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 37184 : mkgNK(GEN N, GEN k, GEN CHI, GEN P) { return mkvec4(N, k, CHI, P); }
82 : static GEN
83 15106 : mkNK(long N, long k, GEN CHI) { return mkgNK(stoi(N), stoi(k), CHI, pol_x(1)); }
84 : GEN
85 8736 : MF_get_CHI(GEN mf) { return gmael(mf,1,3); }
86 : GEN
87 20580 : MF_get_gN(GEN mf) { return gmael(mf,1,1); }
88 : long
89 19439 : MF_get_N(GEN mf) { return itou(MF_get_gN(mf)); }
90 : GEN
91 14875 : MF_get_gk(GEN mf) { return gmael(mf,1,2); }
92 : long
93 7161 : MF_get_k(GEN mf)
94 : {
95 7161 : GEN gk = MF_get_gk(mf);
96 7161 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
97 7161 : 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 14798 : MF_get_space(GEN mf) { return itos(gmael(mf,1,4)); }
108 : GEN
109 4445 : MF_get_E(GEN mf) { return gel(mf,2); }
110 : GEN
111 21259 : MF_get_S(GEN mf) { return gel(mf,3); }
112 : GEN
113 1862 : MF_get_basis(GEN mf) { return shallowconcat(gel(mf,2), gel(mf,3)); }
114 : long
115 5558 : MF_get_dim(GEN mf)
116 : {
117 5558 : switch(MF_get_space(mf))
118 : {
119 714 : case mf_FULL:
120 714 : 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 4704 : default: /* mf_NEW, mf_CUSP, mf_OLD */
124 4704 : return lg(MF_get_S(mf)) - 1;
125 : }
126 : }
127 : GEN
128 7231 : MFnew_get_vj(GEN mf) { return gel(mf,4); }
129 : GEN
130 686 : MFcusp_get_vMjd(GEN mf) { return gel(mf,4); }
131 : GEN
132 6832 : MF_get_M(GEN mf) { return gmael(mf,5,3); }
133 : GEN
134 4816 : MF_get_Minv(GEN mf) { return gmael(mf,5,2); }
135 : GEN
136 10500 : 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 1239 : QabM_init(long n, ulong *p)
147 : {
148 1239 : ulong pinit = 1000000007;
149 : forprime_t T;
150 1239 : 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 1106 : ZabM_pseudoinv_i(GEN M, GEN P, long n, GEN *pv, GEN *den, int ratlift)
186 : {
187 1106 : GEN v = ZabM_indexrank(M, P, n);
188 1106 : if (pv) *pv = v;
189 1106 : M = shallowmatextract(M,gel(v,1),gel(v,2));
190 1106 : return ratlift? ZabM_inv_ratlift(M, P, n, den): ZabM_inv(M, P, n, den);
191 : }
192 :
193 : /* M matrix with coeff in Q(\chi)), where Q(\chi) = Q(X)/(P) for
194 : * P = cyclotomic Phi_n. Assume M rational if n <= 2 */
195 : static GEN
196 1603 : QabM_ker(GEN M, GEN P, long n)
197 : {
198 1603 : if (n <= 2) return QM_ker(M);
199 420 : return ZabM_ker(row_Q_primpart(liftpol_shallow(M)), P, n);
200 : }
201 : /* pseudo-inverse of M. FIXME: should replace QabM_pseudoinv */
202 : static GEN
203 1316 : QabM_pseudoinv_i(GEN M, GEN P, long n, GEN *pv, GEN *pden)
204 : {
205 : GEN cM, Mi;
206 1316 : 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 : {
213 182 : M = Q_primitive_part(liftpol_shallow(M), &cM);
214 182 : Mi = ZabM_pseudoinv(M, P, n, pv, pden);
215 : }
216 1316 : *pden = mul_content(*pden, cM);
217 1316 : return Mi;
218 : }
219 : /* FIXME: delete */
220 : static GEN
221 1057 : QabM_pseudoinv(GEN M, GEN P, long n, GEN *pv, GEN *pden)
222 : {
223 1057 : GEN Mi = QabM_pseudoinv_i(M, P, n, pv, pden);
224 1057 : return P? gmodulo(Mi, P): Mi;
225 : }
226 :
227 : static GEN
228 10444 : QabM_indexrank(GEN M, GEN P, long n)
229 : {
230 : GEN z;
231 10444 : if (n <= 2)
232 : {
233 9247 : M = vec_Q_primpart(M);
234 9247 : z = ZM_indexrank(M); /* M^(-1) = Mi / (cM * den) */
235 : }
236 : else
237 : {
238 1197 : M = vec_Q_primpart(liftpol_shallow(M));
239 1197 : z = ZabM_indexrank(M, P, n);
240 : }
241 10444 : 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 110075 : myeulerphiu(ulong n)
251 : {
252 : pari_sp av;
253 110075 : if (n == 1) return 1;
254 91070 : 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 3003 : mynumdivu(long N)
266 : {
267 : pari_sp av;
268 3003 : if (N == 1) return 1;
269 2877 : av = avma; return gc_long(av, numdivu_fact(myfactoru(N)));
270 : }
271 :
272 : /* N\prod_{p|N} (1+1/p) */
273 : static long
274 395283 : mypsiu(ulong N)
275 : {
276 : pari_sp av;
277 : GEN P;
278 : long j, l, a;
279 395283 : if (N == 1) return 1;
280 311563 : av = avma; P = gel(myfactoru(N), 1); l = lg(P);
281 742007 : for (a = N, j = 1; j < l; j++) a += a / P[j];
282 311563 : return gc_long(av, a);
283 : }
284 : /* write n = mf^2. Return m, set f. */
285 : static ulong
286 73 : mycore(ulong n, long *pf)
287 : {
288 73 : pari_sp av = avma;
289 73 : GEN fa = myfactoru(n), P = gel(fa,1), E = gel(fa,2);
290 73 : long i, l = lg(P), m = 1, f = 1;
291 277 : for (i = 1; i < l; i++)
292 : {
293 204 : long j, p = P[i], e = E[i];
294 204 : if (e & 1) m *= p;
295 463 : for (j = 2; j <= e; j+=2) f *= p;
296 : }
297 73 : *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 6015096 : corediscs_fact(GEN fa)
303 : {
304 6015096 : GEN P = gel(fa,1), E = gel(fa,2);
305 6015096 : long i, l = lg(P), m = 1;
306 19942683 : for (i = 1; i < l; i++)
307 : {
308 13927587 : long p = P[i], e = E[i];
309 13927587 : if (e & 1) m *= p;
310 : }
311 6015096 : if ((m&3L) != 3) m <<= 2;
312 6015096 : return m;
313 : }
314 : static long
315 7028 : mubeta(long n)
316 : {
317 7028 : pari_sp av = avma;
318 7028 : GEN E = gel(myfactoru(n), 2);
319 7028 : long i, s = 1, l = lg(E);
320 14588 : for (i = 1; i < l; i++)
321 : {
322 7560 : long e = E[i];
323 7560 : if (e >= 3) return gc_long(av,0);
324 7560 : if (e == 1) s *= -2;
325 : }
326 7028 : 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 7667084 : mubeta2(long n, long m)
333 : {
334 7667084 : pari_sp av = avma;
335 7667084 : GEN fa = myfactoru(n), P = gel(fa,1), E = gel(fa,2);
336 7667084 : long i, s = 1, l = lg(P);
337 15411430 : for (i = 1; i < l; i++)
338 : {
339 7744346 : long p = P[i], e = E[i];
340 7744346 : if (m % p)
341 : { /* p^e in n1 */
342 6574618 : if (e >= 3) return gc_long(av,0);
343 6574618 : if (e == 1) s *= -2;
344 : }
345 : else
346 : { /* in n2 */
347 1169728 : if (e >= 2) return gc_long(av,0);
348 1169728 : s = -s;
349 : }
350 : }
351 7667084 : 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 1879315 : newt_params(long N, long n, long FC, long *pg, long *pN2)
359 : {
360 1879315 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
361 1879315 : long i, g = 1, N2 = 1, l = lg(P);
362 5015998 : for (i = 1; i < l; i++)
363 : {
364 3136683 : long p = P[i], e = E[i];
365 3136683 : if (e == 1)
366 2739884 : { if (FC % p && n % (p*p) == 0) g *= p; }
367 : else
368 396799 : N2 *= upowuu(p,(n % p)? e-2: e-1);
369 : }
370 1879315 : *pg = g; *pN2 = N2;
371 1879315 : }
372 : /* simplified version of newt_params for n = 1 (newdim) */
373 : static void
374 41706 : newd_params(long N, long *pN2)
375 : {
376 41706 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
377 41706 : long i, N2 = 1, l = lg(P);
378 104391 : for (i = 1; i < l; i++)
379 : {
380 62685 : long p = P[i], e = E[i];
381 62685 : if (e > 2) N2 *= upowuu(p, e-2);
382 : }
383 41706 : *pN2 = N2;
384 41706 : }
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 34034 : 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 34034 : 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 16114 : QabV_tracerel(GEN v, long t, GEN x)
528 : {
529 : GEN z;
530 16114 : 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 2190258 : Qab_Czeta(long k, long o, GEN C, long vt)
544 : {
545 2190258 : if (!k) return C;
546 1456462 : if (!odd(o))
547 : { /* optimization: reduce max degree by a factor 2 for free */
548 1405355 : o >>= 1;
549 1405355 : if (k >= o) { k -= o; C = gneg(C); if (!k) return C; }
550 : }
551 1109710 : return monomial(C, k, vt);
552 : }
553 : /* zeta_o^k */
554 : static GEN
555 200753 : Qab_zeta(long k, long o, long vt) { return Qab_Czeta(k, o, gen_1, vt); }
556 :
557 : /* Operations on Dirichlet characters */
558 :
559 : /* A Dirichlet character can be given in GP in different formats, but in this
560 : * package, it will be a vector CHI=[G,chi,ord], where G is the (Z/MZ)^* to
561 : * which the character belongs, chi is the character in Conrey format, ord is
562 : * the order */
563 :
564 : static GEN
565 3723482 : gmfcharorder(GEN CHI) { return gel(CHI, 3); }
566 : long
567 3685080 : mfcharorder(GEN CHI) { return itou(gmfcharorder(CHI)); }
