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 36967 : mkgNK(GEN N, GEN k, GEN CHI, GEN P) { return mkvec4(N, k, CHI, P); }
82 : static GEN
83 15015 : mkNK(long N, long k, GEN CHI) { return mkgNK(stoi(N), stoi(k), CHI, pol_x(1)); }
84 : GEN
85 8561 : MF_get_CHI(GEN mf) { return gmael(mf,1,3); }
86 : GEN
87 20265 : MF_get_gN(GEN mf) { return gmael(mf,1,1); }
88 : long
89 19173 : MF_get_N(GEN mf) { return itou(MF_get_gN(mf)); }
90 : GEN
91 14574 : MF_get_gk(GEN mf) { return gmael(mf,1,2); }
92 : long
93 7021 : MF_get_k(GEN mf)
94 : {
95 7021 : GEN gk = MF_get_gk(mf);
96 7021 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
97 7021 : 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 14665 : MF_get_space(GEN mf) { return itos(gmael(mf,1,4)); }
108 : GEN
109 4389 : MF_get_E(GEN mf) { return gel(mf,2); }
110 : GEN
111 21077 : MF_get_S(GEN mf) { return gel(mf,3); }
112 : GEN
113 1855 : MF_get_basis(GEN mf) { return shallowconcat(gel(mf,2), gel(mf,3)); }
114 : long
115 5502 : MF_get_dim(GEN mf)
116 : {
117 5502 : switch(MF_get_space(mf))
118 : {
119 693 : case mf_FULL:
120 693 : 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 4669 : default: /* mf_NEW, mf_CUSP, mf_OLD */
124 4669 : return lg(MF_get_S(mf)) - 1;
125 : }
126 : }
127 : GEN
128 7175 : MFnew_get_vj(GEN mf) { return gel(mf,4); }
129 : GEN
130 672 : MFcusp_get_vMjd(GEN mf) { return gel(mf,4); }
131 : GEN
132 6790 : MF_get_M(GEN mf) { return gmael(mf,5,3); }
133 : GEN
134 4774 : MF_get_Minv(GEN mf) { return gmael(mf,5,2); }
135 : GEN
136 10409 : 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 1225 : QabM_init(long n, ulong *p)
147 : {
148 1225 : ulong pinit = 1000000007;
149 : forprime_t T;
150 1225 : if (n <= 1) { *p = pinit; return 0; }
151 1218 : u_forprime_arith_init(&T, pinit, ULONG_MAX, 1, n);
152 1218 : *p = u_forprime_next(&T);
153 1218 : 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 1092 : ZabM_pseudoinv_i(GEN M, GEN P, long n, GEN *pv, GEN *den, int ratlift)
186 : {
187 1092 : GEN v = ZabM_indexrank(M, P, n);
188 1092 : if (pv) *pv = v;
189 1092 : M = shallowmatextract(M,gel(v,1),gel(v,2));
190 1092 : 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 1561 : QabM_ker(GEN M, GEN P, long n)
197 : {
198 1561 : if (n <= 2) return QM_ker(M);
199 378 : 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 1274 : QabM_pseudoinv_i(GEN M, GEN P, long n, GEN *pv, GEN *pden)
204 : {
205 : GEN cM, Mi;
206 1274 : if (n <= 2)
207 : {
208 1134 : M = Q_primitive_part(M, &cM);
209 1134 : Mi = ZM_pseudoinv(M, pv, pden); /* M^(-1) = Mi / (cM * den) */
210 : }
211 : else
212 : {
213 140 : M = Q_primitive_part(liftpol_shallow(M), &cM);
214 140 : Mi = ZabM_pseudoinv(M, P, n, pv, pden);
215 : }
216 1274 : *pden = mul_content(*pden, cM);
217 1274 : return Mi;
218 : }
219 : /* FIXME: delete */
220 : static GEN
221 1015 : QabM_pseudoinv(GEN M, GEN P, long n, GEN *pv, GEN *pden)
222 : {
223 1015 : GEN Mi = QabM_pseudoinv_i(M, P, n, pv, pden);
224 1015 : return P? gmodulo(Mi, P): Mi;
225 : }
226 :
227 : static GEN
228 10388 : QabM_indexrank(GEN M, GEN P, long n)
229 : {
230 : GEN z;
231 10388 : if (n <= 2)
232 : {
233 9233 : M = vec_Q_primpart(M);
234 9233 : z = ZM_indexrank(M); /* M^(-1) = Mi / (cM * den) */
235 : }
236 : else
237 : {
238 1155 : M = vec_Q_primpart(liftpol_shallow(M));
239 1155 : z = ZabM_indexrank(M, P, n);
240 : }
241 10388 : 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 109662 : myeulerphiu(ulong n)
251 : {
252 : pari_sp av;
253 109662 : if (n == 1) return 1;
254 90902 : 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 2975 : mynumdivu(long N)
266 : {
267 : pari_sp av;
268 2975 : if (N == 1) return 1;
269 2870 : av = avma; return gc_long(av, numdivu_fact(myfactoru(N)));
270 : }
271 :
272 : /* N\prod_{p|N} (1+1/p) */
273 : static long
274 393960 : mypsiu(ulong N)
275 : {
276 : pari_sp av;
277 : GEN P;
278 : long j, l, a;
279 393960 : if (N == 1) return 1;
280 310436 : av = avma; P = gel(myfactoru(N), 1); l = lg(P);
281 739452 : for (a = N, j = 1; j < l; j++) a += a / P[j];
282 310436 : return gc_long(av, a);
283 : }
284 : /* write n = mf^2. Return m, set f. */
285 : static ulong
286 70 : mycore(ulong n, long *pf)
287 : {
288 70 : pari_sp av = avma;
289 70 : GEN fa = myfactoru(n), P = gel(fa,1), E = gel(fa,2);
290 70 : long i, l = lg(P), m = 1, f = 1;
291 266 : for (i = 1; i < l; i++)
292 : {
293 196 : long j, p = P[i], e = E[i];
294 196 : if (e & 1) m *= p;
295 455 : for (j = 2; j <= e; j+=2) f *= p;
296 : }
297 70 : *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 5746333 : corediscs_fact(GEN fa)
303 : {
304 5746333 : GEN P = gel(fa,1), E = gel(fa,2);
305 5746333 : long i, l = lg(P), m = 1;
306 19040393 : for (i = 1; i < l; i++)
307 : {
308 13294060 : long p = P[i], e = E[i];
309 13294060 : if (e & 1) m *= p;
310 : }
311 5746333 : if ((m&3L) != 3) m <<= 2;
312 5746333 : return m;
313 : }
314 : static long
315 6993 : mubeta(long n)
316 : {
317 6993 : pari_sp av = avma;
318 6993 : GEN E = gel(myfactoru(n), 2);
319 6993 : long i, s = 1, l = lg(E);
320 14511 : for (i = 1; i < l; i++)
321 : {
322 7518 : long e = E[i];
323 7518 : if (e >= 3) return gc_long(av,0);
324 7518 : if (e == 1) s *= -2;
325 : }
326 6993 : 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 7650494 : mubeta2(long n, long m)
333 : {
334 7650494 : pari_sp av = avma;
335 7650494 : GEN fa = myfactoru(n), P = gel(fa,1), E = gel(fa,2);
336 7650494 : long i, s = 1, l = lg(P);
337 15381428 : for (i = 1; i < l; i++)
338 : {
339 7730934 : long p = P[i], e = E[i];
340 7730934 : if (m % p)
341 : { /* p^e in n1 */
342 6565567 : if (e >= 3) return gc_long(av,0);
343 6565567 : if (e == 1) s *= -2;
344 : }
345 : else
346 : { /* in n2 */
347 1165367 : if (e >= 2) return gc_long(av,0);
348 1165367 : s = -s;
349 : }
350 : }
351 7650494 : 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 1874765 : newt_params(long N, long n, long FC, long *pg, long *pN2)
359 : {
360 1874765 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
361 1874765 : long i, g = 1, N2 = 1, l = lg(P);
362 5004483 : for (i = 1; i < l; i++)
363 : {
364 3129718 : long p = P[i], e = E[i];
365 3129718 : if (e == 1)
366 2732919 : { if (FC % p && n % (p*p) == 0) g *= p; }
367 : else
368 396799 : N2 *= upowuu(p,(n % p)? e-2: e-1);
369 : }
370 1874765 : *pg = g; *pN2 = N2;
371 1874765 : }
372 : /* simplified version of newt_params for n = 1 (newdim) */
373 : static void
374 41510 : newd_params(long N, long *pN2)
375 : {
376 41510 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
377 41510 : long i, N2 = 1, l = lg(P);
378 103999 : for (i = 1; i < l; i++)
379 : {
380 62489 : long p = P[i], e = E[i];
381 62489 : if (e > 2) N2 *= upowuu(p, e-2);
382 : }
383 41510 : *pN2 = N2;
384 41510 : }
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 34013 : 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 34013 : 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 16093 : QabV_tracerel(GEN v, long t, GEN x)
528 : {
529 : GEN z;
530 16093 : 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 2189572 : Qab_Czeta(long k, long o, GEN C, long vt)
544 : {
545 2189572 : if (!k) return C;
546 1456077 : if (!odd(o))
547 : { /* optimization: reduce max degree by a factor 2 for free */
548 1404970 : o >>= 1;
549 1404970 : if (k >= o) { k -= o; C = gneg(C); if (!k) return C; }
550 : }
551 1109451 : return monomial(C, k, vt);
552 : }
553 : /* zeta_o^k */
554 : static GEN
555 200242 : 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 3722761 : gmfcharorder(GEN CHI) { return gel(CHI, 3); }
566 : long
567 3684408 : mfcharorder(GEN CHI) { return itou(gmfcharorder(CHI)); }
568 : static long
569 2709 : mfcharistrivial(GEN CHI) { return !CHI || mfcharorder(CHI) == 1; }
570 : static GEN
571 1560188 : gmfcharmodulus(GEN CHI) { return gmael3(CHI, 1, 1, 1); }
572 : long
573 1560188 : mfcharmodulus(GEN CHI) { return itou(gmfcharmodulus(CHI)); }
574 : GEN
575 564011 : mfcharpol(GEN CHI) { return gel(CHI,4); }
576 :
577 : /* vz[i+1] = image of (zeta_o)^i in Fp */
578 : static ulong
579 221459 : Qab_Czeta_Fl(long k, GEN vz, ulong C, ulong p)
580 : {
581 : long o;
582 221459 : if (!k) return C;
583 148911 : o = lg(vz)-2;
584 148911 : if ((k << 1) == o) return Fl_neg(C,p);
585 123123 : return Fl_mul(C, vz[k+1], p);
586 : }
587 :
588 : static long
589 2508317 : znchareval_i(GEN CHI, long n, GEN ord)
590 2508317 : { 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 14217 : mfchareval(GEN CHI, long n)
595 : {
596 14217 : GEN Pn, C, go = gmfcharorder(CHI);
597 14217 : long k, o = go[2];
598 14217 : if (o == 1) return gen_1;
599 7378 : k = znchareval_i(CHI, n, go);
600 7378 : Pn = mfcharpol(CHI);
601 7378 : C = Qab_zeta(k, o, varn(Pn));
602 7378 : if (typ(C) != t_POL) return C;
603 5320 : 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 3563112 : mfcharevalord(GEN CHI, long n, long d)
609 : {
610 3563112 : if (mfcharorder(CHI) == 1) return 0;
611 2497243 : return znchareval_i(CHI, n, utoi(d));
612 : }
613 :
614 : /* G a znstar, L a Conrey log: return a 'mfchar' */
615 : static GEN
616 373877 : mfcharGL(GEN G, GEN L)
617 : {
618 373877 : GEN o = zncharorder(G,L);
619 373877 : long ord = itou(o), vt = fetch_user_var("t");
620 373877 : return mkvec4(G, L, o, polcyclo(ord,vt));
621 : }
622 : static GEN
623 5775 : mfchartrivial()
624 5775 : { return mfcharGL(znstar0(gen_1,1), cgetg(1,t_COL)); }
625 : /* convert a generic character into an 'mfchar' */
626 : static GEN
627 4039 : get_mfchar(GEN CHI)
628 : {
629 : GEN G, L;
630 4039 : 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 3969 : G = gel(CHI,1);
639 3969 : L = gel(CHI,2); if (typ(L) != t_COL) L = znconreylog(G,L);
640 3969 : 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 9191 : checkCHI(GEN NK, long N, int joker)
646 : {
647 : GEN CHI;
648 9191 : if (lg(NK) == 3)
649 693 : CHI = mfchartrivial();
650 : else
651 : {
652 : long i, l;
653 8498 : CHI = gel(NK,3); l = lg(CHI);
654 8498 : if (isintzero(CHI) && joker)
655 4116 : CHI = NULL; /* all character orbits */
656 4382 : else if (isintm1(CHI) && joker > 1)
657 2373 : CHI = gen_m1; /* sum over all character orbits */
658 2009 : 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 1876 : CHI = get_mfchar(CHI); /* single char */
667 1876 : if (N % mfcharmodulus(CHI)) pari_err_TYPE("checkNF [chi]", NK);
668 : }
669 : }
670 9177 : return CHI;
671 : }
672 : /* support half-integral weight */
673 : static void
674 9198 : checkNK2(GEN NK, long *N, long *nk, long *dk, GEN *CHI, int joker)
675 : {
676 9198 : long l = lg(NK);
677 : GEN T;
678 9198 : if (typ(NK) != t_VEC || l < 3 || l > 4) pari_err_TYPE("checkNK", NK);
679 9198 : T = gel(NK,1); if (typ(T) != t_INT) pari_err_TYPE("checkNF [N]", NK);
680 9198 : *N = itos(T); if (*N <= 0) pari_err_TYPE("checkNF [N <= 0]", NK);
681 9198 : T = gel(NK,2);
682 9198 : switch(typ(T))
683 : {
684 5817 : 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 9191 : *CHI = checkCHI(NK, *N, joker);
691 9177 : }
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 68369 : mfcharparity(GEN CHI)
728 : {
729 68369 : if (!CHI) return 1;
730 68369 : 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 76307 : mfchartoprimitive(GEN CHI, long *pF)
735 : {
736 : pari_sp av;
737 : GEN chi, F;
738 76307 : if (!CHI) { if (pF) *pF = 1; return mfchartrivial(); }
739 76307 : av = avma; F = znconreyconductor(gel(CHI,1), gel(CHI,2), &chi);
740 76307 : if (typ(F) == t_INT) set_avma(av);
741 : else
742 : {
743 7840 : CHI = leafcopy(CHI);
744 7840 : gel(CHI,1) = znstar0(F, 1);
745 7840 : gel(CHI,2) = chi;
746 : }
747 76307 : if (pF) *pF = mfcharmodulus(CHI);
748 76307 : return CHI;
749 : }
750 : static long
751 397327 : mfcharconductor(GEN CHI)
752 : {
753 397327 : pari_sp av = avma;
754 397327 : GEN res = znconreyconductor(gel(CHI,1), gel(CHI,2), NULL);
755 397327 : if (typ(res) == t_VEC) res = gel(res, 1);
756 397327 : return gc_long(av, itos(res));
757 : }
758 :
759 : /* Operations on mf closures */
760 : static GEN
761 62615 : tagparams(long t, GEN NK) { return mkvec2(mkvecsmall(t), NK); }
762 : static GEN
763 1141 : 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 10220 : tag(long t, GEN NK, GEN x) { retmkvec2(tagparams(t,NK), x); }
768 : static GEN
769 36127 : tag2(long t, GEN NK, GEN x, GEN y) { retmkvec3(tagparams(t,NK), x,y); }
770 : static GEN
771 16086 : 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 18627 : checkmf_i(GEN F)
778 18627 : { return typ(F) == t_VEC
779 17808 : && lg(F) > 1 && typ(gel(F,1)) == t_VEC
780 13111 : && lg(gel(F,1)) == 3
781 12950 : && typ(gmael(F,1,1)) == t_VECSMALL
782 36435 : && typ(gmael(F,1,2)) == t_VEC; }
783 231581 : long mf_get_type(GEN F) { return gmael(F,1,1)[1]; }
784 183960 : GEN mf_get_gN(GEN F) { return gmael3(F,1,2,1); }
785 138943 : 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 118860 : long mf_get_N(GEN F) { return itou(mf_get_gN(F)); }
789 71694 : long mf_get_k(GEN F)
790 : {
791 71694 : GEN gk = mf_get_gk(F);
792 71694 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
793 71694 : return itou(gk);
794 : }
795 61894 : GEN mf_get_CHI(GEN F) { return gmael3(F,1,2,3); }
796 24262 : GEN mf_get_field(GEN F) { return gmael3(F,1,2,4); }
797 18767 : GEN mf_get_NK(GEN F) { return gmael(F,1,2); }
798 : static void
799 525 : mf_setfield(GEN f, GEN P)
800 : {
801 525 : gel(f,1) = leafcopy(gel(f,1));
802 525 : gmael(f,1,2) = leafcopy(gmael(f,1,2));
803 525 : gmael3(f,1,2,4) = P;
804 525 : }
805 :
806 : /* UTILITY FUNCTIONS */
807 : GEN
808 9065 : mftocol(GEN F, long lim, long d)
809 9065 : { GEN c = mfcoefs_i(F, lim, d); settyp(c,t_COL); return c; }
810 : GEN
811 2093 : mfvectomat(GEN vF, long lim, long d)
812 : {
813 2093 : long j, l = lg(vF);
814 2093 : GEN M = cgetg(l, t_MAT);
815 10339 : for (j = 1; j < l; j++) gel(M,j) = mftocol(gel(vF,j), lim, d);
816 2093 : 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 3381 : RgV_mul_RgXn(GEN a, GEN b)
836 : {
837 3381 : long n = lg(a)-1;
838 : GEN c;
839 3381 : a = RgV_to_RgX(a,0);
840 3381 : b = RgV_to_RgX(b,0); c = RgXn_mul(a, b, n);
841 3381 : 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 8778 : c_deflate(long n, long d, GEN v)
868 : {
869 8778 : long i, id, l = n+2;
870 : GEN w;
871 8778 : 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 gerepilecopy(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 gerepilecopy(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 gerepilecopy(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 gerepileupto(av, res);
1023 : }
1024 : /* linear combination \sum L[j] vecF[j] */
1025 : static GEN
1026 2975 : c_linear(long n, long d, GEN F, GEN L, GEN dL)
1027 : {
1028 2975 : pari_sp av = avma;
1029 2975 : long j, l = lg(L);
1030 2975 : GEN S = NULL;
1031 10640 : for (j = 1; j < l; j++)
1032 : {
1033 7665 : GEN c = gel(L,j);
1034 7665 : if (gequal0(c)) continue;
1035 6909 : c = gmul(c, mfcoefs_i(gel(F,j), n, d));
1036 6909 : S = S? gadd(S,c): c;
1037 : }
1038 2975 : if (!S) return zerovec(n+1);
1039 2975 : if (!is_pm1(dL)) S = gdiv(S, dL);
1040 2975 : return gerepileupto(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 82481 : bhn_parse(GEN f, long *d, long *j)
1048 : {
1049 82481 : long t = mf_get_type(f);
1050 82481 : *d = *j = 1;
1051 82481 : if (t == t_MF_BD) { *d = itos(gel(f,3)); f = gel(f,2); t = mf_get_type(f); }
1052 82481 : if (t == t_MF_HECKE) { *j = gel(f,2)[1]; f = gel(f,3); }
1053 82481 : return f;
1054 : }
1055 : /* f as above, return the t_MF_NEWTRACE component */
1056 : static GEN
1057 32494 : bhn_newtrace(GEN f)
1058 : {
1059 32494 : long t = mf_get_type(f);
1060 32494 : if (t == t_MF_BD) { f = gel(f,2); t = mf_get_type(f); }
1061 32494 : if (t == t_MF_HECKE) f = gel(f,3);
1062 32494 : return f;
1063 : }
1064 : static int
1065 3976 : ok_bhn_linear(GEN vf)
1066 : {
1067 3976 : long i, N0 = 0, l = lg(vf);
1068 : GEN CHI, gk;
1069 3976 : if (l == 1) return 1;
1070 3976 : gk = mf_get_gk(gel(vf,1));
1071 3976 : CHI = mf_get_CHI(gel(vf,1));
1072 27321 : for (i = 1; i < l; i++)
1073 : {
1074 25676 : GEN f = bhn_newtrace(gel(vf,i));
1075 25676 : long N = mf_get_N(f);
1076 25676 : if (mf_get_type(f) != t_MF_NEWTRACE) return 0;
1077 23345 : if (N < N0) return 0; /* largest level must come last */
1078 23345 : N0 = N;
1079 23345 : if (!gequal(gk,mf_get_gk(f))) return 0; /* same k */
1080 23345 : if (!gequal(gel(mf_get_CHI(f),2), gel(CHI,2))) return 0; /* same CHI */
1081 : }
1082 1645 : return 1;
1083 : }
1084 :
1085 : /* vF not empty, same hypotheses as bhnmat_extend */
1086 : static GEN
1087 6923 : bhnmat_extend_nocache(GEN M, long N, long n, long d, GEN vF)
1088 : {
1089 : cachenew_t cache;
1090 6923 : long l = lg(vF);
1091 : GEN f;
1092 6923 : if (l == 1) return M? M: cgetg(1, t_MAT);
1093 6818 : f = bhn_newtrace(gel(vF,1)); /* N.B. mf_get_N(f) divides N */
1094 6818 : init_cachenew(&cache, n*d, N, f);
1095 6818 : M = bhnmat_extend(M, n, d, vF, &cache);
1096 6818 : dbg_cachenew(&cache); return M;
1097 : }
1098 : /* c_linear of "bhn" mf closures, same hypotheses as bhnmat_extend */
1099 : static GEN
1100 2233 : c_linear_bhn(long n, long d, GEN F)
1101 : {
1102 : pari_sp av;
1103 2233 : GEN M, v, vF = gel(F,2), L = gel(F,3), dL = gel(F,4);
1104 2233 : if (lg(L) == 1) return zerovec(n+1);
1105 2233 : av = avma;
1106 2233 : M = bhnmat_extend_nocache(NULL, mf_get_N(F), n, d, vF);
1107 2233 : v = RgM_RgC_mul(M,L); settyp(v, t_VEC);
1108 2233 : if (!is_pm1(dL)) v = gdiv(v, dL);
1109 2233 : return gerepileupto(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 1673 : RgX_embed(GEN f, GEN E)
1151 : {
1152 : GEN vT;
1153 1673 : if (typ(f) != t_POL || varn(f) != 0) return mfembed(E, f);
1154 1631 : 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 1708 : mfvecembed(GEN E, GEN v)
1165 : {
1166 : GEN vT;
1167 1708 : 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 100670 : mfembed(GEN E, GEN f)
1192 : {
1193 : GEN vT;
1194 100670 : 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 350 : RgX_embedall(GEN f, GEN vE)
1205 : {
1206 350 : long i, l = lg(vE);
1207 : GEN v;
1208 350 : 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 95000 : Rg_embedall(GEN v, GEN vE)
1234 95000 : { 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 gerepilecopy(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 gerepilecopy(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 gerepileupto(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 gerepileupto(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 gerepilecopy(av, c_deflate(n, d, VF));
1323 : }
1324 : }
1325 :
1326 : /* Twist by the character (D/.) */
1327 : static GEN
1328 161 : c_twist(long n, long d, GEN F, GEN D)
1329 : {
1330 161 : pari_sp av = avma;
1331 161 : GEN v = mfcoefs_i(F, n, d), z = cgetg(n+2, t_VEC);
1332 : long i;
1333 707 : for (i = 0; i <= n; i++)
1334 : {
1335 : long s;
1336 546 : GEN a = gel(v, i+1);
1337 546 : 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 546 : switch(s)
1344 : {
1345 147 : case 1: a = gcopy(a); break;
1346 140 : case -1: a = gneg(a); break;
1347 259 : default: a = gen_0; break;
1348 : }
1349 546 : gel(z, i+1) = a;
1350 : }
1351 161 : return gerepileupto(av, z);
1352 : }
1353 :
1354 : /* form F given by closure, compute T(n)(F) as closure */
1355 : static GEN
1356 1050 : c_hecke(long m, long l, GEN DATA, GEN F)
1357 : {
1358 1050 : pari_sp av = avma;
1359 1050 : return gerepilecopy(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 469 : eta3_ZXn(long m)
1374 : {
1375 469 : long l = m+2, n, k;
1376 469 : GEN P = cgetg(l,t_POL);
1377 469 : P[1] = evalsigne(1)|evalvarn(0);
1378 6489 : for (n = 2; n < l; n++) gel(P,n) = gen_0;
1379 469 : for (n = k = 0;; n++)
1380 : {
1381 2611 : if (k + n >= m) { setlg(P, k+3); return P; }
1382 2142 : k += n;
1383 : /* now k = n(n+1) / 2 */
1384 2142 : gel(P, k+2) = odd(n)? utoineg(2*n+1): utoipos(2*n+1);
1385 : }
1386 : }
1387 :
1388 : static GEN
1389 476 : c_delta(long n, long d)
1390 : {
1391 476 : pari_sp ltop = avma;
1392 476 : long N = n*d;
1393 : GEN e;
1394 476 : if (!N) return mkvec(gen_0);
1395 469 : e = eta3_ZXn(N);
1396 469 : e = ZXn_sqr(e,N);
1397 469 : e = ZXn_sqr(e,N);
1398 469 : e = ZXn_sqr(e,N); /* eta(x)^24 */
1399 469 : settyp(e, t_VEC);
1400 469 : gel(e,1) = gen_0; /* Delta(x) = x*eta(x)^24 as a t_VEC */
1401 469 : return gerepilecopy(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) = gerepileupto(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 gerepile 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 gerepilecopy(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 gerepilecopy(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 gerepilecopy(av, res);
1462 : }
1463 :
1464 : static GEN
1465 1897 : c_newtrace(long n, long d, GEN F)
1466 : {
1467 1897 : pari_sp av = avma;
1468 : cachenew_t cache;
1469 1897 : long N = mf_get_N(F);
1470 : GEN v;
1471 1897 : init_cachenew(&cache, n == 1? 1: n*d, N, F);
1472 1897 : v = colnewtrace(0, n, d, N, mf_get_k(F), &cache);
1473 1897 : settyp(v, t_VEC); return gerepilecopy(av, v);
1474 : }
1475 :
1476 : static GEN
1477 7462 : c_Bd(long n, long d, GEN F, GEN A)
1478 : {
1479 7462 : pari_sp av = avma;
1480 7462 : long a = itou(A), ad = ugcd(a,d), aad = a/ad, i, j;
1481 7462 : GEN w, v = mfcoefs_i(F, n/aad, d/ad);
1482 7462 : if (a == 1) return v;
1483 7462 : n++; w = zerovec(n);
1484 213087 : for (i = j = 1; j <= n; i++, j += aad) gel(w,j) = gcopy(gel(v,i));
1485 7462 : return gerepileupto(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 gerepilecopy(av, A);
1497 1043 : return gerepileupto(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 gerepilecopy(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 gerepile 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 gerepileupto(av, v);
1529 : }
1530 :
1531 : static GEN
1532 11256 : c_mfeisen(long n, long d, GEN F)
1533 : {
1534 11256 : pari_sp av = avma;
1535 11256 : GEN v, vchi, E0, P, T, CHI, gk = mf_get_gk(F);
1536 : long i, k;
1537 11256 : if (typ(gk) != t_INT) return c_mfEH(n, d, F);
1538 10913 : k = itou(gk);
1539 10913 : vchi = gel(F,2);
1540 10913 : E0 = gel(vchi,1);
1541 10913 : T = gel(vchi,2);
1542 10913 : P = gel(T,1);
1543 10913 : CHI = gel(vchi,3);
1544 10913 : v = cgetg(n+2, t_VEC);
1545 10913 : gel(v, 1) = gcopy(E0); /* E(0) */
1546 10913 : if (lg(vchi) == 5)
1547 : { /* E_k(chi1,chi2) */
1548 8820 : GEN CHI2 = gel(vchi,4), F3 = gel(F,3);
1549 8820 : long ord = F3[1], j = F3[2];
1550 508634 : for (i = 1; i <= n; i++) gel(v, i+1) = sigchi2(k, CHI, CHI2, i*d, ord);
1551 8820 : 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 10913 : if (degpol(P) != 1 && !RgV_is_QV(v)) return gerepileupto(av, gmodulo(v, P));
1558 7980 : return gerepilecopy(av, v);
1559 : }
1560 :
1561 : /* N^k * (D * B_k)(x/N), set D = denom(B_k) */
1562 : static GEN
1563 1589 : bern_init(long N, long k, GEN *pD)
1564 1589 : { 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 = gerepileuptoint(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 37863 : mfcoefs_i(GEN F, long n, long d)
1593 : {
1594 37863 : if (n < 0) return gen_0;
1595 37863 : switch(mf_get_type(F))
1596 : {
1597 140 : case t_MF_CONST: return c_const(n, d, gel(F,2));
1598 11256 : case t_MF_EISEN: return c_mfeisen(n, d, F);
1599 840 : case t_MF_Ek: return c_Ek(n, d, F);
1600 476 : 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 2975 : case t_MF_LINEAR: return c_linear(n, d, gel(F,2), gel(F,3), gel(F,4));
1608 2233 : 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 161 : case t_MF_TWIST: return c_twist(n, d, gel(F,2), gel(F,3));
1614 1050 : case t_MF_HECKE: return c_hecke(n, d, gel(F,2), gel(F,3));
1615 7462 : 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 1897 : 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 5992 : space_is_cusp(long space) { return space != mf_FULL && space != mf_EISEN; }
1628 : /* safe with flraw mf */
1629 : static GEN
1630 2590 : mfcoefs_mf(GEN mf, long n, long d)
1631 : {
1632 2590 : GEN MS, ME, E = MF_get_E(mf), S = MF_get_S(mf), M = MF_get_M(mf);
1633 2590 : long lE = lg(E), lS = lg(S), l = lE+lS-1;
1634 :
1635 2590 : if (l == 1) return cgetg(1, t_MAT);
1636 2478 : 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 2457 : ME = (lE == 1)? cgetg(1, t_MAT): mfvectomat(E, n, d);
1639 2457 : if (lS == 1)
1640 448 : MS = cgetg(1, t_MAT);
1641 2009 : 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 1645 : 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 1337 : MS = bhnmat_extend_nocache(NULL, MF_get_N(mf), n, d, S);
1657 2457 : return shallowconcat(ME,MS);
1658 : }
1659 : GEN
1660 3864 : mfcoefs(GEN F, long n, long d)
1661 : {
1662 3864 : 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 gerepilecopy(av, mfcoefs_mf(mf,n,d));
1667 : }
1668 3822 : if (d <= 0) pari_err_DOMAIN("mfcoefs", "d", "<=", gen_0, stoi(d));
1669 3822 : if (n < 0) return cgetg(1, t_VEC);
1670 3822 : 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: gerepilecopy(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 307909 : induce(GEN G, GEN CHI)
1698 : {
1699 : GEN o, chi;
1700 307909 : 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 7133 : if (mfcharmodulus(CHI) == itos(znstar_get_N(G))) return CHI;
1709 6510 : CHI = leafcopy(CHI);
1710 6510 : chi = zncharinduce(gel(CHI,1), gel(CHI,2), G);
1711 6510 : gel(CHI,1) = G;
1712 6510 : gel(CHI,2) = chi;
1713 : }
1714 307286 : 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 6615 : char2(GEN *pCHI1, GEN *pCHI2)
1726 : {
1727 6615 : GEN CHI1 = *pCHI1, G1 = gel(CHI1,1), N1 = znstar_get_N(G1);
1728 6615 : GEN CHI2 = *pCHI2, G2 = gel(CHI2,1), N2 = znstar_get_N(G2);
1729 6615 : if (!equalii(N1,N2))
1730 : {
1731 4935 : GEN G, d = gcdii(N1,N2);
1732 4935 : 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 6615 : }
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 5509 : mfchardiv_i(GEN CHI1, GEN CHI2)
1767 : {
1768 5509 : GEN G = gel(CHI1,1), chi3 = znchardiv(G, gel(CHI1,2), gel(CHI2,2));
1769 5509 : return mfcharGL(G, chi3);
1770 : }
1771 : /* mfchar or charinit; outputs a mfchar */
1772 : static GEN
1773 5509 : mfchardiv(GEN CHI1, GEN CHI2)
1774 : {
1775 5509 : 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 3906 : mfsamefield(GEN T, GEN P, GEN Q)
1818 : {
1819 3906 : if (degpol(P) == 1) return Q;
1820 602 : if (degpol(Q) == 1) return P;
1821 511 : if (!gequal(P,Q)) pari_err_TYPE("mfsamefield [different fields]",mkvec2(P,Q));
1822 504 : if (T) err_cyclo();
1823 504 : 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 gerepilecopy(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 gerepilecopy(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 gerepilecopy(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 1820 : mflinear_strip(GEN *pF, GEN *pL)
1882 : {
1883 1820 : pari_sp av = avma;
1884 1820 : GEN F = *pF, L = *pL;
1885 1820 : long i, j, l = lg(L);
1886 1820 : GEN F2 = cgetg(l, t_VEC), L2 = cgetg(l, t_VEC);
1887 10108 : for (i = j = 1; i < l; i++)
1888 : {
1889 8288 : if (gequal0(gel(L,i))) continue;
1890 4207 : gel(F2,j) = gel(F,i);
1891 4207 : gel(L2,j) = gel(L,i); j++;
1892 : }
1893 1820 : if (j == l) set_avma(av);
1894 : else
1895 : {
1896 546 : setlg(F2,j); *pF = F2;
1897 546 : setlg(L2,j); *pL = L2;
1898 : }
1899 1820 : return (j > 1);
1900 : }
1901 : static GEN
1902 6741 : taglinear_i(long t, GEN NK, GEN F, GEN L)
1903 : {
1904 : GEN dL;
1905 6741 : L = Q_remove_denom(L, &dL); if (!dL) dL = gen_1;
1906 6741 : return tag3(t, NK, F, L, dL);
1907 : }
1908 : static GEN
1909 2765 : taglinear(GEN NK, GEN F, GEN L)
1910 : {
1911 2765 : long t = ok_bhn_linear(F)? t_MF_LINEAR_BHN: t_MF_LINEAR;
1912 2765 : return taglinear_i(t, NK, F, L);
1913 : }
1914 : /* assume F has parameters NK = [N,K,CHI] */
1915 : static GEN
1916 476 : mflinear_i(GEN NK, GEN F, GEN L)
1917 : {
1918 476 : if (!mflinear_strip(&F,&L)) return mftrivial();
1919 476 : return taglinear(NK, F,L);
1920 : }
1921 : static GEN
1922 700 : mflinear_bhn(GEN mf, GEN L)
1923 : {
1924 : long i, l;
1925 700 : GEN P, NK, F = MF_get_S(mf);
1926 700 : if (!mflinear_strip(&F,&L)) return mftrivial();
1927 693 : l = lg(L); P = pol_x(1);
1928 3031 : for (i = 1; i < l; i++)
1929 : {
1930 2338 : GEN c = gel(L,i);
1931 2338 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1)
1932 518 : P = mfsamefield(NULL, P, gel(c,1));
1933 : }
1934 693 : NK = mkgNK(MF_get_gN(mf), MF_get_gk(mf), MF_get_CHI(mf), P);
1935 693 : 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 1001 : tobasis(GEN mf, GEN F, GEN L)
2026 : {
2027 1001 : if (checkmf_i(L) && mf) return mftobasis(mf, L, 0);
2028 994 : if (typ(F) != t_VEC) pari_err_TYPE("mflinear",F);
2029 994 : if (!is_vec_t(typ(L))) pari_err_TYPE("mflinear",L);
2030 994 : if (lg(L) != lg(F)) pari_err_DIM("mflinear");
2031 994 : return L;
2032 : }
2033 : GEN
2034 1043 : mflinear(GEN F, GEN L)
2035 : {
2036 1043 : pari_sp av = avma;
2037 1043 : GEN G, NK, P, mf = checkMF_i(F), N = NULL, K = NULL, CHI = NULL;
2038 : long i, l;
2039 1043 : if (mf)
2040 : {
2041 707 : GEN gk = MF_get_gk(mf);
2042 707 : F = MF_get_basis(F);
2043 707 : if (typ(gk) != t_INT)
2044 42 : return gerepilecopy(av, mflineardiv_linear(F, L, 1));
2045 665 : if (itou(gk) > 1 && space_is_cusp(MF_get_space(mf)))
2046 : {
2047 448 : L = tobasis(mf, F, L);
2048 448 : return gerepilecopy(av, mflinear_bhn(mf, L));
2049 : }
2050 : }
2051 553 : L = tobasis(mf, F, L);
2052 553 : if (!mflinear_strip(&F,&L)) return mftrivial();
2053 :
2054 546 : l = lg(F);
2055 546 : if (l == 2 && gequal1(gel(L,1))) return gerepilecopy(av, gel(F,1));
2056 287 : P = pol_x(1);
2057 910 : for (i = 1; i < l; i++)
2058 : {
2059 630 : GEN f = gel(F,i), c = gel(L,i), Ni, Ki;
2060 630 : if (!checkmf_i(f)) pari_err_TYPE("mflinear", f);
2061 630 : Ni = mf_get_gN(f); N = N? lcmii(N, Ni): Ni;
2062 630 : Ki = mf_get_gk(f);
2063 630 : if (!K) K = Ki;
2064 343 : else if (!gequal(K, Ki))
2065 7 : pari_err_TYPE("mflinear [different weights]", mkvec2(K,Ki));
2066 623 : P = mfsamefield(NULL, P, mf_get_field(f));
2067 623 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1)
2068 126 : P = mfsamefield(NULL, P, gel(c,1));
2069 : }
2070 280 : G = znstar0(N,1);
2071 889 : for (i = 1; i < l; i++)
2072 : {
2073 616 : GEN CHI2 = mf_get_CHI(gel(F,i));
2074 616 : CHI2 = induce(G, CHI2);
2075 616 : if (!CHI) CHI = CHI2;
2076 336 : else if (!gequal(CHI, CHI2))
2077 7 : pari_err_TYPE("mflinear [different characters]", mkvec2(CHI,CHI2));
2078 : }
2079 273 : NK = mkgNK(N, K, CHI, P);
2080 273 : return gerepilecopy(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 gerepilecopy(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 gerepilecopy(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 gerepilecopy(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 gerepilecopy(av, tag2(t_MF_DERIVE2, NK, F, stoi(m)));
2157 : }
2158 :
2159 : GEN
2160 21 : mftwist(GEN F, GEN D)
2161 : {
2162 21 : pari_sp av = avma;
2163 : GEN NK, CHI, NT, Da;
2164 : long q;
2165 21 : if (!checkmf_i(F)) pari_err_TYPE("mftwist", F);
2166 21 : if (typ(D) != t_INT) pari_err_TYPE("mftwist", D);
2167 21 : Da = mpabs_shallow(D);
2168 21 : CHI = mf_get_CHI(F); q = mfcharconductor(CHI);
2169 21 : NT = glcm(glcm(mf_get_gN(F), mulsi(q, Da)), sqri(Da));
2170 21 : NK = mkgNK(NT, mf_get_gk(F), CHI, mf_get_field(F));
2171 21 : return gerepilecopy(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 504 : cache_reset(long id) { caches[id].miss = caches[id].maxmiss = 0; }
2202 : static void
2203 9310 : cache_delete(long id) { guncloneNULL(caches[id].cache); }
2204 : static void
2205 518 : cache_set(long id, GEN S)
2206 : {
2207 518 : GEN old = caches[id].cache;
2208 518 : caches[id].cache = gclone(S);
2209 518 : guncloneNULL(old);
2210 518 : }
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 448759827 : cache_get(long id, ulong D)
2217 : {
2218 448759827 : cache *S = &caches[id];
2219 448759827 : const ulong d = S->compressed? D>>1: D;
2220 : ulong max, l;
2221 :
2222 448759827 : if (!S->cache)
2223 : {
2224 377 : max = maxuu(minuu(D, S->maxself), S->minself);
2225 377 : S->init(max);
2226 377 : l = lg(S->cache);
2227 : }
2228 : else
2229 : {
2230 448759450 : l = lg(S->cache);
2231 448759450 : if (l <= d)
2232 : {
2233 367 : if (D > S->maxmiss) S->maxmiss = D;
2234 367 : if (DEBUGLEVEL >= 3)
2235 0 : err_printf("miss in cache %s: %lu, max = %lu\n",
2236 : S->name, D, S->maxmiss);
2237 367 : if (S->miss++ >= 5 && D < S->maxself)
2238 : {
2239 17 : max = minuu(S->maxself, (long)(S->maxmiss * 1.2));
2240 17 : if (max <= S->maxself)
2241 : {
2242 17 : if (DEBUGLEVEL >= 3)
2243 0 : err_printf("resetting cache %s to %lu\n", S->name, max);
2244 17 : S->init(max); l = lg(S->cache);
2245 : }
2246 : }
2247 : }
2248 : }
2249 448759827 : 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 gerepilecopy(av, shallowtrans(M));
2277 : }
2278 :
2279 : void
2280 1862 : pari_close_mf(void)
2281 : {
2282 1862 : cache_delete(cache_FACT);
2283 1862 : cache_delete(cache_DIV);
2284 1862 : cache_delete(cache_H);
2285 1862 : cache_delete(cache_D);
2286 1862 : cache_delete(cache_DIH);
2287 1862 : }
2288 :
2289 : /*************************************************************************/
2290 : /* a odd, update local cache (recycle memory) */
2291 : static GEN
2292 2275 : update_factor_cache(long a, long lim, long *pb)
2293 : {
2294 2275 : const long step = 16000; /* even; don't increase this: RAM cache thrashing */
2295 2275 : if (a + 2*step > lim)
2296 207 : *pb = lim; /* fuse last 2 chunks */
2297 : else
2298 2068 : *pb = a + step;
2299 2275 : 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 36 : av2 = avma;
2313 36 : cachea = cacheb = 0;
2314 5746392 : for (N = 1; N <= lim; N+=2)
2315 : { /* N odd */
2316 : long i, d, d2;
2317 : GEN F;
2318 5746340 : if (N > cacheb)
2319 : {
2320 702 : set_avma(av2); cachea = N;
2321 702 : CACHE = update_factor_cache(N, lim, &cacheb);
2322 : }
2323 5746340 : F = gel(CACHE, ((N-cachea)>>1)+1); /* factoru(N) */
2324 5746340 : D[N] = d = corediscs_fact(F); /* = 3 mod 4 or 4 mod 16 */
2325 5746339 : d2 = odd(d)? d<<3: d<<1;
2326 5746420 : for (i = 1;;)
2327 : {
2328 7662685 : if ((N << i) > lim) break;
2329 3831363 : D[N<<i] = d2; i++;
2330 3831363 : if ((N << i) > lim) break;
2331 1916265 : 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 173 : constfact(long lim)
2340 : {
2341 : pari_sp av;
2342 173 : GEN VFACT = caches[cache_FACT].cache;
2343 173 : long LIM = VFACT? lg(VFACT)-1: 4;
2344 173 : if (lim <= 0) lim = 5;
2345 173 : if (lim <= LIM) return;
2346 152 : cache_reset(cache_FACT); av = avma;
2347 152 : cache_set(cache_FACT, vecfactoru_i(1,lim)); set_avma(av);
2348 : }
2349 : static void
2350 145 : constdiv(long lim)
2351 : {
2352 : pari_sp av;
2353 145 : GEN VFACT, VDIV = caches[cache_DIV].cache;
2354 145 : long N, LIM = VDIV? lg(VDIV)-1: 4;
2355 145 : if (lim <= 0) lim = 5;
2356 145 : if (lim <= LIM) return;
2357 145 : constfact(lim);
2358 145 : VFACT = caches[cache_FACT].cache;
2359 145 : cache_reset(cache_DIV); av = avma;
2360 145 : VDIV = cgetg(lim+1, t_VEC);
2361 6973129 : for (N = 1; N <= lim; N++) gel(VDIV,N) = divisorsu_fact(gel(VFACT,N));
2362 145 : 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 14001648 : lamsig(GEN D, long *pL, long *pS)
2368 : {
2369 14001648 : pari_sp av = avma;
2370 14001648 : long i, l = lg(D), L = 1, S = D[l-1]+1;
2371 50700011 : for (i = 2; i < l; i++) /* skip d = 1 */
2372 : {
2373 51454316 : long d = D[i], nd = D[l-i]; /* nd = n/d */
2374 51454316 : if (d < nd) { L += d; S += d + nd; }
2375 : else
2376 : {
2377 14755953 : L <<= 1; if (d == nd) { L += d; S += d; }
2378 14755953 : break;
2379 : }
2380 : }
2381 14001648 : set_avma(av); *pL = L; *pS = S;
2382 14881663 : }
2383 : /* table of 6 * Hurwitz class numbers D <= lim */
2384 : static void
2385 155 : consttabh(long lim)
2386 : {
2387 155 : pari_sp av = avma, av2;
2388 155 : GEN VHDH0, VDIV, CACHE = NULL;
2389 155 : GEN VHDH = caches[cache_H].cache;
2390 155 : long r, N, cachea, cacheb, lim0 = VHDH? lg(VHDH)-1: 2, LIM = lim0 << 1;
2391 :
2392 155 : if (lim <= 0) lim = 5;
2393 155 : if (lim <= LIM) return;
2394 155 : cache_reset(cache_H);
2395 155 : r = lim&3L; if (r) lim += 4-r;
2396 155 : cache_get(cache_DIV, lim);
2397 155 : VDIV = caches[cache_DIV].cache;
2398 155 : VHDH0 = cgetg(lim/2 + 1, t_VECSMALL);
2399 155 : VHDH0[1] = 2;
2400 155 : VHDH0[2] = 3;
2401 1077019 : for (N = 3; N <= lim0; N++) VHDH0[N] = VHDH[N];
2402 155 : av2 = avma;
2403 155 : cachea = cacheb = 0;
2404 7711386 : for (N = LIM + 3; N <= lim; N += 4)
2405 : {
2406 7736639 : long s = 0, limt = usqrt(N>>2), flsq = 0, ind, t, L, S;
2407 : GEN DN, DN2;
2408 7777261 : if (N + 2 >= lg(VDIV))
2409 : { /* use local cache */
2410 : GEN F;
2411 6015111 : if (N + 2 > cacheb)
2412 : {
2413 1573 : set_avma(av2); cachea = N;
2414 1573 : CACHE = update_factor_cache(N, lim+2, &cacheb);
2415 : }
2416 6015111 : F = gel(CACHE, ((N-cachea)>>1)+1); /* factoru(N) */
2417 6015111 : DN = divisorsu_fact(F);
2418 6001275 : F = gel(CACHE, ((N-cachea)>>1)+2); /* factoru(N+2) */
2419 6001275 : DN2 = divisorsu_fact(F);
2420 : }
2421 : else
2422 : { /* use global cache */
2423 1762150 : DN = gel(VDIV,N);
2424 1762150 : DN2 = gel(VDIV,N+2);
2425 : }
2426 7648198 : ind = N >> 1;
2427 877636020 : for (t = 1; t <= limt; t++)
2428 : {
2429 869987822 : ind -= (t<<2)-2; /* N/2 - 2t^2 */
2430 869987822 : if (ind) s += VHDH0[ind]; else flsq = 1;
2431 : }
2432 7648198 : lamsig(DN, &L,&S);
2433 7611066 : VHDH0[N >> 1] = 2*S - 3*L - 2*s + flsq;
2434 7611066 : s = 0; flsq = 0; limt = (usqrt(N+2) - 1) >> 1;
2435 7431490 : ind = (N+1) >> 1;
2436 879929610 : for (t = 1; t <= limt; t++)
2437 : {
2438 872498120 : ind -= t<<2; /* (N+1)/2 - 2t(t+1) */
2439 872498120 : if (ind) s += VHDH0[ind]; else flsq = 1;
2440 : }
2441 7431490 : lamsig(DN2, &L,&S);
2442 7711231 : 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 33684631 : myfactoru(long N)
2452 : {
2453 33684631 : GEN z = cache_get(cache_FACT, N);
2454 33684631 : return z? gcopy(z): factoru(N);
2455 : }
2456 : static GEN
2457 69087138 : mydivisorsu(long N)
2458 : {
2459 69087138 : GEN z = cache_get(cache_DIV, N);
2460 69087138 : return z? leafcopy(z): divisorsu(N);
2461 : }
2462 : /* write -n = Df^2, D < 0 fundamental discriminant. Return D, set f. */
2463 : static long
2464 176460487 : mycoredisc2neg(ulong n, long *pf)
2465 : {
2466 176460487 : ulong m, D = (ulong)cache_get(cache_D, n);
2467 176460487 : if (D) { *pf = usqrt(n/D); return -(long)D; }
2468 56 : m = mycore(n, pf);
2469 56 : if ((m&3) != 3) { m <<= 2; *pf >>= 1; }
2470 56 : 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 63 : hclassno6_count(long D)
2485 : {
2486 63 : ulong a, b, b2, h = 0, d = -D;
2487 63 : int f = 0;
2488 :
2489 63 : if (d > 500000) return 6 * quadclassnos(D);
2490 : /* this part would work with -d non fundamental */
2491 56 : b = d&1; b2 = (1+d)>>2;
2492 56 : if (!b)
2493 : {
2494 1380 : for (a=1; a*a<b2; a++)
2495 1375 : if (b2%a == 0) h++;
2496 5 : f = (a*a==b2); b=2; b2=(4+d)>>2;
2497 : }
2498 10728 : while (b2*3 < d)
2499 : {
2500 10672 : if (b2%b == 0) h++;
2501 1702387 : for (a=b+1; a*a < b2; a++)
2502 1691715 : if (b2%a == 0) h += 2;
2503 10672 : if (a*a == b2) h++;
2504 10672 : b += 2; b2 = (b*b+d)>>2;
2505 : }
2506 56 : if (b2*3 == d) return 6*h+2;
2507 56 : if (f) return 6*h+3;
2508 56 : return 6*h;
2509 : }
2510 : /* D0 < 0; 6 * hclassno(-D), using D = D0*F^2 */
2511 : static long
2512 95 : hclassno6u_2(long D0, long F)
2513 : {
2514 : long h;
2515 95 : if (F == 1) h = hclassno6_count(D0);
2516 : else
2517 : { /* second chance */
2518 32 : h = (ulong)cache_get(cache_H, -D0);
2519 32 : if (!h) h = hclassno6_count(D0);
2520 32 : h *= uhclassnoF_fact(myfactoru(F), D0);
2521 : }
2522 95 : 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 2427669 : hclassno6u(ulong D)
2528 : {
2529 2427669 : ulong z = (ulong)cache_get(cache_H, D);
2530 : long D0, F;
2531 2427669 : if (z) return z;
2532 95 : D0 = mycoredisc2neg(D, &F);
2533 95 : return hclassno6u_2(D0,F);
2534 : }
2535 : /* same as hclassno6u without creating caches */
2536 : ulong
2537 86913 : hclassno6u_no_cache(ulong D)
2538 : {
2539 86913 : cache *S = &caches[cache_H];
2540 : long D0, F;
2541 86913 : if (S->cache)
2542 : {
2543 79906 : const ulong d = D>>1; /* compressed */
2544 79906 : if ((ulong)lg(S->cache) > d) return S->cache[d];
2545 : }
2546 86642 : S = &caches[cache_D];
2547 86642 : 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 156440563 : hclassno6u_i(ulong D, long D0, long F)
2554 : {
2555 156440563 : ulong z = (ulong)cache_get(cache_H, D);
2556 156440563 : if (z) return z;
2557 0 : return hclassno6u_2(D0,F);
2558 : }
2559 :
2560 : /* D < -4 fundamental, h(D), ordinary class number */
2561 : static long
2562 10646797 : myh(long D)
2563 : {
2564 10646797 : ulong z = (ulong)cache_get(cache_H, -D);
2565 10646797 : return z? z / 6: quadclassnos(D);
2566 : }
2567 :
2568 : /*************************************************************************/
2569 : /* TRACE FORMULAS */
2570 : /* CHIP primitive, initialize for t_POLMOD output */
2571 : static GEN
2572 32690 : mfcharinit(GEN CHIP)
2573 : {
2574 32690 : long n, o, l, vt, N = mfcharmodulus(CHIP);
2575 : GEN c, v, V, G, Pn;
2576 32690 : if (N == 1) return mkvec2(mkvec(gen_1), pol_x(0));
2577 5481 : G = gel(CHIP,1);
2578 5481 : v = ncharvecexpo(G, znconrey_normalized(G, gel(CHIP,2)));
2579 5481 : l = lg(v); V = cgetg(l, t_VEC);
2580 5481 : o = mfcharorder(CHIP);
2581 5481 : Pn = mfcharpol(CHIP); vt = varn(Pn);
2582 5481 : 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 16835 : for (n = 1; n < l; n++)
2593 : {
2594 15883 : if (v[n] < 0) c = gen_0;
2595 : else
2596 : {
2597 8890 : c = Qab_zeta(v[n], o, vt);
2598 8890 : if (typ(c) == t_POL && lg(c) >= lg(Pn)) c = RgX_rem(c, Pn);
2599 : }
2600 15883 : gel(V,n) = c;
2601 : }
2602 : }
2603 5481 : return mkvec2(V, Pn);
2604 : }
2605 : static GEN
2606 410718 : vchip_lift(GEN VCHI, long x, GEN C)
2607 : {
2608 410718 : GEN V = gel(VCHI,1);
2609 410718 : long F = lg(V)-1;
2610 410718 : if (F == 1) return C;
2611 18368 : x %= F;
2612 18368 : if (!x) return C;
2613 18368 : if (x <= 0) x += F;
2614 18368 : return gmul(C, gel(V, x));
2615 : }
2616 : static long
2617 279516120 : vchip_FC(GEN VCHI) { return lg(gel(VCHI,1))-1; }
2618 : static GEN
2619 6440981 : vchip_mod(GEN VCHI, GEN S)
2620 6440981 : { return (typ(S) == t_POL)? RgX_rem(S, gel(VCHI,2)): S; }
2621 : static GEN
2622 1913440 : vchip_polmod(GEN VCHI, GEN S)
2623 1913440 : { return (typ(S) == t_POL)? mkpolmod(S, gel(VCHI,2)): S; }
2624 :
2625 : /* contribution of scalar matrices in dimension formula */
2626 : static GEN
2627 359366 : A1(long N, long k) { return uutoQ(mypsiu(N)*(k-1), 12); }
2628 : static long
2629 7602 : ceilA1(long N, long k) { return ceildivuu(mypsiu(N) * (k-1), 12); }
2630 :
2631 : /* sturm bound, slightly larger than dimension */
2632 : long
2633 21714 : mfsturmNk(long N, long k) { return (mypsiu(N) * k) / 12; }
2634 : long
2635 3262 : mfsturmNgk(long N, GEN k)
2636 : {
2637 3262 : long n,d; Qtoss(k,&n,&d);
2638 3262 : 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 539 : sqrtm3modN(long N)
2646 : {
2647 : pari_sp av;
2648 : GEN fa, P, E, B, mB, A, Q, T, R, v, gen_m3;
2649 539 : long l, i, n, ct, fl3 = 0, Ninit;
2650 539 : if (!odd(N) || (N%9) == 0) return cgetg(1,t_VECSMALL);
2651 511 : Ninit = N;
2652 511 : if ((N%3) == 0) { N /= 3; fl3 = 1; }
2653 511 : fa = myfactoru(N); P = gel(fa, 1); E = gel(fa, 2);
2654 511 : l = lg(P);
2655 707 : for (i = 1; i < l; i++)
2656 518 : 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 10178 : nu3(long N)
2691 : {
2692 : long i, l;
2693 : GEN P;
2694 10178 : if (!odd(N) || (N%9) == 0) return 0;
2695 8953 : if ((N%3) == 0) N /= 3;
2696 8953 : P = gel(myfactoru(N), 1); l = lg(P);
2697 13118 : for (i = 1; i < l; i++) if ((P[i]%3) == 2) return 0;
2698 3983 : return 1L<<(l-1);
2699 : }
2700 : /* number of elliptic points of order 2 in X0(N) */
2701 : static long
2702 17549 : nu2(long N)
2703 : {
2704 : long i, l;
2705 : GEN P;
2706 17549 : if ((N&3L) == 0) return 0;
2707 17549 : if (!odd(N)) N >>= 1;
2708 17549 : P = gel(myfactoru(N), 1); l = lg(P);
2709 21952 : for (i = 1; i < l; i++) if ((P[i]&3L) == 3) return 0;
2710 3941 : 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 43701 : A21(long N, long k, GEN CHI)
2717 : {
2718 : GEN res, G, chi, o;
2719 : long a21, i, limx, S;
2720 43701 : if ((N&1L) == 0) return gen_0;
2721 21105 : a21 = k%3 - 1;
2722 21105 : if (!a21) return gen_0;
2723 20356 : if (N <= 3) return sstoQ(a21, 3);
2724 10717 : if (!CHI) return sstoQ(nu3(N) * a21, 3);
2725 539 : res = sqrtm3modN(N); limx = (N - 1) >> 1;
2726 539 : G = gel(CHI,1); chi = gel(CHI,2);
2727 539 : o = gmfcharorder(CHI);
2728 931 : 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 539 : 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 43701 : A22(long N, long k, GEN CHI)
2788 : {
2789 : GEN G, chi, o, res;
2790 : long S, a22, i, limx, o2;
2791 43701 : if ((N&3L) == 0) return gen_0;
2792 30086 : a22 = (k & 3L) - 1; /* (k % 4) - 1 */
2793 30086 : if (!a22) return gen_0;
2794 30086 : if (N <= 2) return sstoQ(a22, 4);
2795 18354 : 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 38892 : nuinf(long N)
2816 : {
2817 38892 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
2818 38892 : long i, t = 1, l = lg(P);
2819 82656 : for (i=1; i<l; i++)
2820 : {
2821 43764 : long p = P[i], e = E[i];
2822 43764 : if (odd(e))
2823 34972 : t *= upowuu(p,e>>1) << 1;
2824 : else
2825 8792 : t *= upowuu(p,(e>>1)-1) * (p+1);
2826 : }
2827 38892 : return t;
2828 : }
2829 :
2830 : /* contribution of hyperbolic matrices in dimension formula */
2831 : static GEN
2832 44149 : A3(long N, long FC)
2833 : {
2834 : long i, S, NF, l;
2835 : GEN D;
2836 44149 : if (FC == 1) return uutoQ(nuinf(N),2);
2837 5257 : D = mydivisorsu(N); l = lg(D);
2838 5257 : S = 0; NF = N/FC;
2839 41412 : for (i = 1; i < l; i++)
2840 : {
2841 36155 : long g = ugcd(D[i], D[l-i]);
2842 36155 : if (NF%g == 0) S += myeulerphiu(g);
2843 : }
2844 5257 : return uutoQ(S, 2);
2845 : }
2846 :
2847 : /* special contribution in weight 2 in dimension formula */
2848 : static long
2849 43253 : A4(long k, long FC)
2850 43253 : { return (k==2 && FC==1)? 1: 0; }
2851 : /* gcd(x,N) */
2852 : static long
2853 283248301 : myugcd(GEN GCD, ulong x)
2854 : {
2855 283248301 : ulong N = lg(GCD)-1;
2856 283248301 : if (x >= N) x %= N;
2857 283248301 : return GCD[x+1];
2858 : }
2859 : /* 1_{gcd(x,N) = 1} * chi(x), return NULL if 0 */
2860 : static GEN
2861 402384797 : mychicgcd(GEN GCD, GEN VCHI, long x)
2862 : {
2863 402384797 : long N = lg(GCD)-1;
2864 402384797 : if (N == 1) return gen_1;
2865 327947366 : x = umodsu(x, N);
2866 327947366 : if (GCD[x+1] != 1) return NULL;
2867 271862120 : x %= vchip_FC(VCHI); if (!x) return gen_1;
2868 4469213 : return gel(gel(VCHI,1), x);
2869 : }
2870 :
2871 : /* contribution of scalar matrices to trace formula */
2872 : static GEN
2873 6390844 : TA1(long N, long k, GEN VCHI, GEN GCD, long n)
2874 : {
2875 : GEN S;
2876 : ulong m;
2877 6390844 : if (!uissquareall(n, &m)) return gen_0;
2878 385903 : if (m == 1) return A1(N,k); /* common */
2879 346094 : S = mychicgcd(GCD, VCHI, m);
2880 346094 : 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 127169 : mksqr(long N)
2886 : {
2887 127169 : pari_sp av = avma;
2888 127169 : long x, N2 = N << 1, N4 = N << 2;
2889 127169 : GEN v = const_vec(N2, cgetg(1, t_VECSMALL));
2890 127169 : gel(v, N2) = mkvecsmall(0); /* x = 0 */
2891 3490627 : for (x = 1; x <= N; x++)
2892 : {
2893 3363458 : long r = (((x*x - 1)%N4) >> 1) + 1;
2894 3363458 : gel(v,r) = vecsmall_append(gel(v,r), x);
2895 : }
2896 127169 : return gerepilecopy(av, v);
2897 : }
2898 :
2899 : static GEN
2900 127169 : mkgcd(long N)
2901 : {
2902 : GEN GCD, d;
2903 : long i, N2;
2904 127169 : if (N == 1) return mkvecsmall(N);
2905 104755 : GCD = cgetg(N + 1, t_VECSMALL);
2906 104755 : d = GCD+1; /* GCD[i+1] = d[i] = gcd(i,N) = gcd(N-i,N), i = 0..N-1 */
2907 104755 : d[0] = N; d[1] = d[N-1] = 1; N2 = N>>1;
2908 1649809 : for (i = 2; i <= N2; i++) d[i] = d[N-i] = ugcd(N, i);
2909 104755 : 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 15215785 : mutglistall(long t, long N, long NF, GEN VCHI, long n, GEN MUP, GEN li, GEN GCD)
2916 : {
2917 15215785 : long i, lx = lg(li);
2918 15215785 : GEN DNF = mydivisorsu(NF), v = zerovec(NF);
2919 15215785 : long j, g, lDNF = lg(DNF);
2920 42452890 : for (i = 1; i < lx; i++)
2921 : {
2922 27237105 : long x = (li[i] + t) >> 1, y, lD;
2923 27237105 : GEN D, c = mychicgcd(GCD, VCHI, x);
2924 27237105 : if (li[i] && li[i] != N)
2925 : {
2926 18094600 : GEN c2 = mychicgcd(GCD, VCHI, t - x);
2927 18094600 : if (c2) c = c? gadd(c, c2): c2;
2928 : }
2929 27237105 : if (!c) continue;
2930 22096466 : y = (x*(x - t) + n) / N; /* exact division */
2931 22096466 : D = mydivisorsu(ugcd(labs(y), NF)); lD = lg(D);
2932 59531880 : 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 35145437 : for (j=2; j < lDNF; j++) { g = DNF[j]; gel(v,g) = gmulsg(MUP[g], gel(v,g)); }
