041 : z2 = rootsof1powinit(Aig, N, prec);
10373 13041 : data = mkvecsmalln(8, N1/NC1, N2/NC2, NC1, NC2, Cg/NC1, Cg/NC2, u1, u2);
10374 13041 : v = zerovec(lim + 1);
10375 : /* need n*H = alchi (mod cg) */
10376 13041 : gH = mybezout(H, cg, &U);
10377 13041 : if (gH > 1)
10378 : {
10379 511 : if (alchi % gH) return mkvec2(gen_0, v);
10380 511 : alchi /= gH; cg /= gH; H /= gH;
10381 : }
10382 13041 : G1 = gausssumcx(CHI1vec, prec);
10383 13041 : G2 = gausssumcx(CHI2vec, prec);
10384 13041 : if (!alchi)
10385 12446 : gel(v,1) = f00(k, CHI1vec,CHI2vec,G1,G2, mkvecsmall3(NC2,Cg,A*eg), prec);
10386 13041 : n = Fl_mul(alchi,U,cg); if (!n) n = cg;
10387 13041 : m = (n*H - alchi) / cg; /* positive, exact division */
10388 868308 : for (; m <= lim; n+=cg, m+=H)
10389 855267 : gel(v, m+1) = fg1g2n(n, k, CHI1vec, CHI2vec, data, z1,z2);
10390 13041 : t = (2*e)/g; if (odd(k)) t = -t;
10391 13041 : v = gdiv(v, gmul(conj_i(gmul(G1,G2)), mulsi(t, powuu(eg*N2/NC2, k-1))));
10392 13041 : if (k == 2 && N1 == 1 && N2 == 1) v = gsub(mkF2bd(wN,lim), gmulsg(e,v));
10393 :
10394 13041 : Qtoss(ALPHA, &na,&da);
10395 13041 : S = conj_i( mfcharmulcxeval(CHI1vec,CHI2vec,d,prec) ); /* CHI(1/d) */
10396 13041 : if (wN > 1)
10397 : {
10398 11354 : GEN z = rootsof1powinit(-mu, wN, prec);
10399 11354 : long i, l = lg(v);
10400 823739 : for (i = 1; i < l; i++) gel(v,i) = gmul(gel(v,i), rootsof1pow(z,i-1));
10401 : }
10402 13041 : v = RgV_Rg_mul(v, gmul(S, rootsof1q_cx(-mu*na, da, prec)));
10403 13041 : return mkvec2(ALPHA, bdexpand(v, w/wN));
10404 : }
10405 :
10406 : /*****************************************************************/
10407 : /* END EISENSTEIN CUSPS */
10408 : /*****************************************************************/
10409 :
10410 : static GEN
10411 1596 : mfchisimpl(GEN CHI)
10412 : {
10413 : GEN G, chi;
10414 1596 : if (typ(CHI) == t_INT) return CHI;
10415 1596 : G = gel(CHI, 1); chi = gel(CHI, 2);
10416 1596 : switch(mfcharorder(CHI))
10417 : {
10418 1148 : case 1: chi = gen_1; break;
10419 427 : case 2: chi = znchartokronecker(G,chi,1); break;
10420 21 : default:chi = mkintmod(znconreyexp(G,chi), znstar_get_N(G)); break;
10421 : }
10422 1596 : return chi;
10423 : }
10424 :
10425 : GEN
10426 700 : mfparams(GEN F)
10427 : {
10428 700 : pari_sp av = avma;
10429 : GEN z, mf, CHI;
10430 700 : if ((mf = checkMF_i(F)))
10431 : {
10432 14 : long N = MF_get_N(mf);
10433 14 : GEN gk = MF_get_gk(mf);
10434 14 : CHI = MF_get_CHI(mf);
10435 14 : z = mkvec5(utoi(N), gk, CHI, utoi(MF_get_space(mf)), mfcharpol(CHI));
10436 : }
10437 : else
10438 : {
10439 686 : if (!checkmf_i(F)) pari_err_TYPE("mfparams", F);
10440 686 : z = vec_append(mf_get_NK(F), mfcharpol(mf_get_CHI(F)));
10441 : }
10442 700 : gel(z,3) = mfchisimpl(gel(z,3));
10443 700 : return gc_GEN(av, z);
10444 : }
10445 :
10446 : GEN
10447 14 : mfisCM(GEN F)
10448 : {
10449 14 : pari_sp av = avma;
10450 : forprime_t S;
10451 : GEN D, v;
10452 : long N, k, lD, sb, p, i;
10453 14 : if (!checkmf_i(F)) pari_err_TYPE("mfisCM", F);
10454 14 : N = mf_get_N(F);
10455 14 : k = mf_get_k(F); if (N < 0 || k < 0) pari_err_IMPL("mfisCM for this F");
10456 14 : D = mfunram(N, -1);
10457 14 : lD = lg(D);
10458 14 : sb = maxss(mfsturmNk(N, k), 4*N);
10459 14 : v = mfcoefs_i(F, sb, 1);
10460 14 : u_forprime_init(&S, 2, sb);
10461 504 : while ((p = u_forprime_next(&S)))
10462 : {
10463 490 : GEN ap = gel(v, p+1);
10464 490 : if (!gequal0(ap))
10465 406 : for (i = 1; i < lD; i++)
10466 245 : if (kross(D[i], p) == -1) { D = vecsplice(D, i); lD--; }
10467 : }
10468 14 : if (lD == 1) return gc_const(av, gen_0);
10469 14 : if (lD == 2) return gc_stoi(av, D[1]);
10470 7 : if (k > 1) pari_err_BUG("mfisCM");
10471 7 : return gc_upto(av, zv_to_ZV(D));
10472 : }
10473 :
10474 : static long
10475 287 : mfspace_i(GEN mf, GEN F)
10476 : {
10477 : GEN v, vF, gk;
10478 : long n, nE, i, l, s, N;
10479 :
10480 287 : mf = checkMF(mf); s = MF_get_space(mf);
10481 287 : if (!F) return s;
10482 287 : if (!checkmf_i(F)) pari_err_TYPE("mfspace",F);
10483 287 : v = mftobasis(mf, F, 1);
10484 287 : n = lg(v)-1; if (!n) return -1;
10485 224 : nE = lg(MF_get_E(mf))-1;
10486 224 : switch(s)
10487 : {
10488 56 : case mf_NEW: case mf_OLD: case mf_EISEN: return s;
10489 140 : case mf_FULL:
10490 140 : if (mf_get_type(F) == t_MF_THETA) return mf_EISEN;
10491 133 : if (!gequal0(vecslice(v,1,nE)))
10492 63 : return gequal0(vecslice(v,nE+1,n))? mf_EISEN: mf_FULL;
10493 : }
10494 : /* mf is mf_CUSP or mf_FULL, F a cusp form */
10495 98 : gk = mf_get_gk(F);
10496 98 : if (typ(gk) == t_FRAC || equali1(gk)) return mf_CUSP;
10497 84 : vF = mftonew_i(mf, vecslice(v, nE+1, n), &N);
10498 84 : if (N != MF_get_N(mf)) return mf_OLD;
10499 56 : l = lg(vF);
10500 91 : for (i = 1; i < l; i++)
10501 56 : if (itos(gmael(vF,i,1)) != N) return mf_CUSP;
10502 35 : return mf_NEW;
10503 : }
10504 : long
10505 287 : mfspace(GEN mf, GEN F)
10506 287 : { pari_sp av = avma; return gc_long(av, mfspace_i(mf,F)); }
10507 : static GEN
10508 21 : lfunfindchi(GEN ldata, GEN van, long prec)
10509 : {
10510 21 : GEN gN = ldata_get_conductor(ldata), gk = ldata_get_k(ldata);
10511 21 : GEN G = znstar0(gN,1), cyc = znstar_get_conreycyc(G), L, go, vz;
10512 21 : long N = itou(gN), odd = typ(gk) == t_INT && mpodd(gk);
10513 21 : long i, j, o, l, B0 = 2, B = lg(van)-1, bit = 10 - prec2nbits(prec);
10514 :
10515 : /* if van is integral, chi must be trivial */
10516 21 : if (typ(van) == t_VECSMALL) return mfcharGL(G, zerocol(lg(cyc)-1));
10517 14 : L = cyc2elts(cyc); l = lg(L);
10518 42 : for (i = j = 1; i < l; i++)
10519 : {
10520 28 : GEN chi = zc_to_ZC(gel(L,i));
10521 28 : if (zncharisodd(G,chi) == odd) gel(L,j++) = mfcharGL(G,chi);
10522 : }
10523 14 : setlg(L,j); l = j;
10524 14 : if (l <= 2) return gel(L,1);
10525 0 : o = znstar_get_expo(G); go = utoi(o);
10526 0 : vz = grootsof1(o, prec);
10527 : for (;;)
10528 0 : {
10529 : long n;
10530 0 : for (n = B0; n <= B; n++)
10531 : {
10532 : GEN an, r;
10533 : long j;
10534 0 : if (ugcd(n, N) != 1) continue;
10535 0 : an = gel(van,n); if (gexpo(an) < bit) continue;
10536 0 : r = gdiv(an, conj_i(an));
10537 0 : for (i = 1; i < l; i++)
10538 : {
10539 0 : GEN CHI = gel(L,i);
10540 0 : if (gexpo(gsub(r, gel(vz, znchareval_i(CHI,n,go)+1))) > bit)
10541 0 : gel(L,i) = NULL;
10542 : }
10543 0 : for (i = j = 1; i < l; i++)
10544 0 : if (gel(L,i)) gel(L,j++) = gel(L,i);
10545 0 : l = j; setlg(L,l);
10546 0 : if (l == 2) return gel(L,1);
10547 : }
10548 0 : B0 = B+1; B <<= 1;
10549 0 : van = ldata_vecan(ldata_get_an(ldata), B, prec);
10550 : }
10551 : }
10552 :
10553 : GEN
10554 21 : mffromlfun(GEN L, long prec)
10555 : {
10556 21 : pari_sp av = avma;
10557 21 : GEN ldata = lfunmisc_to_ldata_shallow(L), Vga = ldata_get_gammavec(ldata);
10558 21 : GEN van, a0, CHI, NK, gk = ldata_get_k(ldata);
10559 : long N, space;
10560 21 : if (!gequal(Vga, mkvec2(gen_0, gen_1))) pari_err_TYPE("mffromlfun", L);
10561 21 : N = itou(ldata_get_conductor(ldata));
10562 21 : van = ldata_vecan(ldata_get_an(ldata), mfsturmNgk(N,gk) + 2, prec);
10563 21 : CHI = lfunfindchi(ldata, van, prec);
10564 21 : if (typ(van) != t_VEC) van = vecsmall_to_vec_inplace(van);
10565 21 : space = (lg(ldata) == 7)? mf_CUSP: mf_FULL;
10566 21 : a0 = (space == mf_CUSP)? gen_0: gneg(lfun(L, gen_0, prec2nbits(prec)));
10567 21 : NK = mkvec3(utoi(N), gk, mfchisimpl(CHI));
10568 21 : return gc_GEN(av, mkvec3(NK, utoi(space), shallowconcat(a0, van)));
10569 : }
10570 : /*******************************************************************/
10571 : /* */
10572 : /* HALF-INTEGRAL WEIGHT */
10573 : /* */
10574 : /*******************************************************************/
10575 : /* We use the prefix mf2; k represents the weight -1/2, so e.g.
10576 : k = 2 is weight 5/2. N is the level, so 4\mid N, and CHI is the
10577 : character, always even. */
10578 :
10579 : static long
10580 3360 : lamCO(long r, long s, long p)
10581 : {
10582 3360 : if ((s << 1) <= r)
10583 : {
10584 1232 : long rp = r >> 1;
10585 1232 : if (odd(r)) return upowuu(p, rp) << 1;
10586 336 : else return (p + 1)*upowuu(p, rp - 1);
10587 : }
10588 2128 : else return upowuu(p, r - s) << 1;
10589 : }
10590 :
10591 : static int
10592 1568 : condC(GEN faN, GEN valF)
10593 : {
10594 1568 : GEN P = gel(faN, 1), E = gel(faN, 2);
10595 1568 : long l = lg(P), i;
10596 3696 : for (i = 1; i < l; i++)
10597 3024 : if ((P[i] & 3L) == 3)
10598 : {
10599 1120 : long r = E[i];
10600 1120 : if (odd(r) || r < (valF[i] << 1)) return 1;
10601 : }
10602 672 : return 0;
10603 : }
10604 :
10605 : /* returns 2*zetaCO; weight is k + 1/2 */
10606 : static long
10607 3696 : zeta2CO(GEN faN, GEN valF, long r2, long s2, long k)
10608 : {
10609 3696 : if (r2 >= 4) return lamCO(r2, s2, 2) << 1;
10610 2912 : if (r2 == 3) return 6;
10611 1568 : if (condC(faN, valF)) return 4;
10612 672 : if (odd(k)) return s2 ? 3 : 5; else return s2 ? 5: 3;
10613 : }
10614 :
10615 : /* returns 4 times last term in formula */
10616 : static long
10617 3696 : dim22(long N, long F, long k)
10618 : {
10619 3696 : pari_sp av = avma;
10620 3696 : GEN vF, faN = myfactoru(N), P = gel(faN, 1), E = gel(faN, 2);
10621 3696 : long i, D, l = lg(P);
10622 3696 : vF = cgetg(l, t_VECSMALL);
10623 9968 : for (i = 1; i < l; i++) vF[i] = u_lval(F, P[i]);
10624 3696 : D = zeta2CO(faN, vF, E[1], vF[1], k);
10625 6272 : for (i = 2; i < l; i++) D *= lamCO(E[i], vF[i], P[i]);
10626 3696 : return gc_long(av,D);
10627 : }
10628 :
10629 : /* PSI not necessarily primitive, of conductor F */
10630 : static int
10631 13846 : charistotallyeven(GEN PSI, long F)
10632 : {
10633 13846 : pari_sp av = avma;
10634 13846 : GEN P = gel(myfactoru(F), 1);
10635 13846 : GEN G = gel(PSI,1), psi = gel(PSI,2);
10636 : long i;
10637 14350 : for (i = 1; i < lg(P); i++)
10638 : {
10639 532 : GEN psip = znchardecompose(G, psi, utoipos(P[i]));
10640 532 : if (zncharisodd(G, psip)) return gc_bool(av,0);
10641 : }
10642 13818 : return gc_bool(av,1);
10643 : }
10644 :
10645 : static GEN
10646 299775 : get_PSI(GEN CHI, long t)
10647 : {
10648 299775 : long r = t & 3L, t2 = (r == 2 || r == 3) ? t << 2 : t;
10649 299775 : return mfcharmul_i(CHI, induce(gel(CHI,1), utoipos(t2)));
10650 : }
10651 : /* space = mf_CUSP, mf_EISEN or mf_FULL, weight k + 1/2 */
10652 : static long
10653 41363 : mf2dimwt12(long N, GEN CHI, long space)
10654 : {
10655 41363 : pari_sp av = avma;
10656 41363 : GEN D = mydivisorsu(N >> 2);
10657 41363 : long i, l = lg(D), dim3 = 0, dim4 = 0;
10658 :
10659 41363 : CHI = induceN(N, CHI);
10660 341138 : for (i = 1; i < l; i++)
10661 : {
10662 299775 : long rp, t = D[i], Mt = D[l-i];
10663 299775 : GEN PSI = get_PSI(CHI,t);
10664 299775 : rp = mfcharconductor(PSI);
10665 299775 : if (Mt % (rp*rp) == 0) { dim4++; if (charistotallyeven(PSI,rp)) dim3++; }
10666 : }
10667 41363 : set_avma(av);
10668 41363 : switch (space)
10669 : {
10670 40439 : case mf_CUSP: return dim4 - dim3;
10671 462 : case mf_EISEN:return dim3;
10672 462 : case mf_FULL: return dim4;
10673 : }
10674 : return 0; /*LCOV_EXCL_LINE*/
10675 : }
10676 :
10677 : static long
10678 693 : mf2dimwt32(long N, GEN CHI, long F, long space)
10679 : {
10680 : long D;
10681 693 : switch(space)
10682 : {
10683 231 : case mf_CUSP: D = mypsiu(N) - 6*dim22(N, F, 1);
10684 231 : if (D%24) pari_err_BUG("mfdim");
10685 231 : return D/24 + mf2dimwt12(N, CHI, 4);
10686 231 : case mf_FULL: D = mypsiu(N) + 6*dim22(N, F, 0);
10687 231 : if (D%24) pari_err_BUG("mfdim");
10688 231 : return D/24 + mf2dimwt12(N, CHI, 1);
10689 231 : case mf_EISEN: D = dim22(N, F, 0) + dim22(N, F, 1);
10690 231 : if (D & 3L) pari_err_BUG("mfdim");
10691 231 : return (D >> 2) - mf2dimwt12(N, CHI, 3);
10692 : }
10693 : return 0; /*LCOV_EXCL_LINE*/
10694 : }
10695 :
10696 : /* F = conductor(CHI), weight k = r+1/2 */
10697 : static long
10698 43729 : checkmf2(long N, long r, GEN CHI, long F, long space)
10699 : {
10700 43729 : switch(space)
10701 : {
10702 43708 : case mf_FULL: case mf_CUSP: case mf_EISEN: break;
10703 14 : case mf_NEW: case mf_OLD:
10704 14 : pari_err_TYPE("half-integral weight [new/old spaces]", utoi(space));
10705 7 : default:
10706 7 : pari_err_TYPE("half-integral weight [incorrect space]",utoi(space));
10707 : }
10708 43708 : if (N & 3L)
10709 0 : pari_err_DOMAIN("half-integral weight", "N % 4", "!=", gen_0, stoi(N));
10710 43708 : return r >= 0 && mfcharparity(CHI) == 1 && N % F == 0;
10711 : }
10712 :
10713 : /* weight k = r + 1/2 */
10714 : static long
10715 43463 : mf2dim_Nkchi(long N, long r, GEN CHI, ulong space)
10716 : {
10717 43463 : long D, D2, F = mfcharconductor(CHI);
10718 43463 : if (!checkmf2(N, r, CHI, F, space)) return 0;
10719 43442 : if (r == 0) return mf2dimwt12(N, CHI, space);