568 : static long
569 2709 : mfcharistrivial(GEN CHI) { return !CHI || mfcharorder(CHI) == 1; }
570 : static GEN
571 1561567 : gmfcharmodulus(GEN CHI) { return gmael3(CHI, 1, 1, 1); }
572 : long
573 1561567 : mfcharmodulus(GEN CHI) { return itou(gmfcharmodulus(CHI)); }
574 : GEN
575 564382 : mfcharpol(GEN CHI) { return gel(CHI,4); }
576 :
577 : /* vz[i+1] = image of (zeta_o)^i in Fp */
578 : static ulong
579 221585 : Qab_Czeta_Fl(long k, GEN vz, ulong C, ulong p)
580 : {
581 : long o;
582 221585 : 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 2508324 : znchareval_i(GEN CHI, long n, GEN ord)
590 2508324 : { 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 14224 : mfchareval(GEN CHI, long n)
595 : {
596 14224 : GEN Pn, C, go = gmfcharorder(CHI);
597 14224 : long k, o = go[2];
598 14224 : if (o == 1) return gen_1;
599 7385 : k = znchareval_i(CHI, n, go);
600 7385 : Pn = mfcharpol(CHI);
601 7385 : C = Qab_zeta(k, o, varn(Pn));
602 7385 : if (typ(C) != t_POL) return C;
603 5327 : return gmodulo(C, Pn);
604 : }
605 : /* d a multiple of ord(CHI); n coprime with char modulus;
606 : * return x s.t. CHI(n) = \zeta_d^x] */
607 : static long
608 3563462 : mfcharevalord(GEN CHI, long n, long d)
609 : {
610 3563462 : 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 373905 : mfcharGL(GEN G, GEN L)
617 : {
618 373905 : GEN o = zncharorder(G,L);
619 373905 : long ord = itou(o), vt = fetch_user_var("t");
620 373905 : return mkvec4(G, L, o, polcyclo(ord,vt));
621 : }
622 : static GEN
623 5796 : mfchartrivial()
624 5796 : { return mfcharGL(znstar0(gen_1,1), cgetg(1,t_COL)); }
625 : /* convert a generic character into an 'mfchar' */
626 : static GEN
627 4046 : get_mfchar(GEN CHI)
628 : {
629 : GEN G, L;
630 4046 : if (typ(CHI) != t_VEC) CHI = znchar(CHI);
631 : else
632 : {
633 889 : long l = lg(CHI);
634 889 : if ((l != 3 && l != 5) || !checkznstar_i(gel(CHI,1)))
635 7 : pari_err_TYPE("checkNF [chi]", CHI);
636 882 : if (l == 5) return CHI;
637 : }
638 3976 : G = gel(CHI,1);
639 3976 : L = gel(CHI,2); if (typ(L) != t_COL) L = znconreylog(G,L);
640 3976 : return mfcharGL(G, L);
641 : }
642 :
643 : /* parse [N], [N,k], [N,k,CHI]. If 'joker' is set, allow wildcard for CHI */
644 : static GEN
645 9205 : checkCHI(GEN NK, long N, int joker)
646 : {
647 : GEN CHI;
648 9205 : if (lg(NK) == 3)
649 700 : CHI = mfchartrivial();
650 : else
651 : {
652 : long i, l;
653 8505 : CHI = gel(NK,3); l = lg(CHI);
654 8505 : if (isintzero(CHI) && joker)
655 4116 : CHI = NULL; /* all character orbits */
656 4389 : else if (isintm1(CHI) && joker > 1)
657 2373 : CHI = gen_m1; /* sum over all character orbits */
658 2016 : else if ((typ(CHI) == t_VEC &&
659 217 : (l == 1 || l != 3 || !checkznstar_i(gel(CHI,1)))) && joker)
660 : {
661 133 : CHI = shallowtrans(CHI); /* list of characters */
662 952 : for (i = 1; i < l; i++) gel(CHI,i) = get_mfchar(gel(CHI,i));
663 : }
664 : else
665 : {
666 1883 : CHI = get_mfchar(CHI); /* single char */
667 1883 : if (N % mfcharmodulus(CHI)) pari_err_TYPE("checkNF [chi]", NK);
668 : }
669 : }
670 9191 : return CHI;
671 : }
672 : /* support half-integral weight */
673 : static void
674 9212 : checkNK2(GEN NK, long *N, long *nk, long *dk, GEN *CHI, int joker)
675 : {
676 9212 : long l = lg(NK);
677 : GEN T;
678 9212 : if (typ(NK) != t_VEC || l < 3 || l > 4) pari_err_TYPE("checkNK", NK);
679 9212 : T = gel(NK,1); if (typ(T) != t_INT) pari_err_TYPE("checkNF [N]", NK);
680 9212 : *N = itos(T); if (*N <= 0) pari_err_TYPE("checkNF [N <= 0]", NK);
681 9212 : T = gel(NK,2);
682 9212 : switch(typ(T))
683 : {
684 5831 : 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 9205 : *CHI = checkCHI(NK, *N, joker);
691 9191 : }
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 68418 : mfcharparity(GEN CHI)
728 : {
729 68418 : if (!CHI) return 1;
730 68418 : 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 76594 : mfchartoprimitive(GEN CHI, long *pF)
735 : {
736 : pari_sp av;
737 : GEN chi, F;
738 76594 : if (!CHI) { if (pF) *pF = 1; return mfchartrivial(); }
739 76594 : av = avma; F = znconreyconductor(gel(CHI,1), gel(CHI,2), &chi);
740 76594 : if (typ(F) == t_INT) set_avma(av);
741 : else
742 : {
743 7861 : CHI = leafcopy(CHI);
744 7861 : gel(CHI,1) = znstar0(F, 1);
745 7861 : gel(CHI,2) = chi;
746 : }
747 76594 : if (pF) *pF = mfcharmodulus(CHI);
748 76594 : return CHI;
749 : }
750 : static long
751 397516 : mfcharconductor(GEN CHI)
752 : {
753 397516 : pari_sp av = avma;
754 397516 : GEN res = znconreyconductor(gel(CHI,1), gel(CHI,2), NULL);
755 397516 : if (typ(res) == t_VEC) res = gel(res, 1);
756 397516 : return gc_long(av, itos(res));
757 : }
758 :
759 : /* Operations on mf closures */
760 : static GEN
761 63007 : tagparams(long t, GEN NK) { return mkvec2(mkvecsmall(t), NK); }
762 : static GEN
763 1155 : 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 10290 : tag(long t, GEN NK, GEN x) { retmkvec2(tagparams(t,NK), x); }
768 : static GEN
769 36393 : tag2(long t, GEN NK, GEN x, GEN y) { retmkvec3(tagparams(t,NK), x,y); }
770 : static GEN
771 16142 : 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 18844 : checkmf_i(GEN F)
778 18844 : { return typ(F) == t_VEC
779 18018 : && lg(F) > 1 && typ(gel(F,1)) == t_VEC
780 13272 : && lg(gel(F,1)) == 3
781 13111 : && typ(gmael(F,1,1)) == t_VECSMALL
782 36862 : && typ(gmael(F,1,2)) == t_VEC; }
783 232477 : long mf_get_type(GEN F) { return gmael(F,1,1)[1]; }
784 184604 : GEN mf_get_gN(GEN F) { return gmael3(F,1,2,1); }
785 139545 : 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 119406 : long mf_get_N(GEN F) { return itou(mf_get_gN(F)); }
789 72030 : long mf_get_k(GEN F)
790 : {
791 72030 : GEN gk = mf_get_gk(F);
792 72030 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
793 72030 : return itou(gk);
794 : }
795 62188 : GEN mf_get_CHI(GEN F) { return gmael3(F,1,2,3); }
796 24409 : GEN mf_get_field(GEN F) { return gmael3(F,1,2,4); }
797 18942 : GEN mf_get_NK(GEN F) { return gmael(F,1,2); }
798 : static void
799 553 : mf_setfield(GEN f, GEN P)
800 : {
801 553 : gel(f,1) = leafcopy(gel(f,1));
802 553 : gmael(f,1,2) = leafcopy(gmael(f,1,2));
803 553 : gmael3(f,1,2,4) = P;
804 553 : }
805 :
806 : /* UTILITY FUNCTIONS */
807 : GEN
808 9086 : mftocol(GEN F, long lim, long d)
809 9086 : { GEN c = mfcoefs_i(F, lim, d); settyp(c,t_COL); return c; }
810 : GEN
811 2114 : mfvectomat(GEN vF, long lim, long d)
812 : {
813 2114 : long j, l = lg(vF);
814 2114 : GEN M = cgetg(l, t_MAT);
815 10381 : for (j = 1; j < l; j++) gel(M,j) = mftocol(gel(vF,j), lim, d);
816 2114 : 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 8834 : c_deflate(long n, long d, GEN v)
868 : {
869 8834 : long i, id, l = n+2;
870 : GEN w;
871 8834 : if (d == 1) return lg(v) == l ? v: vecslice(v, 1, l-1);
872 574 : w = cgetg(l, typ(v));
873 11123 : for (i = id = 1; i < l; i++, id += d) gel(w, i) = gel(v, id);
874 574 : return w;
875 : }
876 :
877 : static void
878 14 : err_cyclo(void)
879 14 : { pari_err_IMPL("changing cyclotomic fields in mf"); }
880 : /* Q(zeta_a) = Q(zeta_b) ? */
881 : static int
882 616 : same_cyc(long a, long b)
883 616 : { return (a == b) || (odd(a) && b == (a<<1)) || (odd(b) && a == (b<<1)); }
884 : /* need to combine elements in Q(CHI1) and Q(CHI2) with result in Q(CHI),
885 : * CHI = CHI1 * CHI2 or CHI / CHI2 times some character of order 2 */
886 : static GEN
887 2723 : chicompat(GEN CHI, GEN CHI1, GEN CHI2)
888 : {
889 2723 : long o1 = mfcharorder(CHI1);
890 2723 : long o2 = mfcharorder(CHI2), O, o;
891 : GEN T1, T2, P, Po;
892 2723 : if (o1 <= 2 && o2 <= 2) return NULL;
893 623 : o = mfcharorder(CHI);
894 623 : Po = mfcharpol(CHI);
895 623 : P = mfcharpol(CHI1);
896 623 : if (o1 == o2)
897 : {
898 21 : if (o1 == o) return NULL;
899 14 : if (!same_cyc(o1,o)) err_cyclo();
900 0 : return mkvec4(P, gen_1,gen_1, Qab_trace_init(o1, o, P, Po));
901 : }
902 602 : O = ulcm(o1, o2);
903 602 : if (!same_cyc(O,o)) err_cyclo();
904 602 : if (O != o1) P = (O == o2)? mfcharpol(CHI2): polcyclo(O, varn(P));
905 602 : T1 = o1 <= 2? gen_1: utoipos(O / o1);