2936 15215785 : return v;
2937 : }
2938 :
2939 : /* special case (N,F) = 1: easier */
2940 : static GEN
2941 161244593 : mutg1(long t, long N, GEN VCHI, GEN li, GEN GCD)
2942 : { /* (N,F) = 1 */
2943 161244593 : GEN S = NULL;
2944 161244593 : long i, lx = lg(li);
2945 338364024 : for (i = 1; i < lx; i++)
2946 : {
2947 177119431 : long x = (li[i] + t) >> 1;
2948 177119431 : GEN c = mychicgcd(GCD, VCHI, x);
2949 177119431 : if (c) S = S? gadd(S, c): c;
2950 177119431 : if (li[i] && li[i] != N)
2951 : {
2952 97946933 : c = mychicgcd(GCD, VCHI, t - x);
2953 97946933 : if (c) S = S? gadd(S, c): c;
2954 : }
2955 177119431 : if (S && !signe(S)) S = NULL; /* strive hard to add gen_0 */
2956 : }
2957 161244593 : return S; /* single value */
2958 : }
2959 :
2960 : /* Gegenbauer pol; n > 2, P = \sum_{0<=j<=n/2} (-1)^j (n-j)!/j!(n-2*j)! X^j */
2961 : GEN
2962 361541 : mfrhopol(long n)
2963 : {
2964 : #ifdef LONG_IS_64BIT
2965 309936 : const long M = 2642249;
2966 : #else
2967 51605 : const long M = 1629;
2968 : #endif
2969 361541 : long j, d = n >> 1; /* >= 1 */
2970 361541 : GEN P = cgetg(d + 3, t_POL);
2971 :
2972 361541 : if (n > M) pari_err_IMPL("mfrhopol for large weight"); /* avoid overflow */
2973 361541 : P[1] = evalvarn(0)|evalsigne(1);
2974 361541 : gel(P,2) = gen_1;
2975 361541 : gel(P,3) = utoineg(n-1); /* j = 1 */
2976 361541 : if (d > 1) gel(P,4) = utoipos(((n-3)*(n-2)) >> 1); /* j = 2 */
2977 361541 : if (d > 2) gel(P,5) = utoineg(((n-5)*(n-4)*(n-3)) / 6); /* j = 3 */
2978 1516390 : for (j = 4; j <= d; j++)
2979 1154849 : gel(P,j+2) = divis(mulis(gel(P,j+1), (n-2*j+1)*(n-2*j+2)), (n-j+1)*(-j));
2980 361541 : return P;
2981 : }
2982 :
2983 : /* polrecip(Q)(t2), assume Q(0) = 1 */
2984 : GEN
2985 3251065 : mfrhopol_u_eval(GEN Q, ulong t2)
2986 : {
2987 3251065 : GEN T = addiu(gel(Q,3), t2);
2988 3251068 : long l = lg(Q), j;
2989 37887490 : for (j = 4; j < l; j++) T = addii(gel(Q,j), mului(t2, T));
2990 3251067 : return T;
2991 : }
2992 : GEN
2993 56617 : mfrhopol_eval(GEN Q, GEN t2)
2994 : {
2995 : long l, j;
2996 : GEN T;
2997 56617 : if (lgefint(t2) == 3) return mfrhopol_u_eval(Q, t2[2]);
2998 0 : l = lg(Q); T = addii(gel(Q,3), t2);
2999 0 : for (j = 4; j < l; j++) T = addii(gel(Q,j), mulii(t2, T));
3000 0 : return T;
3001 : }
3002 : /* return sh * sqrt(n)^nu * G_nu(t/(2*sqrt(n))) for t != 0
3003 : * else (sh/2) * sqrt(n)^nu * G_nu(0) [ implies nu is even ]
3004 : * G_nu(z) = \sum_{0<=j<=nu/2} (-1)^j (nu-j)!/j!(nu-2*j)! * (2z)^(nu-2*j)) */
3005 : static GEN
3006 168051054 : mfrhopowsimp(GEN Q, GEN sh, long nu, long t, long t2, long n)
3007 : {
3008 : GEN T;
3009 168051054 : switch (nu)
3010 : {
3011 162107834 : case 0: return t? sh: gmul2n(sh,-1);
3012 1125446 : case 1: return gmulsg(t, sh);
3013 1576750 : case 2: return t? gmulsg(t2 - n, sh): gmul(gmul2n(stoi(-n), -1), sh);
3014 427 : case 3: return gmul(mulss(t, t2 - 2*n), sh);
3015 3240597 : default:
3016 3240597 : if (!t) return gmul(gmul2n(gel(Q, lg(Q) - 1), -1), sh);
3017 3194448 : T = mfrhopol_u_eval(Q, t2); if (odd(nu)) T = mulsi(t, T);
3018 3194448 : return gmul(T, sh);
3019 : }
3020 : }
3021 :
3022 : /* contribution of elliptic matrices to trace formula */
3023 : static GEN
3024 6390844 : TA2(long N, long k, GEN VCHI, long n, GEN SQRTS, GEN MUP, GEN GCD)
3025 : {
3026 6390844 : const long n4 = n << 2, N4 = N << 2, nu = k - 2;
3027 6390844 : const long st = (!odd(N) && odd(n)) ? 2 : 1;
3028 : long limt, t;
3029 : GEN S, Q;
3030 :
3031 6390844 : limt = usqrt(n4);
3032 6390844 : if (limt*limt == n4) limt--;
3033 6390844 : Q = nu > 3 ? ZX_z_unscale(mfrhopol(nu), n) : NULL;
3034 6390844 : S = gen_0;
3035 325844922 : for (t = odd(k)? st: 0; t <= limt; t += st) /* t^2 < 4n */
3036 : {
3037 319454078 : pari_sp av = avma;
3038 319454078 : long t2 = t*t, D = n4 - t2, F, D0, NF;
3039 : GEN sh, li;
3040 :
3041 319454078 : li = gel(SQRTS, (umodsu(-D - 1, N4) >> 1) + 1);
3042 327863402 : if (lg(li) == 1) continue;
3043 176460378 : D0 = mycoredisc2neg(D, &F);
3044 176460378 : NF = myugcd(GCD, F);
3045 176460378 : if (NF == 1)
3046 : { /* (N,F) = 1 => single value in mutglistall */
3047 161244593 : GEN mut = mutg1(t, N, VCHI, li, GCD);
3048 161244593 : if (!mut) { set_avma(av); continue; }
3049 156440563 : sh = gmul(uutoQ(hclassno6u_i(D,D0,F),6), mut);
3050 : }
3051 : else
3052 : {
3053 15215785 : GEN v = mutglistall(t, N, NF, VCHI, n, MUP, li, GCD);
3054 15215785 : GEN DF = mydivisorsu(F);
3055 15215785 : long i, lDF = lg(DF);
3056 15215785 : sh = gen_0;
3057 61336626 : for (i = 1; i < lDF; i++)
3058 : {
3059 46120841 : long Ff, f = DF[i], g = myugcd(GCD, f);
3060 46120841 : GEN mut = gel(v, g);
3061 46120841 : if (gequal0(mut)) continue;
3062 31198055 : Ff = DF[lDF-i]; /* F/f */
3063 31198055 : if (Ff == 1) sh = gadd(sh, mut);
3064 : else
3065 : {
3066 22357666 : GEN P = gel(myfactoru(Ff), 1);
3067 22357666 : long j, lP = lg(P);
3068 49309307 : for (j = 1; j < lP; j++) { long p = P[j]; Ff -= kross(D0, p)*Ff/p; }
3069 22357666 : sh = gadd(sh, gmulsg(Ff, mut));
3070 : }
3071 : }
3072 15215785 : if (gequal0(sh)) { set_avma(av); continue; }
3073 11610491 : if (D0 == -3) sh = gdivgu(sh, 3);
3074 11117981 : else if (D0 == -4) sh = gdivgu(sh, 2);
3075 10646797 : else sh = gmulgu(sh, myh(D0));
3076 : }
3077 168051054 : S = gerepileupto(av, gadd(S, mfrhopowsimp(Q,sh,nu,t,t2,n)));
3078 : }
3079 6390844 : return S;
3080 : }
3081 :
3082 : /* compute global auxiliary data for TA3 */
3083 : static GEN
3084 127169 : mkbez(long N, long FC)
3085 : {
3086 127169 : long ct, i, NF = N/FC;
3087 127169 : GEN w, D = mydivisorsu(N);
3088 127169 : long l = lg(D);
3089 :
3090 127169 : w = cgetg(l, t_VEC);
3091 368788 : for (i = ct = 1; i < l; i++)
3092 : {
3093 346374 : long u, v, h, c = D[i], Nc = D[l-i];
3094 346374 : if (c > Nc) break;
3095 241619 : h = cbezout(c, Nc, &u, &v);
3096 241619 : if (h == 1) /* shortcut */
3097 174251 : gel(w, ct++) = mkvecsmall4(1,u*c,1,i);
3098 67368 : else if (!(NF%h))
3099 57456 : gel(w, ct++) = mkvecsmall4(h,u*(c/h),myeulerphiu(h),i);
3100 : }
3101 127169 : setlg(w,ct); stackdummy((pari_sp)(w+ct),(pari_sp)(w+l));
3102 127169 : return w;
3103 : }
3104 :
3105 : /* contribution of hyperbolic matrices to trace formula, d * nd = n,
3106 : * DN = divisorsu(N) */
3107 : static GEN
3108 33177353 : auxsum(GEN VCHI, GEN GCD, long d, long nd, GEN DN, GEN BEZ)
3109 : {
3110 33177353 : GEN S = gen_0;
3111 33177353 : long ct, g = nd - d, lDN = lg(DN), lBEZ = lg(BEZ);
3112 85198288 : for (ct = 1; ct < lBEZ; ct++)
3113 : {
3114 52020935 : GEN y, B = gel(BEZ, ct);
3115 52020935 : long ic, c, Nc, uch, h = B[1];
3116 52020935 : if (g%h) continue;
3117 50813883 : uch = B[2];
3118 50813883 : ic = B[4];
3119 50813883 : c = DN[ic];
3120 50813883 : Nc= DN[lDN - ic]; /* Nc = N/c */
3121 50813883 : if (ugcd(Nc, nd) == 1)
3122 43379701 : y = mychicgcd(GCD, VCHI, d + uch*g); /* 0 if (c,d) > 1 */
3123 : else
3124 7434182 : y = NULL;
3125 50813883 : if (c != Nc && ugcd(Nc, d) == 1)
3126 : {
3127 38260933 : GEN y2 = mychicgcd(GCD, VCHI, nd - uch*g); /* 0 if (c,nd) > 1 */
3128 38260933 : if (y2) y = y? gadd(y, y2): y2;
3129 : }
3130 50813883 : if (y) S = gadd(S, gmulsg(B[3], y));
3131 : }
3132 33177353 : return S;
3133 : }
3134 :
3135 : static GEN
3136 6390844 : TA3(long N, long k, GEN VCHI, GEN GCD, GEN Dn, GEN BEZ)
3137 : {
3138 6390844 : GEN S = gen_0, DN = mydivisorsu(N);
3139 6390844 : long i, l = lg(Dn);
3140 39568197 : for (i = 1; i < l; i++)
3141 : {
3142 39528388 : long d = Dn[i], nd = Dn[l-i]; /* = n/d */
3143 : GEN t, u;
3144 39528388 : if (d > nd) break;
3145 33177353 : t = auxsum(VCHI, GCD, d, nd, DN, BEZ);
3146 33177353 : if (isintzero(t)) continue;
3147 32059621 : u = powuu(d,k-1); if (d == nd) u = gmul2n(u,-1);
3148 32059621 : S = gadd(S, gmul(u,t));
3149 : }
3150 6390844 : return S;
3151 : }
3152 :
3153 : /* special contribution in weight 2 in trace formula */
3154 : static long
3155 6390844 : TA4(long k, GEN VCHIP, GEN Dn, GEN GCD)
3156 : {
3157 : long i, l, S;
3158 6390844 : if (k != 2 || vchip_FC(VCHIP) != 1) return 0;
3159 5687416 : l = lg(Dn); S = 0;
3160 66354498 : for (i = 1; i < l; i++)
3161 : {
3162 60667082 : long d = Dn[i]; /* gcd(N,n/d) == 1? */
3163 60667082 : if (myugcd(GCD, Dn[l-i]) == 1) S += d;
3164 : }
3165 5687416 : return S;
3166 : }
3167 :
3168 : /* precomputation of products occurring im mutg, again to accelerate TA2 */
3169 : static GEN
3170 127169 : mkmup(long N)
3171 : {
3172 127169 : GEN fa = myfactoru(N), P = gel(fa,1), D = divisorsu_fact(fa);
3173 127169 : long i, lP = lg(P), lD = lg(D);
3174 127169 : GEN MUP = zero_zv(N);
3175 127169 : MUP[1] = 1;
3176 445613 : for (i = 2; i < lD; i++)
3177 : {
3178 318444 : long j, g = D[i], Ng = D[lD-i]; /* N/g */
3179 871731 : for (j = 1; j < lP; j++) { long p = P[j]; if (Ng%p) g += g/p; }
3180 318444 : MUP[D[i]] = g;
3181 : }
3182 127169 : return MUP;
3183 : }
3184 :
3185 : /* quadratic nonresidues mod p; p odd prime, p^2 fits in a long */
3186 : static GEN
3187 2814 : non_residues(long p)
3188 : {
3189 2814 : long i, j, p2 = p >> 1;
3190 2814 : GEN v = cgetg(p2+1, t_VECSMALL), w = const_vecsmall(p-1, 1);
3191 4571 : for (i = 2; i <= p2; i++) w[(i*i) % p] = 0; /* no need to check 1 */
3192 9142 : for (i = 2, j = 1; i < p; i++) if (w[i]) v[j++] = i;
3193 2814 : return v;
3194 : }
3195 :
3196 : /* CHIP primitive. Return t_VECSMALL v of length q such that
3197 : * Tr^new_{N,CHIP}(n) = 0 whenever v[(n%q) + 1] is nonzero */
3198 : static GEN
3199 32788 : mfnewzerodata(long N, GEN CHIP)
3200 : {
3201 32788 : GEN V, M, L, faN = myfactoru(N), PN = gel(faN,1), EN = gel(faN,2);
3202 32788 : GEN G = gel(CHIP,1), chi = gel(CHIP,2);
3203 32788 : GEN fa = znstar_get_faN(G), P = ZV_to_zv(gel(fa,1)), E = gel(fa,2);
3204 32788 : long i, mod, j = 1, l = lg(PN);
3205 :
3206 32788 : M = cgetg(l, t_VECSMALL); M[1] = 0;
3207 32788 : V = cgetg(l, t_VEC);
3208 : /* Tr^new(n) = 0 if (n mod M[i]) in V[i] */
3209 32788 : if ((N & 3) == 0)
3210 : {
3211 12929 : long e = EN[1];
3212 12929 : long c = (lg(P) > 1 && P[1] == 2)? E[1]: 0; /* c = v_2(FC) */
3213 : /* e >= 2 */
3214 12929 : if (c == e-1) return NULL; /* Tr^new = 0 */
3215 12824 : if (c == e)
3216 : {
3217 3717 : if (e == 2)
3218 : { /* sc: -4 */
3219 1785 : gel(V,1) = mkvecsmall(3);
3220 1785 : M[1] = 4;
3221 : }
3222 1932 : else if (e == 3)
3223 : { /* sc: -8 (CHI_2(-1)=-1<=>chi[1]=1) and 8 (CHI_2(-1)=1 <=> chi[1]=0) */
3224 1932 : long t = signe(gel(chi,1))? 7: 3;
3225 1932 : gel(V,1) = mkvecsmall2(5, t);
3226 1932 : M[1] = 8;
3227 : }
3228 : }
3229 9107 : else if (e == 5 && c == 3)
3230 154 : { /* sc: -8 (CHI_2(-1)=-1<=>chi[1]=1) and 8 (CHI_2(-1)=1 <=> chi[1]=0) */
3231 154 : long t = signe(gel(chi,1))? 7: 3;
3232 154 : gel(V,1) = mkvecsmalln(6, 2L,4L,5L,6L,8L,t);
3233 154 : M[1] = 8;
3234 : }
3235 8953 : else if ((e == 4 && c == 2) || (e == 5 && c <= 2) || (e == 6 && c <= 2)
3236 7378 : || (e >= 7 && c == e - 3))
3237 : { /* sc: 4 */
3238 1575 : gel(V,1) = mkvecsmall3(0,2,3);
3239 1575 : M[1] = 4;
3240 : }
3241 7378 : else if ((e <= 4 && c == 0) || (e >= 5 && c == e - 2))
3242 : { /* sc: 2 */
3243 7021 : gel(V,1) = mkvecsmall(0);
3244 7021 : M[1] = 2;
3245 : }
3246 357 : else if ((e == 6 && c == 3) || (e >= 7 && c <= e - 4))
3247 : { /* sc: -2 */
3248 357 : gel(V,1) = mkvecsmalln(7, 0L,2L,3L,4L,5L,6L,7L);
3249 357 : M[1] = 8;
3250 : }
3251 : }
3252 32683 : j = M[1]? 2: 1;
3253 69762 : for (i = odd(N)? 1: 2; i < l; i++) /* skip p=2, done above */
3254 : {
3255 37079 : long p = PN[i], e = EN[i];
3256 37079 : long z = zv_search(P, p), c = z? E[z]: 0; /* c = v_p(FC) */
3257 37079 : if ((e <= 2 && c == 1 && itos(gel(chi,z)) == (p>>1)) /* ord(CHI_p)=2 */
3258 34888 : || (e >= 3 && c <= e - 2))
3259 2814 : { /* sc: -p */
3260 2814 : GEN v = non_residues(p);
3261 2814 : if (e != 1) v = vecsmall_prepend(v, 0);
3262 2814 : gel(V,j) = v;
3263 2814 : M[j] = p; j++;
3264 : }
3265 34265 : else if (e >= 2 && c < e)
3266 : { /* sc: p */
3267 2590 : gel(V,j) = mkvecsmall(0);
3268 2590 : M[j] = p; j++;
3269 : }
3270 : }
3271 32683 : if (j == 1) return cgetg(1, t_VECSMALL);
3272 15309 : setlg(V,j); setlg(M,j); mod = zv_prod(M);
3273 15309 : L = zero_zv(mod);
3274 33537 : for (i = 1; i < j; i++)
3275 : {
3276 18228 : GEN v = gel(V,i);
3277 18228 : long s, m = M[i], lv = lg(v);
3278 47481 : for (s = 1; s < lv; s++)
3279 : {
3280 29253 : long a = v[s] + 1;
3281 56322 : do { L[a] = 1; a += m; } while (a <= mod);
3282 : }
3283 : }
3284 15309 : return L;
3285 : }
3286 : /* v=mfnewzerodata(N,CHI); returns TRUE if newtrace(n) must be zero,
3287 : * (but newtrace(n) may still be zero if we return FALSE) */
3288 : static long
3289 2599588 : mfnewchkzero(GEN v, long n) { long q = lg(v)-1; return q && v[(n%q) + 1]; }
3290 :
3291 : /* if (!VCHIP): from mftraceform_cusp;
3292 : * else from initnewtrace and CHI is known to be primitive */
3293 : static GEN
3294 127169 : inittrace(long N, GEN CHI, GEN VCHIP)
3295 : {
3296 : long FC;
3297 127169 : if (VCHIP)
3298 127162 : FC = mfcharmodulus(CHI);
3299 : else
3300 7 : VCHIP = mfcharinit(mfchartoprimitive(CHI, &FC));
3301 127169 : return mkvecn(5, mksqr(N), mkmup(N), mkgcd(N), VCHIP, mkbez(N, FC));
3302 : }
3303 :
3304 : /* p > 2 prime; return a sorted t_VECSMALL of primes s.t Tr^new(p) = 0 for all
3305 : * weights > 2 */
3306 : static GEN
3307 32683 : inittrconj(long N, long FC)
3308 : {
3309 : GEN fa, P, E, v;
3310 : long i, k, l;
3311 :
3312 32683 : if (FC != 1) return cgetg(1,t_VECSMALL);
3313 :
3314 27202 : fa = myfactoru(N >> vals(N));
3315 27202 : P = gel(fa,1); l = lg(P);
3316 27202 : E = gel(fa,2);
3317 27202 : v = cgetg(l, t_VECSMALL);
3318 59318 : for (i = k = 1; i < l; i++)
3319 : {
3320 32116 : long j, p = P[i]; /* > 2 */
3321 77588 : for (j = 1; j < l; j++)
3322 45472 : if (j != i && E[j] == 1 && kross(-p, P[j]) == 1) v[k++] = p;
3323 : }
3324 27202 : setlg(v,k); return v;
3325 : }
3326 :
3327 : /* assume CHIP primitive, f(CHIP) | N; NZ = mfnewzerodata(N,CHIP) */
3328 : static GEN
3329 32683 : initnewtrace_i(long N, GEN CHIP, GEN NZ)
3330 : {
3331 32683 : GEN T = const_vec(N, cgetg(1,t_VEC)), D, VCHIP;
3332 32683 : long FC = mfcharmodulus(CHIP), N1, N2, i, l;
3333 :
3334 32683 : if (!NZ) NZ = mkvecsmall(1); /*Tr^new = 0; initialize data nevertheless*/
3335 32683 : VCHIP = mfcharinit(CHIP);
3336 32683 : N1 = N/FC; newd_params(N1, &N2);
3337 32683 : D = mydivisorsu(N1/N2); l = lg(D);
3338 32683 : N2 *= FC;
3339 159845 : for (i = 1; i < l; i++)
3340 : {
3341 127162 : long M = D[i]*N2;
3342 127162 : gel(T,M) = inittrace(M, CHIP, VCHIP);
3343 : }
3344 32683 : gel(T,N) = shallowconcat(gel(T,N), mkvec2(NZ, inittrconj(N,FC)));
3345 32683 : return T;
3346 : }
3347 : /* don't initialize if Tr^new = 0, return NULL */
3348 : static GEN
3349 32788 : initnewtrace(long N, GEN CHI)
3350 : {
3351 32788 : GEN CHIP = mfchartoprimitive(CHI, NULL), NZ = mfnewzerodata(N,CHIP);
3352 32788 : return NZ? initnewtrace_i(N, CHIP, NZ): NULL;
3353 : }
3354 :
3355 : /* (-1)^k */
3356 : static long
3357 8155 : m1pk(long k) { return odd(k)? -1 : 1; }
3358 : static long
3359 7798 : badchar(long N, long k, GEN CHI)
3360 7798 : { return mfcharparity(CHI) != m1pk(k) || (CHI && N % mfcharconductor(CHI)); }
3361 :
3362 :
3363 : static long
3364 43330 : mfcuspdim_i(long N, long k, GEN CHI, GEN vSP)
3365 : {
3366 43330 : pari_sp av = avma;
3367 : long FC;
3368 : GEN s;
3369 43330 : if (k <= 0) return 0;
3370 43330 : if (k == 1) return CHI? mf1cuspdim(N, CHI, vSP): 0;
3371 43071 : FC = CHI? mfcharconductor(CHI): 1;
3372 43071 : if (FC == 1) CHI = NULL;
3373 43071 : s = gsub(A1(N, k), gadd(A21(N, k, CHI), A22(N, k, CHI)));
3374 43071 : s = gadd(s, gsubsg(A4(k, FC), A3(N, FC)));
3375 43071 : return gc_long(av, itos(s));
3376 : }
3377 : /* dimension of space of cusp forms S_k(\G_0(N),CHI)
3378 : * Only depends on CHIP the primitive char attached to CHI */
3379 : long
3380 3381 : mfcuspdim(long N, long k, GEN CHI) { return mfcuspdim_i(N, k, CHI, NULL); }
3381 :
3382 : /* dimension of whole space M_k(\G_0(N),CHI)
3383 : * Only depends on CHIP the primitive char attached to CHI; assumes !badchar */
3384 : long
3385 847 : mffulldim(long N, long k, GEN CHI)
3386 : {
3387 847 : pari_sp av = avma;
3388 847 : long FC = CHI? mfcharconductor(CHI): 1;
3389 : GEN s;
3390 847 : if (k <= 0) return (k == 0 && FC == 1)? 1: 0;
3391 847 : if (k == 1) return gc_long(av, itos(A3(N, FC)) + mf1cuspdim(N, CHI, NULL));
3392 630 : if (FC == 1) CHI = NULL;
3393 630 : s = gsub(A1(N, k), gadd(A21(N, k, CHI), A22(N, k, CHI)));
3394 630 : s = gadd(s, A3(N, FC));
3395 630 : return gc_long(av, itos(s));
3396 : }
3397 :
3398 : /* Dimension of the space of Eisenstein series */
3399 : long
3400 231 : mfeisensteindim(long N, long k, GEN CHI)
3401 : {
3402 231 : pari_sp av = avma;
3403 231 : long s, FC = CHI? mfcharconductor(CHI): 1;
3404 231 : if (k <= 0) return (k == 0 && FC == 1)? 1: 0;
3405 231 : s = itos(gmul2n(A3(N, FC), 1));
3406 231 : if (k > 1) s -= A4(k, FC); else s >>= 1;
3407 231 : return gc_long(av,s);
3408 : }
3409 :
3410 : enum { _SQRTS = 1, _MUP, _GCD, _VCHIP, _BEZ, _NEWLZ, _TRCONJ };
3411 : /* Trace of T(n) on space of cuspforms; only depends on CHIP the primitive char
3412 : * attached to CHI */
3413 : static GEN
3414 6390844 : mfcusptrace_i(long N, long k, long n, GEN Dn, GEN S)
3415 : {
3416 6390844 : pari_sp av = avma;
3417 : GEN a, b, VCHIP, GCD;
3418 : long t;
3419 6390844 : if (!n) return gen_0;
3420 6390844 : VCHIP = gel(S,_VCHIP);
3421 6390844 : GCD = gel(S,_GCD);
3422 6390844 : t = TA4(k, VCHIP, Dn, GCD);
3423 6390844 : a = TA1(N, k, VCHIP, GCD, n); if (t) a = gaddgs(a,t);
3424 6390844 : b = TA2(N, k, VCHIP, n, gel(S,_SQRTS), gel(S,_MUP), GCD);
3425 6390844 : b = gadd(b, TA3(N, k, VCHIP, GCD, Dn, gel(S,_BEZ)));
3426 6390844 : b = gsub(a,b);
3427 6390844 : if (typ(b) != t_POL) return gerepileupto(av, b);
3428 38675 : return gerepilecopy(av, vchip_polmod(VCHIP, b));
3429 : }
3430 :
3431 : static GEN
3432 7650494 : mfcusptracecache(long N, long k, long n, GEN Dn, GEN S, cachenew_t *cache)
3433 : {
3434 7650494 : GEN C = NULL, T = gel(cache->vfull,N);
3435 7650494 : long lcache = lg(T);
3436 7650494 : if (n < lcache) C = gel(T, n);
3437 7650494 : if (C) cache->cuspHIT++; else C = mfcusptrace_i(N, k, n, Dn, S);
3438 7650494 : cache->cuspTOTAL++;
3439 7650494 : if (n < lcache) gel(T,n) = C;
3440 7650494 : return C;
3441 : }
3442 :
3443 : /* return the divisors of n, known to be among the elements of D */
3444 : static GEN
3445 322077 : div_restrict(GEN D, ulong n)
3446 : {
3447 : long i, j, l;
3448 322077 : GEN v, VDIV = caches[cache_DIV].cache;
3449 322077 : if (lg(VDIV) > n) return gel(VDIV,n);
3450 0 : l = lg(D);
3451 0 : v = cgetg(l, t_VECSMALL);
3452 0 : for (i = j = 1; i < l; i++)
3453 : {
3454 0 : ulong d = D[i];
3455 0 : if (n % d == 0) v[j++] = d;
3456 : }
3457 0 : setlg(v,j); return v;
3458 : }
3459 :
3460 : /* for some prime divisors of N, Tr^new(p) = 0 */
3461 : static int
3462 204061 : trconj(GEN T, long N, long n)
3463 204061 : { return (lg(T) > 1 && N % n == 0 && zv_search(T, n)); }
3464 :
3465 : /* n > 0; trace formula on new space */
3466 : static GEN
3467 2599588 : mfnewtrace_i(long N, long k, long n, cachenew_t *cache)
3468 : {
3469 2599588 : GEN VCHIP, s, Dn, DN1, SN, S = cache->DATA;
3470 : long FC, N1, N2, N1N2, g, i, j, lDN1;
3471 :
3472 2599588 : if (!S) return gen_0;
3473 2599588 : SN = gel(S,N);
3474 2599588 : if (mfnewchkzero(gel(SN,_NEWLZ), n)) return gen_0;
3475 1874793 : if (k > 2 && trconj(gel(SN,_TRCONJ), N, n)) return gen_0;
3476 1874765 : VCHIP = gel(SN, _VCHIP); FC = vchip_FC(VCHIP);
3477 1874765 : N1 = N/FC; newt_params(N1, n, FC, &g, &N2);
3478 1874765 : N1N2 = N1/N2;
3479 1874765 : DN1 = mydivisorsu(N1N2); lDN1 = lg(DN1);
3480 1874765 : N2 *= FC;
3481 1874765 : Dn = mydivisorsu(n); /* this one is probably out of cache */
3482 1874765 : s = gmulsg(mubeta2(N1N2,n), mfcusptracecache(N2, k, n, Dn, gel(S,N2), cache));
3483 7328417 : for (i = 2; i < lDN1; i++)
3484 : { /* skip M1 = 1, done above */
3485 5453652 : long M1 = DN1[i], N1M1 = DN1[lDN1-i];
3486 5453652 : GEN Dg = mydivisorsu(ugcd(M1, g));
3487 5453652 : M1 *= N2;
3488 5453652 : s = gadd(s, gmulsg(mubeta2(N1M1,n),
3489 5453652 : mfcusptracecache(M1, k, n, Dn, gel(S,M1), cache)));
3490 5775729 : for (j = 2; j < lg(Dg); j++) /* skip d = 1, done above */
3491 : {
3492 322077 : long d = Dg[j], ndd = n/(d*d), M = M1/d;
3493 322077 : GEN z = mulsi(mubeta2(N1M1,ndd), powuu(d,k-1)), C = vchip_lift(VCHIP,d,z);
3494 322077 : GEN Dndd = div_restrict(Dn, ndd);
3495 322077 : s = gadd(s, gmul(C, mfcusptracecache(M, k, ndd, Dndd, gel(S,M), cache)));
3496 : }
3497 5453652 : s = vchip_mod(VCHIP, s);
3498 : }
3499 1874765 : return vchip_polmod(VCHIP, s);
3500 : }
3501 :
3502 : static GEN
3503 12355 : get_DIH(long N)
3504 : {
3505 12355 : GEN x = cache_get(cache_DIH, N);
3506 12355 : return x? gcopy(x): mfdihedral(N);
3507 : }
3508 : static GEN
3509 2373 : get_vDIH(long N, GEN D)
3510 : {
3511 2373 : GEN x = const_vec(N, NULL);
3512 : long i, l;
3513 2373 : if (!D) D = mydivisorsu(N);
3514 2373 : l = lg(D);
3515 14504 : for (i = 1; i < l; i++) { long d = D[i]; gel(x, d) = get_DIH(d); }
3516 2373 : return x;
3517 : }
3518 :
3519 : /* divisors of N which are multiple of F */
3520 : static GEN
3521 322 : divisorsNF(long N, long F)
3522 : {
3523 322 : GEN D = mydivisorsu(N / F);
3524 322 : long l = lg(D), i;
3525 833 : for (i = 1; i < l; i++) D[i] = N / D[i];
3526 322 : return D;
3527 : }
3528 : /* mfcuspdim(N,k,CHI) - mfnewdim(N,k,CHI); CHIP primitive (for efficiency) */
3529 : static long
3530 8358 : mfolddim_i(long N, long k, GEN CHIP, GEN vSP)
3531 : {
3532 8358 : long S, i, l, F = mfcharmodulus(CHIP), N1 = N / F, N2;
3533 : GEN D;
3534 8358 : newd_params(N1, &N2); /* will ensure mubeta != 0 */
3535 8358 : D = mydivisorsu(N1/N2); l = lg(D); S = 0;
3536 8358 : if (k == 1 && !vSP) vSP = get_vDIH(N, divisorsNF(N, F));
3537 32480 : for (i = 2; i < l; i++)
3538 : {
3539 24122 : long d = mfcuspdim_i(N / D[i], k, CHIP, vSP);
3540 24122 : if (d) S -= mubeta(D[i]) * d;
3541 : }
3542 8358 : return S;
3543 : }
3544 : long
3545 224 : mfolddim(long N, long k, GEN CHI)
3546 : {
3547 224 : pari_sp av = avma;
3548 224 : GEN CHIP = mfchartoprimitive(CHI, NULL);
3549 224 : return gc_long(av, mfolddim_i(N, k, CHIP, NULL));
3550 : }
3551 : /* Only depends on CHIP the primitive char attached to CHI; assumes !badchar */
3552 : long
3553 15827 : mfnewdim(long N, long k, GEN CHI)
3554 : {
3555 : pari_sp av;
3556 : long S, F;
3557 15827 : GEN vSP, CHIP = mfchartoprimitive(CHI, &F);
3558 15827 : vSP = (k == 1)? get_vDIH(N, divisorsNF(N, F)): NULL;
3559 15827 : S = mfcuspdim_i(N, k, CHIP, vSP); if (!S) return 0;
3560 7861 : av = avma; return gc_long(av, S - mfolddim_i(N, k, CHIP, vSP));
3561 : }
3562 :
3563 : /* trace form, given as closure */
3564 : static GEN
3565 952 : mftraceform_new(long N, long k, GEN CHI)
3566 : {
3567 : GEN T;
3568 952 : if (k == 1) return initwt1newtrace(mfinit_Nkchi(N, 1, CHI, mf_CUSP, 0));
3569 931 : T = initnewtrace(N,CHI); if (!T) return mftrivial();
3570 931 : return tag(t_MF_NEWTRACE, mkNK(N,k,CHI), T);
3571 : }
3572 : static GEN
3573 14 : mftraceform_cusp(long N, long k, GEN CHI)
3574 : {
3575 14 : if (k == 1) return initwt1trace(mfinit_Nkchi(N, 1, CHI, mf_CUSP, 0));
3576 7 : return tag(t_MF_TRACE, mkNK(N,k,CHI), inittrace(N,CHI,NULL));
3577 : }
3578 : static GEN
3579 98 : mftraceform_i(GEN NK, long space)
3580 : {
3581 : GEN CHI;
3582 : long N, k;
3583 98 : checkNK(NK, &N, &k, &CHI, 0);
3584 98 : if (!mfdim_Nkchi(N, k, CHI, space)) return mftrivial();
3585 77 : switch(space)
3586 : {
3587 56 : case mf_NEW: return mftraceform_new(N, k, CHI);
3588 14 : case mf_CUSP:return mftraceform_cusp(N, k, CHI);
3589 : }
3590 7 : pari_err_DOMAIN("mftraceform", "space", "=", utoi(space), NK);
3591 : return NULL;/*LCOV_EXCL_LINE*/
3592 : }
3593 : GEN
3594 98 : mftraceform(GEN NK, long space)
3595 98 : { pari_sp av = avma; return gerepilecopy(av, mftraceform_i(NK,space)); }
3596 :
3597 : static GEN
3598 17542 : hecke_data(long N, long n)
3599 17542 : { return mkvecsmall3(n, u_ppo(n, N), N); }
3600 : /* 1/2-integral weight */
3601 : static GEN
3602 84 : heckef2_data(long N, long n)
3603 : {
3604 : ulong f, fN, fN2;
3605 84 : if (!uissquareall(n, &f)) return NULL;
3606 77 : fN = u_ppo(f, N); fN2 = fN*fN;
3607 77 : return mkvec2(myfactoru(fN), mkvecsmall4(n, N, fN2, n/fN2));
3608 : }
3609 : /* N = mf_get_N(F) or a multiple */
3610 : static GEN
3611 24619 : mfhecke_i(long n, long N, GEN F)
3612 : {
3613 24619 : if (n == 1) return F;
3614 17171 : return tag2(t_MF_HECKE, mf_get_NK(F), hecke_data(N,n), F);
3615 : }
3616 :
3617 : GEN
3618 105 : mfhecke(GEN mf, GEN F, long n)
3619 : {
3620 105 : pari_sp av = avma;
3621 : GEN NK, CHI, gk, DATA;
3622 : long N, nk, dk;
3623 105 : mf = checkMF(mf);
3624 105 : if (!checkmf_i(F)) pari_err_TYPE("mfhecke",F);
3625 105 : if (n <= 0) pari_err_TYPE("mfhecke [n <= 0]", stoi(n));
3626 105 : if (n == 1) return gcopy(F);
3627 105 : gk = mf_get_gk(F);
3628 105 : Qtoss(gk,&nk,&dk);
3629 105 : CHI = mf_get_CHI(F);
3630 105 : N = MF_get_N(mf);
3631 105 : if (dk == 2)
3632 : {
3633 77 : DATA = heckef2_data(N,n);
3634 77 : if (!DATA) return mftrivial();
3635 : }
3636 : else
3637 28 : DATA = hecke_data(N,n);
3638 98 : NK = mkgNK(lcmii(stoi(N), mf_get_gN(F)), gk, CHI, mf_get_field(F));
3639 98 : return gerepilecopy(av, tag2(t_MF_HECKE, NK, DATA, F));
3640 : }
3641 :
3642 : /* form F given by closure, compute B(d)(F) as closure (q -> q^d) */
3643 : static GEN
3644 35805 : mfbd_i(GEN F, long d)
3645 : {
3646 : GEN D, NK, gk, CHI;
3647 35805 : if (d == 1) return F;
3648 13482 : if (d <= 0) pari_err_TYPE("mfbd [d <= 0]", stoi(d));
3649 13482 : if (mf_get_type(F) != t_MF_BD) D = utoi(d);
3650 7 : else { D = mului(d, gel(F,3)); F = gel(F,2); }
3651 13482 : gk = mf_get_gk(F); CHI = mf_get_CHI(F);
3652 13482 : if (typ(gk) != t_INT) CHI = mfcharmul(CHI, get_mfchar(utoi(d << 2)));
3653 13482 : NK = mkgNK(muliu(mf_get_gN(F), d), gk, CHI, mf_get_field(F));
3654 13482 : return tag2(t_MF_BD, NK, F, D);
3655 : }
3656 : GEN
3657 252 : mfbd(GEN F, long d)
3658 : {
3659 252 : pari_sp av = avma;
3660 252 : if (!checkmf_i(F)) pari_err_TYPE("mfbd",F);
3661 252 : return gerepilecopy(av, mfbd_i(F, d));
3662 : }
3663 :
3664 : /* A[i+1] = a(t*i^2) */
3665 : static GEN
3666 105 : RgV_shimura(GEN A, long n, long t, long N, long r, GEN CHI)
3667 : {
3668 105 : GEN R, a0, Pn = mfcharpol(CHI);
3669 105 : long m, st, ord = mfcharorder(CHI), vt = varn(Pn), Nt = t == 1? N: ulcm(N,t);
3670 :
3671 105 : R = cgetg(n + 2, t_VEC);
3672 105 : st = odd(r)? -t: t;
3673 105 : a0 = gel(A, 1);
3674 105 : if (!gequal0(a0))
3675 : {
3676 14 : long o = mfcharorder(CHI);
3677 14 : if (st != 1 && odd(o)) o <<= 1;
3678 14 : a0 = gmul(a0, charLFwtk(Nt, r, CHI, o, st));
3679 : }
3680 105 : gel(R, 1) = a0;
3681 637 : for (m = 1; m <= n; m++)
3682 : {
3683 532 : GEN Dm = mydivisorsu(u_ppo(m, Nt)), S = gel(A, m*m + 1);
3684 532 : long i, l = lg(Dm);
3685 805 : for (i = 2; i < l; i++)
3686 : { /* (e,Nt) = 1; skip i = 1: e = 1, done above */
3687 273 : long e = Dm[i], me = m / e, a = mfcharevalord(CHI, e, ord);
3688 273 : GEN c, C = powuu(e, r - 1);
3689 273 : if (kross(st, e) == -1) C = negi(C);
3690 273 : c = Qab_Czeta(a, ord, C, vt);
3691 273 : S = gadd(S, gmul(c, gel(A, me*me + 1)));
3692 : }
3693 532 : gel(R, m+1) = S;
3694 : }
3695 105 : return degpol(Pn) > 1? gmodulo(R, Pn): R;
3696 : }
3697 :
3698 : static long
3699 28 : mfisinkohnen(GEN mf, GEN F)
3700 : {
3701 28 : GEN v, gk = MF_get_gk(mf), CHI = MF_get_CHI(mf);
3702 28 : long i, eps, N4 = MF_get_N(mf) >> 2, sb = mfsturmNgk(N4 << 4, gk) + 1;
3703 28 : eps = N4 % mfcharconductor(CHI)? -1 : 1;
3704 28 : if (odd(MF_get_r(mf))) eps = -eps;
3705 28 : v = mfcoefs(F, sb, 1);
3706 686 : for (i = 2; i <= sb; i+=4) if (!gequal0(gel(v,i+1))) return 0;
3707 245 : for (i = 2+eps; i <= sb; i+=4) if (!gequal0(gel(v,i+1))) return 0;
3708 14 : return 1;
3709 : }
3710 :
3711 : static long
3712 42 : mfshimura_space_cusp(GEN mf)
3713 : {
3714 : long N4;
3715 42 : if (MF_get_r(mf) == 1 && (N4 = MF_get_N(mf) >> 2) >= 4)
3716 : {
3717 21 : GEN E = gel(myfactoru(N4), 2);
3718 21 : long ma = vecsmall_max(E);
3719 21 : if (ma > 2 || (ma == 2 && !mfcharistrivial(MF_get_CHI(mf)))) return 0;
3720 : }
3721 28 : return 1;
3722 : }
3723 :
3724 : /* D is either a discriminant (not necessarily fundamental) with
3725 : sign(D)=(-1)^{k-1/2}*eps, or a positive squarefree integer t, which is then
3726 : transformed into a fundamental discriminant of the correct sign. */
3727 : GEN
3728 49 : mfshimura(GEN mf, GEN F, long t)
3729 : {
3730 49 : pari_sp av = avma;
3731 : GEN G, res, mf2, CHI;
3732 49 : long sb, M, r, N, space = mf_FULL;
3733 :
3734 49 : if (!checkmf_i(F)) pari_err_TYPE("mfshimura",F);
3735 49 : mf = checkMF(mf);
3736 49 : r = MF_get_r(mf);
3737 49 : if (r <= 0) pari_err_DOMAIN("mfshimura", "weight", "<=", ghalf, mf_get_gk(F));
3738 49 : if (t <= 0 || !uissquarefree(t)) pari_err_TYPE("mfshimura [t]", stoi(t));
3739 42 : N = MF_get_N(mf); M = N >> 1;
3740 42 : if (mfiscuspidal(mf,F))
3741 : {
3742 28 : if (mfshimura_space_cusp(mf)) space = mf_CUSP;
3743 28 : if (mfisinkohnen(mf,F)) M = N >> 2;
3744 : }
3745 42 : CHI = MF_get_CHI(mf);
3746 42 : mf2 = mfinit_Nkchi(M, r << 1, mfcharpow(CHI, gen_2), space, 0);
3747 42 : sb = mfsturm(mf2);
3748 42 : G = RgV_shimura(mfcoefs_i(F, sb*sb, t), sb, t, N, r, CHI);
3749 42 : res = mftobasis_i(mf2, G);
3750 : /* not mflinear(mf2,): we want lowest possible level */
3751 42 : G = mflinear(MF_get_basis(mf2), res);
3752 42 : return gerepilecopy(av, mkvec3(mf2, G, res));
3753 : }
3754 :
3755 : /* W ZabM (ZM if n = 1), a t_INT or NULL, b t_INT, ZXQ mod P or NULL.
3756 : * Write a/b = A/d with d t_INT and A Zab return [W,d,A,P] */
3757 : static GEN
3758 7700 : mkMinv(GEN W, GEN a, GEN b, GEN P)
3759 : {
3760 7700 : GEN A = (b && typ(b) == t_POL)? Q_remove_denom(QXQ_inv(b,P), &b): NULL;
3761 7700 : if (a && b)
3762 : {
3763 1295 : a = Qdivii(a,b);
3764 1295 : if (typ(a) == t_INT) b = gen_1; else { b = gel(a,2); a = gel(a,1); }
3765 1295 : if (is_pm1(a)) a = NULL;
3766 : }
3767 7700 : if (a) A = A? ZX_Z_mul(A,a): a; else if (!A) A = gen_1;
3768 7700 : if (!b) b = gen_1;
3769 7700 : if (!P) P = gen_0;
3770 7700 : return mkvec4(W,b,A,P);
3771 : }
3772 : /* M square invertible QabM, return [M',d], M*M' = d*Id */
3773 : static GEN
3774 588 : QabM_Minv(GEN M, GEN P, long n)
3775 : {
3776 : GEN dW, W, dM;
3777 588 : M = Q_remove_denom(M, &dM);
3778 588 : W = P? ZabM_inv(liftpol_shallow(M), P, n, &dW): ZM_inv(M, &dW);
3779 588 : return mkMinv(W, dM, dW, P);
3780 : }
3781 : /* Simplified form of mfclean, after a QabM_indexrank: M a ZabM with full
3782 : * column rank and z = indexrank(M) is known */
3783 : static GEN
3784 847 : mfclean2(GEN M, GEN z, GEN P, long n)
3785 : {
3786 847 : GEN d, Minv, y = gel(z,1), W = rowpermute(M, y);
3787 847 : W = P? ZabM_inv(liftpol_shallow(W), P, n, &d): ZM_inv(W, &d);
3788 847 : M = rowslice(M, 1, y[lg(y)-1]);
3789 847 : Minv = mkMinv(W, NULL, d, P);
3790 847 : return mkvec3(y, Minv, M);
3791 : }
3792 : /* M QabM, lg(M)>1 and [y,z] its rank profile. Let Minv be the inverse of the
3793 : * invertible square matrix in mkMinv format. Return [y,Minv, M[..y[#y],]]
3794 : * P cyclotomic polynomial of order n > 2 or NULL */
3795 : static GEN
3796 4977 : mfclean(GEN M, GEN P, long n, int ratlift)
3797 : {
3798 4977 : GEN W, v, y, z, d, Minv, dM, MdM = Q_remove_denom(M, &dM);
3799 4977 : if (n <= 2)
3800 3885 : W = ZM_pseudoinv(MdM, &v, &d);
3801 : else
3802 1092 : W = ZabM_pseudoinv_i(liftpol_shallow(MdM), P, n, &v, &d, ratlift);
3803 4977 : y = gel(v,1);
3804 4977 : z = gel(v,2);
3805 4977 : if (lg(z) != lg(MdM)) M = vecpermute(M,z);
3806 4977 : M = rowslice(M, 1, y[lg(y)-1]);
3807 4977 : Minv = mkMinv(W, dM, d, P);
3808 4977 : return mkvec3(y, Minv, M);
3809 : }
3810 : /* call mfclean using only CHI */
3811 : static GEN
3812 4025 : mfcleanCHI(GEN M, GEN CHI, int ratlift)
3813 : {
3814 4025 : long n = mfcharorder(CHI);
3815 4025 : GEN P = (n <= 2)? NULL: mfcharpol(CHI);
3816 4025 : return mfclean(M, P, n, ratlift);
3817 : }
3818 :
3819 : /* DATA component of a t_MF_NEWTRACE. Was it stripped to save memory ? */
3820 : static int
3821 33712 : newtrace_stripped(GEN DATA)
3822 33712 : { return DATA && (lg(DATA) == 5 && typ(gel(DATA,3)) == t_INT); }
3823 : /* f a t_MF_NEWTRACE */
3824 : static GEN
3825 33712 : newtrace_DATA(long N, GEN f)
3826 : {
3827 33712 : GEN DATA = gel(f,2);
3828 33712 : return newtrace_stripped(DATA)? initnewtrace(N, DATA): DATA;
3829 : }
3830 : /* reset cachenew for new level incorporating new DATA, tf a t_MF_NEWTRACE
3831 : * (+ possibly initialize 'full' for new allowed levels) */
3832 : static void
3833 33712 : reset_cachenew(cachenew_t *cache, long N, GEN tf)
3834 : {
3835 : long i, n, l;
3836 33712 : GEN v, DATA = newtrace_DATA(N,tf);
3837 33712 : cache->DATA = DATA;
3838 33712 : if (!DATA) return;
3839 33607 : n = cache->n;
3840 33607 : v = cache->vfull; l = N+1; /* = lg(DATA) */
3841 2198980 : for (i = 1; i < l; i++)
3842 2165373 : if (typ(gel(v,i)) == t_INT && lg(gel(DATA,i)) != 1)
3843 53452 : gel(v,i) = const_vec(n, NULL);
3844 33607 : cache->VCHIP = gel(gel(DATA,N),_VCHIP);
3845 : }
3846 : /* initialize a cache of newtrace / cusptrace up to index n and level | N;
3847 : * DATA may be NULL (<=> Tr^new = 0). tf a t_MF_NEWTRACE */
3848 : static void
3849 13202 : init_cachenew(cachenew_t *cache, long n, long N, GEN tf)
3850 : {
3851 13202 : long i, l = N+1; /* = lg(tf.DATA) when DATA != NULL */
3852 : GEN v;
3853 13202 : cache->n = n;
3854 13202 : cache->vnew = v = cgetg(l, t_VEC);
3855 942872 : for (i = 1; i < l; i++) gel(v,i) = (N % i)? gen_0: const_vec(n, NULL);
3856 13202 : cache->newHIT = cache->newTOTAL = cache->cuspHIT = cache->cuspTOTAL = 0;
3857 13202 : cache->vfull = v = zerovec(N);
3858 13202 : reset_cachenew(cache, N, tf);
3859 13202 : }
3860 : static void
3861 17213 : dbg_cachenew(cachenew_t *C)
3862 : {
3863 17213 : if (DEBUGLEVEL >= 2 && C)
3864 0 : err_printf("newtrace cache hits: new = %ld/%ld, cusp = %ld/%ld\n",
3865 : C->newHIT, C->newTOTAL, C->cuspHIT, C->cuspTOTAL);
3866 17213 : }
3867 :
3868 : /* newtrace_{N,k}(d*i), i = n0, ..., n */
3869 : static GEN
3870 179235 : colnewtrace(long n0, long n, long d, long N, long k, cachenew_t *cache)
3871 : {
3872 179235 : GEN v = cgetg(n-n0+2, t_COL);
3873 : long i;
3874 4680354 : for (i = n0; i <= n; i++) gel(v, i-n0+1) = mfnewtracecache(N, k, i*d, cache);
3875 179235 : return v;
3876 : }
3877 : /* T_n(l*m0, l*(m0+1), ..., l*m) F, F = t_MF_NEWTRACE [N,k],DATA, cache
3878 : * contains DATA != NULL as well as cached values of F */
3879 : static GEN
3880 88697 : heckenewtrace(long m0, long m, long l, long N, long NBIG, long k, long n, cachenew_t *cache)
3881 : {
3882 88697 : long lD, a, k1, nl = n*l;
3883 88697 : GEN D, V, v = colnewtrace(m0, m, nl, N, k, cache); /* d=1 */
3884 : GEN VCHIP;
3885 88697 : if (n == 1) return v;
3886 60886 : VCHIP = cache->VCHIP;
3887 60886 : D = mydivisorsu(u_ppo(n, NBIG)); lD = lg(D);
3888 60886 : k1 = k - 1;
3889 149527 : for (a = 2; a < lD; a++)
3890 : { /* d > 1, (d,NBIG) = 1 */
3891 88641 : long i, j, d = D[a], c = ugcd(l, d), dl = d/c, m0d = ceildivuu(m0, dl);
3892 88641 : GEN C = vchip_lift(VCHIP, d, powuu(d, k1));
3893 : /* m0=0: i = 1 => skip F(0) = 0 */
3894 88641 : if (!m0) { i = 1; j = dl; } else { i = 0; j = m0d*dl; }
3895 88641 : V = colnewtrace(m0d, m/dl, nl/(d*c), N, k, cache);
3896 : /* C = chi(d) d^(k-1) */
3897 1075970 : for (; j <= m; i++, j += dl)
3898 987329 : gel(v,j-m0+1) = gadd(gel(v,j-m0+1), vchip_mod(VCHIP, gmul(C,gel(V,i+1))));
3899 : }
3900 60886 : return v;
3901 : }
3902 :
3903 : /* Given v = an[i], return an[d*i], i=0..n */
3904 : static GEN
3905 2618 : anextract(GEN v, long n, long d)
3906 : {
3907 2618 : long i, id, l = n + 2;
3908 2618 : GEN w = cgetg(l, t_VEC);
3909 2618 : if (d == 1)
3910 7245 : for (i = 1; i < l; i++) gel(w, i) = gel(v, i);
3911 : else
3912 22036 : for (i = id = 1; i < l; i++, id += d) gel(w, i) = gel(v, id);
3913 2618 : return w;
3914 : }
3915 : /* T_n(F)(0, l, ..., l*m) */
3916 : static GEN
3917 2527 : hecke_i(long m, long l, GEN V, GEN F, GEN DATA)
3918 : {
3919 : long k, n, nNBIG, NBIG, lD, M, a, t, nl;
3920 : GEN D, v, CHI;
3921 2527 : if (typ(DATA) == t_VEC)
3922 : { /* 1/2-integral k */
3923 98 : if (!V) { GEN S = gel(DATA,2); V = mfcoefs_i(F, m*l*S[3], S[4]); }
3924 98 : return RgV_heckef2(m, l, V, F, DATA);
3925 : }
3926 2429 : k = mf_get_k(F);
3927 2429 : n = DATA[1]; nl = n*l;
3928 2429 : nNBIG = DATA[2];
3929 2429 : NBIG = DATA[3];
3930 2429 : if (nNBIG == 1) return V? V: mfcoefs_i(F,m,nl);
3931 1673 : if (!V && mf_get_type(F) == t_MF_NEWTRACE)
3932 : { /* inline F to allow cache, T_n at level NBIG acting on Tr^new(N,k,CHI) */
3933 : cachenew_t cache;
3934 364 : long N = mf_get_N(F);
3935 364 : init_cachenew(&cache, m*nl, N, F);
3936 364 : v = heckenewtrace(0, m, l, N, NBIG, k, n, &cache);
3937 364 : dbg_cachenew(&cache);
3938 364 : settyp(v, t_VEC); return v;
3939 : }
3940 1309 : CHI = mf_get_CHI(F);
3941 1309 : D = mydivisorsu(nNBIG); lD = lg(D);
3942 1309 : M = m + 1;
3943 1309 : t = nNBIG * ugcd(nNBIG, l);
3944 1309 : if (!V) V = mfcoefs_i(F, m * t, nl / t); /* usually nl = t */
3945 1309 : v = anextract(V, m, t); /* mfcoefs(F, m, nl); d = 1 */
3946 2618 : for (a = 2; a < lD; a++)
3947 : { /* d > 1, (d, NBIG) = 1 */
3948 1309 : long d = D[a], c = ugcd(l, d), dl = d/c, i, idl;
3949 1309 : GEN C = gmul(mfchareval(CHI, d), powuu(d, k-1));
3950 1309 : GEN w = anextract(V, m/dl, t/(d*c)); /* mfcoefs(F, m/dl, nl/(d*c)) */
3951 7245 : for (i = idl = 1; idl <= M; i++, idl += dl)
3952 5936 : gel(v,idl) = gadd(gel(v,idl), gmul(C, gel(w,i)));
3953 : }
3954 1309 : return v;
3955 : }
3956 :
3957 : static GEN
3958 12306 : mkmf(GEN x1, GEN x2, GEN x3, GEN x4, GEN x5)
3959 : {
3960 12306 : GEN MF = obj_init(5, MF_SPLITN);
3961 12306 : gel(MF,1) = x1;
3962 12306 : gel(MF,2) = x2;
3963 12306 : gel(MF,3) = x3;
3964 12306 : gel(MF,4) = x4;
3965 12306 : gel(MF,5) = x5; return MF;
3966 : }
3967 :
3968 : /* return an integer b such that p | b => T_p^k Tr^new = 0, for all k > 0 */
3969 : static long
3970 7602 : get_badj(long N, long FC)
3971 : {
3972 7602 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
3973 7602 : long i, b = 1, l = lg(P);
3974 20237 : for (i = 1; i < l; i++)
3975 12635 : if (E[i] > 1 && u_lval(FC, P[i]) < E[i]) b *= P[i];
3976 7602 : return b;
3977 : }
3978 : /* in place, assume perm strictly increasing */
3979 : static void
3980 1330 : vecpermute_inplace(GEN v, GEN perm)
3981 : {
3982 1330 : long i, l = lg(perm);
3983 11522 : for (i = 1; i < l; i++) gel(v,i) = gel(v,perm[i]);
3984 1330 : }
3985 :
3986 : /* Find basis of newspace using closures; assume k >= 2 and !badchar.
3987 : * Return NULL if space is empty, else
3988 : * [mf1, list of closures T(j)traceform, list of corresponding j, matrix] */
3989 : static GEN
3990 15582 : mfnewinit(long N, long k, GEN CHI, cachenew_t *cache, long init)
3991 : {
3992 : GEN S, vj, M, CHIP, mf1, listj, P, tf;
3993 : long j, ct, ctlj, dim, jin, SB, sb, two, ord, FC, badj;
3994 :
3995 15582 : dim = mfnewdim(N, k, CHI);
3996 15582 : if (!dim && !init) return NULL;
3997 7602 : sb = mfsturmNk(N, k);
3998 7602 : CHIP = mfchartoprimitive(CHI, &FC);
3999 : /* remove newtrace data from S to save space in output: negligible slowdown */
4000 7602 : tf = tag(t_MF_NEWTRACE, mkNK(N,k,CHIP), CHIP);
4001 7602 : badj = get_badj(N, FC);
4002 : /* try sbsmall first: Sturm bound not sharp for new space */
4003 7602 : SB = ceilA1(N, k);
4004 7602 : listj = cgetg(2*sb + 3, t_VECSMALL);
4005 372533 : for (j = ctlj = 1; ctlj < 2*sb + 3; j++)
4006 364931 : if (ugcd(j, badj) == 1) listj[ctlj++] = j;
4007 7602 : if (init)
4008 : {
4009 4123 : init_cachenew(cache, (SB+1)*listj[dim+1], N, tf);
4010 4123 : if (init == -1 || !dim) return NULL; /* old space or dim = 0 */
4011 : }
4012 : else
4013 3479 : reset_cachenew(cache, N, tf);
4014 : /* cache.DATA is not NULL */
4015 7133 : ord = mfcharorder(CHIP);
4016 7133 : P = ord <= 2? NULL: mfcharpol(CHIP);
4017 7133 : vj = cgetg(dim+1, t_VECSMALL);
4018 7133 : M = cgetg(dim+1, t_MAT);
4019 7140 : for (two = 1, ct = 0, jin = 1; two <= 2; two++)
4020 : {
4021 7140 : long a, jlim = jin + sb;
4022 21952 : for (a = jin; a <= jlim; a++)
4023 : {
4024 : GEN z, vecz;
4025 21945 : ct++; vj[ct] = listj[a];
4026 21945 : gel(M, ct) = heckenewtrace(0, SB, 1, N, N, k, vj[ct], cache);
4027 21945 : if (ct < dim) continue;
4028 :
4029 7798 : z = QabM_indexrank(M, P, ord);
4030 7798 : vecz = gel(z, 2); ct = lg(vecz) - 1;
4031 7798 : if (ct == dim) { M = mkvec3(z, gen_0, M); break; } /*maximal rank, done*/
4032 665 : vecpermute_inplace(M, vecz);
4033 665 : vecpermute_inplace(vj, vecz);
4034 : }
4035 7140 : if (a <= jlim) break;
4036 : /* sbsmall was not sufficient, use Sturm bound: must extend M */
4037 70 : for (j = 1; j <= ct; j++)
4038 : {
4039 63 : GEN t = heckenewtrace(SB + 1, sb, 1, N, N, k, vj[j], cache);
4040 63 : gel(M,j) = shallowconcat(gel(M, j), t);
4041 : }
4042 7 : jin = jlim + 1; SB = sb;
4043 : }
4044 7133 : S = cgetg(dim + 1, t_VEC);
4045 28371 : for (j = 1; j <= dim; j++) gel(S, j) = mfhecke_i(vj[j], N, tf);
4046 7133 : dbg_cachenew(cache);
4047 7133 : mf1 = mkvec4(utoipos(N), utoipos(k), CHI, utoi(mf_NEW));
4048 7133 : return mkmf(mf1, cgetg(1,t_VEC), S, vj, M);
4049 : }
4050 : /* k > 1 integral, mf space is mf_CUSP or mf_FULL */
4051 : static GEN
4052 42 : mfinittonew(GEN mf)
4053 : {
4054 42 : GEN CHI = MF_get_CHI(mf), S = MF_get_S(mf), vMjd = MFcusp_get_vMjd(mf);
4055 42 : GEN M = MF_get_M(mf), vj, mf1;
4056 42 : long i, j, l, l0 = lg(S), N0 = MF_get_N(mf);
4057 203 : for (i = l0-1; i > 0; i--)
4058 : {
4059 189 : long N = gel(vMjd,i)[1];
4060 189 : if (N != N0) break;
4061 : }
4062 42 : if (i == l0-1) return NULL;
4063 35 : S = vecslice(S, i+1, l0-1); /* forms of conductor N0 */
4064 35 : l = lg(S); vj = cgetg(l, t_VECSMALL);
4065 196 : for (j = 1; j < l; j++) vj[j] = gel(vMjd,j+i)[2];
4066 35 : M = vecslice(M, lg(M)-lg(S)+1, lg(M)-1); /* their coefficients */
4067 35 : M = mfcleanCHI(M, CHI, 0);
4068 35 : mf1 = mkvec4(utoipos(N0), MF_get_gk(mf), CHI, utoi(mf_NEW));
4069 35 : return mkmf(mf1, cgetg(1,t_VEC), S, vj, M);
4070 : }
4071 :
4072 : /* Bd(f)[m0..m], v = f[ceil(m0/d)..floor(m/d)], m0d = ceil(m0/d) */
4073 : static GEN
4074 82481 : RgC_Bd_expand(long m0, long m, GEN v, long d, long m0d)
4075 : {
4076 : long i, j;
4077 : GEN w;
4078 82481 : if (d == 1) return v;
4079 23660 : w = zerocol(m-m0+1);
4080 23660 : if (!m0) { i = 1; j = d; } else { i = 0; j = m0d*d; }
4081 469399 : for (; j <= m; i++, j += d) gel(w,j-m0+1) = gel(v,i+1);
4082 23660 : return w;
4083 : }
4084 : /* S a nonempty vector of t_MF_BD(t_MF_HECKE(t_MF_NEWTRACE)); M the matrix
4085 : * of their coefficients r*0, r*1, ..., r*m0 (~ mfvectomat) or NULL (empty),
4086 : * extend it to coeffs up to m > m0. The forms B_d(T_j(tf_N))in S should be
4087 : * sorted by level N, then j, then increasing d. No reordering here. */
4088 : static GEN
4089 9065 : bhnmat_extend(GEN M, long m, long r, GEN S, cachenew_t *cache)
4090 : {
4091 9065 : long i, mr, m0, m0r, Nold = 0, jold = 0, l = lg(S);
4092 9065 : GEN MAT = cgetg(l, t_MAT), v = NULL;
4093 9065 : if (M) { m0 = nbrows(M); m0r = m0 * r; } else m0 = m0r = 0;
4094 9065 : mr = m*r;
4095 91546 : for (i = 1; i < l; i++)
4096 : {
4097 : long d, j, md, N;
4098 82481 : GEN c, f = bhn_parse(gel(S,i), &d,&j); /* t_MF_NEWTRACE */
4099 82481 : N = mf_get_N(f);
4100 82481 : md = ceildivuu(m0r,d);
4101 82481 : if (N != Nold) { reset_cachenew(cache, N, f); Nold = N; jold = 0; }
4102 82481 : if (!cache->DATA) { gel(MAT,i) = zerocol(m+1); continue; }
4103 82481 : if (j != jold || md)
4104 66325 : { v = heckenewtrace(md, mr/d, 1, N, N, mf_get_k(f), j,cache); jold=j; }
4105 82481 : c = RgC_Bd_expand(m0r, mr, v, d, md);
4106 82481 : if (r > 1) c = c_deflate(m-m0, r, c);
4107 82481 : if (M) c = shallowconcat(gel(M,i), c);
4108 82481 : gel(MAT,i) = c;
4109 : }
4110 9065 : return MAT;
4111 : }
4112 :
4113 : /* k > 1 */
4114 : static GEN
4115 3199 : mfinitcusp(long N, long k, GEN CHI, cachenew_t *cache, long space)
4116 : {
4117 : long L, l, lDN1, FC, N1, d1, i, init;
4118 3199 : GEN vS, vMjd, DN1, vmf, CHIP = mfchartoprimitive(CHI, &FC);
4119 :
4120 3199 : d1 = (space == mf_OLD)? mfolddim_i(N, k, CHIP, NULL): mfcuspdim(N, k, CHIP);
4121 3199 : if (!d1) return NULL;
4122 2898 : N1 = N/FC; DN1 = mydivisorsu(N1); lDN1 = lg(DN1);
4123 2898 : init = (space == mf_OLD)? -1: 1;
4124 2898 : vmf = cgetg(lDN1, t_VEC);
4125 17255 : for (i = lDN1 - 1, l = 1; i; i--)
4126 : { /* by decreasing level to allow cache */
4127 14357 : GEN mf = mfnewinit(FC*DN1[i], k, CHIP, cache, init);
4128 14357 : if (mf) gel(vmf, l++) = mf;
4129 14357 : init = 0;
4130 : }
4131 2898 : setlg(vmf,l); vmf = vecreverse(vmf); /* reorder by increasing level */
4132 :
4133 2898 : L = mfsturmNk(N, k)+1;
4134 2898 : vS = vectrunc_init(L);
4135 2898 : vMjd = vectrunc_init(L);
4136 9177 : for (i = 1; i < l; i++)
4137 : {
4138 6279 : GEN DNM, mf = gel(vmf,i), S = MF_get_S(mf), vj = MFnew_get_vj(mf);
4139 6279 : long a, lDNM, lS = lg(S), M = MF_get_N(mf);
4140 6279 : DNM = mydivisorsu(N / M); lDNM = lg(DNM);
4141 25424 : for (a = 1; a < lS; a++)
4142 : {
4143 19145 : GEN tf = gel(S,a);
4144 19145 : long b, j = vj[a];
4145 47635 : for (b = 1; b < lDNM; b++)
4146 : {
4147 28490 : long d = DNM[b];
4148 28490 : vectrunc_append(vS, mfbd_i(tf, d));
4149 28490 : vectrunc_append(vMjd, mkvecsmall3(M, j, d));
4150 : }
4151 : }
4152 : }
4153 2898 : return mkmf(NULL, cgetg(1, t_VEC), vS, vMjd, NULL);
4154 : }
4155 :
4156 : long
4157 4536 : mfsturm_mf(GEN mf)
4158 : {
4159 4536 : GEN Mindex = MF_get_Mindex(mf);
4160 4536 : long n = lg(Mindex)-1;
4161 4536 : return n? Mindex[n]-1: 0;
4162 : }
4163 :
4164 : long
4165 805 : mfsturm(GEN T)
4166 : {
4167 : long N, nk, dk;
4168 805 : GEN CHI, mf = checkMF_i(T);
4169 805 : if (mf) return mfsturm_mf(mf);
4170 7 : checkNK2(T, &N, &nk, &dk, &CHI, 0);
4171 7 : return dk == 1 ? mfsturmNk(N, nk) : mfsturmNk(N, (nk + 1) >> 1);
4172 : }
4173 : long
4174 196 : mfisequal(GEN F, GEN G, long lim)
4175 : {
4176 196 : pari_sp av = avma;
4177 : long b;
4178 196 : if (!checkmf_i(F)) pari_err_TYPE("mfisequal",F);
4179 196 : if (!checkmf_i(G)) pari_err_TYPE("mfisequal",G);
4180 196 : b = lim? lim: maxss(mfsturmmf(F), mfsturmmf(G));
4181 196 : return gc_long(av, gequal(mfcoefs_i(F, b, 1), mfcoefs_i(G, b, 1)));
4182 : }
4183 :
4184 : GEN
4185 35 : mffields(GEN mf)
4186 : {
4187 35 : if (checkmf_i(mf)) return gcopy(mf_get_field(mf));
4188 35 : mf = checkMF(mf); return gcopy(MF_get_fields(mf));
4189 : }
4190 :
4191 : GEN
4192 343 : mfeigenbasis(GEN mf)
4193 : {
4194 343 : pari_sp ltop = avma;
4195 : GEN F, S, v, vP;
4196 : long i, l, k, dS;
4197 :
4198 343 : mf = checkMF(mf);
4199 343 : k = MF_get_k(mf);
4200 343 : S = MF_get_S(mf); dS = lg(S)-1;
4201 343 : if (!dS) return cgetg(1, t_VEC);
4202 336 : F = MF_get_newforms(mf);
4203 336 : vP = MF_get_fields(mf);
4204 336 : if (k == 1)
4205 : {
4206 210 : if (MF_get_space(mf) == mf_FULL)
4207 : {
4208 14 : long dE = lg(MF_get_E(mf)) - 1;
4209 14 : if (dE) F = rowslice(F, dE+1, dE+dS);
4210 : }
4211 210 : v = vecmflineardiv_linear(S, F);
4212 210 : l = lg(v);
4213 : }
4214 : else
4215 : {
4216 126 : GEN (*L)(GEN, GEN) = (MF_get_space(mf) == mf_FULL)? mflinear: mflinear_bhn;
4217 126 : l = lg(F); v = cgetg(l, t_VEC);
4218 427 : for (i = 1; i < l; i++) gel(v,i) = L(mf, gel(F,i));
4219 : }
4220 861 : for (i = 1; i < l; i++) mf_setfield(gel(v,i), gel(vP,i));
4221 336 : return gerepilecopy(ltop, v);
4222 : }
4223 :
4224 : /* Minv = [M, d, A], v a t_COL; A a Zab, d a t_INT; return (A/d) * M*v */
4225 : static GEN
4226 7294 : Minv_RgC_mul(GEN Minv, GEN v)
4227 : {
4228 7294 : GEN M = gel(Minv,1), d = gel(Minv,2), A = gel(Minv,3);
4229 7294 : v = RgM_RgC_mul(M, v);
4230 7294 : if (!equali1(A))
4231 : {
4232 1771 : if (typ(A) == t_POL && degpol(A) > 0) A = mkpolmod(A, gel(Minv,4));
4233 1771 : v = RgC_Rg_mul(v, A);
4234 : }
4235 7294 : if (!equali1(d)) v = RgC_Rg_div(v, d);
4236 7294 : return v;
4237 : }
4238 : static GEN
4239 1281 : Minv_RgM_mul(GEN Minv, GEN B)
4240 : {
4241 1281 : long j, l = lg(B);
4242 1281 : GEN M = cgetg(l, t_MAT);
4243 5915 : for (j = 1; j < l; j++) gel(M,j) = Minv_RgC_mul(Minv, gel(B,j));
4244 1281 : return M;
4245 : }
4246 : /* B * Minv; allow B = NULL for Id */
4247 : static GEN
4248 2436 : RgM_Minv_mul(GEN B, GEN Minv)
4249 : {
4250 2436 : GEN M = gel(Minv,1), d = gel(Minv,2), A = gel(Minv,3);
4251 2436 : if (B) M = RgM_mul(B, M);
4252 2436 : if (!equali1(A))
4253 : {
4254 980 : if (typ(A) == t_POL) A = mkpolmod(A, gel(Minv,4));
4255 980 : M = RgM_Rg_mul(M, A);
4256 : }
4257 2436 : if (!equali1(d)) M = RgM_Rg_div(M,d);
4258 2436 : return M;
4259 : }
4260 :
4261 : /* perm vector of strictly increasing indices, v a vector or arbitrary length;
4262 : * the last r entries of perm fall beyond v.