10720 2772 : if (r == 1) return mf2dimwt32(N, CHI, F, space);
10721 2079 : if (space == mf_EISEN)
10722 : {
10723 693 : D = dim22(N, F, r) + dim22(N, F, 1-r);
10724 693 : if (D & 3L) pari_err_BUG("mfdim");
10725 693 : return D >> 2;
10726 : }
10727 1386 : D2 = space == mf_FULL? dim22(N, F, 1-r): -dim22(N, F, r);
10728 1386 : D = (2*r-1)*mypsiu(N) + 6*D2;
10729 1386 : if (D%24) pari_err_BUG("mfdim");
10730 1386 : return D/24;
10731 : }
10732 :
10733 : /* weight k=r+1/2 */
10734 : static GEN
10735 266 : mf2init_Nkchi(long N, long r, GEN CHI, long space, long flraw)
10736 : {
10737 266 : GEN CHI1, Minv, Minvmat, B, M, gk = gaddsg(r,ghalf);
10738 266 : GEN mf1 = mkvec4(utoi(N),gk,CHI,utoi(space));
10739 : long L;
10740 266 : if (!checkmf2(N, r, CHI, mfcharconductor(CHI), space)) return mfEMPTY(mf1);
10741 266 : if (space==mf_EISEN) pari_err_IMPL("half-integral weight Eisenstein space");
10742 266 : L = mfsturmNgk(N, gk) + 1;
10743 266 : B = mf2basis(N, r, CHI, &CHI1, space);
10744 266 : M = mflineardivtomat(N,B,L); /* defined modulo T = charpol(CHI) */
10745 266 : if (flraw) M = mkvec3(gen_0,gen_0,M);
10746 : else
10747 : {
10748 266 : long o1 = mfcharorder(CHI1), o = mfcharorder(CHI);
10749 266 : M = mfcleanCHI(M, CHI, 0);
10750 266 : Minv = gel(M,2);
10751 266 : Minvmat = RgM_Minv_mul(NULL, Minv); /* mod T */
10752 266 : if (o1 != o)
10753 : {
10754 133 : GEN tr = Qab_trace_init(o, o1, mfcharpol(CHI), mfcharpol(CHI1));
10755 133 : Minvmat = QabM_tracerel(tr, 0, Minvmat);
10756 : }
10757 : /* Minvmat mod T1 = charpol(CHI1) */
10758 266 : B = vecmflineardiv_linear(B, Minvmat);
10759 266 : gel(M,3) = RgM_Minv_mul(gel(M,3), Minv);
10760 266 : gel(M,2) = mkMinv(matid(lg(B)-1), NULL,NULL,NULL);
10761 : }
10762 266 : return mkmf(mf1, cgetg(1,t_VEC), B, gen_0, M);
10763 : }
10764 :
10765 : /**************************************************************************/
10766 : /* Kohnen + space */
10767 : /**************************************************************************/
10768 :
10769 : static GEN
10770 21 : mfkohnenbasis_i(GEN mf, GEN CHI, long eps, long sb)
10771 : {
10772 21 : GEN M = mfcoefs_mf(mf, sb, 1), p, P;
10773 21 : long c, i, n = mfcharorder(CHI), l = sb + 2;
10774 21 : p = cgetg(l, t_VECSMALL);
10775 : /* keep the a_n, n = (2 or 2+eps) mod 4 */
10776 210 : for (i = 3, c = 1; i < l; i+=4) p[c++] = i;
10777 203 : for (i = 3+eps; i < l; i+=4) p[c++] = i;
10778 21 : P = n <= 2? NULL: mfcharpol(CHI);
10779 21 : setlg(p, c);
10780 21 : return QabM_ker(rowpermute(M, p), P, n);
10781 : }
10782 : GEN
10783 21 : mfkohnenbasis(GEN mf)
10784 : {
10785 21 : pari_sp av = avma;
10786 : GEN gk, CHI, CHIP, K;
10787 : long N4, r, eps, sb;
10788 21 : mf = checkMF(mf);
10789 21 : if (MF_get_space(mf) != mf_CUSP)
10790 0 : pari_err_TYPE("mfkohnenbasis [not a cuspidal space", mf);
10791 21 : if (!MF_get_dim(mf)) return cgetg(1, t_MAT);
10792 21 : N4 = MF_get_N(mf) >> 2; gk = MF_get_gk(mf); CHI = MF_get_CHI(mf);
10793 21 : if (typ(gk) == t_INT) pari_err_TYPE("mfkohnenbasis", gk);
10794 21 : r = MF_get_r(mf);
10795 21 : CHIP = mfcharchiliftprim(CHI, N4);
10796 21 : eps = CHIP==CHI? 1: -1;
10797 21 : if (odd(r)) eps = -eps;
10798 21 : if (uissquarefree(N4))
10799 : {
10800 14 : long d = mfdim_Nkchi(N4, 2*r, mfcharpow(CHI, gen_2), mf_CUSP);
10801 14 : sb = mfsturmNgk(N4 << 2, gk) + 1;
10802 14 : K = mfkohnenbasis_i(mf, CHIP, eps, sb);
10803 14 : if (lg(K) - 1 == d) return gc_GEN(av, K);
10804 : }
10805 7 : sb = mfsturmNgk(N4 << 4, gk) + 1;
10806 7 : K = mfkohnenbasis_i(mf, CHIP, eps, sb);
10807 7 : return gc_GEN(av, K);
10808 : }
10809 :
10810 : static GEN
10811 21 : get_Shimura(GEN mf, GEN CHI, GEN vB, long t)
10812 : {
10813 21 : long N = MF_get_N(mf), r = MF_get_k(mf) >> 1;
10814 21 : long i, d = MF_get_dim(mf), sb = mfsturm_mf(mf);
10815 21 : GEN a = cgetg(d+1, t_MAT);
10816 84 : for (i = 1; i <= d; i++)
10817 : {
10818 63 : pari_sp av = avma;
10819 63 : GEN f = c_deflate(sb*sb, t, gel(vB,i));
10820 63 : f = mftobasis_i(mf, RgV_shimura(f, sb, t, N, r, CHI));
10821 63 : gel(a,i) = gc_upto(av, f);
10822 : }
10823 21 : return a;
10824 : }
10825 : static long
10826 35 : QabM_rank(GEN M, GEN P, long n)
10827 : {
10828 35 : GEN z = QabM_indexrank(M, P, n);
10829 35 : return lg(gel(z,2))-1;
10830 : }
10831 : /* discard T[*i] */
10832 : static void
10833 0 : discard_Ti(GEN T, long *i, long *lt)
10834 : {
10835 0 : long j, l = *lt-1;
10836 0 : for (j = *i; j < l; j++) T[j] = T[j+1];
10837 0 : (*i)--; *lt = l;
10838 0 : }
10839 : /* return [mf3, bijection, mfkohnenbasis, codeshi] */
10840 : static GEN
10841 14 : mfkohnenbijection_i(GEN mf)
10842 : {
10843 14 : GEN CHI = MF_get_CHI(mf), K = mfkohnenbasis(mf);
10844 : GEN mres, dMi, Mi, M, C, vB, mf3, SHI, T, P;
10845 14 : long N4 = MF_get_N(mf)>>2, r = MF_get_r(mf), dK = lg(K) - 1;
10846 : long i, c, n, oldr, lt, ltold, sb3, t, limt;
10847 14 : const long MAXlt = 100;
10848 :
10849 14 : mf3 = mfinit_Nkchi(N4, r<<1, mfcharpow(CHI,gen_2), mf_CUSP, 0);
10850 14 : if (MF_get_dim(mf3) != dK)
10851 0 : pari_err_BUG("mfkohnenbijection [different dimensions]");
10852 14 : if (!dK) return mkvec4(mf3, cgetg(1, t_MAT), K, cgetg(1, t_VEC));
10853 14 : CHI = mfcharchiliftprim(CHI, N4);
10854 14 : n = mfcharorder(CHI);
10855 14 : P = n<=2? NULL: mfcharpol(CHI);
10856 14 : SHI = cgetg(MAXlt, t_COL);
10857 14 : T = cgetg(MAXlt, t_VECSMALL);
10858 14 : sb3 = mfsturm_mf(mf3);
10859 14 : limt = 6; oldr = 0; vB = C = M = NULL;
10860 98 : for (t = lt = ltold = 1; lt < MAXlt; t++)
10861 : {
10862 : pari_sp av;
10863 98 : if (!uissquarefree(t)) continue;
10864 84 : T[lt++] = t; if (t <= limt) continue;
10865 14 : av = avma;
10866 14 : if (vB) gunclone(vB);
10867 : /* could improve the rest but 99% of running time is spent here */
10868 14 : vB = gclone( RgM_mul(mfcoefs_mf(mf, t*sb3*sb3, 1), K) );
10869 14 : set_avma(av);
10870 21 : for (i = ltold; i < lt; i++)
10871 : {
10872 : pari_sp av;
10873 : long r;
10874 21 : M = get_Shimura(mf3, CHI, vB, T[i]);
10875 21 : r = QabM_rank(M, P, n); if (!r) { discard_Ti(T, &i, <); continue; }
10876 21 : gel(SHI, i) = M; setlg(SHI, i+1);
10877 21 : if (r >= dK) { C = vecsmall_ei(dK, i); goto DONE; }
10878 14 : if (i == 1) { oldr = r; continue; }
10879 7 : av = avma; M = shallowmatconcat(SHI);
10880 7 : r = QabM_rank(M, P, n); /* >= rank(sum C[j] SHI[j]), probably sharp */
10881 7 : if (r >= dK)
10882 : {
10883 7 : M = RgV_sum(SHI);
10884 7 : if (QabM_rank(M, P, n) >= dK) { C = const_vecsmall(dK, 1); goto DONE; }
10885 0 : C = random_Flv(dK, 16);
10886 0 : M = RgV_zc_mul(SHI, C);
10887 0 : if (QabM_rank(M, P, n) >= dK) goto DONE;
10888 : }
10889 0 : else if (r == oldr) discard_Ti(T, &i, <);
10890 0 : oldr = r; set_avma(av);
10891 : }
10892 0 : limt *= 2; ltold = lt;
10893 : }
10894 0 : pari_err_BUG("mfkohnenbijection");
10895 14 : DONE:
10896 14 : gunclone(vB); lt = lg(SHI);
10897 14 : Mi = QabM_pseudoinv(M,P,n, NULL,&dMi); Mi = RgM_Rg_div(Mi,dMi);
10898 14 : mres = cgetg(lt, t_VEC);
10899 35 : for (i = c = 1; i < lt; i++)
10900 21 : if (C[i]) gel(mres,c++) = mkvec2s(T[i], C[i]);
10901 14 : setlg(mres,c); return mkvec4(mf3, Mi, K, mres);
10902 : }
10903 : GEN
10904 14 : mfkohnenbijection(GEN mf)
10905 : {
10906 14 : pari_sp av = avma;
10907 : long N;
10908 14 : mf = checkMF(mf); N = MF_get_N(mf);
10909 14 : if (!uissquarefree(N >> 2))
10910 0 : pari_err_TYPE("mfkohnenbijection [N/4 not squarefree]", utoi(N));
10911 14 : if (MF_get_space(mf) != mf_CUSP || MF_get_r(mf) == 0 || !mfshimura_space_cusp(mf))
10912 0 : pari_err_TYPE("mfkohnenbijection [incorrect mf for Kohnen]", mf);
10913 14 : return gc_GEN(av, mfkohnenbijection_i(mf));
10914 : }
10915 :
10916 : static int
10917 7 : checkbij_i(GEN b)
10918 : {
10919 7 : return typ(b) == t_VEC && lg(b) == 5 && checkMF_i(gel(b,1))
10920 7 : && typ(gel(b,2)) == t_MAT
10921 7 : && typ(gel(b,3)) == t_MAT
10922 14 : && typ(gel(b,4)) == t_VEC;
10923 : }
10924 :
10925 : /* bij is the output of mfkohnenbijection */
10926 : GEN
10927 7 : mfkohneneigenbasis(GEN mf, GEN bij)
10928 : {
10929 7 : pari_sp av = avma;
10930 : GEN mf3, mf30, B, KM, M, k;
10931 : long r, i, l, N4;
10932 7 : mf = checkMF(mf);
10933 7 : if (!checkbij_i(bij))
10934 0 : pari_err_TYPE("mfkohneneigenbasis [bijection]", bij);
10935 7 : if (MF_get_space(mf) != mf_CUSP)
10936 0 : pari_err_TYPE("mfkohneneigenbasis [not a cuspidal space]", mf);
10937 7 : if (!MF_get_dim(mf))
10938 0 : retmkvec3(cgetg(1, t_VEC), cgetg(1, t_VEC), cgetg(1, t_VEC));
10939 7 : N4 = MF_get_N(mf) >> 2; k = MF_get_gk(mf);
10940 7 : if (typ(k) == t_INT) pari_err_TYPE("mfkohneneigenbasis", k);
10941 7 : if (!uissquarefree(N4))
10942 0 : pari_err_TYPE("mfkohneneigenbasis [N not squarefree]", utoipos(N4));
10943 7 : r = MF_get_r(mf);
10944 7 : KM = RgM_mul(gel(bij,3), gel(bij,2));
10945 7 : mf3 = gel(bij,1);
10946 7 : mf30 = mfinit_Nkchi(N4, 2*r, MF_get_CHI(mf3), mf_NEW, 0);
10947 7 : B = mfcoefs_mf(mf30, mfsturm_mf(mf3), 1); l = lg(B);
10948 7 : M = cgetg(l, t_MAT);
10949 21 : for (i=1; i<l; i++) gel(M,i) = RgM_RgC_mul(KM, mftobasis_i(mf3, gel(B,i)));
10950 7 : return gc_GEN(av, mkvec3(mf30, M, RgM_mul(M, MF_get_newforms(mf30))));
10951 : }
10952 : /*************************** End Kohnen ************************************/
10953 : /***************************************************************************/
10954 :
10955 : static GEN desc(GEN F);
10956 : static GEN
10957 504 : desc_mfeisen(GEN F)
10958 : {
10959 504 : GEN R, gk = mf_get_gk(F);
10960 504 : if (typ(gk) == t_FRAC)
10961 7 : R = gsprintf("H_{%Ps}", gk);
10962 : else
10963 : {
10964 497 : GEN vchi = gel(F, 2), CHI = mfchisimpl(gel(vchi, 3));
10965 497 : long k = itou(gk);
10966 497 : if (lg(vchi) < 5) R = gsprintf("F_%ld(%Ps)", k, CHI);
10967 : else
10968 : {
10969 294 : GEN CHI2 = mfchisimpl(gel(vchi, 4));
10970 294 : R = gsprintf("F_%ld(%Ps, %Ps)", k, CHI, CHI2);
10971 : }
10972 : }
10973 504 : return R;
10974 : }
10975 : static GEN
10976 35 : desc_hecke(GEN F)
10977 : {
10978 : long n, N;
10979 35 : GEN D = gel(F,2);
10980 35 : if (typ(D) == t_VECSMALL) { N = D[3]; n = D[1]; }
10981 14 : else { GEN nN = gel(D,2); n = nN[1]; N = nN[2]; } /* half integer */
10982 35 : return gsprintf("T_%ld(%ld)(%Ps)", N, n, desc(gel(F,3)));
10983 : }
10984 : static GEN
10985 98 : desc_linear(GEN FLD, GEN dL)
10986 : {
10987 98 : GEN F = gel(FLD,2), L = gel(FLD,3), R = strtoGENstr("LIN([");
10988 98 : long n = lg(F) - 1, i;
10989 168 : for (i = 1; i <= n; i++)
10990 : {
10991 168 : R = shallowconcat(R, desc(gel(F,i))); if (i == n) break;
10992 70 : R = shallowconcat(R, strtoGENstr(", "));
10993 : }
10994 98 : return shallowconcat(R, gsprintf("], %Ps)", gdiv(L, dL)));
10995 : }
10996 : static GEN
10997 21 : desc_dihedral(GEN F)
10998 : {
10999 21 : GEN bnr = gel(F,2), D = nf_get_disc(bnr_get_nf(bnr)), f = bnr_get_mod(bnr);
11000 21 : GEN cyc = bnr_get_cyc(bnr);
11001 21 : GEN w = gel(F,3), chin = zv_to_ZV(gel(w,2)), o = utoi(gel(w,1)[1]);
11002 21 : GEN chi = char_denormalize(cyc, o, chin);
11003 21 : if (lg(gel(f,2)) == 1) f = gel(f,1);
11004 21 : return gsprintf("DIH(%Ps, %Ps, %Ps, %Ps)",D,f,cyc,chi);
11005 : }
11006 :
11007 : static void
11008 1043 : unpack0(GEN *U)
11009 1043 : { if (U) *U = mkvec2(cgetg(1, t_VEC), cgetg(1, t_VEC)); }
11010 : static void
11011 42 : unpack2(GEN F, GEN *U)
11012 42 : { if (U) *U = mkvec2(mkvec2(gel(F,2), gel(F,3)), cgetg(1, t_VEC)); }
11013 : static void
11014 308 : unpack23(GEN F, GEN *U)
11015 308 : { if (U) *U = mkvec2(mkvec(gel(F,2)), mkvec(gel(F,3))); }
11016 : static GEN
11017 1540 : desc_i(GEN F, GEN *U)
11018 : {
11019 1540 : switch(mf_get_type(F))
11020 : {
11021 7 : case t_MF_CONST: unpack0(U); return gsprintf("CONST(%Ps)", gel(F,2));
11022 504 : case t_MF_EISEN: unpack0(U); return desc_mfeisen(F);
11023 154 : case t_MF_Ek: unpack0(U); return gsprintf("E_%ld", mf_get_k(F));
11024 63 : case t_MF_DELTA: unpack0(U); return gsprintf("DELTA");
11025 35 : case t_MF_THETA: unpack0(U);
11026 35 : return gsprintf("THETA(%Ps)", mfchisimpl(gel(F,2)));
11027 56 : case t_MF_ETAQUO: unpack0(U);
11028 56 : return gsprintf("ETAQUO(%Ps, %Ps)", gel(F,2), gel(F,3));
11029 56 : case t_MF_ELL: unpack0(U);
11030 56 : return gsprintf("ELL(%Ps)", vecslice(gel(F,2), 1, 5));
11031 7 : case t_MF_TRACE: unpack0(U); return gsprintf("TR(%Ps)", mfparams(F));
11032 140 : case t_MF_NEWTRACE: unpack0(U); return gsprintf("TR^new(%Ps)", mfparams(F));
11033 21 : case t_MF_DIHEDRAL: unpack0(U); return desc_dihedral(F);
11034 28 : case t_MF_MUL: unpack2(F, U);
11035 28 : return gsprintf("MUL(%Ps, %Ps)", desc(gel(F,2)), desc(gel(F,3)));
11036 14 : case t_MF_DIV: unpack2(F, U);
11037 14 : return gsprintf("DIV(%Ps, %Ps)", desc(gel(F,2)), desc(gel(F,3)));
11038 14 : case t_MF_POW: unpack23(F, U);
11039 14 : return gsprintf("POW(%Ps, %ld)", desc(gel(F,2)), itos(gel(F,3)));