906 602 : T2 = o2 <= 2? gen_1: utoipos(O / o2);
907 602 : return mkvec4(P, T1, T2, O == o? gen_1: Qab_trace_init(O, o, P, Po));
908 : }
909 : static GEN
910 49 : inflatemod(GEN f, long o, GEN P)
911 : {
912 49 : f = lift_shallow(f);
913 49 : return gmodulo(typ(f)==t_POL? RgX_inflate(f,o): f, P);
914 : }
915 : static GEN
916 7 : RgV_inflatemod(GEN x, long o, GEN P)
917 56 : { pari_APPLY_same(inflatemod(gel(x,i), o, P)); }
918 : /* *F a vector of cyclotomic numbers */
919 : static void
920 651 : chicompatlift(GEN T, GEN *F, GEN *G)
921 : {
922 651 : long o1 = itou(gel(T,2)), o2 = itou(gel(T,3));
923 651 : GEN P = gel(T,1);
924 651 : if (o1 != 1) *F = RgV_inflatemod(*F, o1, P);
925 651 : if (o2 != 1 && G) *G = RgV_inflatemod(*G, o2, P);
926 651 : }
927 : static GEN
928 651 : chicompatfix(GEN T, GEN F)
929 : {
930 651 : GEN V = gel(T,4);
931 651 : if (typ(V) == t_VEC) F = gmodulo(QabV_tracerel(V, 0, F), gel(V,1));
932 651 : return F;
933 : }
934 :
935 : static GEN
936 637 : c_mul(long n, long d, GEN S)
937 : {
938 637 : pari_sp av = avma;
939 637 : long nd = n*d;
940 637 : GEN F = gel(S,2), G = gel(S,3);
941 637 : F = mfcoefs_i(F, nd, 1);
942 637 : G = mfcoefs_i(G, nd, 1);
943 637 : if (lg(S) == 5) chicompatlift(gel(S,4),&F,&G);
944 637 : F = c_deflate(n, d, RgV_mul_RgXn(F,G));
945 637 : if (lg(S) == 5) F = chicompatfix(gel(S,4), F);
946 637 : return gc_GEN(av, F);
947 : }
948 : static GEN
949 112 : c_pow(long n, long d, GEN S)
950 : {
951 112 : pari_sp av = avma;
952 112 : long nd = n*d;
953 112 : GEN F = gel(S,2), a = gel(S,3), f = mfcoefs_i(F,nd,1);
954 112 : if (lg(S) == 5) chicompatlift(gel(S,4),&F, NULL);
955 112 : f = RgV_pows_RgXn(f, itos(a));
956 112 : f = c_deflate(n, d, f);
957 112 : if (lg(S) == 5) f = chicompatfix(gel(S,4), f);
958 112 : return gc_GEN(av, f);
959 : }
960 :
961 : /* F * Theta */
962 : static GEN
963 448 : mfmultheta(GEN F)
964 : {
965 448 : if (typ(mf_get_gk(F)) == t_FRAC && mf_get_type(F) == t_MF_DIV)
966 : {
967 154 : GEN T = gel(F,3); /* hopefully mfTheta() */
968 154 : if (mf_get_type(T) == t_MF_THETA && mf_get_N(T) == 4) return gel(F,2);
969 : }
970 294 : return mfmul(F, mfTheta(NULL));
971 : }
972 :
973 : static GEN
974 42 : c_bracket(long n, long d, GEN S)
975 : {
976 42 : pari_sp av = avma;
977 42 : long i, nd = n*d;
978 42 : GEN F = gel(S,2), G = gel(S,3), tF, tG, C, mpow, res, gk, gl;
979 42 : GEN VF = mfcoefs_i(F, nd, 1);
980 42 : GEN VG = mfcoefs_i(G, nd, 1);
981 42 : ulong j, m = itou(gel(S,4));
982 :
983 42 : if (!n)
984 : {
985 14 : if (m > 0) { set_avma(av); return mkvec(gen_0); }
986 7 : return gc_GEN(av, mkvec(gmul(gel(VF, 1), gel(VG, 1))));
987 : }
988 28 : tF = cgetg(nd+2, t_VEC);
989 28 : tG = cgetg(nd+2, t_VEC);
990 28 : res = NULL; gk = mf_get_gk(F); gl = mf_get_gk(G);
991 : /* pow[i,j+1] = i^j */
992 28 : if (lg(S) == 6) chicompatlift(gel(S,5),&VF,&VG);
993 28 : mpow = cgetg(m+2, t_MAT);
994 28 : gel(mpow,1) = const_col(nd, gen_1);
995 56 : for (j = 1; j <= m; j++)
996 : {
997 28 : GEN c = cgetg(nd+1, t_COL);
998 28 : gel(mpow,j+1) = c;
999 245 : for (i = 1; i <= nd; i++) gel(c,i) = muliu(gcoeff(mpow,i,j), i);
1000 : }
1001 28 : C = binomial(gaddgs(gk, m-1), m);
1002 28 : if (odd(m)) C = gneg(C);
1003 84 : for (j = 0; j <= m; j++)
1004 : { /* C = (-1)^(m-j) binom(m+l-1, j) binom(m+k-1,m-j) */
1005 : GEN c;
1006 56 : gel(tF,1) = j == 0? gel(VF,1): gen_0;
1007 56 : gel(tG,1) = j == m? gel(VG,1): gen_0;
1008 56 : gel(tF,2) = gel(VF,2); /* assume nd >= 1 */
1009 56 : gel(tG,2) = gel(VG,2);
1010 518 : for (i = 2; i <= nd; i++)
1011 : {
1012 462 : gel(tF, i+1) = gmul(gcoeff(mpow,i,j+1), gel(VF, i+1));
1013 462 : gel(tG, i+1) = gmul(gcoeff(mpow,i,m-j+1), gel(VG, i+1));
1014 : }
1015 56 : c = gmul(C, c_deflate(n, d, RgV_mul_RgXn(tF, tG)));
1016 56 : res = res? gadd(res, c): c;
1017 56 : if (j < m)
1018 56 : C = gdiv(gmul(C, gmulsg(m-j, gaddgs(gl,m-j-1))),
1019 28 : gmulsg(-(j+1), gaddgs(gk,j)));
1020 : }
1021 28 : if (lg(S) == 6) res = chicompatfix(gel(S,5), res);
1022 28 : return gc_upto(av, res);
1023 : }
1024 : /* linear combination \sum L[j] vecF[j] */
1025 : static GEN
1026 3003 : c_linear(long n, long d, GEN F, GEN L, GEN dL)
1027 : {
1028 3003 : pari_sp av = avma;
1029 3003 : long j, l = lg(L);
1030 3003 : GEN S = NULL;
1031 10724 : for (j = 1; j < l; j++)
1032 : {
1033 7721 : GEN c = gel(L,j);
1034 7721 : if (gequal0(c)) continue;
1035 6965 : c = gmul(c, mfcoefs_i(gel(F,j), n, d));
1036 6965 : S = S? gadd(S,c): c;
1037 : }
1038 3003 : if (!S) return zerovec(n+1);
1039 3003 : if (!is_pm1(dL)) S = gdiv(S, dL);
1040 3003 : return gc_upto(av, S);
1041 : }
1042 :
1043 : /* B_d(T_j Trace^new) as t_MF_BD(t_MF_HECKE(t_MF_NEWTRACE)) or
1044 : * t_MF_HECKE(t_MF_NEWTRACE)
1045 : * or t_MF_NEWTRACE in level N. Set d and j, return t_MF_NEWTRACE component*/
1046 : static GEN
1047 82859 : bhn_parse(GEN f, long *d, long *j)
1048 : {
1049 82859 : long t = mf_get_type(f);
1050 82859 : *d = *j = 1;
1051 82859 : if (t == t_MF_BD) { *d = itos(gel(f,3)); f = gel(f,2); t = mf_get_type(f); }
1052 82859 : if (t == t_MF_HECKE) { *j = gel(f,2)[1]; f = gel(f,3); }
1053 82859 : return f;
1054 : }
1055 : /* f as above, return the t_MF_NEWTRACE component */
1056 : static GEN
1057 32571 : bhn_newtrace(GEN f)
1058 : {
1059 32571 : long t = mf_get_type(f);
1060 32571 : if (t == t_MF_BD) { f = gel(f,2); t = mf_get_type(f); }
1061 32571 : if (t == t_MF_HECKE) f = gel(f,3);
1062 32571 : return f;
1063 : }
1064 : static int
1065 3997 : ok_bhn_linear(GEN vf)
1066 : {
1067 3997 : long i, N0 = 0, l = lg(vf);
1068 : GEN CHI, gk;
1069 3997 : if (l == 1) return 1;
1070 3997 : gk = mf_get_gk(gel(vf,1));
1071 3997 : CHI = mf_get_CHI(gel(vf,1));
1072 27370 : for (i = 1; i < l; i++)
1073 : {
1074 25711 : GEN f = bhn_newtrace(gel(vf,i));
1075 25711 : long N = mf_get_N(f);
1076 25711 : if (mf_get_type(f) != t_MF_NEWTRACE) return 0;
1077 23373 : if (N < N0) return 0; /* largest level must come last */
1078 23373 : N0 = N;
1079 23373 : if (!gequal(gk,mf_get_gk(f))) return 0; /* same k */
1080 23373 : if (!gequal(gel(mf_get_CHI(f),2), gel(CHI,2))) return 0; /* same CHI */
1081 : }
1082 1659 : return 1;
1083 : }
1084 :
1085 : /* vF not empty, same hypotheses as bhnmat_extend */
1086 : static GEN
1087 6965 : bhnmat_extend_nocache(GEN M, long N, long n, long d, GEN vF)
1088 : {
1089 : cachenew_t cache;
1090 6965 : long l = lg(vF);
1091 : GEN f;
1092 6965 : if (l == 1) return M? M: cgetg(1, t_MAT);
1093 6860 : f = bhn_newtrace(gel(vF,1)); /* N.B. mf_get_N(f) divides N */
1094 6860 : init_cachenew(&cache, n*d, N, f);
1095 6860 : M = bhnmat_extend(M, n, d, vF, &cache);
1096 6860 : dbg_cachenew(&cache); return M;
1097 : }
1098 : /* c_linear of "bhn" mf closures, same hypotheses as bhnmat_extend */
1099 : static GEN
1100 2261 : c_linear_bhn(long n, long d, GEN F)
1101 : {
1102 : pari_sp av;
1103 2261 : GEN M, v, vF = gel(F,2), L = gel(F,3), dL = gel(F,4);
1104 2261 : if (lg(L) == 1) return zerovec(n+1);
1105 2261 : av = avma;
1106 2261 : M = bhnmat_extend_nocache(NULL, mf_get_N(F), n, d, vF);
1107 2261 : v = RgM_RgC_mul(M,L); settyp(v, t_VEC);
1108 2261 : if (!is_pm1(dL)) v = gdiv(v, dL);
1109 2261 : return gc_upto(av, v);
1110 : }
1111 :
1112 : /* c in K, K := Q[X]/(T) vz = vector of consecutive powers of root z of T
1113 : * attached to an embedding s: K -> C. Return s(c) in C */
1114 : static GEN
1115 84658 : Rg_embed1(GEN c, GEN vz)
1116 : {
1117 84658 : long t = typ(c);
1118 84658 : if (t == t_POLMOD) { c = gel(c,2); t = typ(c); }
1119 84658 : if (t == t_POL) c = RgX_RgV_eval(c, vz);
1120 84658 : return c;
1121 : }
1122 : /* return s(x) in C[X] */
1123 : static GEN
1124 14203 : RgX_embed1(GEN x, GEN vz)
1125 42042 : { pari_APPLY_pol(Rg_embed1(gel(x,i), vz)); }
1126 : /* return s(x) in C^n */
1127 : static GEN
1128 798 : vecembed1(GEN x, GEN vz)
1129 39858 : { pari_APPLY_same(Rg_embed1(gel(x,i), vz)); }
1130 : /* P in L = K[X]/(U), K = Q[t]/T; s an embedding of K -> C attached
1131 : * to a root of T, extended to an embedding of L -> C attached to a root
1132 : * of s(U); vT powers of the root of T, vU powers of the root of s(U).