4263 : * Return v o perm[1..(-r)], discarding the last r entries of v */
4264 : static GEN
4265 1358 : vecpermute_partial(GEN v, GEN perm, long *r)
4266 : {
4267 1358 : long i, n = lg(v)-1, l = lg(perm);
4268 : GEN w;
4269 1358 : if (perm[l-1] <= n) { *r = 0; return vecpermute(v,perm); }
4270 63 : for (i = 1; i < l; i++)
4271 63 : if (perm[i] > n) break;
4272 21 : *r = l - i; l = i;
4273 21 : w = cgetg(l, typ(v));
4274 63 : for (i = 1; i < l; i++) gel(w,i) = gel(v,perm[i]);
4275 21 : return w;
4276 : }
4277 :
4278 : /* given form F, find coeffs of F on mfbasis(mf). If power series, not
4279 : * guaranteed correct if precision less than Sturm bound */
4280 : static GEN
4281 1386 : mftobasis_i(GEN mf, GEN F)
4282 : {
4283 : GEN v, Mindex, Minv;
4284 1386 : if (!MF_get_dim(mf)) return cgetg(1, t_COL);
4285 1386 : Mindex = MF_get_Mindex(mf);
4286 1386 : Minv = MF_get_Minv(mf);
4287 1386 : if (checkmf_i(F))
4288 : {
4289 266 : long n = Mindex[lg(Mindex)-1];
4290 266 : v = vecpermute(mfcoefs_i(F, n, 1), Mindex);
4291 266 : return Minv_RgC_mul(Minv, v);
4292 : }
4293 : else
4294 : {
4295 1120 : GEN A = gel(Minv,1), d = gel(Minv,2);
4296 : long r;
4297 1120 : v = F;
4298 1120 : switch(typ(F))
4299 : {
4300 0 : case t_SER: v = sertocol(v);
4301 1120 : case t_VEC: case t_COL: break;
4302 0 : default: pari_err_TYPE("mftobasis", F);
4303 : }
4304 1120 : if (lg(v) == 1) pari_err_TYPE("mftobasis",v);
4305 1120 : v = vecpermute_partial(v, Mindex, &r);
4306 1120 : if (!r) return Minv_RgC_mul(Minv, v); /* single solution */
4307 : /* affine space of dimension r */
4308 21 : v = RgM_RgC_mul(vecslice(A, 1, lg(v)-1), v);
4309 21 : if (!equali1(d)) v = RgC_Rg_div(v,d);
4310 21 : return mkvec2(v, vecslice(A, lg(A)-r, lg(A)-1));
4311 : }
4312 : }
4313 :
4314 : static GEN
4315 896 : const_mat(long n, GEN x)
4316 : {
4317 896 : long j, l = n+1;
4318 896 : GEN A = cgetg(l,t_MAT);
4319 6832 : for (j = 1; j < l; j++) gel(A,j) = const_col(n, x);
4320 896 : return A;
4321 : }
4322 :
4323 : /* L is the mftobasis of a form on CUSP space. We allow mf_FULL or mf_CUSP */
4324 : static GEN
4325 448 : mftonew_i(GEN mf, GEN L, long *plevel)
4326 : {
4327 : GEN S, listMjd, CHI, res, Aclos, Acoef, D, perm;
4328 448 : long N1, LC, lD, i, l, t, level, N = MF_get_N(mf);
4329 :
4330 448 : if (MF_get_k(mf) == 1) pari_err_IMPL("mftonew in weight 1");
4331 448 : listMjd = MFcusp_get_vMjd(mf);
4332 448 : CHI = MF_get_CHI(mf); LC = mfcharconductor(CHI);
4333 448 : S = MF_get_S(mf);
4334 :
4335 448 : N1 = N/LC;
4336 448 : D = mydivisorsu(N1); lD = lg(D);
4337 448 : perm = cgetg(N1+1, t_VECSMALL);
4338 3416 : for (i = 1; i < lD; i++) perm[D[i]] = i;
4339 448 : Aclos = const_mat(lD-1, cgetg(1,t_VEC));
4340 448 : Acoef = const_mat(lD-1, cgetg(1,t_VEC));
4341 448 : l = lg(listMjd);
4342 4494 : for (i = 1; i < l; i++)
4343 : {
4344 : long M, d;
4345 : GEN v;
4346 4046 : if (gequal0(gel(L,i))) continue;
4347 441 : v = gel(listMjd, i);
4348 441 : M = perm[ v[1]/LC ];
4349 441 : d = perm[ v[3] ];
4350 441 : gcoeff(Aclos,M,d) = vec_append(gcoeff(Aclos,M,d), gel(S,i));
4351 441 : gcoeff(Acoef,M,d) = shallowconcat(gcoeff(Acoef,M,d), gel(L,i));
4352 : }
4353 448 : res = cgetg(l, t_VEC); level = 1;
4354 3416 : for (i = t = 1; i < lD; i++)
4355 : {
4356 2968 : long j, M = D[i]*LC;
4357 2968 : GEN gM = utoipos(M);
4358 26390 : for (j = 1; j < lD; j++)
4359 : {
4360 23422 : GEN f = gcoeff(Aclos,i,j), C, NK;
4361 : long d;
4362 23422 : if (lg(f) == 1) continue;
4363 413 : d = D[j];
4364 413 : C = gcoeff(Acoef,i,j);
4365 413 : NK = mf_get_NK(gel(f, 1));
4366 413 : if (d > 1)
4367 : {
4368 168 : if (lg(f) > 2) pari_err_BUG("should be only one form");
4369 168 : f = gel(f, 1);
4370 168 : if (mf_get_type(f) == t_MF_BD)
4371 : {
4372 168 : if (!equaliu(gel(f,3), d))
4373 0 : pari_err_BUG("inconsistent multiplier");
4374 168 : f = gel(f, 2);
4375 : }
4376 168 : f = mkvec(f);
4377 : }
4378 413 : level = ulcm(level, M*d);
4379 413 : gel(res,t++) = mkvec3(gM, utoipos(d), mflinear_i(NK,f,C));
4380 : }
4381 : }
4382 448 : if (plevel) *plevel = level;
4383 448 : setlg(res, t); return res;
4384 : }
4385 : GEN
4386 210 : mftonew(GEN mf, GEN F)
4387 : {
4388 210 : pari_sp av = avma;
4389 : GEN ES;
4390 : long s;
4391 210 : mf = checkMF(mf);
4392 210 : s = MF_get_space(mf);
4393 210 : if (s != mf_FULL && s != mf_CUSP)
4394 7 : pari_err_TYPE("mftonew [not a full or cuspidal space]", mf);
4395 203 : ES = mftobasisES(mf,F);
4396 196 : if (!gequal0(gel(ES,1)))
4397 0 : pari_err_TYPE("mftonew [not a cuspidal form]", F);
4398 196 : F = gel(ES,2);
4399 196 : return gerepilecopy(av, mftonew_i(mf,F, NULL));
4400 : }
4401 :
4402 : static GEN mfeisenstein_i(long k, GEN CHI1, GEN CHI2);
4403 :
4404 : /* mfinit(F * Theta) */
4405 : static GEN
4406 98 : mf2init(GEN mf)
4407 : {
4408 98 : GEN CHI = MF_get_CHI(mf), gk = gadd(MF_get_gk(mf), ghalf);
4409 98 : long N = MF_get_N(mf);
4410 98 : return mfinit_Nkchi(N, itou(gk), mfchiadjust(CHI, gk, N), mf_FULL, 0);
4411 : }
4412 :
4413 : static long
4414 623 : mfvec_first_cusp(GEN v)
4415 : {
4416 623 : long i, l = lg(v);
4417 1519 : for (i = 1; i < l; i++)
4418 : {
4419 1414 : GEN F = gel(v,i);
4420 1414 : long t = mf_get_type(F);
4421 1414 : if (t == t_MF_BD) { F = gel(F,2); t = mf_get_type(F); }
4422 1414 : if (t == t_MF_HECKE) { F = gel(F,3); t = mf_get_type(F); }
4423 1414 : if (t == t_MF_NEWTRACE) break;
4424 : }
4425 623 : return i;
4426 : }
4427 : /* vF a vector of mf F of type DIV(LINEAR(BAS,L), f) in (lcm) level N,
4428 : * F[2]=LINEAR(BAS,L), F[2][2]=BAS=fixed basis (Eisenstein or bhn type),
4429 : * F[2][3]=L, F[3]=f; mfvectomat(vF, n) */
4430 : static GEN
4431 630 : mflineardivtomat(long N, GEN vF, long n)
4432 : {
4433 630 : GEN F, M, f, fc, ME, dB, B, a0, V = NULL;
4434 630 : long lM, lF = lg(vF), j;
4435 :
4436 630 : if (lF == 1) return cgetg(1,t_MAT);
4437 623 : F = gel(vF,1);
4438 623 : if (lg(F) == 5)
4439 : { /* chicompat */
4440 273 : V = gmael(F,4,4);
4441 273 : if (typ(V) == t_INT) V = NULL;
4442 : }
4443 623 : M = gmael(F,2,2); /* BAS */
4444 623 : lM = lg(M);
4445 623 : j = mfvec_first_cusp(M);
4446 623 : if (j == 1) ME = NULL;
4447 : else
4448 : { /* BAS starts by Eisenstein */
4449 161 : ME = mfvectomat(vecslice(M,1,j-1), n, 1);
4450 161 : M = vecslice(M, j,lM-1);
4451 : }
4452 623 : M = bhnmat_extend_nocache(NULL, N, n, 1, M);
4453 623 : if (ME) M = shallowconcat(ME,M);
4454 : /* M = mfcoefs of BAS */
4455 623 : B = cgetg(lF, t_MAT);
4456 623 : dB= cgetg(lF, t_VEC);
4457 2947 : for (j = 1; j < lF; j++)
4458 : {
4459 2324 : GEN g = gel(vF, j); /* t_MF_DIV */
4460 2324 : gel(B,j) = RgM_RgC_mul(M, gmael(g,2,3));
4461 2324 : gel(dB,j)= gmael(g,2,4);
4462 : }
4463 623 : f = mfcoefsser(gel(F,3),n);
4464 623 : a0 = polcoef_i(f, 0, -1);
4465 623 : if (gequal0(a0) || gequal1(a0))
4466 322 : a0 = NULL;
4467 : else
4468 301 : f = gdiv(ser_unscale(f, a0), a0);
4469 623 : fc = ginv(f);
4470 2947 : for (j = 1; j < lF; j++)
4471 : {
4472 2324 : pari_sp av = avma;
4473 2324 : GEN LISer = RgV_to_ser_full(gel(B,j)), f;
4474 2324 : if (a0) LISer = gdiv(ser_unscale(LISer, a0), a0);
4475 2324 : f = gmul(LISer, fc);
4476 2324 : if (a0) f = ser_unscale(f, ginv(a0));
4477 2324 : f = sertocol(f); setlg(f, n+2);
4478 2324 : if (!gequal1(gel(dB,j))) f = RgC_Rg_div(f, gel(dB,j));
4479 2324 : gel(B,j) = gerepileupto(av,f);
4480 : }
4481 623 : if (V) B = gmodulo(QabM_tracerel(V, 0, B), gel(V,1));
4482 623 : return B;
4483 : }
4484 :
4485 : static GEN
4486 350 : mfheckemat_mfcoefs(GEN mf, GEN B, GEN DATA)
4487 : {
4488 350 : GEN Mindex = MF_get_Mindex(mf), Minv = MF_get_Minv(mf);
4489 350 : long j, l = lg(B), sb = mfsturm_mf(mf);
4490 350 : GEN b = MF_get_basis(mf), Q = cgetg(l, t_VEC);
4491 1827 : for (j = 1; j < l; j++)
4492 : {
4493 1477 : GEN v = hecke_i(sb, 1, gel(B,j), gel(b,j), DATA); /* Tn b[j] */
4494 1477 : settyp(v,t_COL); gel(Q,j) = vecpermute(v, Mindex);
4495 : }
4496 350 : return Minv_RgM_mul(Minv,Q);
4497 : }
4498 : /* T_p^2, p prime, 1/2-integral weight; B = mfcoefs(mf,sb*p^2,1) or (mf,sb,p^2)
4499 : * if p|N */
4500 : static GEN
4501 7 : mfheckemat_mfcoefs_p2(GEN mf, long p, GEN B)
4502 : {
4503 7 : pari_sp av = avma;
4504 7 : GEN DATA = heckef2_data(MF_get_N(mf), p*p);
4505 7 : return gerepileupto(av, mfheckemat_mfcoefs(mf, B, DATA));
4506 : }
4507 : /* convert Mindex from row-index to mfcoef indexation: a(n) is stored in
4508 : * mfcoefs()[n+1], so subtract 1 from all indices */
4509 : static GEN
4510 49 : Mindex_as_coef(GEN mf)
4511 : {
4512 49 : GEN v, Mindex = MF_get_Mindex(mf);
4513 49 : long i, l = lg(Mindex);
4514 49 : v = cgetg(l, t_VECSMALL);
4515 210 : for (i = 1; i < l; i++) v[i] = Mindex[i]-1;
4516 49 : return v;
4517 : }
4518 : /* T_p, p prime; B = mfcoefs(mf,sb*p,1) or (mf,sb,p) if p|N; integral weight */
4519 : static GEN
4520 35 : mfheckemat_mfcoefs_p(GEN mf, long p, GEN B)
4521 : {
4522 35 : pari_sp av = avma;
4523 35 : GEN vm, Q, C, Minv = MF_get_Minv(mf);
4524 35 : long lm, k, i, j, l = lg(B), N = MF_get_N(mf);
4525 :
4526 35 : if (N % p == 0) return Minv_RgM_mul(Minv, rowpermute(B, MF_get_Mindex(mf)));
4527 21 : k = MF_get_k(mf);
4528 21 : C = gmul(mfchareval(MF_get_CHI(mf), p), powuu(p, k-1));
4529 21 : vm = Mindex_as_coef(mf); lm = lg(vm);
4530 21 : Q = cgetg(l, t_MAT);
4531 147 : for (j = 1; j < l; j++) gel(Q,j) = cgetg(lm, t_COL);
4532 147 : for (i = 1; i < lm; i++)
4533 : {
4534 126 : long m = vm[i], mp = m*p;
4535 126 : GEN Cm = (m % p) == 0? C : NULL;
4536 1260 : for (j = 1; j < l; j++)
4537 : {
4538 1134 : GEN S = gel(B,j), s = gel(S, mp + 1);
4539 1134 : if (Cm) s = gadd(s, gmul(C, gel(S, m/p + 1)));
4540 1134 : gcoeff(Q, i, j) = s;
4541 : }
4542 : }
4543 21 : return gerepileupto(av, Minv_RgM_mul(Minv,Q));
4544 : }
4545 : /* Matrix of T(p), p prime, dim(mf) > 0 and integral weight */
4546 : static GEN
4547 343 : mfheckemat_p(GEN mf, long p)
4548 : {
4549 343 : pari_sp av = avma;
4550 343 : long N = MF_get_N(mf), sb = mfsturm_mf(mf);
4551 343 : GEN B = (N % p)? mfcoefs_mf(mf, sb * p, 1): mfcoefs_mf(mf, sb, p);
4552 343 : return gerepileupto(av, mfheckemat_mfcoefs(mf, B, hecke_data(N,p)));
4553 : }
4554 :
4555 : /* mf_NEW != (0), weight > 1, p prime. Use
4556 : * T(p) T(j) = T(j*p) + p^{k-1} \chi(p) 1_{p | j, p \nmid N} T(j/p) */
4557 : static GEN
4558 896 : mfnewmathecke_p(GEN mf, long p)
4559 : {
4560 896 : pari_sp av = avma;
4561 896 : GEN tf, vj = MFnew_get_vj(mf), CHI = MF_get_CHI(mf);
4562 896 : GEN Mindex = MF_get_Mindex(mf), Minv = MF_get_Minv(mf);
4563 896 : long N = MF_get_N(mf), k = MF_get_k(mf);
4564 896 : long i, j, lvj = lg(vj), lim = vj[lvj-1] * p;
4565 896 : GEN M, perm, V, need = zero_zv(lim);
4566 896 : GEN C = (N % p)? gmul(mfchareval(CHI,p), powuu(p,k-1)): NULL;
4567 896 : tf = mftraceform_new(N, k, CHI);
4568 3829 : for (i = 1; i < lvj; i++)
4569 : {
4570 2933 : j = vj[i]; need[j*p] = 1;
4571 2933 : if (N % p && j % p == 0) need[j/p] = 1;
4572 : }
4573 896 : perm = zero_zv(lim);
4574 896 : V = cgetg(lim+1, t_VEC);
4575 12285 : for (i = j = 1; i <= lim; i++)
4576 11389 : if (need[i]) { gel(V,j) = mfhecke_i(i, N, tf); perm[i] = j; j++; }
4577 896 : setlg(V, j);
4578 896 : V = bhnmat_extend_nocache(NULL, N, mfsturm_mf(mf), 1, V);
4579 896 : V = rowpermute(V, Mindex); /* V[perm[i]] = coeffs(T_i newtrace) */
4580 896 : M = cgetg(lvj, t_MAT);
4581 3829 : for (i = 1; i < lvj; i++)
4582 : {
4583 : GEN t;
4584 2933 : j = vj[i]; t = gel(V, perm[j*p]);
4585 2933 : if (C && j % p == 0) t = RgC_add(t, RgC_Rg_mul(gel(V, perm[j/p]),C));
4586 2933 : gel(M,i) = t;
4587 : }
4588 896 : return gerepileupto(av, Minv_RgM_mul(Minv, M));
4589 : }
4590 :
4591 : GEN
4592 77 : mfheckemat(GEN mf, GEN vn)
4593 : {
4594 77 : pari_sp av = avma;
4595 77 : long lv, lvP, i, N, dim, nk, dk, p, sb, flint = (typ(vn)==t_INT);
4596 : GEN CHI, res, vT, FA, B, vP;
4597 :
4598 77 : mf = checkMF(mf);
4599 77 : if (typ(vn) != t_VECSMALL) vn = gtovecsmall(vn);
4600 77 : N = MF_get_N(mf); CHI = MF_get_CHI(mf); Qtoss(MF_get_gk(mf), &nk, &dk);
4601 77 : dim = MF_get_dim(mf);
4602 77 : lv = lg(vn);
4603 77 : res = cgetg(lv, t_VEC);
4604 77 : FA = cgetg(lv, t_VEC);
4605 77 : vP = cgetg(lv, t_VEC);
4606 77 : vT = const_vec(vecsmall_max(vn), NULL);
4607 182 : for (i = 1; i < lv; i++)
4608 : {
4609 105 : ulong n = (ulong)labs(vn[i]);
4610 : GEN fa;
4611 105 : if (!n) pari_err_TYPE("mfheckemat", vn);
4612 105 : if (dk == 1 || uissquareall(n, &n)) fa = myfactoru(n);
4613 0 : else { n = 0; fa = myfactoru(1); } /* dummy: T_{vn[i]} = 0 */
4614 105 : vn[i] = n;
4615 105 : gel(FA,i) = fa;
4616 105 : gel(vP,i) = gel(fa,1);
4617 : }
4618 77 : vP = shallowconcat1(vP); vecsmall_sort(vP);
4619 77 : vP = vecsmall_uniq_sorted(vP); /* all primes occurring in vn */
4620 77 : lvP = lg(vP); if (lvP == 1) goto END;
4621 56 : p = vP[lvP-1];
4622 56 : sb = mfsturm_mf(mf);
4623 56 : if (dk == 1 && nk != 1 && MF_get_space(mf) == mf_NEW)
4624 21 : B = NULL; /* special purpose mfnewmathecke_p is faster */
4625 35 : else if (lvP == 2 && N % p == 0)
4626 21 : B = mfcoefs_mf(mf, sb, dk==2? p*p: p); /* single prime | N, can optimize */
4627 : else
4628 14 : B = mfcoefs_mf(mf, sb * (dk==2? p*p: p), 1); /* general initialization */
4629 126 : for (i = 1; i < lvP; i++)
4630 : {
4631 70 : long j, l, q, e = 1;
4632 : GEN C, Tp, u1, u0;
4633 70 : p = vP[i];
4634 189 : for (j = 1; j < lv; j++) e = maxss(e, z_lval(vn[j], p));
4635 70 : if (!B)
4636 28 : Tp = mfnewmathecke_p(mf, p);
4637 42 : else if (dk == 2)
4638 7 : Tp = mfheckemat_mfcoefs_p2(mf,p, (lvP==2||N%p)? B: matdeflate(sb,p*p,B));
4639 : else
4640 35 : Tp = mfheckemat_mfcoefs_p(mf, p, (lvP==2||N%p)? B: matdeflate(sb,p,B));
4641 70 : gel(vT, p) = Tp;
4642 70 : if (e == 1) continue;
4643 14 : u0 = gen_1;
4644 14 : if (dk == 2)
4645 : {
4646 0 : C = N % p? gmul(mfchareval(CHI,p*p), powuu(p, nk-2)): NULL;
4647 0 : if (e == 2) u0 = uutoQ(p+1,p); /* special case T_{p^4} */
4648 : }
4649 : else
4650 14 : C = N % p? gmul(mfchareval(CHI,p), powuu(p, nk-1)): NULL;
4651 28 : for (u1=Tp, q=p, l=2; l <= e; l++)
4652 : { /* u0 = T_{p^{l-2}}, u1 = T_{p^{l-1}} for l > 2 */
4653 14 : GEN v = gmul(Tp, u1);
4654 14 : if (C) v = gsub(v, gmul(C, u0));
4655 : /* q = p^l, vT[q] = T_q for k integer else T_{q^2} */
4656 14 : q *= p; u0 = u1; gel(vT, q) = u1 = v;
4657 : }
4658 : }
4659 56 : END:
4660 : /* vT[p^e] = T_{p^e} for all p^e occurring below */
4661 182 : for (i = 1; i < lv; i++)
4662 : {
4663 105 : long n = vn[i], j, lP;
4664 : GEN fa, P, E, M;
4665 105 : if (n == 0) { gel(res,i) = zeromat(dim,dim); continue; }
4666 105 : if (n == 1) { gel(res,i) = matid(dim); continue; }
4667 77 : fa = gel(FA,i);
4668 77 : P = gel(fa,1); lP = lg(P);
4669 77 : E = gel(fa,2); M = gel(vT, upowuu(P[1], E[1]));
4670 84 : for (j = 2; j < lP; j++) M = RgM_mul(M, gel(vT, upowuu(P[j], E[j])));
4671 77 : gel(res,i) = M;
4672 : }
4673 77 : if (flint) res = gel(res,1);
4674 77 : return gerepilecopy(av, res);
4675 : }
4676 :
4677 : /* f = \sum_i v[i] T_listj[i] (Trace Form) attached to v; replace by f/a_1(f) */
4678 : static GEN
4679 1477 : mf_normalize(GEN mf, GEN v)
4680 : {
4681 1477 : GEN c, dc = NULL, M = MF_get_M(mf), Mindex = MF_get_Mindex(mf);
4682 1477 : v = Q_primpart(v);
4683 1477 : c = RgMrow_RgC_mul(M, v, 2); /* a_1(f) */
4684 1477 : if (gequal1(c)) return v;
4685 882 : if (typ(c) == t_POL) c = gmodulo(c, mfcharpol(MF_get_CHI(mf)));
4686 882 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1 && degpol(gel(c,1)) >= 40
4687 7 : && Mindex[1] == 2
4688 7 : && mfcharorder(MF_get_CHI(mf)) <= 2)
4689 7 : { /* normalize using expansion at infinity (small coefficients) */
4690 7 : GEN w, P = gel(c,1), a1 = gel(c,2);
4691 7 : long i, l = lg(Mindex);
4692 7 : w = cgetg(l, t_COL);
4693 7 : gel(w,1) = gen_1;
4694 280 : for (i = 2; i < l; i++)
4695 : {
4696 273 : c = liftpol_shallow(RgMrow_RgC_mul(M, v, Mindex[i]));
4697 273 : gel(w,i) = QXQ_div(c, a1, P);
4698 : }
4699 : /* w = expansion at oo of normalized form */
4700 7 : v = Minv_RgC_mul(MF_get_Minv(mf), Q_remove_denom(w, &dc));
4701 7 : v = gmodulo(v, P); /* back to mfbasis coefficients */
4702 : }
4703 : else
4704 : {
4705 875 : c = ginv(c);
4706 875 : if (typ(c) == t_POLMOD) c = Q_remove_denom(c, &dc);
4707 875 : v = RgC_Rg_mul(v, c);
4708 : }
4709 882 : if (dc) v = RgC_Rg_div(v, dc);
4710 882 : return v;
4711 : }
4712 : static void
4713 427 : pol_red(GEN NF, GEN *pP, GEN *pa, long flag)
4714 : {
4715 427 : GEN dP, a, P = *pP;
4716 427 : long d = degpol(P);
4717 :
4718 427 : *pa = a = pol_x(varn(P));
4719 427 : if (d * (NF ? nf_get_degree(NF): 1) > 30) return;
4720 :
4721 420 : dP = RgX_disc(P);
4722 420 : if (typ(dP) != t_INT)
4723 98 : { dP = gnorm(dP); if (typ(dP) != t_INT) pari_err_BUG("mfnewsplit"); }
4724 420 : if (d == 2 || expi(dP) < 62)
4725 : {
4726 385 : if (expi(dP) < 31)
4727 385 : P = NF? rnfpolredabs(NF, P,flag): polredabs0(P,flag);
4728 : else
4729 0 : P = NF? rnfpolredbest(NF,P,flag): polredbest(P,flag);
4730 385 : if (flag)
4731 : {
4732 357 : a = gel(P,2); if (typ(a) == t_POLMOD) a = gel(a,2);
4733 357 : P = gel(P,1);
4734 : }
4735 : }
4736 420 : *pP = P;
4737 420 : *pa = a;
4738 : }
4739 :
4740 : /* Diagonalize and normalize. See mfsplit for meaning of flag. */
4741 : static GEN
4742 1071 : mfspclean(GEN mf, GEN mf0, GEN NF, long ord, GEN simplesp, long flag)
4743 : {
4744 1071 : const long vz = 1;
4745 1071 : long i, l = lg(simplesp), dim = MF_get_dim(mf);
4746 1071 : GEN res = cgetg(l, t_MAT), pols = cgetg(l, t_VEC);
4747 1071 : GEN zeros = (mf == mf0)? NULL: zerocol(dim - MF_get_dim(mf0));
4748 2576 : for (i = 1; i < l; i++)
4749 : {
4750 1505 : GEN ATP = gel(simplesp, i), A = gel(ATP,1), P = gel(ATP,3);
4751 1505 : long d = degpol(P);
4752 1505 : GEN a, v = (flag && d > flag)? NULL: gel(A,1);
4753 1505 : if (d == 1) P = pol_x(vz);
4754 : else
4755 : {
4756 427 : pol_red(NF, &P, &a, !!v);
4757 427 : if (v)
4758 : { /* Mod(a,P) root of charpoly(T), K*gpowers(a) = eigenvector of T */
4759 399 : GEN K, den, M = cgetg(d+1, t_MAT), T = gel(ATP,2);
4760 : long j;
4761 399 : T = shallowtrans(T);
4762 399 : gel(M,1) = vec_ei(d,1); /* basis of cyclic vectors */
4763 1302 : for (j = 2; j <= d; j++) gel(M,j) = RgM_RgC_mul(T, gel(M,j-1));
4764 399 : M = Q_primpart(M);
4765 133 : K = NF? ZabM_inv(liftpol_shallow(M), nf_get_pol(NF), ord, &den)
4766 399 : : ZM_inv(M,&den);
4767 399 : K = shallowtrans(K);
4768 399 : v = gequalX(a)? pol_x_powers(d, vz): RgXQ_powers(a, d-1, P);
4769 399 : v = gmodulo(RgM_RgC_mul(A, RgM_RgC_mul(K,v)), P);
4770 : }
4771 : }
4772 1505 : if (v)
4773 : {
4774 1477 : v = mf_normalize(mf0, v); if (zeros) v = shallowconcat(zeros,v);
4775 1477 : gel(res,i) = v; if (flag) setlg(res,i+1);
4776 : }
4777 : else
4778 28 : gel(res,i) = zerocol(dim);
4779 1505 : gel(pols,i) = P;
4780 : }
4781 1071 : return mkvec2(res, pols);
4782 : }
4783 :
4784 : /* return v = v_{X-r}(P), and set Z = P / (X-r)^v */
4785 : static long
4786 70 : RgX_valrem_root(GEN P, GEN r, GEN *Z)
4787 : {
4788 : long v;
4789 140 : for (v = 0; degpol(P); v++)
4790 : {
4791 140 : GEN t, Q = RgX_div_by_X_x(P, r, &t);
4792 140 : if (!gequal0(t)) break;
4793 70 : P = Q;
4794 : }
4795 70 : *Z = P; return v;
4796 : }
4797 : static GEN
4798 1498 : mynffactor(GEN NF, GEN P, long dimlim)
4799 : {
4800 : long i, l, v;
4801 : GEN R, E;
4802 1498 : if (dimlim != 1)
4803 : {
4804 931 : R = NF? nffactor(NF, P): QX_factor(P);
4805 931 : if (!dimlim) return R;
4806 21 : E = gel(R,2);
4807 21 : R = gel(R,1); l = lg(R);
4808 98 : for (i = 1; i < l; i++)
4809 91 : if (degpol(gel(R,i)) > dimlim) break;
4810 21 : if (i == 1) return NULL;
4811 21 : setlg(E,i);
4812 21 : setlg(R,i); return mkmat2(R, E);
4813 : }
4814 : /* dimlim = 1 */
4815 567 : R = nfroots(NF, P); l = lg(R);
4816 567 : if (l == 1) return NULL;
4817 504 : v = varn(P);
4818 504 : settyp(R, t_COL);
4819 504 : if (degpol(P) == l-1)
4820 448 : E = const_col(l-1, gen_1);
4821 : else
4822 : {
4823 56 : E = cgetg(l, t_COL);
4824 126 : for (i = 1; i < l; i++) gel(E,i) = utoi(RgX_valrem_root(P, gel(R,i), &P));
4825 : }
4826 504 : R = deg1_from_roots(R, v);
4827 504 : return mkmat2(R, E);
4828 : }
4829 :
4830 : /* Let K be a number field attached to NF (Q if NF = NULL). A K-vector
4831 : * space of dimension d > 0 is given by a t_MAT A (n x d, full column rank)
4832 : * giving a K-basis, X a section (d x n: left pseudo-inverse of A). Return a
4833 : * pair (T, fa), where T is an element of the Hecke algebra (a sum of Tp taken
4834 : * from vector vTp) acting on A (a d x d t_MAT) and fa is the factorization of
4835 : * its characteristic polynomial, limited to factors of degree <= dimlim if
4836 : * dimlim != 0 (return NULL if there are no factors of degree <= dimlim) */
4837 : static GEN
4838 1323 : findbestsplit(GEN NF, GEN vTp, GEN A, GEN X, long dimlim, long vz)
4839 : {
4840 1323 : GEN T = NULL, Tkeep = NULL, fakeep = NULL;
4841 1323 : long lmax = 0, i, lT = lg(vTp);
4842 1743 : for (i = 1; i < lT; i++)
4843 : {
4844 1743 : GEN D, P, E, fa, TpA = gel(vTp,i);
4845 : long l;
4846 2758 : if (typ(TpA) == t_INT) break;
4847 1498 : if (lg(TpA) > lg(A)) TpA = RgM_mul(X, RgM_mul(TpA, A)); /* Tp | A */
4848 1498 : T = T ? RgM_add(T, TpA) : TpA;
4849 1498 : if (!NF) { P = QM_charpoly_ZX(T); setvarn(P, vz); }
4850 : else
4851 : {
4852 273 : P = charpoly(Q_remove_denom(T, &D), vz);
4853 273 : if (D) P = gdiv(RgX_unscale(P, D), powiu(D, degpol(P)));
4854 : }
4855 1498 : fa = mynffactor(NF, P, dimlim);
4856 1498 : if (!fa) return NULL;
4857 1435 : E = gel(fa, 2);
4858 : /* characteristic polynomial is separable ? */
4859 1435 : if (isint1(vecmax(E))) { Tkeep = T; fakeep = fa; break; }
4860 420 : l = lg(E);
4861 : /* characteristic polynomial has more factors than before ? */
4862 420 : if (l > lmax) { lmax = l; Tkeep = T; fakeep = fa; }
4863 : }
4864 1260 : return mkvec2(Tkeep, fakeep);
4865 : }
4866 :
4867 : static GEN
4868 210 : nfcontent(GEN nf, GEN v)
4869 : {
4870 210 : long i, l = lg(v);
4871 210 : GEN c = gel(v,1);
4872 1134 : for (i = 2; i < l; i++) c = idealadd(nf, c, gel(v,i));
4873 210 : if (typ(c) == t_MAT && gequal1(gcoeff(c,1,1))) c = gen_1;
4874 210 : return c;
4875 : }
4876 : static GEN
4877 329 : nf_primpart(GEN nf, GEN x)
4878 : {
4879 329 : switch(typ(x))
4880 : {
4881 210 : case t_COL:
4882 : {
4883 210 : GEN A = matalgtobasis(nf, x), c = nfcontent(nf, A);
4884 210 : if (typ(c) == t_INT) return x;
4885 21 : c = idealred_elt(nf,c);
4886 21 : A = Q_primpart( nfC_nf_mul(nf, A, Q_primpart(nfinv(nf,c))) );
4887 21 : A = liftpol_shallow( matbasistoalg(nf, A) );
4888 21 : if (gexpo(A) > gexpo(x)) A = x;
4889 21 : return A;
4890 : }
4891 329 : case t_MAT: pari_APPLY_same(nf_primpart(nf, gel(x,i)));
4892 0 : default:
4893 0 : pari_err_TYPE("nf_primpart", x);
4894 : return NULL; /*LCOV_EXCL_LINE*/
4895 : }
4896 : }
4897 :
4898 : /* rotate entries of v to accomodate new entry 'x' (push out oldest entry) */
4899 : static void
4900 1211 : vecpush(GEN v, GEN x)
4901 : {
4902 : long i;
4903 6055 : for (i = lg(v)-1; i > 1; i--) gel(v,i) = gel(v,i-1);
4904 1211 : gel(v,1) = x;
4905 1211 : }
4906 :
4907 : /* sort t_VEC of vector spaces by increasing dimension */
4908 : static GEN
4909 1071 : sort_by_dim(GEN v)
4910 : {
4911 1071 : long i, l = lg(v);
4912 1071 : GEN D = cgetg(l, t_VECSMALL);
4913 2576 : for (i = 1; i < l; i++) D[i] = lg(gmael(v,i,2));
4914 1071 : return vecpermute(v , vecsmall_indexsort(D));
4915 : }
4916 : static GEN
4917 1071 : split_starting_space(GEN mf)
4918 : {
4919 1071 : long d = MF_get_dim(mf), d2;
4920 1071 : GEN id = matid(d);
4921 1071 : switch(MF_get_space(mf))
4922 : {
4923 1064 : case mf_NEW:
4924 1064 : case mf_CUSP: return mkvec2(id, id);
4925 : }
4926 7 : d2 = lg(MF_get_S(mf))-1;
4927 7 : return mkvec2(vecslice(id, d-d2+1,d),
4928 : shallowconcat(zeromat(d2,d-d2),matid(d2)));
4929 : }
4930 : /* If dimlim > 0, keep only the dimension <= dimlim eigenspaces.
4931 : * See mfsplit for the meaning of flag. */
4932 : static GEN
4933 1470 : split_ii(GEN mf, long dimlim, long flag, GEN vSP, long *pnewd)
4934 : {
4935 : forprime_t iter;
4936 1470 : GEN CHI = MF_get_CHI(mf), empty = cgetg(1, t_VEC), mf0 = mf;
4937 : GEN NF, POLCYC, todosp, Tpbigvec, simplesp;
4938 1470 : long N = MF_get_N(mf), k = MF_get_k(mf);
4939 1470 : long ord, FC, NEWT, dimsimple = 0, newd = -1;
4940 1470 : const long NBH = 5, vz = 1;
4941 : ulong p;
4942 :
4943 1470 : switch(MF_get_space(mf))
4944 : {
4945 1183 : case mf_NEW: break;
4946 280 : case mf_CUSP:
4947 : case mf_FULL:
4948 : {
4949 : GEN CHIP;
4950 280 : if (k > 1) { mf0 = mfinittonew(mf); break; }
4951 259 : CHIP = mfchartoprimitive(CHI, NULL);
4952 259 : newd = lg(MF_get_S(mf))-1 - mfolddim_i(N, k, CHIP, vSP);
4953 259 : break;
4954 : }
4955 7 : default: pari_err_TYPE("mfsplit [space does not contain newspace]", mf);
4956 : return NULL; /*LCOV_EXCL_LINE*/
4957 : }
4958 1463 : if (newd < 0) newd = mf0? MF_get_dim(mf0): 0;
4959 1463 : *pnewd = newd;
4960 1463 : if (!newd) return mkvec2(cgetg(1, t_MAT), empty);
4961 :
4962 1071 : NEWT = (k > 1 && MF_get_space(mf0) == mf_NEW);
4963 1071 : todosp = mkvec( split_starting_space(mf0) );
4964 1071 : simplesp = empty;
4965 1071 : FC = mfcharconductor(CHI);
4966 1071 : ord = mfcharorder(CHI);
4967 1071 : if (ord <= 2) NF = POLCYC = NULL;
4968 : else
4969 : {
4970 203 : POLCYC = mfcharpol(CHI);
4971 203 : NF = nfinit(POLCYC,DEFAULTPREC);
4972 : }
4973 1071 : Tpbigvec = zerovec(NBH);
4974 1071 : u_forprime_init(&iter, 2, ULONG_MAX);
4975 1498 : while (dimsimple < newd && (p = u_forprime_next(&iter)))
4976 : {
4977 : GEN nextsp;
4978 : long ind;
4979 1498 : if (N % (p*p) == 0 && N/p % FC == 0) continue; /* T_p = 0 in this case */
4980 1211 : vecpush(Tpbigvec, NEWT? mfnewmathecke_p(mf0,p): mfheckemat_p(mf0,p));
4981 1211 : nextsp = empty;
4982 1596 : for (ind = 1; ind < lg(todosp); ind++)
4983 : {
4984 1323 : GEN tmp = gel(todosp, ind), fa, P, E, D, Tp, DTp;
4985 1323 : GEN A = gel(tmp, 1);
4986 1323 : GEN X = gel(tmp, 2);
4987 : long lP, i;
4988 1323 : tmp = findbestsplit(NF, Tpbigvec, A, X, dimlim, vz);
4989 1442 : if (!tmp) continue; /* nothing there */
4990 1260 : Tp = gel(tmp, 1);
4991 1260 : fa = gel(tmp, 2);
4992 1260 : P = gel(fa, 1);
4993 1260 : E = gel(fa, 2); lP = lg(P);
4994 : /* lP > 1 */
4995 1260 : if (DEBUGLEVEL) err_printf("Exponents = %Ps\n", E);
4996 1260 : if (lP == 2)
4997 : {
4998 868 : GEN P1 = gel(P,1);
4999 868 : long e1 = itos(gel(E,1)), d1 = degpol(P1);
5000 868 : if (e1 * d1 == lg(Tp)-1)
5001 : {
5002 819 : if (e1 > 1) nextsp = vec_append(nextsp, mkvec2(A,X));
5003 : else
5004 : { /* simple module */
5005 721 : simplesp = vec_append(simplesp, mkvec3(A,Tp,P1));
5006 959 : if ((dimsimple += d1) == newd) goto END;
5007 : }
5008 119 : continue;
5009 : }
5010 : }
5011 : /* Found splitting */
5012 441 : DTp = Q_remove_denom(Tp, &D);
5013 1204 : for (i = 1; i < lP; i++)
5014 : {
5015 1001 : GEN Ai, Xi, dXi, AAi, v, y, Pi = gel(P,i);
5016 1001 : Ai = RgX_RgM_eval(D? RgX_rescale(Pi,D): Pi, DTp);
5017 1001 : Ai = QabM_ker(Ai, POLCYC, ord);
5018 1001 : if (NF) Ai = nf_primpart(NF, Ai);
5019 :
5020 1001 : AAi = RgM_mul(A, Ai);
5021 : /* gives section, works on nonsquare matrices */
5022 1001 : Xi = QabM_pseudoinv(Ai, POLCYC, ord, &v, &dXi);
5023 1001 : Xi = RgM_Rg_div(Xi, dXi);
5024 1001 : y = gel(v,1);
5025 1001 : if (isint1(gel(E,i)))
5026 : {
5027 784 : GEN Tpi = RgM_mul(Xi, RgM_mul(rowpermute(Tp,y), Ai));
5028 784 : simplesp = vec_append(simplesp, mkvec3(AAi, Tpi, Pi));
5029 784 : if ((dimsimple += degpol(Pi)) == newd) goto END;
5030 : }
5031 : else
5032 : {
5033 217 : Xi = RgM_mul(Xi, rowpermute(X,y));
5034 217 : nextsp = vec_append(nextsp, mkvec2(AAi, Xi));
5035 : }
5036 : }
5037 : }
5038 273 : todosp = nextsp; if (lg(todosp) == 1) break;
5039 : }
5040 0 : END:
5041 1071 : if (DEBUGLEVEL) err_printf("end split, need to clean\n");
5042 1071 : return mfspclean(mf, mf0, NF, ord, sort_by_dim(simplesp), flag);
5043 : }
5044 : static GEN
5045 28 : dim_filter(GEN v, long dim)
5046 : {
5047 28 : GEN P = gel(v,2);
5048 28 : long j, l = lg(P);
5049 140 : for (j = 1; j < l; j++)
5050 126 : if (degpol(gel(P,j)) > dim)
5051 : {
5052 14 : v = mkvec2(vecslice(gel(v,1),1,j-1), vecslice(P,1,j-1));
5053 14 : break;
5054 : }
5055 28 : return v;
5056 : }
5057 : static long
5058 287 : dim_sum(GEN v)
5059 : {
5060 287 : GEN P = gel(v,2);
5061 287 : long j, l = lg(P), d = 0;
5062 707 : for (j = 1; j < l; j++) d += degpol(gel(P,j));
5063 287 : return d;
5064 : }
5065 : static GEN
5066 1148 : split_i(GEN mf, long dimlim, long flag)
5067 1148 : { long junk; return split_ii(mf, dimlim, flag, NULL, &junk); }
5068 : /* mf is either already split or output by mfinit. Splitting is done only for
5069 : * newspace except in weight 1. If flag = 0 (default) split completely.
5070 : * If flag = d > 0, only give the Galois polynomials in degree > d
5071 : * Flag is ignored if dimlim = 1. */
5072 : GEN
5073 98 : mfsplit(GEN mf0, long dimlim, long flag)
5074 : {
5075 98 : pari_sp av = avma;
5076 98 : GEN v, mf = checkMF_i(mf0);
5077 98 : if (!mf) pari_err_TYPE("mfsplit", mf0);
5078 98 : if ((v = obj_check(mf, MF_SPLIT)))
5079 28 : { if (dimlim) v = dim_filter(v, dimlim); }
5080 70 : else if (dimlim && (v = obj_check(mf, MF_SPLITN)))
5081 21 : { v = (itos(gel(v,1)) >= dimlim)? dim_filter(gel(v,2), dimlim): NULL; }
5082 98 : if (!v)
5083 : {
5084 : long newd;
5085 70 : v = split_ii(mf, dimlim, flag, NULL, &newd);
5086 70 : if (lg(v) == 1) obj_insert(mf, MF_SPLITN, mkvec2(utoi(dimlim), v));
5087 70 : else if (!flag)
5088 : {
5089 49 : if (dim_sum(v) == newd) obj_insert(mf, MF_SPLIT,v);
5090 21 : else obj_insert(mf, MF_SPLITN, mkvec2(utoi(dimlim), v));
5091 : }
5092 : }
5093 98 : return gerepilecopy(av, v);
5094 : }
5095 : static GEN
5096 231 : split(GEN mf) { return split_i(mf,0,0); }
5097 : GEN
5098 798 : MF_get_newforms(GEN mf) { return gel(obj_checkbuild(mf,MF_SPLIT,&split),1); }
5099 : GEN
5100 595 : MF_get_fields(GEN mf) { return gel(obj_checkbuild(mf,MF_SPLIT,&split),2); }
5101 :
5102 : /*************************************************************************/
5103 : /* Modular forms of Weight 1 */
5104 : /*************************************************************************/
5105 : /* S_1(G_0(N)), small N. Return 1 if definitely empty; return 0 if maybe
5106 : * nonempty */
5107 : static int
5108 16632 : wt1empty(long N)
5109 : {
5110 16632 : if (N <= 100) switch (N)
5111 : { /* nonempty [32/100] */
5112 5453 : case 23: case 31: case 39: case 44: case 46:
5113 : case 47: case 52: case 55: case 56: case 57:
5114 : case 59: case 62: case 63: case 68: case 69:
5115 : case 71: case 72: case 76: case 77: case 78:
5116 : case 79: case 80: case 83: case 84: case 87:
5117 : case 88: case 92: case 93: case 94: case 95:
5118 5453 : case 99: case 100: return 0;
5119 3549 : default: return 1;
5120 : }
5121 7630 : if (N <= 600) switch(N)
5122 : { /* empty [111/500] */
5123 336 : case 101: case 102: case 105: case 106: case 109:
5124 : case 113: case 121: case 122: case 123: case 125:
5125 : case 130: case 134: case 137: case 146: case 149:
5126 : case 150: case 153: case 157: case 162: case 163:
5127 : case 169: case 170: case 173: case 178: case 181:
5128 : case 182: case 185: case 187: case 193: case 194:
5129 : case 197: case 202: case 205: case 210: case 218:
5130 : case 221: case 226: case 233: case 241: case 242:
5131 : case 245: case 246: case 250: case 257: case 265:
5132 : case 267: case 269: case 274: case 277: case 281:
5133 : case 289: case 293: case 298: case 305: case 306:
5134 : case 313: case 314: case 317: case 326: case 337:
5135 : case 338: case 346: case 349: case 353: case 361:
5136 : case 362: case 365: case 369: case 370: case 373:
5137 : case 374: case 377: case 386: case 389: case 394:
5138 : case 397: case 401: case 409: case 410: case 421:
5139 : case 425: case 427: case 433: case 442: case 449:
5140 : case 457: case 461: case 466: case 481: case 482:
5141 : case 485: case 490: case 493: case 509: case 514:
5142 : case 521: case 530: case 533: case 534: case 538:
5143 : case 541: case 545: case 554: case 557: case 562:
5144 : case 565: case 569: case 577: case 578: case 586:
5145 336 : case 593: return 1;
5146 6979 : default: return 0;
5147 : }
5148 315 : return 0;
5149 : }
5150 :
5151 : static GEN
5152 28 : initwt1trace(GEN mf)
5153 : {
5154 28 : GEN S = MF_get_S(mf), v, H;
5155 : long l, i;
5156 28 : if (lg(S) == 1) return mftrivial();
5157 28 : H = mfheckemat(mf, Mindex_as_coef(mf));
5158 28 : l = lg(H); v = cgetg(l, t_VEC);
5159 63 : for (i = 1; i < l; i++) gel(v,i) = gtrace(gel(H,i));
5160 28 : v = Minv_RgC_mul(MF_get_Minv(mf), v);
5161 28 : return mflineardiv_linear(S, v, 1);
5162 : }
5163 : static GEN
5164 21 : initwt1newtrace(GEN mf)
5165 : {
5166 21 : GEN v, D, S, Mindex, CHI = MF_get_CHI(mf);
5167 21 : long FC, lD, i, sb, N1, N2, lM, N = MF_get_N(mf);
5168 21 : CHI = mfchartoprimitive(CHI, &FC);
5169 21 : if (N % FC || mfcharparity(CHI) == 1) return mftrivial();
5170 21 : D = mydivisorsu(N/FC); lD = lg(D);
5171 21 : S = MF_get_S(mf);
5172 21 : if (lg(S) == 1) return mftrivial();
5173 21 : N2 = newd_params2(N);
5174 21 : N1 = N / N2;
5175 21 : Mindex = MF_get_Mindex(mf);
5176 21 : lM = lg(Mindex);
5177 21 : sb = Mindex[lM-1];
5178 21 : v = zerovec(sb+1);
5179 42 : for (i = 1; i < lD; i++)
5180 : {
5181 21 : long M = FC*D[i], j;
5182 21 : GEN tf = initwt1trace(M == N? mf: mfinit_Nkchi(M, 1, CHI, mf_CUSP, 0));
5183 : GEN listd, w;
5184 21 : if (mf_get_type(tf) == t_MF_CONST) continue;
5185 21 : w = mfcoefs_i(tf, sb, 1);
5186 21 : if (M == N) { v = gadd(v, w); continue; }
5187 0 : listd = mydivisorsu(u_ppo(ugcd(N/M, N1), FC));
5188 0 : for (j = 1; j < lg(listd); j++)
5189 : {
5190 0 : long d = listd[j], d2 = d*d; /* coprime to FC */
5191 0 : GEN dk = mfchareval(CHI, d);
5192 0 : long NMd = N/(M*d), m;
5193 0 : for (m = 1; m <= sb/d2; m++)
5194 : {
5195 0 : long be = mubeta2(NMd, m);
5196 0 : if (be)
5197 : {
5198 0 : GEN c = gmul(dk, gmulsg(be, gel(w, m+1)));
5199 0 : long n = m*d2;
5200 0 : gel(v, n+1) = gadd(gel(v, n+1), c);
5201 : }
5202 : }
5203 : }
5204 : }
5205 21 : if (gequal0(gel(v,2))) return mftrivial();
5206 21 : v = vecpermute(v,Mindex);
5207 21 : v = Minv_RgC_mul(MF_get_Minv(mf), v);
5208 21 : return mflineardiv_linear(S, v, 1);
5209 : }
5210 :
5211 : /* i*p + 1, i*p < lim corresponding to a_p(f_j), a_{2p}(f_j)... */
5212 : static GEN
5213 1834 : pindices(long p, long lim)
5214 : {
5215 1834 : GEN v = cgetg(lim, t_VECSMALL);
5216 : long i, ip;
5217 22190 : for (i = 1, ip = p + 1; ip < lim; i++, ip += p) v[i] = ip;
5218 1834 : setlg(v, i); return v;
5219 : }
5220 :
5221 : /* assume !wt1empty(N), in particular N>25 */
5222 : /* Returns [[lim,p], mf (weight 2), p*lim x dim matrix] */
5223 : static GEN
5224 1834 : mf1_pre(long N)
5225 : {
5226 : pari_timer tt;
5227 : GEN mf, v, L, I, M, Minv, den;
5228 : long B, lim, LIM, p;
5229 :
5230 1834 : if (DEBUGLEVEL) timer_start(&tt);
5231 1834 : mf = mfinit_Nkchi(N, 2, mfchartrivial(), mf_CUSP, 0);
5232 1834 : if (DEBUGLEVEL)
5233 0 : timer_printf(&tt, "mf1basis [pre]: S_2(%ld), dim = %ld",
5234 : N, MF_get_dim(mf));
5235 1834 : M = MF_get_M(mf); Minv = MF_get_Minv(mf); den = gel(Minv,2);
5236 1834 : B = mfsturm_mf(mf);
5237 1834 : if (uisprime(N))
5238 : {
5239 392 : lim = 2 * MF_get_dim(mf); /* ensure mfstabiter's first kernel ~ square */
5240 392 : p = 2;
5241 : }
5242 : else
5243 : {
5244 : forprime_t S;
5245 1442 : u_forprime_init(&S, 2, N);
5246 2576 : while ((p = u_forprime_next(&S)))
5247 2576 : if (N % p) break;
5248 1442 : lim = B + 1;
5249 : }
5250 1834 : LIM = (N & (N - 1))? 2 * lim: 3 * lim; /* N power of 2 ? */
5251 1834 : L = mkvecsmall4(lim, LIM, mfsturmNk(N,1), p);
5252 1834 : M = bhnmat_extend_nocache(M, N, LIM-1, 1, MF_get_S(mf));
5253 1834 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis [pre]: bnfmat_extend");
5254 1834 : v = pindices(p, LIM);
5255 1834 : if (!LIM) return mkvec4(L, mf, M, v);
5256 1834 : I = RgM_Rg_div(ZM_mul(rowslice(M, B+2, LIM), gel(Minv,1)), den);
5257 1834 : I = Q_remove_denom(I, &den);
5258 1834 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis [prec]: Iden");
5259 1834 : return mkvec5(L, mf, M, v, mkvec2(I, den));
5260 : }
5261 :
5262 : /* lg(A) > 1, E a t_POL */
5263 : static GEN
5264 686 : mfmatsermul(GEN A, GEN E)
5265 : {
5266 686 : long j, l = lg(A), r = nbrows(A);
5267 686 : GEN M = cgetg(l, t_MAT);
5268 686 : E = RgXn_red_shallow(E, r+1);
5269 5866 : for (j = 1; j < l; j++)
5270 : {
5271 5180 : GEN c = RgV_to_RgX(gel(A,j), 0);
5272 5180 : gel(M, j) = RgX_to_RgC(RgXn_mul(c, E, r+1), r);
5273 : }
5274 686 : return M;
5275 : }
5276 : /* lg(Ap) > 1, Ep an Flxn */
5277 : static GEN
5278 1141 : mfmatsermul_Fl(GEN Ap, GEN Ep, ulong p)
5279 : {
5280 1141 : long j, l = lg(Ap), r = nbrows(Ap);
5281 1141 : GEN M = cgetg(l, t_MAT);
5282 42630 : for (j = 1; j < l; j++)
5283 : {
5284 41489 : GEN c = Flv_to_Flx(gel(Ap,j), 0);
5285 41489 : gel(M,j) = Flx_to_Flv(Flxn_mul(c, Ep, r+1, p), r);
5286 : }
5287 1141 : return M;
5288 : }
5289 :
5290 : /* CHI mod F | N, return mfchar of modulus N.
5291 : * FIXME: wasteful, G should be precomputed */
5292 : static GEN
5293 13048 : mfcharinduce(GEN CHI, long N)
5294 : {
5295 : GEN G, chi;
5296 13048 : if (mfcharmodulus(CHI) == N) return CHI;
5297 1463 : G = znstar0(utoipos(N), 1);
5298 1463 : chi = zncharinduce(gel(CHI,1), gel(CHI,2), G);
5299 1463 : CHI = leafcopy(CHI);
5300 1463 : gel(CHI,1) = G;
5301 1463 : gel(CHI,2) = chi; return CHI;
5302 : }
5303 :
5304 : static GEN
5305 3983 : gmfcharno(GEN CHI)
5306 : {
5307 3983 : GEN G = gel(CHI,1), chi = gel(CHI,2);
5308 3983 : return mkintmod(znconreyexp(G, chi), znstar_get_N(G));
5309 : }
5310 : static long
5311 13678 : mfcharno(GEN CHI)
5312 : {
5313 13678 : GEN n = znconreyexp(gel(CHI,1), gel(CHI,2));
5314 13678 : return itou(n);
5315 : }
5316 :
5317 : /* return k such that minimal mfcharacter in Galois orbit of CHI is CHI^k */
5318 : static long
5319 12138 : mfconreyminimize(GEN CHI)
5320 : {
5321 12138 : GEN G = gel(CHI,1), cyc, chi;
5322 12138 : cyc = ZV_to_zv(znstar_get_cyc(G));
5323 12138 : chi = ZV_to_zv(znconreychar(G, gel(CHI,2)));
5324 12138 : return zv_cyc_minimize(cyc, chi, coprimes_zv(mfcharorder(CHI)));
5325 : }
5326 :
5327 : /* find scalar c such that first nonzero entry of c*v is 1; return c*v */
5328 : static GEN
5329 2065 : RgV_normalize(GEN v, GEN *pc)
5330 : {
5331 2065 : long i, l = lg(v);
5332 5313 : for (i = 1; i < l; i++)
5333 : {
5334 5313 : GEN c = gel(v,i);
5335 5313 : if (!gequal0(c))
5336 : {
5337 2065 : if (gequal1(c)) break;
5338 679 : *pc = ginv(c); return RgV_Rg_mul(v, *pc);
5339 : }
5340 : }
5341 1386 : *pc = gen_1; return v;
5342 : }
5343 : /* pS != NULL; dim > 0 */
5344 : static GEN
5345 784 : mftreatdihedral(long N, GEN DIH, GEN POLCYC, long ordchi, GEN *pS)
5346 : {
5347 784 : long l = lg(DIH), lim = mfsturmNk(N, 1), i;
5348 784 : GEN Minv, C = cgetg(l, t_VEC), M = cgetg(l, t_MAT);
5349 2436 : for (i = 1; i < l; i++)
5350 : {
5351 1652 : GEN c, v = mfcoefs_i(gel(DIH,i), lim, 1);
5352 1652 : gel(M,i) = RgV_normalize(v, &c);
5353 1652 : gel(C,i) = Rg_col_ei(c, l-1, i);
5354 : }
5355 784 : Minv = gel(mfclean(M,POLCYC,ordchi,0),2);
5356 784 : M = RgM_Minv_mul(M, Minv);
5357 784 : C = RgM_Minv_mul(C, Minv);
5358 784 : *pS = vecmflinear(DIH, C); return M;
5359 : }
5360 :
5361 : /* same mode a maximal ideal above q */
5362 : static GEN
5363 2408 : Tpmod(GEN Ap, GEN A, ulong chip, long p, ulong q)
5364 : {
5365 2408 : GEN B = leafcopy(Ap);
5366 2408 : long i, ip, l = lg(B);
5367 86345 : for (i = 1, ip = p; ip < l; i++, ip += p)
5368 83937 : B[ip] = Fl_add(B[ip], Fl_mul(A[i], chip, q), q);
5369 2408 : return B;
5370 : }
5371 : /* Tp(f_1), ..., Tp(f_d) mod q */
5372 : static GEN
5373 301 : matTpmod(GEN xp, GEN x, ulong chip, long p, ulong q)
5374 2709 : { pari_APPLY_same(Tpmod(gel(xp,i), gel(x,i), chip, p, q)); }
5375 :
5376 : /* Ap[i] = a_{p*i}(F), A[i] = a_i(F), i = 1..lim
5377 : * Tp(f)[n] = a_{p*n}(f) + chi(p) a_{n/p}(f) * 1_{p | n} */
5378 : static GEN
5379 469 : Tp(GEN Ap, GEN A, GEN chip, long p)
5380 : {
5381 469 : GEN B = leafcopy(Ap);
5382 469 : long i, ip, l = lg(B);
5383 12915 : for (i = 1, ip = p; ip < l; i++, ip += p)
5384 12446 : gel(B,ip) = gadd(gel(B,ip), gmul(gel(A,i), chip));
5385 469 : return B;
5386 : }
5387 : /* Tp(f_1), ..., Tp(f_d) */
5388 : static GEN
5389 56 : matTp(GEN xp, GEN x, GEN chip, long p)
5390 525 : { pari_APPLY_same(Tp(gel(xp,i), gel(x,i), chip, p)); }
5391 :
5392 : static GEN
5393 378 : _RgXQM_mul(GEN x, GEN y, GEN T)
5394 378 : { return T? RgXQM_mul(x, y, T): RgM_mul(x, y); }
5395 : /* largest T-stable Q(CHI)-subspace of Q(CHI)-vector space spanned by columns
5396 : * of A */
5397 : static GEN
5398 28 : mfstabiter(GEN *pC, GEN A0, GEN chip, GEN TMP, GEN P, long ordchi)
5399 : {
5400 28 : GEN A, Ap, vp = gel(TMP,4), C = NULL;
5401 28 : long i, lA, lim1 = gel(TMP,1)[3], p = gel(TMP,1)[4];
5402 : pari_timer tt;
5403 :
5404 28 : Ap = rowpermute(A0, vp);
5405 28 : A = rowslice(A0, 2, nbrows(Ap)+1); /* remove a0 */
5406 : for(;;)
5407 28 : {
5408 56 : GEN R = shallowconcat(matTp(Ap, A, chip, p), A);
5409 56 : GEN B = QabM_ker(R, P, ordchi);
5410 56 : long lB = lg(B);
5411 56 : if (DEBUGLEVEL)
5412 0 : timer_printf(&tt, "mf1basis: Hecke intersection (dim %ld)", lB-1);
5413 56 : if (lB == 1) return NULL;
5414 56 : lA = lg(A); if (lB == lA) break;
5415 28 : B = rowslice(B, 1, lA-1);
5416 28 : Ap = _RgXQM_mul(Ap, B, P);
5417 28 : A = _RgXQM_mul(A, B, P);
5418 28 : C = C? _RgXQM_mul(C, B, P): B;
5419 : }
5420 28 : if (nbrows(A) < lim1)
5421 : {
5422 14 : A0 = rowslice(A0, 2, lim1);
5423 14 : A = C? _RgXQM_mul(A0, C, P): A0;
5424 : }
5425 : else /* all needed coefs computed */
5426 14 : A = rowslice(A, 1, lim1-1);
5427 28 : if (*pC) C = C? _RgXQM_mul(*pC, C, P): *pC;
5428 : /* put back a0 */
5429 119 : for (i = 1; i < lA; i++) gel(A,i) = vec_prepend(gel(A,i), gen_0);
5430 28 : *pC = C; return A;
5431 : }
5432 :
5433 : static long
5434 252 : mfstabitermod(GEN A, GEN vp, ulong chip, long p, ulong q)
5435 : {
5436 252 : GEN Ap, C = NULL;
5437 252 : Ap = rowpermute(A, vp);
5438 252 : A = rowslice(A, 2, nbrows(Ap)+1);
5439 : while (1)
5440 49 : {
5441 301 : GEN Rp = shallowconcat(matTpmod(Ap, A, chip, p, q), A);
5442 301 : GEN B = Flm_ker(Rp, q);
5443 301 : long lA = lg(A), lB = lg(B);
5444 301 : if (lB == 1) return 0;
5445 266 : if (lB == lA) return lA-1;
5446 49 : B = rowslice(B, 1, lA-1);
5447 49 : Ap = Flm_mul(Ap, B, q);
5448 49 : A = Flm_mul(A, B, q);
5449 49 : C = C? Flm_mul(C, B, q): B;
5450 : }
5451 : }
5452 :
5453 : static GEN
5454 595 : mfcharinv_i(GEN CHI)
5455 : {
5456 595 : GEN G = gel(CHI,1);
5457 595 : CHI = leafcopy(CHI); gel(CHI,2) = zncharconj(G, gel(CHI,2)); return CHI;
5458 : }
5459 :
5460 : /* upper bound dim S_1(Gamma_0(N),chi) performing the linear algebra mod p */
5461 : static long
5462 595 : mf1dimmod(GEN E1, GEN E, GEN chip, long ordchi, long dih, GEN TMP)
5463 : {
5464 595 : GEN E1i, A, vp, mf, C = NULL;
5465 595 : ulong q, r = QabM_init(ordchi, &q);
5466 : long lim, LIM, p;
5467 :
5468 595 : LIM = gel(TMP,1)[2]; lim = gel(TMP,1)[1];
5469 595 : mf= gel(TMP,2);
5470 595 : A = gel(TMP,3);
5471 595 : A = QabM_to_Flm(A, r, q);
5472 595 : E1 = QabX_to_Flx(E1, r, q);
5473 595 : E1i = Flxn_inv(E1, nbrows(A), q);
5474 595 : if (E)
5475 : {
5476 574 : GEN Iden = gel(TMP,5), I = gel(Iden,1), den = gel(Iden,2);
5477 574 : GEN Mindex = MF_get_Mindex(mf), F = rowslice(A, 1, LIM);
5478 574 : GEN E1ip = Flxn_red(E1i, LIM);
5479 574 : ulong d = den? umodiu(den, q): 1;
5480 574 : long i, nE = lg(E) - 1;
5481 : pari_sp av;
5482 :
5483 574 : I = ZM_to_Flm(I, q);
5484 574 : if (d != 1) I = Flm_Fl_mul(I, Fl_inv(d, q), q);
5485 574 : av = avma;
5486 1120 : for (i = 1; i <= nE; i++)
5487 : {
5488 889 : GEN e = Flxn_mul(E1ip, QabX_to_Flx(gel(E,i), r, q), LIM, q);
5489 889 : GEN B = mfmatsermul_Fl(F, e, q), z;
5490 889 : GEN B2 = Flm_mul(I, rowpermute(B, Mindex), q);
5491 889 : B = rowslice(B, lim+1,LIM);
5492 889 : z = Flm_ker(Flm_sub(B2, B, q), q);
5493 889 : if (lg(z)-1 == dih) return dih;
5494 546 : C = C? Flm_mul(C, z, q): z;
5495 546 : F = Flm_mul(F, z, q);
5496 546 : gerepileall(av, 2, &F,&C);
5497 : }
5498 231 : A = F;
5499 : }
5500 : /* use Schaeffer */
5501 252 : p = gel(TMP,1)[4]; vp = gel(TMP,4);
5502 252 : A = mfmatsermul_Fl(A, E1i, q);
5503 252 : return mfstabitermod(A, vp, Qab_to_Fl(chip, r, q), p, q);
5504 : }
5505 :
5506 : static GEN
5507 224 : mf1intermat(GEN A, GEN Mindex, GEN e, GEN Iden, long lim, GEN POLCYC)
5508 : {
5509 224 : long j, l = lg(A), LIM = nbrows(A);
5510 224 : GEN I = gel(Iden,1), den = gel(Iden,2), B = cgetg(l, t_MAT);
5511 :
5512 5257 : for (j = 1; j < l; j++)
5513 : {
5514 5033 : pari_sp av = avma;
5515 5033 : GEN c = RgV_to_RgX(gel(A,j), 0), c1, c2;
5516 5033 : c = RgX_to_RgC(RgXn_mul(c, e, LIM), LIM);
5517 5033 : if (POLCYC) c = liftpol_shallow(c);
5518 5033 : c1 = vecslice(c, lim+1, LIM);
5519 5033 : if (den) c1 = RgC_Rg_mul(c1, den);
5520 5033 : c2 = RgM_RgC_mul(I, vecpermute(c, Mindex));
5521 5033 : gel(B, j) = gerepileupto(av, RgC_sub(c2, c1));
5522 : }
5523 224 : return B;
5524 : }
5525 : /* Compute the full S_1(\G_0(N),\chi); return NULL if space is empty; else
5526 : * if pS is NULL, return stoi(dim), where dim is the dimension; else *pS is
5527 : * set to a vector of forms making up a basis, and return the matrix of their
5528 : * Fourier expansions. pdih gives the dimension of the subspace generated by
5529 : * dihedral forms; TMP is from mf1_pre or NULL. */
5530 : static GEN
5531 11284 : mf1basis(long N, GEN CHI, GEN TMP, GEN vSP, GEN *pS, long *pdih)
5532 : {
5533 11284 : GEN E = NULL, EB, E1, E1i, dE1i, mf, A, C, POLCYC, DIH, Minv, chip;
5534 11284 : long nE = 0, p, LIM, lim, lim1, i, lA, dimp, ordchi, dih;
5535 : pari_timer tt;
5536 : pari_sp av;
5537 :
5538 11284 : if (pdih) *pdih = 0;
5539 11284 : if (pS) *pS = NULL;
5540 11284 : if (wt1empty(N) || mfcharparity(CHI) != -1) return NULL;
5541 10990 : ordchi = mfcharorder(CHI);
5542 10990 : if (uisprime(N) && ordchi > 4) return NULL;
5543 10962 : if (pS)
5544 : {
5545 3857 : DIH = mfdihedralcusp(N, CHI, vSP);
5546 3857 : dih = lg(DIH) - 1;
5547 : }
5548 : else
5549 : {
5550 7105 : DIH = NULL;
5551 7105 : dih = mfdihedralcuspdim(N, CHI, vSP);
5552 : }
5553 10962 : POLCYC = (ordchi <= 2)? NULL: mfcharpol(CHI);
5554 10962 : if (pdih) *pdih = dih;
5555 10962 : if (N <= 600) switch(N)
5556 : {
5557 : long m;
5558 126 : case 219: case 273: case 283: case 331: case 333: case 344: case 416:
5559 : case 438: case 468: case 491: case 504: case 546: case 553: case 563:
5560 : case 566: case 581: case 592:
5561 126 : break; /* one chi with both exotic and dihedral forms */
5562 9499 : default: /* only dihedral forms */
5563 9499 : if (!dih) return NULL;
5564 : /* fall through */
5565 : case 124: case 133: case 148: case 171: case 201: case 209: case 224:
5566 : case 229: case 248: case 261: case 266: case 288: case 296: case 301:
5567 : case 309: case 325: case 342: case 371: case 372: case 380: case 399:
5568 : case 402: case 403: case 404: case 408: case 418: case 432: case 444:
5569 : case 448: case 451: case 453: case 458: case 496: case 497: case 513:
5570 : case 522: case 527: case 532: case 576: case 579:
5571 : /* no chi with both exotic and dihedral; one chi with exotic forms */
5572 3248 : if (dih)
5573 : {
5574 2338 : if (!pS) return utoipos(dih);
5575 728 : return mftreatdihedral(N, DIH, POLCYC, ordchi, pS) ;
5576 : }
5577 910 : m = mfcharno(mfcharinduce(CHI,N));
5578 910 : if (N == 124 && (m != 67 && m != 87)) return NULL;
5579 784 : if (N == 133 && (m != 83 && m !=125)) return NULL;
5580 490 : if (N == 148 && (m !=105 && m !=117)) return NULL;
5581 364 : if (N == 171 && (m != 94 && m !=151)) return NULL;
5582 364 : if (N == 201 && (m != 29 && m !=104)) return NULL;
5583 364 : if (N == 209 && (m != 87 && m !=197)) return NULL;
5584 364 : if (N == 224 && (m != 95 && m !=191)) return NULL;
5585 364 : if (N == 229 && (m !=107 && m !=122)) return NULL;
5586 364 : if (N == 248 && (m != 87 && m !=191)) return NULL;
5587 273 : if (N == 261 && (m != 46 && m !=244)) return NULL;
5588 273 : if (N == 266 && (m != 83 && m !=125)) return NULL;
5589 273 : if (N == 288 && (m != 31 && m !=223)) return NULL;
5590 273 : if (N == 296 && (m !=105 && m !=265)) return NULL;
5591 : }
5592 595 : if (DEBUGLEVEL)
5593 0 : err_printf("mf1basis: start character %Ps, conductor = %ld, order = %ld\n",
5594 : gmfcharno(CHI), mfcharconductor(CHI), ordchi);
5595 595 : if (!TMP) TMP = mf1_pre(N);
5596 595 : lim = gel(TMP,1)[1]; LIM = gel(TMP,1)[2]; lim1 = gel(TMP,1)[3];
5597 595 : p = gel(TMP,1)[4];
5598 595 : mf = gel(TMP,2);
5599 595 : A = gel(TMP,3);
5600 595 : EB = mfeisensteinbasis(N, 1, mfcharinv_i(CHI));
5601 595 : nE = lg(EB) - 1;
5602 595 : E1 = RgV_to_RgX(mftocol(gel(EB,1), LIM-1, 1), 0); /* + O(x^LIM) */
5603 595 : if (--nE)
5604 574 : E = RgM_to_RgXV(mfvectomat(vecslice(EB, 2, nE+1), LIM-1, 1), 0);
5605 595 : chip = mfchareval(CHI, p); /* != 0 */
5606 595 : if (DEBUGLEVEL) timer_start(&tt);
5607 595 : av = avma; dimp = mf1dimmod(E1, E, chip, ordchi, dih, TMP);
5608 595 : set_avma(av);
5609 595 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: dim mod p is %ld", dimp);
5610 595 : if (!dimp) return NULL;
5611 280 : if (!pS) return utoi(dimp);
5612 224 : if (dimp == dih) return mftreatdihedral(N, DIH, POLCYC, ordchi, pS);
5613 168 : E1i = RgXn_inv(E1, LIM); /* E[1] does not vanish at oo */
5614 168 : if (POLCYC) E1i = liftpol_shallow(E1i);
5615 168 : E1i = Q_remove_denom(E1i, &dE1i);
5616 168 : if (DEBUGLEVEL)
5617 : {
5618 0 : GEN a0 = gel(E1,2);
5619 0 : if (typ(a0) == t_POLMOD) a0 = gnorm(a0);
5620 0 : a0 = Q_abs_shallow(a0);
5621 0 : timer_printf(&tt, "mf1basis: invert E; norm(a0(E)) = %Ps", a0);
5622 : }
5623 168 : C = NULL;
5624 168 : if (nE)
5625 : { /* mf attached to S2(N), fi = mfbasis(mf)
5626 : * M = coefs(f1,...,fd) up to LIM
5627 : * F = coefs(F1,...,FD) = M * C, for some matrix C over Q(chi),
5628 : * initially 1, eventually giving \cap_E S2 / E; D <= d.
5629 : * B = coefs(E/E1 F1, .., E/E1 FD); we want X in Q(CHI)^d and
5630 : * Y in Q(CHI)^D such that
5631 : * B * X = M * Y, i.e. Minv * rowpermute(B, Mindex * X) = Y
5632 : *(B - I * rowpermute(B, Mindex)) * X = 0.
5633 : * where I = M * Minv. Rows of (B - I * ...) are 0 up to lim so
5634 : * are not included */
5635 154 : GEN Mindex = MF_get_Mindex(mf), Iden = gel(TMP,5);
5636 : pari_timer TT;
5637 154 : pari_sp av = avma;
5638 154 : if (DEBUGLEVEL) timer_start(&TT);
5639 238 : for (i = 1; i <= nE; i++)
5640 : {
5641 224 : pari_sp av2 = avma;
5642 : GEN e, z, B;
5643 :
5644 224 : e = Q_primpart(RgXn_mul(E1i, gel(E,i), LIM));
5645 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: E[%ld] / E[1]", i+1);
5646 : /* the first time A is over Z and it is more efficient to lift than
5647 : * to let RgXn_mul use Kronecker's trick */
5648 224 : if (POLCYC && i == 1) e = liftpol_shallow(e);
5649 224 : B = mf1intermat(A, Mindex, e, Iden, lim, i == 1? NULL: POLCYC);
5650 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: ... intermat");
5651 224 : z = gerepileupto(av2, QabM_ker(B, POLCYC, ordchi));
5652 224 : if (DEBUGLEVEL)
5653 0 : timer_printf(&TT, "mf1basis: ... kernel (dim %ld)",lg(z)-1);
5654 224 : if (lg(z) == 1) return NULL;
5655 224 : if (lg(z) == lg(A)) { set_avma(av2); continue; } /* no progress */
5656 224 : C = C? _RgXQM_mul(C, z, POLCYC): z;
5657 224 : A = _RgXQM_mul(A, z, POLCYC);
5658 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: ... updates");
5659 224 : if (lg(z)-1 == dimp) break;
5660 84 : if (gc_needed(av, 1))
5661 : {
5662 0 : if (DEBUGMEM > 1) pari_warn(warnmem,"mf1basis i = %ld", i);
5663 0 : gerepileall(av, 2, &A, &C);
5664 : }
5665 : }
5666 154 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: intersection [total]");
5667 : }
5668 168 : lA = lg(A);
5669 168 : if (lA-1 == dimp)
5670 : {
5671 140 : A = mfmatsermul(rowslice(A, 1, lim1), E1i);
5672 140 : if (POLCYC) A = RgXQM_red(A, POLCYC);
5673 140 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: matsermul [1]");
5674 : }
5675 : else
5676 : {
5677 28 : A = mfmatsermul(A, E1i);
5678 28 : if (POLCYC) A = RgXQM_red(A, POLCYC);
5679 28 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: matsermul [2]");
5680 28 : A = mfstabiter(&C, A, chip, TMP, POLCYC, ordchi);
5681 28 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: Hecke stability");
5682 28 : if (!A) return NULL;
5683 : }
5684 168 : if (dE1i) C = RgM_Rg_mul(C, dE1i);
5685 168 : if (POLCYC)
5686 : {
5687 147 : A = QXQM_to_mod_shallow(A, POLCYC);
5688 147 : C = QXQM_to_mod_shallow(C, POLCYC);
5689 : }
5690 168 : lA = lg(A);
5691 581 : for (i = 1; i < lA; i++)
5692 : {
5693 413 : GEN c, v = gel(A,i);
5694 413 : gel(A,i) = RgV_normalize(v, &c);
5695 413 : gel(C,i) = RgC_Rg_mul(gel(C,i), c);
5696 : }
5697 168 : Minv = gel(mfclean(A, POLCYC, ordchi, 0), 2);
5698 168 : A = RgM_Minv_mul(A, Minv);
5699 168 : C = RgM_Minv_mul(C, Minv);
5700 168 : *pS = vecmflineardiv0(MF_get_S(mf), C, gel(EB,1));
5701 168 : return A;
5702 : }
5703 :
5704 : static void
5705 413 : MF_set_space(GEN mf, long x) { gmael(mf,1,4) = utoi(x); }
5706 : static GEN
5707 252 : mf1_cusptonew(GEN mf, GEN vSP)
5708 : {
5709 252 : const long vy = 1;
5710 : long i, lP, dSnew, ct;
5711 252 : GEN vP, F, S, Snew, vF, v = split_ii(mf, 0, 0, vSP, &i);
5712 :
5713 252 : F = gel(v,1);
5714 252 : vP= gel(v,2); lP = lg(vP);
5715 252 : if (lP == 1) { obj_insert(mf, MF_SPLIT, v); return NULL; }
5716 238 : MF_set_space(mf, mf_NEW);
5717 238 : S = MF_get_S(mf);
5718 238 : dSnew = dim_sum(v);
5719 238 : Snew = cgetg(dSnew + 1, t_VEC); ct = 0;
5720 238 : vF = cgetg(lP, t_MAT);
5721 546 : for (i = 1; i < lP; i++)
5722 : {
5723 308 : GEN V, P = gel(vP,i), f = liftpol_shallow(gel(F,i));
5724 308 : long j, d = degpol(P);
5725 308 : gel(vF,i) = V = zerocol(dSnew);
5726 308 : if (d == 1)
5727 : {
5728 140 : gel(Snew, ct+1) = mflineardiv_linear(S, f, 0);
5729 140 : gel(V, ct+1) = gen_1;
5730 : }
5731 : else
5732 : {
5733 168 : f = RgXV_to_RgM(f,d);
5734 511 : for (j = 1; j <= d; j++)
5735 : {
5736 343 : gel(Snew, ct+j) = mflineardiv_linear(S, row(f,j), 0);
5737 343 : gel(V, ct+j) = mkpolmod(pol_xn(j-1,vy), P);
5738 : }
5739 : }
5740 308 : ct += d;
5741 : }
5742 238 : obj_insert(mf, MF_SPLIT, mkvec2(vF, vP));
5743 238 : gel(mf,3) = Snew; return mf;
5744 : }
5745 : static GEN
5746 3969 : mf1init(long N, GEN CHI, GEN TMP, GEN vSP, long space, long flraw)
5747 : {
5748 3969 : GEN mf, mf1, S, M = mf1basis(N, CHI, TMP, vSP, &S, NULL);
5749 3969 : if (!M) return NULL;
5750 952 : mf1 = mkvec4(stoi(N), gen_1, CHI, utoi(mf_CUSP));
5751 952 : mf = mkmf(mf1, cgetg(1,t_VEC), S, gen_0, NULL);
5752 952 : if (space == mf_NEW)
5753 : {
5754 252 : gel(mf,5) = mfcleanCHI(M,CHI, 0);
5755 252 : mf = mf1_cusptonew(mf, vSP); if (!mf) return NULL;
5756 238 : if (!flraw) M = mfcoefs_mf(mf, mfsturmNk(N,1)+1, 1);
5757 : }
5758 938 : gel(mf,5) = flraw? zerovec(3): mfcleanCHI(M, CHI, 0);
5759 938 : return mf;
5760 : }
5761 :
5762 : static GEN
5763 1022 : mfEMPTY(GEN mf1)
5764 : {
5765 1022 : GEN Minv = mkMinv(cgetg(1,t_MAT), NULL,NULL,NULL);
5766 1022 : GEN M = mkvec3(cgetg(1,t_VECSMALL), Minv, cgetg(1,t_MAT));
5767 1022 : return mkmf(mf1, cgetg(1,t_VEC), cgetg(1,t_VEC), cgetg(1,t_VEC), M);
5768 : }
5769 : static GEN
5770 616 : mfEMPTYall(long N, GEN gk, GEN vCHI, long space)
5771 : {
5772 : long i, l;
5773 : GEN v, gN, gs;
5774 616 : if (!vCHI) return cgetg(1, t_VEC);
5775 14 : gN = utoipos(N); gs = utoi(space);
5776 14 : l = lg(vCHI); v = cgetg(l, t_VEC);
5777 42 : for (i = 1; i < l; i++) gel(v,i) = mfEMPTY(mkvec4(gN,gk,gel(vCHI,i),gs));
5778 14 : return v;
5779 : }
5780 :
5781 : static GEN
5782 3983 : fmt_dim(GEN CHI, long d, long dih)
5783 3983 : { return mkvec4(gmfcharorder(CHI), gmfcharno(CHI), utoi(d), stoi(dih)); }
5784 : /* merge two vector of fmt_dim's for the same vector of characters. If CHI
5785 : * is not NULL, remove dim-0 spaces and add character from CHI */
5786 : static GEN
5787 7 : merge_dims(GEN V, GEN W, GEN CHI)
5788 : {
5789 7 : long i, j, id, l = lg(V);
5790 7 : GEN A = cgetg(l, t_VEC);
5791 7 : if (l == 1) return A;
5792 7 : id = CHI? 1: 3;
5793 21 : for (i = j = 1; i < l; i++)
5794 : {
5795 14 : GEN v = gel(V,i), w = gel(W,i);
5796 14 : long dv = itou(gel(v,id)), dvh = itou(gel(v,id+1)), d;
5797 14 : long dw = itou(gel(w,id)), dwh = itou(gel(w,id+1));
5798 14 : d = dv + dw;
5799 14 : if (d || CHI)
5800 14 : gel(A,j++) = CHI? fmt_dim(gel(CHI,i),d, dvh+dwh)
5801 14 : : mkvec2s(d,dvh+dwh);
5802 : }
5803 7 : setlg(A, j); return A;
5804 : }
5805 : static GEN
5806 3010 : mfdim0all(GEN w)
5807 : {
5808 3038 : if (w) retconst_vec(lg(w)-1, zerovec(2));
5809 3003 : return cgetg(1,t_VEC);
5810 : }
5811 : static long
5812 7315 : mf1cuspdim_i(long N, GEN CHI, GEN TMP, GEN vSP, long *dih)
5813 : {
5814 7315 : pari_sp av = avma;
5815 7315 : GEN b = mf1basis(N, CHI, TMP, vSP, NULL, dih);
5816 7315 : return gc_long(av, b? itou(b): 0);
5817 : }
5818 :
5819 : static long
5820 476 : mf1cuspdim(long N, GEN CHI, GEN vSP)
5821 : {
5822 476 : if (!vSP) vSP = get_vDIH(N, divisorsNF(N, mfcharconductor(CHI)));
5823 476 : return mf1cuspdim_i(N, CHI, NULL, vSP, NULL);
5824 : }
5825 : static GEN
5826 4144 : mf1cuspdimall(long N, GEN vCHI)
5827 : {
5828 : GEN z, TMP, w, vSP;
5829 : long i, j, l;
5830 4144 : if (wt1empty(N)) return mfdim0all(vCHI);
5831 1141 : w = mf1chars(N,vCHI);
5832 1141 : l = lg(w); if (l == 1) return cgetg(1,t_VEC);
5833 1141 : z = cgetg(l, t_VEC);
5834 1141 : TMP = mf1_pre(N); vSP = get_vDIH(N, NULL);
5835 7861 : for (i = j = 1; i < l; i++)
5836 : {
5837 6720 : GEN CHI = gel(w,i);
5838 6720 : long dih, d = mf1cuspdim_i(N, CHI, TMP, vSP, &dih);
5839 6720 : if (vCHI)
5840 42 : gel(z,j++) = mkvec2s(d, dih);
5841 6678 : else if (d)
5842 1428 : gel(z,j++) = fmt_dim(CHI, d, dih);
5843 : }
5844 1141 : setlg(z,j); return z;
5845 : }
5846 :
5847 : /* dimension of S_1(Gamma_1(N)) */
5848 : static long
5849 4123 : mf1cuspdimsum(long N)
5850 : {
5851 4123 : pari_sp av = avma;
5852 4123 : GEN v = mf1cuspdimall(N, NULL);
5853 4123 : long i, ct = 0, l = lg(v);
5854 5544 : for (i = 1; i < l; i++)
5855 : {
5856 1421 : GEN w = gel(v,i); /* [ord(CHI),*,dim,*] */
5857 1421 : ct += itou(gel(w,3))*myeulerphiu(itou(gel(w,1)));
5858 : }
5859 4123 : return gc_long(av,ct);
5860 : }
5861 :
5862 : static GEN
5863 56 : mf1newdimall(long N, GEN vCHI)
5864 : {
5865 : GEN z, w, vTMP, vSP, fa, P, E;
5866 : long i, c, l, lw, P1;
5867 56 : if (wt1empty(N)) return mfdim0all(vCHI);
5868 56 : w = mf1chars(N,vCHI);
5869 56 : lw = lg(w); if (lw == 1) return cgetg(1,t_VEC);
5870 56 : vTMP = const_vec(N, NULL);
5871 56 : vSP = get_vDIH(N, NULL);
5872 56 : gel(vTMP,N) = mf1_pre(N);
5873 : /* if p || N and p \nmid F(CHI), S_1^new(G0(N),chi) = 0 */
5874 56 : fa = znstar_get_faN(gmael(w,1,1));
5875 56 : P = gel(fa,1); l = lg(P);
5876 56 : E = gel(fa,2);
5877 154 : for (i = P1 = 1; i < l; i++)
5878 98 : if (E[i] == 1) P1 *= itou(gel(P,i));
5879 : /* P1 = \prod_{v_p(N) = 1} p */
5880 56 : z = cgetg(lw, t_VEC);
5881 182 : for (i = c = 1; i < lw; i++)
5882 : {
5883 : long S, j, l, F, dihnew;
5884 126 : GEN D, CHI = gel(w,i), CHIP = mfchartoprimitive(CHI,&F);
5885 :
5886 126 : S = F % P1? 0: mf1cuspdim_i(N, CHI, gel(vTMP,N), vSP, &dihnew);
5887 126 : if (!S)
5888 : {
5889 56 : if (vCHI) gel(z, c++) = zerovec(2);
5890 56 : continue;
5891 : }
5892 70 : D = mydivisorsu(N/F); l = lg(D);
5893 77 : for (j = l-2; j > 0; j--) /* skip last M = N */
5894 : {
5895 7 : long M = D[j]*F, m, s, dih;
5896 7 : GEN TMP = gel(vTMP,M);
5897 7 : if (wt1empty(M) || !(m = mubeta(D[l-j]))) continue; /*m = mubeta(N/M)*/
5898 7 : if (!TMP) gel(vTMP,M) = TMP = mf1_pre(M);
5899 7 : s = mf1cuspdim_i(M, CHIP, TMP, vSP, &dih);
5900 7 : if (s) { S += m * s; dihnew += m * dih; }
5901 : }
5902 70 : if (vCHI)
5903 63 : gel(z,c++) = mkvec2s(S, dihnew);
5904 7 : else if (S)
5905 7 : gel(z, c++) = fmt_dim(CHI, S, dihnew);
5906 : }
5907 56 : setlg(z,c); return z;
5908 : }
5909 :
5910 : static GEN
5911 28 : mf1olddimall(long N, GEN vCHI)
5912 : {
5913 : long i, j, l;
5914 : GEN z, w;
5915 28 : if (wt1empty(N)) return mfdim0all(vCHI);
5916 28 : w = mf1chars(N,vCHI);
5917 28 : l = lg(w); z = cgetg(l, t_VEC);
5918 84 : for (i = j = 1; i < l; i++)
5919 : {
5920 56 : GEN CHI = gel(w,i);
5921 56 : long d = mfolddim(N, 1, CHI);
5922 56 : if (vCHI)
5923 28 : gel(z,j++) = mkvec2s(d,d?-1:0);
5924 28 : else if (d)
5925 7 : gel(z, j++) = fmt_dim(CHI, d, -1);
5926 : }
5927 28 : setlg(z,j); return z;
5928 : }
5929 :
5930 : static long
5931 469 : mf1olddimsum(long N)
5932 : {
5933 : GEN D;
5934 469 : long N2, i, l, S = 0;
5935 469 : newd_params(N, &N2); /* will ensure mubeta != 0 */
5936 469 : D = mydivisorsu(N/N2); l = lg(D);
5937 2485 : for (i = 2; i < l; i++)
5938 : {
5939 2016 : long M = D[l-i]*N2, d = mf1cuspdimsum(M);
5940 2016 : if (d) S -= mubeta(D[i]) * d;
5941 : }
5942 469 : return S;
5943 : }
5944 : static long
5945 1050 : mf1newdimsum(long N)
5946 : {
5947 1050 : long S = mf1cuspdimsum(N);
5948 1050 : return S? S - mf1olddimsum(N): 0;
5949 : }
5950 :
5951 : /* return the automorphism of a degree-2 nf */
5952 : static GEN
5953 5768 : nf2_get_conj(GEN nf)
5954 : {
5955 5768 : GEN pol = nf_get_pol(nf);
5956 5768 : return deg1pol_shallow(gen_m1, negi(gel(pol,3)), varn(pol));
5957 : }
5958 : static int
5959 42 : foo_stable(GEN foo)
5960 42 : { return lg(foo) != 3 || equalii(gel(foo,1), gel(foo,2)); }
5961 :
5962 : static long
5963 224 : mfisdihedral(GEN vF, GEN DIH)
5964 : {
5965 224 : GEN vG = gel(DIH,1), M = gel(DIH,2), v, G, bnr, w, gen, D, f, nf, tau;
5966 224 : GEN bnr0 = NULL, f0, f0b, xin, foo;
5967 : long i, l, e, j, L, n;
5968 224 : if (lg(M) == 1) return 0;
5969 42 : v = RgM_RgC_invimage(M, vF);
5970 42 : if (!v) return 0;
5971 42 : l = lg(v);
5972 42 : for (i = 1; i < l; i++)
5973 42 : if (!gequal0(gel(v,i))) break;
5974 42 : if (i == l) return 0;
5975 42 : G = gel(vG,i);
5976 42 : bnr = gel(G,2); D = cyc_get_expo(bnr_get_cyc(bnr));
5977 42 : w = gel(G,3);
5978 42 : f = bnr_get_mod(bnr);
5979 42 : nf = bnr_get_nf(bnr);
5980 42 : tau = nf2_get_conj(nf);
5981 42 : f0 = gel(f,1); foo = gel(f,2);
5982 42 : f0b = galoisapply(nf, tau, f0);
5983 42 : xin = zv_to_ZV(gel(w,2)); /* xi(bnr.gen[i]) = e(xin[i] / D) */
5984 42 : if (!foo_stable(foo)) { foo = mkvec2(gen_1, gen_1); bnr0 = bnr; }
5985 42 : if (!gequal(f0, f0b))
5986 : {
5987 21 : f0 = idealmul(nf, f0, idealdivexact(nf, f0b, idealadd(nf, f0, f0b)));
5988 21 : bnr0 = bnr;
5989 : }
5990 42 : if (bnr0)
5991 : { /* conductor not ambiguous */
5992 : GEN S;
5993 28 : bnr = Buchray(bnr_get_bnf(bnr), mkvec2(f0, foo), nf_INIT | nf_GEN);
5994 28 : S = bnrsurjection(bnr, bnr0);
5995 28 : xin = FpV_red(RgV_RgM_mul(xin, gel(S,1)), D);
5996 : /* still xi(gen[i]) = e(xin[i] / D), for the new generators; D stays
5997 : * the same, not exponent(bnr.cyc) ! */
5998 : }
5999 42 : gen = bnr_get_gen(bnr); L = lg(gen);
6000 77 : for (j = 1, e = itou(D); j < L; j++)
6001 : {
6002 63 : GEN Ng = idealnorm(nf, gel(gen,j));
6003 63 : GEN a = shifti(gel(xin,j), 1); /* xi(g_j^2) = e(a/D) */
6004 63 : GEN b = FpV_dotproduct(xin, isprincipalray(bnr,Ng), D);
6005 63 : GEN m = Fp_sub(a, b, D); /* xi(g_j/g_j^\tau) = e(m/D) */
6006 63 : e = ugcd(e, itou(m)); if (e == 1) break;
6007 : }
6008 42 : n = itou(D) / e;
6009 42 : return n == 1? 4: 2*n;
6010 : }
6011 :
6012 : static ulong
6013 119 : myradicalu(ulong n) { return zv_prod(gel(myfactoru(n),1)); }
6014 :
6015 : /* list of fundamental discriminants unramified outside N, with sign s
6016 : * [s = 0 => no sign condition] */
6017 : static GEN
6018 119 : mfunram(long N, long s)
6019 : {
6020 119 : long cN = myradicalu(N >> vals(N)), p = 1, m = 1, l, c, i;
6021 119 : GEN D = mydivisorsu(cN), res;
6022 119 : l = lg(D);
6023 119 : if (s == 1) m = 0; else if (s == -1) p = 0;
6024 119 : res = cgetg(6*l - 5, t_VECSMALL);
6025 119 : c = 1;
6026 119 : if (!odd(N))
6027 : { /* d = 1 */
6028 56 : if (p) res[c++] = 8;
6029 56 : if (m) { res[c++] =-8; res[c++] =-4; }
6030 : }
6031 364 : for (i = 2; i < l; i++)
6032 : { /* skip d = 1, done above */
6033 245 : long d = D[i], d4 = d & 3L; /* d odd, squarefree, d4 = 1 or 3 */
6034 245 : if (d4 == 1) { if (p) res[c++] = d; }
6035 182 : else { if (m) res[c++] =-d; }
6036 245 : if (!odd(N))
6037 : {
6038 56 : if (p) { res[c++] = 8*d; if (d4 == 3) res[c++] = 4*d; }
6039 56 : if (m) { res[c++] =-8*d; if (d4 == 1) res[c++] =-4*d; }
6040 : }
6041 : }
6042 119 : setlg(res, c); return res;
6043 : }
6044 :
6045 : /* Return 1 if F is definitely not S4 type; return 0 on failure. */
6046 : static long
6047 105 : mfisnotS4(long N, GEN w)
6048 : {
6049 105 : GEN D = mfunram(N, 0);
6050 105 : long i, lD = lg(D), lw = lg(w);
6051 616 : for (i = 1; i < lD; i++)
6052 : {
6053 511 : long p, d = D[i], ok = 0;
6054 1442 : for (p = 2; p < lw; p++)
6055 1442 : if (w[p] && kross(d,p) == -1) { ok = 1; break; }
6056 511 : if (!ok) return 0;
6057 : }
6058 105 : return 1;
6059 : }
6060 :
6061 : /* Return 1 if Q(sqrt(5)) \not\subset Q(F), i.e. F is definitely not A5 type;
6062 : * return 0 on failure. */
6063 : static long
6064 105 : mfisnotA5(GEN F)
6065 : {
6066 105 : GEN CHI = mf_get_CHI(F), P = mfcharpol(CHI), T, Q;
6067 :
6068 105 : if (mfcharorder(CHI) % 5 == 0) return 0;
6069 105 : T = mf_get_field(F); if (degpol(T) == 1) return 1;
6070 105 : if (degpol(P) > 1) T = rnfequation(P,T);
6071 105 : Q = gsubgs(pol_xn(2,varn(T)), 5);
6072 105 : return (typ(nfisincl(Q, T)) == t_INT);
6073 : }
6074 :
6075 : /* v[p+1]^2 / chi(p) - 2 = z + 1/z with z primitive root of unity of order n,
6076 : * return n */
6077 : static long
6078 6741 : mffindrootof1(GEN v, long p, GEN CHI)
6079 : {
6080 6741 : GEN ap = gel(v,p+1), u0, u1, u1k, u2;
6081 6741 : long c = 1;
6082 6741 : if (gequal0(ap)) return 2;
6083 5033 : u0 = gen_2; u1k = u1 = gsubgs(gdiv(gsqr(ap), mfchareval(CHI, p)), 2);
6084 14812 : while (!gequalsg(2, liftpol_shallow(u1))) /* u1 = z^c + z^-c */
6085 : {
6086 9779 : u2 = gsub(gmul(u1k, u1), u0);
6087 9779 : u0 = u1; u1 = u2; c++;
6088 : }
6089 5033 : return c;
6090 : }
6091 :
6092 : /* we known that F is not dihedral */
6093 : static long
6094 182 : mfgaloistype_i(long N, GEN CHI, GEN F, GEN v)
6095 : {
6096 : forprime_t iter;
6097 182 : long lim = lg(v)-2;
6098 182 : GEN w = zero_zv(lim);
6099 : pari_sp av;
6100 : ulong p;
6101 182 : u_forprime_init(&iter, 2, lim);
6102 182 : av = avma;
6103 5292 : while((p = u_forprime_next(&iter))) if (N%p) switch(mffindrootof1(v, p, CHI))
6104 : {
6105 1400 : case 1: case 2: continue;
6106 3451 : case 3: w[p] = 1; break;
6107 70 : case 4: return -24; /* S4 */
6108 0 : case 5: return -60; /* A5 */
6109 7 : default: pari_err_DOMAIN("mfgaloistype", "form", "not a",
6110 : strtoGENstr("cuspidal eigenform"), F);
6111 0 : set_avma(av);
6112 : }
6113 105 : if (mfisnotS4(N,w) && mfisnotA5(F)) return -12; /* A4 */
6114 0 : return 0; /* FAILURE */
6115 : }
6116 :
6117 : static GEN
6118 224 : mfgaloistype0(long N, GEN CHI, GEN F, GEN DIH, long lim)
6119 : {
6120 224 : pari_sp av = avma;
6121 224 : GEN vF = mftocol(F, lim, 1);
6122 224 : long t = mfisdihedral(vF, DIH), bound;
6123 224 : if (t) return gc_stoi(av,t);
6124 182 : bound = maxss(200, 5*expu(N)*expu(N));
6125 : for(;;)
6126 : {
6127 182 : t = mfgaloistype_i(N, CHI, F, vF);
6128 175 : set_avma(av); if (t) return stoi(t);
6129 0 : if (lim > bound) return gen_0;
6130 0 : lim += lim >> 1;
6131 0 : vF = mfcoefs_i(F,lim,1);
6132 : }
6133 : }
6134 :
6135 : /* If f is NULL, give all the galoistypes, otherwise just for f */
6136 : /* Return 0 to indicate failure; in this case the type is either -12 or -60,
6137 : * most likely -12. FIXME using the Galois representation. */
6138 : GEN
6139 231 : mfgaloistype(GEN NK, GEN f)
6140 : {
6141 231 : pari_sp av = avma;
6142 231 : GEN CHI, T, F, DIH, SP, mf = checkMF_i(NK);
6143 : long N, k, lL, i, lim, SB;
6144 :
6145 231 : if (f && !checkmf_i(f)) pari_err_TYPE("mfgaloistype", f);
6146 224 : if (mf)
6147 : {
6148 189 : N = MF_get_N(mf);
6149 189 : k = MF_get_k(mf);
6150 189 : CHI = MF_get_CHI(mf);
6151 : }
6152 : else
6153 : {
6154 35 : checkNK(NK, &N, &k, &CHI, 0);
6155 35 : mf = f? NULL: mfinit_i(NK, mf_NEW);
6156 : }
6157 224 : if (k != 1) pari_err_DOMAIN("mfgaloistype", "k", "!=", gen_1, stoi(k));
6158 224 : SB = mf? mfsturm_mf(mf): mfsturmNk(N,1);
6159 224 : SP = get_DIH(N);
6160 224 : DIH = mfdihedralnew(N, CHI, SP);
6161 224 : lim = lg(DIH) == 1? 200: SB;
6162 224 : DIH = mkvec2(DIH, mfvectomat(DIH,SB,1));
6163 224 : if (f) return gerepileuptoint(av, mfgaloistype0(N,CHI, f, DIH, lim));
6164 126 : F = mfeigenbasis(mf); lL = lg(F);
6165 126 : T = cgetg(lL, t_VEC);
6166 252 : for (i=1; i < lL; i++) gel(T,i) = mfgaloistype0(N, CHI, gel(F,i), DIH, lim);
6167 126 : return gerepileupto(av, T);
6168 : }
6169 :
6170 : /******************************************************************/
6171 : /* Find all dihedral forms. */
6172 : /******************************************************************/
6173 : /* lim >= 2 */
6174 : static void
6175 14 : consttabdihedral(long lim) { cache_set(cache_DIH, mfdihedralall(lim)); }
6176 :
6177 : /* a ideal coprime to bnr modulus */
6178 : static long
6179 107611 : mfdiheval(GEN bnr, GEN w, GEN a)
6180 : {
6181 107611 : GEN L, cycn = gel(w,1), chin = gel(w,2);
6182 107611 : long ordmax = cycn[1];
6183 107611 : L = ZV_to_Flv(isprincipalray(bnr,a), ordmax);
6184 107611 : return Flv_dotproduct(chin, L, ordmax);
6185 : }
6186 :
6187 : /* x(t^k) mod T = polcyclo(m), 0 <= k < m */
6188 : static GEN
6189 30331 : Galois(GEN x, long k, GEN T, long m)
6190 : {
6191 : GEN B;
6192 : long i, ik, d;
6193 30331 : if (typ(x) != t_POL) return x;
6194 7455 : if (varn(x) != varn(T)) pari_APPLY_pol_normalized(Galois(gel(x,i), k, T, m));
6195 7420 : if ((d = degpol(x)) <= 0) return x;
6196 7063 : B = cgetg(m + 2, t_POL); B[1] = x[1]; gel(B,2) = gel(x,2);
6197 61565 : for (i = 1; i < m; i++) gel(B, i+2) = gen_0;
6198 23940 : for (i = 1, ik = k; i <= d; i++, ik = Fl_add(ik, k, m))
6199 16877 : gel(B, ik + 2) = gel(x, i+2);
6200 7063 : return QX_ZX_rem(normalizepol(B), T);
6201 : }
6202 : static GEN
6203 1022 : vecGalois(GEN x, long k, GEN T, long m)
6204 31332 : { pari_APPLY_same(Galois(gel(x,i), k, T, m)); }
6205 :
6206 : static GEN
6207 234178 : fix_pol(GEN S, GEN Pn, int *trace)
6208 : {
6209 234178 : if (typ(S) != t_POL) return S;
6210 118069 : S = RgX_rem(S, Pn);
6211 118069 : if (typ(S) == t_POL)
6212 : {
6213 118069 : switch(lg(S))
6214 : {
6215 45108 : case 2: return gen_0;
6216 20517 : case 3: return gel(S,2);
6217 : }
6218 52444 : *trace = 1;
6219 : }
6220 52444 : return S;
6221 : }
6222 :
6223 : static GEN
6224 13573 : dihan(GEN bnr, GEN w, GEN k0j, long m, ulong lim)
6225 : {
6226 13573 : GEN nf = bnr_get_nf(bnr), f = bid_get_ideal(bnr_get_bid(bnr));
6227 13573 : GEN v = zerovec(lim+1), cycn = gel(w,1), Tinit = gel(w,3);
6228 13573 : GEN Pn = gel(Tinit,lg(Tinit)==4? 2: 1);
6229 13573 : long j, ordmax = cycn[1];
6230 13573 : long D = itos(nf_get_disc(nf)), vt = varn(Pn);
6231 13573 : int trace = 0;
6232 : ulong p, n;
6233 : forprime_t T;
6234 :
6235 13573 : if (!lim) return v;
6236 13363 : gel(v,2) = gen_1;
6237 13363 : u_forprime_init(&T, 2, lim);
6238 : /* fill in prime powers first */
6239 116207 : while ((p = u_forprime_next(&T)))
6240 : {
6241 : GEN vP, vchiP, S;
6242 : long k, lP;
6243 : ulong q, qk;
6244 102844 : if (kross(D,p) >= 0) q = p;
6245 45192 : else if (!(q = umuluu_le(p,p,lim))) continue;
6246 : /* q = Norm P */
6247 65856 : vP = idealprimedec(nf, utoipos(p));
6248 65856 : lP = lg(vP);
6249 65856 : vchiP = cgetg(lP, t_VECSMALL);
6250 179081 : for (j = k = 1; j < lP; j++)
6251 : {
6252 113225 : GEN P = gel(vP,j);
6253 113225 : if (!idealval(nf, f, P)) vchiP[k++] = mfdiheval(bnr,w,P);
6254 : }
6255 65856 : if (k == 1) continue;
6256 62188 : setlg(vchiP, k); lP = k;
6257 62188 : if (lP == 2)
6258 : { /* one prime above p not dividing f */
6259 16765 : long s, s0 = vchiP[1];
6260 27069 : for (qk=q, s = s0;; s = Fl_add(s,s0,ordmax))
6261 : {
6262 27069 : S = Qab_zeta(s, ordmax, vt);
6263 27069 : gel(v, qk+1) = fix_pol(S, Pn, &trace);
6264 27069 : if (!(qk = umuluu_le(qk,q,lim))) break;
6265 : }
6266 : }
6267 : else /* two primes above p not dividing f */
6268 : {
6269 45423 : long s, s0 = vchiP[1], s1 = vchiP[2];
6270 45423 : for (qk=q, k = 1;; k++)
6271 18424 : { /* sum over a,b s.t. Norm( P1^a P2^b ) = q^k, i.e. a+b = k */
6272 : long a;
6273 63847 : GEN S = gen_0;
6274 220752 : for (a = 0; a <= k; a++)
6275 : {
6276 156905 : s = Fl_add(Fl_mul(a, s0, ordmax), Fl_mul(k-a, s1, ordmax), ordmax);
6277 156905 : S = gadd(S, Qab_zeta(s, ordmax, vt));
6278 : }
6279 63847 : gel(v, qk+1) = fix_pol(S, Pn, &trace);
6280 63847 : if (!(qk = umuluu_le(qk,q,lim))) break;
6281 : }
6282 : }
6283 : }
6284 : /* complete with nonprime powers */
6285 308098 : for (n = 2; n <= lim; n++)
6286 : {
6287 294735 : GEN S, fa = myfactoru(n), P = gel(fa, 1), E = gel(fa, 2);
6288 : long q;
6289 294735 : if (lg(P) == 2) continue;
6290 : /* not a prime power */
6291 143262 : q = upowuu(P[1],E[1]);
6292 143262 : S = gmul(gel(v, q + 1), gel(v, n/q + 1));
6293 143262 : gel(v, n+1) = fix_pol(S, Pn, &trace);
6294 : }
6295 13363 : if (trace)
6296 : {
6297 7154 : long k0 = k0j[1], jdeg = k0j[2];
6298 7154 : v = QabV_tracerel(Tinit, jdeg, v); /* Apply Galois Mod(k0, ordw) */
6299 7154 : if (k0 > 1) v = vecGalois(v, k0, gel(Tinit,1), m);
6300 : }
6301 13363 : return v;
6302 : }
6303 :
6304 : /* as cyc_normalize for t_VECSMALL cyc */
6305 : static GEN
6306 26810 : cyc_normalize_zv(GEN cyc)
6307 : {
6308 26810 : long i, o = cyc[1], l = lg(cyc); /* > 1 */
6309 26810 : GEN D = cgetg(l, t_VECSMALL);
6310 31185 : D[1] = o; for (i = 2; i < l; i++) D[i] = o / cyc[i];
6311 26810 : return D;
6312 : }
6313 : /* as char_normalize for t_VECSMALLs */
6314 : static GEN
6315 118517 : char_normalize_zv(GEN chi, GEN ncyc)
6316 : {
6317 118517 : long i, l = lg(chi);
6318 118517 : GEN c = cgetg(l, t_VECSMALL);
6319 118517 : if (l > 1) {
6320 118517 : c[1] = chi[1];
6321 160454 : for (i = 2; i < l; i++) c[i] = chi[i] * ncyc[i];
6322 : }
6323 118517 : return c;
6324 : }
6325 :
6326 : static GEN
6327 9331 : dihan_bnf(long D)
6328 : {
6329 9331 : GEN c = getrand(), bnf;
6330 9331 : setrand(gen_1);
6331 9331 : bnf = Buchall(quadpoly_i(stoi(D)), nf_FORCE, LOWDEFAULTPREC);
6332 9331 : setrand(c);
6333 9331 : return bnf;
6334 : }
6335 : static GEN
6336 37758 : dihan_bnr(GEN bnf, GEN A)
6337 : {
6338 37758 : GEN c = getrand(), bnr;
6339 37758 : setrand(gen_1);
6340 37758 : bnr = Buchray(bnf, A, nf_INIT|nf_GEN);
6341 37758 : setrand(c);
6342 37758 : return bnr;
6343 : }
6344 : /* Hecke xi * (D/.) = Dirichlet chi, return v in Q^r st chi(g_i) = e(v[i]).