11040 14 : case t_MF_SHIFT: unpack23(F, U);
11041 14 : return gsprintf("SHIFT(%Ps, %ld)", desc(gel(F,2)), itos(gel(F,3)));
11042 14 : case t_MF_DERIV: unpack23(F, U);
11043 14 : return gsprintf("DER^%ld(%Ps)", itos(gel(F,3)), desc(gel(F,2)));
11044 21 : case t_MF_DERIVE2: unpack23(F, U);
11045 21 : return gsprintf("DERE2^%ld(%Ps)", itos(gel(F,3)), desc(gel(F,2)));
11046 14 : case t_MF_TWIST: unpack23(F, U);
11047 14 : return gsprintf("TWIST(%Ps, %Ps)", desc(gel(F,2)), gel(F,3));
11048 231 : case t_MF_BD: unpack23(F, U);
11049 231 : return gsprintf("B(%ld)(%Ps)", itou(gel(F,3)), desc(gel(F,2)));
11050 14 : case t_MF_BRACKET:
11051 14 : if (U) *U = mkvec2(mkvec2(gel(F,2), gel(F,3)), mkvec(gel(F,4)));
11052 14 : return gsprintf("MULRC_%ld(%Ps, %Ps)", itos(gel(F,4)), desc(gel(F,2)), desc(gel(F,3)));
11053 98 : case t_MF_LINEAR_BHN:
11054 : case t_MF_LINEAR:
11055 98 : if (U) *U = mkvec2(gel(F,2), mkvec(gdiv(gel(F,3), gel(F,4))));
11056 98 : return desc_linear(F,gel(F,4));
11057 35 : case t_MF_HECKE:
11058 35 : if (U) *U = mkvec2(mkvec(gel(F,3)), mkvec(stoi(gel(F,2)[1])));
11059 35 : return desc_hecke(F);
11060 0 : default: pari_err_TYPE("mfdescribe",F);
11061 : return NULL;/*LCOV_EXCL_LINE*/
11062 : }
11063 : }
11064 : static GEN
11065 623 : desc(GEN F) { return desc_i(F, NULL); }
11066 : GEN
11067 966 : mfdescribe(GEN F, GEN *U)
11068 : {
11069 966 : pari_sp av = avma;
11070 : GEN mf;
11071 966 : if ((mf = checkMF_i(F)))
11072 : {
11073 49 : const char *f = NULL;
11074 49 : switch (MF_get_space(mf))
11075 : {
11076 7 : case mf_NEW: f = "S_%Ps^new(G_0(%ld, %Ps))"; break;
11077 14 : case mf_CUSP: f = "S_%Ps(G_0(%ld, %Ps))"; break;
11078 7 : case mf_OLD: f = "S_%Ps^old(G_0(%ld, %Ps))"; break;
11079 7 : case mf_EISEN:f = "E_%Ps(G_0(%ld, %Ps))"; break;
11080 14 : case mf_FULL: f = "M_%Ps(G_0(%ld, %Ps))"; break;
11081 : }
11082 49 : if (U) *U = cgetg(1, t_VEC);
11083 49 : return gsprintf(f, MF_get_gk(mf), MF_get_N(mf), mfchisimpl(MF_get_CHI(mf)));
11084 : }
11085 917 : if (!checkmf_i(F)) pari_err_TYPE("mfdescribe", F);
11086 917 : F = desc_i(F, U); return gc_all(av, U ? 2: 1, &F, U);
11087 : }
11088 :
11089 : /***********************************************************************/
11090 : /* Eisenstein series H_r of weight r+1/2 */
11091 : /***********************************************************************/
11092 : /* radical(u_ppo(g,q)) */
11093 : static long
11094 28 : u_pporad(long g, long q)
11095 : {
11096 28 : GEN F = myfactoru(g), P = gel(F,1);
11097 : long i, l, n;
11098 28 : if (q == 1) return zv_prod(P);
11099 28 : l = lg(P);
11100 35 : for (i = n = 1; i < l; i++)
11101 : {
11102 7 : long p = P[i];
11103 7 : if (q % p) n *= p;
11104 : }
11105 28 : return n;
11106 : }
11107 : static void
11108 266 : c_F2TH4(long n, GEN *pF2, GEN *pTH4)
11109 : {
11110 266 : GEN v = mfcoefs_i(mfEk(2), n, 1), v2 = bdexpand(v,2), v4 = bdexpand(v,4);
11111 266 : GEN F2 = gdivgs(ZC_add(ZC_sub(v, ZC_z_mul(v2,3)), ZC_z_mul(v4,2)), -24);
11112 266 : GEN TH4 = gdivgs(ZC_sub(v, ZC_z_mul(v4,4)), -3);
11113 266 : settyp(F2,t_VEC); *pF2 = F2;
11114 266 : settyp(TH4,t_VEC);*pTH4= TH4;
11115 266 : }
11116 : /* r > 0, N >= 0 */
11117 : static GEN
11118 77 : mfEHcoef(long r, long N)
11119 : {
11120 : long D0, f, i, l, s;
11121 : GEN S, Df;
11122 :
11123 77 : if (r == 1) return hclassno(utoi(N));
11124 77 : if (N == 0) return gdivgs(bernfrac(2*r), -2*r);
11125 56 : s = N & 3L;
11126 56 : if (odd(r))
11127 : {
11128 42 : if (s == 2 || s == 1) return gen_0;
11129 14 : D0 = mycoredisc2neg(N,&f);
11130 : }
11131 : else
11132 : {
11133 14 : if (s == 2 || s == 3) return gen_0;
11134 14 : D0 = mycoredisc2pos(N,&f);
11135 : }
11136 28 : Df = mydivisorsu(u_pporad(f, D0)); l = lg(Df);
11137 28 : S = gen_0;
11138 63 : for (i = 1; i < l; i++)
11139 : {
11140 35 : long d = Df[i], s = mymoebiusu(d)*kross(D0, d); /* != 0 */
11141 35 : GEN c = gmul(powuu(d, r-1), mysumdivku(f/d, 2*r-1));
11142 35 : S = s > 0? addii(S, c): subii(S, c);
11143 : }
11144 28 : return gmul(lfunquadneg_naive(D0, r), S);
11145 : }
11146 : static GEN
11147 266 : mfEHmat(long lim, long r)
11148 : {
11149 266 : long j, l, d = r/2;
11150 : GEN f2, th4, th3, v, vth4, vf2;
11151 266 : c_F2TH4(lim, &f2, &th4);
11152 266 : f2 = RgV_to_ser(f2, 0, lim+3);
11153 266 : th4 = RgV_to_ser(th4, 0, lim+3);
11154 266 : th3 = RgV_to_ser(c_theta(lim, 1, mfchartrivial()), 0, lim+3);
11155 266 : if (odd(r)) th3 = gpowgs(th3, 3);
11156 266 : vth4 = gpowers(th4, d);
11157 266 : vf2 = gpowers0(f2, d, th3); /* th3 f2^j */
11158 266 : l = d+2; v = cgetg(l, t_VEC);
11159 924 : for (j = 1; j < l; j++)
11160 658 : gel(v, j) = ser2rfrac_i(gmul(gel(vth4, l-j), gel(vf2, j)));
11161 266 : return RgXV_to_RgM(v, lim);
11162 : }
11163 : static GEN
11164 7 : Hfind(long r, GEN *pden)
11165 : {
11166 7 : long lim = (r/2)+3, i;
11167 : GEN res, M, B;
11168 :
11169 7 : if (r <= 0) pari_err_DOMAIN("mfEH", "r", "<=", gen_0, stoi(r));
11170 7 : M = mfEHmat(lim, r);
11171 7 : B = cgetg(lim+1, t_COL);
11172 56 : for (i = 1; i <= lim; i++) gel(B, i) = mfEHcoef(r, i-1);
11173 7 : res = QM_gauss(M, B);
11174 7 : if (lg(res) == 1) pari_err_BUG("mfEH");
11175 7 : return Q_remove_denom(res,pden);
11176 : }
11177 : GEN
11178 266 : mfEH(GEN gk)
11179 : {
11180 266 : pari_sp av = avma;
11181 266 : GEN v, d, NK, gr = gsub(gk, ghalf);
11182 : long r;
11183 266 : if (typ(gr) != t_INT) pari_err_TYPE("mfEH", gk);
11184 266 : r = itos(gr);
11185 266 : switch (r)
11186 : {
11187 7 : case 1: v=cgetg(1,t_VEC); d=gen_1; break;
11188 133 : case 2: v=mkvec2s(1,-20); d=utoipos(120); break;
11189 56 : case 3: v=mkvec2s(-1,14); d=utoipos(252); break;
11190 35 : case 4: v=mkvec3s(1,-16,16); d=utoipos(240); break;
11191 7 : case 5: v=mkvec3s(-1,22,-88); d=utoipos(132); break;
11192 14 : case 6: v=mkvec4s(691,-18096,110136,-4160); d=utoipos(32760); break;
11193 7 : case 7: v=mkvec4s(-1,30,-240,224); d=utoipos(12); break;
11194 7 : default: v = Hfind(r, &d); break;
11195 : }
11196 266 : NK = mkgNK(utoipos(4), gaddgs(ghalf,r), mfchartrivial(), pol_x(1));
11197 266 : return gc_GEN(av, tag(t_MF_EISEN, NK, mkvec2(v,d)));
11198 : }
11199 :
11200 : /**********************************************************/
11201 : /* T(f^2) for half-integral weight */
11202 : /**********************************************************/
11203 :
11204 : /* T_p^2 V, p2 = p^2, c1 = chi(p) (-1/p)^r p^(r-1), c2 = chi(p^2)*p^(2r-1) */
11205 : static GEN
11206 70 : tp2apply(GEN V, long p, long p2, GEN c1, GEN c2)
11207 : {
11208 70 : long lw = (lg(V) - 2)/p2 + 1, m, n;
11209 70 : GEN a0 = gel(V,1), W = cgetg(lw + 1, t_VEC);
11210 :
11211 70 : gel(W,1) = gequal0(a0)? gen_0: gmul(a0, gaddsg(1, c2));
11212 11109 : for (n = 1; n < lw; n++)
11213 : {
11214 11039 : GEN c = gel(V, p2*n + 1);
11215 11039 : if (n%p) c = gadd(c, gmulsg(kross(n,p), gmul(gel(V,n+1), c1)));
11216 11039 : gel(W, n+1) = c; /* a(p^2*n) + c1 * (n/p) a(n) */
11217 : }
11218 1253 : for (m = 1, n = p2; n < lw; m++, n += p2)
11219 1183 : gel(W, n+1) = gadd(gel(W,n+1), gmul(gel(V,m+1), c2));
11220 70 : return W;
11221 : }
11222 :
11223 : /* T_{p^{2e}} V; can derecursify [Purkait, Hecke operators in half-integral
11224 : * weight, Prop 4.3], not worth it */
11225 : static GEN
11226 70 : tp2eapply(GEN V, long p, long p2, long e, GEN q, GEN c1, GEN c2)
11227 : {
11228 70 : GEN V4 = NULL;
11229 70 : if (e > 1)
11230 : {
11231 21 : V4 = vecslice(V, 1, (lg(V) - 2)/(p2*p2) + 1);
11232 21 : V = tp2eapply(V, p, p2, e-1, q, c1, c2);
11233 : }
11234 70 : V = tp2apply(V, p, p2, c1, c2);
11235 70 : if (e > 1)
11236 28 : V = gsub(V, (e == 2)? gmul(q, V4)
11237 7 : : gmul(c2, tp2eapply(V4, p, p2, e-2, q, c1, c2)));
11238 70 : return V;
11239 : }
11240 : /* weight k = r+1/2 */
11241 : static GEN
11242 98 : RgV_heckef2(long n, long d, GEN V, GEN F, GEN DATA)
11243 : {
11244 98 : GEN CHI = mf_get_CHI(F), fa = gel(DATA,1), P = gel(fa,1), E = gel(fa,2);
11245 98 : long i, l = lg(P), r = mf_get_r(F), s4 = odd(r)? -4: 4, k2m2 = (r<<1)-1;
11246 98 : if (typ(V) == t_COL) V = shallowtrans(V);
11247 140 : for (i = 1; i < l; i++)
11248 : { /* p does not divide N */
11249 42 : long p = P[i], e = E[i], p2 = p*p;
11250 42 : GEN c1, c2, a, b, q = NULL, C = mfchareval(CHI,p), C2 = gsqr(C);
11251 42 : a = r? powuu(p,r-1): mkfrac(gen_1,utoipos(p)); /* p^(r-1) = p^(k-3/2) */
11252 42 : b = r? mulii(powuu(p,r), a): a; /* p^(2r-1) = p^(2k-2) */
11253 42 : c1 = gmul(C, gmulsg(kross(s4,p),a));
11254 42 : c2 = gmul(C2, b);
11255 42 : if (e > 1)
11256 : {
11257 14 : q = r? powuu(p,k2m2): a;
11258 14 : if (e == 2) q = gmul(q, uutoQ(p+1,p)); /* special case T_{p^4} */
11259 14 : q = gmul(C2, q); /* chi(p^2) [ p^(2k-2) or (p+1)p^(2k-3) ] */
11260 : }
11261 42 : V = tp2eapply(V, p, p2, e, q, c1, c2);
11262 : }
11263 98 : return c_deflate(n, d, V);
11264 : }
11265 :
11266 : static GEN
11267 1428 : GL2toSL2(GEN g, GEN *abd)
11268 : {
11269 : GEN A, B, C, D, u, v, a, b, d, q;
11270 1428 : g = Q_primpart(g);
11271 1428 : if (!check_M2Z(g)) pari_err_TYPE("GL2toSL2", g);
11272 1428 : A = gcoeff(g,1,1); B = gcoeff(g,1,2);
11273 1428 : C = gcoeff(g,2,1); D = gcoeff(g,2,2);
11274 1428 : a = bezout(A, C, &u, &v);
11275 1428 : if (!equali1(a)) { A = diviiexact(A,a); C = diviiexact(C,a); }
11276 1428 : d = subii(mulii(A,D), mulii(B,C));
11277 1428 : if (signe(d) <= 0) pari_err_TYPE("GL2toSL2",g);
11278 1421 : q = dvmdii(addii(mulii(u,B), mulii(v,D)), d, &b);
11279 1421 : *abd = (equali1(a) && equali1(d))? NULL: mkvec3(a, b, d);
11280 1421 : return mkmat22(A, subii(mulii(q,A), v), C, addii(mulii(q,C), u));
11281 : }
11282 :
11283 : static GEN
11284 8582 : Rg_approx(GEN t, long bit)
11285 : {
11286 8582 : GEN a = real_i(t), b = imag_i(t);
11287 8582 : long e1 = gexpo(a), e2 = gexpo(b);
11288 8582 : if (e2 < -bit) { t = e1 < -bit? gen_0: a; }
11289 6510 : else if (e1 < -bit) t = gmul(b, gen_I());
11290 8582 : return t;
11291 : }
11292 : static GEN
11293 126 : RgV_approx(GEN x, long bit)
11294 840 : { pari_APPLY_same(Rg_approx(gel(x,i), bit)); }
11295 : /* m != 2 (mod 4), D t_INT; V has "denominator" D, recognize in Q(zeta_m) */
11296 : static GEN
11297 126 : bestapprnf2(GEN V, long m, GEN D, long prec)
11298 : {
11299 126 : long i, j, f, vt = fetch_user_var("t"), bit = prec2nbits_mul(prec, 0.8);
11300 126 : GEN Tinit, Vl, H, Pf, P = polcyclo(m, vt);
11301 :
11302 126 : V = liftpol_shallow(V);
11303 126 : V = gmul(RgV_approx(V, bit), D);
11304 126 : V = bestapprnf(V, P, NULL, prec);
11305 126 : Vl = liftpol_shallow(V);
11306 126 : H = coprimes_zv(m);
11307 672 : for (i = 2; i < m; i++)
11308 : {
11309 546 : if (H[i] != 1) continue;
11310 280 : if (!gequal(Vl, vecGalois(Vl, i, P, m))) H[i] = 0;
11311 14 : else for (j = i; j < m; j *= i) H[i] = 3;
11312 : }
11313 126 : f = znstar_conductor_bits(Flv_to_F2v(H));
11314 126 : if (f == 1) return gdiv(V, D);
11315 98 : if (f == m) return gmodulo(gdiv(V, D), P);
11316 7 : Pf = polcyclo(f, vt);
11317 7 : Tinit = Qab_trace_init(m, f, P, Pf);
11318 7 : return gmodulo(gdiv(QabV_tracerel(Tinit, 0, Vl), D), Pf);
11319 : }
11320 :
11321 : /* f | ga expansion; [f, mf_eisendec(f)]~ allowed */
11322 : GEN
11323 1365 : mfslashexpansion(GEN mf, GEN f, GEN ga, long n, long flrat, GEN *params, long prec)
11324 : {
11325 1365 : pari_sp av = avma;
11326 1365 : GEN a, b, d, res, al, V, M, ad, abd, gk, A, awd = NULL;
11327 : long i, w;
11328 :
11329 1365 : mf = checkMF(mf);
11330 1365 : gk = MF_get_gk(mf);
11331 1365 : M = GL2toSL2(ga, &abd);
11332 1358 : if (abd) { a = gel(abd,1); b = gel(abd,2); d = gel(abd,3); }
11333 903 : else { a = d = gen_1; b = gen_0; }
11334 1358 : ad = gdiv(a,d);
11335 1358 : res = mfgaexpansion(mf, f, M, n, prec);
11336 1358 : al = gel(res,1);
11337 1358 : w = itou(gel(res,2));
11338 1358 : V = gel(res,3);
11339 1358 : if (flrat)
11340 : {
11341 126 : GEN CHI = MF_get_CHI(mf);
11342 126 : long N = MF_get_N(mf), F = mfcharconductor(CHI);
11343 126 : long ord = mfcharorder(CHI), k, deg;
11344 126 : long B = umodiu(gcoeff(M,1,2), N);
11345 126 : long C = umodiu(gcoeff(M,2,1), N);
11346 126 : long D = umodiu(gcoeff(M,2,2), N);
11347 126 : long CD = (C * D) % N, BC = (B * C) % F;
11348 : GEN CV, t;
11349 : /* weight of f * Theta in 1/2-integral weight */
11350 126 : k = typ(gk) == t_INT? (long) itou(gk): MF_get_r(mf)+1;
11351 126 : CV = odd(k) ? powuu(N, k - 1) : powuu(N, k >> 1);
11352 126 : deg = ulcm(ulcm(ord, N/ugcd(N,CD)), F/ugcd(F,BC));
11353 126 : if ((deg & 3) == 2) deg >>= 1;
11354 126 : if (typ(gk) != t_INT && odd(deg) && mfthetaI(C,D)) deg <<= 2;
11355 126 : V = bestapprnf2(V, deg, CV, prec);
11356 126 : if (abd && !signe(b))
11357 : { /* can [a,0; 0,d] be simplified to id ? */
11358 7 : long nk, dk; Qtoss(gk, &nk, &dk);
11359 7 : if (ispower(ad, utoipos(2*dk), &t)) /* t^(2*dk) = a/d or t = NULL */