1133 : * Return s(P) in C^n */
1134 : static GEN
1135 13328 : Rg_embed2(GEN P, long vt, GEN vT, GEN vU)
1136 : {
1137 13328 : P = liftpol_shallow(P);
1138 13328 : if (typ(P) != t_POL) return P;
1139 13300 : if (varn(P) == vt) return Rg_embed1(P, vT);
1140 13293 : return Rg_embed1(RgX_embed1(P, vT), vU); /* varn(P) == vx */
1141 : }
1142 : static GEN
1143 42 : vecembed2(GEN x, long vt, GEN vT, GEN vU)
1144 1050 : { pari_APPLY_same(Rg_embed2(gel(x,i), vt, vT, vU)); }
1145 : static GEN
1146 532 : RgX_embed2(GEN x, long vt, GEN vT, GEN vU)
1147 3724 : { pari_APPLY_pol(Rg_embed2(gel(x,i), vt, vT, vU)); }
1148 : /* embed polynomial f in variable 0 [ may be a scalar ], E from getembed */
1149 : static GEN
1150 1687 : RgX_embed(GEN f, GEN E)
1151 : {
1152 : GEN vT;
1153 1687 : if (typ(f) != t_POL || varn(f) != 0) return mfembed(E, f);
1154 1645 : if (lg(E) == 1) return f;
1155 1407 : vT = gel(E,2);
1156 1407 : if (lg(E) == 3)
1157 875 : f = RgX_embed1(f, vT);
1158 : else
1159 532 : f = RgX_embed2(f, varn(gel(E,1)), vT, gel(E,3));
1160 1407 : return f;
1161 : }
1162 : /* embed vector, E from getembed */
1163 : GEN
1164 1715 : mfvecembed(GEN E, GEN v)
1165 : {
1166 : GEN vT;
1167 1715 : 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 100831 : mfembed(GEN E, GEN f)
1192 : {
1193 : GEN vT;
1194 100831 : if (lg(E) == 1) return f;
1195 13587 : vT = gel(E,2);
1196 13587 : if (lg(E) == 3)
1197 4459 : f = Rg_embed1(f, vT);
1198 : else
1199 9128 : f = Rg_embed2(f, varn(gel(E,1)), vT, gel(E,3));
1200 13587 : return f;
1201 : }
1202 : /* vector of the sigma(f), sigma in vE */
1203 : static GEN
1204 364 : RgX_embedall(GEN f, GEN vE)
1205 : {
1206 364 : long i, l = lg(vE);
1207 : GEN v;
1208 364 : if (l == 2) return RgX_embed(f, gel(vE,1));
1209 35 : v = cgetg(l, t_VEC);
1210 105 : for (i = 1; i < l; i++) gel(v,i) = RgX_embed(f, gel(vE,i));
1211 35 : return v;
1212 : }
1213 : /* matrix whose colums are the sigma(v), sigma in vE */
1214 : static GEN
1215 350 : RgC_embedall(GEN v, GEN vE)
1216 : {
1217 350 : long j, l = lg(vE);
1218 350 : GEN M = cgetg(l, t_MAT);
1219 875 : for (j = 1; j < l; j++) gel(M,j) = mfvecembed(gel(vE,j), v);
1220 350 : return M;
1221 : }
1222 : /* vector of the sigma(v), sigma in vE */
1223 : static GEN
1224 4907 : Rg_embedall_i(GEN v, GEN vE)
1225 : {
1226 4907 : long j, l = lg(vE);
1227 4907 : GEN M = cgetg(l, t_VEC);
1228 14735 : for (j = 1; j < l; j++) gel(M,j) = mfembed(gel(vE,j), v);
1229 4907 : return M;
1230 : }
1231 : /* vector of the sigma(v), sigma in vE; if #vE == 1, return v */
1232 : static GEN
1233 95154 : Rg_embedall(GEN v, GEN vE)
1234 95154 : { return (lg(vE) == 2)? mfembed(gel(vE,1), v): Rg_embedall_i(v, vE); }
1235 :
1236 : static GEN
1237 833 : c_div_i(long n, GEN S)
1238 : {
1239 833 : GEN F = gel(S,2), G = gel(S,3);
1240 : GEN a0, a0i, H;
1241 833 : F = mfcoefs_i(F, n, 1);
1242 833 : G = mfcoefs_i(G, n, 1);
1243 833 : if (lg(S) == 5) chicompatlift(gel(S,4),&F,&G);
1244 833 : F = RgV_to_ser_full(F);
1245 833 : G = RgV_to_ser_full(G);
1246 833 : a0 = polcoef_i(G, 0, -1); /* != 0 */
1247 833 : if (gequal1(a0)) a0 = a0i = NULL;
1248 : else
1249 : {
1250 602 : a0i = ginv(a0);
1251 602 : G = gmul(ser_unscale(G,a0), a0i);
1252 602 : F = gmul(ser_unscale(F,a0), a0i);
1253 : }
1254 833 : H = gdiv(F, G);
1255 833 : if (a0) H = ser_unscale(H,a0i);
1256 833 : H = sertovecslice(H, n);
1257 833 : if (lg(S) == 5) H = chicompatfix(gel(S,4), H);
1258 833 : return H;
1259 : }
1260 : static GEN
1261 833 : c_div(long n, long d, GEN S)
1262 : {
1263 833 : pari_sp av = avma;
1264 833 : GEN D = (d==1)? c_div_i(n, S): c_deflate(n, d, c_div_i(n*d, S));
1265 833 : return gc_GEN(av, D);
1266 : }
1267 :
1268 : static GEN
1269 35 : c_shift(long n, long d, GEN F, GEN gsh)
1270 : {
1271 35 : pari_sp av = avma;
1272 : GEN vF;
1273 35 : long sh = itos(gsh), n1 = n*d + sh;
1274 35 : if (n1 < 0) return zerovec(n+1);
1275 35 : vF = mfcoefs_i(F, n1, 1);
1276 35 : if (sh < 0) vF = shallowconcat(zerovec(-sh), vF);
1277 35 : else vF = vecslice(vF, sh+1, n1+1);
1278 35 : return gc_GEN(av, c_deflate(n, d, vF));
1279 : }
1280 :
1281 : static GEN
1282 175 : c_deriv(long n, long d, GEN F, GEN gm)
1283 : {
1284 175 : pari_sp av = avma;
1285 175 : GEN V = mfcoefs_i(F, n, d), res;
1286 175 : long i, m = itos(gm);
1287 175 : if (!m) return V;
1288 175 : res = cgetg(n+2, t_VEC); gel(res,1) = gen_0;
1289 175 : if (m < 0)
1290 49 : { for (i=1; i <= n; i++) gel(res, i+1) = gdiv(gel(V, i+1), powuu(i,-m)); }
1291 : else
1292 2457 : { for (i=1; i <= n; i++) gel(res, i+1) = gmul(gel(V,i+1), powuu(i,m)); }
1293 175 : return gc_upto(av, res);
1294 : }
1295 :
1296 : static GEN
1297 14 : c_derivE2(long n, long d, GEN F, GEN gm)
1298 : {
1299 14 : pari_sp av = avma;
1300 : GEN VF, VE, res, tmp, gk;
1301 14 : long i, m = itos(gm), nd;
1302 14 : if (m == 0) return mfcoefs_i(F, n, d);
1303 14 : nd = n*d;
1304 14 : VF = mfcoefs_i(F, nd, 1); VE = mfcoefs_i(mfEk(2), nd, 1);
1305 14 : gk = mf_get_gk(F);
1306 14 : if (m == 1)
1307 : {
1308 7 : res = cgetg(n+2, t_VEC);
1309 56 : for (i = 0; i <= n; i++) gel(res, i+1) = gmulsg(i, gel(VF, i*d+1));
1310 7 : tmp = c_deflate(n, d, RgV_mul_RgXn(VF, VE));
1311 7 : return gc_upto(av, gsub(res, gmul(gdivgu(gk, 12), tmp)));
1312 : }
1313 : else
1314 : {
1315 : long j;
1316 35 : for (j = 1; j <= m; j++)
1317 : {
1318 28 : tmp = RgV_mul_RgXn(VF, VE);
1319 140 : for (i = 0; i <= nd; i++) gel(VF, i+1) = gmulsg(i, gel(VF, i+1));
1320 28 : VF = gsub(VF, gmul(gdivgu(gaddgs(gk, 2*(j-1)), 12), tmp));
1321 : }
1322 7 : return gc_GEN(av, c_deflate(n, d, VF));
1323 : }
1324 : }
1325 :
1326 : /* Twist by the character (D/.) */
1327 : static GEN
1328 168 : c_twist(long n, long d, GEN F, GEN D)
1329 : {
1330 168 : pari_sp av = avma;
1331 168 : GEN v = mfcoefs_i(F, n, d), z = cgetg(n+2, t_VEC);
1332 : long i;
1333 994 : for (i = 0; i <= n; i++)
1334 : {
1335 : long s;
1336 826 : GEN a = gel(v, i+1);
1337 826 : if (d == 1) s = krois(D, i);
1338 : else
1339 : {
1340 266 : pari_sp av2 = avma;
1341 266 : s = kronecker(D, muluu(i, d)); set_avma(av2);
1342 : }
1343 826 : switch(s)
1344 : {
1345 259 : case 1: a = gcopy(a); break;
1346 252 : case -1: a = gneg(a); break;
1347 315 : default: a = gen_0; break;
1348 : }
1349 826 : gel(z, i+1) = a;
1350 : }
1351 168 : return gc_upto(av, z);
1352 : }
1353 :
1354 : /* form F given by closure, compute T(n)(F) as closure */
1355 : static GEN
1356 1050 : c_hecke(long m, long l, GEN DATA, GEN F)
1357 : {
1358 1050 : pari_sp av = avma;
1359 1050 : return gc_GEN(av, hecke_i(m, l, NULL, F, DATA));
1360 : }
1361 : static GEN
1362 140 : c_const(long n, long d, GEN C)
1363 : {
1364 140 : GEN V = zerovec(n+1);
1365 140 : long i, j, l = lg(C);
1366 140 : if (l > d*n+2) l = d*n+2;
1367 189 : for (i = j = 1; i < l; i+=d, j++) gel(V, j) = gcopy(gel(C,i));
1368 140 : return V;
1369 : }
1370 :
1371 : /* m > 0 */
1372 : static GEN
1373 525 : eta3_ZXn(long m)
1374 : {
1375 525 : long l = m+2, n, k;
1376 525 : GEN P = cgetg(l,t_POL);
1377 525 : P[1] = evalsigne(1)|evalvarn(0);
1378 7245 : for (n = 2; n < l; n++) gel(P,n) = gen_0;
1379 525 : for (n = k = 0;; n++)
1380 : {
1381 2891 : if (k + n >= m) { setlg(P, k+3); return P; }
1382 2366 : k += n;
1383 : /* now k = n(n+1) / 2 */
1384 2366 : gel(P, k+2) = odd(n)? utoineg(2*n+1): utoipos(2*n+1);
1385 : }
1386 : }
1387 :
1388 : static GEN
1389 539 : c_delta(long n, long d)
1390 : {
1391 539 : pari_sp ltop = avma;
1392 539 : long N = n*d;
1393 : GEN e;
1394 539 : if (!N) return mkvec(gen_0);
1395 525 : e = eta3_ZXn(N);
1396 525 : e = ZXn_sqr(e,N);
1397 525 : e = ZXn_sqr(e,N);
1398 525 : e = ZXn_sqr(e,N); /* eta(x)^24 */
1399 525 : settyp(e, t_VEC);
1400 525 : gel(e,1) = gen_0; /* Delta(x) = x*eta(x)^24 as a t_VEC */
1401 525 : return gc_GEN(ltop, c_deflate(n, d, e));
1402 : }
1403 :
1404 : /* return s(d) such that s|f <=> d | f^2 */
1405 : static long
1406 56 : mysqrtu(ulong d)
1407 : {
1408 56 : GEN fa = myfactoru(d), P = gel(fa,1), E = gel(fa,2);
1409 56 : long l = lg(P), i, s = 1;
1410 140 : for (i = 1; i < l; i++) s *= upowuu(P[i], (E[i]+1)>>1);
1411 56 : return s;
1412 : }
1413 : static GEN
1414 1911 : c_theta(long n, long d, GEN psi)
1415 : {
1416 1911 : long lim = usqrt(n*d), F = mfcharmodulus(psi), par = mfcharparity(psi);
1417 1911 : long f, d2 = d == 1? 1: mysqrtu(d);
1418 1911 : GEN V = zerovec(n + 1);
1419 8414 : for (f = d2; f <= lim; f += d2)
1420 6503 : if (ugcd(F, f) == 1)
1421 : {
1422 6496 : pari_sp av = avma;
1423 6496 : GEN c = mfchareval(psi, f);
1424 6496 : gel(V, f*f/d + 1) = gc_upto(av, par < 0? gmulgu(c,2*f): gmul2n(c,1));
1425 : }
1426 1911 : if (F == 1) gel(V, 1) = gen_1;
1427 1911 : return V; /* no GC needed */
1428 : }
1429 :
1430 : static GEN
1431 203 : c_etaquo(long n, long d, GEN eta, GEN gs)
1432 : {