6345 : * cycn = cyc_normalize_zv(bnr.cyc), chin = char_normalize_zv(chi,cyc) */
6346 : static GEN
6347 34489 : bnrchartwist2conrey(GEN chin, GEN cycn, GEN bnrconreyN, GEN kroconreyN)
6348 : {
6349 34489 : long l = lg(bnrconreyN), c1 = cycn[1], i;
6350 34489 : GEN v = cgetg(l, t_COL);
6351 125363 : for (i = 1; i < l; i++)
6352 : {
6353 90874 : GEN d = sstoQ(zv_dotproduct(chin, gel(bnrconreyN,i)), c1);
6354 90874 : if (kroconreyN[i] < 0) d = gadd(d, ghalf);
6355 90874 : gel(v,i) = d;
6356 : }
6357 34489 : return v;
6358 : }
6359 :
6360 : /* chi(g_i) = e(v[i]) denormalize wrt Conrey generators orders */
6361 : static GEN
6362 34489 : conreydenormalize(GEN znN, GEN v)
6363 : {
6364 34489 : GEN gcyc = znstar_get_conreycyc(znN), w;
6365 34489 : long l = lg(v), i;
6366 34489 : w = cgetg(l, t_COL);
6367 125363 : for (i = 1; i < l; i++)
6368 90874 : gel(w,i) = modii(gmul(gel(v,i), gel(gcyc,i)), gel(gcyc,i));
6369 34489 : return w;
6370 : }
6371 :
6372 : static long
6373 84028 : Miyake(GEN vchi, GEN gb, GEN cycn)
6374 : {
6375 84028 : long i, e = cycn[1], lb = lg(gb);
6376 84028 : GEN v = char_normalize_zv(vchi, cycn);
6377 124992 : for (i = 1; i < lb; i++)
6378 100268 : if ((zv_dotproduct(v, gel(gb,i)) - v[i]) % e) return 1;
6379 24724 : return 0;
6380 : }
6381 :
6382 : /* list of Hecke characters not induced by a Dirichlet character up to Galois
6383 : * conjugation, whose conductor is bnr.cond; cycn = cyc_normalize(bnr.cyc)*/
6384 : static GEN
6385 26810 : mklvchi(GEN bnr, GEN cycn, GEN gb)
6386 : {
6387 26810 : GEN cyc = bnr_get_cyc(bnr), cycsmall = ZV_to_zv(cyc);
6388 26810 : GEN vchi = cyc2elts(cycsmall);
6389 26810 : long ordmax = cycsmall[1], c, i, l;
6390 26810 : l = lg(vchi);
6391 304024 : for (i = c = 1; i < l; i++)
6392 : {
6393 277214 : GEN chi = gel(vchi,i);
6394 277214 : if (!gb || Miyake(chi, gb, cycn)) gel(vchi, c++) = Flv_to_ZV(chi);
6395 : }
6396 26810 : setlg(vchi, c); l = c;
6397 279300 : for (i = 1; i < l; i++)
6398 : {
6399 252490 : GEN chi = gel(vchi,i);
6400 : long n;
6401 252490 : if (!chi) continue;
6402 1055754 : for (n = 2; n < ordmax; n++)
6403 966476 : if (ugcd(n, ordmax) == 1)
6404 : {
6405 397670 : GEN tmp = ZV_ZV_mod(gmulsg(n, chi), cyc);
6406 : long j;
6407 7623539 : for (j = i+1; j < l; j++)
6408 7225869 : if (gel(vchi,j) && gequal(gel(vchi,j), tmp)) gel(vchi,j) = NULL;
6409 : }
6410 : }
6411 279300 : for (i = c = 1; i < l; i++)
6412 : {
6413 252490 : GEN chi = gel(vchi,i);
6414 252490 : if (chi && bnrisconductor(bnr, chi)) gel(vchi, c++) = chi;
6415 : }
6416 26810 : setlg(vchi, c); return vchi;
6417 : }
6418 :
6419 : static GEN
6420 7805 : get_gb(GEN bnr, GEN con)
6421 : {
6422 7805 : GEN gb, g = bnr_get_gen(bnr), nf = bnr_get_nf(bnr);
6423 7805 : long i, l = lg(g);
6424 7805 : gb = cgetg(l, t_VEC);
6425 18326 : for (i = 1; i < l; i++)
6426 10521 : gel(gb,i) = ZV_to_zv(isprincipalray(bnr, galoisapply(nf, con, gel(g,i))));
6427 7805 : return gb;
6428 : }
6429 : static GEN
6430 15862 : get_bnrconreyN(GEN bnr, GEN znN)
6431 : {
6432 15862 : GEN z, g = znstar_get_conreygen(znN);
6433 15862 : long i, l = lg(g);
6434 15862 : z = cgetg(l, t_VEC);
6435 57134 : for (i = 1; i < l; i++) gel(z,i) = ZV_to_zv(isprincipalray(bnr,gel(g,i)));
6436 15862 : return z;
6437 : }
6438 : /* con = NULL if D > 0 or if D < 0 and id != idcon. */
6439 : static GEN
6440 33698 : mfdihedralcommon(GEN bnf, GEN id, GEN znN, GEN kroconreyN, long vt,
6441 : long N, long D, GEN con)
6442 : {
6443 33698 : GEN bnr = dihan_bnr(bnf, id), cyc = ZV_to_zv( bnr_get_cyc(bnr) );
6444 : GEN bnrconreyN, cycn, cycN, Lvchi, res, P, vT;
6445 : long j, ordmax, l, lc, deghecke;
6446 :
6447 33698 : lc = lg(cyc); if (lc == 1) return NULL;
6448 26810 : cycn = cyc_normalize_zv(cyc);
6449 26810 : Lvchi = mklvchi(bnr, cycn, con? get_gb(bnr, con): NULL);
6450 26810 : l = lg(Lvchi);
6451 26810 : if (l == 1) return NULL;
6452 :
6453 15862 : bnrconreyN = get_bnrconreyN(bnr, znN);
6454 15862 : cycN = ZV_to_zv(znstar_get_cyc(znN));
6455 15862 : ordmax = cyc[1];
6456 15862 : vT = const_vec(odd(ordmax)? ordmax << 1: ordmax, NULL);
6457 15862 : P = polcyclo(ordmax, vt);
6458 15862 : gel(vT,ordmax) = Qab_trace_init(ordmax, ordmax, P, P);
6459 15862 : deghecke = myeulerphiu(ordmax);
6460 15862 : res = cgetg(l, t_VEC);
6461 50351 : for (j = 1; j < l; j++)
6462 : {
6463 34489 : GEN T, v, vchi = ZV_to_zv(gel(Lvchi,j));
6464 34489 : GEN chi, chin = char_normalize_zv(vchi, cycn);
6465 : long o, vnum, k0, degrel;
6466 34489 : v = bnrchartwist2conrey(chin, cycn, bnrconreyN, kroconreyN);
6467 34489 : o = itou(Q_denom(v));
6468 34489 : T = gel(vT, o);
6469 34489 : if (!T) gel(vT,o) = T = Qab_trace_init(ordmax, o, P, polcyclo(o,vt));
6470 34489 : chi = conreydenormalize(znN, v);
6471 34489 : vnum = itou(znconreyexp(znN, chi));
6472 34489 : chi = ZV_to_zv(znconreychar(znN,chi));
6473 34489 : degrel = deghecke / degpol(gel(T,1));
6474 34489 : k0 = zv_cyc_minimize(cycN, chi, coprimes_zv(o));
6475 34489 : vnum = Fl_powu(vnum, k0, N);
6476 : /* encodes degrel forms: jdeg = 0..degrel-1 */
6477 34489 : gel(res,j) = mkvec3(mkvecsmalln(5, N, k0 % o, vnum, D, degrel),
6478 : id, mkvec3(cycn,chin,T));
6479 : }
6480 15862 : return res;
6481 : }
6482 :
6483 : static long
6484 49364 : is_cond(long D, long n)
6485 : {
6486 49364 : if (D > 0) return n != 4 || (D&7L) == 1;
6487 30114 : return n != 2 && n != 3 && (n != 4 || (D&7L)!=1);
6488 : }
6489 : /* Append to v all dihedral weight 1 forms coming from D, if fundamental.
6490 : * level in [l1, l2] */
6491 : static void
6492 18718 : append_dihedral(GEN v, long D, long l1, long l2, long vt)
6493 : {
6494 18718 : long Da = labs(D), no, i, numi, ct, min, max;
6495 : GEN bnf, con, vI, resall, arch1, arch2;
6496 : pari_sp av;
6497 :
6498 : /* min <= Nf <= max */
6499 18718 : max = l2 / Da;
6500 18718 : if (l1 == l2)
6501 : { /* assume Da | l2 */
6502 140 : min = max;
6503 140 : if (D > 0 && min < 3) return;
6504 : }
6505 : else /* assume l1 < l2 */
6506 18578 : min = (l1 + Da-1)/Da;
6507 18718 : if (!sisfundamental(D)) return;
6508 :
6509 5726 : av = avma;
6510 5726 : bnf = dihan_bnf(D);
6511 5726 : con = nf2_get_conj(bnf_get_nf(bnf));
6512 5726 : vI = ideallist(bnf, max);
6513 55090 : numi = 0; for (i = min; i <= max; i++) numi += lg(gel(vI, i)) - 1;
6514 5726 : if (D > 0)
6515 : {
6516 1428 : numi <<= 1;
6517 1428 : arch1 = mkvec2(gen_1,gen_0);
6518 1428 : arch2 = mkvec2(gen_0,gen_1);
6519 : }
6520 : else
6521 4298 : arch1 = arch2 = NULL;
6522 5726 : resall = cgetg(numi+1, t_VEC); ct = 1;
6523 55090 : for (no = min; no <= max; no++) if (is_cond(D, no))
6524 : {
6525 44646 : long N = Da*no, lc, lI;
6526 44646 : GEN I = gel(vI, no), znN = znstar0(utoipos(N), 1), conreyN, kroconreyN;
6527 :
6528 44646 : conreyN = znstar_get_conreygen(znN); lc = lg(conreyN);
6529 44646 : kroconreyN = cgetg(lc, t_VECSMALL);
6530 166054 : for (i = 1; i < lc; i++) kroconreyN[i] = krosi(D, gel(conreyN, i));
6531 44646 : lI = lg(I);
6532 87822 : for (i = 1; i < lI; i++)
6533 : {
6534 43176 : GEN id = gel(I, i), idcon, z;
6535 : long j;
6536 43176 : if (typ(id) == t_INT) continue;
6537 28182 : idcon = galoisapply(bnf, con, id);
6538 51408 : for (j = i; j < lI; j++)
6539 51408 : if (gequal(idcon, gel(I, j))) { gel(I, j) = gen_0; break; }
6540 28182 : if (D < 0)
6541 : {
6542 17479 : GEN conk = i == j ? con : NULL;
6543 17479 : z = mfdihedralcommon(bnf, id, znN, kroconreyN, vt, N, D, conk);
6544 17479 : if (z) gel(resall, ct++) = z;
6545 : }
6546 : else
6547 : {
6548 : GEN ide;
6549 10703 : ide = mkvec2(id, arch1);
6550 10703 : z = mfdihedralcommon(bnf, ide, znN, kroconreyN, vt, N, D, NULL);
6551 10703 : if (z) gel(resall, ct++) = z;
6552 10703 : if (gequal(idcon,id)) continue;
6553 5516 : ide = mkvec2(id, arch2);
6554 5516 : z = mfdihedralcommon(bnf, ide, znN, kroconreyN, vt, N, D, NULL);
6555 5516 : if (z) gel(resall, ct++) = z;
6556 : }
6557 : }
6558 : }
6559 5726 : if (ct == 1) set_avma(av);
6560 : else
6561 : {
6562 4816 : setlg(resall, ct);
6563 4816 : vectrunc_append(v, gerepilecopy(av, shallowconcat1(resall)));
6564 : }
6565 : }
6566 :
6567 : static long
6568 42042 : di_N(GEN a) { return gel(a,1)[1]; }
6569 : static GEN
6570 14 : mfdihedral(long N)
6571 : {
6572 14 : GEN D = mydivisorsu(N), res = vectrunc_init(2*N);
6573 14 : long j, l = lg(D), vt = fetch_user_var("t");
6574 105 : for (j = 2; j < l; j++)
6575 : { /* skip d = 1 */
6576 91 : long d = D[j];
6577 91 : if (d == 2) continue;
6578 84 : append_dihedral(res, -d, N,N, vt);
6579 84 : if (d >= 5 && D[l-j] >= 3) append_dihedral(res, d, N,N, vt);/* Nf >= 3 */
6580 : }
6581 14 : if (lg(res) > 1) res = shallowconcat1(res);
6582 14 : return res;
6583 : }
6584 : /* All primitive dihedral weight 1 forms of leven in [1, N], N > 1 */
6585 : static GEN
6586 14 : mfdihedralall(long N)
6587 : {
6588 14 : GEN res = vectrunc_init(2*N), z;
6589 14 : long D, ct, i, vt = fetch_user_var("t");
6590 :
6591 13986 : for (D = -3; D >= -N; D--) append_dihedral(res, D, 1,N, vt);
6592 : /* Nf >= 3 (GTM 193, prop 3.3.18) */
6593 4620 : for (D = N / 3; D >= 5; D--) append_dihedral(res, D, 1,N, vt);
6594 14 : ct = lg(res);
6595 14 : if (ct > 1)
6596 : { /* sort wrt N */
6597 14 : res = shallowconcat1(res);
6598 14 : res = vecpermute(res, indexvecsort(res, mkvecsmall(1)));
6599 14 : ct = lg(res);
6600 : }
6601 14 : z = const_vec(N, cgetg(1,t_VEC));
6602 7658 : for (i = 1; i < ct;)
6603 : { /* regroup result sharing the same N */
6604 7644 : long n = di_N(gel(res,i)), j = i+1, k;
6605 : GEN v;
6606 34412 : while (j < ct && di_N(gel(res,j)) == n) j++;
6607 7644 : gel(z, n) = v = cgetg(j-i+1, t_VEC);
6608 42056 : for (k = 1; i < j; k++,i++) gel(v,k) = gel(res,i);
6609 : }
6610 14 : return z;
6611 : }
6612 :
6613 : /* return [vF, index], where vecpermute(vF,index) generates dihedral forms
6614 : * for character CHI */
6615 : static GEN
6616 24969 : mfdihedralnew_i(long N, GEN CHI, GEN SP)
6617 : {
6618 : GEN bnf, Tinit, Pm, vf, M, V, NK;
6619 : long Dold, d, ordw, i, SB, c, l, k0, k1, chino, chinoorig, lv;
6620 :
6621 24969 : lv = lg(SP); if (lv == 1) return NULL;
6622 12138 : CHI = mfcharinduce(CHI,N);
6623 12138 : ordw = mfcharorder(CHI);
6624 12138 : chinoorig = mfcharno(CHI);
6625 12138 : k0 = mfconreyminimize(CHI);
6626 12138 : chino = Fl_powu(chinoorig, k0, N);
6627 12138 : k1 = Fl_inv(k0 % ordw, ordw);
6628 12138 : V = cgetg(lv, t_VEC);
6629 12138 : d = 0;
6630 39039 : for (i = l = 1; i < lv; i++)
6631 : {
6632 26901 : GEN sp = gel(SP,i), T = gel(sp,1);
6633 26901 : if (T[3] != chino) continue;
6634 4060 : d += T[5];
6635 4060 : if (k1 != 1)
6636 : {
6637 77 : GEN t = leafcopy(T);
6638 77 : t[3] = chinoorig;
6639 77 : t[2] = (t[2]*k1) % ordw;
6640 77 : sp = mkvec4(t, gel(sp,2), gel(sp,3), gel(sp,4));
6641 : }
6642 4060 : gel(V, l++) = sp;
6643 : }
6644 12138 : setlg(V, l); /* dihedral forms of level N and character CHI */
6645 12138 : if (l == 1) return NULL;
6646 :
6647 2555 : SB = mfsturmNk(N,1) + 1;
6648 2555 : M = cgetg(d+1, t_MAT);
6649 2555 : vf = cgetg(d+1, t_VEC);
6650 2555 : NK = mkNK(N, 1, CHI);
6651 2555 : bnf = NULL; Dold = 0;
6652 6615 : for (i = c = 1; i < l; i++)
6653 : { /* T = [N, k0, conreyno, D, degrel] */
6654 4060 : GEN bnr, Vi = gel(V,i), T = gel(Vi,1), id = gel(Vi,2), w = gel(Vi,3);
6655 4060 : long jdeg, k0i = T[2], D = T[4], degrel = T[5];
6656 :
6657 4060 : if (D != Dold) { Dold = D; bnf = dihan_bnf(D); }
6658 4060 : bnr = dihan_bnr(bnf, id);
6659 12054 : for (jdeg = 0; jdeg < degrel; jdeg++,c++)
6660 : {
6661 7994 : GEN k0j = mkvecsmall2(k0i, jdeg), an = dihan(bnr, w, k0j, ordw, SB);
6662 7994 : settyp(an, t_COL); gel(M,c) = an;
6663 7994 : gel(vf,c) = tag3(t_MF_DIHEDRAL, NK, bnr, w, k0j);
6664 : }
6665 : }
6666 2555 : Tinit = gmael3(V,1,3,3); Pm = gel(Tinit,1);
6667 2555 : V = QabM_indexrank(M, degpol(Pm)==1? NULL: Pm, ordw);
6668 2555 : return mkvec2(vf,gel(V,2));
6669 : }
6670 : static long
6671 16149 : mfdihedralnewdim(long N, GEN CHI, GEN SP)
6672 : {
6673 16149 : pari_sp av = avma;
6674 16149 : GEN S = mfdihedralnew_i(N, CHI, SP);
6675 16149 : return gc_long(av, S? lg(gel(S,2))-1: 0);
6676 : }
6677 : static GEN
6678 8820 : mfdihedralnew(long N, GEN CHI, GEN SP)
6679 : {
6680 8820 : pari_sp av = avma;
6681 8820 : GEN S = mfdihedralnew_i(N, CHI, SP);
6682 8820 : if (!S) { set_avma(av); return cgetg(1, t_VEC); }
6683 917 : return vecpermute(gel(S,1), gel(S,2));
6684 : }
6685 :
6686 : static long
6687 7105 : mfdihedralcuspdim(long N, GEN CHI, GEN vSP)
6688 : {
6689 7105 : pari_sp av = avma;
6690 : GEN D, CHIP;
6691 : long F, i, lD, dim;
6692 :
6693 7105 : CHIP = mfchartoprimitive(CHI, &F);
6694 7105 : D = mydivisorsu(N/F); lD = lg(D);
6695 7105 : dim = mfdihedralnewdim(N, CHI, gel(vSP,N)); /* d = 1 */
6696 16149 : for (i = 2; i < lD; i++)
6697 : {
6698 9044 : long d = D[i], a = mfdihedralnewdim(N/d, CHIP, gel(vSP, N/d));
6699 9044 : if (a) dim += a * mynumdivu(d);
6700 : }
6701 7105 : return gc_long(av,dim);
6702 : }
6703 :
6704 : static GEN
6705 7343 : mfbdall(GEN E, long N)
6706 : {
6707 7343 : GEN v, D = mydivisorsu(N);
6708 7343 : long i, j, nD = lg(D) - 1, nE = lg(E) - 1;
6709 7343 : v = cgetg(nD*nE + 1, t_VEC);
6710 10416 : for (j = 1; j <= nE; j++)
6711 : {
6712 3073 : GEN Ej = gel(E, j);
6713 9415 : for (i = 0; i < nD; i++) gel(v, i*nE + j) = mfbd_i(Ej, D[i+1]);
6714 : }
6715 7343 : return v;
6716 : }
6717 : static GEN
6718 3857 : mfdihedralcusp(long N, GEN CHI, GEN vSP)
6719 : {
6720 3857 : pari_sp av = avma;
6721 : GEN D, CHIP, z;
6722 : long F, i, lD;
6723 :
6724 3857 : CHIP = mfchartoprimitive(CHI, &F);
6725 3857 : D = mydivisorsu(N/F); lD = lg(D);
6726 3857 : z = cgetg(lD, t_VEC);
6727 3857 : gel(z,1) = mfdihedralnew(N, CHI, gel(vSP,N));
6728 8596 : for (i = 2; i < lD; i++) /* skip 1 */
6729 : {
6730 4739 : GEN LF = mfdihedralnew(N / D[i], CHIP, gel(vSP, N / D[i]));
6731 4739 : gel(z,i) = mfbdall(LF, D[i]);
6732 : }
6733 3857 : return gerepilecopy(av, shallowconcat1(z));
6734 : }
6735 :
6736 : /* used to decide between ratlift and comatrix for ZM_inv; ratlift is better
6737 : * when N has many divisors */
6738 : static int
6739 2548 : abundant(ulong N) { return mynumdivu(N) >= 8; }
6740 :
6741 : /* CHI an mfchar */
6742 : static int
6743 371 : cmp_ord(void *E, GEN a, GEN b)
6744 : {
6745 371 : GEN chia = MF_get_CHI(a), chib = MF_get_CHI(b);
6746 371 : (void)E; return cmpii(gmfcharorder(chia), gmfcharorder(chib));
6747 : }
6748 : /* mfinit structure.
6749 : -- mf[1] contains [N,k,CHI,space],
6750 : -- mf[2] contains vector of closures of Eisenstein series, empty if not
6751 : full space.
6752 : -- mf[3] contains vector of closures, so #mf[3] = dimension of cusp/new space.
6753 : -- mf[4] contains the corresponding indices: either j for T(j)tf if newspace,
6754 : or [M,j,d] for B(d)T(j)tf_M if cuspspace or oldspace.
6755 : -- mf[5] contains the matrix M of first coefficients of basis, never cleaned.
6756 : * NK is either [N,k] or [N,k,CHI].
6757 : * mfinit does not do the splitting, only the basis generation. */
6758 :
6759 : /* Set flraw to 1 if do not need mf[5]: no mftobasis etc..., only the
6760 : expansions of the basis elements are needed. */
6761 :
6762 : static GEN
6763 4991 : mfinit_Nkchi(long N, long k, GEN CHI, long space, long flraw)
6764 : {
6765 4991 : GEN M = NULL, mf = NULL, mf1 = mkvec4(utoi(N), stoi(k), CHI, utoi(space));
6766 4991 : long sb = mfsturmNk(N, k);
6767 4991 : if (k < 0 || badchar(N, k, CHI)) return mfEMPTY(mf1);
6768 4956 : if (k == 0 || space == mf_EISEN) /*nothing*/;
6769 4795 : else if (k == 1)
6770 : {
6771 364 : switch (space)
6772 : {
6773 350 : case mf_NEW:
6774 : case mf_FULL:
6775 350 : case mf_CUSP: mf = mf1init(N, CHI, NULL, get_vDIH(N,NULL), space, flraw);
6776 350 : break;
6777 7 : case mf_OLD: pari_err_IMPL("mfinit in weight 1 for old space");
6778 7 : default: pari_err_FLAG("mfinit");
6779 : }
6780 : }
6781 : else /* k >= 2 */
6782 : {
6783 4431 : long ord = mfcharorder(CHI);
6784 4431 : GEN z = NULL, P = (ord <= 2)? NULL: mfcharpol(CHI);
6785 : cachenew_t cache;
6786 4431 : switch(space)
6787 : {
6788 1225 : case mf_NEW:
6789 1225 : mf = mfnewinit(N, k, CHI, &cache, 1);
6790 1225 : if (mf && !flraw) { M = MF_get_M(mf); z = MF_get_Mindex(mf); }
6791 1225 : break;
6792 3199 : case mf_OLD:
6793 : case mf_CUSP:
6794 : case mf_FULL:
6795 3199 : if (!(mf = mfinitcusp(N, k, CHI, &cache, space))) break;
6796 2898 : if (!flraw)
6797 : {
6798 2247 : M = bhnmat_extend(M, sb+1, 1, MF_get_S(mf), &cache);
6799 2247 : if (space != mf_FULL) gel(mf,5) = mfcleanCHI(M, CHI, abundant(N));
6800 : }
6801 2898 : dbg_cachenew(&cache); break;
6802 7 : default: pari_err_FLAG("mfinit");
6803 : }
6804 4424 : if (z) gel(mf,5) = mfclean2(M, z, P, ord);
6805 : }
6806 4935 : if (!mf) mf = mfEMPTY(mf1);
6807 : else
6808 : {
6809 3976 : gel(mf,1) = mf1;
6810 3976 : if (flraw) gel(mf,5) = zerovec(3);
6811 : }
6812 4935 : if (!space_is_cusp(space))
6813 : {
6814 819 : GEN E = mfeisensteinbasis(N, k, CHI);
6815 819 : gel(mf,2) = E;
6816 819 : if (!flraw)
6817 : {
6818 497 : if (M)
6819 196 : M = shallowconcat(mfvectomat(E, sb+1, 1), M);
6820 : else
6821 301 : M = mfcoefs_mf(mf, sb+1, 1);
6822 497 : gel(mf,5) = mfcleanCHI(M, CHI, abundant(N));
6823 : }
6824 : }
6825 4935 : return mf;
6826 : }
6827 :
6828 : /* mfinit for k = nk/dk */
6829 : static GEN
6830 2695 : mfinit_Nndkchi(long N, long nk, long dk, GEN CHI, long space, long flraw)
6831 266 : { return (dk == 2)? mf2init_Nkchi(N, nk >> 1, CHI, space, flraw)
6832 2961 : : mfinit_Nkchi(N, nk, CHI, space, flraw); }
6833 : static GEN
6834 3360 : mfinit_i(GEN NK, long space)
6835 : {
6836 : GEN CHI, mf;
6837 : long N, k, dk, joker;
6838 3360 : if (checkmf_i(NK))
6839 : {
6840 147 : N = mf_get_N(NK);
6841 147 : Qtoss(mf_get_gk(NK), &k, &dk);
6842 147 : CHI = mf_get_CHI(NK);
6843 : }
6844 3213 : else if ((mf = checkMF_i(NK)))
6845 : {
6846 21 : long s = MF_get_space(mf);
6847 21 : if (s == space) return mf;
6848 21 : Qtoss(MF_get_gk(mf), &k, &dk);
6849 21 : if (dk == 1 && k > 1 && space == mf_NEW && (s == mf_CUSP || s == mf_FULL))
6850 21 : return mfinittonew(mf);
6851 0 : N = MF_get_N(mf);
6852 0 : CHI = MF_get_CHI(mf);
6853 : }
6854 : else
6855 3192 : checkNK2(NK, &N, &k, &dk, &CHI, 1);
6856 3318 : joker = !CHI || typ(CHI) == t_COL;
6857 3318 : if (joker)
6858 : {
6859 1162 : GEN mf, vCHI = CHI;
6860 : long i, j, l;
6861 1162 : if (CHI && lg(CHI) == 1) return cgetg(1,t_VEC);
6862 1155 : if (k < 0) return mfEMPTYall(N, uutoQ(k,dk), CHI, space);
6863 1141 : if (k == 1 && dk == 1 && space != mf_EISEN)
6864 504 : {
6865 : GEN TMP, vSP, gN, gs;
6866 : pari_timer tt;
6867 1106 : if (space != mf_CUSP && space != mf_NEW)
6868 0 : pari_err_IMPL("mfinit([N,1,wildcard], space != cusp or new space)");
6869 1106 : if (wt1empty(N)) return mfEMPTYall(N, gen_1, CHI, space);
6870 504 : vCHI = mf1chars(N,vCHI);
6871 504 : l = lg(vCHI); mf = cgetg(l, t_VEC); if (l == 1) return mf;
6872 504 : TMP = mf1_pre(N); vSP = get_vDIH(N, NULL);
6873 504 : gN = utoipos(N); gs = utoi(space);
6874 504 : if (DEBUGLEVEL) timer_start(&tt);
6875 4123 : for (i = j = 1; i < l; i++)
6876 : {
6877 3619 : pari_sp av = avma;
6878 3619 : GEN c = gel(vCHI,i), z = mf1init(N, c, TMP, vSP, space, 0);
6879 3619 : if (z) z = gerepilecopy(av, z);
6880 : else
6881 : {
6882 2905 : set_avma(av);
6883 2905 : if (CHI) z = mfEMPTY(mkvec4(gN,gen_1,c,gs));
6884 : }
6885 3619 : if (z) gel(mf, j++) = z;
6886 3619 : if (DEBUGLEVEL)
6887 0 : timer_printf(&tt, "mf1basis: character %ld / %ld (order = %ld)",
6888 : i, l-1, mfcharorder(c));
6889 : }
6890 : }
6891 : else
6892 : {
6893 35 : vCHI = mfchars(N,k,dk,vCHI);
6894 35 : l = lg(vCHI); mf = cgetg(l, t_VEC);
6895 119 : for (i = j = 1; i < l; i++)
6896 : {
6897 84 : pari_sp av = avma;
6898 84 : GEN v = mfinit_Nndkchi(N, k, dk, gel(vCHI,i), space, 0);
6899 84 : if (MF_get_dim(v) || CHI) gel(mf, j++) = v; else set_avma(av);
6900 : }
6901 : }
6902 539 : setlg(mf,j);
6903 539 : if (!CHI) gen_sort_inplace(mf, NULL, &cmp_ord, NULL);
6904 539 : return mf;
6905 : }
6906 2156 : return mfinit_Nndkchi(N, k, dk, CHI, space, 0);
6907 : }
6908 : GEN
6909 2394 : mfinit(GEN NK, long space)
6910 : {
6911 2394 : pari_sp av = avma;
6912 2394 : return gerepilecopy(av, mfinit_i(NK, space));
6913 : }
6914 :
6915 : /* UTILITY FUNCTIONS */
6916 : static void
6917 364 : cusp_canon(GEN cusp, long N, long *pA, long *pC)
6918 : {
6919 364 : pari_sp av = avma;
6920 : long A, C, tc, cg;
6921 364 : if (N <= 0) pari_err_DOMAIN("mfcuspwidth","N","<=",gen_0,stoi(N));
6922 357 : if (!cusp || (tc = typ(cusp)) == t_INFINITY) { *pA = 1; *pC = N; return; }
6923 350 : if (tc != t_INT && tc != t_FRAC) pari_err_TYPE("checkcusp", cusp);
6924 350 : Qtoss(cusp, &A,&C);
6925 350 : if (N % C)
6926 : {
6927 : ulong uC;
6928 14 : long u = Fl_invgen((C-1)%N + 1, N, &uC);
6929 14 : A = Fl_mul(A, u, N);
6930 14 : C = (long)uC;
6931 : }
6932 350 : cg = ugcd(C, N/C);
6933 420 : while (ugcd(A, N) > 1) A += cg;
6934 350 : *pA = A % N; *pC = C; set_avma(av);
6935 : }
6936 : static long
6937 952 : mfcuspcanon_width(long N, long C)
6938 952 : { return (!C || C == N)? 1 : N / ugcd(N, Fl_sqr(umodsu(C,N),N)); }
6939 : /* v = [a,c] a ZC, width of cusp (a:c) */
6940 : static long
6941 9667 : mfZC_width(long N, GEN v)
6942 : {
6943 9667 : ulong C = umodiu(gel(v,2), N);
6944 9667 : return (C == 0)? 1: N / ugcd(N, Fl_sqr(C,N));
6945 : }
6946 : long
6947 161 : mfcuspwidth(GEN gN, GEN cusp)
6948 : {
6949 161 : long N = 0, A, C;
6950 : GEN mf;
6951 161 : if (typ(gN) == t_INT) N = itos(gN);
6952 42 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
6953 0 : else pari_err_TYPE("mfcuspwidth", gN);
6954 161 : cusp_canon(cusp, N, &A, &C);
6955 154 : return mfcuspcanon_width(N, C);
6956 : }
6957 :
6958 : /* Q a t_INT */
6959 : static GEN
6960 14 : findq(GEN al, GEN Q)
6961 : {
6962 : long n;
6963 14 : if (typ(al) == t_FRAC && cmpii(gel(al,2), Q) <= 0)
6964 0 : return mkvec(mkvec2(gel(al,1), gel(al,2)));
6965 14 : n = 1 + (long)ceil(2.0781*gtodouble(glog(Q, LOWDEFAULTPREC)));
6966 14 : return contfracpnqn(gboundcf(al,n), n);
6967 : }
6968 : static GEN
6969 91 : findqga(long N, GEN z)
6970 : {
6971 91 : GEN Q, LDC, CK = NULL, DK = NULL, ma, x, y = imag_i(z);
6972 : long j, l;
6973 91 : if (gcmpgs(gmulsg(2*N, y), 1) >= 0) return NULL;
6974 14 : x = real_i(z);
6975 14 : Q = ground(ginv(gsqrt(gmulsg(N, y), LOWDEFAULTPREC)));
6976 14 : LDC = findq(gmulsg(-N,x), Q);
6977 14 : ma = gen_1; l = lg(LDC);
6978 35 : for (j = 1; j < l; j++)
6979 : {
6980 21 : GEN D, DC = gel(LDC,j), C1 = gel(DC,2);
6981 21 : if (cmpii(C1,Q) > 0) break;
6982 21 : D = gel(DC,1);
6983 21 : if (ugcdiu(D,N) == 1)
6984 : {
6985 7 : GEN C = mului(N, C1), den;
6986 7 : den = gadd(gsqr(gmul(C,y)), gsqr(gadd(D, gmul(C,x))));
6987 7 : if (gcmp(den, ma) < 0) { ma = den; CK = C; DK = D; }
6988 : }
6989 : }
6990 14 : return DK? mkvec2(CK, DK): NULL;
6991 : }
6992 :
6993 : static long
6994 168 : valNC2(GEN P, GEN E, long e)
6995 : {
6996 168 : long i, d = 1, l = lg(P);
6997 504 : for (i = 1; i < l; i++)
6998 : {
6999 336 : long v = u_lval(e, P[i]) << 1;
7000 336 : if (v == E[i] + 1) v--;
7001 336 : d *= upowuu(P[i], v);
7002 : }
7003 168 : return d;
7004 : }
7005 :
7006 : static GEN
7007 49 : findqganew(long N, GEN z)
7008 : {
7009 49 : GEN MI, DI, x = real_i(z), y = imag_i(z), Ck = gen_0, Dk = gen_1, fa, P, E;
7010 : long i;
7011 49 : MI = uutoQ(1,N);
7012 49 : DI = mydivisorsu(mysqrtu(N));
7013 49 : fa = myfactoru(N); P = gel(fa,1); E = gel(fa,2);
7014 217 : for (i = 1; i < lg(DI); i++)
7015 : {
7016 168 : long e = DI[i], g;
7017 : GEN U, C, D, m;
7018 168 : (void)cxredsl2(gmulsg(e, z), &U);
7019 168 : C = gcoeff(U,2,1); if (!signe(C)) continue;
7020 168 : D = gcoeff(U,2,2);
7021 168 : g = ugcdiu(D,e);
7022 168 : if (g > 1) { C = muliu(C,e/g); D = diviuexact(D,g); } else C = muliu(C,e);
7023 168 : m = gadd(gsqr(gadd(gmul(C, x), D)), gsqr(gmul(C, y)));
7024 168 : m = gdivgu(m, valNC2(P, E, e));
7025 168 : if (gcmp(m, MI) < 0) { MI = m; Ck = C; Dk = D; }
7026 : }
7027 49 : return signe(Ck)? mkvec2(Ck, Dk): NULL;
7028 : }
7029 :
7030 : /* Return z' and U = [a,b;c,d] \in SL_2(Z), z' = U*z,
7031 : * Im(z')/width(U.oo) > sqrt(3)/(2N). Set *pczd = c*z+d */
7032 : static GEN
7033 175 : cxredga0N(long N, GEN z, GEN *pU, GEN *pczd, long flag)
7034 : {
7035 175 : GEN v = NULL, A, B, C, D;
7036 : long e;
7037 175 : if (N == 1) return cxredsl2_i(z, pU, pczd);
7038 140 : e = gexpo(gel(z,2));
7039 140 : if (e < 0) z = gprec_wensure(z, precision(z) + nbits2extraprec(-e));
7040 140 : v = flag? findqganew(N,z): findqga(N,z);
7041 140 : if (!v) { *pU = matid(2); *pczd = gen_1; return z; }
7042 56 : C = gel(v,1);
7043 56 : D = gel(v,2);
7044 56 : if (!is_pm1(bezout(C,D, &B,&A))) pari_err_BUG("cxredga0N [gcd > 1]");
7045 56 : B = negi(B);
7046 56 : *pU = mkmat2(mkcol2(A,C), mkcol2(B,D));
7047 56 : *pczd = gadd(gmul(C,z), D);
7048 56 : return gdiv(gadd(gmul(A,z), B), *pczd);
7049 : }
7050 :
7051 : static GEN
7052 154 : lfunthetaall(GEN b, GEN vL, GEN t, long bitprec)
7053 : {
7054 154 : long i, l = lg(vL);
7055 154 : GEN v = cgetg(l, t_VEC);
7056 336 : for (i = 1; i < l; i++)
7057 : {
7058 182 : GEN T, L = gel(vL,i), a0 = gel(L,1), ldata = gel(L,2);
7059 182 : GEN van = gel(ldata_get_an(ldata),2);
7060 182 : if (lg(van) == 1)
7061 : {
7062 0 : T = gmul(b, a0);
7063 0 : if (isexactzero(T)) { GEN z = real_0_bit(-bitprec); T = mkcomplex(z,z); }
7064 : }
7065 : else
7066 : {
7067 182 : T = gmul2n(lfuntheta(ldata, t, 0, bitprec), -1);
7068 182 : T = gmul(b, gadd(a0, T));
7069 : }
7070 182 : gel(v,i) = T;
7071 : }
7072 154 : return l == 2? gel(v,1): v;
7073 : }
7074 :
7075 : /* P in ZX, irreducible */
7076 : static GEN
7077 182 : ZX_roots(GEN P, long prec)
7078 : {
7079 182 : long d = degpol(P);
7080 182 : if (d == 1) return mkvec(gen_0);
7081 182 : if (d == 2 && isint1(gel(P,2)) && isintzero(gel(P,3)) && isint1(gel(P,4)))
7082 7 : return mkvec2(powIs(3), gen_I()); /* order as polroots */
7083 294 : return (ZX_sturm_irred(P) == d)? ZX_realroots_irred(P, prec)
7084 294 : : QX_complex_roots(P, prec);
7085 : }
7086 : /* initializations for RgX_RgV_eval / RgC_embed */
7087 : static GEN
7088 217 : rootspowers(GEN v)
7089 : {
7090 217 : long i, l = lg(v);
7091 217 : GEN w = cgetg(l, t_VEC);
7092 868 : for (i = 1; i < l; i++) gel(w,i) = gpowers(gel(v,i), l-2);
7093 217 : return w;
7094 : }
7095 : /* mf embeddings attached to Q(chi)/(T), chi attached to cyclotomic P */
7096 : static GEN
7097 903 : getembed(GEN P, GEN T, GEN zcyclo, long prec)
7098 : {
7099 : long i, l;
7100 : GEN v;
7101 903 : if (degpol(P) == 1) P = NULL; /* mfcharpol for quadratic char */
7102 903 : if (degpol(T) == 1) T = NULL; /* dim 1 orbit */
7103 903 : if (T && P)
7104 35 : { /* K(y) / (T(y)), K = Q(t)/(P) cyclotomic */
7105 35 : GEN vr = RgX_is_ZX(T)? ZX_roots(T,prec): roots(RgX_embed1(T,zcyclo), prec);
7106 35 : v = rootspowers(vr); l = lg(v);
7107 105 : for (i = 1; i < l; i++) gel(v,i) = mkcol3(P,zcyclo,gel(v,i));
7108 : }
7109 868 : else if (T)
7110 : { /* Q(y) / (T(y)), T noncyclotomic */
7111 182 : GEN vr = ZX_roots(T, prec);
7112 182 : v = rootspowers(vr); l = lg(v);
7113 763 : for (i = 1; i < l; i++) gel(v,i) = mkcol2(T, gel(v,i));
7114 : }
7115 : else /* cyclotomic or rational */
7116 686 : v = mkvec(P? mkvec2(P, zcyclo): cgetg(1,t_VEC));
7117 903 : return v;
7118 : }
7119 : static GEN
7120 756 : grootsof1_CHI(GEN CHI, long prec)
7121 756 : { return grootsof1(mfcharorder(CHI), prec); }
7122 : /* return the [Q(F):Q(chi)] embeddings of F */
7123 : static GEN
7124 588 : mfgetembed(GEN F, long prec)
7125 : {
7126 588 : GEN T = mf_get_field(F), CHI = mf_get_CHI(F), P = mfcharpol(CHI);
7127 588 : return getembed(P, T, grootsof1_CHI(CHI, prec), prec);
7128 : }
7129 : static GEN
7130 7 : mfchiembed(GEN mf, long prec)
7131 : {
7132 7 : GEN CHI = MF_get_CHI(mf), P = mfcharpol(CHI);
7133 7 : return getembed(P, pol_x(0), grootsof1_CHI(CHI, prec), prec);
7134 : }
7135 : /* mfgetembed for the successive eigenforms in MF_get_newforms */
7136 : static GEN
7137 161 : mfeigenembed(GEN mf, long prec)
7138 : {
7139 161 : GEN vP = MF_get_fields(mf), vF = MF_get_newforms(mf);
7140 161 : GEN zcyclo, vE, CHI = MF_get_CHI(mf), P = mfcharpol(CHI);
7141 161 : long i, l = lg(vP);
7142 161 : vF = Q_remove_denom(liftpol_shallow(vF), NULL);
7143 161 : prec += nbits2extraprec(gexpo(vF));
7144 161 : zcyclo = grootsof1_CHI(CHI, prec);
7145 161 : vE = cgetg(l, t_VEC);
7146 469 : for (i = 1; i < l; i++) gel(vE,i) = getembed(P, gel(vP,i), zcyclo, prec);
7147 161 : return vE;
7148 : }
7149 :
7150 : static int
7151 28 : checkPv(GEN P, GEN v)
7152 28 : { return typ(P) == t_POL && is_vec_t(typ(v)) && lg(v)-1 >= degpol(P); }
7153 : static int
7154 28 : checkemb_i(GEN E)
7155 : {
7156 28 : long t = typ(E), l = lg(E);
7157 28 : if (t == t_VEC) return l == 1 || (l == 3 && checkPv(gel(E,1), gel(E,2)));
7158 21 : if (t != t_COL) return 0;
7159 21 : if (l == 3) return checkPv(gel(E,1), gel(E,2));
7160 21 : return l == 4 && is_vec_t(typ(gel(E,2))) && checkPv(gel(E,1), gel(E,3));
7161 : }
7162 : static GEN
7163 28 : anyembed(GEN v, GEN E)
7164 : {
7165 28 : switch(typ(v))
7166 : {
7167 21 : case t_VEC: case t_COL: return mfvecembed(E, v);
7168 7 : case t_MAT: return mfmatembed(E, v);
7169 : }
7170 0 : return mfembed(E, v);
7171 : }
7172 : GEN
7173 49 : mfembed0(GEN E, GEN v, long prec)
7174 : {
7175 49 : pari_sp av = avma;
7176 49 : GEN mf, vE = NULL;
7177 49 : if (checkmf_i(E)) vE = mfgetembed(E, prec);
7178 35 : else if ((mf = checkMF_i(E))) vE = mfchiembed(mf, prec);
7179 49 : if (vE)
7180 : {
7181 21 : long i, l = lg(vE);
7182 : GEN w;
7183 21 : if (!v) return gerepilecopy(av, l == 2? gel(vE,1): vE);
7184 0 : w = cgetg(l, t_VEC);
7185 0 : for (i = 1; i < l; i++) gel(w,i) = anyembed(v, gel(vE,i));
7186 0 : return gerepilecopy(av, l == 2? gel(w,1): w);
7187 : }
7188 28 : if (!checkemb_i(E) || !v) pari_err_TYPE("mfembed", E);
7189 28 : return gerepilecopy(av, anyembed(v,E));
7190 : }
7191 :
7192 : /* dummy lfun create for theta evaluation */
7193 : static GEN
7194 938 : mfthetaancreate(GEN van, GEN N, GEN k)
7195 : {
7196 938 : GEN L = zerovec(6);
7197 938 : gel(L,1) = lfuntag(t_LFUN_GENERIC, van);
7198 938 : gel(L,3) = mkvec2(gen_0, gen_1);
7199 938 : gel(L,4) = k;
7200 938 : gel(L,5) = N; return L;
7201 : }
7202 : /* destroy van and prepare to evaluate theta(sigma(van)), for all sigma in
7203 : * embeddings vector vE */
7204 : static GEN
7205 336 : van_embedall(GEN van, GEN vE, GEN gN, GEN gk)
7206 : {
7207 336 : GEN a0 = gel(van,1), vL;
7208 336 : long i, lE = lg(vE), l = lg(van);
7209 336 : van++; van[0] = evaltyp(t_VEC) | _evallg(l-1); /* remove a0 */
7210 336 : vL = cgetg(lE, t_VEC);
7211 903 : for (i = 1; i < lE; i++)
7212 : {
7213 567 : GEN E = gel(vE,i), v = mfvecembed(E, van);
7214 567 : gel(vL,i) = mkvec2(mfembed(E,a0), mfthetaancreate(v, gN, gk));
7215 : }
7216 336 : return vL;
7217 : }
7218 :
7219 : static int
7220 1120 : cusp_AC(GEN cusp, long *A, long *C)
7221 : {
7222 1120 : switch(typ(cusp))
7223 : {
7224 140 : case t_INFINITY: *A = 1; *C = 0; break;
7225 294 : case t_INT: *A = itos(cusp); *C = 1; break;
7226 448 : case t_FRAC: *A = itos(gel(cusp, 1)); *C = itos(gel(cusp, 2)); break;
7227 238 : case t_REAL: case t_COMPLEX:
7228 238 : *A = 0; *C = 0;
7229 238 : if (gsigne(imag_i(cusp)) <= 0)
7230 7 : pari_err_DOMAIN("mfeval","imag(tau)","<=",gen_0,cusp);
7231 231 : return 0;
7232 0 : default: pari_err_TYPE("cusp_AC", cusp);
7233 : }
7234 882 : return 1;
7235 : }
7236 : static GEN
7237 518 : cusp2mat(long A, long C)
7238 : { long B, D;
7239 518 : cbezout(A, C, &D, &B);
7240 518 : return mkmat22s(A, -B, C, D);
7241 : }
7242 : static GEN
7243 21 : mkS(void) { return mkmat22s(0,-1,1,0); }
7244 :
7245 : /* if t is a cusp, return F(t), else NULL */
7246 : static GEN
7247 350 : evalcusp(GEN mf, GEN F, GEN t, long prec)
7248 : {
7249 : long A, C;
7250 : GEN R;
7251 350 : if (!cusp_AC(t, &A,&C)) return NULL;
7252 189 : if (C % mf_get_N(F) == 0) return gel(mfcoefs_i(F, 0, 1), 1);
7253 175 : R = mfgaexpansion(mf, F, cusp2mat(A,C), 0, prec);
7254 175 : return gequal0(gel(R,1))? gmael(R,3,1): gen_0;
7255 : }
7256 : /* Evaluate an mf closure numerically, i.e., in the usual sense, either for a
7257 : * single tau or a vector of tau; for each, return a vector of results
7258 : * corresponding to all complex embeddings of F. If flag is nonzero, allow
7259 : * replacing F by F | gamma to increase imag(gamma^(-1).tau) [ expensive if
7260 : * MF_EISENSPACE not present ] */
7261 : static GEN
7262 161 : mfeval_i(GEN mf, GEN F, GEN vtau, long flag, long bitprec)
7263 : {
7264 : GEN L0, vL, vb, sqN, vczd, vTAU, vs, van, vE;
7265 161 : long N = MF_get_N(mf), N0, ta, lv, i, prec = nbits2prec(bitprec);
7266 161 : GEN gN = utoipos(N), gk = mf_get_gk(F), gk1 = gsubgs(gk,1), vgk;
7267 161 : long flscal = 0;
7268 :
7269 : /* gen_0 is ignored, second component assumes Ramanujan-Petersson in
7270 : * 1/2-integer weight */
7271 161 : vgk = mkvec2(gen_0, mfiscuspidal(mf,F)? gmul2n(gk1,-1): gk1);
7272 161 : ta = typ(vtau);
7273 161 : if (!is_vec_t(ta)) { flscal = 1; vtau = mkvec(vtau); ta = t_VEC; }
7274 161 : lv = lg(vtau);
7275 161 : sqN = sqrtr_abs(utor(N, prec));
7276 161 : vs = const_vec(lv-1, NULL);
7277 161 : vb = const_vec(lv-1, NULL);
7278 161 : vL = cgetg(lv, t_VEC);
7279 161 : vTAU = cgetg(lv, t_VEC);
7280 161 : vczd = cgetg(lv, t_VEC);
7281 161 : L0 = mfthetaancreate(NULL, gN, vgk); /* only for thetacost */
7282 161 : vE = mfgetembed(F, prec);
7283 161 : N0 = 0;
7284 343 : for (i = 1; i < lv; i++)
7285 : {
7286 189 : GEN z = gel(vtau,i), tau, U;
7287 : long w, n;
7288 :
7289 189 : gel(vs,i) = evalcusp(mf, F, z, prec);
7290 182 : if (gel(vs,i)) continue;
7291 154 : tau = cxredga0N(N, z, &U, &gel(vczd,i), flag);
7292 154 : if (!flag) w = 0; else { w = mfZC_width(N, gel(U,1)); tau = gdivgu(tau,w); }
7293 154 : gel(vTAU,i) = mulcxmI(gmul(tau, sqN));
7294 154 : n = lfunthetacost(L0, real_i(gel(vTAU,i)), 0, bitprec);
7295 154 : if (N0 < n) N0 = n;
7296 154 : if (flag)
7297 : {
7298 42 : GEN A, al, v = mfslashexpansion(mf, F, ZM_inv(U,NULL), n, 0, &A, prec);
7299 42 : gel(vL,i) = van_embedall(v, vE, gN, vgk);
7300 42 : al = gel(A,1);
7301 42 : if (!gequal0(al))
7302 7 : gel(vb,i) = gexp(gmul(gmul(gmulsg(w,al),PiI2(prec)), tau), prec);
7303 : }
7304 : }
7305 154 : if (!flag)
7306 : {
7307 112 : van = mfcoefs_i(F, N0, 1);
7308 112 : vL = const_vec(lv-1, van_embedall(van, vE, gN, vgk));
7309 : }
7310 336 : for (i = 1; i < lv; i++)
7311 : {
7312 : GEN T;
7313 182 : if (gel(vs,i)) continue;
7314 154 : T = gpow(gel(vczd,i), gneg(gk), prec);
7315 154 : if (flag && gel(vb,i)) T = gmul(T, gel(vb,i));
7316 154 : gel(vs,i) = lfunthetaall(T, gel(vL,i), gel(vTAU,i), bitprec);
7317 : }
7318 154 : return flscal? gel(vs,1): vs;
7319 : }
7320 :
7321 : static long
7322 1323 : mfistrivial(GEN F)
7323 : {
7324 1323 : switch(mf_get_type(F))
7325 : {
7326 7 : case t_MF_CONST: return lg(gel(F,2)) == 1;
7327 266 : case t_MF_LINEAR: case t_MF_LINEAR_BHN: return gequal0(gel(F,3));
7328 1050 : default: return 0;
7329 : }
7330 : }
7331 :
7332 : static long
7333 1141 : mf_same_k(GEN mf, GEN f) { return gequal(MF_get_gk(mf), mf_get_gk(f)); }
7334 : static long
7335 1099 : mf_same_CHI(GEN mf, GEN f)
7336 : {
7337 1099 : GEN F1, F2, chi1, chi2, CHI1 = MF_get_CHI(mf), CHI2 = mf_get_CHI(f);
7338 : /* are the primitive chars attached to CHI1 and CHI2 equal ? */
7339 1099 : F1 = znconreyconductor(gel(CHI1,1), gel(CHI1,2), &chi1);
7340 1099 : if (typ(F1) == t_VEC) F1 = gel(F1,1);
7341 1099 : F2 = znconreyconductor(gel(CHI2,1), gel(CHI2,2), &chi2);
7342 1099 : if (typ(F2) == t_VEC) F2 = gel(F2,1);
7343 1099 : return equalii(F1,F2) && ZV_equal(chi1,chi2);
7344 : }
7345 : /* check k and CHI rigorously, but not coefficients nor N */
7346 : static long
7347 238 : mfisinspace_i(GEN mf, GEN F)
7348 : {
7349 238 : return mfistrivial(F) || (mf_same_k(mf,F) && mf_same_CHI(mf,F));
7350 : }
7351 : static void
7352 7 : err_space(GEN F)
7353 7 : { pari_err_DOMAIN("mftobasis", "form", "does not belong to",
7354 0 : strtoGENstr("space"), F); }
7355 :
7356 : static long
7357 147 : mfcheapeisen(GEN mf)
7358 : {
7359 147 : long k, L, N = MF_get_N(mf);
7360 : GEN P;
7361 147 : if (N <= 70) return 1;
7362 84 : k = itos(gceil(MF_get_gk(mf)));
7363 84 : if (odd(k)) k--;
7364 84 : switch (k)
7365 : {
7366 0 : case 2: L = 190; break;
7367 14 : case 4: L = 162; break;
7368 70 : case 6:
7369 70 : case 8: L = 88; break;
7370 0 : case 10: L = 78; break;
7371 0 : default: L = 66; break;
7372 : }
7373 84 : P = gel(myfactoru(N), 1);
7374 84 : return P[lg(P)-1] <= L;
7375 : }
7376 :
7377 : static GEN
7378 182 : myimag_i(GEN x)
7379 : {
7380 182 : long tc = typ(x);
7381 182 : if (tc == t_INFINITY || tc == t_INT || tc == t_FRAC) return gen_1;
7382 189 : if (tc == t_VEC) pari_APPLY_same(myimag_i(gel(x,i)));
7383 147 : return imag_i(x);
7384 : }
7385 :
7386 : static GEN
7387 147 : mintau(GEN vtau)
7388 : {
7389 147 : if (!is_vec_t(typ(vtau))) return myimag_i(vtau);
7390 7 : return (lg(vtau) == 1)? gen_1: vecmin(myimag_i(vtau));
7391 : }
7392 :
7393 : /* initialization for mfgaexpansion: what does not depend on cusp */
7394 : static GEN
7395 1001 : mf_eisendec(GEN mf, GEN F, long prec)
7396 : {
7397 1001 : GEN B = liftpol_shallow(mfeisensteindec(mf, F)), v = variables_vecsmall(B);
7398 1001 : GEN Mvecj = obj_check(mf, MF_EISENSPACE);
7399 1001 : long l = lg(v), i, ord;
7400 1001 : if (lg(Mvecj) < 5) Mvecj = gel(Mvecj,1);
7401 1001 : ord = itou(gel(Mvecj,4));
7402 1057 : for (i = 1; i < l; i++)
7403 721 : if (v[i] != 1)
7404 : {
7405 : GEN d;
7406 : long e;
7407 665 : B = Q_remove_denom(B, &d);
7408 665 : e = gexpo(B);
7409 665 : if (e > 0) prec += nbits2prec(e);
7410 665 : B = gsubst(B, v[i], rootsof1u_cx(ord, prec));
7411 665 : if (d) B = gdiv(B, d);
7412 665 : break;
7413 : }
7414 1001 : return B;
7415 : }
7416 :
7417 : GEN
7418 161 : mfeval(GEN mf0, GEN F, GEN vtau, long bitprec)
7419 : {
7420 161 : pari_sp av = avma;
7421 161 : long flnew = 1;
7422 161 : GEN mf = checkMF_i(mf0);
7423 161 : if (!mf) pari_err_TYPE("mfeval", mf0);
7424 161 : if (!checkmf_i(F)) pari_err_TYPE("mfeval", F);
7425 161 : if (!mfisinspace_i(mf, F)) err_space(F);
7426 161 : if (!obj_check(mf, MF_EISENSPACE)) flnew = mfcheapeisen(mf);
7427 161 : if (flnew && gcmpgs(gmulsg(2*MF_get_N(mf), mintau(vtau)), 1) >= 0) flnew = 0;
7428 161 : return gerepilecopy(av, mfeval_i(mf, F, vtau, flnew, bitprec));
7429 : }
7430 :
7431 : static long
7432 189 : val(GEN v, long bit)
7433 : {
7434 189 : long c, l = lg(v);
7435 392 : for (c = 1; c < l; c++)
7436 378 : if (gexpo(gel(v,c)) > -bit) return c-1;
7437 14 : return -1;
7438 : }
7439 : GEN
7440 203 : mfcuspval(GEN mf, GEN F, GEN cusp, long bitprec)
7441 : {
7442 203 : pari_sp av = avma;
7443 203 : long lvE, w, N, sb, n, A, C, prec = nbits2prec(bitprec);
7444 : GEN ga, gk, vE;
7445 203 : mf = checkMF(mf);
7446 203 : if (!checkmf_i(F)) pari_err_TYPE("mfcuspval",F);
7447 203 : N = MF_get_N(mf);
7448 203 : cusp_canon(cusp, N, &A, &C);
7449 203 : gk = mf_get_gk(F);
7450 203 : if (typ(gk) != t_INT)
7451 : {
7452 42 : GEN FT = mfmultheta(F), mf2 = obj_checkbuild(mf, MF_MF2INIT, &mf2init);
7453 42 : GEN r = mfcuspval(mf2, FT, cusp, bitprec);
7454 42 : if ((C & 3L) == 2)
7455 : {
7456 14 : GEN z = uutoQ(1,4);
7457 14 : r = gsub(r, typ(r) == t_VEC? const_vec(lg(r)-1, z): z);
7458 : }
7459 42 : return gerepileupto(av, r);
7460 : }
7461 161 : vE = mfgetembed(F, prec);
7462 161 : lvE = lg(vE);
7463 161 : w = mfcuspcanon_width(N, C);
7464 161 : sb = w * mfsturmNk(N, itos(gk));
7465 161 : ga = cusp2mat(A,C);
7466 168 : for (n = 8;; n = minss(sb, n << 1))
7467 7 : {
7468 168 : GEN R = mfgaexpansion(mf, F, ga, n, prec), res = liftpol_shallow(gel(R,3));
7469 168 : GEN v = cgetg(lvE-1, t_VECSMALL);
7470 168 : long j, ok = 1;
7471 168 : res = RgC_embedall(res, vE);
7472 357 : for (j = 1; j < lvE; j++)
7473 : {
7474 189 : v[j] = val(gel(res,j), bitprec/2);
7475 189 : if (v[j] < 0) ok = 0;
7476 : }
7477 168 : if (ok)
7478 : {
7479 154 : res = cgetg(lvE, t_VEC);
7480 329 : for (j = 1; j < lvE; j++) gel(res,j) = gadd(gel(R,1), uutoQ(v[j], w));
7481 154 : return gerepilecopy(av, lvE==2? gel(res,1): res);
7482 : }
7483 14 : if (n == sb) return lvE==2? mkoo(): const_vec(lvE-1, mkoo()); /* 0 */
7484 : }
7485 : }
7486 :
7487 : long
7488 224 : mfiscuspidal(GEN mf, GEN F)
7489 : {
7490 224 : pari_sp av = avma;
7491 : GEN mf2;
7492 224 : if (space_is_cusp(MF_get_space(mf))) return 1;
7493 98 : if (typ(mf_get_gk(F)) == t_INT)
7494 : {
7495 56 : GEN v = mftobasis(mf,F,0), vE = vecslice(v, 1, lg(MF_get_E(mf))-1);
7496 56 : return gc_long(av, gequal0(vE));
7497 : }
7498 42 : if (!gequal0(mfak_i(F, 0))) return 0;
7499 21 : mf2 = obj_checkbuild(mf, MF_MF2INIT, &mf2init);
7500 21 : return mfiscuspidal(mf2, mfmultheta(F));
7501 : }
7502 :
7503 : /* F = vector of newforms in mftobasis format */
7504 : static GEN
7505 105 : mffrickeeigen_i(GEN mf, GEN F, GEN vE, long prec)
7506 : {
7507 105 : GEN M, Z, L0, gN = MF_get_gN(mf), gk = MF_get_gk(mf);
7508 105 : long N0, i, lM, bit = prec2nbits(prec), k = itou(gk);
7509 105 : long LIM = 5; /* Sturm bound is enough */
7510 :
7511 105 : L0 = mfthetaancreate(NULL, gN, gk); /* only for thetacost */
7512 105 : START:
7513 105 : N0 = lfunthetacost(L0, gen_1, LIM, bit);
7514 105 : M = mfcoefs_mf(mf, N0, 1);
7515 105 : lM = lg(F);
7516 105 : Z = cgetg(lM, t_VEC);
7517 287 : for (i = 1; i < lM; i++)
7518 : { /* expansion of D * F[i] */
7519 182 : GEN D, z, van = RgM_RgC_mul(M, Q_remove_denom(gel(F,i), &D));
7520 182 : GEN L = van_embedall(van, gel(vE,i), gN, gk);
7521 182 : long l = lg(L), j, bit_add = D? expi(D): 0;
7522 182 : gel(Z,i) = z = cgetg(l, t_VEC);
7523 567 : for (j = 1; j < l; j++)
7524 : {
7525 : GEN v, C, C0;
7526 : long m, e;
7527 518 : for (m = 0; m <= LIM; m++)
7528 : {
7529 518 : v = lfuntheta(gmael(L,j,2), gen_1, m, bit);
7530 518 : if (gexpo(v) > bit_add - bit/2) break;
7531 : }
7532 385 : if (m > LIM) { LIM <<= 1; goto START; }
7533 385 : C = mulcxpowIs(gdiv(v,conj_i(v)), 2*m - k);
7534 385 : C0 = grndtoi(C, &e); if (e < 5-prec2nbits(precision(C))) C = C0;
7535 385 : gel(z,j) = C;
7536 : }
7537 : }
7538 105 : return Z;
7539 : }
7540 : static GEN
7541 84 : mffrickeeigen(GEN mf, GEN vE, long prec)
7542 : {
7543 84 : GEN D = obj_check(mf, MF_FRICKE);
7544 84 : if (D) { long p = gprecision(D); if (!p || p >= prec) return D; }
7545 77 : D = mffrickeeigen_i(mf, MF_get_newforms(mf), vE, prec);
7546 77 : return obj_insert(mf, MF_FRICKE, D);
7547 : }
7548 :
7549 : /* integral weight, new space for primitive quadratic character CHIP;
7550 : * MF = vector of embedded eigenforms coefs on mfbasis, by orbit.