11360 : {
11361 7 : V = RgV_Rg_mul(V, powiu(t,nk));
11362 7 : awd = gdiv(a, muliu(d,w));
11363 : }
11364 : }
11365 : }
11366 1232 : else if (abd)
11367 : { /* ga = M * [a,b;0,d] * rational, F := f | M = q^al * \sum V[j] q^(j/w) */
11368 448 : GEN u, t = NULL, wd = muliu(d,w);
11369 : /* a > 0, 0 <= b < d; f | ga = (a/d)^(k/2) * F(tau + b/d) */
11370 448 : if (signe(b))
11371 : {
11372 : long ns, ds;
11373 : GEN z;
11374 0 : Qtoss(gdiv(b, wd), &ns, &ds); z = rootsof1powinit(ns, ds, prec);
11375 0 : for (i = 1; i <= n+1; i++) gel(V,i) = gmul(gel(V,i), rootsof1pow(z, i-1));
11376 0 : if (!gequal0(al)) t = gexp(gmul(PiI2(prec), gmul(al, gdiv(b,d))), prec);
11377 : }
11378 448 : awd = gdiv(a, wd);
11379 448 : u = gpow(ad, gmul2n(gk,-1), prec);
11380 448 : t = t? gmul(t, u): u;
11381 448 : V = RgV_Rg_mul(V, t);
11382 : }
11383 1358 : if (!awd) A = mkmat22(a, b, gen_0, d);
11384 : else
11385 : { /* rescale and update w from [a,0; 0,d] */
11386 : long ns;
11387 455 : Qtoss(awd, &ns, &w); /* update w */
11388 455 : V = bdexpand(V, ns);
11389 455 : if (!gequal0(al))
11390 : {
11391 0 : GEN adal = gmul(ad, al), sh = gfloor(adal);
11392 0 : al = gsub(adal, sh);
11393 0 : V = RgV_shift(V, sh);
11394 : }
11395 455 : A = matid(2);
11396 : }
11397 1358 : if (params) *params = mkvec3(al, utoipos(w), A);
11398 1358 : return gc_all(av,params?2:1,&V,params);
11399 : }
11400 :
11401 : /**************************************************************/
11402 : /* Alternative method for 1/2-integral weight */
11403 : /**************************************************************/
11404 : static GEN
11405 266 : mf2basis(long N, long r, GEN CHI, GEN *pCHI1, long space)
11406 : {
11407 : GEN CHI1, CHI2, mf1, mf2, B1, B2, BT, M1, M2, M, M2i, T, Th, v, den;
11408 266 : long sb, N2, o1, o2, k1 = r + 1;
11409 :
11410 266 : if (odd(k1))
11411 : {
11412 154 : CHI1 = mfcharmul(CHI, get_mfchar(stoi(-4)));
11413 154 : CHI2 = mfcharmul(CHI, get_mfchar(stoi(-8)));
11414 : }
11415 : else
11416 : {
11417 112 : CHI1 = CHI;
11418 112 : CHI2 = mfcharmul(CHI, get_mfchar(utoi(8)));
11419 : }
11420 266 : mf1 = mfinit_Nkchi(N, k1, CHI1, space, 1);
11421 266 : if (pCHI1) *pCHI1 = CHI1;
11422 266 : B1 = MF_get_basis(mf1); if (lg(B1) == 1) return cgetg(1,t_VEC);
11423 259 : N2 = ulcm(8, N);
11424 259 : mf2 = mfinit_Nkchi(N2, k1, CHI2, space, 1);
11425 259 : B2 = MF_get_basis(mf2); if (lg(B2) == 1) return cgetg(1,t_VEC);
11426 259 : sb = mfsturmNgk(N2, gaddsg(k1, ghalf));
11427 259 : M1 = mfcoefs_mf(mf1, sb, 1);
11428 259 : M2 = mfcoefs_mf(mf2, sb, 1);
11429 259 : Th = mfTheta(NULL);
11430 259 : BT = mfcoefs_i(Th, sb, 1);
11431 259 : M1 = mfmatsermul(M1, RgV_to_RgX(expandbd(BT,2),0));
11432 259 : M2 = mfmatsermul(M2, RgV_to_RgX(BT,0));
11433 259 : o1= mfcharorder(CHI1);
11434 259 : T = (o1 <= 2)? NULL: mfcharpol(CHI1);
11435 259 : if (o1 > 2) M1 = liftpol_shallow(M1);
11436 259 : o2= mfcharorder(CHI2);
11437 259 : if (T)
11438 : {
11439 14 : if (o2 == o1) M2 = liftpol_shallow(M2);
11440 : else
11441 : {
11442 0 : GEN tr = Qab_trace_init(o2, o1, mfcharpol(CHI2), mfcharpol(CHI1));
11443 0 : M2 = QabM_tracerel(tr, 0, M2);
11444 : }
11445 : }
11446 : /* now everything is defined mod T = mfcharpol(CHI1) */
11447 259 : M2i = QabM_pseudoinv_i(M2, T, o1, &v, &den);
11448 259 : M = RgM_mul(M2i, rowpermute(M1, gel(v,1)));
11449 259 : M = RgM_mul(M2, M);
11450 259 : M1 = RgM_Rg_mul(M1, den);
11451 259 : M = RgM_sub(M1, M); if (T) M = RgXQM_red(M, T);
11452 259 : return vecmflineardiv0(B1, QabM_ker(M, T, o1), Th);
11453 : }
11454 :
11455 : /*******************************************************************/
11456 : /* Integration */
11457 : /*******************************************************************/
11458 : static GEN
11459 490 : vanembed(GEN F, GEN v, long prec)
11460 : {
11461 490 : GEN CHI = mf_get_CHI(F);
11462 490 : long o = mfcharorder(CHI);
11463 490 : if (o > 2 || degpol(mf_get_field(F)) > 1) v = liftpol_shallow(v);
11464 490 : if (o > 2) v = gsubst(v, varn(mfcharpol(CHI)), rootsof1u_cx(o, prec));
11465 490 : return v;
11466 : }
11467 :
11468 : static long
11469 1253 : mfperiod_prelim_double(double t0, long k, long bitprec)
11470 : {
11471 1253 : double nlim, c = 2*M_PI*t0;
11472 1253 : nlim = ceil(bitprec * M_LN2 / c);
11473 1253 : c -= (k - 1)/(2*nlim); if (c < 1) c = 1.;
11474 1253 : nlim += ceil((0.7 + (k-1)/2*log(nlim))/c);
11475 1253 : return (long)nlim;
11476 : }
11477 : static long
11478 301 : mfperiod_prelim(GEN t0, long k, long bitprec)
11479 301 : { return mfperiod_prelim_double(gtodouble(t0), k, bitprec); }
11480 :
11481 : /* (-X)^(k-2) * P(-1/X) = (-1)^{k-2} P|_{k-2} S */
11482 : static GEN
11483 1288 : RgX_act_S(GEN P, long k)
11484 : {
11485 1288 : P = RgX_unscale(RgX_recipspec_shallow(P+2, lgpol(P), k-1), gen_m1);
11486 1288 : setvarn(P,0); return P;
11487 : }
11488 : static int
11489 2842 : RgX_act_typ(GEN P, long k)
11490 : {
11491 2842 : switch(typ(P))
11492 : {
11493 35 : case t_RFRAC: return t_RFRAC;
11494 2807 : case t_POL:
11495 2807 : if (varn(P) == 0)
11496 : {
11497 2807 : long d = degpol(P);
11498 2807 : if (d > k-2) return t_RFRAC;
11499 2653 : if (d) return t_POL;
11500 : }
11501 : }
11502 1211 : return 0;
11503 : }
11504 : static GEN
11505 2576 : act_S(GEN P, long k)
11506 : {
11507 : GEN X;
11508 2576 : switch(RgX_act_typ(P, k))
11509 : {
11510 140 : case t_RFRAC:
11511 140 : X = gneg(pol_x(0));
11512 140 : return gmul(gsubst(P, 0, ginv(X)), gpowgs(X, k - 2));
11513 1288 : case t_POL: return RgX_act_S(P, k);
11514 : }
11515 1148 : return P;
11516 : }
11517 :
11518 : static GEN
11519 203 : AX_B(GEN M)
11520 203 : { GEN A = gcoeff(M,1,1), B = gcoeff(M,1,2); return deg1pol_shallow(A,B,0); }
11521 : static GEN
11522 203 : CX_D(GEN M)
11523 203 : { GEN C = gcoeff(M,2,1), D = gcoeff(M,2,2); return deg1pol_shallow(C,D,0); }
11524 :
11525 : /* P|_{2-k}M = (CX+D)^{k-2}P((AX+B)/(CX+D)) */
11526 : static GEN
11527 154 : RgX_act_gen(GEN P, GEN M, long k)
11528 : {
11529 154 : GEN S = gen_0, PCD, PAB;
11530 : long i;
11531 154 : PCD = gpowers(CX_D(M), k-2);
11532 154 : PAB = gpowers(AX_B(M), k-2);
11533 833 : for (i = 0; i <= k-2; i++)
11534 : {
11535 679 : GEN t = RgX_coeff(P, i);
11536 679 : if (!gequal0(t)) S = gadd(S, gmul(t, gmul(gel(PCD, k-i-1), gel(PAB, i+1))));
11537 : }
11538 154 : return S;
11539 : }
11540 : static GEN
11541 266 : act_GL2(GEN P, GEN M, long k)
11542 : {
11543 266 : switch(RgX_act_typ(P, k))
11544 : {
11545 49 : case t_RFRAC:
11546 : {
11547 49 : GEN AB = AX_B(M), CD = CX_D(M);
11548 49 : return gmul(gsubst(P, 0, gdiv(AB, CD)), gpowgs(CD, k - 2));
11549 : }
11550 154 : case t_POL: return RgX_act_gen(P, M, k);
11551 : }
11552 63 : return P;
11553 : }
11554 : static GEN
11555 7 : vecact_GL2(GEN x, GEN M, long k)
11556 21 : { pari_APPLY_same(act_GL2(gel(x,i), M, k)); }
11557 :
11558 : static GEN
11559 2863 : RgX_approx(GEN x, long bit)
11560 10731 : { pari_APPLY_pol_normalized(Rg_approx(gel(x,i),bit)); }
11561 :
11562 : static GEN normalizeapprox(GEN x, long bit);
11563 : static GEN
11564 2898 : normalizeapprox_i(GEN x, long bit)
11565 : {
11566 2898 : GEN D = gen_1;
11567 2954 : if (is_vec_t(typ(x))) pari_APPLY_same(normalizeapprox(gel(x,i), bit));
11568 2870 : if (typ(x) == t_RFRAC && varn(gel(x,2)) == 0) { D = gel(x,2); x = gel(x,1); }
11569 2870 : if (typ(x) != t_POL || varn(x) != 0) return gdiv(x, D);
11570 2863 : return gdiv(RgX_approx(x, bit), D);
11571 : }
11572 : static GEN
11573 56 : normalizeapprox(GEN x, long bit)
11574 : {
11575 56 : pari_sp av = avma;
11576 56 : return gc_upto(av, normalizeapprox_i(x, bit));
11577 : }
11578 :
11579 : /* make sure T is a t_POL in variable 0 */
11580 : static GEN
11581 2863 : toRgX0(GEN T)
11582 2863 : { return typ(T) == t_POL && varn(T) == 0? T: scalarpol_shallow(T,0); }
11583 :
11584 : /* integrate by summing nlim+1 terms of van [may be < lg(van)]
11585 : * van can be an expansion with vector coefficients
11586 : * \int_A^oo \sum_n van[n] * q^(n/w + al) * P(z-A) dz, q = e(z) */
11587 : static GEN
11588 945 : intAoo(GEN van, long nlim, GEN al, long w, GEN P, GEN A, long k, long prec)
11589 : {
11590 : GEN alw, P1, piI2A, q, S, van0;
11591 945 : long n, vz = varn(gel(P,2));
11592 :
11593 945 : if (nlim < 1) nlim = 1;
11594 945 : alw = gmulsg(w, al);
11595 945 : P1 = RgX_Rg_translate(P, gneg(A));
11596 945 : piI2A = gmul(PiI2n(1, prec), A);
11597 945 : q = gexp(gdivgu(piI2A, w), prec);
11598 945 : S = gen_0;
11599 121674 : for (n = nlim; n >= 1; n--)
11600 : {
11601 120729 : GEN t = gsubst(P1, vz, gdivsg(w, gaddsg(n, alw)));
11602 120729 : S = gadd(gmul(gel(van, n+1), t), gmul(q, S));
11603 : }
11604 945 : S = gmul(q, S);
11605 945 : van0 = gel(van, 1);
11606 945 : if (!gequal0(al))
11607 : {
11608 42 : S = gadd(S, gmul(gsubst(P1, vz, ginv(al)), van0));
11609 42 : S = gmul(S, gexp(gmul(piI2A, al), prec));
11610 : }
11611 903 : else if (!gequal0(van0))
11612 231 : S = gsub(S, gdivgu(gmul(van0, gpowgs(gsub(pol_x(0), A), k-1)), k-1));
11613 945 : if (is_vec_t(typ(S)))
11614 : {
11615 637 : long j, l = lg(S);
11616 3192 : for (j = 1; j < l; j++) gel(S,j) = toRgX0(gel(S,j));
11617 : }
11618 : else
11619 308 : S = toRgX0(S);
11620 945 : return gneg(S);
11621 : }
11622 :
11623 : /* \sum_{j <= k} X^j * (Y / (2I\pi))^{k+1-j} k! / j! */
11624 : static GEN
11625 259 : get_P(long k, long v, long prec)
11626 : {
11627 259 : GEN a, S = cgetg(k + 1, t_POL), u = invr(Pi2n(1, prec+EXTRAPREC64));
11628 259 : long j, K = k-2;
11629 259 : S[1] = evalsigne(1)|evalvarn(0); a = u;
11630 259 : gel(S,K+2) = monomial(mulcxpowIs(a,3), 1, v); /* j = K */
11631 1176 : for(j = K-1; j >= 0; j--)
11632 : {
11633 917 : a = mulrr(mulru(a,j+1), u);
11634 917 : gel(S,j+2) = monomial(mulcxpowIs(a,3*(K+1-j)), K+1-j, v);
11635 : }
11636 259 : return S;
11637 : }
11638 :
11639 : static GEN
11640 2555 : getw1w2(long N, GEN ga)
11641 2555 : { return mkvecsmall2(mfZC_width(N, gel(ga,1)),
11642 2555 : mfZC_width(N, gel(ga,2))); }
11643 :
11644 : static GEN
11645 147 : intAoowithvanall(GEN mf, GEN vanall, GEN P, GEN cosets, long bitprec)
11646 : {
11647 147 : GEN vvan = gel(vanall,1), vaw = gel(vanall,2), W1W2, resall;
11648 147 : long prec = nbits2prec(bitprec), N, k, lco, j;
11649 :
11650 147 : N = MF_get_N(mf); k = MF_get_k(mf);
11651 147 : lco = lg(cosets);
11652 147 : W1W2 = cgetg(lco, t_VEC); resall = cgetg(lco, t_VEC);
11653 2702 : for (j = 1; j < lco; j++) gel(W1W2,j) = getw1w2(N, gel(cosets, j));
11654 2702 : for (j = 1; j < lco; j++)
11655 : {
11656 2555 : GEN w1w2j = gel(W1W2,j), alj, M, VAN, RES, AR, Q;
11657 : long jq, c, w1, w2, w;
11658 2555 : if (!w1w2j) continue;
11659 637 : alj = gel(vaw,j);
11660 637 : w1 = w1w2j[1]; Q = cgetg(lco, t_VECSMALL);
11661 637 : w2 = w1w2j[2]; M = cgetg(lco, t_COL);
11662 8267 : for (c = 1, jq = j; jq < lco; jq++)
11663 : {
11664 7630 : GEN W = gel(W1W2, jq);
11665 7630 : if (jq == j || (W && gequal(W, w1w2j) && gequal(gel(vaw, jq), alj)))
11666 : {
11667 2555 : Q[c] = jq; gel(W1W2, jq) = NULL;
11668 2555 : gel(M, c) = gel(vvan, jq); c++;
11669 : }
11670 : }
11671 637 : setlg(M,c); VAN = shallowmatconcat(M);
11672 637 : AR = mkcomplex(gen_0, sqrtr_abs(divru(utor(w1, prec+EXTRAPREC64), w2)));
11673 637 : w = itos(gel(alj,2));
11674 637 : RES = intAoo(VAN, lg(VAN)-2, gel(alj,1),w, P, AR, k, prec);
11675 3192 : for (jq = 1; jq < c; jq++) gel(resall, Q[jq]) = gel(RES, jq);
11676 : }
11677 147 : return resall;
11678 : }
11679 :
11680 : GEN
11681 539 : mftobasisES(GEN mf, GEN F)
11682 : {
11683 539 : GEN v = mftobasis(mf, F, 0);
11684 532 : long nE = lg(MF_get_E(mf))-1;
11685 532 : return mkvec2(vecslice(v,1,nE), vecslice(v,nE+1,lg(v)-1));
11686 : }
11687 :
11688 : static long
11689 0 : wt1mulcond(GEN F, long D, long space)
11690 : {
11691 0 : GEN E = mfeisenstein_i(1, mfchartrivial(), get_mfchar(stoi(D))), mf;
11692 0 : F = mfmul(F, E);
11693 0 : mf = mfinit_Nkchi(mf_get_N(F), mf_get_k(F), mf_get_CHI(F), space, 0);
11694 0 : return mfconductor(mf, F);
11695 : }
11696 : static int
11697 7 : wt1newlevel(long N)
11698 : {
11699 7 : GEN P = gel(myfactoru(N),1);
11700 7 : long l = lg(P), i;
11701 14 : for (i = 1; i < l; i++)
11702 7 : if (!wt1empty(N/P[i])) return 0;
11703 7 : return 1;
11704 : }
11705 : long
11706 175 : mfconductor(GEN mf, GEN F)
11707 : {
11708 175 : pari_sp av = avma;
11709 : GEN gk;
11710 : long space, N, M;
11711 :
11712 175 : mf = checkMF(mf);
11713 175 : if (!checkmf_i(F)) pari_err_TYPE("mfconductor",F);
11714 175 : if (mfistrivial(F)) return 1;
11715 175 : space = MF_get_space(mf);
11716 175 : if (space == mf_NEW) return mf_get_N(F);
11717 175 : gk = MF_get_gk(mf);
11718 175 : if (isint1(gk))
11719 : {
11720 7 : N = mf_get_N(F);
11721 7 : if (!wt1newlevel(N))
11722 : {
11723 0 : long s = space_is_cusp(space)? mf_CUSP: mf_FULL;