1433 203 : pari_sp av = avma;
1434 203 : long s = itos(gs), nd = n*d, nds = nd - s + 1;
1435 : GEN c;
1436 203 : if (nds <= 0) return zerovec(n+1);
1437 182 : c = RgX_to_RgC(eta_product_ZXn(eta, nds), nds); settyp(c, t_VEC);
1438 182 : if (s > 0) c = shallowconcat(zerovec(s), c);
1439 182 : return gc_GEN(av, c_deflate(n, d, c));
1440 : }
1441 :
1442 : static GEN
1443 77 : c_ell(long n, long d, GEN E)
1444 : {
1445 77 : pari_sp av = avma;
1446 : GEN v;
1447 77 : if (d == 1) return gconcat(gen_0, ellan(E, n));
1448 7 : v = vec_prepend(ellan(E, n*d), gen_0);
1449 7 : return gc_GEN(av, c_deflate(n, d, v));
1450 : }
1451 :
1452 : static GEN
1453 21 : c_cusptrace(long n, long d, GEN F)
1454 : {
1455 21 : pari_sp av = avma;
1456 21 : GEN D = gel(F,2), res = cgetg(n+2, t_VEC);
1457 21 : long i, N = mf_get_N(F), k = mf_get_k(F);
1458 21 : gel(res, 1) = gen_0;
1459 140 : for (i = 1; i <= n; i++)
1460 119 : gel(res, i+1) = mfcusptrace_i(N, k, i*d, mydivisorsu(i*d), D);
1461 21 : return gc_GEN(av, res);
1462 : }
1463 :
1464 : static GEN
1465 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 gc_GEN(av, v);
1474 : }
1475 :
1476 : static GEN
1477 7469 : c_Bd(long n, long d, GEN F, GEN A)
1478 : {
1479 7469 : pari_sp av = avma;
1480 7469 : long a = itou(A), ad = ugcd(a,d), aad = a/ad, i, j;
1481 7469 : GEN w, v = mfcoefs_i(F, n/aad, d/ad);
1482 7469 : if (a == 1) return v;
1483 7469 : n++; w = zerovec(n);
1484 213115 : for (i = j = 1; j <= n; i++, j += aad) gel(w,j) = gcopy(gel(v,i));
1485 7469 : return gc_upto(av, w);
1486 : }
1487 :
1488 : static GEN
1489 5579 : c_dihedral(long n, long d, GEN F)
1490 : {
1491 5579 : pari_sp av = avma;
1492 5579 : GEN CHI = mf_get_CHI(F);
1493 5579 : GEN w = gel(F,3), V = dihan(gel(F,2), w, gel(F,4), mfcharorder(CHI), n*d);
1494 5579 : GEN Tinit = gel(w,3), Pm = gel(Tinit,1);
1495 5579 : GEN A = c_deflate(n, d, V);
1496 5579 : if (degpol(Pm) == 1 || RgV_is_ZV(A)) return gc_GEN(av, A);
1497 1043 : return gc_upto(av, gmodulo(A, Pm));
1498 : }
1499 :
1500 : static GEN
1501 343 : c_mfEH(long n, long d, GEN F)
1502 : {
1503 343 : pari_sp av = avma;
1504 : GEN v, M, A;
1505 343 : long i, r = mf_get_r(F);
1506 343 : if (n == 1)
1507 14 : return gc_GEN(av, mkvec2(mfEHcoef(r,0),mfEHcoef(r,d)));
1508 : /* speedup mfcoef */
1509 329 : if (r == 1)
1510 : {
1511 70 : v = cgetg(n+2, t_VEC);
1512 70 : gel(v,1) = sstoQ(-1,12);
1513 83258 : for (i = 1; i <= n; i++)
1514 : {
1515 83188 : long id = i*d, a = id & 3;
1516 83188 : gel(v,i+1) = (a==1 || a==2)? gen_0: uutoQ(hclassno6u(id), 6);
1517 : }
1518 70 : return v; /* no GC needed */
1519 : }
1520 259 : M = mfEHmat(n*d+1,r);
1521 259 : if (d > 1)
1522 : {
1523 35 : long l = lg(M);
1524 119 : for (i = 1; i < l; i++) gel(M,i) = c_deflate(n, d, gel(M,i));
1525 : }
1526 259 : A = gel(F,2); /* [num(B), den(B)] */
1527 259 : v = RgC_Rg_div(RgM_RgC_mul(M, gel(A,1)), gel(A,2));
1528 259 : settyp(v,t_VEC); return gc_upto(av, v);
1529 : }
1530 :
1531 : static GEN
1532 11277 : c_mfeisen(long n, long d, GEN F)
1533 : {
1534 11277 : pari_sp av = avma;
1535 11277 : GEN v, vchi, E0, P, T, CHI, gk = mf_get_gk(F);
1536 : long i, k;
1537 11277 : if (typ(gk) != t_INT) return c_mfEH(n, d, F);
1538 10934 : k = itou(gk);
1539 10934 : vchi = gel(F,2);
1540 10934 : E0 = gel(vchi,1);
1541 10934 : T = gel(vchi,2);
1542 10934 : P = gel(T,1);
1543 10934 : CHI = gel(vchi,3);
1544 10934 : v = cgetg(n+2, t_VEC);
1545 10934 : gel(v, 1) = gcopy(E0); /* E(0) */
1546 10934 : if (lg(vchi) == 5)
1547 : { /* E_k(chi1,chi2) */
1548 8841 : GEN CHI2 = gel(vchi,4), F3 = gel(F,3);
1549 8841 : long ord = F3[1], j = F3[2];
1550 508690 : for (i = 1; i <= n; i++) gel(v, i+1) = sigchi2(k, CHI, CHI2, i*d, ord);
1551 8841 : 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 10934 : if (degpol(P) != 1 && !RgV_is_QV(v)) return gc_upto(av, gmodulo(v, P));
1558 8001 : return gc_GEN(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 = gc_INT(av, S);
1585 : }
1586 : }
1587 28 : return gdiv(S, dS);
1588 : }
1589 :
1590 : /* Returns vector of coeffs from F[0], F[d], ..., F[d*n] */
1591 : static GEN
1592 38059 : mfcoefs_i(GEN F, long n, long d)
1593 : {
1594 38059 : if (n < 0) return gen_0;
1595 38059 : switch(mf_get_type(F))
1596 : {
1597 140 : case t_MF_CONST: return c_const(n, d, gel(F,2));
1598 11277 : case t_MF_EISEN: return c_mfeisen(n, d, F);
1599 882 : case t_MF_Ek: return c_Ek(n, d, F);
1600 539 : case t_MF_DELTA: return c_delta(n, d);
1601 1645 : case t_MF_THETA: return c_theta(n, d, gel(F,2));
1602 203 : case t_MF_ETAQUO: return c_etaquo(n, d, gel(F,2), gel(F,3));
1603 77 : case t_MF_ELL: return c_ell(n, d, gel(F,2));
1604 637 : case t_MF_MUL: return c_mul(n, d, F);
1605 112 : case t_MF_POW: return c_pow(n, d, F);
1606 42 : case t_MF_BRACKET: return c_bracket(n, d, F);
1607 3003 : case t_MF_LINEAR: return c_linear(n, d, gel(F,2), gel(F,3), gel(F,4));
1608 2261 : case t_MF_LINEAR_BHN: return c_linear_bhn(n, d, F);
1609 833 : case t_MF_DIV: return c_div(n, d, F);
1610 35 : case t_MF_SHIFT: return c_shift(n, d, gel(F,2), gel(F,3));
1611 175 : case t_MF_DERIV: return c_deriv(n, d, gel(F,2), gel(F,3));
1612 14 : case t_MF_DERIVE2: return c_derivE2(n, d, gel(F,2), gel(F,3));
1613 168 : case t_MF_TWIST: return c_twist(n, d, gel(F,2), gel(F,3));
1614 1050 : case t_MF_HECKE: return c_hecke(n, d, gel(F,2), gel(F,3));
1615 7469 : 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 6034 : space_is_cusp(long space) { return space != mf_FULL && space != mf_EISEN; }
1628 : /* safe with flraw mf */
1629 : static GEN
1630 2597 : mfcoefs_mf(GEN mf, long n, long d)
1631 : {
1632 2597 : GEN MS, ME, E = MF_get_E(mf), S = MF_get_S(mf), M = MF_get_M(mf);
1633 2597 : long lE = lg(E), lS = lg(S), l = lE+lS-1;
1634 :
1635 2597 : if (l == 1) return cgetg(1, t_MAT);
1636 2485 : 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 2464 : ME = (lE == 1)? cgetg(1, t_MAT): mfvectomat(E, n, d);
1639 2464 : if (lS == 1)
1640 455 : 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 2464 : return shallowconcat(ME,MS);
1658 : }
1659 : GEN
1660 3892 : mfcoefs(GEN F, long n, long d)
1661 : {
1662 3892 : if (!checkmf_i(F))
1663 : {
1664 42 : pari_sp av = avma;
1665 42 : GEN mf = checkMF_i(F); if (!mf) pari_err_TYPE("mfcoefs", F);
1666 42 : return gc_GEN(av, mfcoefs_mf(mf,n,d));
1667 : }
1668 3850 : if (d <= 0) pari_err_DOMAIN("mfcoefs", "d", "<=", gen_0, stoi(d));
1669 3850 : if (n < 0) return cgetg(1, t_VEC);
1670 3850 : return mfcoefs_i(F, n, d);
1671 : }
1672 :
1673 : /* assume k >= 0 */
1674 : static GEN
1675 455 : mfak_i(GEN F, long k)
1676 : {
1677 455 : if (!k) return gel(mfcoefs_i(F,0,1), 1);
1678 294 : return gel(mfcoefs_i(F,1,k), 2);
1679 : }
1680 : GEN
1681 301 : mfcoef(GEN F, long n)
1682 : {
1683 301 : pari_sp av = avma;
1684 301 : if (!checkmf_i(F)) pari_err_TYPE("mfcoef",F);
1685 301 : return n < 0? gen_0: gc_GEN(av, mfak_i(F, n));
1686 : }
1687 :
1688 : static GEN
1689 126 : paramconst() { return tagparams(t_MF_CONST, mkNK(1,0,mfchartrivial())); }
1690 : static GEN
1691 84 : mftrivial(void) { retmkvec2(paramconst(), cgetg(1,t_VEC)); }
1692 : static GEN
1693 42 : mf1(void) { retmkvec2(paramconst(), mkvec(gen_1)); }
1694 :
1695 : /* induce mfchar CHI to G */
1696 : static GEN
1697 307937 : induce(GEN G, GEN CHI)
1698 : {
1699 : GEN o, chi;
1700 307937 : 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 7161 : 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 4004 : mfsamefield(GEN T, GEN P, GEN Q)
1818 : {
1819 4004 : if (degpol(P) == 1) return Q;
1820 672 : if (degpol(Q) == 1) return P;
1821 581 : if (!gequal(P,Q)) pari_err_TYPE("mfsamefield [different fields]",mkvec2(P,Q));
1822 574 : if (T) err_cyclo();
1823 574 : return P;
1824 : }
1825 :
1826 : GEN
1827 455 : mfmul(GEN f, GEN g)
1828 : {
1829 455 : pari_sp av = avma;
1830 : GEN T, N, K, NK, CHI, CHIf, CHIg;
1831 455 : if (!checkmf_i(f)) pari_err_TYPE("mfmul",f);
1832 455 : if (!checkmf_i(g)) pari_err_TYPE("mfmul",g);
1833 455 : N = lcmii(mf_get_gN(f), mf_get_gN(g));
1834 455 : K = gadd(mf_get_gk(f), mf_get_gk(g));
1835 455 : CHIf = mf_get_CHI(f);
1836 455 : CHIg = mf_get_CHI(g);
1837 455 : CHI = mfchiadjust(mfcharmul(CHIf,CHIg), K, itos(N));
1838 455 : T = chicompat(CHI, CHIf, CHIg);
1839 455 : NK = mkgNK(N, K, CHI, mfsamefield(T, mf_get_field(f), mf_get_field(g)));
1840 448 : return gc_GEN(av, T? tag3(t_MF_MUL,NK,f,g,T): tag2(t_MF_MUL,NK,f,g));
1841 : }
1842 : GEN
1843 77 : mfpow(GEN f, long n)
1844 : {
1845 77 : pari_sp av = avma;
1846 : GEN T, KK, NK, gn, CHI, CHIf;
1847 77 : if (!checkmf_i(f)) pari_err_TYPE("mfpow",f);
1848 77 : if (!n) return mf1();
1849 77 : if (n == 1) return gcopy(f);
1850 77 : KK = gmulsg(n,mf_get_gk(f));
1851 77 : gn = stoi(n);
1852 77 : CHIf = mf_get_CHI(f);
1853 77 : CHI = mfchiadjust(mfcharpow(CHIf,gn), KK, mf_get_N(f));