7551 : * Assume N > Q > 1 and (Q,f(CHIP)) = 1 */
7552 : static GEN
7553 56 : mfatkineigenquad(GEN mf, GEN CHIP, long Q, GEN MF, long bitprec)
7554 : {
7555 : GEN L0, la2, S, F, vP, tau, wtau, Z, va, vb, den, coe, sqrtQ, sqrtN;
7556 56 : GEN M, gN, gk = MF_get_gk(mf);
7557 56 : long N0, x, yq, i, j, lF, dim, muQ, prec = nbits2prec(bitprec);
7558 56 : long N = MF_get_N(mf), k = itos(gk), NQ = N / Q;
7559 :
7560 : /* Q coprime to FC */
7561 56 : F = MF_get_newforms(mf);
7562 56 : vP = MF_get_fields(mf);
7563 56 : lF = lg(F);
7564 56 : Z = cgetg(lF, t_VEC);
7565 56 : S = MF_get_S(mf); dim = lg(S) - 1;
7566 56 : muQ = mymoebiusu(Q);
7567 56 : if (muQ)
7568 : {
7569 42 : GEN SQ = cgetg(dim+1,t_VEC), Qk = gpow(stoi(Q), sstoQ(k-2, 2), prec);
7570 42 : long i, bit2 = bitprec >> 1;
7571 154 : for (j = 1; j <= dim; j++) gel(SQ,j) = mfak_i(gel(S,j), Q);
7572 84 : for (i = 1; i < lF; i++)
7573 : {
7574 42 : GEN S = RgV_dotproduct(gel(F,i), SQ), T = gel(vP,i);
7575 : long e;
7576 42 : if (degpol(T) > 1 && typ(S) != t_POLMOD) S = gmodulo(S, T);
7577 42 : S = grndtoi(gdiv(conjvec(S, prec), Qk), &e);
7578 42 : if (e > -bit2) pari_err_PREC("mfatkineigenquad");
7579 42 : if (muQ == -1) S = gneg(S);
7580 42 : gel(Z,i) = S;
7581 : }
7582 42 : return Z;
7583 : }
7584 14 : la2 = mfchareval(CHIP, Q); /* 1 or -1 */
7585 14 : (void)cbezout(Q, NQ, &x, &yq);
7586 14 : sqrtQ = sqrtr_abs(utor(Q,prec));
7587 14 : tau = mkcomplex(gadd(sstoQ(-1, NQ), uutoQ(1, 1000)),
7588 : divru(sqrtQ, N));
7589 14 : den = gaddgs(gmulsg(NQ, tau), 1);
7590 14 : wtau = gdiv(gsub(gmulsg(x, tau), sstoQ(yq, Q)), den);
7591 14 : coe = gpowgs(gmul(sqrtQ, den), k);
7592 :
7593 14 : sqrtN = sqrtr_abs(utor(N,prec));
7594 14 : tau = mulcxmI(gmul(tau, sqrtN));
7595 14 : wtau = mulcxmI(gmul(wtau, sqrtN));
7596 14 : gN = utoipos(N);
7597 14 : L0 = mfthetaancreate(NULL, gN, gk); /* only for thetacost */
7598 14 : N0 = maxss(lfunthetacost(L0,real_i(tau), 0,bitprec),
7599 : lfunthetacost(L0,real_i(wtau),0,bitprec));
7600 14 : M = mfcoefs_mf(mf, N0, 1);
7601 14 : va = cgetg(dim+1, t_VEC);
7602 14 : vb = cgetg(dim+1, t_VEC);
7603 105 : for (j = 1; j <= dim; j++)
7604 : {
7605 91 : GEN L, v = vecslice(gel(M,j), 2, N0+1); /* remove a0 */
7606 91 : settyp(v, t_VEC); L = mfthetaancreate(v, gN, gk);
7607 91 : gel(va,j) = lfuntheta(L, tau,0,bitprec);
7608 91 : gel(vb,j) = lfuntheta(L,wtau,0,bitprec);
7609 : }
7610 84 : for (i = 1; i < lF; i++)
7611 : {
7612 70 : GEN z, FE = gel(MF,i);
7613 70 : long l = lg(FE);
7614 70 : z = cgetg(l, t_VEC);
7615 70 : for (j = 1; j < l; j++)
7616 : {
7617 70 : GEN f = gel(FE,j), a = RgV_dotproduct(va,f), b = RgV_dotproduct(vb,f);
7618 70 : GEN la = ground( gdiv(b, gmul(a,coe)) );
7619 70 : if (!gequal(gsqr(la), la2)) pari_err_PREC("mfatkineigenquad");
7620 70 : if (typ(la) == t_INT)
7621 : {
7622 70 : if (j != 1) pari_err_BUG("mfatkineigenquad");
7623 70 : z = const_vec(l-1, la); break;
7624 : }
7625 0 : gel(z,j) = la;
7626 : }
7627 70 : gel(Z,i) = z;
7628 : }
7629 14 : return Z;
7630 : }
7631 :
7632 : static GEN
7633 84 : myusqrt(ulong a, long prec)
7634 : {
7635 84 : if (a == 1UL) return gen_1;
7636 70 : if (uissquareall(a, &a)) return utoipos(a);
7637 49 : return sqrtr_abs(utor(a, prec));
7638 : }
7639 : /* Assume mf is a nontrivial new space, rational primitive character CHIP
7640 : * and (Q,FC) = 1 */
7641 : static GEN
7642 112 : mfatkinmatnewquad(GEN mf, GEN CHIP, long Q, long flag, long PREC)
7643 : {
7644 112 : GEN cM, M, D, MF, den, vE, F = MF_get_newforms(mf);
7645 112 : long i, c, e, prec, bitprec, lF = lg(F), N = MF_get_N(mf), k = MF_get_k(mf);
7646 :
7647 112 : if (Q == 1) return mkvec4(gen_0, matid(MF_get_dim(mf)), gen_1, mf);
7648 112 : den = gel(MF_get_Minv(mf), 2);
7649 112 : bitprec = expi(den) + 64;
7650 112 : if (!flag) bitprec = maxss(bitprec, prec2nbits(PREC));
7651 :
7652 35 : START:
7653 112 : prec = nbits2prec(bitprec);
7654 112 : vE = mfeigenembed(mf, prec);
7655 112 : M = cgetg(lF, t_VEC);
7656 294 : for (i = 1; i < lF; i++) gel(M,i) = RgC_embedall(gel(F,i), gel(vE,i));
7657 112 : if (Q != N)
7658 : {
7659 56 : D = mfatkineigenquad(mf, CHIP, Q, M, bitprec);
7660 56 : c = odd(k)? Q: 1;
7661 : }
7662 : else
7663 : {
7664 56 : D = mffrickeeigen(mf, vE, prec);
7665 56 : c = mfcharmodulus(CHIP); if (odd(k)) c = -Q/c;
7666 : }
7667 112 : D = shallowconcat1(D);
7668 112 : if (vec_isconst(D)) { MF = diagonal_shallow(D); flag = 0; }
7669 : else
7670 : {
7671 63 : M = shallowconcat1(M);
7672 63 : MF = RgM_mul(matmuldiagonal(M,D), ginv(M));
7673 : }
7674 112 : if (!flag) return mkvec4(gen_0, MF, gen_1, mf);
7675 :
7676 21 : if (c > 0)
7677 21 : cM = myusqrt(c, PREC);
7678 : else
7679 : {
7680 0 : MF = imag_i(MF); c = -c;
7681 0 : cM = mkcomplex(gen_0, myusqrt(c,PREC));
7682 : }
7683 21 : if (c != 1) MF = RgM_Rg_mul(MF, myusqrt(c,prec));
7684 21 : MF = grndtoi(RgM_Rg_mul(MF,den), &e);
7685 21 : if (e > -32) { bitprec <<= 1; goto START; }
7686 21 : MF = RgM_Rg_div(MF, den);
7687 21 : if (is_rational_t(typ(cM)) && !isint1(cM))
7688 0 : { MF = RgM_Rg_div(MF, cM); cM = gen_1; }
7689 21 : return mkvec4(gen_0, MF, cM, mf);
7690 : }
7691 :
7692 : /* let CHI mod N, Q || N, return \bar{CHI_Q} * CHI_{N/Q} */
7693 : static GEN
7694 105 : mfcharAL(GEN CHI, long Q)
7695 : {
7696 105 : GEN G = gel(CHI,1), c = gel(CHI,2), cycc, d, P, E, F;
7697 105 : long l = lg(c), N = mfcharmodulus(CHI), i;
7698 105 : if (N == Q) return mfcharconj(CHI);
7699 49 : if (N == 1) return CHI;
7700 42 : CHI = leafcopy(CHI);
7701 42 : gel(CHI,2) = d = leafcopy(c);
7702 42 : F = znstar_get_faN(G);
7703 42 : P = gel(F,1);
7704 42 : E = gel(F,2);
7705 42 : cycc = znstar_get_conreycyc(G);
7706 42 : if (!odd(Q) && equaliu(gel(P,1), 2) && E[1] >= 3)
7707 14 : gel(d,2) = Fp_neg(gel(d,2), gel(cycc,2));
7708 56 : else for (i = 1; i < l; i++)
7709 28 : if (!umodui(Q, gel(P,i))) gel(d,i) = Fp_neg(gel(d,i), gel(cycc,i));
7710 42 : return CHI;
7711 : }
7712 : static long
7713 238 : atkin_get_NQ(long N, long Q, const char *f)
7714 : {
7715 238 : long NQ = N / Q;
7716 238 : if (N % Q) pari_err_DOMAIN(f,"N % Q","!=",gen_0,utoi(Q));
7717 238 : if (ugcd(NQ, Q) > 1) pari_err_DOMAIN(f,"gcd(Q,N/Q)","!=",gen_1,utoi(Q));
7718 238 : return NQ;
7719 : }
7720 :
7721 : /* transform mf to new_NEW if possible */
7722 : static GEN
7723 1379 : MF_set_new(GEN mf)
7724 : {
7725 1379 : GEN vMjd, vj, gk = MF_get_gk(mf);
7726 : long l, j;
7727 1379 : if (MF_get_space(mf) != mf_CUSP
7728 1379 : || typ(gk) != t_INT || itou(gk) == 1) return mf;
7729 182 : vMjd = MFcusp_get_vMjd(mf); l = lg(vMjd);
7730 182 : if (l > 1 && gel(vMjd,1)[1] != MF_get_N(mf)) return mf; /* oldspace != 0 */
7731 175 : mf = shallowcopy(mf);
7732 175 : gel(mf,1) = shallowcopy(gel(mf,1));
7733 175 : MF_set_space(mf, mf_NEW);
7734 175 : vj = cgetg(l, t_VECSMALL);
7735 938 : for (j = 1; j < l; j++) vj[j] = gel(vMjd, j)[2];
7736 175 : gel(mf,4) = vj; return mf;
7737 : }
7738 :
7739 : /* if flag = 1, rationalize, else don't */
7740 : static GEN
7741 217 : mfatkininit_i(GEN mf, long Q, long flag, long prec)
7742 : {
7743 : GEN M, B, C, CHI, CHIAL, G, chi, P, z, g, mfB, s, Mindex, Minv;
7744 217 : long j, l, lim, ord, FC, NQ, cQ, nk, dk, N = MF_get_N(mf);
7745 :
7746 217 : B = MF_get_basis(mf); l = lg(B);
7747 217 : M = cgetg(l, t_MAT); if (l == 1) return mkvec4(gen_0,M,gen_1,mf);
7748 217 : Qtoss(MF_get_gk(mf), &nk,&dk);
7749 217 : Q = labs(Q);
7750 217 : NQ = atkin_get_NQ(N, Q, "mfatkininit");
7751 217 : CHI = MF_get_CHI(mf);
7752 217 : CHI = mfchartoprimitive(CHI, &FC);
7753 217 : ord = mfcharorder(CHI);
7754 217 : mf = MF_set_new(mf);
7755 217 : if (MF_get_space(mf) == mf_NEW && ord <= 2 && NQ % FC == 0 && dk == 1)
7756 112 : return mfatkinmatnewquad(mf, CHI, Q, flag, prec);
7757 : /* now flag != 0 */
7758 105 : G = gel(CHI,1);
7759 105 : chi = gel(CHI,2);
7760 105 : if (Q == N) { g = mkmat22s(0, -1, N, 0); cQ = NQ; } /* Fricke */
7761 : else
7762 : {
7763 28 : GEN F, gQP = utoi(ugcd(Q, FC));
7764 : long t, v;
7765 28 : chi = znchardecompose(G, chi, gQP);
7766 28 : F = znconreyconductor(G, chi, &chi);
7767 28 : G = znstar0(F,1);
7768 28 : (void)cbezout(Q, NQ, &t, &v);
7769 28 : g = mkmat22s(Q*t, 1, -N*v, Q);
7770 28 : cQ = -NQ*v;
7771 : }
7772 105 : C = s = gen_1;
7773 : /* N.B. G,chi are G_Q,chi_Q [primitive] at this point */
7774 105 : if (lg(chi) != 1) C = ginv( znchargauss(G, chi, gen_1, prec2nbits(prec)) );
7775 105 : if (dk == 1)
7776 84 : { if (odd(nk)) s = myusqrt(Q,prec); }
7777 : else
7778 : {
7779 21 : long r = nk >> 1; /* k-1/2 */
7780 21 : s = gpow(utoipos(Q), mkfracss(odd(r)? 1: 3, 4), prec);
7781 21 : if (odd(cQ))
7782 : {
7783 21 : long t = r + ((cQ-1) >> 1);
7784 21 : s = mkcomplex(s, odd(t)? gneg(s): s);
7785 : }
7786 : }
7787 105 : if (!isint1(s)) C = gmul(C, s);
7788 105 : CHIAL = mfcharAL(CHI, Q);
7789 105 : if (dk == 2)
7790 : {
7791 21 : ulong q = odd(Q)? Q << 2: Q, Nq = ulcm(q, mfcharmodulus(CHIAL));
7792 21 : CHIAL = induceN(Nq, CHIAL);
7793 21 : CHIAL = mfcharmul(CHIAL, induce(gel(CHIAL,1), utoipos(q)));
7794 : }
7795 105 : CHIAL = mfchartoprimitive(CHIAL,NULL);
7796 105 : mfB = gequal(CHIAL,CHI)? mf: mfinit_Nndkchi(N,nk,dk,CHIAL,MF_get_space(mf),0);
7797 105 : Mindex = MF_get_Mindex(mfB);
7798 105 : Minv = MF_get_Minv(mfB);
7799 105 : P = z = NULL;
7800 105 : if (ord > 2) { P = mfcharpol(CHI); z = rootsof1u_cx(ord, prec); }
7801 105 : lim = maxss(mfsturm(mfB), mfsturm(mf)) + 1;
7802 343 : for (j = 1; j < l; j++)
7803 : {
7804 238 : GEN v = mfslashexpansion(mf, gel(B,j), g, lim, 0, NULL, prec+EXTRAPREC64);
7805 : long junk;
7806 238 : if (!isint1(C)) v = RgV_Rg_mul(v, C);
7807 238 : v = bestapprnf(v, P, z, prec);
7808 238 : v = vecpermute_partial(v, Mindex, &junk);
7809 238 : v = Minv_RgC_mul(Minv, v); /* cf mftobasis_i */
7810 238 : gel(M, j) = v;
7811 : }
7812 105 : if (is_rational_t(typ(C)) && !gequal1(C)) { M = gdiv(M, C); C = gen_1; }
7813 105 : if (mfB == mf) mfB = gen_0;
7814 105 : return mkvec4(mfB, M, C, mf);
7815 : }
7816 : GEN
7817 91 : mfatkininit(GEN mf, long Q, long prec)
7818 : {
7819 91 : pari_sp av = avma;
7820 91 : mf = checkMF(mf); return gerepilecopy(av, mfatkininit_i(mf, Q, 1, prec));
7821 : }
7822 : static void
7823 56 : checkmfa(GEN z)
7824 : {
7825 56 : if (typ(z) != t_VEC || lg(z) != 5 || typ(gel(z,2)) != t_MAT
7826 56 : || !checkMF_i(gel(z,4))
7827 56 : || (!isintzero(gel(z,1)) && !checkMF_i(gel(z,1))))
7828 0 : pari_err_TYPE("mfatkin [please apply mfatkininit()]",z);
7829 56 : }
7830 :
7831 : /* Apply atkin Q to closure F */
7832 : GEN
7833 56 : mfatkin(GEN mfa, GEN F)
7834 : {
7835 56 : pari_sp av = avma;
7836 : GEN z, mfB, MQ, mf;
7837 56 : checkmfa(mfa);
7838 56 : mfB= gel(mfa,1);
7839 56 : MQ = gel(mfa,2);
7840 56 : mf = gel(mfa,4);
7841 56 : if (typ(mfB) == t_INT) mfB = mf;
7842 56 : z = RgM_RgC_mul(MQ, mftobasis_i(mf,F));
7843 56 : return gerepileupto(av, mflinear(mfB, z));
7844 : }
7845 :
7846 : GEN
7847 49 : mfatkineigenvalues(GEN mf, long Q, long prec)
7848 : {
7849 49 : pari_sp av = avma;
7850 : GEN vF, L, CHI, M, mfatk, C, MQ, vE, mfB;
7851 : long N, NQ, l, i;
7852 :
7853 49 : mf = checkMF(mf); N = MF_get_N(mf);
7854 49 : vF = MF_get_newforms(mf); l = lg(vF);
7855 : /* N.B. k is integral */
7856 49 : if (l == 1) { set_avma(av); return cgetg(1, t_VEC); }
7857 49 : L = cgetg(l, t_VEC);
7858 49 : if (Q == 1)
7859 : {
7860 7 : GEN vP = MF_get_fields(mf);
7861 21 : for (i = 1; i < l; i++) gel(L,i) = const_vec(degpol(gel(vP,i)), gen_1);
7862 7 : return L;
7863 : }
7864 42 : vE = mfeigenembed(mf,prec);
7865 42 : if (Q == N) return gerepileupto(av, mffrickeeigen(mf, vE, prec));
7866 21 : Q = labs(Q);
7867 21 : NQ = atkin_get_NQ(N, Q, "mfatkineigenvalues"); /* != 1 */
7868 21 : mfatk = mfatkininit(mf, Q, prec);
7869 21 : mfB= gel(mfatk,1); if (typ(mfB) != t_VEC) mfB = mf;
7870 21 : MQ = gel(mfatk,2);
7871 21 : C = gel(mfatk,3);
7872 21 : M = row(mfcoefs_mf(mfB,1,1), 2); /* vec of a_1(b_i) for mfbasis functions */
7873 56 : for (i = 1; i < l; i++)
7874 : {
7875 35 : GEN c = RgV_dotproduct(RgM_RgC_mul(MQ,gel(vF,i)), M); /* C * eigen_i */
7876 35 : gel(L,i) = Rg_embedall_i(c, gel(vE,i));
7877 : }
7878 21 : if (!gequal1(C)) L = gdiv(L, C);
7879 21 : CHI = MF_get_CHI(mf);
7880 21 : if (mfcharorder(CHI) <= 2 && NQ % mfcharconductor(CHI) == 0) L = ground(L);
7881 21 : return gerepilecopy(av, L);
7882 : }
7883 :
7884 : /* expand B_d V, keeping same length */
7885 : static GEN
7886 6370 : bdexpand(GEN V, long d)
7887 : {
7888 : GEN W;
7889 : long N, n;
7890 6370 : if (d == 1) return V;
7891 2352 : N = lg(V)-1; W = zerovec(N);
7892 46634 : for (n = 0; n <= (N-1)/d; n++) gel(W, n*d+1) = gel(V, n+1);
7893 2352 : return W;
7894 : }
7895 : /* expand B_d V, increasing length up to lim */
7896 : static GEN
7897 294 : bdexpandall(GEN V, long d, long lim)
7898 : {
7899 : GEN W;
7900 : long N, n;
7901 294 : if (d == 1) return V;
7902 35 : N = lg(V)-1; W = zerovec(lim);
7903 259 : for (n = 0; n <= N-1 && n*d <= lim; n++) gel(W, n*d+1) = gel(V, n+1);
7904 35 : return W;
7905 : }
7906 :
7907 : static void
7908 9366 : parse_vecj(GEN T, GEN *E1, GEN *E2)
7909 : {
7910 9366 : if (lg(T)==3) { *E1 = gel(T,1); *E2 = gel(T,2); }
7911 5180 : else { *E1 = T; *E2 = NULL; }
7912 9366 : }
7913 :
7914 : /* g in M_2(Z) ? */
7915 : static int
7916 2982 : check_M2Z(GEN g)
7917 2982 : { return typ(g) == t_MAT && lg(g) == 3 && lgcols(g) == 3 && RgM_is_ZM(g); }
7918 : /* g in SL_2(Z) ? */
7919 : static int
7920 1806 : check_SL2Z(GEN g) { return check_M2Z(g) && equali1(ZM_det(g)); }
7921 :
7922 : static GEN
7923 9436 : mfcharcxeval(GEN CHI, long n, long prec)
7924 : {
7925 9436 : ulong ord, N = mfcharmodulus(CHI);
7926 : GEN ordg;
7927 9436 : if (N == 1) return gen_1;
7928 3696 : if (ugcd(N, labs(n)) > 1) return gen_0;
7929 3696 : ordg = gmfcharorder(CHI);
7930 3696 : ord = itou(ordg);
7931 3696 : return rootsof1q_cx(znchareval_i(CHI,n,ordg), ord, prec);
7932 : }
7933 :
7934 : static GEN
7935 5159 : RgV_shift(GEN V, GEN gn)
7936 : {
7937 : long i, n, l;
7938 : GEN W;
7939 5159 : if (typ(gn) != t_INT) pari_err_BUG("RgV_shift [n not integral]");
7940 5159 : n = itos(gn);
7941 5159 : if (n < 0) pari_err_BUG("RgV_shift [n negative]");
7942 5159 : if (!n) return V;
7943 112 : W = cgetg_copy(V, &l); if (n > l-1) n = l-1;
7944 308 : for (i=1; i <= n; i++) gel(W,i) = gen_0;
7945 4900 : for ( ; i < l; i++) gel(W,i) = gel(V, i-n);
7946 112 : return W;
7947 : }
7948 : static GEN
7949 7868 : hash_eisengacx(hashtable *H, void *E, long w, GEN ga, long n, long prec)
7950 : {
7951 7868 : ulong h = H->hash(E);
7952 7868 : hashentry *e = hash_search2(H, E, h);
7953 : GEN v;
7954 7868 : if (e) v = (GEN)e->val;
7955 : else
7956 : {
7957 5390 : v = mfeisensteingacx((GEN)E, w, ga, n, prec);
7958 5390 : hash_insert2(H, E, (void*)v, h);
7959 : }
7960 7868 : return v;
7961 : }
7962 : static GEN
7963 5159 : vecj_expand(GEN B, hashtable *H, long w, GEN ga, long n, long prec)
7964 : {
7965 : GEN E1, E2, v;
7966 5159 : parse_vecj(B, &E1, &E2);
7967 5159 : v = hash_eisengacx(H, (void*)E1, w, ga, n, prec);
7968 5159 : if (E2)
7969 : {
7970 2653 : GEN u = hash_eisengacx(H, (void*)E2, w, ga, n, prec);
7971 2653 : GEN a = gadd(gel(v,1), gel(u,1));
7972 2653 : GEN b = RgV_mul_RgXn(gel(v,2), gel(u,2));
7973 2653 : v = mkvec2(a,b);
7974 : }
7975 5159 : return v;
7976 : }
7977 : static GEN
7978 1085 : shift_M(GEN M, GEN Valpha, long w)
7979 : {
7980 1085 : long i, l = lg(Valpha);
7981 1085 : GEN almin = vecmin(Valpha);
7982 6244 : for (i = 1; i < l; i++)
7983 : {
7984 5159 : GEN alpha = gel(Valpha, i), gsh = gmulsg(w, gsub(alpha,almin));
7985 5159 : gel(M,i) = RgV_shift(gel(M,i), gsh);
7986 : }
7987 1085 : return almin;
7988 : }
7989 : static GEN mfeisensteinspaceinit(GEN NK);
7990 : #if 0
7991 : /* ga in M_2^+(Z)), n >= 0 */
7992 : static GEN
7993 : mfgaexpansion_init(GEN mf, GEN ga, long n, long prec)
7994 : {
7995 : GEN M, Mvecj, vecj, almin, Valpha;
7996 : long i, w, l, N = MF_get_N(mf), c = itos(gcoeff(ga,2,1));
7997 : hashtable *H;
7998 :
7999 : if (c % N == 0)
8000 : { /* ga in G_0(N), trivial case; w = 1 */
8001 : GEN chid = mfcharcxeval(MF_get_CHI(mf), itos(gcoeff(ga,2,2)), prec);
8002 : return mkvec2(chid, utoi(n));
8003 : }
8004 :
8005 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
8006 : if (lg(Mvecj) < 5) pari_err_IMPL("mfgaexpansion_init in this case");
8007 : w = mfcuspcanon_width(N, c);
8008 : vecj = gel(Mvecj, 3);
8009 : l = lg(vecj);
8010 : M = cgetg(l, t_VEC);
8011 : Valpha = cgetg(l, t_VEC);
8012 : H = hash_create(l, (ulong(*)(void*))&hash_GEN,
8013 : (int(*)(void*,void*))&gidentical, 1);
8014 : for (i = 1; i < l; i++)
8015 : {
8016 : GEN v = vecj_expand(gel(vecj,i), H, w, ga, n, prec);
8017 : gel(Valpha,i) = gel(v,1);
8018 : gel(M,i) = gel(v,2);
8019 : }
8020 : almin = shift_M(M, Valpha, w);
8021 : return mkvec3(almin, utoi(w), M);
8022 : }
8023 : /* half-integer weight not supported; vF = [F,eisendec(F)].
8024 : * Minit = mfgaexpansion_init(mf, ga, n, prec) */
8025 : static GEN
8026 : mfgaexpansion_with_init(GEN Minit, GEN vF)
8027 : {
8028 : GEN v;
8029 : if (lg(Minit) == 3)
8030 : { /* ga in G_0(N) */
8031 : GEN chid = gel(Minit,1), gn = gel(Minit,2);
8032 : v = mfcoefs_i(gel(vF,1), itou(gn), 1);
8033 : v = mkvec3(gen_0, gen_1, RgV_Rg_mul(v,chid));
8034 : }
8035 : else
8036 : {
8037 : GEN V = RgM_RgC_mul(gel(Minit,3), gel(vF,2));
8038 : v = mkvec3(gel(Minit,1), gel(Minit,2), V);
8039 : }
8040 : return v;
8041 : }
8042 : #endif
8043 :
8044 : /* B = mfeisensteindec(F) already embedded, ga in M_2^+(Z)), n >= 0 */
8045 : static GEN
8046 1085 : mfgaexpansion_i(GEN mf, GEN B0, GEN ga, long n, long prec)
8047 : {
8048 1085 : GEN M, Mvecj, vecj, almin, Valpha, B, E = NULL;
8049 1085 : long i, j, w, nw, l, N = MF_get_N(mf), bit = prec2nbits(prec) / 2;
8050 : hashtable *H;
8051 :
8052 1085 : Mvecj = obj_check(mf, MF_EISENSPACE);
8053 1085 : if (lg(Mvecj) < 5) { E = gel(Mvecj, 2); Mvecj = gel(Mvecj, 1); }
8054 1085 : vecj = gel(Mvecj, 3);
8055 1085 : l = lg(vecj);
8056 1085 : B = cgetg(l, t_COL);
8057 1085 : M = cgetg(l, t_VEC);
8058 1085 : Valpha = cgetg(l, t_VEC);
8059 1085 : w = mfZC_width(N, gel(ga,1));
8060 1085 : nw = E ? n + w : n;
8061 1085 : H = hash_create(l, (ulong(*)(void*))&hash_GEN,
8062 : (int(*)(void*,void*))&gidentical, 1);
8063 9177 : for (i = j = 1; i < l; i++)
8064 : {
8065 : GEN v;
8066 8092 : if (gequal0(gel(B0,i))) continue;
8067 5159 : v = vecj_expand(gel(vecj,i), H, w, ga, nw, prec);
8068 5159 : gel(B,j) = gel(B0,i);
8069 5159 : gel(Valpha,j) = gel(v,1);
8070 5159 : gel(M,j) = gel(v,2); j++;
8071 : }
8072 1085 : setlg(Valpha, j);
8073 1085 : setlg(B, j);
8074 1085 : setlg(M, j); l = j;
8075 1085 : if (l == 1) return mkvec3(gen_0, utoi(w), zerovec(n+1));
8076 1085 : almin = shift_M(M, Valpha, w);
8077 1085 : B = RgM_RgC_mul(M, B); l = lg(B);
8078 151445 : for (i = 1; i < l; i++)
8079 150360 : if (gexpo(gel(B,i)) < -bit) gel(B,i) = gen_0;
8080 1085 : settyp(B, t_VEC);
8081 1085 : if (E)
8082 : {
8083 : GEN v, e;
8084 56 : long ell = 0, vB, ve;
8085 126 : for (i = 1; i < l; i++)
8086 126 : if (!gequal0(gel(B,i))) break;
8087 56 : vB = i-1;
8088 56 : v = hash_eisengacx(H, (void*)E, w, ga, n + vB, prec);
8089 56 : e = gel(v,2); l = lg(e);
8090 56 : for (i = 1; i < l; i++)
8091 56 : if (!gequal0(gel(e,i))) break;
8092 56 : ve = i-1;
8093 56 : almin = gsub(almin, gel(v,1));
8094 56 : if (gsigne(almin) < 0)
8095 : {
8096 0 : GEN gell = gceil(gmulsg(-w, almin));
8097 0 : ell = itos(gell);
8098 0 : almin = gadd(almin, gdivgu(gell, w));
8099 0 : if (nw < ell) pari_err_IMPL("alpha < 0 in mfgaexpansion");
8100 : }
8101 56 : if (ve) { ell += ve; e = vecslice(e, ve+1, l-1); }
8102 56 : B = vecslice(B, ell + 1, minss(n + ell + 1, lg(B)-1));
8103 56 : B = RgV_div_RgXn(B, e);
8104 : }
8105 1085 : return mkvec3(almin, utoi(w), B);
8106 : }
8107 :
8108 : /* Theta multiplier: assume 4 | C, (C,D)=1 */
8109 : static GEN
8110 343 : mfthetamultiplier(GEN C, GEN D)
8111 : {
8112 343 : long s = kronecker(C, D);
8113 343 : if (Mod4(D) == 1) return s > 0 ? gen_1: gen_m1;
8114 84 : return s > 0? powIs(3): gen_I();
8115 : }
8116 : /* theta | [*,*;C,D] defined over Q(i) [else over Q] */
8117 : static int
8118 56 : mfthetaI(long C, long D) { return odd(C) || (D & 3) == 3; }
8119 : /* (theta | M) [0..n], assume (C,D) = 1 */
8120 : static GEN
8121 343 : mfthetaexpansion(GEN M, long n)
8122 : {
8123 343 : GEN w, s, al, sla, E, V = zerovec(n+1), C = gcoeff(M,2,1), D = gcoeff(M,2,2);
8124 343 : long lim, la, f, C4 = Mod4(C);
8125 343 : switch (C4)
8126 : {
8127 70 : case 0: al = gen_0; w = gen_1;
8128 70 : s = mfthetamultiplier(C,D);
8129 70 : lim = usqrt(n); gel(V, 1) = s;
8130 70 : s = gmul2n(s, 1);
8131 756 : for (f = 1; f <= lim; f++) gel(V, f*f + 1) = s;
8132 70 : break;
8133 105 : case 2: al = uutoQ(1,4); w = gen_1;
8134 105 : E = subii(C, shifti(D,1)); /* (E, D) = 1 */
8135 105 : s = gmul2n(mfthetamultiplier(E, D), 1);
8136 105 : if ((!signe(E) && equalim1(D)) || (signe(E) > 0 && signe(C) < 0))
8137 14 : s = gneg(s);
8138 105 : lim = (usqrt(n << 2) - 1) >> 1;
8139 966 : for (f = 0; f <= lim; f++) gel(V, f*(f+1) + 1) = s;
8140 105 : break;
8141 168 : default: al = gen_0; w = utoipos(4);
8142 168 : la = (-Mod4(D)*C4) & 3L;
8143 168 : E = negi(addii(D, mului(la, C)));
8144 168 : s = mfthetamultiplier(E, C); /* (E,C) = 1 */
8145 168 : if (signe(C) < 0 && signe(E) >= 0) s = gneg(s);
8146 168 : s = gsub(s, mulcxI(s));
8147 168 : sla = gmul(s, powIs(-la));
8148 168 : lim = usqrt(n); gel(V, 1) = gmul2n(s, -1);
8149 1708 : for (f = 1; f <= lim; f++) gel(V, f*f + 1) = odd(f) ? sla : s;
8150 168 : break;
8151 : }
8152 343 : return mkvec3(al, w, V);
8153 : }
8154 :
8155 : /* F 1/2 integral weight */
8156 : static GEN
8157 343 : mf2gaexpansion(GEN mf2, GEN F, GEN ga, long n, long prec)
8158 : {
8159 343 : GEN FT = mfmultheta(F), mf = obj_checkbuild(mf2, MF_MF2INIT, &mf2init);
8160 343 : GEN res, V1, Tres, V2, al, V, gsh, C = gcoeff(ga,2,1);
8161 343 : long w2, N = MF_get_N(mf), w = mfcuspcanon_width(N, umodiu(C,N));
8162 343 : long ext = (Mod4(C) != 2)? 0: (w+3) >> 2;
8163 343 : long prec2 = prec + nbits2extraprec((long)M_PI/(2*M_LN2)*sqrt(n + ext));
8164 343 : res = mfgaexpansion(mf, FT, ga, n + ext, prec2);
8165 343 : Tres = mfthetaexpansion(ga, n + ext);
8166 343 : V1 = gel(res,3);
8167 343 : V2 = gel(Tres,3);
8168 343 : al = gsub(gel(res,1), gel(Tres,1));
8169 343 : w2 = itos(gel(Tres,2));
8170 343 : if (w != itos(gel(res,2)) || w % w2)
8171 0 : pari_err_BUG("mf2gaexpansion [incorrect w2 or w]");
8172 343 : if (w2 != w) V2 = bdexpand(V2, w/w2);
8173 343 : V = RgV_div_RgXn(V1, V2);
8174 343 : gsh = gfloor(gmulsg(w, al));
8175 343 : if (!gequal0(gsh))
8176 : {
8177 35 : al = gsub(al, gdivgu(gsh, w));
8178 35 : if (gsigne(gsh) > 0)
8179 : {
8180 0 : V = RgV_shift(V, gsh);
8181 0 : V = vecslice(V, 1, n + 1);
8182 : }
8183 : else
8184 : {
8185 35 : long sh = -itos(gsh), i;
8186 35 : if (sh > ext) pari_err_BUG("mf2gaexpansion [incorrect sh]");
8187 154 : for (i = 1; i <= sh; i++)
8188 119 : if (!gequal0(gel(V,i))) pari_err_BUG("mf2gaexpansion [sh too large]");
8189 35 : V = vecslice(V, sh+1, n + sh+1);
8190 : }
8191 : }
8192 343 : obj_free(mf); return mkvec3(al, stoi(w), gprec_wtrunc(V, prec));
8193 : }
8194 :
8195 : static GEN
8196 77 : mfgaexpansionatkin(GEN mf, GEN F, GEN C, GEN D, long Q, long n, long prec)
8197 : {
8198 77 : GEN mfa = mfatkininit_i(mf, Q, 0, prec), MQ = gel(mfa,2);
8199 77 : long i, FC, k = MF_get_k(mf);
8200 77 : GEN x, v, V, z, s, CHI = mfchartoprimitive(MF_get_CHI(mf), &FC);
8201 :
8202 : /* V = mfcoefs(F | w_Q, n), can't use mfatkin because MQ nonrational */
8203 77 : V = RgM_RgC_mul(mfcoefs_mf(mf,n,1), RgM_RgC_mul(MQ, mftobasis_i(mf,F)));
8204 77 : (void)bezout(utoipos(Q), C, &x, &v);
8205 77 : s = mfchareval(CHI, (umodiu(x, FC) * umodiu(D, FC)) % FC);
8206 77 : s = gdiv(s, gpow(utoipos(Q), uutoQ(k,2), prec));
8207 77 : V = RgV_Rg_mul(V, s);
8208 77 : z = rootsof1powinit(umodiu(D,Q)*umodiu(v,Q) % Q, Q, prec);
8209 11613 : for (i = 1; i <= n+1; i++) gel(V,i) = gmul(gel(V,i), rootsof1pow(z, i-1));
8210 77 : return mkvec3(gen_0, utoipos(Q), V);
8211 : }
8212 :
8213 : static long
8214 70 : inveis_extraprec(long N, GEN ga, GEN Mvecj, long n)
8215 : {
8216 70 : long e, w = mfZC_width(N, gel(ga,1));
8217 70 : GEN f, E = gel(Mvecj,2), v = mfeisensteingacx(E, w, ga, n, DEFAULTPREC);
8218 70 : v = gel(v,2);
8219 70 : f = RgV_to_RgX(v,0); n -= RgX_valrem(f, &f);
8220 70 : e = gexpo(RgXn_inv(f, n+1));
8221 70 : return (e > 0)? nbits2extraprec(e): 0;
8222 : }
8223 : /* allow F of the form [F, mf_eisendec(F)]~ */
8224 : static GEN
8225 1799 : mfgaexpansion(GEN mf, GEN F, GEN ga, long n, long prec)
8226 : {
8227 1799 : GEN v, EF = NULL, res, Mvecj, c, d;
8228 : long precnew, N;
8229 :
8230 1799 : if (n < 0) pari_err_DOMAIN("mfgaexpansion", "n", "<", gen_0, stoi(n));
8231 1799 : if (typ(F) == t_COL && lg(F) == 3) { EF = gel(F,2); F = gel(F,1); }
8232 1799 : if (!checkmf_i(F)) pari_err_TYPE("mfgaexpansion", F);
8233 1799 : if (!check_SL2Z(ga)) pari_err_TYPE("mfgaexpansion",ga);
8234 1799 : if (typ(mf_get_gk(F)) != t_INT) return mf2gaexpansion(mf, F, ga, n, prec);
8235 1456 : c = gcoeff(ga,2,1);
8236 1456 : d = gcoeff(ga,2,2);
8237 1456 : N = MF_get_N(mf);
8238 1456 : if (!umodiu(c, mf_get_N(F)))
8239 : { /* trivial case: ga in Gamma_0(N) */
8240 294 : long w = mfcuspcanon_width(N, umodiu(c,N));
8241 294 : GEN CHI = mf_get_CHI(F);
8242 294 : GEN chid = mfcharcxeval(CHI, umodiu(d,mfcharmodulus(CHI)), prec);
8243 294 : v = mfcoefs_i(F, n/w, 1); if (!isint1(chid)) v = RgV_Rg_mul(v,chid);
8244 294 : return mkvec3(gen_0, stoi(w), bdexpandall(v,w,n+1));
8245 : }
8246 1162 : mf = MF_set_new(mf);
8247 1162 : if (MF_get_space(mf) == mf_NEW)
8248 : {
8249 483 : long cN = umodiu(c,N), g = ugcd(cN,N), Q = N/g;
8250 483 : GEN CHI = MF_get_CHI(mf);
8251 483 : if (ugcd(cN, Q)==1 && mfcharorder(CHI) <= 2
8252 231 : && g % mfcharconductor(CHI) == 0
8253 119 : && degpol(mf_get_field(F)) == 1)
8254 77 : return mfgaexpansionatkin(mf, F, c, d, Q, n, prec);
8255 : }
8256 1085 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
8257 1085 : precnew = prec;
8258 1085 : if (lg(Mvecj) < 5) precnew += inveis_extraprec(N, ga, Mvecj, n);
8259 1085 : if (!EF) EF = mf_eisendec(mf, F, precnew);
8260 1085 : res = mfgaexpansion_i(mf, EF, ga, n, precnew);
8261 1085 : return precnew == prec ? res : gprec_wtrunc(res, prec);
8262 : }
8263 :
8264 : /* parity = -1 or +1 */
8265 : static GEN
8266 217 : findd(long N, long parity)
8267 : {
8268 217 : GEN L, D = mydivisorsu(N);
8269 217 : long i, j, l = lg(D);
8270 217 : L = cgetg(l, t_VEC);
8271 1218 : for (i = j = 1; i < l; i++)
8272 : {
8273 1001 : long d = D[i];
8274 1001 : if (parity == -1) d = -d;
8275 1001 : if (sisfundamental(d)) gel(L,j++) = stoi(d);
8276 : }
8277 217 : setlg(L,j); return L;
8278 : }
8279 : /* does ND contain a divisor of N ? */
8280 : static int
8281 413 : seenD(long N, GEN ND)
8282 : {
8283 413 : long j, l = lg(ND);
8284 427 : for (j = 1; j < l; j++)
8285 14 : if (N % ND[j] == 0) return 1;
8286 413 : return 0;
8287 : }
8288 : static GEN
8289 63 : search_levels(GEN vN, const char *f)
8290 : {
8291 63 : switch(typ(vN))
8292 : {
8293 28 : case t_INT: vN = mkvecsmall(itos(vN)); break;
8294 35 : case t_VEC: case t_COL: vN = ZV_to_zv(vN); break;
8295 0 : case t_VECSMALL: vN = leafcopy(vN); break;
8296 0 : default: pari_err_TYPE(f, vN);
8297 : }
8298 63 : vecsmall_sort(vN); return vN;
8299 : }
8300 : GEN
8301 28 : mfsearch(GEN NK, GEN V, long space)
8302 : {
8303 28 : pari_sp av = avma;
8304 : GEN F, gk, NbyD, vN;
8305 : long n, nk, dk, parity, nV, i, lvN;
8306 :
8307 28 : if (typ(NK) != t_VEC || lg(NK) != 3) pari_err_TYPE("mfsearch", NK);
8308 28 : gk = gel(NK,2);
8309 28 : if (typ(gmul2n(gk, 1)) != t_INT) pari_err_TYPE("mfsearch [k]", gk);
8310 28 : switch(typ(V))
8311 : {
8312 28 : case t_VEC: V = shallowtrans(V);
8313 28 : case t_COL: break;
8314 0 : default: pari_err_TYPE("mfsearch [V]", V);
8315 : }
8316 28 : vN = search_levels(gel(NK,1), "mfsearch [N]");
8317 28 : if (gequal0(V)) { set_avma(av); retmkvec(mftrivial()); }
8318 14 : lvN = lg(vN);
8319 :
8320 14 : Qtoss(gk, &nk,&dk);
8321 14 : parity = (dk == 1 && odd(nk)) ? -1 : 1;
8322 14 : nV = lg(V)-2;
8323 14 : F = cgetg(1, t_VEC);
8324 14 : NbyD = const_vec(vN[lvN-1], cgetg(1,t_VECSMALL));
8325 231 : for (n = 1; n < lvN; n++)
8326 : {
8327 217 : long N = vN[n];
8328 : GEN L;
8329 217 : if (N <= 0 || (dk == 2 && (N & 3))) continue;
8330 217 : L = findd(N, parity);
8331 630 : for (i = 1; i < lg(L); i++)
8332 : {
8333 413 : GEN mf, M, CO, gD = gel(L,i);
8334 413 : GEN *ND = (GEN*)NbyD + itou(gD); /* points to NbyD[|D|] */
8335 :
8336 413 : if (seenD(N, *ND)) continue;
8337 413 : mf = mfinit_Nndkchi(N, nk, dk, get_mfchar(gD), space, 1);
8338 413 : M = mfcoefs_mf(mf, nV, 1);
8339 413 : CO = inverseimage(M, V); if (lg(CO) == 1) continue;
8340 :
8341 42 : F = vec_append(F, mflinear(mf,CO));
8342 42 : *ND = vecsmall_append(*ND, N); /* add to NbyD[|D|] */
8343 : }
8344 : }
8345 14 : return gerepilecopy(av, F);
8346 : }
8347 :
8348 : static GEN
8349 889 : search_from_split(GEN mf, GEN vap, GEN vlp)
8350 : {
8351 889 : pari_sp av = avma;
8352 889 : long lvlp = lg(vlp), j, jv, l1;
8353 889 : GEN v, NK, S1, S, M = NULL;
8354 :
8355 889 : S1 = gel(split_i(mf, 1, 0), 1); /* rational newforms */
8356 889 : l1 = lg(S1);
8357 889 : if (l1 == 1) return gc_NULL(av);
8358 455 : v = cgetg(l1, t_VEC);
8359 455 : S = MF_get_S(mf);
8360 455 : NK = mf_get_NK(gel(S,1));
8361 455 : if (lvlp > 1) M = rowpermute(mfcoefs_mf(mf, vlp[lvlp-1], 1), vlp);
8362 980 : for (j = jv = 1; j < l1; j++)
8363 : {
8364 525 : GEN vF = gel(S1,j);
8365 : long t;
8366 658 : for (t = lvlp-1; t > 0; t--)
8367 : { /* lhs = vlp[j]-th coefficient of eigenform */
8368 595 : GEN rhs = gel(vap,t), lhs = RgMrow_RgC_mul(M, vF, t);
8369 595 : if (!gequal(lhs, rhs)) break;
8370 : }
8371 525 : if (!t) gel(v,jv++) = mflinear_i(NK,S,vF);
8372 : }
8373 455 : if (jv == 1) return gc_NULL(av);
8374 63 : setlg(v,jv); return v;
8375 : }
8376 : GEN
8377 35 : mfeigensearch(GEN NK, GEN AP)
8378 : {
8379 35 : pari_sp av = avma;
8380 35 : GEN k, vN, vap, vlp, vres = cgetg(1, t_VEC), D;
8381 : long n, lvN, i, l, even;
8382 :
8383 35 : if (!AP) l = 1;
8384 : else
8385 : {
8386 28 : l = lg(AP);
8387 28 : if (typ(AP) != t_VEC) pari_err_TYPE("mfeigensearch",AP);
8388 : }
8389 35 : vap = cgetg(l, t_VEC);
8390 35 : vlp = cgetg(l, t_VECSMALL);
8391 35 : if (l > 1)
8392 : {
8393 28 : GEN perm = indexvecsort(AP, mkvecsmall(1));
8394 77 : for (i = 1; i < l; i++)
8395 : {
8396 49 : GEN v = gel(AP,perm[i]), gp, ap;
8397 49 : if (typ(v) != t_VEC || lg(v) != 3) pari_err_TYPE("mfeigensearch", AP);
8398 49 : gp = gel(v,1);
8399 49 : ap = gel(v,2);
8400 49 : if (typ(gp) != t_INT || (typ(ap) != t_INT && typ(ap) != t_INTMOD))
8401 0 : pari_err_TYPE("mfeigensearch", AP);
8402 49 : gel(vap,i) = ap;
8403 49 : vlp[i] = itos(gp)+1; if (vlp[i] < 0) pari_err_TYPE("mfeigensearch", AP);
8404 : }
8405 : }
8406 35 : l = lg(NK);
8407 35 : if (typ(NK) != t_VEC || l != 3) pari_err_TYPE("mfeigensearch",NK);
8408 35 : k = gel(NK,2);
8409 35 : vN = search_levels(gel(NK,1), "mfeigensearch [N]");
8410 35 : lvN = lg(vN);
8411 35 : vecsmall_sort(vlp);
8412 35 : even = !mpodd(k);
8413 980 : for (n = 1; n < lvN; n++)
8414 : {
8415 945 : pari_sp av2 = avma;
8416 : GEN mf, L;
8417 945 : long N = vN[n];
8418 945 : if (even) D = gen_1;
8419 : else
8420 : {
8421 112 : long r = (N&3L);
8422 112 : if (r == 1 || r == 2) continue;
8423 56 : D = stoi( corediscs(-N, NULL) ); /* < 0 */
8424 : }
8425 889 : mf = mfinit_i(mkvec3(utoipos(N), k, D), mf_NEW);
8426 889 : L = search_from_split(mf, vap, vlp);
8427 889 : if (L) vres = shallowconcat(vres, L); else set_avma(av2);
8428 : }
8429 35 : return gerepilecopy(av, vres);
8430 : }
8431 :
8432 : /* tf_{N,k}(n) */
8433 : static GEN
8434 4501119 : mfnewtracecache(long N, long k, long n, cachenew_t *cache)
8435 : {
8436 4501119 : GEN C = NULL, S;
8437 : long lcache;
8438 4501119 : if (!n) return gen_0;
8439 4363093 : S = gel(cache->vnew,N);
8440 4363093 : lcache = lg(S);
8441 4363093 : if (n < lcache) C = gel(S, n);
8442 4363093 : if (C) cache->newHIT++;
8443 2599588 : else C = mfnewtrace_i(N,k,n,cache);
8444 4363093 : cache->newTOTAL++;
8445 4363093 : if (n < lcache) gel(S,n) = C;
8446 4363093 : return C;
8447 : }
8448 :
8449 : static long
8450 1393 : mfdim_Nkchi(long N, long k, GEN CHI, long space)
8451 : {
8452 1393 : if (k < 0 || badchar(N,k,CHI)) return 0;
8453 1092 : if (k == 0)
8454 35 : return mfcharistrivial(CHI) && !space_is_cusp(space)? 1: 0;
8455 1057 : switch(space)
8456 : {
8457 245 : case mf_NEW: return mfnewdim(N,k,CHI);
8458 196 : case mf_CUSP:return mfcuspdim(N,k,CHI);
8459 168 : case mf_OLD: return mfolddim(N,k,CHI);
8460 217 : case mf_FULL:return mffulldim(N,k,CHI);
8461 231 : case mf_EISEN: return mfeisensteindim(N,k,CHI);
8462 0 : default: pari_err_FLAG("mfdim");
8463 : }
8464 : return 0;/*LCOV_EXCL_LINE*/
8465 : }
8466 : static long
8467 2114 : mf1dimsum(long N, long space)
8468 : {
8469 2114 : switch(space)
8470 : {
8471 1050 : case mf_NEW: return mf1newdimsum(N);
8472 1057 : case mf_CUSP: return mf1cuspdimsum(N);
8473 7 : case mf_OLD: return mf1olddimsum(N);
8474 : }
8475 0 : pari_err_FLAG("mfdim");
8476 : return 0; /*LCOV_EXCL_LINE*/
8477 : }
8478 : /* mfdim for k = nk/dk */
8479 : static long
8480 44744 : mfdim_Nndkchi(long N, long nk, long dk, GEN CHI, long space)
8481 43463 : { return (dk == 2)? mf2dim_Nkchi(N, nk >> 1, CHI, space)
8482 88186 : : mfdim_Nkchi(N, nk, CHI, space); }
8483 : /* FIXME: use direct dim Gamma1(N) formula, don't compute individual spaces */
8484 : static long
8485 252 : mfkdimsum(long N, long k, long dk, long space)
8486 : {
8487 252 : GEN w = mfchars(N, k, dk, NULL);
8488 252 : long i, j, D = 0, l = lg(w);
8489 1239 : for (i = j = 1; i < l; i++)
8490 : {
8491 987 : GEN CHI = gel(w,i);
8492 987 : long d = mfdim_Nndkchi(N,k,dk,CHI,space);
8493 987 : if (d) D += d * myeulerphiu(mfcharorder(CHI));
8494 : }
8495 252 : return D;
8496 : }
8497 : static GEN
8498 105 : mf1dims(long N, GEN vCHI, long space)
8499 : {
8500 105 : GEN D = NULL;
8501 105 : switch(space)
8502 : {
8503 56 : case mf_NEW: D = mf1newdimall(N, vCHI); break;
8504 21 : case mf_CUSP:D = mf1cuspdimall(N, vCHI); break;
8505 28 : case mf_OLD: D = mf1olddimall(N, vCHI); break;
8506 0 : default: pari_err_FLAG("mfdim");
8507 : }
8508 105 : return D;
8509 : }
8510 : static GEN
8511 2961 : mfkdims(long N, long k, long dk, GEN vCHI, long space)
8512 : {
8513 2961 : GEN D, w = mfchars(N, k, dk, vCHI);
8514 2961 : long i, j, l = lg(w);
8515 2961 : D = cgetg(l, t_VEC);
8516 46592 : for (i = j = 1; i < l; i++)
8517 : {
8518 43631 : GEN CHI = gel(w,i);
8519 43631 : long d = mfdim_Nndkchi(N,k,dk,CHI,space);
8520 43631 : if (vCHI)
8521 574 : gel(D, j++) = mkvec2s(d, 0);
8522 43057 : else if (d)
8523 2520 : gel(D, j++) = fmt_dim(CHI, d, 0);
8524 : }
8525 2961 : setlg(D,j); return D;
8526 : }
8527 : GEN
8528 5719 : mfdim(GEN NK, long space)
8529 : {
8530 5719 : pari_sp av = avma;
8531 : long N, k, dk, joker;
8532 : GEN CHI, mf;
8533 5719 : if ((mf = checkMF_i(NK))) return utoi(MF_get_dim(mf));
8534 5586 : checkNK2(NK, &N, &k, &dk, &CHI, 2);
8535 5586 : if (!CHI) joker = 1;
8536 : else
8537 2611 : switch(typ(CHI))
8538 : {
8539 2373 : case t_INT: joker = 2; break;
8540 112 : case t_COL: joker = 3; break;
8541 126 : default: joker = 0; break;
8542 : }
8543 5586 : if (joker)
8544 : {
8545 : long d;
8546 : GEN D;
8547 5460 : if (k < 0) switch(joker)
8548 : {
8549 0 : case 1: return cgetg(1,t_VEC);
8550 7 : case 2: return gen_0;
8551 0 : case 3: return mfdim0all(CHI);
8552 : }
8553 5453 : if (k == 0)
8554 : {
8555 28 : if (space_is_cusp(space)) switch(joker)
8556 : {
8557 7 : case 1: return cgetg(1,t_VEC);
8558 0 : case 2: return gen_0;
8559 7 : case 3: return mfdim0all(CHI);
8560 : }
8561 14 : switch(joker)
8562 : {
8563 : long i, l;
8564 7 : case 1: retmkvec(fmt_dim(mfchartrivial(),0,0));
8565 0 : case 2: return gen_1;
8566 7 : case 3: l = lg(CHI); D = cgetg(l,t_VEC);
8567 35 : for (i = 1; i < l; i++)
8568 : {
8569 28 : long t = mfcharistrivial(gel(CHI,i));
8570 28 : gel(D,i) = mkvec2(t? gen_1: gen_0, gen_0);
8571 : }
8572 7 : return D;
8573 : }
8574 : }
8575 5425 : if (dk == 1 && k == 1 && space != mf_EISEN)
8576 105 : {
8577 2219 : long fix = 0, space0 = space;
8578 2219 : if (space == mf_FULL) space = mf_CUSP; /* remove Eisenstein part */
8579 2219 : if (joker == 2)
8580 : {
8581 2114 : d = mf1dimsum(N, space);
8582 2114 : if (space0 == mf_FULL) d += mfkdimsum(N,k,dk,mf_EISEN);/*add it back*/
8583 2114 : return gc_utoi(av, d);
8584 : }
8585 : /* must initialize explicitly: trivial spaces for E_k/S_k differ */
8586 105 : if (space0 == mf_FULL)
8587 : {
8588 7 : if (!CHI) fix = 1; /* must remove 0 spaces */
8589 7 : CHI = mfchars(N, k, dk, CHI);
8590 : }
8591 105 : D = mf1dims(N, CHI, space);
8592 105 : if (space0 == mf_FULL)
8593 : {
8594 7 : GEN D2 = mfkdims(N, k, dk, CHI, mf_EISEN);
8595 7 : D = merge_dims(D, D2, fix? CHI: NULL);
8596 : }
8597 : }
8598 : else
8599 : {
8600 3206 : if (joker==2) { d = mfkdimsum(N,k,dk,space); return gc_utoi(av,d); }
8601 2954 : D = mfkdims(N, k, dk, CHI, space);
8602 : }
8603 3059 : if (!CHI) return gerepileupto(av, vecsort(D, mkvecsmall(1)));
8604 105 : return gerepilecopy(av, D);
8605 : }
8606 126 : return utoi( mfdim_Nndkchi(N, k, dk, CHI, space) );
8607 : }
8608 :
8609 : GEN
8610 357 : mfbasis(GEN NK, long space)
8611 : {
8612 357 : pari_sp av = avma;
8613 : long N, k, dk;
8614 : GEN mf, CHI;
8615 357 : if ((mf = checkMF_i(NK))) return gconcat(gel(mf,2), gel(mf,3));
8616 14 : checkNK2(NK, &N, &k, &dk, &CHI, 0);
8617 14 : if (dk == 2) return gerepilecopy(av, mf2basis(N, k>>1, CHI, NULL, space));
8618 14 : mf = mfinit_Nkchi(N, k, CHI, space, 1);
8619 14 : return gerepilecopy(av, MF_get_basis(mf));
8620 : }
8621 :
8622 : static GEN
8623 49 : deg1ser_shallow(GEN a1, GEN a0, long v, long e)
8624 49 : { return RgX_to_ser(deg1pol_shallow(a1, a0, v), e+2); }
8625 : /* r / x + O(1) */
8626 : static GEN
8627 49 : simple_pole(GEN r)
8628 : {
8629 49 : GEN S = deg1ser_shallow(gen_0, r, 0, 1);
8630 49 : setvalser(S, -1); return S;
8631 : }
8632 :
8633 : /* F form, E embedding; mfa = mfatkininit or root number (eigenform case) */
8634 : static GEN
8635 161 : mflfuncreate(GEN mfa, GEN F, GEN E, GEN N, GEN gk)
8636 : {
8637 161 : GEN LF = cgetg(8,t_VEC), polar = cgetg(1,t_COL), eps;
8638 161 : long k = itou(gk);
8639 161 : gel(LF,1) = lfuntag(t_LFUN_MFCLOS, mkvec3(F,E,gen_1));
8640 161 : if (typ(mfa) != t_VEC)
8641 98 : eps = mfa; /* cuspidal eigenform: root number; no poles */
8642 : else
8643 : { /* mfatkininit */
8644 63 : GEN a0, b0, vF, vG, G = NULL;
8645 63 : GEN M = gel(mfa,2), C = gel(mfa,3), mf = gel(mfa,4);
8646 63 : M = gdiv(mfmatembed(E, M), C);
8647 63 : vF = mfvecembed(E, mftobasis_i(mf, F));
8648 63 : vG = RgM_RgC_mul(M, vF);
8649 63 : if (gequal(vF,vG)) eps = gen_1;
8650 49 : else if (gequal(vF,gneg(vG))) eps = gen_m1;
8651 : else
8652 : { /* not self-dual */
8653 42 : eps = NULL;
8654 42 : G = mfatkin(mfa, F);
8655 42 : gel(LF,2) = lfuntag(t_LFUN_MFCLOS, mkvec3(G,E,ginv(C)));
8656 42 : gel(LF,6) = powIs(k);
8657 : }
8658 : /* polar part */
8659 63 : a0 = mfembed(E, mfcoef(F,0));
8660 63 : b0 = eps? gmul(eps,a0): gdiv(mfembed(E, mfcoef(G,0)), C);
8661 63 : if (!gequal0(b0))
8662 : {
8663 28 : b0 = mulcxpowIs(gmul2n(b0,1), k);
8664 28 : polar = vec_append(polar, mkvec2(gk, simple_pole(b0)));
8665 : }
8666 63 : if (!gequal0(a0))
8667 : {
8668 21 : a0 = gneg(gmul2n(a0,1));
8669 21 : polar = vec_append(polar, mkvec2(gen_0, simple_pole(a0)));
8670 : }
8671 : }
8672 161 : if (eps) /* self-dual */
8673 : {
8674 119 : gel(LF,2) = mfcharorder(mf_get_CHI(F)) <= 2? gen_0: gen_1;
8675 119 : gel(LF,6) = mulcxpowIs(eps,k);
8676 : }
8677 161 : gel(LF,3) = mkvec2(gen_0, gen_1);
8678 161 : gel(LF,4) = gk;
8679 161 : gel(LF,5) = N;
8680 161 : if (lg(polar) == 1) setlg(LF,7); else gel(LF,7) = polar;
8681 161 : return LF;
8682 : }
8683 : static GEN
8684 133 : mflfuncreateall(long sd, GEN mfa, GEN F, GEN vE, GEN gN, GEN gk)
8685 : {
8686 133 : long i, l = lg(vE);
8687 133 : GEN L = cgetg(l, t_VEC);
8688 294 : for (i = 1; i < l; i++)
8689 161 : gel(L,i) = mflfuncreate(sd? gel(mfa,i): mfa, F, gel(vE,i), gN, gk);
8690 133 : return L;
8691 : }
8692 : GEN
8693 84 : lfunmf(GEN mf, GEN F, long bitprec)
8694 : {
8695 84 : pari_sp av = avma;
8696 84 : long i, l, prec = nbits2prec(bitprec);
8697 : GEN L, gk, gN;
8698 84 : mf = checkMF(mf);
8699 84 : gk = MF_get_gk(mf);
8700 84 : gN = MF_get_gN(mf);
8701 84 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
8702 84 : if (F)
8703 : {
8704 : GEN v;
8705 77 : long s = MF_get_space(mf);
8706 77 : if (!checkmf_i(F)) pari_err_TYPE("lfunmf", F);
8707 77 : if (!mfisinspace_i(mf, F)) err_space(F);
8708 77 : L = NULL;
8709 77 : if ((s == mf_NEW || s == mf_CUSP || s == mf_FULL)
8710 63 : && gequal(mfcoefs_i(F,1,1), mkvec2(gen_0,gen_1)))
8711 : { /* check if eigenform */
8712 35 : GEN vP, vF, b = mftobasis_i(mf, F);
8713 35 : long lF, d = degpol(mf_get_field(F));
8714 35 : v = mfsplit(mf, d, 0);
8715 35 : vF = gel(v,1);
8716 35 : vP = gel(v,2); lF = lg(vF);
8717 35 : for (i = 1; i < lF; i++)
8718 28 : if (degpol(gel(vP,i)) == d && gequal(gel(vF,i), b))
8719 : {
8720 28 : GEN vE = mfgetembed(F, prec);
8721 28 : GEN Z = mffrickeeigen_i(mf, mkvec(b), mkvec(vE), prec);
8722 28 : L = mflfuncreateall(1, gel(Z,1), F, vE, gN, gk);
8723 28 : break;
8724 : }
8725 : }
8726 77 : if (!L)
8727 : { /* not an eigenform: costly general case */
8728 49 : GEN mfa = mfatkininit_i(mf, itou(gN), 1, prec);
8729 49 : L = mflfuncreateall(0,mfa, F, mfgetembed(F,prec), gN, gk);
8730 : }
8731 77 : if (lg(L) == 2) L = gel(L,1);
8732 : }
8733 : else
8734 : {
8735 7 : GEN M = mfeigenbasis(mf), vE = mfeigenembed(mf, prec);
8736 7 : GEN v = mffrickeeigen(mf, vE, prec);
8737 7 : l = lg(vE); L = cgetg(l, t_VEC);
8738 63 : for (i = 1; i < l; i++)
8739 56 : gel(L,i) = mflfuncreateall(1,gel(v,i), gel(M,i), gel(vE,i), gN, gk);
8740 : }
8741 84 : return gerepilecopy(av, L);
8742 : }
8743 :
8744 : GEN
8745 28 : mffromell(GEN E)
8746 : {
8747 28 : pari_sp av = avma;
8748 : GEN mf, F, z, v, S;
8749 : long N, i, l;
8750 :
8751 28 : checkell(E);
8752 28 : if (ell_get_type(E) != t_ELL_Q) pari_err_TYPE("mfffromell [E not over Q]", E);
8753 28 : N = itos(ellQ_get_N(E));
8754 28 : mf = mfinit_i(mkvec2(utoi(N), gen_2), mf_NEW);
8755 28 : v = split_i(mf, 1, 0);
8756 28 : S = gel(v,1); l = lg(S); /* rational newforms */
8757 28 : F = tag(t_MF_ELL, mkNK(N,2,mfchartrivial()), E);
8758 28 : z = mftobasis_i(mf, F);
8759 28 : for(i = 1; i < l; i++)
8760 28 : if (gequal(z, gel(S,i))) break;
8761 28 : if (i == l) pari_err_BUG("mffromell [E is not modular]");
8762 28 : return gerepilecopy(av, mkvec3(mf, F, z));
8763 : }
8764 :
8765 : /* returns -1 if not, degree otherwise */
8766 : long
8767 140 : polishomogeneous(GEN P)
8768 : {
8769 : long i, D, l;
8770 140 : if (typ(P) != t_POL) return 0;
8771 77 : D = -1; l = lg(P);
8772 322 : for (i = 2; i < l; i++)
8773 : {
8774 245 : GEN c = gel(P,i);
8775 : long d;
8776 245 : if (gequal0(c)) continue;
8777 112 : d = polishomogeneous(c);
8778 112 : if (d < 0) return -1;
8779 112 : if (D < 0) D = d + i-2; else if (D != d + i-2) return -1;
8780 : }
8781 77 : return D;
8782 : }
8783 :
8784 : /* M a pp((Gram q)^(-1)) ZM; P a homogeneous t_POL, is P spherical ? */
8785 : static int
8786 28 : RgX_isspherical(GEN M, GEN P)
8787 : {
8788 28 : pari_sp av = avma;
8789 28 : GEN S, v = variables_vecsmall(P);
8790 28 : long i, j, l = lg(v);
8791 28 : if (l > lg(M)) pari_err(e_MISC, "too many variables in mffromqf");
8792 21 : S = gen_0;
8793 63 : for (j = 1; j < l; j++)
8794 : {
8795 42 : GEN Mj = gel(M, j), Pj = deriv(P, v[j]);
8796 105 : for (i = 1; i <= j; i++)
8797 : {
8798 63 : GEN c = gel(Mj, i);
8799 63 : if (!signe(c)) continue;
8800 42 : if (i != j) c = shifti(c, 1);
8801 42 : S = gadd(S, gmul(c, deriv(Pj, v[i])));
8802 : }
8803 : }
8804 21 : return gc_bool(av, gequal0(S));
8805 : }
8806 :
8807 : static GEN
8808 49 : c_QFsimple_i(long n, GEN Q, GEN P)
8809 : {
8810 49 : GEN V, v = qfrep0(Q, utoi(n), 1);
8811 49 : long i, l = lg(v);
8812 49 : V = cgetg(l+1, t_VEC);
8813 49 : if (!P || equali1(P))
8814 : {
8815 42 : gel(V,1) = gen_1;
8816 420 : for (i = 2; i <= l; i++) gel(V,i) = utoi(v[i-1] << 1);
8817 : }
8818 : else
8819 : {
8820 7 : gel(V,1) = gcopy(P);
8821 7 : for (i = 2; i <= l; i++) gel(V,i) = gmulgu(P, v[i-1] << 1);
8822 : }
8823 49 : return V;
8824 : }
8825 :
8826 : /* v a t_VECSMALL of variable numbers, lg(r) >= lg(v), r is a vector of
8827 : * scalars [not involving any variable in v] */
8828 : static GEN
8829 14 : gsubstvec_i(GEN e, GEN v, GEN r)
8830 : {
8831 14 : long i, l = lg(v);
8832 42 : for(i = 1; i < l; i++) e = gsubst(e, v[i], gel(r,i));
8833 14 : return e;
8834 : }
8835 : static GEN
8836 56 : c_QF_i(long n, GEN Q, GEN P)
8837 : {
8838 56 : pari_sp av = avma;
8839 : GEN V, v, va;
8840 : long i, l;
8841 56 : if (!P || typ(P) != t_POL) return gerepileupto(av, c_QFsimple_i(n, Q, P));
8842 7 : v = gel(minim(Q, utoi(2*n), NULL), 3);
8843 7 : va = variables_vecsmall(P);
8844 7 : V = zerovec(n + 1); l = lg(v);
8845 21 : for (i = 1; i < l; i++)
8846 : {
8847 14 : pari_sp av = avma;
8848 14 : GEN X = gel(v,i);
8849 14 : long c = (itos(qfeval(Q, X)) >> 1) + 1;
8850 14 : gel(V, c) = gerepileupto(av, gadd(gel(V, c), gsubstvec_i(P, va, X)));
8851 : }
8852 7 : return gmul2n(V, 1);
8853 : }
8854 :
8855 : GEN
8856 77 : mffromqf(GEN Q, GEN P)
8857 : {
8858 77 : pari_sp av = avma;
8859 : GEN G, Qi, F, D, N, mf, v, gk, chi;
8860 : long m, d, space;
8861 77 : if (typ(Q) != t_MAT) pari_err_TYPE("mffromqf", Q);
8862 77 : if (!RgM_is_ZM(Q) || !qfiseven(Q))
8863 0 : pari_err_TYPE("mffromqf [not integral or even]", Q);
8864 77 : m = lg(Q)-1;
8865 77 : Qi = ZM_inv(Q, &N);
8866 77 : if (!qfiseven(Qi)) N = shifti(N, 1);
8867 77 : d = 0;
8868 77 : if (!P || gequal1(P)) P = NULL;
8869 : else
8870 : {
8871 35 : P = simplify_shallow(P);
8872 35 : if (typ(P) == t_POL)
8873 : {
8874 28 : d = polishomogeneous(P);
8875 28 : if (d < 0) pari_err_TYPE("mffromqf [not homogeneous t_POL]", P);
8876 28 : if (!RgX_isspherical(Qi, P))
8877 7 : pari_err_TYPE("mffromqf [not a spherical t_POL]", P);
8878 : }
8879 : }
8880 63 : gk = uutoQ(m + 2*d, 2);
8881 63 : D = ZM_det(Q);
8882 63 : if (!odd(m)) { if ((m & 3) == 2) D = negi(D); } else D = shifti(D, 1);
8883 63 : space = d > 0 ? mf_CUSP : mf_FULL;
8884 63 : G = znstar0(N,1);
8885 63 : chi = mkvec2(G, znchar_quad(G,D));
8886 63 : mf = mfinit(mkvec3(N, gk, chi), space);
8887 63 : if (odd(d))
8888 : {
8889 7 : F = mftrivial();
8890 7 : v = zerocol(MF_get_dim(mf));
8891 : }
8892 : else
8893 : {
8894 56 : F = c_QF_i(mfsturm(mf), Q, P);
8895 56 : v = mftobasis_i(mf, F);
8896 56 : F = mflinear(mf, v);
8897 : }
8898 63 : return gerepilecopy(av, mkvec3(mf, F, v));
8899 : }
8900 :
8901 : /***********************************************************************/
8902 : /* Eisenstein Series */
8903 : /***********************************************************************/
8904 : /* \sigma_{k-1}(\chi,n) */
8905 : static GEN
8906 24192 : sigchi(long k, GEN CHI, long n)
8907 : {
8908 24192 : pari_sp av = avma;
8909 24192 : GEN S = gen_1, D = mydivisorsu(u_ppo(n,mfcharmodulus(CHI)));
8910 24192 : long i, l = lg(D), ord = mfcharorder(CHI), vt = varn(mfcharpol(CHI));
8911 83671 : for (i = 2; i < l; i++) /* skip D[1] = 1 */
8912 : {
8913 59479 : long d = D[i], a = mfcharevalord(CHI, d, ord);
8914 59479 : S = gadd(S, Qab_Czeta(a, ord, powuu(d, k-1), vt));
8915 : }
8916 24192 : return gerepileupto(av,S);
8917 : }
8918 :
8919 : /* write n = n0*n1*n2, (n0,N1*N2) = 1, n1 | N1^oo, n2 | N2^oo;
8920 : * return NULL if (n,N1,N2) > 1, else return factoru(n0) */
8921 : static GEN
8922 649215 : sigchi2_dec(long n, long N1, long N2, long *pn1, long *pn2)
8923 : {
8924 649215 : GEN P0, E0, P, E, fa = myfactoru(n);
8925 : long i, j, l;
8926 649215 : *pn1 = 1;
8927 649215 : *pn2 = 1;
8928 649215 : if (N1 == 1 && N2 == 1) return fa;
8929 635593 : P = gel(fa,1); l = lg(P);
8930 635593 : E = gel(fa,2);
8931 635593 : P0 = cgetg(l, t_VECSMALL);
8932 635593 : E0 = cgetg(l, t_VECSMALL);
8933 1472373 : for (i = j = 1; i < l; i++)
8934 : {
8935 942088 : long p = P[i], e = E[i];
8936 942088 : if (N1 % p == 0)
8937 : {
8938 142520 : if (N2 % p == 0) return NULL;
8939 37212 : *pn1 *= upowuu(p,e);
8940 : }
8941 799568 : else if (N2 % p == 0)
8942 129059 : *pn2 *= upowuu(p,e);
8943 670509 : else { P0[j] = p; E0[j] = e; j++; }
8944 : }
8945 530285 : setlg(P0, j);
8946 530285 : setlg(E0, j); return mkvec2(P0,E0);
8947 : }
8948 :
8949 : /* sigma_{k-1}(\chi_1,\chi_2,n), ord multiple of lcm(ord(CHI1),ord(CHI2)) */
8950 : static GEN
8951 593684 : sigchi2(long k, GEN CHI1, GEN CHI2, long n, long ord)
8952 : {
8953 593684 : pari_sp av = avma;
8954 : GEN S, D;
8955 593684 : long i, l, n1, n2, vt, N1 = mfcharmodulus(CHI1), N2 = mfcharmodulus(CHI2);
8956 593684 : D = sigchi2_dec(n, N1, N2, &n1, &n2); if (!D) return gc_const(av, gen_0);
8957 493059 : D = divisorsu_fact(D); l = lg(D);
8958 493059 : vt = varn(mfcharpol(CHI1));
8959 2122848 : for (i = 1, S = gen_0; i < l; i++)
8960 : { /* S += d^(k-1)*chi1(d)*chi2(n/d) */
8961 1629789 : long a, d = n2*D[i], nd = n1*D[l-i]; /* (d,N1)=1; (n/d,N2) = 1 */
8962 1629789 : a = mfcharevalord(CHI1, d, ord) + mfcharevalord(CHI2, nd, ord);
8963 1629789 : if (a >= ord) a -= ord;
8964 1629789 : S = gadd(S, Qab_Czeta(a, ord, powuu(d, k-1), vt));
8965 : }
8966 493059 : return gerepileupto(av, S);
8967 : }
8968 :
8969 : /**************************************************************************/
8970 : /** Dirichlet characters with precomputed values **/
8971 : /**************************************************************************/
8972 : /* CHI mfchar */
8973 : static GEN
8974 14266 : mfcharcxinit(GEN CHI, long prec)
8975 : {
8976 14266 : GEN G = gel(CHI,1), chi = gel(CHI,2), z, V;
8977 14266 : GEN v = ncharvecexpo(G, znconrey_normalized(G,chi));
8978 14266 : long n, l = lg(v), o = mfcharorder(CHI);
8979 14266 : V = cgetg(l, t_VEC);
8980 14266 : z = grootsof1(o, prec); /* Mod(t, Phi_o(t)) -> e(1/o) */
8981 123893 : for (n = 1; n < l; n++) gel(V,n) = v[n] < 0? gen_0: gel(z, v[n]+1);
8982 14266 : return mkvecn(6, G, chi, gmfcharorder(CHI), v, V, mfcharpol(CHI));
8983 : }
8984 : /* v a "CHIvec" */
8985 : static long
8986 24694754 : CHIvec_N(GEN v) { return itou(znstar_get_N(gel(v,1))); }
8987 : static GEN
8988 15372 : CHIvec_CHI(GEN v)
8989 15372 : { return mkvec4(gel(v,1), gel(v,2), gel(v,3), gel(v,6)); }
8990 : /* character order */
8991 : static long
8992 35763 : CHIvec_ord(GEN v) { return itou(gel(v,3)); }
8993 : /* character exponents, i.e. t such that chi(n) = e(t) */
8994 : static GEN
8995 422177 : CHIvec_expo(GEN v) { return gel(v,4); }
8996 : /* character values chi(n) */
8997 : static GEN
8998 24048745 : CHIvec_val(GEN v) { return gel(v,5); }
8999 : /* CHI(n) */
9000 : static GEN
9001 24038714 : mychareval(GEN v, long n)
9002 : {
9003 24038714 : long N = CHIvec_N(v), ind = n%N;
9004 24038714 : if (ind <= 0) ind += N;
9005 24038714 : return gel(CHIvec_val(v), ind);
9006 : }
9007 : /* return c such that CHI(n) = e(c / ordz) or -1 if (n,N) > 1 */
9008 : static long
9009 422177 : mycharexpo(GEN v, long n)
9010 : {
9011 422177 : long N = CHIvec_N(v), ind = n%N;
9012 422177 : if (ind <= 0) ind += N;
9013 422177 : return CHIvec_expo(v)[ind];
9014 : }
9015 : /* faster than mfcharparity */
9016 : static long
9017 54054 : CHIvec_parity(GEN v) { return mycharexpo(v,-1) ? -1: 1; }
9018 : /**************************************************************************/
9019 :
9020 : static ulong
9021 55531 : sigchi2_Fl(long k, GEN CHI1vec, GEN CHI2vec, long n, GEN vz, ulong p)
9022 : {
9023 55531 : pari_sp av = avma;
9024 55531 : long ordz = lg(vz)-2, i, l, n1, n2;
9025 55531 : ulong S = 0;
9026 55531 : GEN D = sigchi2_dec(n, CHIvec_N(CHI1vec), CHIvec_N(CHI2vec), &n1, &n2);
9027 55531 : if (!D) return gc_ulong(av,S);
9028 50848 : D = divisorsu_fact(D);
9029 50848 : l = lg(D);
9030 172053 : for (i = 1; i < l; i++)
9031 : { /* S += d^(k-1)*chi1(d)*chi2(n/d) */
9032 121205 : long a, d = n2*D[i], nd = n1*D[l-i]; /* (d,N1)=1, (n/d,N2)=1 */
9033 121205 : a = mycharexpo(CHI2vec, nd) + mycharexpo(CHI1vec, d);
9034 121205 : if (a >= ordz) a -= ordz;
9035 121205 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, Fl_powu(d,k-1,p), p), p);
9036 : }
9037 50848 : return gc_ulong(av,S);
9038 : }
9039 :
9040 : /**********************************************************************/
9041 : /* Fourier expansions of Eisenstein series */
9042 : /**********************************************************************/
9043 : /* L(CHI_t,0) / 2, CHI_t(n) = CHI(n)(t/n) as a character modulo N*t,
9044 : * order(CHI) | ord != 0 */
9045 : static GEN
9046 2618 : charLFwt1(long N, GEN CHI, long ord, long t)
9047 : {
9048 : GEN S;
9049 : long r, vt;
9050 :
9051 2618 : if (N == 1 && t == 1) return mkfrac(gen_m1,stoi(4));
9052 2618 : S = gen_0; vt = varn(mfcharpol(CHI));
9053 295435 : for (r = 1; r < N; r++)
9054 : { /* S += r*chi(r) */
9055 : long a, c;
9056 292817 : if (ugcd(N,r) != 1) continue;
9057 233310 : a = mfcharevalord(CHI,r,ord);
9058 233310 : c = (t != 1 && kross(t, r) < 0)? -r: r;
9059 233310 : S = gadd(S, Qab_Czeta(a, ord, stoi(c), vt));
9060 : }
9061 2618 : return gdivgs(S, -2*N);
9062 : }
9063 : /* L(CHI,0) / 2, mod p */
9064 : static ulong
9065 2002 : charLFwt1_Fl(GEN CHIvec, GEN vz, ulong p)
9066 : {
9067 2002 : long r, m = CHIvec_N(CHIvec);
9068 : ulong S;
9069 2002 : if (m == 1) return Rg_to_Fl(mkfrac(gen_m1,stoi(4)), p);
9070 2002 : S = 0;
9071 95977 : for (r = 1; r < m; r++)
9072 : { /* S += r*chi(r) */
9073 93975 : long a = mycharexpo(CHIvec,r);
9074 93975 : if (a < 0) continue;
9075 91616 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, r, p), p);
9076 : }
9077 2002 : return Fl_div(Fl_neg(S,p), 2*m, p);
9078 : }
9079 : /* L(CHI_t,1-k) / 2, CHI_t(n) = CHI(n) * (t/n), order(CHI) | ord != 0;
9080 : * assume conductor of CHI_t divides N */
9081 : static GEN
9082 4298 : charLFwtk(long N, long k, GEN CHI, long ord, long t)
9083 : {
9084 : GEN S, P, dS;
9085 : long r, vt;
9086 :
9087 4298 : if (k == 1) return charLFwt1(N, CHI, ord, t);
9088 1680 : if (N == 1 && t == 1) return gdivgs(bernfrac(k),-2*k);
9089 1057 : vt = varn(mfcharpol(CHI));
9090 1057 : P = bern_init(N, k, &dS);
9091 1057 : dS = mul_denom(dS, stoi(-2*N*k));
9092 14126 : for (r = 1, S = gen_0; r < N; r++)
9093 : { /* S += P(r)*chi(r) */
9094 : long a;
9095 : GEN C;
9096 13069 : if (ugcd(r,N) != 1) continue;
9097 10472 : a = mfcharevalord(CHI,r,ord);
9098 10472 : C = ZX_Z_eval(P, utoi(r));
9099 10472 : if (t != 1 && kross(t, r) < 0) C = gneg(C);
9100 10472 : S = gadd(S, Qab_Czeta(a, ord, C, vt));
9101 : }
9102 1057 : return gdiv(S, dS);
9103 : }
9104 : /* L(CHI,1-k) / 2, mod p */
9105 : static ulong
9106 2849 : charLFwtk_Fl(long k, GEN CHIvec, GEN vz, ulong p)
9107 : {
9108 : GEN P, dS;
9109 : long r, m;
9110 : ulong S, d;
9111 2849 : if (k == 1) return charLFwt1_Fl(CHIvec, vz, p);
9112 847 : m = CHIvec_N(CHIvec);
9113 847 : if (m == 1) return Rg_to_Fl(gdivgs(bernfrac(k),-2*k), p);
9114 504 : P = ZX_to_Flx(bern_init(m, k, &dS), p);
9115 10402 : for (r = 1, S = 0; r < m; r++)
9116 : { /* S += P(r)*chi(r) */
9117 9898 : long a = mycharexpo(CHIvec,r);
9118 9898 : if (a < 0) continue;
9119 8638 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, Flx_eval(P,r,p), p), p);
9120 : }
9121 504 : d = (2 * k * m) % p; if (dS) d = Fl_mul(d, umodiu(dS, p), p);
9122 504 : return Fl_div(Fl_neg(S,p), d, p);
9123 : }
9124 :
9125 : static GEN
9126 7854 : mfeisenstein2_0(long k, GEN CHI1, GEN CHI2, long ord)
9127 : {
9128 7854 : long N1 = mfcharmodulus(CHI1), N2 = mfcharmodulus(CHI2);
9129 7854 : if (k == 1 && N1 == 1) return charLFwtk(N2, 1, CHI2, ord, 1);
9130 5243 : if (N2 == 1) return charLFwtk(N1, k, CHI1, ord, 1);
9131 3773 : return gen_0;
9132 : }
9133 : static ulong
9134 4340 : mfeisenstein2_0_Fl(long k, GEN CHI1vec, GEN CHI2vec, GEN vz, ulong p)
9135 : {
9136 4340 : if (k == 1 && CHIvec_N(CHI1vec) == 1)
9137 2002 : return charLFwtk_Fl(k, CHI2vec, vz, p);
9138 2338 : else if (CHIvec_N(CHI2vec) == 1)
9139 847 : return charLFwtk_Fl(k, CHI1vec, vz, p);
9140 1491 : else return 0;
9141 : }
9142 : static GEN
9143 133 : NK_eisen2(long k, GEN CHI1, GEN CHI2, long ord)
9144 : {
9145 133 : long o, N = mfcharmodulus(CHI1)*mfcharmodulus(CHI2);
9146 133 : GEN CHI = mfcharmul(CHI1, CHI2);
9147 133 : o = mfcharorder(CHI);
9148 133 : if ((ord & 3) == 2) ord >>= 1;
9149 133 : if ((o & 3) == 2) o >>= 1;
9150 133 : if (ord != o) pari_err_IMPL("mfeisenstein for these characters");
9151 126 : return mkNK(N, k, CHI);
9152 : }
9153 : static GEN
9154 371 : mfeisenstein_i(long k, GEN CHI1, GEN CHI2)
9155 : {
9156 371 : long s = 1, ord, vt;
9157 : GEN E0, NK, vchi, T;
9158 371 : if (CHI2) { CHI2 = get_mfchar(CHI2); if (mfcharparity(CHI2) < 0) s = -s; }
9159 371 : if (CHI1) { CHI1 = get_mfchar(CHI1); if (mfcharparity(CHI1) < 0) s = -s; }
9160 357 : if (s != m1pk(k)) return mftrivial();
9161 336 : if (!CHI1) CHI1 = mfchartrivial();
9162 336 : if (!CHI2)
9163 : { /* E_k(chi1) */
9164 203 : vt = varn(mfcharpol(CHI1));
9165 203 : ord = mfcharorder(CHI1);
9166 203 : NK = mkNK(mfcharmodulus(CHI1), k, CHI1);
9167 203 : E0 = charLFwtk(mfcharmodulus(CHI1), k, CHI1, ord, 1);
9168 203 : vchi = mkvec3(E0, mkvec(mfcharpol(CHI1)), CHI1);
9169 203 : return tag(t_MF_EISEN, NK, vchi);
9170 : }
9171 : /* E_k(chi1,chi2) */
9172 133 : vt = varn(mfcharpol(CHI1));
9173 133 : ord = ulcm(mfcharorder(CHI1), mfcharorder(CHI2));
9174 133 : NK = NK_eisen2(k, CHI1, CHI2, ord);
9175 126 : E0 = mfeisenstein2_0(k, CHI1, CHI2, ord);
9176 126 : T = mkvec(polcyclo(ord, vt));
9177 126 : vchi = mkvec4(E0, T, CHI1, CHI2);
9178 126 : return tag2(t_MF_EISEN, NK, vchi, mkvecsmall2(ord,0));
9179 : }
9180 : GEN
9181 371 : mfeisenstein(long k, GEN CHI1, GEN CHI2)
9182 : {
9183 371 : pari_sp av = avma;
9184 371 : if (k < 1) pari_err_DOMAIN("mfeisenstein", "k", "<", gen_1, stoi(k));
9185 371 : return gerepilecopy(av, mfeisenstein_i(k, CHI1, CHI2));
9186 : }
9187 :
9188 : static GEN
9189 2604 : mfeisenstein2all(long N0, GEN NK, long k, GEN CHI1, GEN CHI2, GEN T, long o)
9190 : {
9191 2604 : GEN E, E0 = mfeisenstein2_0(k, CHI1,CHI2, o), vchi = mkvec4(E0, T, CHI1,CHI2);
9192 2604 : long j, d = (lg(T)==4)? itou(gmael(T,3,1)): 1;
9193 2604 : E = cgetg(d+1, t_VEC);
9194 5327 : for (j=1; j<=d; j++) gel(E,j) = tag2(t_MF_EISEN, NK,vchi,mkvecsmall2(o,j-1));
9195 2604 : return mfbdall(E, N0 / mf_get_N(gel(E,1)));
9196 : }
9197 :
9198 : /* list of characters on G = (Z/NZ)^*, v[i] = NULL if (i,N) > 1, else
9199 : * the conductor of Conrey label i, [conductor, primitive char].