11724 0 : N = ugcd(N, wt1mulcond(F,-3,s));
11725 0 : if (!wt1newlevel(N)) N = ugcd(N, wt1mulcond(F,-4,s));
11726 : }
11727 7 : return gc_long(av,N);
11728 : }
11729 168 : if (typ(gk) != t_INT)
11730 : {
11731 42 : F = mfmultheta(F);
11732 42 : mf = obj_checkbuild(mf, MF_MF2INIT, &mf2init); /* mf_FULL */
11733 : }
11734 168 : N = 1;
11735 168 : if (space_is_cusp(space))
11736 : {
11737 7 : F = mftobasis_i(mf, F);
11738 7 : if (typ(gk) != t_INT) F = vecslice(F, lg(MF_get_E(mf)), lg(F) - 1);
11739 : }
11740 : else
11741 : {
11742 161 : GEN EF = mftobasisES(mf, F), vE = gel(EF,1), B = MF_get_E(mf);
11743 161 : long i, l = lg(B);
11744 1267 : for (i = 1; i < l; i++)
11745 1106 : if (!gequal0(gel(vE,i))) N = ulcm(N, mf_get_N(gel(B, i)));
11746 161 : F = gel(EF,2);
11747 : }
11748 168 : (void)mftonew_i(mf, F, &M); /* M = conductor of cuspidal part */
11749 168 : return gc_long(av, ulcm(M, N));
11750 : }
11751 :
11752 : static GEN
11753 1463 : fs_get_MF(GEN fs) { return gel(fs,1); }
11754 : static GEN
11755 847 : fs_get_vES(GEN fs) { return gel(fs,2); }
11756 : static GEN
11757 1596 : fs_get_pols(GEN fs) { return gel(fs,3); }
11758 : static GEN
11759 2191 : fs_get_cosets(GEN fs) { return gel(fs,4); }
11760 : static long
11761 630 : fs_get_bitprec(GEN fs) { return itou(gel(fs,5)); }
11762 : static GEN
11763 1246 : fs_get_vE(GEN fs) { return gel(fs,6); }
11764 : static GEN
11765 70 : fs_get_EF(GEN fs) { return gel(fs,7); }
11766 : static GEN
11767 1890 : fs_get_expan(GEN fs) { return gel(fs,8); }
11768 : static GEN
11769 28 : fs_set_expan(GEN fs, GEN vanall)
11770 28 : { GEN f = shallowcopy(fs); gel(f,8) = vanall; return f; }
11771 : static int
11772 49 : mfs_checkmf(GEN fs, GEN mf)
11773 49 : { GEN mfF = fs_get_MF(fs); return gequal(gel(mfF,1), gel(mf,1)); }
11774 : static long
11775 798 : checkfs_i(GEN v)
11776 798 : { return typ(v) == t_VEC && lg(v) == 9 && checkMF_i(fs_get_MF(v))
11777 567 : && typ(fs_get_vES(v)) == t_VEC
11778 567 : && typ(fs_get_pols(v)) == t_VEC
11779 567 : && typ(fs_get_cosets(v)) == t_VEC
11780 567 : && typ(fs_get_vE(v)) == t_VEC
11781 567 : && lg(fs_get_pols(v)) == lg(fs_get_cosets(v))
11782 567 : && typ(fs_get_expan(v)) == t_VEC
11783 567 : && lg(fs_get_expan(v)) == 3
11784 567 : && lg(gel(fs_get_expan(v), 1)) == lg(fs_get_cosets(v))
11785 1596 : && typ(gel(v,5)) == t_INT; }
11786 : GEN
11787 19187 : checkMF_i(GEN mf)
11788 : {
11789 19187 : long l = lg(mf);
11790 : GEN v;
11791 19187 : if (typ(mf) != t_VEC) return NULL;
11792 19159 : if (l == 9) return checkMF_i(fs_get_MF(mf));
11793 19159 : if (l != 7) return NULL;
11794 7924 : v = gel(mf,1);
11795 7924 : if (typ(v) != t_VEC || lg(v) != 5) return NULL;
11796 7924 : return (typ(gel(v,1)) == t_INT
11797 7924 : && typ(gmul2n(gel(v,2), 1)) == t_INT
11798 7924 : && typ(gel(v,3)) == t_VEC
11799 15848 : && typ(gel(v,4)) == t_INT)? mf: NULL; }
11800 : GEN
11801 4193 : checkMF(GEN T)
11802 : {
11803 4193 : GEN mf = checkMF_i(T);
11804 4193 : if (!mf) pari_err_TYPE("checkMF [please use mfinit]", T);
11805 4193 : return mf;
11806 : }
11807 :
11808 : /* c,d >= 0; c * Nc = N, find coset whose image in P1(Z/NZ) ~ (c, d + k(N/c)) */
11809 : static GEN
11810 11963 : coset_complete(long c, long d, long Nc)
11811 : {
11812 : long a, b;
11813 13307 : while (ugcd(c, d) > 1) d += Nc;
11814 11963 : (void)cbezout(c, d, &b, &a);
11815 11963 : return mkmat22s(a, -b, c, d);
11816 : }
11817 : /* right cosets of $\G_0(N)$: $\G=\bigsqcup_j \G_0(N)\ga_j$. */
11818 : /* We choose them with c\mid N and d mod N/c, not the reverse */
11819 : GEN
11820 168 : mfcosets(GEN gN)
11821 : {
11822 168 : pari_sp av = avma;
11823 : GEN V, D, mf;
11824 168 : long l, i, ct, N = 0;
11825 168 : if (typ(gN) == t_INT) N = itos(gN);
11826 14 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
11827 7 : else pari_err_TYPE("mfcosets", gN);
11828 161 : if (N <= 0) pari_err_DOMAIN("mfcosets", "N", "<=", gen_0, stoi(N));
11829 161 : V = cgetg(mypsiu(N) + 1, t_VEC);
11830 161 : D = mydivisorsu(N); l = lg(D);
11831 588 : for (i = ct = 1; i < l; i++)
11832 : {
11833 427 : long d, c = D[i], Nc = D[l-i], e = ugcd(Nc, c);
11834 3332 : for (d = 0; d < Nc; d++)
11835 2905 : if (ugcd(d,e) == 1) gel(V, ct++) = coset_complete(c, d, Nc);
11836 : }
11837 161 : return gc_GEN(av, V);
11838 : }
11839 : static int
11840 35469 : cmp_coset(void *E, GEN A, GEN B)
11841 : {
11842 35469 : ulong N = (ulong)E, Nc, c = itou(gcoeff(A,2,1));
11843 35469 : int r = cmpuu(c, itou(gcoeff(B,2,1)));
11844 35469 : if (r) return r;
11845 30660 : Nc = N / c;
11846 30660 : return cmpuu(umodiu(gcoeff(A,2,2), Nc), umodiu(gcoeff(B,2,2), Nc));
11847 : }
11848 : /* M in SL_2(Z) */
11849 : static long
11850 9198 : mftocoset_i(ulong N, GEN M, GEN cosets)
11851 : {
11852 9198 : pari_sp av = avma;
11853 9198 : long A = itos(gcoeff(M,1,1)), c, u, v, Nc, i;
11854 9198 : long C = itos(gcoeff(M,2,1)), D = itos(gcoeff(M,2,2));
11855 : GEN ga;
11856 9198 : c = cbezout(N*A, C, &u, &v); Nc = N/c;
11857 9198 : ga = coset_complete(c, umodsu(v*D, Nc), Nc);
11858 9198 : i = gen_search(cosets, ga, (void*)N, &cmp_coset);
11859 9198 : if (i < 0) pari_err_BUG("mftocoset [no coset found]");
11860 9198 : return gc_long(av,i);
11861 : }
11862 : /* (U * V^(-1))[2,2] mod N, assuming V in SL2(Z) */
11863 : static long
11864 9177 : SL2_div_D(ulong N, GEN U, GEN V)
11865 : {
11866 9177 : long c = umodiu(gcoeff(U,2,1), N), d = umodiu(gcoeff(U,2,2), N);
11867 9177 : long a2 = umodiu(gcoeff(V,1,1), N), b2 = umodiu(gcoeff(V,1,2), N);
11868 9177 : return (a2*d - b2*c) % (long)N;
11869 : }
11870 : static long
11871 9177 : mftocoset_iD(ulong N, GEN M, GEN cosets, long *D)
11872 : {
11873 9177 : long i = mftocoset_i(N, M, cosets);
11874 9177 : *D = SL2_div_D(N, M, gel(cosets,i)); return i;
11875 : }
11876 : GEN
11877 7 : mftocoset(ulong N, GEN M, GEN cosets)
11878 : {
11879 : long i;
11880 7 : if (!check_SL2Z(M)) pari_err_TYPE("mftocoset",M);
11881 7 : i = mftocoset_i(N, M, cosets);
11882 7 : retmkvec2(gdiv(M,gel(cosets,i)), utoipos(i));
11883 : }
11884 :
11885 : static long
11886 2555 : getnlim2(long N, long w1, long w2, long nlim, long k, long bitprec)
11887 : {
11888 2555 : if (w2 == N) return nlim;
11889 483 : return mfperiod_prelim_double(1./sqrt((double)w1*w2), k, bitprec + 32);
11890 : }
11891 :
11892 : /* g * S, g 2x2 */
11893 : static GEN
11894 1337 : ZM_mulS(GEN g)
11895 1337 : { return mkmat2(gel(g,2), ZC_neg(gel(g,1))); }
11896 : /* g * T, g 2x2 */
11897 : static GEN
11898 4634 : ZM_mulT(GEN g)
11899 4634 : { return mkmat2(gel(g,1), ZC_add(gel(g,2), gel(g,1))); }
11900 : /* g * T^(-1), g 2x2 */
11901 : static GEN
11902 2352 : ZM_mulTi(GEN g)
11903 2352 : { return mkmat2(gel(g,1), ZC_sub(gel(g,2), gel(g,1))); }
11904 :
11905 : /* Compute all slashexpansions for all cosets */
11906 : static GEN
11907 175 : mfgaexpansionall(GEN mf, GEN FE, GEN cosets, double height, long prec)
11908 : {
11909 175 : GEN CHI = MF_get_CHI(mf), vres, vresaw;
11910 175 : long lco, j, k = MF_get_k(mf), N = MF_get_N(mf), bitprec = prec2nbits(prec);
11911 :
11912 175 : lco = lg(cosets);
11913 175 : vres = const_vec(lco-1, NULL);
11914 175 : vresaw = cgetg(lco, t_VEC);
11915 2912 : for (j = 1; j < lco; j++) if (!gel(vres,j))
11916 : {
11917 455 : GEN ga = gel(cosets, j), van, aw, al, z, gai;
11918 455 : long w1 = mfZC_width(N, gel(ga,1));
11919 455 : long w2 = mfZC_width(N, gel(ga,2));
11920 : long nlim, nlim2, daw, da, na, i;
11921 455 : double sqNinvdbl = height ? height/w1 : 1./sqrt((double)w1*N);
11922 455 : nlim = mfperiod_prelim_double(sqNinvdbl, k, bitprec + 32);
11923 455 : van = mfslashexpansion(mf, FE, ga, nlim, 0, &aw, prec + EXTRAPREC64);
11924 455 : van = vanembed(gel(FE, 1), van, prec + EXTRAPREC64);
11925 455 : al = gel(aw, 1);
11926 455 : nlim2 = height? nlim: getnlim2(N, w1, w2, nlim, k, bitprec);
11927 455 : gel(vres, j) = vecslice(van, 1, nlim2 + 1);
11928 455 : gel(vresaw, j) = aw;
11929 455 : Qtoss(al, &na, &da); daw = da*w1;
11930 455 : z = rootsof1powinit(1, daw, prec + EXTRAPREC64);
11931 455 : gai = ga;
11932 2737 : for (i = 1; i < w1; i++)
11933 : {
11934 : GEN V, coe;
11935 2282 : long Di, n, ind, w2, s = ((i*na) % da) * w1, t = i*da;
11936 2282 : gai = ZM_mulT(gai);
11937 2282 : ind = mftocoset_iD(N, gai, cosets, &Di);
11938 2282 : w2 = mfZC_width(N, gel(gel(cosets,ind), 2));
11939 2282 : nlim2 = height? nlim: getnlim2(N, w1, w2, nlim, k, bitprec);
11940 2282 : gel(vresaw, ind) = aw;
11941 2282 : V = cgetg(nlim2 + 2, t_VEC);
11942 909034 : for (n = 0; n <= nlim2; n++, s = Fl_add(s, t, daw))
11943 906752 : gel(V, n+1) = gmul(gel(van, n+1), rootsof1pow(z, s));
11944 2282 : coe = mfcharcxeval(CHI, Di, prec + EXTRAPREC64);
11945 2282 : if (!gequal1(coe)) V = RgV_Rg_mul(V, conj_i(coe));
11946 2282 : gel(vres, ind) = V;
11947 : }
11948 : }
11949 175 : return mkvec2(vres, vresaw);
11950 : }
11951 :
11952 : /* Compute all period pols of F|_k\ga_j, vF = mftobasis(F_S) */
11953 : static GEN
11954 168 : mfperiodpols_i(GEN mf, GEN FE, GEN cosets, GEN *pvan, long bit)
11955 : {
11956 168 : long N, i, prec = nbits2prec(bit), k = MF_get_k(mf);
11957 168 : GEN vP, P, CHI, intall = gen_0;
11958 :
11959 168 : *pvan = gen_0;
11960 168 : if (k == 0 && gequal0(gel(FE,2)))
11961 0 : return cosets? const_vec(lg(cosets)-1, pol_0(0)): pol_0(0);
11962 168 : N = MF_get_N(mf);
11963 168 : CHI = MF_get_CHI(mf);
11964 168 : P = get_P(k, fetch_var(), prec);
11965 168 : if (!cosets)
11966 : { /* ga = id */
11967 21 : long nlim, PREC = prec + EXTRAPREC64;
11968 21 : GEN F = gel(FE,1), sqNinv = invr(sqrtr_abs(utor(N, PREC))); /* A/w */
11969 : GEN AR, v, van, T1, T2;
11970 :
11971 21 : nlim = mfperiod_prelim(sqNinv, k, bit + 32);
11972 : /* F|id: al = 0, w = 1 */
11973 21 : v = mfcoefs_i(F, nlim, 1);
11974 21 : van = vanembed(F, v, PREC);
11975 21 : AR = mkcomplex(gen_0, sqNinv);
11976 21 : T1 = intAoo(van, nlim, gen_0,1, P, AR, k, prec);
11977 21 : if (N == 1) T2 = T1;
11978 : else
11979 : { /* F|S: al = 0, w = N */
11980 7 : v = mfgaexpansion(mf, FE, mkS(), nlim, PREC);
11981 7 : van = vanembed(F, gel(v,3), PREC);
11982 7 : AR = mkcomplex(gen_0, mulur(N,sqNinv));
11983 7 : T2 = intAoo(van, nlim, gen_0,N, P, AR, k, prec);
11984 : }
11985 21 : T1 = gsub(T1, act_S(T2, k));
11986 21 : T1 = normalizeapprox_i(T1, bit-20);
11987 21 : vP = gprec_wtrunc(T1, prec);
11988 : }
11989 : else
11990 : {
11991 147 : long lco = lg(cosets);
11992 147 : GEN vanall = mfgaexpansionall(mf, FE, cosets, 0, prec);
11993 147 : *pvan = vanall;
11994 147 : intall = intAoowithvanall(mf, vanall, P, cosets, bit);
11995 147 : vP = const_vec(lco-1, NULL);
11996 2702 : for (i = 1; i < lco; i++)
11997 : {
11998 2555 : GEN P, P1, P2, c, ga = gel(cosets, i);
11999 : long iS, DS;
12000 2646 : if (gel(vP,i)) continue;
12001 1323 : P1 = gel(intall, i);
12002 1323 : iS = mftocoset_iD(N, ZM_mulS(ga), cosets, &DS);
12003 1323 : c = mfcharcxeval(CHI, DS, prec + EXTRAPREC64);
12004 1323 : P2 = gel(intall, iS);
12005 :
12006 1323 : P = act_S(isint1(c)? P2: gmul(c, P2), k);
12007 1323 : P = normalizeapprox_i(gsub(P1, P), bit-20);
12008 1323 : gel(vP,i) = gprec_wtrunc(P, prec);
12009 1323 : if (iS == i) continue;
12010 :
12011 1232 : P = act_S(isint1(c)? P1: gmul(conj_i(c), P1), k);
12012 1232 : if (!odd(k)) P = gneg(P);
12013 1232 : P = normalizeapprox_i(gadd(P, P2), bit-20);
12014 1232 : gel(vP,iS) = gprec_wtrunc(P, prec);
12015 : }
12016 : }
12017 168 : delete_var(); return vP;
12018 : }
12019 :
12020 : /* when cosets = NULL, return a "fake" symbol containing only fs(oo->0) */
12021 : static GEN
12022 168 : mfsymbol_i(GEN mf, GEN F, GEN cosets, long bit)
12023 : {
12024 168 : GEN FE, van, vP, vE, Mvecj, vES = mftobasisES(mf,F);
12025 168 : long precnew, prec = nbits2prec(bit), k = MF_get_k(mf);
12026 168 : vE = mfgetembed(F, prec);
12027 168 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
12028 168 : if (lg(Mvecj) >= 5) precnew = prec;
12029 : else
12030 : {
12031 14 : long N = MF_get_N(mf), n = mfperiod_prelim_double(1/(double)N, k, bit + 32);
12032 14 : precnew = prec + inveis_extraprec(N, mkS(), Mvecj, n);
12033 : }
12034 168 : FE = mkcol2(F, mf_eisendec(mf,F,precnew));
12035 168 : vP = mfperiodpols_i(mf, FE, cosets, &van, bit);
12036 168 : return mkvecn(8, mf, vES, vP, cosets, utoi(bit), vE, FE, van);
12037 : }
12038 :
12039 : static GEN
12040 56 : fs2_get_cusps(GEN f) { return gel(f,3); }
12041 : static GEN
12042 56 : fs2_get_MF(GEN f) { return gel(f,1); }
12043 : static GEN
12044 56 : fs2_get_W(GEN f) { return gel(f,2); }
12045 : static GEN
12046 56 : fs2_get_F(GEN f) { return gel(f,4); }
12047 : static long
12048 0 : fs2_get_bitprec(GEN f) { return itou(gel(f,5)); }
12049 : static GEN
12050 56 : fs2_get_al0(GEN f) { return gel(f,6); }
12051 : static GEN
12052 21 : fs2_get_den(GEN f) { return gel(f,7); }