1854 77 : T = chicompat(CHI, CHIf, CHIf);
1855 70 : NK = mkgNK(mf_get_gN(f), KK, CHI, mf_get_field(f));
1856 70 : return gc_GEN(av, T? tag3(t_MF_POW,NK,f,gn,T): tag2(t_MF_POW,NK,f,gn));
1857 : }
1858 : GEN
1859 28 : mfbracket(GEN f, GEN g, long m)
1860 : {
1861 28 : pari_sp av = avma;
1862 : GEN T, N, K, NK, CHI, CHIf, CHIg;
1863 28 : if (!checkmf_i(f)) pari_err_TYPE("mfbracket",f);
1864 28 : if (!checkmf_i(g)) pari_err_TYPE("mfbracket",g);
1865 28 : if (m < 0) pari_err_TYPE("mfbracket [m<0]",stoi(m));
1866 28 : K = gaddgs(gadd(mf_get_gk(f), mf_get_gk(g)), 2*m);
1867 28 : if (gsigne(K) < 0) pari_err_IMPL("mfbracket for this form");
1868 28 : N = lcmii(mf_get_gN(f), mf_get_gN(g));
1869 28 : CHIf = mf_get_CHI(f);
1870 28 : CHIg = mf_get_CHI(g);
1871 28 : CHI = mfcharmul(CHIf, CHIg);
1872 28 : CHI = mfchiadjust(CHI, K, itou(N));
1873 28 : T = chicompat(CHI, CHIf, CHIg);
1874 28 : NK = mkgNK(N, K, CHI, mfsamefield(T, mf_get_field(f), mf_get_field(g)));
1875 56 : return gc_GEN(av, T? tag4(t_MF_BRACKET, NK, f, g, utoi(m), T)
1876 28 : : tag3(t_MF_BRACKET, NK, f, g, utoi(m)));
1877 : }
1878 :
1879 : /* remove 0 entries in L */
1880 : static int
1881 1876 : mflinear_strip(GEN *pF, GEN *pL)
1882 : {
1883 1876 : pari_sp av = avma;
1884 1876 : GEN F = *pF, L = *pL;
1885 1876 : long i, j, l = lg(L);
1886 1876 : GEN F2 = cgetg(l, t_VEC), L2 = cgetg(l, t_VEC);
1887 11046 : for (i = j = 1; i < l; i++)
1888 : {
1889 9170 : if (gequal0(gel(L,i))) continue;
1890 4445 : gel(F2,j) = gel(F,i);
1891 4445 : gel(L2,j) = gel(L,i); j++;
1892 : }
1893 1876 : if (j == l) set_avma(av);
1894 : else
1895 : {
1896 581 : setlg(F2,j); *pF = F2;
1897 581 : setlg(L2,j); *pL = L2;
1898 : }
1899 1876 : return (j > 1);
1900 : }
1901 : static GEN
1902 6797 : taglinear_i(long t, GEN NK, GEN F, GEN L)
1903 : {
1904 : GEN dL;
1905 6797 : L = Q_remove_denom(L, &dL); if (!dL) dL = gen_1;
1906 6797 : return tag3(t, NK, F, L, dL);
1907 : }
1908 : static GEN
1909 2786 : taglinear(GEN NK, GEN F, GEN L)
1910 : {
1911 2786 : long t = ok_bhn_linear(F)? t_MF_LINEAR_BHN: t_MF_LINEAR;
1912 2786 : return taglinear_i(t, NK, F, L);
1913 : }
1914 : /* assume F has parameters NK = [N,K,CHI] */
1915 : static GEN
1916 490 : mflinear_i(GEN NK, GEN F, GEN L)
1917 : {
1918 490 : if (!mflinear_strip(&F,&L)) return mftrivial();
1919 490 : return taglinear(NK, F,L);
1920 : }
1921 : static GEN
1922 735 : mflinear_bhn(GEN mf, GEN L)
1923 : {
1924 : long i, l;
1925 735 : GEN P, NK, F = MF_get_S(mf);
1926 735 : if (!mflinear_strip(&F,&L)) return mftrivial();
1927 728 : l = lg(L); P = pol_x(1);
1928 3262 : for (i = 1; i < l; i++)
1929 : {
1930 2534 : GEN c = gel(L,i);
1931 2534 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1)
1932 602 : P = mfsamefield(NULL, P, gel(c,1));
1933 : }
1934 728 : NK = mkgNK(MF_get_gN(mf), MF_get_gk(mf), MF_get_CHI(mf), P);
1935 728 : 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 1015 : tobasis(GEN mf, GEN F, GEN L)
2026 : {
2027 1015 : if (checkmf_i(L) && mf) return mftobasis(mf, L, 0);
2028 1008 : if (typ(F) != t_VEC) pari_err_TYPE("mflinear",F);
2029 1008 : if (!is_vec_t(typ(L))) pari_err_TYPE("mflinear",L);
2030 1008 : if (lg(L) != lg(F)) pari_err_DIM("mflinear");
2031 1008 : return L;
2032 : }
2033 : GEN
2034 1057 : mflinear(GEN F, GEN L)
2035 : {
2036 1057 : pari_sp av = avma;
2037 1057 : GEN G, NK, P, mf = checkMF_i(F), N = NULL, K = NULL, CHI = NULL;
2038 : long i, l;
2039 1057 : if (mf)
2040 : {
2041 714 : GEN gk = MF_get_gk(mf);
2042 714 : F = MF_get_basis(F);
2043 714 : if (typ(gk) != t_INT)
2044 42 : return gc_GEN(av, mflineardiv_linear(F, L, 1));
2045 672 : if (itou(gk) > 1 && space_is_cusp(MF_get_space(mf)))
2046 : {
2047 455 : L = tobasis(mf, F, L);
2048 455 : return gc_GEN(av, mflinear_bhn(mf, L));
2049 : }
2050 : }
2051 560 : L = tobasis(mf, F, L);
2052 560 : if (!mflinear_strip(&F,&L)) return mftrivial();
2053 :
2054 553 : l = lg(F);
2055 553 : if (l == 2 && gequal1(gel(L,1))) return gc_GEN(av, gel(F,1));
2056 294 : P = pol_x(1);
2057 931 : for (i = 1; i < l; i++)
2058 : {
2059 644 : GEN f = gel(F,i), c = gel(L,i), Ni, Ki;
2060 644 : if (!checkmf_i(f)) pari_err_TYPE("mflinear", f);
2061 644 : Ni = mf_get_gN(f); N = N? lcmii(N, Ni): Ni;
2062 644 : Ki = mf_get_gk(f);
2063 644 : if (!K) K = Ki;
2064 350 : else if (!gequal(K, Ki))
2065 7 : pari_err_TYPE("mflinear [different weights]", mkvec2(K,Ki));
2066 637 : P = mfsamefield(NULL, P, mf_get_field(f));
2067 637 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1)
2068 126 : P = mfsamefield(NULL, P, gel(c,1));
2069 : }
2070 287 : G = znstar0(N,1);
2071 910 : for (i = 1; i < l; i++)
2072 : {
2073 630 : GEN CHI2 = mf_get_CHI(gel(F,i));
2074 630 : CHI2 = induce(G, CHI2);
2075 630 : if (!CHI) CHI = CHI2;
2076 343 : else if (!gequal(CHI, CHI2))
2077 7 : pari_err_TYPE("mflinear [different characters]", mkvec2(CHI,CHI2));
2078 : }
2079 280 : NK = mkgNK(N, K, CHI, P);
2080 280 : return gc_GEN(av, taglinear(NK,F,L));
2081 : }
2082 :
2083 : GEN
2084 42 : mfshift(GEN F, long sh)
2085 : {
2086 42 : pari_sp av = avma;
2087 42 : if (!checkmf_i(F)) pari_err_TYPE("mfshift",F);
2088 42 : return gc_GEN(av, tag2(t_MF_SHIFT, mf_get_NK(F), F, stoi(sh)));
2089 : }
2090 : static long
2091 49 : mfval(GEN F)
2092 : {
2093 49 : pari_sp av = avma;
2094 49 : long i = 0, n, sb;
2095 : GEN gk, gN;
2096 49 : if (!checkmf_i(F)) pari_err_TYPE("mfval", F);
2097 49 : gN = mf_get_gN(F);
2098 49 : gk = mf_get_gk(F);
2099 49 : sb = mfsturmNgk(itou(gN), gk);
2100 70 : for (n = 1; n <= sb;)
2101 : {
2102 : GEN v;
2103 63 : if (n > 0.5*sb) n = sb+1;
2104 63 : v = mfcoefs_i(F, n, 1);
2105 119 : for (; i <= n; i++)
2106 98 : if (!gequal0(gel(v, i+1))) return gc_long(av,i);
2107 21 : n <<= 1;
2108 : }
2109 7 : return gc_long(av,-1);
2110 : }
2111 :
2112 : GEN
2113 2163 : mfdiv_val(GEN f, GEN g, long vg)
2114 : {
2115 : GEN T, N, K, NK, CHI, CHIf, CHIg;
2116 2163 : if (vg) { f = mfshift(f,vg); g = mfshift(g,vg); }
2117 2163 : N = lcmii(mf_get_gN(f), mf_get_gN(g));
2118 2163 : K = gsub(mf_get_gk(f), mf_get_gk(g));
2119 2163 : CHIf = mf_get_CHI(f);
2120 2163 : CHIg = mf_get_CHI(g);
2121 2163 : CHI = mfchiadjust(mfchardiv(CHIf, CHIg), K, itos(N));
2122 2163 : T = chicompat(CHI, CHIf, CHIg);
2123 2156 : NK = mkgNK(N, K, CHI, mfsamefield(T, mf_get_field(f), mf_get_field(g)));
2124 2156 : return T? tag3(t_MF_DIV, NK, f, g, T): tag2(t_MF_DIV, NK, f, g);
2125 : }
2126 : GEN
2127 49 : mfdiv(GEN F, GEN G)
2128 : {
2129 49 : pari_sp av = avma;
2130 49 : long v = mfval(G);
2131 49 : if (!checkmf_i(F)) pari_err_TYPE("mfdiv", F);
2132 42 : if (v < 0 || (v && !gequal0(mfcoefs(F, v-1, 1))))
2133 14 : pari_err_DOMAIN("mfdiv", "ord(G)", ">", strtoGENstr("ord(F)"),
2134 : mkvec2(F, G));
2135 28 : return gc_GEN(av, mfdiv_val(F, G, v));
2136 : }
2137 : GEN
2138 182 : mfderiv(GEN F, long m)
2139 : {
2140 182 : pari_sp av = avma;
2141 : GEN NK, gk;
2142 182 : if (!checkmf_i(F)) pari_err_TYPE("mfderiv",F);
2143 182 : gk = gaddgs(mf_get_gk(F), 2*m);
2144 182 : NK = mkgNK(mf_get_gN(F), gk, mf_get_CHI(F), mf_get_field(F));
2145 182 : return gc_GEN(av, tag2(t_MF_DERIV, NK, F, stoi(m)));
2146 : }
2147 : GEN
2148 21 : mfderivE2(GEN F, long m)
2149 : {
2150 21 : pari_sp av = avma;
2151 : GEN NK, gk;
2152 21 : if (!checkmf_i(F)) pari_err_TYPE("mfderivE2",F);
2153 21 : if (m < 0) pari_err_DOMAIN("mfderivE2","m","<",gen_0,stoi(m));
2154 21 : gk = gaddgs(mf_get_gk(F), 2*m);
2155 21 : NK = mkgNK(mf_get_gN(F), gk, mf_get_CHI(F), mf_get_field(F));
2156 21 : return gc_GEN(av, tag2(t_MF_DERIVE2, NK, F, stoi(m)));
2157 : }
2158 :
2159 : GEN
2160 28 : mftwist(GEN F, GEN D)
2161 : {
2162 28 : pari_sp av = avma;
2163 : GEN NK, CHI, NT, Da;
2164 : long q;
2165 28 : if (!checkmf_i(F)) pari_err_TYPE("mftwist", F);
2166 28 : if (typ(D) != t_INT) pari_err_TYPE("mftwist", D);
2167 28 : Da = mpabs_shallow(D);
2168 28 : CHI = mf_get_CHI(F); q = mfcharconductor(CHI);
2169 28 : NT = glcm(glcm(mf_get_gN(F), mulsi(q, Da)), sqri(Da));
2170 28 : NK = mkgNK(NT, mf_get_gk(F), CHI, mf_get_field(F));
2171 28 : return gc_GEN(av, tag2(t_MF_TWIST, NK, F, D));
2172 : }
2173 :
2174 : /***************************************************************/
2175 : /* Generic cache handling */
2176 : /***************************************************************/
2177 : enum { cache_FACT, cache_DIV, cache_H, cache_D, cache_DIH };
2178 : typedef struct {
2179 : const char *name;
2180 : GEN cache;
2181 : ulong minself, maxself;
2182 : void (*init)(long);
2183 : ulong miss, maxmiss;
2184 : long compressed;
2185 : } cache;
2186 :
2187 : static void constfact(long lim);
2188 : static void constdiv(long lim);
2189 : static void consttabh(long lim);
2190 : static void consttabdihedral(long lim);
2191 : static void constcoredisc(long lim);