9200 : * Trivial chi (label 1) comes first */
9201 : static GEN
9202 1169 : zncharsG(GEN G)
9203 : {
9204 1169 : long i, l, N = itou(znstar_get_N(G));
9205 : GEN vCHI, V;
9206 1169 : if (N == 1) return mkvec2(gen_1,cgetg(1,t_COL));
9207 1169 : vCHI = const_vec(N,NULL);
9208 1169 : V = cyc2elts(znstar_get_conreycyc(G));
9209 1169 : l = lg(V);
9210 207739 : for (i = 1; i < l; i++)
9211 : {
9212 206570 : GEN chi0, chi = zc_to_ZC(gel(V,i)), n, F;
9213 206570 : F = znconreyconductor(G, chi, &chi0);
9214 206570 : if (typ(F) != t_INT) F = gel(F,1);
9215 206570 : n = znconreyexp(G, chi);
9216 206570 : gel(vCHI, itos(n)) = mkvec2(chi0, F);
9217 : }
9218 1169 : return vCHI;
9219 : }
9220 :
9221 : /* CHI primitive, f(CHI) | N. Return pairs (CHI1,CHI2) both primitive
9222 : * such that f(CHI1)*f(CHI2) | N and CHI1 * CHI2 = CHI;
9223 : * if k = 1, CHI1 is even; if k = 2, omit (1,1) if CHI = 1 */
9224 : static GEN
9225 1386 : mfeisensteinbasis_i(long N0, long k, GEN CHI)
9226 : {
9227 1386 : GEN G = gel(CHI,1), chi = gel(CHI,2), vT = const_vec(myeulerphiu(N0), NULL);
9228 1386 : GEN CHI0, GN, chiN, Lchi, LG, V, RES, NK, T, C = mfcharpol(CHI);
9229 1386 : long i, j, l, n, n1, N, ord = mfcharorder(CHI);
9230 1386 : long F = mfcharmodulus(CHI), vt = varn(mfcharpol(CHI));
9231 :
9232 1386 : CHI0 = (F == 1)? CHI: mfchartrivial();
9233 1386 : j = 1; RES = cgetg(N0+1, t_VEC);
9234 1386 : T = gel(vT,ord) = Qab_trace_init(ord, ord, C, C);
9235 1386 : if (F != 1 || k != 2)
9236 : { /* N1 = 1 */
9237 1232 : NK = mkNK(F, k, CHI);
9238 1232 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI0, CHI, T, ord);
9239 1232 : if (F != 1 && k != 1)
9240 329 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI, CHI0, T, ord);
9241 : }
9242 1386 : if (N0 == 1) { setlg(RES,j); return RES; }
9243 1316 : GN = G; chiN = chi;
9244 1316 : if (F == N0) N = N0;
9245 : else
9246 : {
9247 714 : GEN faN = myfactoru(N0), P = gel(faN,1), E = gel(faN,2);
9248 714 : long lP = lg(P);
9249 1848 : for (i = N = 1; i < lP; i++)
9250 : {
9251 1134 : long p = P[i];
9252 1134 : N *= upowuu(p, maxuu(E[i]/2, z_lval(F,p)));
9253 : }
9254 714 : if ((N & 3) == 2) N >>= 1;
9255 714 : if (N == 1) { setlg(RES,j); return RES; }
9256 567 : if (F != N)
9257 : {
9258 133 : GN = znstar0(utoipos(N),1);
9259 133 : chiN = zncharinduce(G, chi, GN);
9260 : }
9261 : }
9262 1169 : LG = const_vec(N, NULL); /* LG[d] = znstar(d,1) or NULL */
9263 1169 : gel(LG,1) = gel(CHI0,1);
9264 1169 : gel(LG,F) = G;
9265 1169 : gel(LG,N) = GN;
9266 1169 : Lchi = coprimes_zv(N);
9267 1169 : n = itou(znconreyexp(GN,chiN));
9268 1169 : V = zncharsG(GN); l = lg(V);
9269 263305 : for (n1 = 2; n1 < l; n1++) /* skip 1 (trivial char) */
9270 : {
9271 262136 : GEN v = gel(V,n1), w, chi1, chi2, G1, G2, CHI1, CHI2;
9272 : long N12, N1, N2, no, o12, t, m;
9273 262136 : if (!Lchi[n1] || n1 == n) continue; /* skip trivial chi2 */
9274 204197 : chi1 = gel(v,1); N1 = itou(gel(v,2)); /* conductor of chi1 */
9275 204197 : w = gel(V, Fl_div(n,n1,N));
9276 204197 : chi2 = gel(w,1); N2 = itou(gel(w,2)); /* conductor of chi2 */
9277 204197 : N12 = N1 * N2;
9278 204197 : if (N0 % N12) continue;
9279 :
9280 1771 : G1 = gel(LG,N1); if (!G1) gel(LG,N1) = G1 = znstar0(utoipos(N1), 1);
9281 1771 : if (k == 1 && zncharisodd(G1,chi1)) continue;
9282 1043 : G2 = gel(LG,N2); if (!G2) gel(LG,N2) = G2 = znstar0(utoipos(N2), 1);
9283 1043 : CHI1 = mfcharGL(G1, chi1);
9284 1043 : CHI2 = mfcharGL(G2, chi2);
9285 1043 : o12 = ulcm(mfcharorder(CHI1), mfcharorder(CHI2));
9286 : /* remove Galois orbit: same trace */
9287 1043 : no = Fl_powu(n1, ord, N);
9288 1414 : for (t = 1+ord, m = n1; t <= o12; t += ord)
9289 : { /* m <-> CHI1^t, if t in Gal(Q(chi1,chi2)/Q), omit (CHI1^t,CHI2^t) */
9290 371 : m = Fl_mul(m, no, N); if (!m) break;
9291 371 : if (ugcd(t, o12) == 1) Lchi[m] = 0;
9292 : }
9293 1043 : T = gel(vT,o12);
9294 1043 : if (!T) T = gel(vT,o12) = Qab_trace_init(o12, ord, polcyclo(o12,vt), C);
9295 1043 : NK = mkNK(N12, k, CHI);
9296 1043 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI1, CHI2, T, o12);
9297 : }
9298 1169 : setlg(RES,j); return RES;
9299 : }
9300 :
9301 : static GEN
9302 721 : mfbd_E2(GEN E2, long d, GEN CHI)
9303 : {
9304 721 : GEN E2d = mfbd_i(E2, d);
9305 721 : GEN F = mkvec2(E2, E2d), L = mkvec2(gen_1, utoineg(d));
9306 : /* cannot use mflinear_i: E2 and E2d do not have the same level */
9307 721 : return tag3(t_MF_LINEAR, mkNK(d,2,CHI), F, L, gen_1);
9308 : }
9309 : /* C-basis of E_k(Gamma_0(N),chi). If k = 1, the first basis element must not
9310 : * vanish at oo [used in mf1basis]. Here E_1(CHI), whose q^0 coefficient
9311 : * does not vanish (since L(CHI,0) does not) *if* CHI is not trivial; which
9312 : * must be the case in weight 1.
9313 : *
9314 : * (k>=3): In weight k >= 3, basis is B(d) E(CHI1,(CHI/CHI1)_prim), where
9315 : * CHI1 is primitive modulo N1, and if N2 is the conductor of CHI/CHI1
9316 : * then d*N1*N2 | N.
9317 : * (k=2): In weight k=2, same if CHI is nontrivial. If CHI is trivial, must
9318 : * not take CHI1 trivial, and must add E_2(tau)-dE_2(d tau)), where
9319 : * d|N, d > 1.
9320 : * (k=1): In weight k=1, same as k >= 3 except that we restrict to CHI1 even */
9321 : static GEN
9322 1414 : mfeisensteinbasis(long N, long k, GEN CHI)
9323 : {
9324 : long i, F;
9325 : GEN L;
9326 1414 : if (badchar(N, k, CHI)) return cgetg(1, t_VEC);
9327 1414 : if (k == 0) return mfcharistrivial(CHI)? mkvec(mf1()): cgetg(1, t_VEC);
9328 1386 : CHI = mfchartoprimitive(CHI, &F);
9329 1386 : L = mfeisensteinbasis_i(N, k, CHI);
9330 1386 : if (F == 1 && k == 2)
9331 : {
9332 154 : GEN v, E2 = mfeisenstein(2, NULL, NULL), D = mydivisorsu(N);
9333 154 : long nD = lg(D)-1;
9334 154 : v = cgetg(nD, t_VEC); L = vec_append(L,v);
9335 868 : for (i = 1; i < nD; i++) gel(v,i) = mfbd_E2(E2, D[i+1], CHI);
9336 : }
9337 1386 : return lg(L) == 1? L: shallowconcat1(L);
9338 : }
9339 :
9340 : static GEN
9341 77 : not_in_space(GEN F, long flag)
9342 : {
9343 77 : if (!flag) err_space(F);
9344 70 : return cgetg(1, t_COL);
9345 : }
9346 : /* when flag set, no error */
9347 : GEN
9348 1001 : mftobasis(GEN mf, GEN F, long flag)
9349 : {
9350 1001 : pari_sp av2, av = avma;
9351 : GEN G, v, y, gk;
9352 1001 : long N, B, ismf = checkmf_i(F);
9353 :
9354 1001 : mf = checkMF(mf);
9355 1001 : if (ismf)
9356 : {
9357 910 : if (mfistrivial(F)) return zerocol(MF_get_dim(mf));
9358 903 : if (!mf_same_k(mf, F) || !mf_same_CHI(mf, F)) return not_in_space(F, flag);
9359 : }
9360 952 : N = MF_get_N(mf);
9361 952 : gk = MF_get_gk(mf);
9362 952 : if (ismf)
9363 : {
9364 861 : long NF = mf_get_N(F);
9365 861 : B = maxuu(mfsturmNgk(NF,gk), mfsturmNgk(N,gk)) + 1;
9366 861 : v = mfcoefs_i(F,B,1);
9367 : }
9368 : else
9369 : {
9370 91 : B = mfsturmNgk(N, gk) + 1;
9371 91 : switch(typ(F))
9372 : { /* F(0),...,F(lg(v)-2) */
9373 63 : case t_SER: v = sertocol(F); settyp(v,t_VEC); break;
9374 14 : case t_VEC: v = F; break;
9375 7 : case t_COL: v = shallowtrans(F); break;
9376 7 : default: pari_err_TYPE("mftobasis",F);
9377 : v = NULL;/*LCOV_EXCL_LINE*/
9378 : }
9379 84 : if (flag) B = minss(B, lg(v)-2);
9380 : }
9381 945 : y = mftobasis_i(mf, v);
9382 945 : if (typ(y) == t_VEC)
9383 : {
9384 21 : if (flag) return gerepilecopy(av, y);
9385 0 : pari_err(e_MISC, "not enough coefficients in mftobasis");
9386 : }
9387 924 : av2 = avma;
9388 924 : if (MF_get_space(mf) == mf_FULL || mfsturm(mf)+1 == B) return y;
9389 469 : G = mflinear(mf, y);
9390 469 : if (!gequal(v, mfcoefs_i(G, lg(v)-2,1))) y = NULL;
9391 469 : if (!y) { set_avma(av); return not_in_space(F, flag); }
9392 434 : set_avma(av2); return gerepileupto(av, y);
9393 : }
9394 :
9395 : /* assume N > 0; first cusp is always 0 */
9396 : static GEN
9397 49 : mfcusps_i(long N)
9398 : {
9399 : long i, c, l;
9400 : GEN D, v;
9401 :
9402 49 : if (N == 1) return mkvec(gen_0);
9403 49 : D = mydivisorsu(N); l = lg(D); /* left on stack */
9404 49 : c = mfnumcuspsu_fact(myfactoru(N));
9405 49 : v = cgetg(c + 1, t_VEC);
9406 350 : for (i = c = 1; i < l; i++)
9407 : {
9408 301 : long C = D[i], NC = D[l-i], lima = ugcd(C, NC), A0, A;
9409 889 : for (A0 = 0; A0 < lima; A0++)
9410 588 : if (ugcd(A0, lima) == 1)
9411 : {
9412 539 : A = A0; while (ugcd(A,C) > 1) A += lima;
9413 392 : gel(v, c++) = uutoQ(A, C);
9414 : }
9415 : }
9416 49 : return v;
9417 : }
9418 : /* List of cusps of Gamma_0(N) */
9419 : GEN
9420 28 : mfcusps(GEN gN)
9421 : {
9422 : long N;
9423 : GEN mf;
9424 28 : if (typ(gN) == t_INT) N = itos(gN);
9425 14 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
9426 0 : else { pari_err_TYPE("mfcusps", gN); N = 0; }
9427 28 : if (N <= 0) pari_err_DOMAIN("mfcusps", "N", "<=", gen_0, stoi(N));
9428 28 : return mfcusps_i(N);
9429 : }
9430 :
9431 : long
9432 315 : mfcuspisregular(GEN NK, GEN cusp)
9433 : {
9434 : long v, N, dk, nk, t, o;
9435 : GEN mf, CHI, go, A, C, g, c, d;
9436 315 : if ((mf = checkMF_i(NK)))
9437 : {
9438 49 : GEN gk = MF_get_gk(mf);
9439 49 : N = MF_get_N(mf);
9440 49 : CHI = MF_get_CHI(mf);
9441 49 : Qtoss(gk, &nk, &dk);
9442 : }
9443 : else
9444 266 : checkNK2(NK, &N, &nk, &dk, &CHI, 0);
9445 315 : if (typ(cusp) == t_INFINITY) return 1;
9446 315 : if (typ(cusp) == t_FRAC) { A = gel(cusp,1); C = gel(cusp,2); }
9447 28 : else { A = cusp; C = gen_1; }
9448 315 : g = diviuexact(mului(N,C), ugcd(N, Fl_sqr(umodiu(C,N), N)));
9449 315 : c = mulii(negi(C),g);
9450 315 : d = addiu(mulii(A,g), 1);
9451 315 : if (!CHI) return 1;
9452 315 : go = gmfcharorder(CHI);
9453 315 : v = vali(go); if (v < 2) go = shifti(go, 2-v);
9454 315 : t = itou( znchareval(gel(CHI,1), gel(CHI,2), d, go) );
9455 315 : if (dk == 1) return t == 0;
9456 154 : o = itou(go);
9457 154 : if (kronecker(c,d) < 0) t = Fl_add(t, o/2, o);
9458 154 : if (Mod4(d) == 1) return t == 0;
9459 14 : t = Fl_sub(t, Fl_mul(o/4, nk, o), o);
9460 14 : return t == 0;
9461 : }
9462 :
9463 : /* Some useful closures */
9464 :
9465 : /* sum_{d|n} d^k */
9466 : static GEN
9467 47600 : mysumdivku(ulong n, ulong k)
9468 : {
9469 47600 : GEN fa = myfactoru(n);
9470 47600 : return k == 1? usumdiv_fact(fa): usumdivk_fact(fa,k);
9471 : }
9472 : static GEN
9473 840 : c_Ek(long n, long d, GEN F)
9474 : {
9475 840 : GEN E = cgetg(n + 2, t_VEC), C = gel(F,2);
9476 840 : long i, k = mf_get_k(F);
9477 840 : gel (E, 1) = gen_1;
9478 25802 : for (i = 1; i <= n; i++)
9479 : {
9480 24962 : pari_sp av = avma;
9481 24962 : gel(E, i+1) = gerepileupto(av, gmul(C, mysumdivku(i*d, k-1)));
9482 : }
9483 840 : return E;
9484 : }
9485 :
9486 : GEN
9487 392 : mfEk(long k)
9488 : {
9489 392 : pari_sp av = avma;
9490 : GEN E0, NK;
9491 392 : if (k < 0 || odd(k)) pari_err_TYPE("mfEk [incorrect k]", stoi(k));
9492 392 : if (!k) return mf1();
9493 385 : E0 = gdivsg(-2*k, bernfrac(k));
9494 385 : NK = mkNK(1,k,mfchartrivial());
9495 385 : return gerepilecopy(av, tag(t_MF_Ek, NK, E0));
9496 : }
9497 :
9498 : GEN
9499 56 : mfDelta(void)
9500 : {
9501 56 : pari_sp av = avma;
9502 56 : return gerepilecopy(av, tag0(t_MF_DELTA, mkNK(1,12,mfchartrivial())));
9503 : }
9504 :
9505 : GEN
9506 798 : mfTheta(GEN psi)
9507 : {
9508 798 : pari_sp av = avma;
9509 : GEN N, gk, psi2;
9510 : long par;
9511 798 : if (!psi) { psi = mfchartrivial(); N = utoipos(4); par = 1; }
9512 : else
9513 : {
9514 : long FC;
9515 21 : psi = get_mfchar(psi);
9516 21 : FC = mfcharconductor(psi);
9517 21 : if (mfcharmodulus(psi) != FC)
9518 0 : pari_err_TYPE("mfTheta [nonprimitive character]", psi);
9519 21 : par = mfcharparity(psi);
9520 21 : N = shifti(sqru(FC),2);
9521 : }
9522 798 : if (par > 0) { gk = ghalf; psi2 = psi; }
9523 7 : else { gk = gsubsg(2, ghalf); psi2 = mfcharmul(psi, get_mfchar(stoi(-4))); }
9524 798 : return gerepilecopy(av, tag(t_MF_THETA, mkgNK(N, gk, psi2, pol_x(1)), psi));
9525 : }
9526 :
9527 : /* Output 0 if not desired eta product: if flag=0 (default) require
9528 : * holomorphic at cusps. If flag set, accept meromorphic, but sill in some
9529 : * modular function space */
9530 : GEN
9531 210 : mffrometaquo(GEN eta, long flag)
9532 : {
9533 210 : pari_sp av = avma;
9534 : GEN NK, N, k, BR, P;
9535 210 : long v, cusp = 0;
9536 210 : if (!etaquotype(&eta, &N,&k,&P, &v, NULL, flag? NULL: &cusp) || cusp < 0)
9537 14 : return gc_const(av, gen_0);
9538 196 : if (lg(gel(eta,1)) == 1) { set_avma(av); return mf1(); }
9539 189 : BR = mkvec2(ZV_to_zv(gel(eta,1)), ZV_to_zv(gel(eta,2)));
9540 189 : if (v < 0) v = 0;
9541 189 : NK = mkgNK(N, k, get_mfchar(P), pol_x(1));
9542 189 : return gerepilecopy(av, tag2(t_MF_ETAQUO, NK, BR, utoi(v)));
9543 : }
9544 :
9545 : /* Q^(-r) */
9546 : static GEN
9547 375 : RgXn_negpow(GEN Q, long r, long L)
9548 : {
9549 375 : if (r < 0) r = -r; else Q = RgXn_inv_i(Q, L);
9550 375 : if (r != 1) Q = RgXn_powu_i(Q, r, L);
9551 375 : return Q;
9552 : }
9553 : /* flag same as in mffrometaquo: if set, accept meromorphic. */
9554 : static GEN
9555 49 : mfisetaquo_i(GEN F, long flag)
9556 : {
9557 : GEN gk, P, E, M, S, G, CHI, v, w;
9558 : long b, l, L, N, vS, m, j;
9559 49 : const long bextra = 10;
9560 :
9561 49 : if (!checkmf_i(F)) pari_err_TYPE("mfisetaquo",F);
9562 49 : CHI = mf_get_CHI(F); if (mfcharorder(CHI) > 2) return NULL;
9563 49 : N = mf_get_N(F);
9564 49 : gk = mf_get_gk(F);
9565 49 : b = mfsturmNgk(N, gk);
9566 49 : L = maxss(N, b) + bextra;
9567 49 : S = mfcoefs_i(F, L, 1);
9568 49 : if (!RgV_is_ZV(S)) return NULL;
9569 889 : for (vS = 1; vS <= L+1; vS++)
9570 889 : if (signe(gel(S,vS))) break;
9571 49 : vS--;
9572 49 : if (vS >= bextra - 1) { L += vS; S = mfcoefs_i(F, L, 1); }
9573 49 : if (vS) { S = vecslice(S, vS+1, L+1); L -= vS; }
9574 49 : S = RgV_to_RgX(S, 0); l = lg(S)-2;
9575 49 : P = cgetg(l, t_COL);
9576 49 : E = cgetg(l, t_COL); w = v = gen_0; /* w = weight, v = valuation */
9577 1908 : for (m = j = 1; m+2 < lg(S); m++)
9578 : {
9579 1866 : GEN c = gel(S,m+2);
9580 : long r;
9581 1866 : if (is_bigint(c)) return NULL;
9582 1859 : r = -itos(c);
9583 1859 : if (r)
9584 : {
9585 375 : S = ZXn_mul(S, RgXn_negpow(eta_ZXn(m, L), r, L), L);
9586 375 : gel(P,j) = utoipos(m);
9587 375 : gel(E,j) = stoi(r);
9588 375 : v = addmuliu(v, gel(E,j), m);
9589 375 : w = addis(w, r);
9590 375 : j++;
9591 : }
9592 : }
9593 42 : if (!equalii(w, gmul2n(gk, 1)) || (!flag && !equalii(v, muluu(24,vS))))
9594 7 : return NULL;
9595 35 : setlg(P, j);
9596 35 : setlg(E, j); M = mkmat2(P, E); G = mffrometaquo(M, flag);
9597 35 : return (typ(G) != t_INT
9598 35 : && (mfsturmmf(G) <= b + bextra || mfisequal(F, G, b)))? M: NULL;
9599 : }
9600 : GEN
9601 49 : mfisetaquo(GEN F, long flag)
9602 : {
9603 49 : pari_sp av = avma;
9604 49 : GEN M = mfisetaquo_i(F, flag);
9605 49 : return M? gerepilecopy(av, M): gc_const(av, gen_0);
9606 : }
9607 :
9608 : #if 0
9609 : /* number of primitive characters modulo N */
9610 : static ulong
9611 : numprimchars(ulong N)
9612 : {
9613 : GEN fa, P, E;
9614 : long i, l;
9615 : ulong n;
9616 : if ((N & 3) == 2) return 0;
9617 : fa = myfactoru(N);
9618 : P = gel(fa,1); l = lg(P);
9619 : E = gel(fa,2);
9620 : for (i = n = 1; i < l; i++)
9621 : {
9622 : ulong p = P[i], e = E[i];
9623 : if (e == 2) n *= p-2; else n *= (p-1)*(p-1)*upowuu(p,e-2);
9624 : }
9625 : return n;
9626 : }
9627 : #endif
9628 :
9629 : /* Space generated by products of two Eisenstein series */
9630 :
9631 : static int
9632 73759 : cmp_small_priority(void *E, GEN a, GEN b)
9633 : {
9634 73759 : GEN prio = (GEN)E;
9635 73759 : return cmpss(prio[(long)a], prio[(long)b]);
9636 : }
9637 : static long
9638 1260 : znstar_get_expo(GEN G) { return itou(cyc_get_expo(znstar_get_cyc(G))); }
9639 :
9640 : /* Return [vchi, bymod, vG]:
9641 : * vG[f] = znstar(f,1) for f a conductor of (at least) a char mod N; else NULL
9642 : * bymod[f] = vecsmall of conrey indexes of chars modulo f | N; else NULL
9643 : * vchi[n] = a list of CHIvec [G0,chi0,o,ncharvecexpo(G0,nchi0),...]:
9644 : * chi0 = primitive char attached to Conrey Mod(n,N)
9645 : * (resp. NULL if (n,N) > 1) */
9646 : static GEN
9647 630 : charsmodN(long N)
9648 : {
9649 630 : GEN D, G, prio, phio, dummy = cgetg(1,t_VEC);
9650 630 : GEN vP, vG = const_vec(N,NULL), vCHI = const_vec(N,NULL);
9651 630 : GEN bymod = const_vec(N,NULL);
9652 630 : long pn, i, l, vt = fetch_user_var("t");
9653 630 : D = mydivisorsu(N); l = lg(D);
9654 3892 : for (i = 1; i < l; i++)
9655 3262 : gel(bymod, D[i]) = vecsmalltrunc_init(myeulerphiu(D[i])+1);
9656 630 : gel(vG,N) = G = znstar0(utoipos(N),1);
9657 630 : pn = znstar_get_expo(G); /* exponent(Z/NZ)^* */
9658 630 : vP = const_vec(pn,NULL);
9659 26817 : for (i = 1; i <= N; i++)
9660 : {
9661 : GEN P, gF, G0, chi0, nchi0, chi, v, go;
9662 : long j, F, o;
9663 26187 : if (ugcd(i,N) != 1) continue;
9664 13923 : chi = znconreylog(G, utoipos(i));
9665 13923 : gF = znconreyconductor(G, chi, &chi0);
9666 13923 : F = (typ(gF) == t_INT)? itou(gF): itou(gel(gF,1));
9667 13923 : G0 = gel(vG, F); if (!G0) G0 = gel(vG,F) = znstar0(gF, 1);
9668 13923 : nchi0 = znconreylog_normalize(G0,chi0);
9669 13923 : go = gel(nchi0,1); o = itou(go); /* order(chi0) */
9670 13923 : v = ncharvecexpo(G0, nchi0);
9671 13923 : if (!equaliu(go, pn)) v = zv_z_mul(v, pn / o);
9672 13923 : P = gel(vP, o); if (!P) P = gel(vP,o) = polcyclo(o,vt);
9673 : /* mfcharcxinit with dummy complex powers */
9674 13923 : gel(vCHI,i) = mkvecn(6, G0, chi0, go, v, dummy, P);
9675 13923 : D = mydivisorsu(N / F); l = lg(D);
9676 40110 : for (j = 1; j < l; j++) vecsmalltrunc_append(gel(bymod, F*D[j]), i);
9677 : }
9678 630 : phio = zero_zv(pn); l = lg(vCHI); prio = cgetg(l, t_VEC);
9679 26817 : for (i = 1; i < l; i++)
9680 : {
9681 26187 : GEN CHI = gel(vCHI,i);
9682 : long o;
9683 26187 : if (!CHI) continue;
9684 13923 : o = CHIvec_ord(CHI);
9685 13923 : if (!phio[o]) phio[o] = myeulerphiu(o);
9686 13923 : prio[i] = phio[o];
9687 : }
9688 630 : l = lg(bymod);
9689 : /* sort characters by increasing value of phi(order) */
9690 26817 : for (i = 1; i < l; i++)
9691 : {
9692 26187 : GEN z = gel(bymod,i);
9693 26187 : if (z) gen_sort_inplace(z, (void*)prio, &cmp_small_priority, NULL);
9694 : }
9695 630 : return mkvec3(vCHI, bymod, vG);
9696 : }
9697 :
9698 : static GEN
9699 5124 : mfeisenstein2pure(long k, GEN CHI1, GEN CHI2, long ord, GEN P, long lim)
9700 : {
9701 5124 : GEN c, V = cgetg(lim+2, t_COL);
9702 : long n;
9703 5124 : c = mfeisenstein2_0(k, CHI1, CHI2, ord);
9704 5124 : if (P) c = grem(c, P);
9705 5124 : gel(V,1) = c;
9706 98994 : for (n=1; n <= lim; n++)
9707 : {
9708 93870 : c = sigchi2(k, CHI1, CHI2, n, ord);
9709 93870 : if (P) c = grem(c, P);
9710 93870 : gel(V,n+1) = c;
9711 : }
9712 5124 : return V;
9713 : }
9714 : static GEN
9715 4340 : mfeisenstein2pure_Fl(long k, GEN CHI1vec, GEN CHI2vec, GEN vz, ulong p, long lim)
9716 : {
9717 4340 : GEN V = cgetg(lim+2, t_VECSMALL);
9718 : long n;
9719 4340 : V[1] = mfeisenstein2_0_Fl(k, CHI1vec, CHI2vec, vz, p);
9720 59871 : for (n=1; n <= lim; n++) V[n+1] = sigchi2_Fl(k, CHI1vec, CHI2vec, n, vz, p);
9721 4340 : return V;
9722 : }
9723 :
9724 : static GEN
9725 252 : getcolswt2(GEN M, GEN D, ulong p)
9726 : {
9727 252 : GEN R, v = gel(M,1);
9728 252 : long i, l = lg(M) - 1;
9729 252 : R = cgetg(l, t_MAT); /* skip D[1] = 1 */
9730 1008 : for (i = 1; i < l; i++)
9731 : {
9732 756 : GEN w = Flv_Fl_mul(gel(M,i+1), D[i+1], p);
9733 756 : gel(R,i) = Flv_sub(v, w, p);
9734 : }
9735 252 : return R;
9736 : }
9737 : static GEN
9738 5383 : expandbd(GEN V, long d)
9739 : {
9740 : long L, n, nd;
9741 : GEN W;
9742 5383 : if (d == 1) return V;
9743 2086 : L = lg(V)-1; W = zerocol(L); /* nd = n/d */
9744 17871 : for (n = nd = 0; n < L; n += d, nd++) gel(W, n+1) = gel(V, nd+1);
9745 2086 : return W;
9746 : }
9747 : static GEN
9748 6979 : expandbd_Fl(GEN V, long d)
9749 : {
9750 : long L, n, nd;
9751 : GEN W;
9752 6979 : if (d == 1) return V;
9753 2639 : L = lg(V)-1; W = zero_Flv(L); /* nd = n/d */
9754 16366 : for (n = nd = 0; n < L; n += d, nd++) W[n+1] = V[nd+1];
9755 2639 : return W;
9756 : }
9757 : static void
9758 4340 : getcols_i(GEN *pM, GEN *pvj, GEN gk, GEN CHI1vec, GEN CHI2vec, long NN1, GEN vz,
9759 : ulong p, long lim)
9760 : {
9761 4340 : GEN CHI1 = CHIvec_CHI(CHI1vec), CHI2 = CHIvec_CHI(CHI2vec);
9762 4340 : long N2 = CHIvec_N(CHI2vec);
9763 4340 : GEN vj, M, D = mydivisorsu(NN1/N2);
9764 4340 : long i, l = lg(D), k = gk[2];
9765 4340 : GEN V = mfeisenstein2pure_Fl(k, CHI1vec, CHI2vec, vz, p, lim);
9766 4340 : M = cgetg(l, t_MAT);
9767 11319 : for (i = 1; i < l; i++) gel(M,i) = expandbd_Fl(V, D[i]);
9768 4340 : if (k == 2 && N2 == 1 && CHIvec_N(CHI1vec) == 1)
9769 : {
9770 252 : M = getcolswt2(M, D, p); l--;
9771 252 : D = vecslice(D, 2, l);
9772 : }
9773 4340 : *pM = M;
9774 4340 : *pvj = vj = cgetg(l, t_VEC);
9775 11067 : for (i = 1; i < l; i++) gel(vj,i) = mkvec4(gk, CHI1, CHI2, utoipos(D[i]));
9776 4340 : }
9777 :
9778 : /* find all CHI1, CHI2 mod N such that CHI1*CHI2 = CHI, f(CHI1)*f(CHI2) | N.
9779 : * set M = mfcoefs(B_e E(CHI1,CHI2), lim), vj = [e,i1,i2] */
9780 : static void
9781 1778 : getcols(GEN *pM, GEN *pv, long k, long nCHI, GEN allN, GEN vz, ulong p,
9782 : long lim)
9783 : {
9784 1778 : GEN vCHI = gel(allN,1), gk = utoi(k);
9785 1778 : GEN M = cgetg(1,t_MAT), v = cgetg(1,t_VEC);
9786 1778 : long i1, N = lg(vCHI)-1;
9787 86079 : for (i1 = 1; i1 <= N; i1++)
9788 : {
9789 84301 : GEN CHI1vec = gel(vCHI, i1), CHI2vec, M1, v1;
9790 : long NN1, i2;
9791 147301 : if (!CHI1vec) continue;
9792 66192 : if (k == 1 && CHIvec_parity(CHI1vec) == -1) continue;
9793 41433 : NN1 = N/CHIvec_N(CHI1vec); /* N/f(chi1) */;
9794 41433 : i2 = Fl_div(nCHI,i1, N);
9795 41433 : if (!i2) i2 = 1;
9796 41433 : CHI2vec = gel(vCHI,i2);
9797 41433 : if (NN1 % CHIvec_N(CHI2vec)) continue; /* f(chi1)f(chi2) | N ? */
9798 3192 : getcols_i(&M1, &v1, gk, CHI1vec, CHI2vec, NN1, vz, p, lim);
9799 3192 : M = shallowconcat(M, M1);
9800 3192 : v = shallowconcat(v, v1);
9801 : }
9802 1778 : *pM = M;
9803 1778 : *pv = v;
9804 1778 : }
9805 :
9806 : static void
9807 1176 : update_Mj(GEN *M, GEN *vecj, GEN *pz, ulong p)
9808 : {
9809 : GEN perm;
9810 1176 : *pz = Flm_indexrank(*M, p); perm = gel(*pz,2);
9811 1176 : *M = vecpermute(*M, perm);
9812 1176 : *vecj = vecpermute(*vecj, perm);
9813 1176 : }
9814 : static int
9815 413 : getcolsgen(long dim, GEN *pM, GEN *pvj, GEN *pz, long k, long ell, long nCHI,
9816 : GEN allN, GEN vz, ulong p, long lim)
9817 : {
9818 413 : GEN vCHI = gel(allN,1), bymod = gel(allN,2), gell = utoi(ell);
9819 413 : long i1, N = lg(vCHI)-1;
9820 413 : long L = lim+1;
9821 413 : if (lg(*pvj)-1 >= dim) update_Mj(pM, pvj, pz, p);
9822 413 : if (lg(*pvj)-1 == dim) return 1;
9823 1561 : for (i1 = 1; i1 <= N; i1++)
9824 : {
9825 1540 : GEN CHI1vec = gel(vCHI, i1), T;
9826 : long par1, j, l, N1, NN1;
9827 :
9828 1540 : if (!CHI1vec) continue;
9829 1519 : par1 = CHIvec_parity(CHI1vec);
9830 1519 : if (ell == 1 && par1 == -1) continue;
9831 938 : if (odd(ell)) par1 = -par1;
9832 938 : N1 = CHIvec_N(CHI1vec);
9833 938 : NN1 = N/N1;
9834 938 : T = gel(bymod, NN1); l = lg(T);
9835 3598 : for (j = 1; j < l; j++)
9836 : {
9837 3017 : long i2 = T[j], l1, l2, j1, s, nC;
9838 3017 : GEN M, M1, M2, vj, vj1, vj2, CHI2vec = gel(vCHI, i2);
9839 3017 : if (CHIvec_parity(CHI2vec) != par1) continue;
9840 1148 : nC = Fl_div(nCHI, Fl_mul(i1,i2,N), N);
9841 1148 : getcols(&M2, &vj2, k-ell, nC, allN, vz, p, lim);
9842 1148 : l2 = lg(M2); if (l2 == 1) continue;
9843 1148 : getcols_i(&M1, &vj1, gell, CHI1vec, CHI2vec, NN1, vz, p, lim);
9844 1148 : l1 = lg(M1);
9845 1148 : M1 = Flm_to_FlxV(M1, 0);
9846 1148 : M2 = Flm_to_FlxV(M2, 0);
9847 1148 : M = cgetg((l1-1)*(l2-1) + 1, t_MAT);
9848 1148 : vj = cgetg((l1-1)*(l2-1) + 1, t_VEC);
9849 2835 : for (j1 = s = 1; j1 < l1; j1++)
9850 : {
9851 1687 : GEN E = gel(M1,j1), v = gel(vj1,j1);
9852 : long j2;
9853 7084 : for (j2 = 1; j2 < l2; j2++, s++)
9854 : {
9855 5397 : GEN c = Flx_to_Flv(Flxn_mul(E, gel(M2,j2), L, p), L);
9856 5397 : gel(M,s) = c;
9857 5397 : gel(vj,s) = mkvec2(v, gel(vj2,j2));
9858 : }
9859 : }
9860 1148 : *pM = shallowconcat(*pM, M);
9861 1148 : *pvj = shallowconcat(*pvj, vj);
9862 1148 : if (lg(*pvj)-1 >= dim) update_Mj(pM, pvj, pz, p);
9863 1148 : if (lg(*pvj)-1 == dim) return 1;
9864 : }
9865 : }
9866 21 : if (ell == 1)
9867 : {
9868 21 : update_Mj(pM, pvj, pz, p);
9869 21 : return (lg(*pvj)-1 == dim);
9870 : }
9871 0 : return 0;
9872 : }
9873 :
9874 : static GEN
9875 1645 : mkF2bd(long d, long lim)
9876 : {
9877 1645 : GEN V = zerovec(lim + 1);
9878 : long n;
9879 1645 : gel(V, 1) = sstoQ(-1, 24);
9880 24248 : for (n = 1; n <= lim/d; n++) gel(V, n*d + 1) = mysumdivku(n, 1);
9881 1645 : return V;
9882 : }
9883 :
9884 : static GEN
9885 5740 : mkeisen(GEN E, long ord, GEN P, long lim)
9886 : {
9887 5740 : long k = itou(gel(E,1)), e = itou(gel(E,4));
9888 5740 : GEN CHI1 = gel(E,2), CHI2 = gel(E,3);
9889 5740 : if (k == 2 && mfcharistrivial(CHI1) && mfcharistrivial(CHI2))
9890 616 : return gsub(mkF2bd(1,lim), gmulgu(mkF2bd(e,lim), e));
9891 : else
9892 : {
9893 5124 : GEN V = mfeisenstein2pure(k, CHI1, CHI2, ord, P, lim);
9894 5124 : return expandbd(V, e);
9895 : }
9896 : }
9897 : static GEN
9898 588 : mkM(GEN vj, long pn, GEN P, long lim)
9899 : {
9900 588 : long j, l = lg(vj), L = lim+1;
9901 588 : GEN M = cgetg(l, t_MAT);
9902 4795 : for (j = 1; j < l; j++)
9903 : {
9904 : GEN E1, E2;
9905 4207 : parse_vecj(gel(vj,j), &E1,&E2);
9906 4207 : E1 = RgV_to_RgX(mkeisen(E1, pn, P, lim), 0);
9907 4207 : if (E2)
9908 : {
9909 1533 : E2 = RgV_to_RgX(mkeisen(E2, pn, P, lim), 0);
9910 1533 : E1 = RgXn_mul(E1, E2, L);
9911 : }
9912 4207 : E1 = RgX_to_RgC(E1, L);
9913 4207 : if (P && E2) E1 = RgXQV_red(E1, P);
9914 4207 : gel(M,j) = E1;
9915 : }
9916 588 : return M;
9917 : }
9918 :
9919 : /* assume N > 2 */
9920 : static GEN
9921 35 : mffindeisen1(long N)
9922 : {
9923 35 : GEN G = znstar0(utoipos(N), 1), L = chargalois(G, NULL), chi0 = NULL;
9924 35 : long j, m = N, l = lg(L);
9925 259 : for (j = 1; j < l; j++)
9926 : {
9927 245 : GEN chi = gel(L,j);
9928 245 : long r = myeulerphiu(itou(zncharorder(G,chi)));
9929 245 : if (r >= m) continue;
9930 182 : chi = znconreyfromchar(G, chi);
9931 182 : if (zncharisodd(G,chi)) { m = r; chi0 = chi; if (r == 1) break; }
9932 : }
9933 35 : if (!chi0) pari_err_BUG("mffindeisen1 [no Eisenstein series found]");
9934 35 : chi0 = znchartoprimitive(G,chi0);
9935 35 : return mfcharGL(gel(chi0,1), gel(chi0,2));
9936 : }
9937 :
9938 : static GEN
9939 630 : mfeisensteinspaceinit_i(long N, long k, GEN CHI)
9940 : {
9941 630 : GEN M, Minv, vj, vG, GN, allN, P, vz, z = NULL;
9942 630 : long nCHI, lim, ell, ord, dim = mffulldim(N, k, CHI);
9943 : ulong r, p;
9944 :
9945 630 : if (!dim) retmkvec3(cgetg(1,t_VECSMALL),
9946 : mkvec2(cgetg(1,t_MAT),gen_1),cgetg(1,t_VEC));
9947 630 : lim = mfsturmNk(N, k) + 1;
9948 630 : allN = charsmodN(N);
9949 630 : vG = gel(allN,3);
9950 630 : GN = gel(vG,N);
9951 630 : ord = znstar_get_expo(GN);
9952 630 : P = ord <= 2? NULL: polcyclo(ord, varn(mfcharpol(CHI)));
9953 630 : CHI = induce(GN, CHI); /* lift CHI mod N before mfcharno*/
9954 630 : nCHI = mfcharno(CHI);
9955 630 : r = QabM_init(ord, &p);
9956 630 : vz = Fl_powers(r, ord, p);
9957 630 : getcols(&M, &vj, k, nCHI, allN, vz, p, lim);
9958 651 : for (ell = k>>1; ell >= 1; ell--)
9959 413 : if (getcolsgen(dim, &M, &vj, &z, k, ell, nCHI, allN, vz, p, lim)) break;
9960 630 : if (!z) update_Mj(&M, &vj, &z, p);
9961 630 : if (lg(vj) - 1 < dim) return NULL;
9962 588 : M = mkM(vj, ord, P, lim);
9963 588 : Minv = QabM_Minv(rowpermute(M, gel(z,1)), P, ord);
9964 588 : return mkvec4(gel(z,1), Minv, vj, utoi(ord));
9965 : }
9966 : /* true mf */
9967 : static GEN
9968 588 : mfeisensteinspaceinit(GEN mf)
9969 : {
9970 588 : pari_sp av = avma;
9971 588 : GEN z, CHI = MF_get_CHI(mf);
9972 588 : long N = MF_get_N(mf), k = MF_get_k(mf);
9973 588 : if (!CHI) CHI = mfchartrivial();
9974 588 : z = mfeisensteinspaceinit_i(N, k, CHI);
9975 588 : if (!z)
9976 : {
9977 35 : GEN E, CHIN = mffindeisen1(N), CHI0 = mfchartrivial();
9978 35 : z = mfeisensteinspaceinit_i(N, k+1, mfcharmul(CHI, CHIN));
9979 35 : if (z) E = mkvec4(gen_1, CHI0, CHIN, gen_1);
9980 : else
9981 : {
9982 7 : z = mfeisensteinspaceinit_i(N, k+2, CHI);
9983 7 : E = mkvec4(gen_2, CHI0, CHI0, utoipos(N));
9984 : }
9985 35 : z = mkvec2(z, E);
9986 : }
9987 588 : return gerepilecopy(av, z);
9988 : }
9989 :
9990 : /* decomposition of modular form on eisenspace */
9991 : static GEN
9992 1001 : mfeisensteindec(GEN mf, GEN F)
9993 : {
9994 1001 : pari_sp av = avma;
9995 : GEN M, Mindex, Mvecj, V, B, CHI;
9996 : long o, ord;
9997 :
9998 1001 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
9999 1001 : if (lg(Mvecj) < 5)
10000 : {
10001 56 : GEN E, e = gel(Mvecj,2), gkE = gel(e,1);
10002 56 : long dE = itou(gel(e,4));
10003 56 : Mvecj = gel(Mvecj,1);
10004 56 : E = mfeisenstein(itou(gkE), NULL, gel(e,3));
10005 56 : if (dE != 1) E = mfbd_E2(E, dE, gel(e,2)); /* here k = 2 */
10006 56 : F = mfmul(F, E);
10007 : }
10008 1001 : M = gel(Mvecj, 2);
10009 1001 : if (lg(M) == 1) return cgetg(1, t_VEC);
10010 1001 : Mindex = gel(Mvecj, 1);
10011 1001 : ord = itou(gel(Mvecj,4));
10012 1001 : V = mfcoefs(F, Mindex[lg(Mindex)-1]-1, 1); settyp(V, t_COL);
10013 1001 : CHI = mf_get_CHI(F);
10014 1001 : o = mfcharorder(CHI);
10015 1001 : if (o > 2 && o != ord)
10016 : { /* convert Mod(.,polcyclo(o)) to Mod(., polcyclo(N)) for o | N,
10017 : * o and N both != 2 (mod 4) */
10018 84 : GEN z, P = gel(M,4); /* polcyclo(ord) */
10019 84 : long vt = varn(P);
10020 84 : z = gmodulo(pol_xn(ord/o, vt), P);
10021 84 : if (ord % o) pari_err_TYPE("mfeisensteindec", V);
10022 84 : V = gsubst(liftpol_shallow(V), vt, z);
10023 : }
10024 1001 : B = Minv_RgC_mul(M, vecpermute(V, Mindex));
10025 1001 : return gerepileupto(av, B);
10026 : }
10027 :
10028 : /*********************************************************************/
10029 : /* END EISENSPACE */
10030 : /*********************************************************************/
10031 :
10032 : static GEN
10033 70 : sertocol2(GEN S, long l)
10034 : {
10035 70 : GEN C = cgetg(l + 2, t_COL);
10036 : long i;
10037 420 : for (i = 0; i <= l; i++) gel(C, i+1) = polcoef_i(S, i, -1);
10038 70 : return C;
10039 : }
10040 :
10041 : /* Compute polynomial P0 such that F=E4^(k/4)P0(E6/E4^(3/2)). */
10042 : static GEN
10043 14 : mfcanfindp0(GEN F, long k)
10044 : {
10045 14 : pari_sp ltop = avma;
10046 : GEN E4, E6, V, V1, Q, W, res, M, B;
10047 : long l, j;
10048 14 : l = k/6 + 2;
10049 14 : V = mfcoefsser(F,l);
10050 14 : E4 = mfcoefsser(mfEk(4),l);
10051 14 : E6 = mfcoefsser(mfEk(6),l);
10052 14 : V1 = gdiv(V, gpow(E4, uutoQ(k,4), 0));
10053 14 : Q = gdiv(E6, gpow(E4, uutoQ(3,2), 0));
10054 14 : W = gpowers(Q, l - 1);
10055 14 : M = cgetg(l + 1, t_MAT);
10056 70 : for (j = 1; j <= l; j++) gel(M,j) = sertocol2(gel(W,j), l);
10057 14 : B = sertocol2(V1, l);
10058 14 : res = inverseimage(M, B);
10059 14 : if (lg(res) == 1) err_space(F);
10060 14 : return gerepilecopy(ltop, gtopolyrev(res, 0));
10061 : }
10062 :
10063 : /* Compute the first n+1 Taylor coeffs at tau=I of a modular form
10064 : * on SL_2(Z). */
10065 : GEN
10066 14 : mftaylor(GEN F, long n, long flreal, long prec)
10067 : {
10068 14 : pari_sp ltop = avma;
10069 14 : GEN P0, Pm1 = gen_0, v;
10070 14 : GEN X2 = mkpoln(3, ghalf,gen_0,gneg(ghalf)); /* (x^2-1) / 2 */
10071 : long k, m;
10072 14 : if (!checkmf_i(F)) pari_err_TYPE("mftaylor",F);
10073 14 : k = mf_get_k(F);
10074 14 : if (mf_get_N(F) != 1 || k < 0) pari_err_IMPL("mftaylor for this form");
10075 14 : P0 = mfcanfindp0(F, k);
10076 14 : v = cgetg(n+2, t_VEC); gel(v, 1) = RgX_coeff(P0,0);
10077 154 : for (m = 0; m < n; m++)
10078 : {
10079 140 : GEN P1 = gdivgu(gmulsg(-(k + 2*m), RgX_shift(P0,1)), 12);
10080 140 : P1 = gadd(P1, gmul(X2, RgX_deriv(P0)));
10081 140 : if (m) P1 = gsub(P1, gdivgu(gmulsg(m*(m+k-1), Pm1), 144));
10082 140 : Pm1 = P0; P0 = P1;
10083 140 : gel(v, m+2) = RgX_coeff(P0, 0);
10084 : }
10085 14 : if (flreal)
10086 : {
10087 7 : GEN pi2 = Pi2n(1, prec), pim4 = gmulsg(-2, pi2), VPC;
10088 7 : GEN C = gmulsg(3, gdiv(gpowgs(ggamma(uutoQ(1,4), prec), 8), gpowgs(pi2, 6)));
10089 : /* E_4(i): */
10090 7 : GEN facn = gen_1;
10091 7 : VPC = gpowers(gmul(pim4, gsqrt(C, prec)), n);
10092 7 : C = gpow(C, uutoQ(k,4), prec);
10093 84 : for (m = 0; m <= n; m++)
10094 : {
10095 77 : gel(v, m+1) = gdiv(gmul(C, gmul(gel(v, m+1), gel(VPC, m+1))), facn);
10096 77 : facn = gmulgu(facn, m+1);
10097 : }
10098 : }
10099 14 : return gerepilecopy(ltop, v);
10100 : }
10101 :
10102 : #if 0
10103 : /* To be used in mfeigensearch() */
10104 : GEN
10105 : mfreadratfile()
10106 : {
10107 : GEN eqn;
10108 : pariFILE *F = pari_fopengz("rateigen300.gp");
10109 : eqn = gp_readvec_stream(F->file);
10110 : pari_fclose(F);
10111 : return eqn;
10112 : }
10113 : #endif
10114 : /*****************************************************************/
10115 : /* EISENSTEIN CUSPS: COMPLEX DIRECTLY: one F_k */
10116 : /*****************************************************************/
10117 :
10118 : /* CHIvec = charinit(CHI); data = [N1g/g1,N2g/g2,g1/g,g2/g,C/g1,C/g2,
10119 : * (N1g/g1)^{-1},(N2g/g2)^{-1}] */
10120 :
10121 : /* nm = n/m;
10122 : * z1 = powers of \z_{C/g}^{(Ae/g)^{-1}},
10123 : * z2 = powers of \z_N^{A^{-1}(g1g2/C)}]
10124 : * N.B. : we compute value and conjugate at the end, so it is (Ae/g)^{-1}
10125 : * and not -(Ae/g)^{-1} */
10126 : static GEN
10127 7830900 : eiscnm(long nm, long m, GEN CHI1vec, GEN CHI2vec, GEN data, GEN z1)
10128 : {
10129 7830900 : long Cg1 = data[5], s10 = (nm*data[7]) % Cg1, r10 = (nm - data[1]*s10) / Cg1;
10130 7830900 : long Cg2 = data[6], s20 = (m *data[8]) % Cg2, r20 = (m - data[2]*s20) / Cg2;
10131 : long j1, r1, s1;
10132 7830900 : GEN T = gen_0;
10133 18857104 : for (j1 = 0, r1 = r10, s1 = s10; j1 < data[3]; j1++, r1 -= data[1], s1 += Cg1)
10134 : {
10135 11026204 : GEN c1 = mychareval(CHI1vec, r1);
10136 11026204 : if (!gequal0(c1))
10137 : {
10138 : long j2, r2, s2;
10139 8127560 : GEN S = gen_0;
10140 21136570 : for (j2 = 0, r2 = r20, s2 = s20; j2 < data[4]; j2++, r2 -= data[2], s2 += Cg2)
10141 : {
10142 13009010 : GEN c2 = mychareval(CHI2vec, r2);
10143 13009010 : if (!gequal0(c2)) S = gadd(S, gmul(c2, rootsof1pow(z1, s1*s2)));
10144 : }
10145 8127560 : T = gadd(T, gmul(c1, S));
10146 : }
10147 : }
10148 7830900 : return conj_i(T);
10149 : }
10150 :
10151 : static GEN
10152 612675 : fg1g2n(long n, long k, GEN CHI1vec, GEN CHI2vec, GEN data, GEN z1, GEN z2)
10153 : {
10154 612675 : pari_sp av = avma;
10155 612675 : GEN S = gen_0, D = mydivisorsu(n);
10156 612675 : long i, l = lg(D);
10157 4528125 : for (i = 1; i < l; i++)
10158 : {
10159 3915450 : long m = D[i], nm = D[l-i]; /* n/m */
10160 3915450 : GEN u = eiscnm( nm, m, CHI1vec, CHI2vec, data, z1);
10161 3915450 : GEN v = eiscnm(-nm, -m, CHI1vec, CHI2vec, data, z1);
10162 3915450 : GEN w = odd(k) ? gsub(u, v) : gadd(u, v);
10163 3915450 : S = gadd(S, gmul(powuu(m, k-1), w));
10164 : }
10165 612675 : return gerepileupto(av, gmul(S, rootsof1pow(z2, n)));
10166 : }
10167 :
10168 : static GEN
10169 14266 : gausssumcx(GEN CHIvec, long prec)
10170 : {
10171 : GEN z, S, V;
10172 14266 : long m, N = CHIvec_N(CHIvec);
10173 14266 : if (N == 1) return gen_1;
10174 7693 : V = CHIvec_val(CHIvec);
10175 7693 : z = rootsof1u_cx(N, prec);
10176 7693 : S = gmul(z, gel(V, N));
10177 103054 : for (m = N-1; m >= 1; m--) S = gmul(z, gadd(gel(V, m), S));
10178 7693 : return S;
10179 : }
10180 :
10181 : /* Computation of Q_k(\z_N^s) as a polynomial in \z_N^s. FIXME: explicit
10182 : * formula ? */
10183 : static GEN
10184 2338 : mfqk(long k, long N)
10185 : {
10186 : GEN X, P, ZI, Q, Xm1, invden;
10187 : long i;
10188 2338 : ZI = gdivgu(RgX_shift_shallow(RgV_to_RgX(identity_ZV(N-1), 0), 1), N);
10189 2338 : if (k == 1) return ZI;
10190 1176 : P = gsubgs(pol_xn(N,0), 1);
10191 1176 : invden = RgXQ_powu(ZI, k, P);
10192 1176 : X = pol_x(0); Q = gneg(X); Xm1 = gsubgs(X, 1);
10193 2884 : for (i = 2; i < k; i++)
10194 1708 : Q = RgX_shift_shallow(ZX_add(gmul(Xm1, ZX_deriv(Q)), gmulsg(-i, Q)), 1);
10195 1176 : return RgXQ_mul(Q, invden, P);
10196 : }
10197 :
10198 : /* CHI mfchar; M is a multiple of the conductor of CHI, but is NOT
10199 : * necessarily its modulus */
10200 : static GEN
10201 3346 : mfskcx(long k, GEN CHI, long M, long prec)
10202 : {
10203 : GEN S, CHIvec, P;
10204 : long F, m, i, l;
10205 3346 : CHI = mfchartoprimitive(CHI, &F);
10206 3346 : CHIvec = mfcharcxinit(CHI, prec);
10207 3346 : if (F == 1) S = gdivgu(bernfrac(k), k);
10208 : else
10209 : {
10210 2338 : GEN Q = mfqk(k, F), V = CHIvec_val(CHIvec);
10211 2338 : S = gmul(gel(V, F), RgX_coeff(Q, 0));
10212 39431 : for (m = 1; m < F; m++) S = gadd(S, gmul(gel(V, m), RgX_coeff(Q, m)));
10213 2338 : S = conj_i(S);
10214 : }
10215 : /* prime divisors of M not dividing f(chi) */
10216 3346 : P = gel(myfactoru(u_ppo(M/F,F)), 1); l = lg(P);
10217 3500 : for (i = 1; i < l; i++)
10218 : {
10219 154 : long p = P[i];
10220 154 : S = gmul(S, gsubsg(1, gdiv(mychareval(CHIvec, p), powuu(p, k))));
10221 : }
10222 3346 : return gmul(gmul(gausssumcx(CHIvec, prec), S), powuu(M/F, k));
10223 : }
10224 :
10225 : static GEN
10226 6146 : f00_i(long k, GEN CHI1vec, GEN CHI2vec, GEN G2, GEN S, long prec)
10227 : {
10228 : GEN c, a;
10229 6146 : long N1 = CHIvec_N(CHI1vec), N2 = CHIvec_N(CHI2vec);
10230 6146 : if (S[2] != N1) return gen_0;
10231 3346 : c = mychareval(CHI1vec, S[3]);
10232 3346 : if (isintzero(c)) return gen_0;
10233 3346 : a = mfskcx(k, mfchardiv(CHIvec_CHI(CHI2vec), CHIvec_CHI(CHI1vec)), N1*N2, prec);
10234 3346 : a = gmul(a, conj_i(gmul(c,G2)));
10235 3346 : return gdiv(a, mulsi(-N2, powuu(S[1], k-1)));
10236 : }
10237 :
10238 : static GEN
10239 4865 : f00(long k, GEN CHI1vec,GEN CHI2vec, GEN G1,GEN G2, GEN data, long prec)
10240 : {
10241 : GEN T1, T2;
10242 4865 : T2 = f00_i(k, CHI1vec, CHI2vec, G2, data, prec);
10243 4865 : if (k > 1) return T2;
10244 1281 : T1 = f00_i(k, CHI2vec, CHI1vec, G1, data, prec);
10245 1281 : return gadd(T1, T2);
10246 : }
10247 :
10248 : /* ga in SL_2(Z), find beta [a,b;c,d] in Gamma_0(N) and mu in Z such that
10249 : * beta * ga * T^u = [A',B';C',D'] with C' | N and N | B', C' > 0 */
10250 : static void
10251 5460 : mfgatogap(GEN ga, long N, long *pA, long *pC, long *pD, long *pd, long *pmu)
10252 : {
10253 5460 : GEN A = gcoeff(ga,1,1), B = gcoeff(ga,1,2);
10254 5460 : GEN C = gcoeff(ga,2,1), D = gcoeff(ga,2,2), a, b, c, d;
10255 : long t, Ap, Cp, B1, D1, mu;
10256 5460 : Cp = itou(bezout(muliu(A,N), C, &c, &d)); /* divides N */
10257 5460 : t = 0;
10258 5460 : if (Cp > 1)
10259 : { /* (d, N/Cp) = 1, find t such that (d - t*(A*N/Cp), N) = 1 */
10260 2604 : long dN = umodiu(d,Cp), Q = (N/Cp * umodiu(A,Cp)) % Cp;
10261 2989 : while (ugcd(dN, Cp) > 1) { t++; dN = Fl_sub(dN, Q, Cp); }
10262 : }
10263 5460 : if (t)
10264 : {
10265 385 : c = addii(c, mului(t, diviuexact(C,Cp)));
10266 385 : d = subii(d, mului(t, muliu(A, N/Cp))); /* (d,N) = 1 */
10267 : }
10268 5460 : D1 = umodiu(mulii(d,D), N);
10269 5460 : (void)bezout(d, mulis(c,-N), &a, &b); /* = 1 */
10270 5460 : t = 0; Ap = umodiu(addii(mulii(a,A), mulii(b,C)), N); /* (Ap,Cp) = 1 */
10271 7483 : while (ugcd(Ap, N) > 1) { t++; Ap = Fl_add(Ap, Cp, N); }
10272 5460 : B1 = umodiu(a,N)*umodiu(B,N) + umodiu(b,N)*umodiu(D,N) + t*D1;
10273 5460 : B1 %= N;
10274 5460 : *pmu = mu = Fl_neg(Fl_div(B1, Ap, N), N);
10275 : /* A', D' and d only needed modulo N */
10276 5460 : *pd = umodiu(d, N);
10277 5460 : *pA = Ap;
10278 5460 : *pC = Cp; *pD = (D1 + Cp*mu) % N;
10279 5460 : }
10280 :
10281 : #if 0
10282 : /* CHI is a mfchar, return alpha(CHI) */
10283 : static long
10284 : mfalchi(GEN CHI, long AN, long cg)
10285 : {
10286 : GEN G = gel(CHI,1), chi = gel(CHI,2), go = gmfcharorder(CHI);
10287 : long o = itou(go), a = itos( znchareval(G, chi, stoi(1 + AN/cg), go) );
10288 : if (a < 0 || (cg * a) % o) pari_err_BUG("mfalchi");
10289 : return (cg * a) / o;
10290 : }
10291 : #endif
10292 : /* return A such that CHI1(t) * CHI2(t) = e(A) or NULL if (t,N1*N2) > 1 */
10293 : static GEN
10294 10920 : mfcharmuleval(GEN CHI1vec, GEN CHI2vec, long t)
10295 : {
10296 10920 : long a1 = mycharexpo(CHI1vec, t), o1 = CHIvec_ord(CHI1vec);
10297 10920 : long a2 = mycharexpo(CHI2vec, t), o2 = CHIvec_ord(CHI2vec);;
10298 10920 : if (a1 < 0 || a2 < 0) return NULL;
10299 10920 : return sstoQ(a1*o2 + a2*o1, o1*o2);
10300 : }
10301 : static GEN
10302 5460 : mfcharmulcxeval(GEN CHI1vec, GEN CHI2vec, long t, long prec)
10303 : {
10304 5460 : GEN A = mfcharmuleval(CHI1vec, CHI2vec, t);
10305 : long n, d;
10306 5460 : if (!A) return gen_0;
10307 5460 : Qtoss(A, &n,&d); return rootsof1q_cx(n, d, prec);
10308 : }
10309 : /* alpha(CHI1 * CHI2) */
10310 : static long
10311 5460 : mfalchi2(GEN CHI1vec, GEN CHI2vec, long AN, long cg)
10312 : {
10313 5460 : GEN A = mfcharmuleval(CHI1vec, CHI2vec, 1 + AN/cg);
10314 : long a;
10315 5460 : if (!A) pari_err_BUG("mfalchi2");
10316 5460 : A = gmulsg(cg, A);
10317 5460 : if (typ(A) != t_INT) pari_err_BUG("mfalchi2");
10318 5460 : a = itos(A) % cg; if (a < 0) a += cg;
10319 5460 : return a;
10320 : }
10321 :
10322 : /* return g = (a,b), set u >= 0 s.t. g = a * u (mod b) */
10323 : static long
10324 21840 : mybezout(long a, long b, long *pu)
10325 : {
10326 21840 : long junk, g = cbezout(a, b, pu, &junk);
10327 21840 : if (*pu < 0) *pu += b/g;
10328 21840 : return g;
10329 : }
10330 :
10331 : /* E = [k, CHI1,CHI2, e], CHI1 and CHI2 primitive mfchars such that,
10332 : * CHI1(-1)*CHI2(-1) = (-1)^k; expansion of (B_e (E_k(CHI1,CHI2))) | ga.