12053 : static int
12054 210 : checkfs2_i(GEN f)
12055 : {
12056 : GEN W, C, F, al0;
12057 : long l;
12058 210 : if (typ(f) != t_VEC || lg(f) != 8 || typ(gel(f,5)) != t_INT) return 0;
12059 35 : C = fs2_get_cusps(f); l = lg(C);
12060 35 : W = fs2_get_W(f);
12061 35 : F = fs2_get_F(f);
12062 35 : al0 = fs2_get_al0(f);
12063 35 : return checkMF_i(fs2_get_MF(f))
12064 35 : && typ(W) == t_VEC && typ(F) == t_VEC && typ(al0) == t_VECSMALL
12065 70 : && lg(W) == l && lg(F) == l && lg(al0) == l;
12066 : }
12067 : static GEN fs2_init(GEN mf, GEN F, long bit);
12068 : GEN
12069 175 : mfsymbol(GEN mf, GEN F, long bit)
12070 : {
12071 175 : pari_sp av = avma;
12072 175 : GEN cosets = NULL;
12073 175 : if (!F)
12074 : {
12075 35 : F = mf;
12076 35 : if (!checkmf_i(F)) pari_err_TYPE("mfsymbol", F);
12077 35 : mf = mfinit_i(F, mf_FULL);
12078 : }
12079 140 : else if (!checkmf_i(F)) pari_err_TYPE("mfsymbol", F);
12080 175 : if (checkfs2_i(mf)) return fs2_init(mf, F, bit);
12081 175 : if (checkfs_i(mf))
12082 : {
12083 0 : cosets = fs_get_cosets(mf);
12084 0 : mf = fs_get_MF(mf);
12085 : }
12086 175 : else if (checkMF_i(mf))
12087 : {
12088 175 : GEN gk = MF_get_gk(mf);
12089 175 : if (typ(gk) != t_INT || equali1(gk)) return fs2_init(mf, F, bit);
12090 154 : if (signe(gk) <= 0) pari_err_TYPE("mfsymbol [k <= 0]", mf);
12091 147 : cosets = mfcosets(MF_get_gN(mf));
12092 : }
12093 0 : else pari_err_TYPE("mfsymbol",mf);
12094 147 : return gc_GEN(av, mfsymbol_i(mf, F, cosets, bit));
12095 : }
12096 :
12097 : static GEN
12098 14 : RgX_by_parity(GEN P, long odd)
12099 : {
12100 14 : long i, l = lg(P);
12101 : GEN Q;
12102 14 : if (l < 4) return odd ? pol_x(0): P;
12103 14 : Q = cgetg(l, t_POL); Q[1] = P[1];
12104 91 : for (i = odd? 2: 3; i < l; i += 2) gel(Q,i) = gen_0;
12105 91 : for (i = odd? 3: 2; i < l; i += 2) gel(Q,i) = gel(P,i);
12106 14 : return normalizepol_lg(Q, l);
12107 : }
12108 : /* flag 0: period polynomial of F, >0 or <0 with corresponding parity */
12109 : GEN
12110 35 : mfperiodpol(GEN mf0, GEN F, long flag, long bit)
12111 : {
12112 35 : pari_sp av = avma;
12113 35 : GEN pol, mf = checkMF_i(mf0);
12114 35 : if (!mf) pari_err_TYPE("mfperiodpol",mf0);
12115 35 : if (checkfs_i(F))
12116 : {
12117 14 : GEN mfpols = fs_get_pols(F);
12118 14 : if (!mfs_checkmf(F, mf)) pari_err_TYPE("mfperiodpol [different mf]",F);
12119 14 : pol = veclast(mfpols); /* trivial coset is last */
12120 : }
12121 : else
12122 : {
12123 21 : GEN gk = MF_get_gk(mf);
12124 21 : if (typ(gk) != t_INT) pari_err_TYPE("mfperiodpol [half-integral k]", mf);
12125 21 : if (equali1(gk)) pari_err_TYPE("mfperiodpol [k = 1]", mf);
12126 21 : F = mfsymbol_i(mf, F, NULL, bit);
12127 21 : pol = fs_get_pols(F);
12128 : }
12129 35 : if (flag) pol = RgX_by_parity(pol, flag < 0);
12130 35 : return gc_GEN(av, RgX_embedall(pol, fs_get_vE(F)));
12131 : }
12132 :
12133 : static int
12134 35 : mfs_iscusp(GEN mfs) { return gequal0(gmael(mfs,2,1)); }
12135 : /* given cusps s1 and s2 (rationals or oo)
12136 : * compute $\int_{s1}^{s2}(X-\tau)^{k-2}F|_k\ga_j(\tau)\,d\tau$ */
12137 : /* If flag = 1, do not give an error message if divergent, but
12138 : give the rational function as result. */
12139 :
12140 : static GEN
12141 126 : col2cusp(GEN v)
12142 : {
12143 : GEN A, C;
12144 126 : if (lg(v) != 3 || !RgV_is_ZV(v)) pari_err_TYPE("col2cusp",v);
12145 126 : A = gel(v,1);
12146 126 : C = gel(v,2);
12147 126 : if (gequal0(C))
12148 : {
12149 0 : if (gequal0(A)) pari_err_TYPE("mfsymboleval", mkvec2(A, C));
12150 0 : return mkoo();
12151 : }
12152 126 : return gdiv(A, C);
12153 : }
12154 : /* g.oo */
12155 : static GEN
12156 112 : mat2cusp(GEN g) { return col2cusp(gel(g,1)); }
12157 :
12158 : static GEN
12159 7 : pathmattovec(GEN path)
12160 7 : { return mkvec2(col2cusp(gel(path,1)), col2cusp(gel(path,2))); }
12161 :
12162 : static void
12163 546 : get_mf_F(GEN fs, GEN *mf, GEN *F)
12164 : {
12165 546 : if (lg(fs) == 3) { *mf = gel(fs,1); *F = gel(fs,2); }
12166 546 : else { *mf = fs_get_MF(fs); *F = NULL; }
12167 546 : }
12168 : static GEN
12169 189 : mfgetvan(GEN fs, GEN ga, GEN *pal, long nlim, long prec)
12170 : {
12171 : GEN van, mf, F, W;
12172 : long PREC;
12173 189 : get_mf_F(fs, &mf, &F);
12174 189 : if (!F)
12175 : {
12176 189 : GEN vanall = fs_get_expan(fs), cosets = fs_get_cosets(fs);
12177 189 : long D, jga = mftocoset_iD(MF_get_N(mf), ga, cosets, &D);
12178 189 : van = gmael(vanall, 1, jga);
12179 189 : W = gmael(vanall, 2, jga);
12180 189 : if (lg(van) >= nlim + 2)
12181 : {
12182 182 : GEN z = mfcharcxeval(MF_get_CHI(mf), D, prec);
12183 182 : if (!gequal1(z)) van = RgV_Rg_mul(van, z);
12184 182 : *pal = gel(W,1); return van;
12185 : }
12186 7 : F = gel(fs_get_EF(fs), 1);
12187 : }
12188 7 : PREC = prec + EXTRAPREC64;
12189 7 : van = mfslashexpansion(mf, F, ga, nlim, 0, &W, PREC);
12190 7 : van = vanembed(F, van, PREC);
12191 7 : *pal = gel(W,1); return van;
12192 : }
12193 : /* Computation of int_A^oo (f | ga)(t)(X-t)^{k-2} dt, assuming convergence;
12194 : * fs is either a symbol or a triple [mf,F,bitprec]. A != oo and im(A) > 0 */
12195 : static GEN
12196 77 : intAoo0(GEN fs, GEN A, GEN ga, GEN P, long bit)
12197 : {
12198 77 : long nlim, N, k, w, prec = nbits2prec(bit);
12199 : GEN van, mf, F, al;
12200 77 : get_mf_F(fs, &mf,&F); N = MF_get_N(mf); k = MF_get_k(mf);
12201 77 : w = mfZC_width(N, gel(ga,1));
12202 77 : nlim = mfperiod_prelim(gdivgu(imag_i(A), w), k, bit + 32);
12203 77 : van = mfgetvan(fs, ga, &al, nlim, prec);
12204 77 : return intAoo(van, nlim, al,w, P, A, k, prec);
12205 : }
12206 :
12207 : /* fs symbol, naive summation, A != oo, im(A) > 0 and B = oo or im(B) > 0 */
12208 : static GEN
12209 112 : mfsymboleval_direct(GEN fs, GEN path, GEN ga, GEN P)
12210 : {
12211 112 : GEN A, B, van, S, al, mf = fs_get_MF(fs);
12212 112 : long w, nlimA, nlimB = 0, N = MF_get_N(mf), k = MF_get_k(mf);
12213 112 : long bit = fs_get_bitprec(fs), prec = nbits2prec(bit);
12214 :
12215 112 : A = gel(path, 1);
12216 112 : B = gel(path, 2); if (typ(B) == t_INFINITY) B = NULL;
12217 112 : w = mfZC_width(N, gel(ga,1));
12218 112 : nlimA = mfperiod_prelim(gdivgu(imag_i(A),w), k, bit + 32);
12219 112 : if (B) nlimB = mfperiod_prelim(gdivgu(imag_i(B),w), k, bit + 32);
12220 112 : van = mfgetvan(fs, ga, &al, maxss(nlimA,nlimB), prec);
12221 112 : S = intAoo(van, nlimA, al,w, P, A, k, prec);
12222 112 : if (B) S = gsub(S, intAoo(van, nlimB, al,w, P, B, k, prec));
12223 112 : return RgX_embedall(S, fs_get_vE(fs));
12224 : }
12225 :
12226 : /* Computation of int_A^oo (f | ga)(t)(X-t)^{k-2} dt, assuming convergence;
12227 : * fs is either a symbol or a pair [mf,F]. */
12228 : static GEN
12229 77 : mfsymbolevalpartial(GEN fs, GEN A, GEN ga, long bit)
12230 : {
12231 : GEN Y, F, S, P, mf;
12232 77 : long N, k, w, prec = nbits2prec(bit);
12233 :
12234 77 : get_mf_F(fs, &mf, &F);
12235 77 : N = MF_get_N(mf); w = mfZC_width(N, gel(ga,1));
12236 77 : k = MF_get_k(mf);
12237 77 : Y = gdivgu(imag_i(A), w);
12238 77 : P = get_P(k, fetch_var(), prec);
12239 77 : if (lg(fs) != 3 && gtodouble(Y)*(2*N) < 1)
12240 21 : { /* true symbol + low imaginary part: use GL_2 action to improve */
12241 21 : GEN U, ga2, czd, A2 = cxredga0N(N, A, &U, &czd, 1);
12242 21 : GEN vE = fs_get_vE(fs);
12243 21 : ga2 = ZM_mul(ga, ZM_inv(U, NULL));
12244 21 : S = RgX_embedall(intAoo0(fs, A2, ga2, P, bit), vE);
12245 21 : S = gsub(S, mfsymboleval(fs, mkvec2(mat2cusp(U), mkoo()), ga2, bit));
12246 21 : S = typ(S) == t_VEC? vecact_GL2(S, U, k): act_GL2(S, U, k);
12247 : }
12248 : else
12249 : {
12250 56 : S = intAoo0(fs, A, ga, P, bit);
12251 56 : S = RgX_embedall(S, F? mfgetembed(F,prec): fs_get_vE(fs));
12252 : }
12253 77 : delete_var(); return normalizeapprox_i(S, bit-20);
12254 : }
12255 :
12256 : static GEN
12257 42 : actal(GEN x, GEN vabd)
12258 : {
12259 42 : if (typ(x) == t_INFINITY) return x;
12260 35 : return gdiv(gadd(gmul(gel(vabd,1), x), gel(vabd,2)), gel(vabd,3));
12261 : }
12262 :
12263 : static GEN
12264 14 : unact(GEN z, GEN vabd, long k, long prec)
12265 : {
12266 14 : GEN res = gsubst(z, 0, actal(pol_x(0), vabd));
12267 14 : GEN CO = gpow(gdiv(gel(vabd,3), gel(vabd,1)), sstoQ(k-2, 2), prec);
12268 14 : return gmul(CO, res);
12269 : }
12270 :
12271 : GEN
12272 210 : mfsymboleval(GEN fs, GEN path, GEN ga, long bitprec)
12273 : {
12274 210 : pari_sp av = avma;
12275 210 : GEN tau, V, LM, S, CHI, mfpols, cosets, al, be, mf, F, vabd = NULL;
12276 : long D, B, m, u, v, a, b, c, d, j, k, N, prec, tsc1, tsc2;
12277 :
12278 210 : if (checkfs_i(fs))
12279 : {
12280 203 : get_mf_F(fs, &mf, &F);
12281 203 : bitprec = minss(bitprec, fs_get_bitprec(fs));
12282 : }
12283 : else
12284 : {
12285 7 : if (checkfs2_i(fs)) pari_err_TYPE("mfsymboleval [need integral k > 1]",fs);
12286 0 : if (typ(fs) != t_VEC || lg(fs) != 3) pari_err_TYPE("mfsymboleval",fs);
12287 0 : get_mf_F(fs, &mf, &F);
12288 0 : mf = checkMF_i(mf);
12289 0 : if (!mf ||!checkmf_i(F)) pari_err_TYPE("mfsymboleval",fs);
12290 : }
12291 203 : if (lg(path) != 3) pari_err_TYPE("mfsymboleval",path);
12292 203 : if (typ(path) == t_MAT) path = pathmattovec(path);
12293 203 : if (typ(path) != t_VEC) pari_err_TYPE("mfsymboleval",path);
12294 203 : al = gel(path,1);
12295 203 : be = gel(path,2);
12296 203 : ga = ga? GL2toSL2(ga, &vabd): matid(2);
12297 203 : if (vabd)
12298 : {
12299 14 : al = actal(al, vabd);
12300 14 : be = actal(be, vabd); path = mkvec2(al, be);
12301 : }
12302 203 : tsc1 = cusp_AC(al, &a, &c);
12303 203 : tsc2 = cusp_AC(be, &b, &d);
12304 203 : prec = nbits2prec(bitprec);
12305 203 : k = MF_get_k(mf);
12306 203 : if (!tsc1)
12307 : {
12308 42 : GEN z2, z = mfsymbolevalpartial(fs, al, ga, bitprec);
12309 42 : if (tsc2)
12310 28 : z2 = d? mfsymboleval(fs, mkvec2(be, mkoo()), ga, bitprec): gen_0;
12311 : else
12312 14 : z2 = mfsymbolevalpartial(fs, be, ga, bitprec);
12313 42 : z = gsub(z, z2);
12314 42 : if (vabd) z = unact(z, vabd, k, prec);
12315 42 : return gc_upto(av, normalizeapprox_i(z, bitprec-20));
12316 : }
12317 161 : else if (!tsc2)
12318 : {
12319 21 : GEN z = mfsymbolevalpartial(fs, be, ga, bitprec);
12320 21 : if (c) z = gsub(mfsymboleval(fs, mkvec2(al, mkoo()), ga, bitprec), z);
12321 7 : else z = gneg(z);
12322 21 : if (vabd) z = unact(z, vabd, k, prec);
12323 21 : return gc_upto(av, normalizeapprox_i(z, bitprec-20));
12324 : }
12325 140 : if (F) pari_err_TYPE("mfsymboleval", fs);
12326 140 : D = a*d-b*c;
12327 140 : if (!D) { set_avma(av); return RgX_embedall(gen_0, fs_get_vE(fs)); }
12328 126 : mfpols = fs_get_pols(fs);
12329 126 : cosets = fs_get_cosets(fs);
12330 126 : CHI = MF_get_CHI(mf); N = MF_get_N(mf);
12331 126 : cbezout(a, c, &u, &v); B = u*b + v*d; tau = mkmat22s(a, -v, c, u);
12332 126 : V = gcf(sstoQ(B, D));
12333 126 : LM = shallowconcat(mkcol2(gen_1, gen_0), contfracpnqn(V, lg(V)));
12334 126 : S = gen_0; m = lg(LM) - 2;
12335 364 : for (j = 0; j < m; j++)
12336 : {
12337 : GEN M, P;
12338 : long D, iN;
12339 238 : M = mkmat2(gel(LM, j+2), gel(LM, j+1));
12340 238 : if (!odd(j)) gel(M,1) = ZC_neg(gel(M,1));
12341 238 : M = ZM_mul(tau, M);
12342 238 : iN = mftocoset_iD(N, ZM_mul(ga, M), cosets, &D);
12343 238 : P = gmul(gel(mfpols,iN), mfcharcxeval(CHI,D,prec));
12344 238 : S = gadd(S, act_GL2(P, ZM_inv(M, NULL), k));
12345 : }
12346 126 : if (typ(S) == t_RFRAC)
12347 : {
12348 : GEN R, S1, co;
12349 21 : gel(S,2) = primitive_part(gel(S,2), &co);
12350 21 : if (co) gel(S,1) = gdiv(gel(S,1), gtofp(co,prec));
12351 21 : S1 = poldivrem(gel(S,1), gel(S,2), &R);
12352 21 : if (gexpo(R) < -bitprec + 20) S = S1;
12353 : }
12354 126 : if (vabd) S = unact(S, vabd, k, prec);
12355 126 : S = RgX_embedall(S, fs_get_vE(fs));
12356 126 : return gc_upto(av, normalizeapprox_i(S, bitprec-20));
12357 : }
12358 :
12359 : /* v a scalar or t_POL; set *pw = a if expo(a) > E for some coefficient;
12360 : * take the 'a' with largest exponent */
12361 : static void
12362 5740 : improve(GEN v, GEN *pw, long *E)
12363 : {
12364 5740 : if (typ(v) != t_POL)
12365 : {
12366 4270 : long e = gexpo(v);
12367 4270 : if (e > *E) { *E = e; *pw = v; }
12368 : }
12369 : else
12370 : {
12371 1470 : long j, l = lg(v);
12372 5740 : for (j = 2; j < l; j++) improve(gel(v,j), pw, E);
12373 : }
12374 5740 : }
12375 : static GEN
12376 518 : polabstorel(GEN rnfeq, GEN x)
12377 : {
12378 518 : if (typ(x) != t_POL) return x;
12379 3500 : pari_APPLY_pol_normalized(eltabstorel(rnfeq, gel(x,i)));
12380 : }
12381 : static GEN
12382 1519 : bestapprnfrel(GEN x, GEN polabs, GEN roabs, GEN rnfeq, long prec)
12383 : {
12384 1519 : x = bestapprnf(x, polabs, roabs, prec);
12385 1519 : if (rnfeq) x = polabstorel(rnfeq, liftpol_shallow(x));
12386 1519 : return x;
12387 : }
12388 : /* v vector of polynomials polynomial in C[X] (possibly scalar).