2192 : static THREAD cache caches[] = {
2193 : { "Factors", NULL, 50000, 50000, &constfact, 0, 0, 0 },
2194 : { "Divisors", NULL, 50000, 50000, &constdiv, 0, 0, 0 },
2195 : { "H", NULL, 100000, 10000000, &consttabh, 0, 0, 1 },
2196 : { "CorediscF",NULL, 100000, 10000000, &constcoredisc, 0, 0, 0 },
2197 : { "Dihedral", NULL, 1000, 3000, &consttabdihedral, 0, 0, 0 },
2198 : };
2199 :
2200 : static void
2201 519 : cache_reset(long id) { caches[id].miss = caches[id].maxmiss = 0; }
2202 : static void
2203 9450 : cache_delete(long id) { guncloneNULL(caches[id].cache); }
2204 : static void
2205 533 : cache_set(long id, GEN S)
2206 : {
2207 533 : GEN old = caches[id].cache;
2208 533 : caches[id].cache = gclone(S);
2209 533 : guncloneNULL(old);
2210 533 : }
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 448980563 : cache_get(long id, ulong D)
2217 : {
2218 448980563 : cache *S = &caches[id];
2219 448980563 : const ulong d = S->compressed? D>>1: D;
2220 : ulong max, l;
2221 :
2222 448980563 : if (!S->cache)
2223 : {
2224 391 : max = maxuu(minuu(D, S->maxself), S->minself);
2225 391 : S->init(max);
2226 392 : l = lg(S->cache);
2227 : }
2228 : else
2229 : {
2230 448980172 : l = lg(S->cache);
2231 448980172 : if (l <= d)
2232 : {
2233 372 : if (D > S->maxmiss) S->maxmiss = D;
2234 372 : if (DEBUGLEVEL >= 3)
2235 0 : err_printf("miss in cache %s: %lu, max = %lu\n",
2236 : S->name, D, S->maxmiss);
2237 372 : 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 448980564 : return (l <= d)? NULL: gel(S->cache, d);
2250 : }
2251 : static GEN
2252 70 : cache_report(long id)
2253 : {
2254 70 : cache *S = &caches[id];
2255 70 : GEN v = zerocol(5);
2256 70 : gel(v,1) = strtoGENstr(S->name);
2257 70 : if (S->cache)
2258 : {
2259 35 : gel(v,2) = utoi(lg(S->cache)-1);
2260 35 : gel(v,3) = utoi(S->miss);
2261 35 : gel(v,4) = utoi(S->maxmiss);
2262 35 : gel(v,5) = utoi(gsizebyte(S->cache));
2263 : }
2264 70 : return v;
2265 : }
2266 : GEN
2267 14 : getcache(void)
2268 : {
2269 14 : pari_sp av = avma;
2270 14 : GEN M = cgetg(6, t_MAT);
2271 14 : gel(M,1) = cache_report(cache_FACT);
2272 14 : gel(M,2) = cache_report(cache_DIV);
2273 14 : gel(M,3) = cache_report(cache_H);
2274 14 : gel(M,4) = cache_report(cache_D);
2275 14 : gel(M,5) = cache_report(cache_DIH);
2276 14 : return gc_GEN(av, shallowtrans(M));
2277 : }
2278 :
2279 : void
2280 1890 : pari_close_mf(void)
2281 : {
2282 1890 : cache_delete(cache_FACT);
2283 1890 : cache_delete(cache_DIV);
2284 1890 : cache_delete(cache_H);
2285 1890 : cache_delete(cache_D);
2286 1890 : cache_delete(cache_DIH);
2287 1890 : }
2288 :
2289 : /*************************************************************************/
2290 : /* a odd, update local cache (recycle memory) */
2291 : static GEN
2292 2289 : update_factor_cache(long a, long lim, long *pb)
2293 : {
2294 2289 : const long step = 16000; /* even; don't increase this: RAM cache thrashing */
2295 2289 : if (a + 2*step > lim)
2296 208 : *pb = lim; /* fuse last 2 chunks */
2297 : else
2298 2081 : *pb = a + step;
2299 2289 : return vecfactoroddu_i(a, *pb);
2300 : }
2301 : /* assume lim < MAX_LONG/8 */
2302 : static void
2303 53 : constcoredisc(long lim)
2304 : {
2305 53 : pari_sp av2, av = avma;
2306 53 : GEN D = caches[cache_D].cache, CACHE = NULL;
2307 53 : long cachea, cacheb, N, LIM = !D ? 4 : lg(D)-1;
2308 53 : if (lim <= 0) lim = 5;
2309 53 : if (lim <= LIM) return;
2310 53 : cache_reset(cache_D);
2311 53 : D = zero_zv(lim);
2312 53 : av2 = avma;
2313 53 : cachea = cacheb = 0;
2314 6015149 : for (N = 1; N <= lim; N+=2)
2315 : { /* N odd */
2316 : long i, d, d2;
2317 : GEN F;
2318 6015096 : if (N > cacheb)
2319 : {
2320 734 : set_avma(av2); cachea = N;
2321 734 : CACHE = update_factor_cache(N, lim, &cacheb);
2322 : }
2323 6015096 : F = gel(CACHE, ((N-cachea)>>1)+1); /* factoru(N) */
2324 6015096 : D[N] = d = corediscs_fact(F); /* = 3 mod 4 or 4 mod 16 */
2325 6015096 : d2 = odd(d)? d<<3: d<<1;
2326 6015096 : for (i = 1;;)
2327 : {
2328 8020106 : if ((N << i) > lim) break;
2329 4010075 : D[N<<i] = d2; i++;
2330 4010075 : if ((N << i) > lim) break;
2331 2005010 : D[N<<i] = d; i++;
2332 : }
2333 : }
2334 53 : cache_set(cache_D, D);
2335 53 : set_avma(av);
2336 : }
2337 :
2338 : static void
2339 187 : constfact(long lim)
2340 : {
2341 : pari_sp av;
2342 187 : GEN VFACT = caches[cache_FACT].cache;
2343 187 : long LIM = VFACT? lg(VFACT)-1: 4;
2344 187 : if (lim <= 0) lim = 5;
2345 187 : if (lim <= LIM) return;
2346 159 : cache_reset(cache_FACT); av = avma;
2347 159 : cache_set(cache_FACT, vecfactoru_i(1,lim)); set_avma(av);
2348 : }
2349 : static void
2350 152 : constdiv(long lim)
2351 : {
2352 : pari_sp av;
2353 152 : GEN VFACT, VDIV = caches[cache_DIV].cache;
2354 152 : long N, LIM = VDIV? lg(VDIV)-1: 4;
2355 152 : if (lim <= 0) lim = 5;
2356 152 : if (lim <= LIM) return;
2357 152 : constfact(lim);
2358 152 : VFACT = caches[cache_FACT].cache;
2359 152 : cache_reset(cache_DIV); av = avma;
2360 152 : VDIV = cgetg(lim+1, t_VEC);
2361 7418691 : for (N = 1; N <= lim; N++) gel(VDIV,N) = divisorsu_fact(gel(VFACT,N));
2362 152 : cache_set(cache_DIV, VDIV); set_avma(av);
2363 : }
2364 :
2365 : /* n > 1, D = divisors(n); sets L = 2*lambda(n), S = sigma(n) */
2366 : static void
2367 14346234 : lamsig(GEN D, long *pL, long *pS)
2368 : {
2369 14346234 : pari_sp av = avma;
2370 14346234 : long i, l = lg(D), L = 1, S = D[l-1]+1;
2371 51545390 : for (i = 2; i < l; i++) /* skip d = 1 */
2372 : {
2373 52059888 : long d = D[i], nd = D[l-i]; /* nd = n/d */
2374 52059888 : if (d < nd) { L += d; S += d + nd; }
2375 : else
2376 : {
2377 14860732 : L <<= 1; if (d == nd) { L += d; S += d; }
2378 14860732 : break;
2379 : }
2380 : }
2381 14346234 : set_avma(av); *pL = L; *pS = S;
2382 14912681 : }
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 1048505 : for (N = 3; N <= lim0; N++) VHDH0[N] = VHDH[N];
2402 155 : av2 = avma;
2403 155 : cachea = cacheb = 0;
2404 7678510 : for (N = LIM + 3; N <= lim; N += 4)
2405 : {
2406 7723706 : long s = 0, limt = usqrt(N>>2), flsq = 0, ind, t, L, S;
2407 : GEN DN, DN2;
2408 7690757 : if (N + 2 >= lg(VDIV))
2409 : { /* use local cache */
2410 : GEN F;
2411 5899931 : if (N + 2 > cacheb)
2412 : {
2413 1555 : set_avma(av2); cachea = N;
2414 1555 : CACHE = update_factor_cache(N, lim+2, &cacheb);
2415 : }
2416 5899931 : F = gel(CACHE, ((N-cachea)>>1)+1); /* factoru(N) */
2417 5899931 : DN = divisorsu_fact(F);
2418 6147232 : F = gel(CACHE, ((N-cachea)>>1)+2); /* factoru(N+2) */
2419 6147232 : DN2 = divisorsu_fact(F);
2420 : }
2421 : else
2422 : { /* use global cache */
2423 1790826 : DN = gel(VDIV,N);
2424 1790826 : DN2 = gel(VDIV,N+2);
2425 : }
2426 7883008 : ind = N >> 1;
2427 937653158 : for (t = 1; t <= limt; t++)
2428 : {
2429 929770150 : ind -= (t<<2)-2; /* N/2 - 2t^2 */
2430 929770150 : if (ind) s += VHDH0[ind]; else flsq = 1;
2431 : }
2432 7883008 : lamsig(DN, &L,&S);
2433 7607069 : VHDH0[N >> 1] = 2*S - 3*L - 2*s + flsq;
2434 7607069 : s = 0; flsq = 0; limt = (usqrt(N+2) - 1) >> 1;
2435 7678956 : ind = (N+1) >> 1;
2436 928668036 : for (t = 1; t <= limt; t++)
2437 : {
2438 920989080 : ind -= t<<2; /* (N+1)/2 - 2t(t+1) */
2439 920989080 : if (ind) s += VHDH0[ind]; else flsq = 1;
2440 : }
2441 7678956 : lamsig(DN2, &L,&S);
2442 7678355 : 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 33718058 : myfactoru(long N)
2452 : {
2453 33718058 : GEN z = cache_get(cache_FACT, N);
2454 33718058 : return z? gcopy(z): factoru(N);
2455 : }
2456 : static GEN
2457 69162115 : mydivisorsu(long N)
2458 : {
2459 69162115 : GEN z = cache_get(cache_DIV, N);
2460 69162115 : return z? leafcopy(z): divisorsu(N);
2461 : }
2462 : /* write -n = Df^2, D < 0 fundamental discriminant. Return D, set f. */
2463 : static long
2464 176538437 : mycoredisc2neg(ulong n, long *pf)
2465 : {
2466 176538437 : ulong m, D = (ulong)cache_get(cache_D, n);
2467 176538437 : if (D) { *pf = usqrt(n/D); return -(long)D; }
2468 59 : m = mycore(n, pf);
2469 59 : if ((m&3) != 3) { m <<= 2; *pf >>= 1; }
2470 59 : 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 2129 : for (a=1; a*a<b2; a++)
2495 2122 : if (b2%a == 0) h++;
2496 7 : f = (a*a==b2); b=2; b2=(4+d)>>2;
2497 : }
2498 10819 : while (b2*3 < d)
2499 : {
2500 10763 : if (b2%b == 0) h++;
2501 1729110 : for (a=b+1; a*a < b2; a++)
2502 1718347 : if (b2%a == 0) h += 2;
2503 10763 : if (a*a == b2) h++;
2504 10763 : 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 93 : hclassno6u_2(long D0, long F)
2513 : {
2514 : long h;
2515 93 : if (F == 1) h = hclassno6_count(D0);
2516 : else
2517 : { /* second chance */
2518 31 : h = (ulong)cache_get(cache_H, -D0);
2519 31 : if (!h) h = hclassno6_count(D0);
2520 31 : h *= uhclassnoF_fact(myfactoru(F), D0);
2521 : }