10333 : * w is the width for the space of the calling function. */
10334 : static GEN
10335 5460 : mfeisensteingacx(GEN E, long w, GEN ga, long lim, long prec)
10336 : {
10337 5460 : GEN CHI1vec, CHI2vec, CHI1 = gel(E,2), CHI2 = gel(E,3), v, S, ALPHA;
10338 : GEN G1, G2, z1, z2, data;
10339 5460 : long k = itou(gel(E,1)), e = itou(gel(E,4));
10340 5460 : long N1 = mfcharmodulus(CHI1);
10341 5460 : long N2 = mfcharmodulus(CHI2), N = e * N1 * N2;
10342 : long NsurC, cg, wN, A, C, Ai, d, mu, alchi, na, da;
10343 : long eg, g, gH, U, u0, u1, u2, Aig, H, m, n, t, Cg, NC1, NC2;
10344 :
10345 5460 : mfgatogap(ga, N, &A, &C, &Ai, &d, &mu);
10346 5460 : CHI1vec = mfcharcxinit(CHI1, prec);
10347 5460 : CHI2vec = mfcharcxinit(CHI2, prec);
10348 5460 : NsurC = N/C; cg = ugcd(C, NsurC); wN = NsurC / cg;
10349 5460 : if (w%wN) pari_err_BUG("mfeisensteingacx [wN does not divide w]");
10350 5460 : alchi = mfalchi2(CHI1vec, CHI2vec, A*N, cg);
10351 5460 : ALPHA = sstoQ(alchi, NsurC);
10352 :
10353 5460 : g = mybezout(A*e, C, &u0); Cg = C/g; eg = e/g;
10354 5460 : NC1 = mybezout(N1, Cg, &u1);
10355 5460 : NC2 = mybezout(N2, Cg, &u2);
10356 5460 : H = (NC1*NC2*g)/Cg;
10357 5460 : Aig = (Ai*H)%N; if (Aig < 0) Aig += N;
10358 5460 : z1 = rootsof1powinit(u0, Cg, prec);
10359 5460 : z2 = rootsof1powinit(Aig, N, prec);
10360 5460 : data = mkvecsmalln(8, N1/NC1, N2/NC2, NC1, NC2, Cg/NC1, Cg/NC2, u1, u2);
10361 5460 : v = zerovec(lim + 1);
10362 : /* need n*H = alchi (mod cg) */
10363 5460 : gH = mybezout(H, cg, &U);
10364 5460 : if (gH > 1)
10365 : {
10366 511 : if (alchi % gH) return mkvec2(gen_0, v);
10367 511 : alchi /= gH; cg /= gH; H /= gH;
10368 : }
10369 5460 : G1 = gausssumcx(CHI1vec, prec);
10370 5460 : G2 = gausssumcx(CHI2vec, prec);
10371 5460 : if (!alchi)
10372 4865 : gel(v,1) = f00(k, CHI1vec,CHI2vec,G1,G2, mkvecsmall3(NC2,Cg,A*eg), prec);
10373 5460 : n = Fl_mul(alchi,U,cg); if (!n) n = cg;
10374 5460 : m = (n*H - alchi) / cg; /* positive, exact division */
10375 618135 : for (; m <= lim; n+=cg, m+=H)
10376 612675 : gel(v, m+1) = fg1g2n(n, k, CHI1vec, CHI2vec, data, z1,z2);
10377 5460 : t = (2*e)/g; if (odd(k)) t = -t;
10378 5460 : v = gdiv(v, gmul(conj_i(gmul(G1,G2)), mulsi(t, powuu(eg*N2/NC2, k-1))));
10379 5460 : if (k == 2 && N1 == 1 && N2 == 1) v = gsub(mkF2bd(wN,lim), gmulsg(e,v));
10380 :
10381 5460 : Qtoss(ALPHA, &na,&da);
10382 5460 : S = conj_i( mfcharmulcxeval(CHI1vec,CHI2vec,d,prec) ); /* CHI(1/d) */
10383 5460 : if (wN > 1)
10384 : {
10385 4151 : GEN z = rootsof1powinit(-mu, wN, prec);
10386 4151 : long i, l = lg(v);
10387 578837 : for (i = 1; i < l; i++) gel(v,i) = gmul(gel(v,i), rootsof1pow(z,i-1));
10388 : }
10389 5460 : v = RgV_Rg_mul(v, gmul(S, rootsof1q_cx(-mu*na, da, prec)));
10390 5460 : return mkvec2(ALPHA, bdexpand(v, w/wN));
10391 : }
10392 :
10393 : /*****************************************************************/
10394 : /* END EISENSTEIN CUSPS */
10395 : /*****************************************************************/
10396 :
10397 : static GEN
10398 1596 : mfchisimpl(GEN CHI)
10399 : {
10400 : GEN G, chi;
10401 1596 : if (typ(CHI) == t_INT) return CHI;
10402 1596 : G = gel(CHI, 1); chi = gel(CHI, 2);
10403 1596 : switch(mfcharorder(CHI))
10404 : {
10405 1148 : case 1: chi = gen_1; break;
10406 427 : case 2: chi = znchartokronecker(G,chi,1); break;
10407 21 : default:chi = mkintmod(znconreyexp(G,chi), znstar_get_N(G)); break;
10408 : }
10409 1596 : return chi;
10410 : }
10411 :
10412 : GEN
10413 700 : mfparams(GEN F)
10414 : {
10415 700 : pari_sp av = avma;
10416 : GEN z, mf, CHI;
10417 700 : if ((mf = checkMF_i(F)))
10418 : {
10419 14 : long N = MF_get_N(mf);
10420 14 : GEN gk = MF_get_gk(mf);
10421 14 : CHI = MF_get_CHI(mf);
10422 14 : z = mkvec5(utoi(N), gk, CHI, utoi(MF_get_space(mf)), mfcharpol(CHI));
10423 : }
10424 : else
10425 : {
10426 686 : if (!checkmf_i(F)) pari_err_TYPE("mfparams", F);
10427 686 : z = vec_append(mf_get_NK(F), mfcharpol(mf_get_CHI(F)));
10428 : }
10429 700 : gel(z,3) = mfchisimpl(gel(z,3));
10430 700 : return gerepilecopy(av, z);
10431 : }
10432 :
10433 : GEN
10434 14 : mfisCM(GEN F)
10435 : {
10436 14 : pari_sp av = avma;
10437 : forprime_t S;
10438 : GEN D, v;
10439 : long N, k, lD, sb, p, i;
10440 14 : if (!checkmf_i(F)) pari_err_TYPE("mfisCM", F);
10441 14 : N = mf_get_N(F);
10442 14 : k = mf_get_k(F); if (N < 0 || k < 0) pari_err_IMPL("mfisCM for this F");
10443 14 : D = mfunram(N, -1);
10444 14 : lD = lg(D);
10445 14 : sb = maxss(mfsturmNk(N, k), 4*N);
10446 14 : v = mfcoefs_i(F, sb, 1);
10447 14 : u_forprime_init(&S, 2, sb);
10448 504 : while ((p = u_forprime_next(&S)))
10449 : {
10450 490 : GEN ap = gel(v, p+1);
10451 490 : if (!gequal0(ap))
10452 406 : for (i = 1; i < lD; i++)
10453 245 : if (kross(D[i], p) == -1) { D = vecsplice(D, i); lD--; }
10454 : }
10455 14 : if (lD == 1) return gc_const(av, gen_0);
10456 14 : if (lD == 2) return gc_stoi(av, D[1]);
10457 7 : if (k > 1) pari_err_BUG("mfisCM");
10458 7 : return gerepileupto(av, zv_to_ZV(D));
10459 : }
10460 :
10461 : static long
10462 287 : mfspace_i(GEN mf, GEN F)
10463 : {
10464 : GEN v, vF, gk;
10465 : long n, nE, i, l, s, N;
10466 :
10467 287 : mf = checkMF(mf); s = MF_get_space(mf);
10468 287 : if (!F) return s;
10469 287 : if (!checkmf_i(F)) pari_err_TYPE("mfspace",F);
10470 287 : v = mftobasis(mf, F, 1);
10471 287 : n = lg(v)-1; if (!n) return -1;
10472 224 : nE = lg(MF_get_E(mf))-1;
10473 224 : switch(s)
10474 : {
10475 56 : case mf_NEW: case mf_OLD: case mf_EISEN: return s;
10476 140 : case mf_FULL:
10477 140 : if (mf_get_type(F) == t_MF_THETA) return mf_EISEN;
10478 133 : if (!gequal0(vecslice(v,1,nE)))
10479 63 : return gequal0(vecslice(v,nE+1,n))? mf_EISEN: mf_FULL;
10480 : }
10481 : /* mf is mf_CUSP or mf_FULL, F a cusp form */
10482 98 : gk = mf_get_gk(F);
10483 98 : if (typ(gk) == t_FRAC || equali1(gk)) return mf_CUSP;
10484 84 : vF = mftonew_i(mf, vecslice(v, nE+1, n), &N);
10485 84 : if (N != MF_get_N(mf)) return mf_OLD;
10486 56 : l = lg(vF);
10487 91 : for (i = 1; i < l; i++)
10488 56 : if (itos(gmael(vF,i,1)) != N) return mf_CUSP;
10489 35 : return mf_NEW;
10490 : }
10491 : long
10492 287 : mfspace(GEN mf, GEN F)
10493 287 : { pari_sp av = avma; return gc_long(av, mfspace_i(mf,F)); }
10494 : static GEN
10495 21 : lfunfindchi(GEN ldata, GEN van, long prec)
10496 : {
10497 21 : GEN gN = ldata_get_conductor(ldata), gk = ldata_get_k(ldata);
10498 21 : GEN G = znstar0(gN,1), cyc = znstar_get_conreycyc(G), L, go, vz;
10499 21 : long N = itou(gN), odd = typ(gk) == t_INT && mpodd(gk);
10500 21 : long i, j, o, l, B0 = 2, B = lg(van)-1, bit = 10 - prec2nbits(prec);
10501 :
10502 : /* if van is integral, chi must be trivial */
10503 21 : if (typ(van) == t_VECSMALL) return mfcharGL(G, zerocol(lg(cyc)-1));
10504 14 : L = cyc2elts(cyc); l = lg(L);
10505 42 : for (i = j = 1; i < l; i++)
10506 : {
10507 28 : GEN chi = zc_to_ZC(gel(L,i));
10508 28 : if (zncharisodd(G,chi) == odd) gel(L,j++) = mfcharGL(G,chi);
10509 : }
10510 14 : setlg(L,j); l = j;
10511 14 : if (l <= 2) return gel(L,1);
10512 0 : o = znstar_get_expo(G); go = utoi(o);
10513 0 : vz = grootsof1(o, prec);
10514 : for (;;)
10515 0 : {
10516 : long n;
10517 0 : for (n = B0; n <= B; n++)
10518 : {
10519 : GEN an, r;
10520 : long j;
10521 0 : if (ugcd(n, N) != 1) continue;
10522 0 : an = gel(van,n); if (gexpo(an) < bit) continue;
10523 0 : r = gdiv(an, conj_i(an));
10524 0 : for (i = 1; i < l; i++)
10525 : {
10526 0 : GEN CHI = gel(L,i);
10527 0 : if (gexpo(gsub(r, gel(vz, znchareval_i(CHI,n,go)+1))) > bit)
10528 0 : gel(L,i) = NULL;
10529 : }
10530 0 : for (i = j = 1; i < l; i++)
10531 0 : if (gel(L,i)) gel(L,j++) = gel(L,i);
10532 0 : l = j; setlg(L,l);
10533 0 : if (l == 2) return gel(L,1);
10534 : }
10535 0 : B0 = B+1; B <<= 1;
10536 0 : van = ldata_vecan(ldata_get_an(ldata), B, prec);
10537 : }
10538 : }
10539 :
10540 : GEN
10541 21 : mffromlfun(GEN L, long prec)
10542 : {
10543 21 : pari_sp av = avma;
10544 21 : GEN ldata = lfunmisc_to_ldata_shallow(L), Vga = ldata_get_gammavec(ldata);
10545 21 : GEN van, a0, CHI, NK, gk = ldata_get_k(ldata);
10546 : long N, space;
10547 21 : if (!gequal(Vga, mkvec2(gen_0, gen_1))) pari_err_TYPE("mffromlfun", L);
10548 21 : N = itou(ldata_get_conductor(ldata));
10549 21 : van = ldata_vecan(ldata_get_an(ldata), mfsturmNgk(N,gk) + 2, prec);
10550 21 : CHI = lfunfindchi(ldata, van, prec);
10551 21 : if (typ(van) != t_VEC) van = vecsmall_to_vec_inplace(van);
10552 21 : space = (lg(ldata) == 7)? mf_CUSP: mf_FULL;
10553 21 : a0 = (space == mf_CUSP)? gen_0: gneg(lfun(L, gen_0, prec2nbits(prec)));
10554 21 : NK = mkvec3(utoi(N), gk, mfchisimpl(CHI));
10555 21 : return gerepilecopy(av, mkvec3(NK, utoi(space), shallowconcat(a0, van)));
10556 : }
10557 : /*******************************************************************/
10558 : /* */
10559 : /* HALF-INTEGRAL WEIGHT */
10560 : /* */
10561 : /*******************************************************************/
10562 : /* We use the prefix mf2; k represents the weight -1/2, so e.g.
10563 : k = 2 is weight 5/2. N is the level, so 4\mid N, and CHI is the
10564 : character, always even. */
10565 :
10566 : static long
10567 3360 : lamCO(long r, long s, long p)
10568 : {
10569 3360 : if ((s << 1) <= r)
10570 : {
10571 1232 : long rp = r >> 1;
10572 1232 : if (odd(r)) return upowuu(p, rp) << 1;
10573 336 : else return (p + 1)*upowuu(p, rp - 1);
10574 : }
10575 2128 : else return upowuu(p, r - s) << 1;
10576 : }
10577 :
10578 : static int
10579 1568 : condC(GEN faN, GEN valF)
10580 : {
10581 1568 : GEN P = gel(faN, 1), E = gel(faN, 2);
10582 1568 : long l = lg(P), i;
10583 3696 : for (i = 1; i < l; i++)
10584 3024 : if ((P[i] & 3L) == 3)
10585 : {
10586 1120 : long r = E[i];
10587 1120 : if (odd(r) || r < (valF[i] << 1)) return 1;
10588 : }
10589 672 : return 0;
10590 : }
10591 :
10592 : /* returns 2*zetaCO; weight is k + 1/2 */
10593 : static long
10594 3696 : zeta2CO(GEN faN, GEN valF, long r2, long s2, long k)
10595 : {
10596 3696 : if (r2 >= 4) return lamCO(r2, s2, 2) << 1;
10597 2912 : if (r2 == 3) return 6;
10598 1568 : if (condC(faN, valF)) return 4;
10599 672 : if (odd(k)) return s2 ? 3 : 5; else return s2 ? 5: 3;
10600 : }
10601 :
10602 : /* returns 4 times last term in formula */
10603 : static long
10604 3696 : dim22(long N, long F, long k)
10605 : {
10606 3696 : pari_sp av = avma;
10607 3696 : GEN vF, faN = myfactoru(N), P = gel(faN, 1), E = gel(faN, 2);
10608 3696 : long i, D, l = lg(P);
10609 3696 : vF = cgetg(l, t_VECSMALL);
10610 9968 : for (i = 1; i < l; i++) vF[i] = u_lval(F, P[i]);
10611 3696 : D = zeta2CO(faN, vF, E[1], vF[1], k);
10612 6272 : for (i = 2; i < l; i++) D *= lamCO(E[i], vF[i], P[i]);
10613 3696 : return gc_long(av,D);
10614 : }
10615 :
10616 : /* PSI not necessarily primitive, of conductor F */
10617 : static int
10618 13846 : charistotallyeven(GEN PSI, long F)
10619 : {
10620 13846 : pari_sp av = avma;
10621 13846 : GEN P = gel(myfactoru(F), 1);
10622 13846 : GEN G = gel(PSI,1), psi = gel(PSI,2);
10623 : long i;
10624 14350 : for (i = 1; i < lg(P); i++)
10625 : {
10626 532 : GEN psip = znchardecompose(G, psi, utoipos(P[i]));
10627 532 : if (zncharisodd(G, psip)) return gc_bool(av,0);
10628 : }
10629 13818 : return gc_bool(av,1);
10630 : }
10631 :
10632 : static GEN
10633 299775 : get_PSI(GEN CHI, long t)
10634 : {
10635 299775 : long r = t & 3L, t2 = (r == 2 || r == 3) ? t << 2 : t;
10636 299775 : return mfcharmul_i(CHI, induce(gel(CHI,1), utoipos(t2)));
10637 : }
10638 : /* space = mf_CUSP, mf_EISEN or mf_FULL, weight k + 1/2 */
10639 : static long
10640 41363 : mf2dimwt12(long N, GEN CHI, long space)
10641 : {
10642 41363 : pari_sp av = avma;
10643 41363 : GEN D = mydivisorsu(N >> 2);
10644 41363 : long i, l = lg(D), dim3 = 0, dim4 = 0;
10645 :
10646 41363 : CHI = induceN(N, CHI);
10647 341138 : for (i = 1; i < l; i++)
10648 : {
10649 299775 : long rp, t = D[i], Mt = D[l-i];
10650 299775 : GEN PSI = get_PSI(CHI,t);
10651 299775 : rp = mfcharconductor(PSI);
10652 299775 : if (Mt % (rp*rp) == 0) { dim4++; if (charistotallyeven(PSI,rp)) dim3++; }
10653 : }
10654 41363 : set_avma(av);
10655 41363 : switch (space)
10656 : {
10657 40439 : case mf_CUSP: return dim4 - dim3;
10658 462 : case mf_EISEN:return dim3;
10659 462 : case mf_FULL: return dim4;
10660 : }
10661 : return 0; /*LCOV_EXCL_LINE*/
10662 : }
10663 :
10664 : static long
10665 693 : mf2dimwt32(long N, GEN CHI, long F, long space)
10666 : {
10667 : long D;
10668 693 : switch(space)
10669 : {
10670 231 : case mf_CUSP: D = mypsiu(N) - 6*dim22(N, F, 1);
10671 231 : if (D%24) pari_err_BUG("mfdim");
10672 231 : return D/24 + mf2dimwt12(N, CHI, 4);
10673 231 : case mf_FULL: D = mypsiu(N) + 6*dim22(N, F, 0);
10674 231 : if (D%24) pari_err_BUG("mfdim");
10675 231 : return D/24 + mf2dimwt12(N, CHI, 1);
10676 231 : case mf_EISEN: D = dim22(N, F, 0) + dim22(N, F, 1);
10677 231 : if (D & 3L) pari_err_BUG("mfdim");
10678 231 : return (D >> 2) - mf2dimwt12(N, CHI, 3);
10679 : }
10680 : return 0; /*LCOV_EXCL_LINE*/
10681 : }
10682 :
10683 : /* F = conductor(CHI), weight k = r+1/2 */
10684 : static long
10685 43729 : checkmf2(long N, long r, GEN CHI, long F, long space)
10686 : {
10687 43729 : switch(space)
10688 : {
10689 43708 : case mf_FULL: case mf_CUSP: case mf_EISEN: break;
10690 14 : case mf_NEW: case mf_OLD:
10691 14 : pari_err_TYPE("half-integral weight [new/old spaces]", utoi(space));
10692 7 : default:
10693 7 : pari_err_TYPE("half-integral weight [incorrect space]",utoi(space));
10694 : }
10695 43708 : if (N & 3L)
10696 0 : pari_err_DOMAIN("half-integral weight", "N % 4", "!=", gen_0, stoi(N));
10697 43708 : return r >= 0 && mfcharparity(CHI) == 1 && N % F == 0;
10698 : }
10699 :
10700 : /* weight k = r + 1/2 */
10701 : static long
10702 43463 : mf2dim_Nkchi(long N, long r, GEN CHI, ulong space)
10703 : {
10704 43463 : long D, D2, F = mfcharconductor(CHI);
10705 43463 : if (!checkmf2(N, r, CHI, F, space)) return 0;
10706 43442 : if (r == 0) return mf2dimwt12(N, CHI, space);
10707 2772 : if (r == 1) return mf2dimwt32(N, CHI, F, space);
10708 2079 : if (space == mf_EISEN)
10709 : {
10710 693 : D = dim22(N, F, r) + dim22(N, F, 1-r);
10711 693 : if (D & 3L) pari_err_BUG("mfdim");
10712 693 : return D >> 2;
10713 : }
10714 1386 : D2 = space == mf_FULL? dim22(N, F, 1-r): -dim22(N, F, r);
10715 1386 : D = (2*r-1)*mypsiu(N) + 6*D2;
10716 1386 : if (D%24) pari_err_BUG("mfdim");
10717 1386 : return D/24;
10718 : }
10719 :
10720 : /* weight k=r+1/2 */
10721 : static GEN
10722 266 : mf2init_Nkchi(long N, long r, GEN CHI, long space, long flraw)
10723 : {
10724 266 : GEN CHI1, Minv, Minvmat, B, M, gk = gaddsg(r,ghalf);
10725 266 : GEN mf1 = mkvec4(utoi(N),gk,CHI,utoi(space));
10726 : long L;
10727 266 : if (!checkmf2(N, r, CHI, mfcharconductor(CHI), space)) return mfEMPTY(mf1);
10728 266 : if (space==mf_EISEN) pari_err_IMPL("half-integral weight Eisenstein space");
10729 266 : L = mfsturmNgk(N, gk) + 1;
10730 266 : B = mf2basis(N, r, CHI, &CHI1, space);
10731 266 : M = mflineardivtomat(N,B,L); /* defined modulo T = charpol(CHI) */
10732 266 : if (flraw) M = mkvec3(gen_0,gen_0,M);
10733 : else
10734 : {
10735 266 : long o1 = mfcharorder(CHI1), o = mfcharorder(CHI);
10736 266 : M = mfcleanCHI(M, CHI, 0);
10737 266 : Minv = gel(M,2);
10738 266 : Minvmat = RgM_Minv_mul(NULL, Minv); /* mod T */
10739 266 : if (o1 != o)
10740 : {
10741 133 : GEN tr = Qab_trace_init(o, o1, mfcharpol(CHI), mfcharpol(CHI1));
10742 133 : Minvmat = QabM_tracerel(tr, 0, Minvmat);
10743 : }
10744 : /* Minvmat mod T1 = charpol(CHI1) */
10745 266 : B = vecmflineardiv_linear(B, Minvmat);
10746 266 : gel(M,3) = RgM_Minv_mul(gel(M,3), Minv);
10747 266 : gel(M,2) = mkMinv(matid(lg(B)-1), NULL,NULL,NULL);
10748 : }
10749 266 : return mkmf(mf1, cgetg(1,t_VEC), B, gen_0, M);
10750 : }
10751 :
10752 : /**************************************************************************/
10753 : /* Kohnen + space */
10754 : /**************************************************************************/
10755 :
10756 : static GEN
10757 21 : mfkohnenbasis_i(GEN mf, GEN CHI, long eps, long sb)
10758 : {
10759 21 : GEN M = mfcoefs_mf(mf, sb, 1), p, P;
10760 21 : long c, i, n = mfcharorder(CHI), l = sb + 2;
10761 21 : p = cgetg(l, t_VECSMALL);
10762 : /* keep the a_n, n = (2 or 2+eps) mod 4 */
10763 210 : for (i = 3, c = 1; i < l; i+=4) p[c++] = i;
10764 203 : for (i = 3+eps; i < l; i+=4) p[c++] = i;
10765 21 : P = n <= 2? NULL: mfcharpol(CHI);
10766 21 : setlg(p, c);
10767 21 : return QabM_ker(rowpermute(M, p), P, n);
10768 : }
10769 : GEN
10770 21 : mfkohnenbasis(GEN mf)
10771 : {
10772 21 : pari_sp av = avma;
10773 : GEN gk, CHI, CHIP, K;
10774 : long N4, r, eps, sb;
10775 21 : mf = checkMF(mf);
10776 21 : if (MF_get_space(mf) != mf_CUSP)
10777 0 : pari_err_TYPE("mfkohnenbasis [not a cuspidal space", mf);
10778 21 : if (!MF_get_dim(mf)) return cgetg(1, t_MAT);
10779 21 : N4 = MF_get_N(mf) >> 2; gk = MF_get_gk(mf); CHI = MF_get_CHI(mf);
10780 21 : if (typ(gk) == t_INT) pari_err_TYPE("mfkohnenbasis", gk);
10781 21 : r = MF_get_r(mf);
10782 21 : CHIP = mfcharchiliftprim(CHI, N4);
10783 21 : eps = CHIP==CHI? 1: -1;
10784 21 : if (odd(r)) eps = -eps;
10785 21 : if (uissquarefree(N4))
10786 : {
10787 14 : long d = mfdim_Nkchi(N4, 2*r, mfcharpow(CHI, gen_2), mf_CUSP);
10788 14 : sb = mfsturmNgk(N4 << 2, gk) + 1;
10789 14 : K = mfkohnenbasis_i(mf, CHIP, eps, sb);
10790 14 : if (lg(K) - 1 == d) return gerepilecopy(av, K);
10791 : }
10792 7 : sb = mfsturmNgk(N4 << 4, gk) + 1;
10793 7 : K = mfkohnenbasis_i(mf, CHIP, eps, sb);
10794 7 : return gerepilecopy(av, K);
10795 : }
10796 :
10797 : static GEN
10798 21 : get_Shimura(GEN mf, GEN CHI, GEN vB, long t)
10799 : {
10800 21 : long N = MF_get_N(mf), r = MF_get_k(mf) >> 1;
10801 21 : long i, d = MF_get_dim(mf), sb = mfsturm_mf(mf);
10802 21 : GEN a = cgetg(d+1, t_MAT);
10803 84 : for (i = 1; i <= d; i++)
10804 : {
10805 63 : pari_sp av = avma;
10806 63 : GEN f = c_deflate(sb*sb, t, gel(vB,i));
10807 63 : f = mftobasis_i(mf, RgV_shimura(f, sb, t, N, r, CHI));
10808 63 : gel(a,i) = gerepileupto(av, f);
10809 : }
10810 21 : return a;
10811 : }
10812 : static long
10813 35 : QabM_rank(GEN M, GEN P, long n)
10814 : {
10815 35 : GEN z = QabM_indexrank(M, P, n);
10816 35 : return lg(gel(z,2))-1;
10817 : }
10818 : /* discard T[*i] */
10819 : static void
10820 0 : discard_Ti(GEN T, long *i, long *lt)
10821 : {
10822 0 : long j, l = *lt-1;
10823 0 : for (j = *i; j < l; j++) T[j] = T[j+1];
10824 0 : (*i)--; *lt = l;
10825 0 : }
10826 : /* return [mf3, bijection, mfkohnenbasis, codeshi] */
10827 : static GEN
10828 14 : mfkohnenbijection_i(GEN mf)
10829 : {
10830 14 : GEN CHI = MF_get_CHI(mf), K = mfkohnenbasis(mf);
10831 : GEN mres, dMi, Mi, M, C, vB, mf3, SHI, T, P;
10832 14 : long N4 = MF_get_N(mf)>>2, r = MF_get_r(mf), dK = lg(K) - 1;
10833 : long i, c, n, oldr, lt, ltold, sb3, t, limt;
10834 14 : const long MAXlt = 100;
10835 :
10836 14 : mf3 = mfinit_Nkchi(N4, r<<1, mfcharpow(CHI,gen_2), mf_CUSP, 0);
10837 14 : if (MF_get_dim(mf3) != dK)
10838 0 : pari_err_BUG("mfkohnenbijection [different dimensions]");
10839 14 : if (!dK) return mkvec4(mf3, cgetg(1, t_MAT), K, cgetg(1, t_VEC));
10840 14 : CHI = mfcharchiliftprim(CHI, N4);
10841 14 : n = mfcharorder(CHI);
10842 14 : P = n<=2? NULL: mfcharpol(CHI);
10843 14 : SHI = cgetg(MAXlt, t_COL);
10844 14 : T = cgetg(MAXlt, t_VECSMALL);
10845 14 : sb3 = mfsturm_mf(mf3);
10846 14 : limt = 6; oldr = 0; vB = C = M = NULL;
10847 98 : for (t = lt = ltold = 1; lt < MAXlt; t++)
10848 : {
10849 : pari_sp av;
10850 98 : if (!uissquarefree(t)) continue;
10851 84 : T[lt++] = t; if (t <= limt) continue;
10852 14 : av = avma;
10853 14 : if (vB) gunclone(vB);
10854 : /* could improve the rest but 99% of running time is spent here */
10855 14 : vB = gclone( RgM_mul(mfcoefs_mf(mf, t*sb3*sb3, 1), K) );
10856 14 : set_avma(av);
10857 21 : for (i = ltold; i < lt; i++)
10858 : {
10859 : pari_sp av;
10860 : long r;
10861 21 : M = get_Shimura(mf3, CHI, vB, T[i]);
10862 21 : r = QabM_rank(M, P, n); if (!r) { discard_Ti(T, &i, <); continue; }
10863 21 : gel(SHI, i) = M; setlg(SHI, i+1);
10864 21 : if (r >= dK) { C = vecsmall_ei(dK, i); goto DONE; }
10865 14 : if (i == 1) { oldr = r; continue; }
10866 7 : av = avma; M = shallowmatconcat(SHI);
10867 7 : r = QabM_rank(M, P, n); /* >= rank(sum C[j] SHI[j]), probably sharp */
10868 7 : if (r >= dK)
10869 : {
10870 7 : M = RgV_sum(SHI);
10871 7 : if (QabM_rank(M, P, n) >= dK) { C = const_vecsmall(dK, 1); goto DONE; }
10872 0 : C = random_Flv(dK, 16);
10873 0 : M = RgV_zc_mul(SHI, C);
10874 0 : if (QabM_rank(M, P, n) >= dK) goto DONE;
10875 : }
10876 0 : else if (r == oldr) discard_Ti(T, &i, <);
10877 0 : oldr = r; set_avma(av);
10878 : }
10879 0 : limt *= 2; ltold = lt;
10880 : }
10881 0 : pari_err_BUG("mfkohnenbijection");
10882 14 : DONE:
10883 14 : gunclone(vB); lt = lg(SHI);
10884 14 : Mi = QabM_pseudoinv(M,P,n, NULL,&dMi); Mi = RgM_Rg_div(Mi,dMi);
10885 14 : mres = cgetg(lt, t_VEC);
10886 35 : for (i = c = 1; i < lt; i++)
10887 21 : if (C[i]) gel(mres,c++) = mkvec2s(T[i], C[i]);
10888 14 : setlg(mres,c); return mkvec4(mf3, Mi, K, mres);
10889 : }
10890 : GEN
10891 14 : mfkohnenbijection(GEN mf)
10892 : {
10893 14 : pari_sp av = avma;
10894 : long N;
10895 14 : mf = checkMF(mf); N = MF_get_N(mf);
10896 14 : if (!uissquarefree(N >> 2))
10897 0 : pari_err_TYPE("mfkohnenbijection [N/4 not squarefree]", utoi(N));
10898 14 : if (MF_get_space(mf) != mf_CUSP || MF_get_r(mf) == 0 || !mfshimura_space_cusp(mf))
10899 0 : pari_err_TYPE("mfkohnenbijection [incorrect mf for Kohnen]", mf);
10900 14 : return gerepilecopy(av, mfkohnenbijection_i(mf));
10901 : }
10902 :
10903 : static int
10904 7 : checkbij_i(GEN b)
10905 : {
10906 7 : return typ(b) == t_VEC && lg(b) == 5 && checkMF_i(gel(b,1))
10907 7 : && typ(gel(b,2)) == t_MAT
10908 7 : && typ(gel(b,3)) == t_MAT
10909 14 : && typ(gel(b,4)) == t_VEC;
10910 : }
10911 :
10912 : /* bij is the output of mfkohnenbijection */
10913 : GEN
10914 7 : mfkohneneigenbasis(GEN mf, GEN bij)
10915 : {
10916 7 : pari_sp av = avma;
10917 : GEN mf3, mf30, B, KM, M, k;
10918 : long r, i, l, N4;
10919 7 : mf = checkMF(mf);
10920 7 : if (!checkbij_i(bij))
10921 0 : pari_err_TYPE("mfkohneneigenbasis [bijection]", bij);
10922 7 : if (MF_get_space(mf) != mf_CUSP)
10923 0 : pari_err_TYPE("mfkohneneigenbasis [not a cuspidal space]", mf);
10924 7 : if (!MF_get_dim(mf))
10925 0 : retmkvec3(cgetg(1, t_VEC), cgetg(1, t_VEC), cgetg(1, t_VEC));
10926 7 : N4 = MF_get_N(mf) >> 2; k = MF_get_gk(mf);
10927 7 : if (typ(k) == t_INT) pari_err_TYPE("mfkohneneigenbasis", k);
10928 7 : if (!uissquarefree(N4))
10929 0 : pari_err_TYPE("mfkohneneigenbasis [N not squarefree]", utoipos(N4));
10930 7 : r = MF_get_r(mf);
10931 7 : KM = RgM_mul(gel(bij,3), gel(bij,2));
10932 7 : mf3 = gel(bij,1);
10933 7 : mf30 = mfinit_Nkchi(N4, 2*r, MF_get_CHI(mf3), mf_NEW, 0);
10934 7 : B = mfcoefs_mf(mf30, mfsturm_mf(mf3), 1); l = lg(B);
10935 7 : M = cgetg(l, t_MAT);
10936 21 : for (i=1; i<l; i++) gel(M,i) = RgM_RgC_mul(KM, mftobasis_i(mf3, gel(B,i)));
10937 7 : return gerepilecopy(av, mkvec3(mf30, M, RgM_mul(M, MF_get_newforms(mf30))));
10938 : }
10939 : /*************************** End Kohnen ************************************/
10940 : /***************************************************************************/
10941 :
10942 : static GEN desc(GEN F);
10943 : static GEN
10944 504 : desc_mfeisen(GEN F)
10945 : {
10946 504 : GEN R, gk = mf_get_gk(F);
10947 504 : if (typ(gk) == t_FRAC)
10948 7 : R = gsprintf("H_{%Ps}", gk);
10949 : else
10950 : {
10951 497 : GEN vchi = gel(F, 2), CHI = mfchisimpl(gel(vchi, 3));
10952 497 : long k = itou(gk);
10953 497 : if (lg(vchi) < 5) R = gsprintf("F_%ld(%Ps)", k, CHI);
10954 : else
10955 : {
10956 294 : GEN CHI2 = mfchisimpl(gel(vchi, 4));
10957 294 : R = gsprintf("F_%ld(%Ps, %Ps)", k, CHI, CHI2);
10958 : }
10959 : }
10960 504 : return R;
10961 : }
10962 : static GEN
10963 35 : desc_hecke(GEN F)
10964 : {
10965 : long n, N;
10966 35 : GEN D = gel(F,2);
10967 35 : if (typ(D) == t_VECSMALL) { N = D[3]; n = D[1]; }
10968 14 : else { GEN nN = gel(D,2); n = nN[1]; N = nN[2]; } /* half integer */
10969 35 : return gsprintf("T_%ld(%ld)(%Ps)", N, n, desc(gel(F,3)));
10970 : }
10971 : static GEN
10972 98 : desc_linear(GEN FLD, GEN dL)
10973 : {
10974 98 : GEN F = gel(FLD,2), L = gel(FLD,3), R = strtoGENstr("LIN([");
10975 98 : long n = lg(F) - 1, i;
10976 168 : for (i = 1; i <= n; i++)
10977 : {
10978 168 : R = shallowconcat(R, desc(gel(F,i))); if (i == n) break;
10979 70 : R = shallowconcat(R, strtoGENstr(", "));
10980 : }
10981 98 : return shallowconcat(R, gsprintf("], %Ps)", gdiv(L, dL)));
10982 : }
10983 : static GEN
10984 21 : desc_dihedral(GEN F)
10985 : {
10986 21 : GEN bnr = gel(F,2), D = nf_get_disc(bnr_get_nf(bnr)), f = bnr_get_mod(bnr);
10987 21 : GEN cyc = bnr_get_cyc(bnr);
10988 21 : GEN w = gel(F,3), chin = zv_to_ZV(gel(w,2)), o = utoi(gel(w,1)[1]);
10989 21 : GEN chi = char_denormalize(cyc, o, chin);
10990 21 : if (lg(gel(f,2)) == 1) f = gel(f,1);
10991 21 : return gsprintf("DIH(%Ps, %Ps, %Ps, %Ps)",D,f,cyc,chi);
10992 : }
10993 :
10994 : static void
10995 1043 : unpack0(GEN *U)
10996 1043 : { if (U) *U = mkvec2(cgetg(1, t_VEC), cgetg(1, t_VEC)); }
10997 : static void
10998 42 : unpack2(GEN F, GEN *U)
10999 42 : { if (U) *U = mkvec2(mkvec2(gel(F,2), gel(F,3)), cgetg(1, t_VEC)); }
11000 : static void
11001 308 : unpack23(GEN F, GEN *U)
11002 308 : { if (U) *U = mkvec2(mkvec(gel(F,2)), mkvec(gel(F,3))); }
11003 : static GEN
11004 1540 : desc_i(GEN F, GEN *U)
11005 : {
11006 1540 : switch(mf_get_type(F))
11007 : {
11008 7 : case t_MF_CONST: unpack0(U); return gsprintf("CONST(%Ps)", gel(F,2));
11009 504 : case t_MF_EISEN: unpack0(U); return desc_mfeisen(F);
11010 154 : case t_MF_Ek: unpack0(U); return gsprintf("E_%ld", mf_get_k(F));
11011 63 : case t_MF_DELTA: unpack0(U); return gsprintf("DELTA");
11012 35 : case t_MF_THETA: unpack0(U);
11013 35 : return gsprintf("THETA(%Ps)", mfchisimpl(gel(F,2)));
11014 56 : case t_MF_ETAQUO: unpack0(U);
11015 56 : return gsprintf("ETAQUO(%Ps, %Ps)", gel(F,2), gel(F,3));
11016 56 : case t_MF_ELL: unpack0(U);
11017 56 : return gsprintf("ELL(%Ps)", vecslice(gel(F,2), 1, 5));
11018 7 : case t_MF_TRACE: unpack0(U); return gsprintf("TR(%Ps)", mfparams(F));
11019 140 : case t_MF_NEWTRACE: unpack0(U); return gsprintf("TR^new(%Ps)", mfparams(F));
11020 21 : case t_MF_DIHEDRAL: unpack0(U); return desc_dihedral(F);
11021 28 : case t_MF_MUL: unpack2(F, U);
11022 28 : return gsprintf("MUL(%Ps, %Ps)", desc(gel(F,2)), desc(gel(F,3)));
11023 14 : case t_MF_DIV: unpack2(F, U);
11024 14 : return gsprintf("DIV(%Ps, %Ps)", desc(gel(F,2)), desc(gel(F,3)));
11025 14 : case t_MF_POW: unpack23(F, U);
11026 14 : return gsprintf("POW(%Ps, %ld)", desc(gel(F,2)), itos(gel(F,3)));
11027 14 : case t_MF_SHIFT: unpack23(F, U);
11028 14 : return gsprintf("SHIFT(%Ps, %ld)", desc(gel(F,2)), itos(gel(F,3)));
11029 14 : case t_MF_DERIV: unpack23(F, U);
11030 14 : return gsprintf("DER^%ld(%Ps)", itos(gel(F,3)), desc(gel(F,2)));
11031 21 : case t_MF_DERIVE2: unpack23(F, U);
11032 21 : return gsprintf("DERE2^%ld(%Ps)", itos(gel(F,3)), desc(gel(F,2)));
11033 14 : case t_MF_TWIST: unpack23(F, U);
11034 14 : return gsprintf("TWIST(%Ps, %Ps)", desc(gel(F,2)), gel(F,3));
11035 231 : case t_MF_BD: unpack23(F, U);
11036 231 : return gsprintf("B(%ld)(%Ps)", itou(gel(F,3)), desc(gel(F,2)));
11037 14 : case t_MF_BRACKET:
11038 14 : if (U) *U = mkvec2(mkvec2(gel(F,2), gel(F,3)), mkvec(gel(F,4)));
11039 14 : return gsprintf("MULRC_%ld(%Ps, %Ps)", itos(gel(F,4)), desc(gel(F,2)), desc(gel(F,3)));
11040 98 : case t_MF_LINEAR_BHN:
11041 : case t_MF_LINEAR:
11042 98 : if (U) *U = mkvec2(gel(F,2), mkvec(gdiv(gel(F,3), gel(F,4))));
11043 98 : return desc_linear(F,gel(F,4));
11044 35 : case t_MF_HECKE:
11045 35 : if (U) *U = mkvec2(mkvec(gel(F,3)), mkvec(stoi(gel(F,2)[1])));
11046 35 : return desc_hecke(F);
11047 0 : default: pari_err_TYPE("mfdescribe",F);
11048 : return NULL;/*LCOV_EXCL_LINE*/
11049 : }
11050 : }
11051 : static GEN
11052 623 : desc(GEN F) { return desc_i(F, NULL); }
11053 : GEN
11054 966 : mfdescribe(GEN F, GEN *U)
11055 : {
11056 966 : pari_sp av = avma;
11057 : GEN mf;
11058 966 : if ((mf = checkMF_i(F)))
11059 : {
11060 49 : const char *f = NULL;
11061 49 : switch (MF_get_space(mf))
11062 : {
11063 7 : case mf_NEW: f = "S_%Ps^new(G_0(%ld, %Ps))"; break;
11064 14 : case mf_CUSP: f = "S_%Ps(G_0(%ld, %Ps))"; break;
11065 7 : case mf_OLD: f = "S_%Ps^old(G_0(%ld, %Ps))"; break;
11066 7 : case mf_EISEN:f = "E_%Ps(G_0(%ld, %Ps))"; break;
11067 14 : case mf_FULL: f = "M_%Ps(G_0(%ld, %Ps))"; break;
11068 : }
11069 49 : if (U) *U = cgetg(1, t_VEC);
11070 49 : return gsprintf(f, MF_get_gk(mf), MF_get_N(mf), mfchisimpl(MF_get_CHI(mf)));
11071 : }
11072 917 : if (!checkmf_i(F)) pari_err_TYPE("mfdescribe", F);
11073 917 : F = desc_i(F, U); return gc_all(av, U ? 2: 1, &F, U);
11074 : }
11075 :
11076 : /***********************************************************************/
11077 : /* Eisenstein series H_r of weight r+1/2 */
11078 : /***********************************************************************/
11079 : /* radical(u_ppo(g,q)) */
11080 : static long
11081 28 : u_pporad(long g, long q)
11082 : {
11083 28 : GEN F = myfactoru(g), P = gel(F,1);
11084 : long i, l, n;
11085 28 : if (q == 1) return zv_prod(P);
11086 28 : l = lg(P);
11087 35 : for (i = n = 1; i < l; i++)
11088 : {
11089 7 : long p = P[i];
11090 7 : if (q % p) n *= p;
11091 : }
11092 28 : return n;
11093 : }
11094 : static void
11095 266 : c_F2TH4(long n, GEN *pF2, GEN *pTH4)
11096 : {
11097 266 : GEN v = mfcoefs_i(mfEk(2), n, 1), v2 = bdexpand(v,2), v4 = bdexpand(v,4);
11098 266 : GEN F2 = gdivgs(ZC_add(ZC_sub(v, ZC_z_mul(v2,3)), ZC_z_mul(v4,2)), -24);
11099 266 : GEN TH4 = gdivgs(ZC_sub(v, ZC_z_mul(v4,4)), -3);
11100 266 : settyp(F2,t_VEC); *pF2 = F2;
11101 266 : settyp(TH4,t_VEC);*pTH4= TH4;
11102 266 : }
11103 : /* r > 0, N >= 0 */
11104 : static GEN
11105 77 : mfEHcoef(long r, long N)
11106 : {
11107 : long D0, f, i, l, s;
11108 : GEN S, Df;
11109 :
11110 77 : if (r == 1) return hclassno(utoi(N));
11111 77 : if (N == 0) return gdivgs(bernfrac(2*r), -2*r);
11112 56 : s = N & 3L;
11113 56 : if (odd(r))
11114 : {
11115 42 : if (s == 2 || s == 1) return gen_0;
11116 14 : D0 = mycoredisc2neg(N,&f);
11117 : }
11118 : else
11119 : {
11120 14 : if (s == 2 || s == 3) return gen_0;
11121 14 : D0 = mycoredisc2pos(N,&f);
11122 : }
11123 28 : Df = mydivisorsu(u_pporad(f, D0)); l = lg(Df);
11124 28 : S = gen_0;
11125 63 : for (i = 1; i < l; i++)
11126 : {
11127 35 : long d = Df[i], s = mymoebiusu(d)*kross(D0, d); /* != 0 */
11128 35 : GEN c = gmul(powuu(d, r-1), mysumdivku(f/d, 2*r-1));
11129 35 : S = s > 0? addii(S, c): subii(S, c);
11130 : }
11131 28 : return gmul(lfunquadneg_naive(D0, r), S);
11132 : }
11133 : static GEN
11134 266 : mfEHmat(long lim, long r)
11135 : {
11136 266 : long j, l, d = r/2;
11137 : GEN f2, th4, th3, v, vth4, vf2;
11138 266 : c_F2TH4(lim, &f2, &th4);
11139 266 : f2 = RgV_to_ser(f2, 0, lim+3);
11140 266 : th4 = RgV_to_ser(th4, 0, lim+3);
11141 266 : th3 = RgV_to_ser(c_theta(lim, 1, mfchartrivial()), 0, lim+3);
11142 266 : if (odd(r)) th3 = gpowgs(th3, 3);
11143 266 : vth4 = gpowers(th4, d);
11144 266 : vf2 = gpowers0(f2, d, th3); /* th3 f2^j */
11145 266 : l = d+2; v = cgetg(l, t_VEC);
11146 924 : for (j = 1; j < l; j++)
11147 658 : gel(v, j) = ser2rfrac_i(gmul(gel(vth4, l-j), gel(vf2, j)));
11148 266 : return RgXV_to_RgM(v, lim);
11149 : }
11150 : static GEN
11151 7 : Hfind(long r, GEN *pden)
11152 : {
11153 7 : long lim = (r/2)+3, i;
11154 : GEN res, M, B;
11155 :
11156 7 : if (r <= 0) pari_err_DOMAIN("mfEH", "r", "<=", gen_0, stoi(r));
11157 7 : M = mfEHmat(lim, r);
11158 7 : B = cgetg(lim+1, t_COL);
11159 56 : for (i = 1; i <= lim; i++) gel(B, i) = mfEHcoef(r, i-1);
11160 7 : res = QM_gauss(M, B);
11161 7 : if (lg(res) == 1) pari_err_BUG("mfEH");
11162 7 : return Q_remove_denom(res,pden);
11163 : }
11164 : GEN
11165 266 : mfEH(GEN gk)
11166 : {
11167 266 : pari_sp av = avma;
11168 266 : GEN v, d, NK, gr = gsub(gk, ghalf);
11169 : long r;
11170 266 : if (typ(gr) != t_INT) pari_err_TYPE("mfEH", gk);
11171 266 : r = itos(gr);
11172 266 : switch (r)
11173 : {
11174 7 : case 1: v=cgetg(1,t_VEC); d=gen_1; break;
11175 133 : case 2: v=mkvec2s(1,-20); d=utoipos(120); break;
11176 56 : case 3: v=mkvec2s(-1,14); d=utoipos(252); break;
11177 35 : case 4: v=mkvec3s(1,-16,16); d=utoipos(240); break;
11178 7 : case 5: v=mkvec3s(-1,22,-88); d=utoipos(132); break;
11179 14 : case 6: v=mkvec4s(691,-18096,110136,-4160); d=utoipos(32760); break;
11180 7 : case 7: v=mkvec4s(-1,30,-240,224); d=utoipos(12); break;
11181 7 : default: v = Hfind(r, &d); break;
11182 : }
11183 266 : NK = mkgNK(utoipos(4), gaddgs(ghalf,r), mfchartrivial(), pol_x(1));
11184 266 : return gerepilecopy(av, tag(t_MF_EISEN, NK, mkvec2(v,d)));
11185 : }
11186 :
11187 : /**********************************************************/
11188 : /* T(f^2) for half-integral weight */
11189 : /**********************************************************/
11190 :
11191 : /* T_p^2 V, p2 = p^2, c1 = chi(p) (-1/p)^r p^(r-1), c2 = chi(p^2)*p^(2r-1) */
11192 : static GEN
11193 70 : tp2apply(GEN V, long p, long p2, GEN c1, GEN c2)
11194 : {
11195 70 : long lw = (lg(V) - 2)/p2 + 1, m, n;
11196 70 : GEN a0 = gel(V,1), W = cgetg(lw + 1, t_VEC);
11197 :
11198 70 : gel(W,1) = gequal0(a0)? gen_0: gmul(a0, gaddsg(1, c2));
11199 11109 : for (n = 1; n < lw; n++)
11200 : {
11201 11039 : GEN c = gel(V, p2*n + 1);
11202 11039 : if (n%p) c = gadd(c, gmulsg(kross(n,p), gmul(gel(V,n+1), c1)));
11203 11039 : gel(W, n+1) = c; /* a(p^2*n) + c1 * (n/p) a(n) */
11204 : }
11205 1253 : for (m = 1, n = p2; n < lw; m++, n += p2)
11206 1183 : gel(W, n+1) = gadd(gel(W,n+1), gmul(gel(V,m+1), c2));
11207 70 : return W;
11208 : }
11209 :
11210 : /* T_{p^{2e}} V; can derecursify [Purkait, Hecke operators in half-integral
11211 : * weight, Prop 4.3], not worth it */
11212 : static GEN
11213 70 : tp2eapply(GEN V, long p, long p2, long e, GEN q, GEN c1, GEN c2)
11214 : {
11215 70 : GEN V4 = NULL;
11216 70 : if (e > 1)
11217 : {
11218 21 : V4 = vecslice(V, 1, (lg(V) - 2)/(p2*p2) + 1);
11219 21 : V = tp2eapply(V, p, p2, e-1, q, c1, c2);
11220 : }
11221 70 : V = tp2apply(V, p, p2, c1, c2);
11222 70 : if (e > 1)
11223 28 : V = gsub(V, (e == 2)? gmul(q, V4)
11224 7 : : gmul(c2, tp2eapply(V4, p, p2, e-2, q, c1, c2)));
11225 70 : return V;
11226 : }
11227 : /* weight k = r+1/2 */
11228 : static GEN
11229 98 : RgV_heckef2(long n, long d, GEN V, GEN F, GEN DATA)
11230 : {
11231 98 : GEN CHI = mf_get_CHI(F), fa = gel(DATA,1), P = gel(fa,1), E = gel(fa,2);
11232 98 : long i, l = lg(P), r = mf_get_r(F), s4 = odd(r)? -4: 4, k2m2 = (r<<1)-1;
11233 98 : if (typ(V) == t_COL) V = shallowtrans(V);
11234 140 : for (i = 1; i < l; i++)
11235 : { /* p does not divide N */
11236 42 : long p = P[i], e = E[i], p2 = p*p;
11237 42 : GEN c1, c2, a, b, q = NULL, C = mfchareval(CHI,p), C2 = gsqr(C);
11238 42 : a = r? powuu(p,r-1): mkfrac(gen_1,utoipos(p)); /* p^(r-1) = p^(k-3/2) */
11239 42 : b = r? mulii(powuu(p,r), a): a; /* p^(2r-1) = p^(2k-2) */
11240 42 : c1 = gmul(C, gmulsg(kross(s4,p),a));
11241 42 : c2 = gmul(C2, b);
11242 42 : if (e > 1)
11243 : {
11244 14 : q = r? powuu(p,k2m2): a;
11245 14 : if (e == 2) q = gmul(q, uutoQ(p+1,p)); /* special case T_{p^4} */
11246 14 : q = gmul(C2, q); /* chi(p^2) [ p^(2k-2) or (p+1)p^(2k-3) ] */
11247 : }
11248 42 : V = tp2eapply(V, p, p2, e, q, c1, c2);
11249 : }
11250 98 : return c_deflate(n, d, V);
11251 : }
11252 :
11253 : static GEN
11254 1176 : GL2toSL2(GEN g, GEN *abd)
11255 : {
11256 : GEN A, B, C, D, u, v, a, b, d, q;
11257 1176 : g = Q_primpart(g);
11258 1176 : if (!check_M2Z(g)) pari_err_TYPE("GL2toSL2", g);
11259 1176 : A = gcoeff(g,1,1); B = gcoeff(g,1,2);
11260 1176 : C = gcoeff(g,2,1); D = gcoeff(g,2,2);
11261 1176 : a = bezout(A, C, &u, &v);
11262 1176 : if (!equali1(a)) { A = diviiexact(A,a); C = diviiexact(C,a); }
11263 1176 : d = subii(mulii(A,D), mulii(B,C));
11264 1176 : if (signe(d) <= 0) pari_err_TYPE("GL2toSL2",g);
11265 1169 : q = dvmdii(addii(mulii(u,B), mulii(v,D)), d, &b);
11266 1169 : *abd = (equali1(a) && equali1(d))? NULL: mkvec3(a, b, d);
11267 1169 : return mkmat22(A, subii(mulii(q,A), v), C, addii(mulii(q,C), u));
11268 : }
11269 :
11270 : static GEN
11271 8414 : Rg_approx(GEN t, long bit)
11272 : {
11273 8414 : GEN a = real_i(t), b = imag_i(t);
11274 8414 : long e1 = gexpo(a), e2 = gexpo(b);
11275 8414 : if (e2 < -bit) { t = e1 < -bit? gen_0: a; }
11276 6426 : else if (e1 < -bit) t = gmul(b, gen_I());
11277 8414 : return t;
11278 : }
11279 : static GEN
11280 126 : RgV_approx(GEN x, long bit)
11281 840 : { pari_APPLY_same(Rg_approx(gel(x,i), bit)); }
11282 : /* m != 2 (mod 4), D t_INT; V has "denominator" D, recognize in Q(zeta_m) */
11283 : static GEN
11284 126 : bestapprnf2(GEN V, long m, GEN D, long prec)
11285 : {
11286 126 : long i, j, f, vt = fetch_user_var("t"), bit = prec2nbits_mul(prec, 0.8);
11287 126 : GEN Tinit, Vl, H, Pf, P = polcyclo(m, vt);
11288 :
11289 126 : V = liftpol_shallow(V);
11290 126 : V = gmul(RgV_approx(V, bit), D);
11291 126 : V = bestapprnf(V, P, NULL, prec);
11292 126 : Vl = liftpol_shallow(V);
11293 126 : H = coprimes_zv(m);
11294 672 : for (i = 2; i < m; i++)
11295 : {
11296 546 : if (H[i] != 1) continue;
11297 280 : if (!gequal(Vl, vecGalois(Vl, i, P, m))) H[i] = 0;
11298 14 : else for (j = i; j < m; j *= i) H[i] = 3;
11299 : }
11300 126 : f = znstar_conductor_bits(Flv_to_F2v(H));
11301 126 : if (f == 1) return gdiv(V, D);
11302 98 : if (f == m) return gmodulo(gdiv(V, D), P);
11303 7 : Pf = polcyclo(f, vt);
11304 7 : Tinit = Qab_trace_init(m, f, P, Pf);
11305 7 : return gmodulo(gdiv(QabV_tracerel(Tinit, 0, Vl), D), Pf);
11306 : }
11307 :
11308 : /* f | ga expansion; [f, mf_eisendec(f)]~ allowed */
11309 : GEN
11310 1113 : mfslashexpansion(GEN mf, GEN f, GEN ga, long n, long flrat, GEN *params, long prec)
11311 : {
11312 1113 : pari_sp av = avma;
11313 1113 : GEN a, b, d, res, al, V, M, ad, abd, gk, A, awd = NULL;
11314 : long i, w;
11315 :
11316 1113 : mf = checkMF(mf);
11317 1113 : gk = MF_get_gk(mf);
11318 1113 : M = GL2toSL2(ga, &abd);
11319 1106 : if (abd) { a = gel(abd,1); b = gel(abd,2); d = gel(abd,3); }
11320 868 : else { a = d = gen_1; b = gen_0; }
11321 1106 : ad = gdiv(a,d);
11322 1106 : res = mfgaexpansion(mf, f, M, n, prec);
11323 1106 : al = gel(res,1);
11324 1106 : w = itou(gel(res,2));
11325 1106 : V = gel(res,3);
11326 1106 : if (flrat)
11327 : {
11328 126 : GEN CHI = MF_get_CHI(mf);
11329 126 : long N = MF_get_N(mf), F = mfcharconductor(CHI);
11330 126 : long ord = mfcharorder(CHI), k, deg;
11331 126 : long B = umodiu(gcoeff(M,1,2), N);
11332 126 : long C = umodiu(gcoeff(M,2,1), N);
11333 126 : long D = umodiu(gcoeff(M,2,2), N);
11334 126 : long CD = (C * D) % N, BC = (B * C) % F;
11335 : GEN CV, t;
11336 : /* weight of f * Theta in 1/2-integral weight */
11337 126 : k = typ(gk) == t_INT? (long) itou(gk): MF_get_r(mf)+1;
11338 126 : CV = odd(k) ? powuu(N, k - 1) : powuu(N, k >> 1);
11339 126 : deg = ulcm(ulcm(ord, N/ugcd(N,CD)), F/ugcd(F,BC));
11340 126 : if ((deg & 3) == 2) deg >>= 1;
11341 126 : if (typ(gk) != t_INT && odd(deg) && mfthetaI(C,D)) deg <<= 2;
11342 126 : V = bestapprnf2(V, deg, CV, prec);
11343 126 : if (abd && !signe(b))
11344 : { /* can [a,0; 0,d] be simplified to id ? */
11345 7 : long nk, dk; Qtoss(gk, &nk, &dk);
11346 7 : if (ispower(ad, utoipos(2*dk), &t)) /* t^(2*dk) = a/d or t = NULL */
11347 : {
11348 7 : V = RgV_Rg_mul(V, powiu(t,nk));
11349 7 : awd = gdiv(a, muliu(d,w));
11350 : }
11351 : }
11352 : }
11353 980 : else if (abd)
11354 : { /* ga = M * [a,b;0,d] * rational, F := f | M = q^al * \sum V[j] q^(j/w) */
11355 231 : GEN u, t = NULL, wd = muliu(d,w);
11356 : /* a > 0, 0 <= b < d; f | ga = (a/d)^(k/2) * F(tau + b/d) */
11357 231 : if (signe(b))
11358 : {
11359 : long ns, ds;
11360 : GEN z;
11361 0 : Qtoss(gdiv(b, wd), &ns, &ds); z = rootsof1powinit(ns, ds, prec);
11362 0 : for (i = 1; i <= n+1; i++) gel(V,i) = gmul(gel(V,i), rootsof1pow(z, i-1));
11363 0 : if (!gequal0(al)) t = gexp(gmul(PiI2(prec), gmul(al, gdiv(b,d))), prec);
11364 : }
11365 231 : awd = gdiv(a, wd);
11366 231 : u = gpow(ad, gmul2n(gk,-1), prec);
11367 231 : t = t? gmul(t, u): u;
11368 231 : V = RgV_Rg_mul(V, t);
11369 : }
11370 1106 : if (!awd) A = mkmat22(a, b, gen_0, d);
11371 : else
11372 : { /* rescale and update w from [a,0; 0,d] */
11373 : long ns;
11374 238 : Qtoss(awd, &ns, &w); /* update w */
11375 238 : V = bdexpand(V, ns);
11376 238 : if (!gequal0(al))
11377 : {
11378 0 : GEN adal = gmul(ad, al), sh = gfloor(adal);
11379 0 : al = gsub(adal, sh);
11380 0 : V = RgV_shift(V, sh);
11381 : }
11382 238 : A = matid(2);
11383 : }
11384 1106 : if (params) *params = mkvec3(al, utoipos(w), A);
11385 1106 : return gc_all(av,params?2:1,&V,params);
11386 : }
11387 :
11388 : /**************************************************************/
11389 : /* Alternative method for 1/2-integral weight */
11390 : /**************************************************************/
11391 : static GEN
11392 266 : mf2basis(long N, long r, GEN CHI, GEN *pCHI1, long space)
11393 : {
11394 : GEN CHI1, CHI2, mf1, mf2, B1, B2, BT, M1, M2, M, M2i, T, Th, v, den;
11395 266 : long sb, N2, o1, o2, k1 = r + 1;
11396 :
11397 266 : if (odd(k1))
11398 : {
11399 154 : CHI1 = mfcharmul(CHI, get_mfchar(stoi(-4)));
11400 154 : CHI2 = mfcharmul(CHI, get_mfchar(stoi(-8)));
11401 : }
11402 : else
11403 : {
11404 112 : CHI1 = CHI;
11405 112 : CHI2 = mfcharmul(CHI, get_mfchar(utoi(8)));
11406 : }
11407 266 : mf1 = mfinit_Nkchi(N, k1, CHI1, space, 1);
11408 266 : if (pCHI1) *pCHI1 = CHI1;
11409 266 : B1 = MF_get_basis(mf1); if (lg(B1) == 1) return cgetg(1,t_VEC);
11410 259 : N2 = ulcm(8, N);
11411 259 : mf2 = mfinit_Nkchi(N2, k1, CHI2, space, 1);
11412 259 : B2 = MF_get_basis(mf2); if (lg(B2) == 1) return cgetg(1,t_VEC);
11413 259 : sb = mfsturmNgk(N2, gaddsg(k1, ghalf));
11414 259 : M1 = mfcoefs_mf(mf1, sb, 1);
11415 259 : M2 = mfcoefs_mf(mf2, sb, 1);
11416 259 : Th = mfTheta(NULL);
11417 259 : BT = mfcoefs_i(Th, sb, 1);
11418 259 : M1 = mfmatsermul(M1, RgV_to_RgX(expandbd(BT,2),0));
11419 259 : M2 = mfmatsermul(M2, RgV_to_RgX(BT,0));
11420 259 : o1= mfcharorder(CHI1);
11421 259 : T = (o1 <= 2)? NULL: mfcharpol(CHI1);
11422 259 : if (o1 > 2) M1 = liftpol_shallow(M1);
11423 259 : o2= mfcharorder(CHI2);
11424 259 : if (T)
11425 : {
11426 14 : if (o2 == o1) M2 = liftpol_shallow(M2);
11427 : else
11428 : {
11429 0 : GEN tr = Qab_trace_init(o2, o1, mfcharpol(CHI2), mfcharpol(CHI1));
11430 0 : M2 = QabM_tracerel(tr, 0, M2);
11431 : }
11432 : }
11433 : /* now everything is defined mod T = mfcharpol(CHI1) */
11434 259 : M2i = QabM_pseudoinv_i(M2, T, o1, &v, &den);
11435 259 : M = RgM_mul(M2i, rowpermute(M1, gel(v,1)));
11436 259 : M = RgM_mul(M2, M);
11437 259 : M1 = RgM_Rg_mul(M1, den);
11438 259 : M = RgM_sub(M1, M); if (T) M = RgXQM_red(M, T);
11439 259 : return vecmflineardiv0(B1, QabM_ker(M, T, o1), Th);
11440 : }
11441 :
11442 : /*******************************************************************/
11443 : /* Integration */
11444 : /*******************************************************************/
11445 : static GEN
11446 462 : vanembed(GEN F, GEN v, long prec)
11447 : {
11448 462 : GEN CHI = mf_get_CHI(F);
11449 462 : long o = mfcharorder(CHI);
11450 462 : if (o > 2 || degpol(mf_get_field(F)) > 1) v = liftpol_shallow(v);
11451 462 : if (o > 2) v = gsubst(v, varn(mfcharpol(CHI)), rootsof1u_cx(o, prec));
11452 462 : return v;
11453 : }
11454 :
11455 : static long
11456 1204 : mfperiod_prelim_double(double t0, long k, long bitprec)
11457 : {
11458 1204 : double nlim, c = 2*M_PI*t0;
11459 1204 : nlim = ceil(bitprec * M_LN2 / c);
11460 1204 : c -= (k - 1)/(2*nlim); if (c < 1) c = 1.;
11461 1204 : nlim += ceil((0.7 + (k-1)/2*log(nlim))/c);
11462 1204 : return (long)nlim;
11463 : }
11464 : static long
11465 280 : mfperiod_prelim(GEN t0, long k, long bitprec)
11466 280 : { return mfperiod_prelim_double(gtodouble(t0), k, bitprec); }
11467 :
11468 : /* (-X)^(k-2) * P(-1/X) = (-1)^{k-2} P|_{k-2} S */
11469 : static GEN
11470 1281 : RgX_act_S(GEN P, long k)
11471 : {
11472 1281 : P = RgX_unscale(RgX_recipspec_shallow(P+2, lgpol(P), k-1), gen_m1);
11473 1281 : setvarn(P,0); return P;
11474 : }
11475 : static int
11476 2828 : RgX_act_typ(GEN P, long k)
11477 : {
11478 2828 : switch(typ(P))
11479 : {
11480 35 : case t_RFRAC: return t_RFRAC;
11481 2793 : case t_POL:
11482 2793 : if (varn(P) == 0)
11483 : {
11484 2793 : long d = degpol(P);
11485 2793 : if (d > k-2) return t_RFRAC;
11486 2646 : if (d) return t_POL;
11487 : }
11488 : }
11489 1211 : return 0;
11490 : }
11491 : static GEN
11492 2562 : act_S(GEN P, long k)
11493 : {
11494 : GEN X;
11495 2562 : switch(RgX_act_typ(P, k))
11496 : {
11497 133 : case t_RFRAC:
11498 133 : X = gneg(pol_x(0));
11499 133 : return gmul(gsubst(P, 0, ginv(X)), gpowgs(X, k - 2));
11500 1281 : case t_POL: return RgX_act_S(P, k);
11501 : }
11502 1148 : return P;
11503 : }
11504 :
11505 : static GEN
11506 203 : AX_B(GEN M)
11507 203 : { GEN A = gcoeff(M,1,1), B = gcoeff(M,1,2); return deg1pol_shallow(A,B,0); }
11508 : static GEN
11509 203 : CX_D(GEN M)
11510 203 : { GEN C = gcoeff(M,2,1), D = gcoeff(M,2,2); return deg1pol_shallow(C,D,0); }
11511 :
11512 : /* P|_{2-k}M = (CX+D)^{k-2}P((AX+B)/(CX+D)) */
11513 : static GEN
11514 154 : RgX_act_gen(GEN P, GEN M, long k)
11515 : {
11516 154 : GEN S = gen_0, PCD, PAB;
11517 : long i;
11518 154 : PCD = gpowers(CX_D(M), k-2);
11519 154 : PAB = gpowers(AX_B(M), k-2);
11520 833 : for (i = 0; i <= k-2; i++)
11521 : {
11522 679 : GEN t = RgX_coeff(P, i);
11523 679 : if (!gequal0(t)) S = gadd(S, gmul(t, gmul(gel(PCD, k-i-1), gel(PAB, i+1))));
11524 : }
11525 154 : return S;
11526 : }
11527 : static GEN
11528 266 : act_GL2(GEN P, GEN M, long k)
11529 : {
11530 266 : switch(RgX_act_typ(P, k))
11531 : {
11532 49 : case t_RFRAC:
11533 : {
11534 49 : GEN AB = AX_B(M), CD = CX_D(M);
11535 49 : return gmul(gsubst(P, 0, gdiv(AB, CD)), gpowgs(CD, k - 2));
11536 : }
11537 154 : case t_POL: return RgX_act_gen(P, M, k);
11538 : }
11539 63 : return P;
11540 : }
11541 : static GEN
11542 7 : vecact_GL2(GEN x, GEN M, long k)
11543 21 : { pari_APPLY_same(act_GL2(gel(x,i), M, k)); }
11544 :
11545 : static GEN
11546 2849 : RgX_approx(GEN x, long bit)
11547 10549 : { pari_APPLY_pol_normalized(Rg_approx(gel(x,i),bit)); }
11548 :
11549 : static GEN normalizeapprox(GEN x, long bit);
11550 : static GEN
11551 2884 : normalizeapprox_i(GEN x, long bit)
11552 : {
11553 2884 : GEN D = gen_1;
11554 2940 : if (is_vec_t(typ(x))) pari_APPLY_same(normalizeapprox(gel(x,i), bit));
11555 2856 : if (typ(x) == t_RFRAC && varn(gel(x,2)) == 0) { D = gel(x,2); x = gel(x,1); }
11556 2856 : if (typ(x) != t_POL || varn(x) != 0) return gdiv(x, D);
11557 2849 : return gdiv(RgX_approx(x, bit), D);
11558 : }
11559 : static GEN
11560 56 : normalizeapprox(GEN x, long bit)
11561 : {
11562 56 : pari_sp av = avma;
11563 56 : return gerepileupto(av, normalizeapprox_i(x, bit));
11564 : }
11565 :
11566 : /* make sure T is a t_POL in variable 0 */
11567 : static GEN
11568 2828 : toRgX0(GEN T)
11569 2828 : { return typ(T) == t_POL && varn(T) == 0? T: scalarpol_shallow(T,0); }
11570 :
11571 : /* integrate by summing nlim+1 terms of van [may be < lg(van)]
11572 : * van can be an expansion with vector coefficients
11573 : * \int_A^oo \sum_n van[n] * q^(n/w + al) * P(z-A) dz, q = e(z) */
11574 : static GEN
11575 910 : intAoo(GEN van, long nlim, GEN al, long w, GEN P, GEN A, long k, long prec)
11576 : {
11577 : GEN alw, P1, piI2A, q, S, van0;
11578 910 : long n, vz = varn(gel(P,2));
11579 :
11580 910 : if (nlim < 1) nlim = 1;
11581 910 : alw = gmulsg(w, al);
11582 910 : P1 = RgX_translate(P, gneg(A));
11583 910 : piI2A = gmul(PiI2n(1, prec), A);
11584 910 : q = gexp(gdivgu(piI2A, w), prec);
11585 910 : S = gen_0;
11586 120876 : for (n = nlim; n >= 1; n--)
11587 : {
11588 119966 : GEN t = gsubst(P1, vz, gdivsg(w, gaddsg(n, alw)));
11589 119966 : S = gadd(gmul(gel(van, n+1), t), gmul(q, S));
11590 : }
11591 910 : S = gmul(q, S);
11592 910 : van0 = gel(van, 1);
11593 910 : if (!gequal0(al))
11594 : {
11595 42 : S = gadd(S, gmul(gsubst(P1, vz, ginv(al)), van0));
11596 42 : S = gmul(S, gexp(gmul(piI2A, al), prec));
11597 : }
11598 868 : else if (!gequal0(van0))
11599 210 : S = gsub(S, gdivgu(gmul(van0, gpowgs(gsub(pol_x(0), A), k-1)), k-1));
11600 910 : if (is_vec_t(typ(S)))
11601 : {
11602 623 : long j, l = lg(S);
11603 3164 : for (j = 1; j < l; j++) gel(S,j) = toRgX0(gel(S,j));
11604 : }
11605 : else
11606 287 : S = toRgX0(S);
11607 910 : return gneg(S);
11608 : }
11609 :
11610 : /* \sum_{j <= k} X^j * (Y / (2I\pi))^{k+1-j} k! / j! */
11611 : static GEN
11612 238 : get_P(long k, long v, long prec)
11613 : {
11614 238 : GEN a, S = cgetg(k + 1, t_POL), u = invr(Pi2n(1, prec+EXTRAPREC64));
11615 238 : long j, K = k-2;
11616 238 : S[1] = evalsigne(1)|evalvarn(0); a = u;
11617 238 : gel(S,K+2) = monomial(mulcxpowIs(a,3), 1, v); /* j = K */
11618 945 : for(j = K-1; j >= 0; j--)
11619 : {
11620 707 : a = mulrr(mulru(a,j+1), u);
11621 707 : gel(S,j+2) = monomial(mulcxpowIs(a,3*(K+1-j)), K+1-j, v);
11622 : }
11623 238 : return S;
11624 : }
11625 :
11626 : static GEN
11627 2541 : getw1w2(long N, GEN ga)
11628 2541 : { return mkvecsmall2(mfZC_width(N, gel(ga,1)),
11629 2541 : mfZC_width(N, gel(ga,2))); }
11630 :
11631 : static GEN
11632 133 : intAoowithvanall(GEN mf, GEN vanall, GEN P, GEN cosets, long bitprec)
11633 : {
11634 133 : GEN vvan = gel(vanall,1), vaw = gel(vanall,2), W1W2, resall;
11635 133 : long prec = nbits2prec(bitprec), N, k, lco, j;
11636 :
11637 133 : N = MF_get_N(mf); k = MF_get_k(mf);
11638 133 : lco = lg(cosets);
11639 133 : W1W2 = cgetg(lco, t_VEC); resall = cgetg(lco, t_VEC);
11640 2674 : for (j = 1; j < lco; j++) gel(W1W2,j) = getw1w2(N, gel(cosets, j));
11641 2674 : for (j = 1; j < lco; j++)
11642 : {
11643 2541 : GEN w1w2j = gel(W1W2,j), alj, M, VAN, RES, AR, Q;
11644 : long jq, c, w1, w2, w;
11645 2541 : if (!w1w2j) continue;
11646 623 : alj = gel(vaw,j);
11647 623 : w1 = w1w2j[1]; Q = cgetg(lco, t_VECSMALL);
11648 623 : w2 = w1w2j[2]; M = cgetg(lco, t_COL);
11649 8239 : for (c = 1, jq = j; jq < lco; jq++)
11650 : {
11651 7616 : GEN W = gel(W1W2, jq);
11652 7616 : if (jq == j || (W && gequal(W, w1w2j) && gequal(gel(vaw, jq), alj)))
11653 : {
11654 2541 : Q[c] = jq; gel(W1W2, jq) = NULL;
11655 2541 : gel(M, c) = gel(vvan, jq); c++;
11656 : }
11657 : }
11658 623 : setlg(M,c); VAN = shallowmatconcat(M);
11659 623 : AR = mkcomplex(gen_0, sqrtr_abs(divru(utor(w1, prec+EXTRAPREC64), w2)));
11660 623 : w = itos(gel(alj,2));
11661 623 : RES = intAoo(VAN, lg(VAN)-2, gel(alj,1),w, P, AR, k, prec);
11662 3164 : for (jq = 1; jq < c; jq++) gel(resall, Q[jq]) = gel(RES, jq);
11663 : }
11664 133 : return resall;
11665 : }
11666 :
11667 : GEN
11668 518 : mftobasisES(GEN mf, GEN F)
11669 : {
11670 518 : GEN v = mftobasis(mf, F, 0);
11671 511 : long nE = lg(MF_get_E(mf))-1;
11672 511 : return mkvec2(vecslice(v,1,nE), vecslice(v,nE+1,lg(v)-1));
11673 : }
11674 :
11675 : static long
11676 0 : wt1mulcond(GEN F, long D, long space)
11677 : {
11678 0 : GEN E = mfeisenstein_i(1, mfchartrivial(), get_mfchar(stoi(D))), mf;
11679 0 : F = mfmul(F, E);
11680 0 : mf = mfinit_Nkchi(mf_get_N(F), mf_get_k(F), mf_get_CHI(F), space, 0);
11681 0 : return mfconductor(mf, F);
11682 : }
11683 : static int
11684 7 : wt1newlevel(long N)
11685 : {
11686 7 : GEN P = gel(myfactoru(N),1);
11687 7 : long l = lg(P), i;
11688 14 : for (i = 1; i < l; i++)
11689 7 : if (!wt1empty(N/P[i])) return 0;
11690 7 : return 1;
11691 : }
11692 : long