12389 : * Set *w = coeff with largest exponent and return T / *w, rationalized */
12390 : static GEN
12391 98 : normal(GEN v, GEN polabs, GEN roabs, GEN rnfeq, GEN *w, long prec)
12392 : {
12393 98 : long i, l = lg(v), E = -(long)HIGHEXPOBIT;
12394 : GEN dv;
12395 1568 : for (i = 1; i < l; i++) improve(gel(v,i), w, &E);
12396 98 : v = RgV_Rg_mul(v, ginv(*w));
12397 1568 : for (i = 1; i < l; i++)
12398 1470 : gel(v,i) = bestapprnfrel(gel(v,i), polabs,roabs,rnfeq,prec);
12399 98 : v = Q_primitive_part(v,&dv);
12400 98 : if (dv) *w = gmul(*w,dv);
12401 98 : return v;
12402 : }
12403 :
12404 : static GEN mfpetersson_i(GEN FS, GEN GS);
12405 :
12406 : GEN
12407 42 : mfmanin(GEN FS, long bitprec)
12408 : {
12409 42 : pari_sp av = avma;
12410 : GEN mf, M, vp, vm, cosets, CHI, vpp, vmm, f, T, P, vE, polabs, roabs, rnfeq;
12411 : GEN pet;
12412 : long N, k, lco, i, prec, lvE;
12413 :
12414 42 : if (!checkfs_i(FS))
12415 : {
12416 7 : if (checkfs2_i(FS)) pari_err_TYPE("mfmanin [need integral k > 1]",FS);
12417 0 : pari_err_TYPE("mfmanin",FS);
12418 : }
12419 35 : if (!mfs_iscusp(FS)) pari_err_TYPE("mfmanin [noncuspidal]",FS);
12420 35 : mf = fs_get_MF(FS);
12421 35 : vp = fs_get_pols(FS);
12422 35 : cosets = fs_get_cosets(FS);
12423 35 : bitprec = fs_get_bitprec(FS);
12424 35 : N = MF_get_N(mf); k = MF_get_k(mf); CHI = MF_get_CHI(mf);
12425 35 : lco = lg(cosets); vm = cgetg(lco, t_VEC);
12426 35 : prec = nbits2prec(bitprec);
12427 476 : for (i = 1; i < lco; i++)
12428 : {
12429 441 : GEN g = gel(cosets, i), c;
12430 441 : long A = itos(gcoeff(g,1,1)), B = itos(gcoeff(g,1,2));
12431 441 : long C = itos(gcoeff(g,2,1)), D = itos(gcoeff(g,2,2));
12432 441 : long Dbar, ibar = mftocoset_iD(N, mkmat22s(-B,-A,D,C), cosets, &Dbar);
12433 :
12434 441 : c = mfcharcxeval(CHI, Dbar, prec); if (odd(k)) c = gneg(c);
12435 441 : T = RgX_Rg_mul(gel(vp,ibar), c);
12436 441 : if (typ(T) == t_POL && varn(T) == 0) T = RgX_recip(T);
12437 441 : gel(vm,i) = T;
12438 : }
12439 35 : vpp = gadd(vp,vm);
12440 35 : vmm = gsub(vp,vm);
12441 :
12442 35 : vE = fs_get_vE(FS); lvE = lg(vE);
12443 35 : f = gel(fs_get_EF(FS), 1);
12444 35 : P = mf_get_field(f); if (degpol(P) == 1) P = NULL;
12445 35 : T = mfcharpol(CHI); if (degpol(T) == 1) T = NULL;
12446 35 : if (T && P)
12447 : {
12448 7 : rnfeq = nf_rnfeqsimple(T, P);
12449 7 : polabs = gel(rnfeq,1);
12450 7 : roabs = gel(QX_complex_roots(polabs,prec), 1);
12451 : }
12452 : else
12453 : {
12454 28 : rnfeq = roabs = NULL;
12455 28 : polabs = P? P: T;
12456 : }
12457 35 : pet = mfpetersson_i(FS, NULL);
12458 35 : M = cgetg(lvE, t_VEC);
12459 84 : for (i = 1; i < lvE; i++)
12460 : {
12461 49 : GEN p, m, wp, wm, petdiag, r, E = gel(vE,i);
12462 49 : p = normal(RgXV_embed(vpp, E), polabs, roabs, rnfeq, &wp, prec);
12463 49 : m = normal(RgXV_embed(vmm, E), polabs, roabs, rnfeq, &wm, prec);
12464 49 : petdiag = typ(pet)==t_MAT? gcoeff(pet,i,i): pet;
12465 49 : r = gdiv(mulimag(wp, conj_i(wm)), petdiag);
12466 49 : r = bestapprnfrel(r, polabs, roabs, rnfeq, prec);
12467 49 : gel(M,i) = mkvec2(mkvec2(p,m), mkvec3(wp,wm,r));
12468 : }
12469 35 : return gc_GEN(av, lvE == 2? gel(M,1): M);
12470 : }
12471 :
12472 : /* flag = 0: full, flag = +1 or -1, odd/even */
12473 : /* Basis of period polynomials in level 1. */
12474 : GEN
12475 49 : mfperiodpolbasis(long k, long flag)
12476 : {
12477 49 : pari_sp av = avma;
12478 49 : long i, j, n = k - 2;
12479 : GEN M, C, v;
12480 49 : if (k <= 4) return cgetg(1,t_VEC);
12481 35 : M = cgetg(k, t_MAT);
12482 35 : C = matpascal(n);
12483 35 : if (!flag)
12484 392 : for (j = 0; j <= n; j++)
12485 : {
12486 371 : gel(M, j+1) = v = cgetg(k, t_COL);
12487 4767 : for (i = 0; i <= j; i++) gel(v, i+1) = gcoeff(C, j+1, i+1);
12488 4396 : for (; i <= n; i++) gel(v, i+1) = gcoeff(C, n-j+1, i-j+1);
12489 : }
12490 : else
12491 168 : for (j = 0; j <= n; j++)
12492 : {
12493 154 : gel(M, j+1) = v = cgetg(k, t_COL);
12494 1848 : for (i = 0; i <= n; i++)
12495 : {
12496 1694 : GEN a = i < j ? gcoeff(C, j+1, i+1) : gen_0;
12497 1694 : if (i + j >= n)
12498 : {
12499 924 : GEN b = gcoeff(C, j+1, i+j-n+1);
12500 924 : a = flag < 0 ? addii(a,b) : subii(a,b);
12501 : }
12502 1694 : gel(v, i+1) = a;
12503 : }
12504 : }
12505 35 : return gc_GEN(av, RgM_to_RgXV(ZM_ker(M), 0));
12506 : }
12507 :
12508 : static int
12509 168 : zero_at_cusp(GEN mf, GEN F, GEN c)
12510 : {
12511 168 : GEN v = evalcusp(mf, F, c, LOWDEFAULTPREC);
12512 168 : return (gequal0(v) || gexpo(v) <= -62);
12513 : }
12514 : /* Compute list E of j such that F|_k g_j vanishes at oo: return [E, s(E)] */
12515 : static void
12516 14 : mffvanish(GEN mf, GEN F, GEN G, GEN cosets, GEN *pres, GEN *press)
12517 : {
12518 14 : long j, lc = lg(cosets), N = MF_get_N(mf);
12519 : GEN v, vs;
12520 14 : *pres = v = zero_zv(lc-1);
12521 14 : *press= vs = zero_zv(lc-1);
12522 105 : for (j = 1; j < lc; j++)
12523 : {
12524 91 : GEN ga = gel(cosets,j), c = mat2cusp(ga);
12525 91 : if (zero_at_cusp(mf, F, c))
12526 14 : v[j] = vs[ mftocoset_i(N, ZM_mulS(ga), cosets) ] = 1;
12527 77 : else if (!zero_at_cusp(mf, G, c))
12528 0 : pari_err_IMPL("divergent Petersson product");
12529 : }
12530 14 : }
12531 : static GEN
12532 140 : Haberland(GEN PF, GEN PG, GEN vEF, GEN vEG, long k)
12533 : {
12534 140 : GEN S = gen_0, vC = vecbinomial(k-2); /* vC[n+1] = (-1)^n binom(k-2,n) */
12535 140 : long n, j, l = lg(PG);
12536 406 : for (n = 2; n < k; n+=2) gel(vC,n) = negi(gel(vC,n));
12537 2583 : for (j = 1; j < l; j++)
12538 : {
12539 2443 : GEN PFj = gel(PF,j), PGj = gel(PG,j);
12540 10038 : for (n = 0; n <= k-2; n++)
12541 : {
12542 7595 : GEN a = RgX_coeff(PGj, k-2-n), b = RgX_coeff(PFj, n);
12543 7595 : a = Rg_embedall(a, vEG);
12544 7595 : b = Rg_embedall(b, vEF);
12545 7595 : a = conj_i(a); if (typ(a) == t_VEC) settyp(a, t_COL);
12546 : /* a*b = scalar or t_VEC or t_COL or t_MAT */
12547 7595 : S = gadd(S, gdiv(gmul(a,b), gel(vC,n+1)));
12548 : }
12549 : }
12550 140 : S = mulcxpowIs(gmul2n(S, 1-k), 1+k);
12551 140 : return vEF==vEG? real_i(S): S;
12552 : }
12553 : /* F1S, F2S both symbols, same mf */
12554 : static GEN
12555 14 : mfpeterssonnoncusp(GEN F1S, GEN F2S)
12556 : {
12557 14 : pari_sp av = avma;
12558 : GEN mf, F1, F2, GF1, GF2, P2, cosets, vE1, vE2, FE1, FE2, P;
12559 : GEN I, IP1, RHO, RHOP1, INF, res, ress;
12560 14 : const double height = sqrt(3.)/2;
12561 : long k, r, j, bitprec, prec;
12562 :
12563 14 : mf = fs_get_MF(F1S);
12564 14 : FE1 = fs_get_EF(F1S); F1 = gel(FE1, 1);
12565 14 : FE2 = fs_get_EF(F2S); F2 = gel(FE2, 1);
12566 14 : cosets = fs_get_cosets(F1S);
12567 14 : bitprec = minuu(fs_get_bitprec(F1S), fs_get_bitprec(F2S));
12568 14 : prec = nbits2prec(bitprec);
12569 14 : F1S = fs_set_expan(F1S, mfgaexpansionall(mf, FE1, cosets, height, prec));
12570 14 : if (F2S != F1S)
12571 14 : F2S = fs_set_expan(F2S, mfgaexpansionall(mf, FE2, cosets, height, prec));
12572 14 : k = MF_get_k(mf); r = lg(cosets) - 1;
12573 14 : vE1 = fs_get_vE(F1S);
12574 14 : vE2 = fs_get_vE(F2S);
12575 14 : I = gen_I();
12576 14 : IP1 = mkcomplex(gen_1,gen_1);
12577 14 : RHO = rootsof1u_cx(3, prec+EXTRAPREC64);
12578 14 : RHOP1 = gaddsg(1, RHO);
12579 14 : INF = mkoo();
12580 14 : mffvanish(mf, F1, F2, cosets, &res, &ress);
12581 14 : P2 = fs_get_pols(F2S);
12582 14 : GF1 = cgetg(r+1, t_VEC);
12583 14 : GF2 = cgetg(r+1, t_VEC); P = get_P(k, fetch_var(), prec);
12584 105 : for (j = 1; j <= r; j++)
12585 : {
12586 91 : GEN g = gel(cosets,j);
12587 91 : if (res[j]) {
12588 14 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHOP1,INF), g, P);
12589 14 : gel(GF2,j) = mfsymboleval_direct(F2S, mkvec2(I,IP1), g, P);
12590 77 : } else if (ress[j]) {
12591 7 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHOP1,RHO), g, P);
12592 7 : gel(GF2,j) = mfsymboleval_direct(F2S, mkvec2(I,INF), g, P);
12593 : } else {
12594 70 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHO,I), g, P);
12595 70 : gel(GF2,j) = gneg(gel(P2,j)); /* - symboleval(F2S, [0,oo] */
12596 : }
12597 : }
12598 14 : delete_var();
12599 14 : return gc_upto(av, gdivgu(Haberland(GF1,GF2, vE1,vE2, k), r));
12600 : }
12601 :
12602 : /* Petersson product of F and G, given by mfsymbol's [k > 1 integral] */
12603 : static GEN
12604 140 : mfpetersson_i(GEN FS, GEN GS)
12605 : {
12606 140 : pari_sp av = avma;
12607 : GEN mf, ESF, ESG, PF, PG, PH, CHI, cosets, vEF, vEG;
12608 : long k, r, j, N, bitprec, prec;
12609 :
12610 140 : if (!checkfs_i(FS)) pari_err_TYPE("mfpetersson",FS);
12611 140 : mf = fs_get_MF(FS);
12612 140 : ESF = fs_get_vES(FS);
12613 140 : if (!GS) GS = FS;
12614 : else
12615 : {
12616 35 : if (!checkfs_i(GS)) pari_err_TYPE("mfpetersson",GS);
12617 35 : if (!mfs_checkmf(GS, mf))
12618 0 : pari_err_TYPE("mfpetersson [different mf]", mkvec2(FS,GS));
12619 : }
12620 140 : ESG = fs_get_vES(GS);
12621 140 : if (!gequal0(gel(ESF,1)) || !gequal0(gel(ESG,1)))
12622 14 : return mfpeterssonnoncusp(FS, GS);
12623 126 : if (gequal0(gel(ESF,2)) || gequal0(gel(ESG,2))) return gc_const(av, gen_0);
12624 126 : N = MF_get_N(mf);
12625 126 : k = MF_get_k(mf);
12626 126 : CHI = MF_get_CHI(mf);
12627 126 : PF = fs_get_pols(FS); vEF = fs_get_vE(FS);
12628 126 : PG = fs_get_pols(GS); vEG = fs_get_vE(GS);
12629 126 : cosets = fs_get_cosets(FS);
12630 126 : bitprec = minuu(fs_get_bitprec(FS), fs_get_bitprec(GS));
12631 126 : prec = nbits2prec(bitprec);
12632 126 : r = lg(PG)-1;
12633 126 : PH = cgetg(r+1, t_VEC);
12634 2478 : for (j = 1; j <= r; j++)
12635 : {
12636 2352 : GEN ga = gel(cosets,j), PGj1, PGjm1;
12637 : long iT, D;
12638 2352 : iT = mftocoset_iD(N, ZM_mulTi(ga), cosets, &D);
12639 2352 : PGj1 = RgX_Rg_translate(gel(PG, iT), gen_1);
12640 2352 : PGj1 = RgX_Rg_mul(PGj1, mfcharcxeval(CHI, D, prec));
12641 2352 : iT = mftocoset_iD(N, ZM_mulT(ga), cosets, &D);
12642 2352 : PGjm1 = RgX_Rg_translate(gel(PG,iT), gen_m1);
12643 2352 : PGjm1 = RgX_Rg_mul(PGjm1, mfcharcxeval(CHI, D, prec));
12644 2352 : gel(PH,j) = gsub(PGj1, PGjm1);
12645 : }
12646 126 : return gc_upto(av, gdivgu(Haberland(PF, PH, vEF, vEG, k), 6*r));
12647 : }
12648 :
12649 : /****************************************************************/
12650 : /* Petersson products using Nelson-Collins */
12651 : /****************************************************************/
12652 : /* Compute W(k,z) = sum_{m >= 1} (mz)^{k-1}(mzK_{k-2}(mz)-K_{k-1}(mz))
12653 : * for z>0 and absolute accuracy < 2^{-B}.
12654 : * K_k(x) ~ (Pi/(2x))^{1/2} e^{-x} */
12655 :
12656 : static void
12657 10304 : Wparams(GEN *ph, long *pN, long k, double x, long prec)
12658 : {
12659 10304 : double B = prec2nbits(prec) + 10;
12660 10304 : double C = B + k*log(x)/M_LN2 + 1, D = C*M_LN2 + 2.065;
12661 10304 : double F = 2 * M_LN2 * (C - 1 + dbllog2(mpfact(k))) / x;
12662 10304 : double T = log(F) * (1 + 2*k/x/F), PI2 = M_PI*M_PI;
12663 10304 : *pN = (long)ceil((T/PI2) * (D + log(D/PI2)));
12664 10304 : *ph = gprec_w(dbltor(T / *pN), prec);
12665 10304 : }
12666 :
12667 : static void
12668 10304 : Wcoshall(GEN *pCH, GEN *pCHK, GEN *pCHK1, GEN h, long N, long k, long prec)
12669 : {
12670 10304 : GEN CH, CHK, CHK1, z = gexp(h, prec);
12671 10304 : GEN PO = gpowers(z, N), POK1 = gpowers(gpowgs(z, k-1), N);
12672 10304 : GEN E = ginv(gel(PO, N + 1)); /* exp(-hN) */
12673 10304 : GEN E1 = ginv(gel(POK1, N + 1)); /* exp(-(k-1)h) */
12674 : long j;
12675 10304 : *pCH = CH = cgetg(N+1, t_VEC);
12676 10304 : *pCHK = CHK = cgetg(N+1, t_VEC);
12677 10304 : *pCHK1 = CHK1 = cgetg(N+1, t_VEC);
12678 146048 : for (j = 1; j <= N; j++)
12679 : {
12680 135744 : GEN eh = gel(PO, j+1), emh = gmul(gel(PO, N-j+1), E); /* e^{jh}, e^{-jh} */
12681 135744 : GEN ek1h = gel(POK1, j+1), ek1mh = gmul(gel(POK1, N-j+1), E1);
12682 135744 : gel(CH, j) = gmul2n(gadd(eh, emh), -1); /* cosh(jh) */
12683 135744 : gel(CHK1,j) = gmul2n(gadd(ek1h, ek1mh), -1); /* cosh((k-1)jh) */
12684 135744 : gel(CHK, j) = gmul2n(gadd(gmul(eh, ek1h), gmul(emh, ek1mh)), -1);
12685 : }
12686 10304 : }
12687 :
12688 : /* computing W(k,x) via integral */
12689 : static GEN
12690 10304 : Wint(long k, GEN vP, GEN x, long prec)
12691 : {
12692 : GEN P, P1, S1, S, h, CH, CHK, CHK1;
12693 : long N, j;
12694 10304 : Wparams(&h, &N, k, gtodouble(x), prec);
12695 10304 : Wcoshall(&CH, &CHK, &CHK1, h, N, k, prec);
12696 10304 : P = gel(vP, k+1); P1 = gel(vP, k); S = S1 = NULL;
12697 156352 : for (j = 0; j <= N; j++)
12698 : {
12699 146048 : GEN eh = gexp(j? gmul(x, gel(CH, j)): x, prec);
12700 146048 : GEN eh1 = gsubgs(eh, 1), eh1k = gpowgs(eh1, k), t1, t;
12701 146048 : t = gdiv(poleval(P, eh), gmul(eh1, eh1k));
12702 146048 : t1 = gdiv(poleval(P1, eh), eh1k);
12703 146048 : if (j)
12704 : {
12705 135744 : S = gadd(S, gmul(t, gel(CHK, j)));
12706 135744 : S1 = gadd(S1, gmul(t1, gel(CHK1, j)));
12707 : }
12708 : else
12709 : {
12710 10304 : S = gmul2n(t, -1);
12711 10304 : S1 = gmul2n(t1, -1);
12712 : }
12713 : }
12714 10304 : return gmul(gmul(h, gpowgs(x, k-1)), gsub(gmul(x, S), gmulsg(2*k-1, S1)));
12715 : }
12716 :
12717 : static GEN
12718 21 : get_vP(long k)
12719 : {
12720 21 : GEN P, v = cgetg(k+2, t_VEC), Q = deg1pol_shallow(gen_1,gen_m1,0);
12721 : long j;
12722 21 : gel(v,1) = gen_1;
12723 21 : gel(v,2) = P = pol_x(0);
12724 28 : for (j = 2; j <= k; j++)
12725 7 : gel(v,j+1) = P = RgX_shift_shallow(gsub(gmulsg(j, P),
12726 : gmul(Q, ZX_deriv(P))), 1);
12727 21 : return v;
12728 : }
12729 : /* vector of (-1)^j(1/(exp(x)-1))^(j) [x = z] * z^j for 0<=j<=r */
12730 : static GEN
12731 63742 : VS(long r, GEN z, GEN V, long prec)
12732 : {
12733 63742 : GEN e = gexp(z, prec), c = ginv(gsubgs(e,1));
12734 63742 : GEN T = gpowers0(gmul(c, z), r, c);
12735 : long j;
12736 63742 : V = gsubst(V, 0, e);
12737 143864 : for (j = 1; j <= r + 1; j++) gel(V,j) = gmul(gel(V,j), gel(T,j));
12738 63742 : return V;
12739 : }
12740 :
12741 : /* U(r,x)=sum_{m >= 1} (mx)^k K_k(mx), k = r+1/2 */
12742 : static GEN
12743 71932 : Unelson(long r, GEN V)
12744 : {
12745 71932 : GEN S = gel(V,r+1), C = gen_1; /* (r+j)! / j! / (r-j)! */
12746 : long j;
12747 71932 : if (!r) return S;
12748 40950 : for (j = 1; j <= r; j++)
12749 : {
12750 24570 : C = gdivgu(gmulgu(C, (r+j)*(r-j+1)), j);
12751 24570 : S = gadd(S, gmul2n(gmul(C, gel(V, r-j+1)), -j));
12752 : }
12753 16380 : return S;
12754 : }
12755 : /* W(r+1/2,z) / sqrt(Pi/2) */
12756 : static GEN
12757 63742 : Wint2(long r, GEN vP, GEN z, long prec)
12758 : {
12759 63742 : GEN R, V = VS(r, z, vP, prec);
12760 63742 : R = Unelson(r, V);
12761 63742 : if (r) R = gsub(R, gmulsg(2*r, Unelson(r-1, V)));
12762 63742 : return R;
12763 : }
12764 : typedef GEN(*Wfun_t)(long, GEN, GEN, long);
12765 : static GEN
12766 74046 : WfromZ(GEN Z, GEN vP, GEN gkm1, Wfun_t W, long k, GEN pi4, long prec)
12767 : {
12768 74046 : pari_sp av = avma;
12769 74046 : GEN Zk = gpow(Z, gkm1, prec), z = gmul(pi4, gsqrt(Z,prec));
12770 74046 : return gc_upto(av, gdiv(W(k, vP, z, prec), Zk));
12771 : }
12772 : /* mf a true mf or an fs2 */
12773 : static GEN
12774 21 : fs2_init(GEN mf, GEN F, long bit)
12775 : {
12776 21 : pari_sp av = avma;
12777 21 : long i, l, lim, N, k, k2, prec = nbits2prec(bit);
12778 : GEN DEN, cusps, tab, gk, gkm1, W0, vW, vVW, vVF, vP, al0;
12779 21 : GEN vE = mfgetembed(F, prec), pi4 = Pi2n(2, prec);
12780 : Wfun_t Wf;
12781 :
12782 21 : if (lg(mf) == 7)
12783 : {
12784 21 : vW = cusps = NULL; /* true mf */
12785 21 : DEN = tab = NULL; /* -Wall */
12786 : }
12787 : else
12788 : { /* mf already an fs2, reset if its precision is too low */
12789 0 : vW = (fs2_get_bitprec(mf) < bit)? NULL: fs2_get_W(mf);
12790 0 : cusps = fs2_get_cusps(mf);
12791 0 : DEN = fs2_get_den(mf);
12792 0 : mf = fs2_get_MF(mf);
12793 : }
12794 21 : N = MF_get_N(mf);
12795 21 : gk = MF_get_gk(mf); gkm1 = gsubgs(gk, 1);
12796 21 : k2 = itos(gmul2n(gk,1));
12797 21 : Wf = odd(k2)? Wint2: Wint;
12798 21 : k = k2 >> 1; vP = get_vP(k);
12799 21 : if (vW) lim = (lg(gel(vW,1)) - 2) / N; /* vW[1] attached to cusp 0, width N */
12800 : else
12801 : { /* true mf */
12802 21 : double B = (bit + 10)*M_LN2;
12803 21 : double L = (B + k2*log(B)/2 + k2*k2*log(B)/(4*B)) / (4*M_PI);
12804 : long n, Lw;
12805 21 : lim = ((long)ceil(L*L));
12806 21 : Lw = N*lim;
12807 21 : tab = cgetg(Lw+1,t_VEC);
12808 59157 : for (n = 1; n <= Lw; n++)
12809 59136 : gel(tab,n) = WfromZ(uutoQ(n,N), vP, gkm1, Wf, k, pi4, prec);
12810 21 : if (!cusps) cusps = mfcusps_i(N);
12811 21 : DEN = gmul2n(gmulgu(gpow(Pi2n(3, prec), gkm1, prec), mypsiu(N)), -2);
12812 21 : if (odd(k2)) DEN = gdiv(DEN, sqrtr_abs(Pi2n(-1,prec)));
12813 : }
12814 21 : l = lg(cusps);
12815 21 : vVF = cgetg(l, t_VEC);
12816 21 : vVW = cgetg(l, t_VEC);
12817 21 : al0 = cgetg(l, t_VECSMALL);
12818 21 : W0 = k2==1? ginv(pi4): gen_0;
12819 203 : for (i = 1; i < l; i++)
12820 : {
12821 : long A, C, w, wi, Lw, n;
12822 : GEN VF, W, paramsF, al;
12823 182 : (void)cusp_AC(gel(cusps,i), &A,&C);
12824 182 : wi = ugcd(N, C*C); w = N / wi; Lw = w * lim;
12825 182 : VF = mfslashexpansion(mf, F, cusp2mat(A,C), Lw, 0, ¶msF, prec);
12826 : /* paramsF[2] = w */
12827 182 : al = gel(paramsF, 1); if (gequal0(al)) al = NULL;
12828 100240 : for (n = 0; n <= Lw; n++)
12829 : {
12830 100058 : GEN a = gel(VF,n+1);
12831 100058 : gel(VF,n+1) = gequal0(a)? gen_0: Rg_embedall(a, vE);
12832 : }
12833 182 : if (vW)
12834 0 : W = gel(vW, i);
12835 : else
12836 : {
12837 182 : W = cgetg(Lw+2, t_VEC);
12838 100240 : for (n = 0; n <= Lw; n++)
12839 100058 : gel(W, n+1) = al? WfromZ(gadd(al,uutoQ(n,w)),vP,gkm1,Wf,k,pi4, prec)
12840 100058 : : (n? gel(tab, n * wi): W0);
12841 : }
12842 182 : al0[i] = !al;
12843 182 : gel(vVF, i) = VF;
12844 182 : gel(vVW, i) = W;
12845 : }
12846 21 : if (k2 <= 1) al0 = zero_zv(l-1); /* no need to test for convergence */
12847 21 : return gc_GEN(av, mkvecn(7, mf,vVW,cusps,vVF,utoipos(bit),al0,DEN));
12848 : }
12849 :
12850 : static GEN
12851 21 : mfpetersson2(GEN Fs, GEN Gs)
12852 : {
12853 21 : pari_sp av = avma;
12854 21 : GEN VC, RES, vF, vG, vW = fs2_get_W(Fs), al0 = fs2_get_al0(Fs);
12855 21 : long N = MF_get_N(fs2_get_MF(Fs)), j, lC;
12856 :
12857 21 : VC = fs2_get_cusps(Fs); lC = lg(VC);
12858 21 : vF = fs2_get_F(Fs);
12859 21 : vG = Gs? fs2_get_F(Gs): vF;
12860 21 : RES = gen_0;
12861 203 : for (j = 1; j < lC; j++)
12862 : {
12863 182 : GEN W = gel(vW,j), VF = gel(vF,j), VG = gel(vG,j), T = gen_0;
12864 182 : long A, C, w, n, L = lg(W);
12865 182 : pari_sp av = avma;
12866 182 : (void)cusp_AC(gel(VC,j), &A,&C); w = N/ugcd(N, C*C);
12867 182 : if (al0[j] && !isintzero(gel(VF,1)) && !isintzero(gel(VG,1)))
12868 0 : pari_err_IMPL("divergent Petersson product");
12869 100240 : for (n = 1; n < L; n++)
12870 : {
12871 100058 : GEN b = gel(VF,n), a = gel(VG,n);
12872 100058 : if (!isintzero(a) && !isintzero(b))
12873 : {
12874 79964 : T = gadd(T, gmul(gel(W,n), gmul(conj_i(a),b)));
12875 79964 : if (gc_needed(av,2)) T = gc_upto(av,T);
12876 : }
12877 : }
12878 182 : if (w != 1) T = gmulgu(T,w);
12879 182 : RES = gc_upto(av, gadd(RES, T));
12880 : }
12881 21 : if (!Gs) RES = real_i(RES);
12882 21 : return gc_upto(av, gdiv(RES, fs2_get_den(Fs)));
12883 : }
12884 :
12885 : static long
12886 161 : symbol_type(GEN F)
12887 : {
12888 161 : if (checkfs_i(F)) return 1;
12889 21 : if (checkfs2_i(F)) return 2;
12890 0 : return 0;
12891 : }
12892 : static int
12893 35 : symbol_same_mf(GEN F, GEN G) { return gequal(gmael(F,1,1), gmael(G,1,1)); }
12894 : GEN
12895 126 : mfpetersson(GEN F, GEN G)
12896 : {
12897 126 : long tF = symbol_type(F);
12898 126 : if (!tF) pari_err_TYPE("mfpetersson",F);
12899 126 : if (G)
12900 : {
12901 35 : long tG = symbol_type(G);
12902 35 : if (!tG) pari_err_TYPE("mfpetersson",F);
12903 35 : if (tF != tG || !symbol_same_mf(F,G))
12904 0 : pari_err_TYPE("mfpetersson [incompatible symbols]", mkvec2(F,G));
12905 : }
12906 126 : return (tF == 1)? mfpetersson_i(F, G): mfpetersson2(F, G);
12907 : }
12908 :
12909 : /****************************************************************/
12910 : /* projective Galois representation, weight 1 */
12911 : /****************************************************************/
12912 : static void
12913 392 : moreorders(long N, GEN CHI, GEN F, GEN *pP, GEN *pO, ulong *bound)
12914 : {
12915 392 : pari_sp av = avma;
12916 : forprime_t iter;
12917 392 : ulong a = *bound+1, b = 2*(*bound), p;
12918 392 : long i = 1;
12919 392 : GEN P, O, V = mfcoefs_i(F, b, 1);
12920 392 : *bound = b;
12921 392 : P = cgetg(b-a+2, t_VECSMALL);
12922 392 : O = cgetg(b-a+2, t_VECSMALL);
12923 392 : u_forprime_init(&iter, a, b);
12924 2310 : while((p = u_forprime_next(&iter))) if (N % p)
12925 : {
12926 1813 : O[i] = mffindrootof1(V, p, CHI);
12927 1813 : P[i++] = p;
12928 : }
12929 392 : setlg(P, i); *pP = shallowconcat(*pP, P);
12930 392 : setlg(O, i); *pO = shallowconcat(*pO, O);
12931 392 : (void)gc_all(av, 2, pP, pO);
12932 392 : }
12933 :
12934 : static GEN
12935 182 : search_abelian(GEN nf, long n, long k, GEN N, GEN CHI, GEN F,
12936 : GEN *pP, GEN *pO, ulong *bound, long prec)
12937 : {
12938 182 : pari_sp av = avma;
12939 : GEN bnr, cond, H, cyc, gn, T, Bquo, P, E;
12940 182 : long sN = itos(N), r1 = nf_get_r1(nf), i, j, d;
12941 :
12942 182 : cond = idealfactor(nf, N);
12943 182 : P = gel(cond,1);
12944 182 : E = gel(cond,2);
12945 679 : for (i = j = 1; i < lg(P); i++)
12946 : {
12947 497 : GEN pr = gel(P,i), Ej = gen_1;
12948 497 : long p = itos(pr_get_p(pr));
12949 497 : if (p == n)
12950 : {
12951 98 : long e = pr_get_e(pr); /* 1 + [e*p/(p-1)] */
12952 98 : Ej = utoipos(1 + (e*p) / (p-1));
12953 : }
12954 : else
12955 : {
12956 399 : long f = pr_get_f(pr);
12957 399 : if (Fl_powu(p % n, f, n) != 1) continue;
12958 : }
12959 462 : gel(P,j) = pr;
12960 462 : gel(E,j) = Ej; j++;
12961 : }
12962 182 : setlg(P,j);
12963 182 : setlg(E,j);
12964 182 : cond = mkvec2(cond, const_vec(r1, gen_1));
12965 182 : bnr = Buchraymod(Buchall(nf, nf_FORCE, prec), cond, nf_INIT, utoipos(n));
12966 182 : cyc = bnr_get_cyc(bnr);
12967 182 : d = lg(cyc)-1;
12968 182 : H = zv_diagonal(ZV_to_Flv(cyc, n));
12969 182 : gn = utoi(n);
12970 182 : for (i = 1;;)
12971 : {
12972 2646 : for(j = 2; i < lg(*pO); i++)
12973 : {
12974 2072 : long o, q = (*pP)[i];
12975 2072 : GEN pr = idealprimedec_galois(nf, stoi(q));
12976 2072 : o = ((*pO)[i] / pr_get_f(pr)) % n;
12977 2072 : if (o)
12978 : {
12979 1442 : GEN v = ZV_to_Flv(isprincipalray(bnr, pr), n);
12980 1442 : H = vec_append(H, Flv_Fl_mul(v, o, n));
12981 : }
12982 : }
12983 574 : H = Flm_image(H, n); if (lg(cyc)-lg(H) <= k) break;
12984 392 : moreorders(sN, CHI, F, pP, pO, bound);
12985 : }
12986 182 : H = hnfmodid(shallowconcat(zm_to_ZM(H), diagonal_shallow(cyc)), gn);
12987 :
12988 182 : Bquo = cgetg(k+1, t_MAT);
12989 812 : for (i = j = 1; i <= d; i++)
12990 630 : if (!equali1(gcoeff(H,i,i))) gel(Bquo,j++) = col_ei(d,i);
12991 :
12992 441 : for (i = 1, T = NULL; i<=k; i++)
12993 : {
12994 259 : GEN Hi = hnfmodid(shallowconcat(H, vecsplice(Bquo,i)), gn);
12995 259 : GEN pol = rnfkummer(bnr, Hi, prec);
12996 259 : T = T? nfcompositum(nf, T, pol, 2): pol;
12997 : }
12998 182 : T = rnfequation(nf, T); return gc_all(av, 3, &T, pP, pO);
12999 : }
13000 :
13001 : static GEN
13002 77 : search_solvable(GEN LG, GEN mf, GEN F, long prec)
13003 : {
13004 77 : GEN N = MF_get_gN(mf), CHI = MF_get_CHI(mf), pol, O, P, nf, Nfa;
13005 77 : long i, l = lg(LG), v = fetch_var();
13006 77 : ulong bound = 1;
13007 77 : O = cgetg(1, t_VECSMALL); /* projective order of rho(Frob_p) */
13008 77 : P = cgetg(1, t_VECSMALL);
13009 77 : Nfa = Z_factor(N);
13010 77 : pol = pol_x(v);
13011 259 : for (i = 1; i < l; i++)
13012 : { /* n prime, find a (Z/nZ)^k - extension */
13013 182 : GEN G = gel(LG,i);
13014 182 : long n = G[1], k = G[2];
13015 182 : nf = nfinitred(mkvec2(pol,Nfa), prec);
13016 182 : pol = search_abelian(nf, n, k, N, CHI, F, &P, &O, &bound, prec);
13017 182 : setvarn(pol,v);
13018 : }
13019 77 : delete_var(); setvarn(pol,0); return pol;
13020 : }
13021 :
13022 : static GEN
13023 0 : search_A5(GEN mf, GEN F)
13024 : {
13025 0 : GEN CHI = MF_get_CHI(mf), O, P, L;
13026 0 : long N = MF_get_N(mf), i, j, lL, nd, r;
13027 0 : ulong bound = 1;
13028 0 : r = radicalu(N);
13029 0 : L = veccond_to_A5(zv_z_mul(divisorsu(N/r),r), 2); lL = lg(L); nd = lL-1;
13030 0 : if (nd == 1) return gmael(L,1,1);
13031 0 : O = cgetg(1, t_VECSMALL); /* projective order of rho(Frob_p) */
13032 0 : P = cgetg(1, t_VECSMALL);
13033 0 : for(i = 1; nd > 1; )
13034 : {
13035 : long l;
13036 0 : moreorders(N, CHI, F, &P, &O, &bound);
13037 0 : l = lg(P);
13038 0 : for ( ; i < l; i++)
13039 : {
13040 0 : ulong p = P[i], f = O[i];
13041 0 : for (j = 1; j < lL; j++)
13042 0 : if (gel(L,j))
13043 : {
13044 0 : GEN FE = ZpX_primedec(gmael(L,j,1), utoi(p)), F = gel(FE,1);
13045 0 : long nF = lg(F)-1;
13046 0 : if (!equaliu(gel(F, nF), f)) { gel(L,j) = NULL; nd--; }
13047 : }
13048 0 : if (nd <= 1) break;
13049 : }
13050 : }
13051 0 : for (j = 1; j < lL; j++)
13052 0 : if (gel(L,j)) return gmael(L,j,1);
13053 0 : return NULL;
13054 : }
13055 :
13056 : GEN
13057 77 : mfgaloisprojrep(GEN mf, GEN F, long prec)
13058 : {
13059 77 : pari_sp av = avma;
13060 77 : GEN LG = NULL;
13061 77 : if (!checkMF_i(mf) && !checkmf_i(F)) pari_err_TYPE("mfgaloisrep", F);
13062 77 : switch( itos(mfgaloistype(mf,F)) )
13063 : {
13064 49 : case 0: case -12:
13065 49 : LG = mkvec2(mkvecsmall2(3,1), mkvecsmall2(2,2)); break;
13066 28 : case -24:
13067 28 : LG = mkvec3(mkvecsmall2(2,1), mkvecsmall2(3,1), mkvecsmall2(2,2)); break;
13068 0 : case -60: return gc_GEN(av, search_A5(mf, F));
13069 0 : default: pari_err_IMPL("mfgaloisprojrep for types D_n");
13070 : }
13071 77 : return gc_GEN(av, search_solvable(LG, mf, F, prec));
13072 : }
|