11693 175 : mfconductor(GEN mf, GEN F)
11694 : {
11695 175 : pari_sp av = avma;
11696 : GEN gk;
11697 : long space, N, M;
11698 :
11699 175 : mf = checkMF(mf);
11700 175 : if (!checkmf_i(F)) pari_err_TYPE("mfconductor",F);
11701 175 : if (mfistrivial(F)) return 1;
11702 175 : space = MF_get_space(mf);
11703 175 : if (space == mf_NEW) return mf_get_N(F);
11704 175 : gk = MF_get_gk(mf);
11705 175 : if (isint1(gk))
11706 : {
11707 7 : N = mf_get_N(F);
11708 7 : if (!wt1newlevel(N))
11709 : {
11710 0 : long s = space_is_cusp(space)? mf_CUSP: mf_FULL;
11711 0 : N = ugcd(N, wt1mulcond(F,-3,s));
11712 0 : if (!wt1newlevel(N)) N = ugcd(N, wt1mulcond(F,-4,s));
11713 : }
11714 7 : return gc_long(av,N);
11715 : }
11716 168 : if (typ(gk) != t_INT)
11717 : {
11718 42 : F = mfmultheta(F);
11719 42 : mf = obj_checkbuild(mf, MF_MF2INIT, &mf2init); /* mf_FULL */
11720 : }
11721 168 : N = 1;
11722 168 : if (space_is_cusp(space))
11723 : {
11724 7 : F = mftobasis_i(mf, F);
11725 7 : if (typ(gk) != t_INT) F = vecslice(F, lg(MF_get_E(mf)), lg(F) - 1);
11726 : }
11727 : else
11728 : {
11729 161 : GEN EF = mftobasisES(mf, F), vE = gel(EF,1), B = MF_get_E(mf);
11730 161 : long i, l = lg(B);
11731 1267 : for (i = 1; i < l; i++)
11732 1106 : if (!gequal0(gel(vE,i))) N = ulcm(N, mf_get_N(gel(B, i)));
11733 161 : F = gel(EF,2);
11734 : }
11735 168 : (void)mftonew_i(mf, F, &M); /* M = conductor of cuspidal part */
11736 168 : return gc_long(av, ulcm(M, N));
11737 : }
11738 :
11739 : static GEN
11740 1386 : fs_get_MF(GEN fs) { return gel(fs,1); }
11741 : static GEN
11742 805 : fs_get_vES(GEN fs) { return gel(fs,2); }
11743 : static GEN
11744 1533 : fs_get_pols(GEN fs) { return gel(fs,3); }
11745 : static GEN
11746 2086 : fs_get_cosets(GEN fs) { return gel(fs,4); }
11747 : static long
11748 602 : fs_get_bitprec(GEN fs) { return itou(gel(fs,5)); }
11749 : static GEN
11750 1190 : fs_get_vE(GEN fs) { return gel(fs,6); }
11751 : static GEN
11752 56 : fs_get_EF(GEN fs) { return gel(fs,7); }
11753 : static GEN
11754 1792 : fs_get_expan(GEN fs) { return gel(fs,8); }
11755 : static GEN
11756 14 : fs_set_expan(GEN fs, GEN vanall)
11757 14 : { GEN f = shallowcopy(fs); gel(f,8) = vanall; return f; }
11758 : static int
11759 42 : mfs_checkmf(GEN fs, GEN mf)
11760 42 : { GEN mfF = fs_get_MF(fs); return gequal(gel(mfF,1), gel(mf,1)); }
11761 : static long
11762 756 : checkfs_i(GEN v)
11763 756 : { return typ(v) == t_VEC && lg(v) == 9 && checkMF_i(fs_get_MF(v))
11764 539 : && typ(fs_get_vES(v)) == t_VEC
11765 539 : && typ(fs_get_pols(v)) == t_VEC
11766 539 : && typ(fs_get_cosets(v)) == t_VEC
11767 539 : && typ(fs_get_vE(v)) == t_VEC
11768 539 : && lg(fs_get_pols(v)) == lg(fs_get_cosets(v))
11769 539 : && typ(fs_get_expan(v)) == t_VEC
11770 539 : && lg(fs_get_expan(v)) == 3
11771 539 : && lg(gel(fs_get_expan(v), 1)) == lg(fs_get_cosets(v))
11772 1512 : && typ(gel(v,5)) == t_INT; }
11773 : GEN
11774 18697 : checkMF_i(GEN mf)
11775 : {
11776 18697 : long l = lg(mf);
11777 : GEN v;
11778 18697 : if (typ(mf) != t_VEC) return NULL;
11779 18669 : if (l == 9) return checkMF_i(fs_get_MF(mf));
11780 18669 : if (l != 7) return NULL;
11781 7483 : v = gel(mf,1);
11782 7483 : if (typ(v) != t_VEC || lg(v) != 5) return NULL;
11783 7483 : return (typ(gel(v,1)) == t_INT
11784 7483 : && typ(gmul2n(gel(v,2), 1)) == t_INT
11785 7483 : && typ(gel(v,3)) == t_VEC
11786 14966 : && typ(gel(v,4)) == t_INT)? mf: NULL; }
11787 : GEN
11788 3864 : checkMF(GEN T)
11789 : {
11790 3864 : GEN mf = checkMF_i(T);
11791 3864 : if (!mf) pari_err_TYPE("checkMF [please use mfinit]", T);
11792 3864 : return mf;
11793 : }
11794 :
11795 : /* c,d >= 0; c * Nc = N, find coset whose image in P1(Z/NZ) ~ (c, d + k(N/c)) */
11796 : static GEN
11797 11914 : coset_complete(long c, long d, long Nc)
11798 : {
11799 : long a, b;
11800 13258 : while (ugcd(c, d) > 1) d += Nc;
11801 11914 : (void)cbezout(c, d, &b, &a);
11802 11914 : return mkmat22s(a, -b, c, d);
11803 : }
11804 : /* right cosets of $\G_0(N)$: $\G=\bigsqcup_j \G_0(N)\ga_j$. */
11805 : /* We choose them with c\mid N and d mod N/c, not the reverse */
11806 : GEN
11807 154 : mfcosets(GEN gN)
11808 : {
11809 154 : pari_sp av = avma;
11810 : GEN V, D, mf;
11811 154 : long l, i, ct, N = 0;
11812 154 : if (typ(gN) == t_INT) N = itos(gN);
11813 14 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
11814 7 : else pari_err_TYPE("mfcosets", gN);
11815 147 : if (N <= 0) pari_err_DOMAIN("mfcosets", "N", "<=", gen_0, stoi(N));
11816 147 : V = cgetg(mypsiu(N) + 1, t_VEC);
11817 147 : D = mydivisorsu(N); l = lg(D);
11818 560 : for (i = ct = 1; i < l; i++)
11819 : {
11820 413 : long d, c = D[i], Nc = D[l-i], e = ugcd(Nc, c);
11821 3304 : for (d = 0; d < Nc; d++)
11822 2891 : if (ugcd(d,e) == 1) gel(V, ct++) = coset_complete(c, d, Nc);
11823 : }
11824 147 : return gerepilecopy(av, V);
11825 : }
11826 : static int
11827 35434 : cmp_coset(void *E, GEN A, GEN B)
11828 : {
11829 35434 : ulong N = (ulong)E, Nc, c = itou(gcoeff(A,2,1));
11830 35434 : int r = cmpuu(c, itou(gcoeff(B,2,1)));
11831 35434 : if (r) return r;
11832 30625 : Nc = N / c;
11833 30625 : return cmpuu(umodiu(gcoeff(A,2,2), Nc), umodiu(gcoeff(B,2,2), Nc));
11834 : }
11835 : /* M in SL_2(Z) */
11836 : static long
11837 9163 : mftocoset_i(ulong N, GEN M, GEN cosets)
11838 : {
11839 9163 : pari_sp av = avma;
11840 9163 : long A = itos(gcoeff(M,1,1)), c, u, v, Nc, i;
11841 9163 : long C = itos(gcoeff(M,2,1)), D = itos(gcoeff(M,2,2));
11842 : GEN ga;
11843 9163 : c = cbezout(N*A, C, &u, &v); Nc = N/c;
11844 9163 : ga = coset_complete(c, umodsu(v*D, Nc), Nc);
11845 9163 : i = gen_search(cosets, ga, (void*)N, &cmp_coset);
11846 9163 : if (i < 0) pari_err_BUG("mftocoset [no coset found]");
11847 9163 : return gc_long(av,i);
11848 : }
11849 : /* (U * V^(-1))[2,2] mod N, assuming V in SL2(Z) */
11850 : static long
11851 9149 : SL2_div_D(ulong N, GEN U, GEN V)
11852 : {
11853 9149 : long c = umodiu(gcoeff(U,2,1), N), d = umodiu(gcoeff(U,2,2), N);
11854 9149 : long a2 = umodiu(gcoeff(V,1,1), N), b2 = umodiu(gcoeff(V,1,2), N);
11855 9149 : return (a2*d - b2*c) % (long)N;
11856 : }
11857 : static long
11858 9149 : mftocoset_iD(ulong N, GEN M, GEN cosets, long *D)
11859 : {
11860 9149 : long i = mftocoset_i(N, M, cosets);
11861 9149 : *D = SL2_div_D(N, M, gel(cosets,i)); return i;
11862 : }
11863 : GEN
11864 7 : mftocoset(ulong N, GEN M, GEN cosets)
11865 : {
11866 : long i;
11867 7 : if (!check_SL2Z(M)) pari_err_TYPE("mftocoset",M);
11868 7 : i = mftocoset_i(N, M, cosets);
11869 7 : retmkvec2(gdiv(M,gel(cosets,i)), utoipos(i));
11870 : }
11871 :
11872 : static long
11873 2541 : getnlim2(long N, long w1, long w2, long nlim, long k, long bitprec)
11874 : {
11875 2541 : if (w2 == N) return nlim;
11876 483 : return mfperiod_prelim_double(1./sqrt((double)w1*w2), k, bitprec + 32);
11877 : }
11878 :
11879 : /* g * S, g 2x2 */
11880 : static GEN
11881 1316 : ZM_mulS(GEN g)
11882 1316 : { return mkmat2(gel(g,2), ZC_neg(gel(g,1))); }
11883 : /* g * T, g 2x2 */
11884 : static GEN
11885 4634 : ZM_mulT(GEN g)
11886 4634 : { return mkmat2(gel(g,1), ZC_add(gel(g,2), gel(g,1))); }
11887 : /* g * T^(-1), g 2x2 */
11888 : static GEN
11889 2352 : ZM_mulTi(GEN g)
11890 2352 : { return mkmat2(gel(g,1), ZC_sub(gel(g,2), gel(g,1))); }
11891 :
11892 : /* Compute all slashexpansions for all cosets */
11893 : static GEN
11894 147 : mfgaexpansionall(GEN mf, GEN FE, GEN cosets, double height, long prec)
11895 : {
11896 147 : GEN CHI = MF_get_CHI(mf), vres, vresaw;
11897 147 : long lco, j, k = MF_get_k(mf), N = MF_get_N(mf), bitprec = prec2nbits(prec);
11898 :
11899 147 : lco = lg(cosets);
11900 147 : vres = const_vec(lco-1, NULL);
11901 147 : vresaw = cgetg(lco, t_VEC);
11902 2856 : for (j = 1; j < lco; j++) if (!gel(vres,j))
11903 : {
11904 427 : GEN ga = gel(cosets, j), van, aw, al, z, gai;
11905 427 : long w1 = mfZC_width(N, gel(ga,1));
11906 427 : long w2 = mfZC_width(N, gel(ga,2));
11907 : long nlim, nlim2, daw, da, na, i;
11908 427 : double sqNinvdbl = height ? height/w1 : 1./sqrt((double)w1*N);
11909 427 : nlim = mfperiod_prelim_double(sqNinvdbl, k, bitprec + 32);
11910 427 : van = mfslashexpansion(mf, FE, ga, nlim, 0, &aw, prec + EXTRAPREC64);
11911 427 : van = vanembed(gel(FE, 1), van, prec + EXTRAPREC64);
11912 427 : al = gel(aw, 1);
11913 427 : nlim2 = height? nlim: getnlim2(N, w1, w2, nlim, k, bitprec);
11914 427 : gel(vres, j) = vecslice(van, 1, nlim2 + 1);
11915 427 : gel(vresaw, j) = aw;
11916 427 : Qtoss(al, &na, &da); daw = da*w1;
11917 427 : z = rootsof1powinit(1, daw, prec + EXTRAPREC64);
11918 427 : gai = ga;
11919 2709 : for (i = 1; i < w1; i++)
11920 : {
11921 : GEN V, coe;
11922 2282 : long Di, n, ind, w2, s = ((i*na) % da) * w1, t = i*da;
11923 2282 : gai = ZM_mulT(gai);
11924 2282 : ind = mftocoset_iD(N, gai, cosets, &Di);
11925 2282 : w2 = mfZC_width(N, gel(gel(cosets,ind), 2));
11926 2282 : nlim2 = height? nlim: getnlim2(N, w1, w2, nlim, k, bitprec);
11927 2282 : gel(vresaw, ind) = aw;
11928 2282 : V = cgetg(nlim2 + 2, t_VEC);
11929 909034 : for (n = 0; n <= nlim2; n++, s = Fl_add(s, t, daw))
11930 906752 : gel(V, n+1) = gmul(gel(van, n+1), rootsof1pow(z, s));
11931 2282 : coe = mfcharcxeval(CHI, Di, prec + EXTRAPREC64);
11932 2282 : if (!gequal1(coe)) V = RgV_Rg_mul(V, conj_i(coe));
11933 2282 : gel(vres, ind) = V;
11934 : }
11935 : }
11936 147 : return mkvec2(vres, vresaw);
11937 : }
11938 :
11939 : /* Compute all period pols of F|_k\ga_j, vF = mftobasis(F_S) */
11940 : static GEN
11941 154 : mfperiodpols_i(GEN mf, GEN FE, GEN cosets, GEN *pvan, long bit)
11942 : {
11943 154 : long N, i, prec = nbits2prec(bit), k = MF_get_k(mf);
11944 154 : GEN vP, P, CHI, intall = gen_0;
11945 :
11946 154 : *pvan = gen_0;
11947 154 : if (k == 0 && gequal0(gel(FE,2)))
11948 0 : return cosets? const_vec(lg(cosets)-1, pol_0(0)): pol_0(0);
11949 154 : N = MF_get_N(mf);
11950 154 : CHI = MF_get_CHI(mf);
11951 154 : P = get_P(k, fetch_var(), prec);
11952 154 : if (!cosets)
11953 : { /* ga = id */
11954 21 : long nlim, PREC = prec + EXTRAPREC64;
11955 21 : GEN F = gel(FE,1), sqNinv = invr(sqrtr_abs(utor(N, PREC))); /* A/w */
11956 : GEN AR, v, van, T1, T2;
11957 :
11958 21 : nlim = mfperiod_prelim(sqNinv, k, bit + 32);
11959 : /* F|id: al = 0, w = 1 */
11960 21 : v = mfcoefs_i(F, nlim, 1);
11961 21 : van = vanembed(F, v, PREC);
11962 21 : AR = mkcomplex(gen_0, sqNinv);
11963 21 : T1 = intAoo(van, nlim, gen_0,1, P, AR, k, prec);
11964 21 : if (N == 1) T2 = T1;
11965 : else
11966 : { /* F|S: al = 0, w = N */
11967 7 : v = mfgaexpansion(mf, FE, mkS(), nlim, PREC);
11968 7 : van = vanembed(F, gel(v,3), PREC);
11969 7 : AR = mkcomplex(gen_0, mulur(N,sqNinv));
11970 7 : T2 = intAoo(van, nlim, gen_0,N, P, AR, k, prec);
11971 : }
11972 21 : T1 = gsub(T1, act_S(T2, k));
11973 21 : T1 = normalizeapprox_i(T1, bit-20);
11974 21 : vP = gprec_wtrunc(T1, prec);
11975 : }
11976 : else
11977 : {
11978 133 : long lco = lg(cosets);
11979 133 : GEN vanall = mfgaexpansionall(mf, FE, cosets, 0, prec);
11980 133 : *pvan = vanall;
11981 133 : intall = intAoowithvanall(mf, vanall, P, cosets, bit);
11982 133 : vP = const_vec(lco-1, NULL);
11983 2674 : for (i = 1; i < lco; i++)
11984 : {
11985 2541 : GEN P, P1, P2, c, ga = gel(cosets, i);
11986 : long iS, DS;
11987 2618 : if (gel(vP,i)) continue;
11988 1309 : P1 = gel(intall, i);
11989 1309 : iS = mftocoset_iD(N, ZM_mulS(ga), cosets, &DS);
11990 1309 : c = mfcharcxeval(CHI, DS, prec + EXTRAPREC64);
11991 1309 : P2 = gel(intall, iS);
11992 :
11993 1309 : P = act_S(isint1(c)? P2: gmul(c, P2), k);
11994 1309 : P = normalizeapprox_i(gsub(P1, P), bit-20);
11995 1309 : gel(vP,i) = gprec_wtrunc(P, prec);
11996 1309 : if (iS == i) continue;
11997 :
11998 1232 : P = act_S(isint1(c)? P1: gmul(conj_i(c), P1), k);
11999 1232 : if (!odd(k)) P = gneg(P);
12000 1232 : P = normalizeapprox_i(gadd(P, P2), bit-20);
12001 1232 : gel(vP,iS) = gprec_wtrunc(P, prec);
12002 : }
12003 : }
12004 154 : delete_var(); return vP;
12005 : }
12006 :
12007 : /* when cosets = NULL, return a "fake" symbol containing only fs(oo->0) */
12008 : static GEN
12009 154 : mfsymbol_i(GEN mf, GEN F, GEN cosets, long bit)
12010 : {
12011 154 : GEN FE, van, vP, vE, Mvecj, vES = mftobasisES(mf,F);
12012 154 : long precnew, prec = nbits2prec(bit), k = MF_get_k(mf);
12013 154 : vE = mfgetembed(F, prec);
12014 154 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
12015 154 : if (lg(Mvecj) >= 5) precnew = prec;
12016 : else
12017 : {
12018 14 : long N = MF_get_N(mf), n = mfperiod_prelim_double(1/(double)N, k, bit + 32);
12019 14 : precnew = prec + inveis_extraprec(N, mkS(), Mvecj, n);
12020 : }
12021 154 : FE = mkcol2(F, mf_eisendec(mf,F,precnew));
12022 154 : vP = mfperiodpols_i(mf, FE, cosets, &van, bit);
12023 154 : return mkvecn(8, mf, vES, vP, cosets, utoi(bit), vE, FE, van);
12024 : }
12025 :
12026 : static GEN
12027 56 : fs2_get_cusps(GEN f) { return gel(f,3); }
12028 : static GEN
12029 56 : fs2_get_MF(GEN f) { return gel(f,1); }
12030 : static GEN
12031 56 : fs2_get_W(GEN f) { return gel(f,2); }
12032 : static GEN
12033 56 : fs2_get_F(GEN f) { return gel(f,4); }
12034 : static long
12035 0 : fs2_get_bitprec(GEN f) { return itou(gel(f,5)); }
12036 : static GEN
12037 56 : fs2_get_al0(GEN f) { return gel(f,6); }
12038 : static GEN
12039 21 : fs2_get_den(GEN f) { return gel(f,7); }
12040 : static int
12041 196 : checkfs2_i(GEN f)
12042 : {
12043 : GEN W, C, F, al0;
12044 : long l;
12045 196 : if (typ(f) != t_VEC || lg(f) != 8 || typ(gel(f,5)) != t_INT) return 0;
12046 35 : C = fs2_get_cusps(f); l = lg(C);
12047 35 : W = fs2_get_W(f);
12048 35 : F = fs2_get_F(f);
12049 35 : al0 = fs2_get_al0(f);
12050 35 : return checkMF_i(fs2_get_MF(f))
12051 35 : && typ(W) == t_VEC && typ(F) == t_VEC && typ(al0) == t_VECSMALL
12052 70 : && lg(W) == l && lg(F) == l && lg(al0) == l;
12053 : }
12054 : static GEN fs2_init(GEN mf, GEN F, long bit);
12055 : GEN
12056 161 : mfsymbol(GEN mf, GEN F, long bit)
12057 : {
12058 161 : pari_sp av = avma;
12059 161 : GEN cosets = NULL;
12060 161 : if (!F)
12061 : {
12062 21 : F = mf;
12063 21 : if (!checkmf_i(F)) pari_err_TYPE("mfsymbol", F);
12064 21 : mf = mfinit_i(F, mf_FULL);
12065 : }
12066 140 : else if (!checkmf_i(F)) pari_err_TYPE("mfsymbol", F);
12067 161 : if (checkfs2_i(mf)) return fs2_init(mf, F, bit);
12068 161 : if (checkfs_i(mf))
12069 : {
12070 0 : cosets = fs_get_cosets(mf);
12071 0 : mf = fs_get_MF(mf);
12072 : }
12073 161 : else if (checkMF_i(mf))
12074 : {
12075 161 : GEN gk = MF_get_gk(mf);
12076 161 : if (typ(gk) != t_INT || equali1(gk)) return fs2_init(mf, F, bit);
12077 140 : if (signe(gk) <= 0) pari_err_TYPE("mfsymbol [k <= 0]", mf);
12078 133 : cosets = mfcosets(MF_get_gN(mf));
12079 : }
12080 0 : else pari_err_TYPE("mfsymbol",mf);
12081 133 : return gerepilecopy(av, mfsymbol_i(mf, F, cosets, bit));
12082 : }
12083 :
12084 : static GEN
12085 14 : RgX_by_parity(GEN P, long odd)
12086 : {
12087 14 : long i, l = lg(P);
12088 : GEN Q;
12089 14 : if (l < 4) return odd ? pol_x(0): P;
12090 14 : Q = cgetg(l, t_POL); Q[1] = P[1];
12091 91 : for (i = odd? 2: 3; i < l; i += 2) gel(Q,i) = gen_0;
12092 91 : for (i = odd? 3: 2; i < l; i += 2) gel(Q,i) = gel(P,i);
12093 14 : return normalizepol_lg(Q, l);
12094 : }
12095 : /* flag 0: period polynomial of F, >0 or <0 with corresponding parity */
12096 : GEN
12097 35 : mfperiodpol(GEN mf0, GEN F, long flag, long bit)
12098 : {
12099 35 : pari_sp av = avma;
12100 35 : GEN pol, mf = checkMF_i(mf0);
12101 35 : if (!mf) pari_err_TYPE("mfperiodpol",mf0);
12102 35 : if (checkfs_i(F))
12103 : {
12104 14 : GEN mfpols = fs_get_pols(F);
12105 14 : if (!mfs_checkmf(F, mf)) pari_err_TYPE("mfperiodpol [different mf]",F);
12106 14 : pol = veclast(mfpols); /* trivial coset is last */
12107 : }
12108 : else
12109 : {
12110 21 : GEN gk = MF_get_gk(mf);
12111 21 : if (typ(gk) != t_INT) pari_err_TYPE("mfperiodpol [half-integral k]", mf);
12112 21 : if (equali1(gk)) pari_err_TYPE("mfperiodpol [k = 1]", mf);
12113 21 : F = mfsymbol_i(mf, F, NULL, bit);
12114 21 : pol = fs_get_pols(F);
12115 : }
12116 35 : if (flag) pol = RgX_by_parity(pol, flag < 0);
12117 35 : return gerepilecopy(av, RgX_embedall(pol, fs_get_vE(F)));
12118 : }
12119 :
12120 : static int
12121 35 : mfs_iscusp(GEN mfs) { return gequal0(gmael(mfs,2,1)); }
12122 : /* given cusps s1 and s2 (rationals or oo)
12123 : * compute $\int_{s1}^{s2}(X-\tau)^{k-2}F|_k\ga_j(\tau)\,d\tau$ */
12124 : /* If flag = 1, do not give an error message if divergent, but
12125 : give the rational function as result. */
12126 :
12127 : static GEN
12128 119 : col2cusp(GEN v)
12129 : {
12130 : GEN A, C;
12131 119 : if (lg(v) != 3 || !RgV_is_ZV(v)) pari_err_TYPE("col2cusp",v);
12132 119 : A = gel(v,1);
12133 119 : C = gel(v,2);
12134 119 : if (gequal0(C))
12135 : {
12136 0 : if (gequal0(A)) pari_err_TYPE("mfsymboleval", mkvec2(A, C));
12137 0 : return mkoo();
12138 : }
12139 119 : return gdiv(A, C);
12140 : }
12141 : /* g.oo */
12142 : static GEN
12143 105 : mat2cusp(GEN g) { return col2cusp(gel(g,1)); }
12144 :
12145 : static GEN
12146 7 : pathmattovec(GEN path)
12147 7 : { return mkvec2(col2cusp(gel(path,1)), col2cusp(gel(path,2))); }
12148 :
12149 : static void
12150 532 : get_mf_F(GEN fs, GEN *mf, GEN *F)
12151 : {
12152 532 : if (lg(fs) == 3) { *mf = gel(fs,1); *F = gel(fs,2); }
12153 532 : else { *mf = fs_get_MF(fs); *F = NULL; }
12154 532 : }
12155 : static GEN
12156 175 : mfgetvan(GEN fs, GEN ga, GEN *pal, long nlim, long prec)
12157 : {
12158 : GEN van, mf, F, W;
12159 : long PREC;
12160 175 : get_mf_F(fs, &mf, &F);
12161 175 : if (!F)
12162 : {
12163 175 : GEN vanall = fs_get_expan(fs), cosets = fs_get_cosets(fs);
12164 175 : long D, jga = mftocoset_iD(MF_get_N(mf), ga, cosets, &D);
12165 175 : van = gmael(vanall, 1, jga);
12166 175 : W = gmael(vanall, 2, jga);
12167 175 : if (lg(van) >= nlim + 2)
12168 : {
12169 168 : GEN z = mfcharcxeval(MF_get_CHI(mf), D, prec);
12170 168 : if (!gequal1(z)) van = RgV_Rg_mul(van, z);
12171 168 : *pal = gel(W,1); return van;
12172 : }
12173 7 : F = gel(fs_get_EF(fs), 1);
12174 : }
12175 7 : PREC = prec + EXTRAPREC64;
12176 7 : van = mfslashexpansion(mf, F, ga, nlim, 0, &W, PREC);
12177 7 : van = vanembed(F, van, PREC);
12178 7 : *pal = gel(W,1); return van;
12179 : }
12180 : /* Computation of int_A^oo (f | ga)(t)(X-t)^{k-2} dt, assuming convergence;
12181 : * fs is either a symbol or a triple [mf,F,bitprec]. A != oo and im(A) > 0 */
12182 : static GEN
12183 77 : intAoo0(GEN fs, GEN A, GEN ga, GEN P, long bit)
12184 : {
12185 77 : long nlim, N, k, w, prec = nbits2prec(bit);
12186 : GEN van, mf, F, al;
12187 77 : get_mf_F(fs, &mf,&F); N = MF_get_N(mf); k = MF_get_k(mf);
12188 77 : w = mfZC_width(N, gel(ga,1));
12189 77 : nlim = mfperiod_prelim(gdivgu(imag_i(A), w), k, bit + 32);
12190 77 : van = mfgetvan(fs, ga, &al, nlim, prec);
12191 77 : return intAoo(van, nlim, al,w, P, A, k, prec);
12192 : }
12193 :
12194 : /* fs symbol, naive summation, A != oo, im(A) > 0 and B = oo or im(B) > 0 */
12195 : static GEN
12196 98 : mfsymboleval_direct(GEN fs, GEN path, GEN ga, GEN P)
12197 : {
12198 98 : GEN A, B, van, S, al, mf = fs_get_MF(fs);
12199 98 : long w, nlimA, nlimB = 0, N = MF_get_N(mf), k = MF_get_k(mf);
12200 98 : long bit = fs_get_bitprec(fs), prec = nbits2prec(bit);
12201 :
12202 98 : A = gel(path, 1);
12203 98 : B = gel(path, 2); if (typ(B) == t_INFINITY) B = NULL;
12204 98 : w = mfZC_width(N, gel(ga,1));
12205 98 : nlimA = mfperiod_prelim(gdivgu(imag_i(A),w), k, bit + 32);
12206 98 : if (B) nlimB = mfperiod_prelim(gdivgu(imag_i(B),w), k, bit + 32);
12207 98 : van = mfgetvan(fs, ga, &al, maxss(nlimA,nlimB), prec);
12208 98 : S = intAoo(van, nlimA, al,w, P, A, k, prec);
12209 98 : if (B) S = gsub(S, intAoo(van, nlimB, al,w, P, B, k, prec));
12210 98 : return RgX_embedall(S, fs_get_vE(fs));
12211 : }
12212 :
12213 : /* Computation of int_A^oo (f | ga)(t)(X-t)^{k-2} dt, assuming convergence;
12214 : * fs is either a symbol or a pair [mf,F]. */
12215 : static GEN
12216 77 : mfsymbolevalpartial(GEN fs, GEN A, GEN ga, long bit)
12217 : {
12218 : GEN Y, F, S, P, mf;
12219 77 : long N, k, w, prec = nbits2prec(bit);
12220 :
12221 77 : get_mf_F(fs, &mf, &F);
12222 77 : N = MF_get_N(mf); w = mfZC_width(N, gel(ga,1));
12223 77 : k = MF_get_k(mf);
12224 77 : Y = gdivgu(imag_i(A), w);
12225 77 : P = get_P(k, fetch_var(), prec);
12226 77 : if (lg(fs) != 3 && gtodouble(Y)*(2*N) < 1)
12227 21 : { /* true symbol + low imaginary part: use GL_2 action to improve */
12228 21 : GEN U, ga2, czd, A2 = cxredga0N(N, A, &U, &czd, 1);
12229 21 : GEN vE = fs_get_vE(fs);
12230 21 : ga2 = ZM_mul(ga, ZM_inv(U, NULL));
12231 21 : S = RgX_embedall(intAoo0(fs, A2, ga2, P, bit), vE);
12232 21 : S = gsub(S, mfsymboleval(fs, mkvec2(mat2cusp(U), mkoo()), ga2, bit));
12233 21 : S = typ(S) == t_VEC? vecact_GL2(S, U, k): act_GL2(S, U, k);
12234 : }
12235 : else
12236 : {
12237 56 : S = intAoo0(fs, A, ga, P, bit);
12238 56 : S = RgX_embedall(S, F? mfgetembed(F,prec): fs_get_vE(fs));
12239 : }
12240 77 : delete_var(); return normalizeapprox_i(S, bit-20);
12241 : }
12242 :
12243 : static GEN
12244 42 : actal(GEN x, GEN vabd)
12245 : {
12246 42 : if (typ(x) == t_INFINITY) return x;
12247 35 : return gdiv(gadd(gmul(gel(vabd,1), x), gel(vabd,2)), gel(vabd,3));
12248 : }
12249 :
12250 : static GEN
12251 14 : unact(GEN z, GEN vabd, long k, long prec)
12252 : {
12253 14 : GEN res = gsubst(z, 0, actal(pol_x(0), vabd));
12254 14 : GEN CO = gpow(gdiv(gel(vabd,3), gel(vabd,1)), sstoQ(k-2, 2), prec);
12255 14 : return gmul(CO, res);
12256 : }
12257 :
12258 : GEN
12259 210 : mfsymboleval(GEN fs, GEN path, GEN ga, long bitprec)
12260 : {
12261 210 : pari_sp av = avma;
12262 210 : GEN tau, V, LM, S, CHI, mfpols, cosets, al, be, mf, F, vabd = NULL;
12263 : long D, B, m, u, v, a, b, c, d, j, k, N, prec, tsc1, tsc2;
12264 :
12265 210 : if (checkfs_i(fs))
12266 : {
12267 203 : get_mf_F(fs, &mf, &F);
12268 203 : bitprec = minss(bitprec, fs_get_bitprec(fs));
12269 : }
12270 : else
12271 : {
12272 7 : if (checkfs2_i(fs)) pari_err_TYPE("mfsymboleval [need integral k > 1]",fs);
12273 0 : if (typ(fs) != t_VEC || lg(fs) != 3) pari_err_TYPE("mfsymboleval",fs);
12274 0 : get_mf_F(fs, &mf, &F);
12275 0 : mf = checkMF_i(mf);
12276 0 : if (!mf ||!checkmf_i(F)) pari_err_TYPE("mfsymboleval",fs);
12277 : }
12278 203 : if (lg(path) != 3) pari_err_TYPE("mfsymboleval",path);
12279 203 : if (typ(path) == t_MAT) path = pathmattovec(path);
12280 203 : if (typ(path) != t_VEC) pari_err_TYPE("mfsymboleval",path);
12281 203 : al = gel(path,1);
12282 203 : be = gel(path,2);
12283 203 : ga = ga? GL2toSL2(ga, &vabd): matid(2);
12284 203 : if (vabd)
12285 : {
12286 14 : al = actal(al, vabd);
12287 14 : be = actal(be, vabd); path = mkvec2(al, be);
12288 : }
12289 203 : tsc1 = cusp_AC(al, &a, &c);
12290 203 : tsc2 = cusp_AC(be, &b, &d);
12291 203 : prec = nbits2prec(bitprec);
12292 203 : k = MF_get_k(mf);
12293 203 : if (!tsc1)
12294 : {
12295 42 : GEN z2, z = mfsymbolevalpartial(fs, al, ga, bitprec);
12296 42 : if (tsc2)
12297 28 : z2 = d? mfsymboleval(fs, mkvec2(be, mkoo()), ga, bitprec): gen_0;
12298 : else
12299 14 : z2 = mfsymbolevalpartial(fs, be, ga, bitprec);
12300 42 : z = gsub(z, z2);
12301 42 : if (vabd) z = unact(z, vabd, k, prec);
12302 42 : return gerepileupto(av, normalizeapprox_i(z, bitprec-20));
12303 : }
12304 161 : else if (!tsc2)
12305 : {
12306 21 : GEN z = mfsymbolevalpartial(fs, be, ga, bitprec);
12307 21 : if (c) z = gsub(mfsymboleval(fs, mkvec2(al, mkoo()), ga, bitprec), z);
12308 7 : else z = gneg(z);
12309 21 : if (vabd) z = unact(z, vabd, k, prec);
12310 21 : return gerepileupto(av, normalizeapprox_i(z, bitprec-20));
12311 : }
12312 140 : if (F) pari_err_TYPE("mfsymboleval", fs);
12313 140 : D = a*d-b*c;
12314 140 : if (!D) { set_avma(av); return RgX_embedall(gen_0, fs_get_vE(fs)); }
12315 126 : mfpols = fs_get_pols(fs);
12316 126 : cosets = fs_get_cosets(fs);
12317 126 : CHI = MF_get_CHI(mf); N = MF_get_N(mf);
12318 126 : cbezout(a, c, &u, &v); B = u*b + v*d; tau = mkmat22s(a, -v, c, u);
12319 126 : V = gcf(sstoQ(B, D));
12320 126 : LM = shallowconcat(mkcol2(gen_1, gen_0), contfracpnqn(V, lg(V)));
12321 126 : S = gen_0; m = lg(LM) - 2;
12322 364 : for (j = 0; j < m; j++)
12323 : {
12324 : GEN M, P;
12325 : long D, iN;
12326 238 : M = mkmat2(gel(LM, j+2), gel(LM, j+1));
12327 238 : if (!odd(j)) gel(M,1) = ZC_neg(gel(M,1));
12328 238 : M = ZM_mul(tau, M);
12329 238 : iN = mftocoset_iD(N, ZM_mul(ga, M), cosets, &D);
12330 238 : P = gmul(gel(mfpols,iN), mfcharcxeval(CHI,D,prec));
12331 238 : S = gadd(S, act_GL2(P, ZM_inv(M, NULL), k));
12332 : }
12333 126 : if (typ(S) == t_RFRAC)
12334 : {
12335 : GEN R, S1, co;
12336 21 : gel(S,2) = primitive_part(gel(S,2), &co);
12337 21 : if (co) gel(S,1) = gdiv(gel(S,1), gtofp(co,prec));
12338 21 : S1 = poldivrem(gel(S,1), gel(S,2), &R);
12339 21 : if (gexpo(R) < -bitprec + 20) S = S1;
12340 : }
12341 126 : if (vabd) S = unact(S, vabd, k, prec);
12342 126 : S = RgX_embedall(S, fs_get_vE(fs));
12343 126 : return gerepileupto(av, normalizeapprox_i(S, bitprec-20));
12344 : }
12345 :
12346 : /* v a scalar or t_POL; set *pw = a if expo(a) > E for some coefficient;
12347 : * take the 'a' with largest exponent */
12348 : static void
12349 5740 : improve(GEN v, GEN *pw, long *E)
12350 : {
12351 5740 : if (typ(v) != t_POL)
12352 : {
12353 4270 : long e = gexpo(v);
12354 4270 : if (e > *E) { *E = e; *pw = v; }
12355 : }
12356 : else
12357 : {
12358 1470 : long j, l = lg(v);
12359 5740 : for (j = 2; j < l; j++) improve(gel(v,j), pw, E);
12360 : }
12361 5740 : }
12362 : static GEN
12363 518 : polabstorel(GEN rnfeq, GEN x)
12364 : {
12365 518 : if (typ(x) != t_POL) return x;
12366 3500 : pari_APPLY_pol_normalized(eltabstorel(rnfeq, gel(x,i)));
12367 : }
12368 : static GEN
12369 1519 : bestapprnfrel(GEN x, GEN polabs, GEN roabs, GEN rnfeq, long prec)
12370 : {
12371 1519 : x = bestapprnf(x, polabs, roabs, prec);
12372 1519 : if (rnfeq) x = polabstorel(rnfeq, liftpol_shallow(x));
12373 1519 : return x;
12374 : }
12375 : /* v vector of polynomials polynomial in C[X] (possibly scalar).
12376 : * Set *w = coeff with largest exponent and return T / *w, rationalized */
12377 : static GEN
12378 98 : normal(GEN v, GEN polabs, GEN roabs, GEN rnfeq, GEN *w, long prec)
12379 : {
12380 98 : long i, l = lg(v), E = -(long)HIGHEXPOBIT;
12381 : GEN dv;
12382 1568 : for (i = 1; i < l; i++) improve(gel(v,i), w, &E);
12383 98 : v = RgV_Rg_mul(v, ginv(*w));
12384 1568 : for (i = 1; i < l; i++)
12385 1470 : gel(v,i) = bestapprnfrel(gel(v,i), polabs,roabs,rnfeq,prec);
12386 98 : v = Q_primitive_part(v,&dv);
12387 98 : if (dv) *w = gmul(*w,dv);
12388 98 : return v;
12389 : }
12390 :
12391 : static GEN mfpetersson_i(GEN FS, GEN GS);
12392 :
12393 : GEN
12394 42 : mfmanin(GEN FS, long bitprec)
12395 : {
12396 42 : pari_sp av = avma;
12397 : GEN mf, M, vp, vm, cosets, CHI, vpp, vmm, f, T, P, vE, polabs, roabs, rnfeq;
12398 : GEN pet;
12399 : long N, k, lco, i, prec, lvE;
12400 :
12401 42 : if (!checkfs_i(FS))
12402 : {
12403 7 : if (checkfs2_i(FS)) pari_err_TYPE("mfmanin [need integral k > 1]",FS);
12404 0 : pari_err_TYPE("mfmanin",FS);
12405 : }
12406 35 : if (!mfs_iscusp(FS)) pari_err_TYPE("mfmanin [noncuspidal]",FS);
12407 35 : mf = fs_get_MF(FS);
12408 35 : vp = fs_get_pols(FS);
12409 35 : cosets = fs_get_cosets(FS);
12410 35 : bitprec = fs_get_bitprec(FS);
12411 35 : N = MF_get_N(mf); k = MF_get_k(mf); CHI = MF_get_CHI(mf);
12412 35 : lco = lg(cosets); vm = cgetg(lco, t_VEC);
12413 35 : prec = nbits2prec(bitprec);
12414 476 : for (i = 1; i < lco; i++)
12415 : {
12416 441 : GEN g = gel(cosets, i), c;
12417 441 : long A = itos(gcoeff(g,1,1)), B = itos(gcoeff(g,1,2));
12418 441 : long C = itos(gcoeff(g,2,1)), D = itos(gcoeff(g,2,2));
12419 441 : long Dbar, ibar = mftocoset_iD(N, mkmat22s(-B,-A,D,C), cosets, &Dbar);
12420 :
12421 441 : c = mfcharcxeval(CHI, Dbar, prec); if (odd(k)) c = gneg(c);
12422 441 : T = RgX_Rg_mul(gel(vp,ibar), c);
12423 441 : if (typ(T) == t_POL && varn(T) == 0) T = RgX_recip(T);
12424 441 : gel(vm,i) = T;
12425 : }
12426 35 : vpp = gadd(vp,vm);
12427 35 : vmm = gsub(vp,vm);
12428 :
12429 35 : vE = fs_get_vE(FS); lvE = lg(vE);
12430 35 : f = gel(fs_get_EF(FS), 1);
12431 35 : P = mf_get_field(f); if (degpol(P) == 1) P = NULL;
12432 35 : T = mfcharpol(CHI); if (degpol(T) == 1) T = NULL;
12433 35 : if (T && P)
12434 : {
12435 7 : rnfeq = nf_rnfeqsimple(T, P);
12436 7 : polabs = gel(rnfeq,1);
12437 7 : roabs = gel(QX_complex_roots(polabs,prec), 1);
12438 : }
12439 : else
12440 : {
12441 28 : rnfeq = roabs = NULL;
12442 28 : polabs = P? P: T;
12443 : }
12444 35 : pet = mfpetersson_i(FS, NULL);
12445 35 : M = cgetg(lvE, t_VEC);
12446 84 : for (i = 1; i < lvE; i++)
12447 : {
12448 49 : GEN p, m, wp, wm, petdiag, r, E = gel(vE,i);
12449 49 : p = normal(RgXV_embed(vpp, E), polabs, roabs, rnfeq, &wp, prec);
12450 49 : m = normal(RgXV_embed(vmm, E), polabs, roabs, rnfeq, &wm, prec);
12451 49 : petdiag = typ(pet)==t_MAT? gcoeff(pet,i,i): pet;
12452 49 : r = gdiv(imag_i(gmul(wp, conj_i(wm))), petdiag);
12453 49 : r = bestapprnfrel(r, polabs, roabs, rnfeq, prec);
12454 49 : gel(M,i) = mkvec2(mkvec2(p,m), mkvec3(wp,wm,r));
12455 : }
12456 35 : return gerepilecopy(av, lvE == 2? gel(M,1): M);
12457 : }
12458 :
12459 : /* flag = 0: full, flag = +1 or -1, odd/even */
12460 : /* Basis of period polynomials in level 1. */
12461 : GEN
12462 49 : mfperiodpolbasis(long k, long flag)
12463 : {
12464 49 : pari_sp av = avma;
12465 49 : long i, j, n = k - 2;
12466 : GEN M, C, v;
12467 49 : if (k <= 4) return cgetg(1,t_VEC);
12468 35 : M = cgetg(k, t_MAT);
12469 35 : C = matpascal(n);
12470 35 : if (!flag)
12471 392 : for (j = 0; j <= n; j++)
12472 : {
12473 371 : gel(M, j+1) = v = cgetg(k, t_COL);
12474 4767 : for (i = 0; i <= j; i++) gel(v, i+1) = gcoeff(C, j+1, i+1);
12475 4396 : for (; i <= n; i++) gel(v, i+1) = gcoeff(C, n-j+1, i-j+1);
12476 : }
12477 : else
12478 168 : for (j = 0; j <= n; j++)
12479 : {
12480 154 : gel(M, j+1) = v = cgetg(k, t_COL);
12481 1848 : for (i = 0; i <= n; i++)
12482 : {
12483 1694 : GEN a = i < j ? gcoeff(C, j+1, i+1) : gen_0;
12484 1694 : if (i + j >= n)
12485 : {
12486 924 : GEN b = gcoeff(C, j+1, i+j-n+1);
12487 924 : a = flag < 0 ? addii(a,b) : subii(a,b);
12488 : }
12489 1694 : gel(v, i+1) = a;
12490 : }
12491 : }
12492 35 : return gerepilecopy(av, RgM_to_RgXV(ZM_ker(M), 0));
12493 : }
12494 :
12495 : static int
12496 161 : zero_at_cusp(GEN mf, GEN F, GEN c)
12497 : {
12498 161 : GEN v = evalcusp(mf, F, c, LOWDEFAULTPREC);
12499 161 : return (gequal0(v) || gexpo(v) <= -62);
12500 : }
12501 : /* Compute list E of j such that F|_k g_j vanishes at oo: return [E, s(E)] */
12502 : static void
12503 7 : mffvanish(GEN mf, GEN F, GEN G, GEN cosets, GEN *pres, GEN *press)
12504 : {
12505 7 : long j, lc = lg(cosets), N = MF_get_N(mf);
12506 : GEN v, vs;
12507 7 : *pres = v = zero_zv(lc-1);
12508 7 : *press= vs = zero_zv(lc-1);
12509 91 : for (j = 1; j < lc; j++)
12510 : {
12511 84 : GEN ga = gel(cosets,j), c = mat2cusp(ga);
12512 84 : if (zero_at_cusp(mf, F, c))
12513 7 : v[j] = vs[ mftocoset_i(N, ZM_mulS(ga), cosets) ] = 1;
12514 77 : else if (!zero_at_cusp(mf, G, c))
12515 0 : pari_err_IMPL("divergent Petersson product");
12516 : }
12517 7 : }
12518 : static GEN
12519 133 : Haberland(GEN PF, GEN PG, GEN vEF, GEN vEG, long k)
12520 : {
12521 133 : GEN S = gen_0, vC = vecbinomial(k-2); /* vC[n+1] = (-1)^n binom(k-2,n) */
12522 133 : long n, j, l = lg(PG);
12523 364 : for (n = 2; n < k; n+=2) gel(vC,n) = negi(gel(vC,n));
12524 2569 : for (j = 1; j < l; j++)
12525 : {
12526 2436 : GEN PFj = gel(PF,j), PGj = gel(PG,j);
12527 9954 : for (n = 0; n <= k-2; n++)
12528 : {
12529 7518 : GEN a = RgX_coeff(PGj, k-2-n), b = RgX_coeff(PFj, n);
12530 7518 : a = Rg_embedall(a, vEG);
12531 7518 : b = Rg_embedall(b, vEF);
12532 7518 : a = conj_i(a); if (typ(a) == t_VEC) settyp(a, t_COL);
12533 : /* a*b = scalar or t_VEC or t_COL or t_MAT */
12534 7518 : S = gadd(S, gdiv(gmul(a,b), gel(vC,n+1)));
12535 : }
12536 : }
12537 133 : S = mulcxpowIs(gmul2n(S, 1-k), 1+k);
12538 133 : return vEF==vEG? real_i(S): S;
12539 : }
12540 : /* F1S, F2S both symbols, same mf */
12541 : static GEN
12542 7 : mfpeterssonnoncusp(GEN F1S, GEN F2S)
12543 : {
12544 7 : pari_sp av = avma;
12545 : GEN mf, F1, F2, GF1, GF2, P2, cosets, vE1, vE2, FE1, FE2, P;
12546 : GEN I, IP1, RHO, RHOP1, INF, res, ress;
12547 7 : const double height = sqrt(3.)/2;
12548 : long k, r, j, bitprec, prec;
12549 :
12550 7 : mf = fs_get_MF(F1S);
12551 7 : FE1 = fs_get_EF(F1S); F1 = gel(FE1, 1);
12552 7 : FE2 = fs_get_EF(F2S); F2 = gel(FE2, 1);
12553 7 : cosets = fs_get_cosets(F1S);
12554 7 : bitprec = minuu(fs_get_bitprec(F1S), fs_get_bitprec(F2S));
12555 7 : prec = nbits2prec(bitprec);
12556 7 : F1S = fs_set_expan(F1S, mfgaexpansionall(mf, FE1, cosets, height, prec));
12557 7 : if (F2S != F1S)
12558 7 : F2S = fs_set_expan(F2S, mfgaexpansionall(mf, FE2, cosets, height, prec));
12559 7 : k = MF_get_k(mf); r = lg(cosets) - 1;
12560 7 : vE1 = fs_get_vE(F1S);
12561 7 : vE2 = fs_get_vE(F2S);
12562 7 : I = gen_I();
12563 7 : IP1 = mkcomplex(gen_1,gen_1);
12564 7 : RHO = rootsof1u_cx(3, prec+EXTRAPREC64);
12565 7 : RHOP1 = gaddsg(1, RHO);
12566 7 : INF = mkoo();
12567 7 : mffvanish(mf, F1, F2, cosets, &res, &ress);
12568 7 : P2 = fs_get_pols(F2S);
12569 7 : GF1 = cgetg(r+1, t_VEC);
12570 7 : GF2 = cgetg(r+1, t_VEC); P = get_P(k, fetch_var(), prec);
12571 91 : for (j = 1; j <= r; j++)
12572 : {
12573 84 : GEN g = gel(cosets,j);
12574 84 : if (res[j]) {
12575 7 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHOP1,INF), g, P);
12576 7 : gel(GF2,j) = mfsymboleval_direct(F2S, mkvec2(I,IP1), g, P);
12577 77 : } else if (ress[j]) {
12578 7 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHOP1,RHO), g, P);
12579 7 : gel(GF2,j) = mfsymboleval_direct(F2S, mkvec2(I,INF), g, P);
12580 : } else {
12581 70 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHO,I), g, P);
12582 70 : gel(GF2,j) = gneg(gel(P2,j)); /* - symboleval(F2S, [0,oo] */
12583 : }
12584 : }
12585 7 : delete_var();
12586 7 : return gerepileupto(av, gdivgu(Haberland(GF1,GF2, vE1,vE2, k), r));
12587 : }
12588 :
12589 : /* Petersson product of F and G, given by mfsymbol's [k > 1 integral] */
12590 : static GEN
12591 133 : mfpetersson_i(GEN FS, GEN GS)
12592 : {
12593 133 : pari_sp av = avma;
12594 : GEN mf, ESF, ESG, PF, PG, PH, CHI, cosets, vEF, vEG;
12595 : long k, r, j, N, bitprec, prec;
12596 :
12597 133 : if (!checkfs_i(FS)) pari_err_TYPE("mfpetersson",FS);
12598 133 : mf = fs_get_MF(FS);
12599 133 : ESF = fs_get_vES(FS);
12600 133 : if (!GS) GS = FS;
12601 : else
12602 : {
12603 28 : if (!checkfs_i(GS)) pari_err_TYPE("mfpetersson",GS);
12604 28 : if (!mfs_checkmf(GS, mf))
12605 0 : pari_err_TYPE("mfpetersson [different mf]", mkvec2(FS,GS));
12606 : }
12607 133 : ESG = fs_get_vES(GS);
12608 133 : if (!gequal0(gel(ESF,1)) && !gequal0(gel(ESG,1)))
12609 7 : return mfpeterssonnoncusp(FS, GS);
12610 126 : if (gequal0(gel(ESF,2)) || gequal0(gel(ESG,2))) return gc_const(av, gen_0);
12611 126 : N = MF_get_N(mf);
12612 126 : k = MF_get_k(mf);
12613 126 : CHI = MF_get_CHI(mf);
12614 126 : PF = fs_get_pols(FS); vEF = fs_get_vE(FS);
12615 126 : PG = fs_get_pols(GS); vEG = fs_get_vE(GS);
12616 126 : cosets = fs_get_cosets(FS);
12617 126 : bitprec = minuu(fs_get_bitprec(FS), fs_get_bitprec(GS));
12618 126 : prec = nbits2prec(bitprec);
12619 126 : r = lg(PG)-1;
12620 126 : PH = cgetg(r+1, t_VEC);
12621 2478 : for (j = 1; j <= r; j++)
12622 : {
12623 2352 : GEN ga = gel(cosets,j), PGj1, PGjm1;
12624 : long iT, D;
12625 2352 : iT = mftocoset_iD(N, ZM_mulTi(ga), cosets, &D);
12626 2352 : PGj1 = RgX_translate(gel(PG, iT), gen_1);
12627 2352 : PGj1 = RgX_Rg_mul(PGj1, mfcharcxeval(CHI, D, prec));
12628 2352 : iT = mftocoset_iD(N, ZM_mulT(ga), cosets, &D);
12629 2352 : PGjm1 = RgX_translate(gel(PG,iT), gen_m1);
12630 2352 : PGjm1 = RgX_Rg_mul(PGjm1, mfcharcxeval(CHI, D, prec));
12631 2352 : gel(PH,j) = gsub(PGj1, PGjm1);
12632 : }
12633 126 : return gerepileupto(av, gdivgu(Haberland(PF, PH, vEF, vEG, k), 6*r));
12634 : }
12635 :
12636 : /****************************************************************/
12637 : /* Petersson products using Nelson-Collins */
12638 : /****************************************************************/
12639 : /* Compute W(k,z) = sum_{m >= 1} (mz)^{k-1}(mzK_{k-2}(mz)-K_{k-1}(mz))
12640 : * for z>0 and absolute accuracy < 2^{-B}.
12641 : * K_k(x) ~ (Pi/(2x))^{1/2} e^{-x} */
12642 :
12643 : static void
12644 10304 : Wparams(GEN *ph, long *pN, long k, double x, long prec)
12645 : {
12646 10304 : double B = prec2nbits(prec) + 10;
12647 10304 : double C = B + k*log(x)/M_LN2 + 1, D = C*M_LN2 + 2.065;
12648 10304 : double F = 2 * ((C - 1) * M_LN2 + log(gtodouble(mpfact(k)))) / x;
12649 10304 : double T = log(F) * (1 + 2*k/x/F), PI2 = M_PI*M_PI;
12650 10304 : *pN = (long)ceil((T/PI2) * (D + log(D/PI2)));
12651 10304 : *ph = gprec_w(dbltor(T / *pN), prec);
12652 10304 : }
12653 :
12654 : static void
12655 10304 : Wcoshall(GEN *pCH, GEN *pCHK, GEN *pCHK1, GEN h, long N, long k, long prec)
12656 : {
12657 10304 : GEN CH, CHK, CHK1, z = gexp(h, prec);
12658 10304 : GEN PO = gpowers(z, N), POK1 = gpowers(gpowgs(z, k-1), N);
12659 10304 : GEN E = ginv(gel(PO, N + 1)); /* exp(-hN) */
12660 10304 : GEN E1 = ginv(gel(POK1, N + 1)); /* exp(-(k-1)h) */
12661 : long j;
12662 10304 : *pCH = CH = cgetg(N+1, t_VEC);
12663 10304 : *pCHK = CHK = cgetg(N+1, t_VEC);
12664 10304 : *pCHK1 = CHK1 = cgetg(N+1, t_VEC);
12665 146048 : for (j = 1; j <= N; j++)
12666 : {
12667 135744 : GEN eh = gel(PO, j+1), emh = gmul(gel(PO, N-j+1), E); /* e^{jh}, e^{-jh} */
12668 135744 : GEN ek1h = gel(POK1, j+1), ek1mh = gmul(gel(POK1, N-j+1), E1);
12669 135744 : gel(CH, j) = gmul2n(gadd(eh, emh), -1); /* cosh(jh) */
12670 135744 : gel(CHK1,j) = gmul2n(gadd(ek1h, ek1mh), -1); /* cosh((k-1)jh) */
12671 135744 : gel(CHK, j) = gmul2n(gadd(gmul(eh, ek1h), gmul(emh, ek1mh)), -1);
12672 : }
12673 10304 : }
12674 :
12675 : /* computing W(k,x) via integral */
12676 : static GEN
12677 10304 : Wint(long k, GEN vP, GEN x, long prec)
12678 : {
12679 : GEN P, P1, S1, S, h, CH, CHK, CHK1;
12680 : long N, j;
12681 10304 : Wparams(&h, &N, k, gtodouble(x), prec);
12682 10304 : Wcoshall(&CH, &CHK, &CHK1, h, N, k, prec);
12683 10304 : P = gel(vP, k+1); P1 = gel(vP, k); S = S1 = NULL;
12684 156352 : for (j = 0; j <= N; j++)
12685 : {
12686 146048 : GEN eh = gexp(j? gmul(x, gel(CH, j)): x, prec);
12687 146048 : GEN eh1 = gsubgs(eh, 1), eh1k = gpowgs(eh1, k), t1, t;
12688 146048 : t = gdiv(poleval(P, eh), gmul(eh1, eh1k));
12689 146048 : t1 = gdiv(poleval(P1, eh), eh1k);
12690 146048 : if (j)
12691 : {
12692 135744 : S = gadd(S, gmul(t, gel(CHK, j)));
12693 135744 : S1 = gadd(S1, gmul(t1, gel(CHK1, j)));
12694 : }
12695 : else
12696 : {
12697 10304 : S = gmul2n(t, -1);
12698 10304 : S1 = gmul2n(t1, -1);
12699 : }
12700 : }
12701 10304 : return gmul(gmul(h, gpowgs(x, k-1)), gsub(gmul(x, S), gmulsg(2*k-1, S1)));
12702 : }
12703 :
12704 : static GEN
12705 21 : get_vP(long k)
12706 : {
12707 21 : GEN P, v = cgetg(k+2, t_VEC), Q = deg1pol_shallow(gen_1,gen_m1,0);
12708 : long j;
12709 21 : gel(v,1) = gen_1;
12710 21 : gel(v,2) = P = pol_x(0);
12711 28 : for (j = 2; j <= k; j++)
12712 7 : gel(v,j+1) = P = RgX_shift_shallow(gsub(gmulsg(j, P),
12713 : gmul(Q, ZX_deriv(P))), 1);
12714 21 : return v;
12715 : }
12716 : /* vector of (-1)^j(1/(exp(x)-1))^(j) [x = z] * z^j for 0<=j<=r */
12717 : static GEN
12718 63742 : VS(long r, GEN z, GEN V, long prec)
12719 : {
12720 63742 : GEN e = gexp(z, prec), c = ginv(gsubgs(e,1));
12721 63742 : GEN T = gpowers0(gmul(c, z), r, c);
12722 : long j;
12723 63742 : V = gsubst(V, 0, e);
12724 143864 : for (j = 1; j <= r + 1; j++) gel(V,j) = gmul(gel(V,j), gel(T,j));
12725 63742 : return V;
12726 : }
12727 :
12728 : /* U(r,x)=sum_{m >= 1} (mx)^k K_k(mx), k = r+1/2 */
12729 : static GEN
12730 71932 : Unelson(long r, GEN V)
12731 : {
12732 71932 : GEN S = gel(V,r+1), C = gen_1; /* (r+j)! / j! / (r-j)! */
12733 : long j;
12734 71932 : if (!r) return S;
12735 40950 : for (j = 1; j <= r; j++)
12736 : {
12737 24570 : C = gdivgu(gmulgu(C, (r+j)*(r-j+1)), j);
12738 24570 : S = gadd(S, gmul2n(gmul(C, gel(V, r-j+1)), -j));
12739 : }
12740 16380 : return S;
12741 : }
12742 : /* W(r+1/2,z) / sqrt(Pi/2) */
12743 : static GEN
12744 63742 : Wint2(long r, GEN vP, GEN z, long prec)
12745 : {
12746 63742 : GEN R, V = VS(r, z, vP, prec);
12747 63742 : R = Unelson(r, V);
12748 63742 : if (r) R = gsub(R, gmulsg(2*r, Unelson(r-1, V)));
12749 63742 : return R;
12750 : }
12751 : typedef GEN(*Wfun_t)(long, GEN, GEN, long);
12752 : static GEN
12753 74046 : WfromZ(GEN Z, GEN vP, GEN gkm1, Wfun_t W, long k, GEN pi4, long prec)
12754 : {
12755 74046 : pari_sp av = avma;
12756 74046 : GEN Zk = gpow(Z, gkm1, prec), z = gmul(pi4, gsqrt(Z,prec));
12757 74046 : return gerepileupto(av, gdiv(W(k, vP, z, prec), Zk));
12758 : }
12759 : /* mf a true mf or an fs2 */
12760 : static GEN
12761 21 : fs2_init(GEN mf, GEN F, long bit)
12762 : {
12763 21 : pari_sp av = avma;
12764 21 : long i, l, lim, N, k, k2, prec = nbits2prec(bit);
12765 : GEN DEN, cusps, tab, gk, gkm1, W0, vW, vVW, vVF, vP, al0;
12766 21 : GEN vE = mfgetembed(F, prec), pi4 = Pi2n(2, prec);
12767 : Wfun_t Wf;
12768 :
12769 21 : if (lg(mf) == 7)
12770 : {
12771 21 : vW = cusps = NULL; /* true mf */
12772 21 : DEN = tab = NULL; /* -Wall */
12773 : }
12774 : else
12775 : { /* mf already an fs2, reset if its precision is too low */
12776 0 : vW = (fs2_get_bitprec(mf) < bit)? NULL: fs2_get_W(mf);
12777 0 : cusps = fs2_get_cusps(mf);
12778 0 : DEN = fs2_get_den(mf);
12779 0 : mf = fs2_get_MF(mf);
12780 : }
12781 21 : N = MF_get_N(mf);
12782 21 : gk = MF_get_gk(mf); gkm1 = gsubgs(gk, 1);
12783 21 : k2 = itos(gmul2n(gk,1));
12784 21 : Wf = odd(k2)? Wint2: Wint;
12785 21 : k = k2 >> 1; vP = get_vP(k);
12786 21 : if (vW) lim = (lg(gel(vW,1)) - 2) / N; /* vW[1] attached to cusp 0, width N */
12787 : else
12788 : { /* true mf */
12789 21 : double B = (bit + 10)*M_LN2;
12790 21 : double L = (B + k2*log(B)/2 + k2*k2*log(B)/(4*B)) / (4*M_PI);
12791 : long n, Lw;
12792 21 : lim = ((long)ceil(L*L));
12793 21 : Lw = N*lim;
12794 21 : tab = cgetg(Lw+1,t_VEC);
12795 59157 : for (n = 1; n <= Lw; n++)
12796 59136 : gel(tab,n) = WfromZ(uutoQ(n,N), vP, gkm1, Wf, k, pi4, prec);
12797 21 : if (!cusps) cusps = mfcusps_i(N);
12798 21 : DEN = gmul2n(gmulgu(gpow(Pi2n(3, prec), gkm1, prec), mypsiu(N)), -2);
12799 21 : if (odd(k2)) DEN = gdiv(DEN, sqrtr_abs(Pi2n(-1,prec)));
12800 : }
12801 21 : l = lg(cusps);
12802 21 : vVF = cgetg(l, t_VEC);
12803 21 : vVW = cgetg(l, t_VEC);
12804 21 : al0 = cgetg(l, t_VECSMALL);
12805 21 : W0 = k2==1? ginv(pi4): gen_0;
12806 203 : for (i = 1; i < l; i++)
12807 : {
12808 : long A, C, w, wi, Lw, n;
12809 : GEN VF, W, paramsF, al;
12810 182 : (void)cusp_AC(gel(cusps,i), &A,&C);
12811 182 : wi = ugcd(N, C*C); w = N / wi; Lw = w * lim;
12812 182 : VF = mfslashexpansion(mf, F, cusp2mat(A,C), Lw, 0, ¶msF, prec);
12813 : /* paramsF[2] = w */
12814 182 : al = gel(paramsF, 1); if (gequal0(al)) al = NULL;
12815 100240 : for (n = 0; n <= Lw; n++)
12816 : {
12817 100058 : GEN a = gel(VF,n+1);
12818 100058 : gel(VF,n+1) = gequal0(a)? gen_0: Rg_embedall(a, vE);
12819 : }
12820 182 : if (vW)
12821 0 : W = gel(vW, i);
12822 : else
12823 : {
12824 182 : W = cgetg(Lw+2, t_VEC);
12825 100240 : for (n = 0; n <= Lw; n++)
12826 100058 : gel(W, n+1) = al? WfromZ(gadd(al,uutoQ(n,w)),vP,gkm1,Wf,k,pi4, prec)
12827 100058 : : (n? gel(tab, n * wi): W0);
12828 : }
12829 182 : al0[i] = !al;
12830 182 : gel(vVF, i) = VF;
12831 182 : gel(vVW, i) = W;
12832 : }
12833 21 : if (k2 <= 1) al0 = zero_zv(l-1); /* no need to test for convergence */
12834 21 : return gerepilecopy(av, mkvecn(7, mf,vVW,cusps,vVF,utoipos(bit),al0,DEN));
12835 : }
12836 :
12837 : static GEN
12838 21 : mfpetersson2(GEN Fs, GEN Gs)
12839 : {
12840 21 : pari_sp av = avma;
12841 21 : GEN VC, RES, vF, vG, vW = fs2_get_W(Fs), al0 = fs2_get_al0(Fs);
12842 21 : long N = MF_get_N(fs2_get_MF(Fs)), j, lC;
12843 :
12844 21 : VC = fs2_get_cusps(Fs); lC = lg(VC);
12845 21 : vF = fs2_get_F(Fs);
12846 21 : vG = Gs? fs2_get_F(Gs): vF;
12847 21 : RES = gen_0;
12848 203 : for (j = 1; j < lC; j++)
12849 : {
12850 182 : GEN W = gel(vW,j), VF = gel(vF,j), VG = gel(vG,j), T = gen_0;
12851 182 : long A, C, w, n, L = lg(W);
12852 182 : pari_sp av = avma;
12853 182 : (void)cusp_AC(gel(VC,j), &A,&C); w = N/ugcd(N, C*C);
12854 182 : if (al0[j] && !isintzero(gel(VF,1)) && !isintzero(gel(VG,1)))
12855 0 : pari_err_IMPL("divergent Petersson product");
12856 100240 : for (n = 1; n < L; n++)
12857 : {
12858 100058 : GEN b = gel(VF,n), a = gel(VG,n);
12859 100058 : if (!isintzero(a) && !isintzero(b))
12860 : {
12861 79964 : T = gadd(T, gmul(gel(W,n), gmul(conj_i(a),b)));
12862 79964 : if (gc_needed(av,2)) T = gerepileupto(av,T);
12863 : }
12864 : }
12865 182 : if (w != 1) T = gmulgu(T,w);
12866 182 : RES = gerepileupto(av, gadd(RES, T));
12867 : }
12868 21 : if (!Gs) RES = real_i(RES);
12869 21 : return gerepileupto(av, gdiv(RES, fs2_get_den(Fs)));
12870 : }
12871 :
12872 : static long
12873 147 : symbol_type(GEN F)
12874 : {
12875 147 : if (checkfs_i(F)) return 1;
12876 21 : if (checkfs2_i(F)) return 2;
12877 0 : return 0;
12878 : }
12879 : static int
12880 28 : symbol_same_mf(GEN F, GEN G) { return gequal(gmael(F,1,1), gmael(G,1,1)); }
12881 : GEN
12882 119 : mfpetersson(GEN F, GEN G)
12883 : {
12884 119 : long tF = symbol_type(F);
12885 119 : if (!tF) pari_err_TYPE("mfpetersson",F);
12886 119 : if (G)
12887 : {
12888 28 : long tG = symbol_type(G);
12889 28 : if (!tG) pari_err_TYPE("mfpetersson",F);
12890 28 : if (tF != tG || !symbol_same_mf(F,G))
12891 0 : pari_err_TYPE("mfpetersson [incompatible symbols]", mkvec2(F,G));
12892 : }
12893 119 : return (tF == 1)? mfpetersson_i(F, G): mfpetersson2(F, G);
12894 : }
12895 :
12896 : /****************************************************************/
12897 : /* projective Galois representation, weight 1 */
12898 : /****************************************************************/
12899 : static void
12900 392 : moreorders(long N, GEN CHI, GEN F, GEN *pP, GEN *pO, ulong *bound)
12901 : {
12902 392 : pari_sp av = avma;
12903 : forprime_t iter;
12904 392 : ulong a = *bound+1, b = 2*(*bound), p;
12905 392 : long i = 1;
12906 392 : GEN P, O, V = mfcoefs_i(F, b, 1);
12907 392 : *bound = b;
12908 392 : P = cgetg(b-a+2, t_VECSMALL);
12909 392 : O = cgetg(b-a+2, t_VECSMALL);
12910 392 : u_forprime_init(&iter, a, b);
12911 2310 : while((p = u_forprime_next(&iter))) if (N % p)
12912 : {
12913 1813 : O[i] = mffindrootof1(V, p, CHI);
12914 1813 : P[i++] = p;
12915 : }
12916 392 : setlg(P, i); *pP = shallowconcat(*pP, P);
12917 392 : setlg(O, i); *pO = shallowconcat(*pO, O);
12918 392 : gerepileall(av, 2, pP, pO);
12919 392 : }
12920 :
12921 : static GEN
12922 182 : search_abelian(GEN nf, long n, long k, GEN N, GEN CHI, GEN F,
12923 : GEN *pP, GEN *pO, ulong *bound, long prec)
12924 : {
12925 182 : pari_sp av = avma;
12926 : GEN bnr, cond, H, cyc, gn, T, Bquo, P, E;
12927 182 : long sN = itos(N), r1 = nf_get_r1(nf), i, j, d;
12928 :
12929 182 : cond = idealfactor(nf, N);
12930 182 : P = gel(cond,1);
12931 182 : E = gel(cond,2);
12932 679 : for (i = j = 1; i < lg(P); i++)
12933 : {
12934 497 : GEN pr = gel(P,i), Ej = gen_1;
12935 497 : long p = itos(pr_get_p(pr));
12936 497 : if (p == n)
12937 : {
12938 98 : long e = pr_get_e(pr); /* 1 + [e*p/(p-1)] */
12939 98 : Ej = utoipos(1 + (e*p) / (p-1));
12940 : }
12941 : else
12942 : {
12943 399 : long f = pr_get_f(pr);
12944 399 : if (Fl_powu(p % n, f, n) != 1) continue;
12945 : }
12946 462 : gel(P,j) = pr;
12947 462 : gel(E,j) = Ej; j++;
12948 : }
12949 182 : setlg(P,j);
12950 182 : setlg(E,j);
12951 182 : cond = mkvec2(cond, const_vec(r1, gen_1));
12952 182 : bnr = Buchraymod(Buchall(nf, nf_FORCE, prec), cond, nf_INIT, utoipos(n));
12953 182 : cyc = bnr_get_cyc(bnr);
12954 182 : d = lg(cyc)-1;
12955 182 : H = zv_diagonal(ZV_to_Flv(cyc, n));
12956 182 : gn = utoi(n);
12957 182 : for (i = 1;;)
12958 : {
12959 2646 : for(j = 2; i < lg(*pO); i++)
12960 : {
12961 2072 : long o, q = (*pP)[i];
12962 2072 : GEN pr = idealprimedec_galois(nf, stoi(q));
12963 2072 : o = ((*pO)[i] / pr_get_f(pr)) % n;
12964 2072 : if (o)
12965 : {
12966 1442 : GEN v = ZV_to_Flv(isprincipalray(bnr, pr), n);
12967 1442 : H = vec_append(H, Flv_Fl_mul(v, o, n));
12968 : }
12969 : }
12970 574 : H = Flm_image(H, n); if (lg(cyc)-lg(H) <= k) break;
12971 392 : moreorders(sN, CHI, F, pP, pO, bound);
12972 : }
12973 182 : H = hnfmodid(shallowconcat(zm_to_ZM(H), diagonal_shallow(cyc)), gn);
12974 :
12975 182 : Bquo = cgetg(k+1, t_MAT);
12976 812 : for (i = j = 1; i <= d; i++)
12977 630 : if (!equali1(gcoeff(H,i,i))) gel(Bquo,j++) = col_ei(d,i);
12978 :
12979 441 : for (i = 1, T = NULL; i<=k; i++)
12980 : {
12981 259 : GEN Hi = hnfmodid(shallowconcat(H, vecsplice(Bquo,i)), gn);
12982 259 : GEN pol = rnfkummer(bnr, Hi, prec);
12983 259 : T = T? nfcompositum(nf, T, pol, 2): pol;
12984 : }
12985 182 : T = rnfequation(nf, T); return gc_all(av, 3, &T, pP, pO);
12986 : }
12987 :
12988 : static GEN
12989 77 : search_solvable(GEN LG, GEN mf, GEN F, long prec)
12990 : {
12991 77 : GEN N = MF_get_gN(mf), CHI = MF_get_CHI(mf), pol, O, P, nf, Nfa;
12992 77 : long i, l = lg(LG), v = fetch_var();
12993 77 : ulong bound = 1;
12994 77 : O = cgetg(1, t_VECSMALL); /* projective order of rho(Frob_p) */
12995 77 : P = cgetg(1, t_VECSMALL);
12996 77 : Nfa = Z_factor(N);
12997 77 : pol = pol_x(v);
12998 259 : for (i = 1; i < l; i++)
12999 : { /* n prime, find a (Z/nZ)^k - extension */
13000 182 : GEN G = gel(LG,i);
13001 182 : long n = G[1], k = G[2];
13002 182 : nf = nfinitred(mkvec2(pol,Nfa), prec);
13003 182 : pol = search_abelian(nf, n, k, N, CHI, F, &P, &O, &bound, prec);
13004 182 : setvarn(pol,v);
13005 : }
13006 77 : delete_var(); setvarn(pol,0); return pol;
13007 : }
13008 :
13009 : static GEN
13010 0 : search_A5(GEN mf, GEN F)
13011 : {
13012 0 : GEN CHI = MF_get_CHI(mf), O, P, L;
13013 0 : long N = MF_get_N(mf), i, j, lL, nd, r;
13014 0 : ulong bound = 1;
13015 0 : r = radicalu(N);
13016 0 : L = veccond_to_A5(zv_z_mul(divisorsu(N/r),r), 2); lL = lg(L); nd = lL-1;
13017 0 : if (nd == 1) return gmael(L,1,1);
13018 0 : O = cgetg(1, t_VECSMALL); /* projective order of rho(Frob_p) */
13019 0 : P = cgetg(1, t_VECSMALL);
13020 0 : for(i = 1; nd > 1; )
13021 : {
13022 : long l;
13023 0 : moreorders(N, CHI, F, &P, &O, &bound);
13024 0 : l = lg(P);
13025 0 : for ( ; i < l; i++)
13026 : {
13027 0 : ulong p = P[i], f = O[i];
13028 0 : for (j = 1; j < lL; j++)
13029 0 : if (gel(L,j))
13030 : {
13031 0 : GEN FE = ZpX_primedec(gmael(L,j,1), utoi(p)), F = gel(FE,1);
13032 0 : long nF = lg(F)-1;
13033 0 : if (!equaliu(gel(F, nF), f)) { gel(L,j) = NULL; nd--; }
13034 : }
13035 0 : if (nd <= 1) break;
13036 : }
13037 : }
13038 0 : for (j = 1; j < lL; j++)
13039 0 : if (gel(L,j)) return gmael(L,j,1);
13040 0 : return NULL;
13041 : }
13042 :
13043 : GEN
13044 77 : mfgaloisprojrep(GEN mf, GEN F, long prec)
13045 : {
13046 77 : pari_sp av = avma;
13047 77 : GEN LG = NULL;
13048 77 : if (!checkMF_i(mf) && !checkmf_i(F)) pari_err_TYPE("mfgaloisrep", F);
13049 77 : switch( itos(mfgaloistype(mf,F)) )
13050 : {
13051 49 : case 0: case -12:
13052 49 : LG = mkvec2(mkvecsmall2(3,1), mkvecsmall2(2,2)); break;
13053 28 : case -24:
13054 28 : LG = mkvec3(mkvecsmall2(2,1), mkvecsmall2(3,1), mkvecsmall2(2,2)); break;
13055 0 : case -60: return gerepilecopy(av, search_A5(mf, F));
13056 0 : default: pari_err_IMPL("mfgaloisprojrep for types D_n");
13057 : }
13058 77 : return gerepilecopy(av, search_solvable(LG, mf, F, prec));
13059 : }
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