2522 93 : 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 2412548 : hclassno6u(ulong D)
2528 : {
2529 2412548 : ulong z = (ulong)cache_get(cache_H, D);
2530 : long D0, F;
2531 2412549 : if (z) return z;
2532 93 : D0 = mycoredisc2neg(D, &F);
2533 93 : return hclassno6u_2(D0,F);
2534 : }
2535 : /* same as hclassno6u without creating caches */
2536 : ulong
2537 86921 : hclassno6u_no_cache(ulong D)
2538 : {
2539 86921 : cache *S = &caches[cache_H];
2540 : long D0, F;
2541 86921 : if (S->cache)
2542 : {
2543 79911 : const ulong d = D>>1; /* compressed */
2544 79911 : if ((ulong)lg(S->cache) > d) return S->cache[d];
2545 : }
2546 86651 : S = &caches[cache_D];
2547 86651 : 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 156484775 : hclassno6u_i(ulong D, long D0, long F)
2554 : {
2555 156484775 : ulong z = (ulong)cache_get(cache_H, D);
2556 156484775 : 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 10652089 : myh(long D)
2563 : {
2564 10652089 : ulong z = (ulong)cache_get(cache_H, -D);
2565 10652089 : return z? z / 6: quadclassnos(D);
2566 : }
2567 :
2568 : /*************************************************************************/
2569 : /* TRACE FORMULAS */
2570 : /* CHIP primitive, initialize for t_POLMOD output */
2571 : static GEN
2572 32844 : mfcharinit(GEN CHIP)
2573 : {
2574 32844 : long n, o, l, vt, N = mfcharmodulus(CHIP);
2575 : GEN c, v, V, G, Pn;
2576 32844 : if (N == 1) return mkvec2(mkvec(gen_1), pol_x(0));
2577 5607 : G = gel(CHIP,1);
2578 5607 : v = ncharvecexpo(G, znconrey_normalized(G, gel(CHIP,2)));
2579 5607 : l = lg(v); V = cgetg(l, t_VEC);
2580 5607 : o = mfcharorder(CHIP);
2581 5607 : Pn = mfcharpol(CHIP); vt = varn(Pn);
2582 5607 : if (o <= 2)
2583 : {
2584 59248 : for (n = 1; n < l; n++)
2585 : {
2586 54719 : if (v[n] < 0) c = gen_0; else c = v[n]? gen_m1: gen_1;
2587 54719 : gel(V,n) = c;
2588 : }
2589 : }
2590 : else
2591 : {
2592 17591 : for (n = 1; n < l; n++)
2593 : {
2594 16513 : if (v[n] < 0) c = gen_0;
2595 : else
2596 : {
2597 9394 : c = Qab_zeta(v[n], o, vt);
2598 9394 : if (typ(c) == t_POL && lg(c) >= lg(Pn)) c = RgX_rem(c, Pn);
2599 : }
2600 16513 : gel(V,n) = c;
2601 : }
2602 : }
2603 5607 : return mkvec2(V, Pn);
2604 : }
2605 : static GEN
2606 413406 : vchip_lift(GEN VCHI, long x, GEN C)
2607 : {
2608 413406 : GEN V = gel(VCHI,1);
2609 413406 : long F = lg(V)-1;
2610 413406 : if (F == 1) return C;
2611 21056 : x %= F;
2612 21056 : if (!x) return C;
2613 21056 : if (x <= 0) x += F;
2614 21056 : return gmul(C, gel(V, x));
2615 : }
2616 : static long
2617 279730397 : vchip_FC(GEN VCHI) { return lg(gel(VCHI,1))-1; }
2618 : static GEN
2619 6452636 : vchip_mod(GEN VCHI, GEN S)
2620 6452636 : { return (typ(S) == t_POL)? RgX_rem(S, gel(VCHI,2)): S; }
2621 : static GEN
2622 1930149 : vchip_polmod(GEN VCHI, GEN S)
2623 1930149 : { return (typ(S) == t_POL)? mkpolmod(S, gel(VCHI,2)): S; }
2624 :
2625 : /* contribution of scalar matrices in dimension formula */
2626 : static GEN
2627 360472 : A1(long N, long k) { return uutoQ(mypsiu(N)*(k-1), 12); }
2628 : static long
2629 7644 : ceilA1(long N, long k) { return ceildivuu(mypsiu(N) * (k-1), 12); }
2630 :
2631 : /* sturm bound, slightly larger than dimension */
2632 : long
2633 21819 : mfsturmNk(long N, long k) { return (mypsiu(N) * k) / 12; }
2634 : long
2635 3318 : mfsturmNgk(long N, GEN k)
2636 : {
2637 3318 : long n,d; Qtoss(k,&n,&d);
2638 3318 : return 1 + (mypsiu(N)*n)/(d == 1? 12: 24);
2639 : }
2640 : static long
2641 427 : mfsturmmf(GEN F) { return mfsturmNgk(mf_get_N(F), mf_get_gk(F)); }
2642 :
2643 : /* List of all solutions of x^2 + x + 1 = 0 modulo N, x modulo N */
2644 : static GEN
2645 581 : sqrtm3modN(long N)
2646 : {
2647 : pari_sp av;
2648 : GEN fa, P, E, B, mB, A, Q, T, R, v, gen_m3;
2649 581 : long l, i, n, ct, fl3 = 0, Ninit;
2650 581 : if (!odd(N) || (N%9) == 0) return cgetg(1,t_VECSMALL);
2651 553 : Ninit = N;
2652 553 : if ((N%3) == 0) { N /= 3; fl3 = 1; }
2653 553 : fa = myfactoru(N); P = gel(fa, 1); E = gel(fa, 2);
2654 553 : l = lg(P);
2655 749 : for (i = 1; i < l; i++)
2656 560 : if ((P[i]%3) == 2) return cgetg(1,t_VECSMALL);
2657 189 : A = cgetg(l, t_VECSMALL);
2658 189 : B = cgetg(l, t_VECSMALL);
2659 189 : mB= cgetg(l, t_VECSMALL);
2660 189 : Q = cgetg(l, t_VECSMALL); gen_m3 = utoineg(3);
2661 385 : for (i = 1; i < l; i++)
2662 : {
2663 196 : long p = P[i], e = E[i];
2664 196 : Q[i] = upowuu(p,e);
2665 196 : B[i] = itou( Zp_sqrt(gen_m3, utoipos(p), e) );
2666 196 : mB[i]= Q[i] - B[i];
2667 : }
2668 189 : ct = 1 << (l-1);
2669 189 : T = ZV_producttree(Q);
2670 189 : R = ZV_chinesetree(Q,T);
2671 189 : v = cgetg(ct+1, t_VECSMALL);
2672 189 : av = avma;
2673 581 : for (n = 1; n <= ct; n++)
2674 : {
2675 392 : long m = n-1, r;
2676 812 : for (i = 1; i < l; i++)
2677 : {
2678 420 : A[i] = (m&1L)? mB[i]: B[i];
2679 420 : m >>= 1;
2680 : }
2681 392 : r = itou( ZV_chinese_tree(A, Q, T, R) );
2682 462 : if (fl3) while (r%3) r += N;
2683 392 : set_avma(av); v[n] = odd(r) ? (r-1) >> 1 : (r+Ninit-1) >> 1;
2684 : }
2685 189 : return v;
2686 : }
2687 :
2688 : /* number of elliptic points of order 3 in X0(N) */
2689 : static long
2690 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 43820 : A21(long N, long k, GEN CHI)
2717 : {
2718 : GEN res, G, chi, o;
2719 : long a21, i, limx, S;
2720 43820 : if ((N&1L) == 0) return gen_0;
2721 21196 : a21 = k%3 - 1;
2722 21196 : if (!a21) return gen_0;
2723 20440 : if (N <= 3) return sstoQ(a21, 3);
2724 10759 : if (!CHI) return sstoQ(nu3(N) * a21, 3);
2725 581 : res = sqrtm3modN(N); limx = (N - 1) >> 1;
2726 581 : G = gel(CHI,1); chi = gel(CHI,2);
2727 581 : o = gmfcharorder(CHI);
2728 973 : for (S = 0, i = 1; i < lg(res); i++)
2729 : { /* (x,N) = 1; S += chi(x) + chi(x^2) */
2730 392 : long x = res[i];
2731 392 : if (x <= limx)
2732 : { /* CHI(x)=e(c/o), 3rd-root of 1 */
2733 196 : GEN c = znchareval(G, chi, utoi(x), o);
2734 196 : if (!signe(c)) S += 2; else S--;
2735 : }
2736 : }
2737 581 : return sstoQ(a21 * S, 3);
2738 : }
2739 :
2740 : /* List of all square roots of -1 modulo N */
2741 : static GEN
2742 595 : sqrtm1modN(long N)
2743 : {
2744 : pari_sp av;
2745 : GEN fa, P, E, B, mB, A, Q, T, R, v;
2746 595 : long l, i, n, ct, fleven = 0;
2747 595 : if ((N&3L) == 0) return cgetg(1,t_VECSMALL);
2748 595 : if ((N&1L) == 0) { N >>= 1; fleven = 1; }
2749 595 : fa = myfactoru(N); P = gel(fa,1); E = gel(fa,2);
2750 595 : l = lg(P);
2751 945 : for (i = 1; i < l; i++)
2752 665 : if ((P[i]&3L) == 3) return cgetg(1,t_VECSMALL);
2753 280 : A = cgetg(l, t_VECSMALL);
2754 280 : B = cgetg(l, t_VECSMALL);
2755 280 : mB= cgetg(l, t_VECSMALL);
2756 280 : Q = cgetg(l, t_VECSMALL);
2757 574 : for (i = 1; i < l; i++)
2758 : {
2759 294 : long p = P[i], e = E[i];
2760 294 : Q[i] = upowuu(p,e);
2761 294 : B[i] = itou( Zp_sqrt(gen_m1, utoipos(p), e) );
2762 294 : mB[i]= Q[i] - B[i];
2763 : }
2764 280 : ct = 1 << (l-1);
2765 280 : T = ZV_producttree(Q);
2766 280 : R = ZV_chinesetree(Q,T);
2767 280 : v = cgetg(ct+1, t_VECSMALL);
2768 280 : av = avma;
2769 868 : for (n = 1; n <= ct; n++)
2770 : {
2771 588 : long m = n-1, r;
2772 1232 : for (i = 1; i < l; i++)
2773 : {
2774 644 : A[i] = (m&1L)? mB[i]: B[i];
2775 644 : m >>= 1;
2776 : }
2777 588 : r = itou( ZV_chinese_tree(A, Q, T, R) );
2778 588 : if (fleven && !odd(r)) r += N;
2779 588 : set_avma(av); v[n] = r;
2780 : }
2781 280 : return v;
2782 : }
2783 :
2784 : /* contribution of elliptic matrices of order 4 in dimension formula.
2785 : * Only depends on CHIP the primitive char attached to CHI */
2786 : static GEN
2787 43820 : A22(long N, long k, GEN CHI)
2788 : {
2789 : GEN G, chi, o, res;
2790 : long S, a22, i, limx, o2;
2791 43820 : if ((N&3L) == 0) return gen_0;
2792 30205 : a22 = (k & 3L) - 1; /* (k % 4) - 1 */
2793 30205 : if (!a22) return gen_0;
2794 30135 : 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 38941 : nuinf(long N)
2816 : {
2817 38941 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
2818 38941 : long i, t = 1, l = lg(P);
2819 82705 : 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 38941 : return t;
2828 : }
2829 :
2830 : /* contribution of hyperbolic matrices in dimension formula */
2831 : static GEN
2832 44268 : A3(long N, long FC)
2833 : {
2834 : long i, S, NF, l;
2835 : GEN D;
2836 44268 : if (FC == 1) return uutoQ(nuinf(N),2);
2837 5327 : D = mydivisorsu(N); l = lg(D);
2838 5327 : S = 0; NF = N/FC;
2839 41720 : for (i = 1; i < l; i++)
2840 : {
2841 36393 : long g = ugcd(D[i], D[l-i]);
2842 36393 : if (NF%g == 0) S += myeulerphiu(g);
2843 : }
2844 5327 : return uutoQ(S, 2);
2845 : }
2846 :
2847 : /* special contribution in weight 2 in dimension formula */
2848 : static long
2849 43358 : A4(long k, long FC)
2850 43358 : { return (k==2 && FC==1)? 1: 0; }
2851 : /* gcd(x,N) */
2852 : static long
2853 283363493 : myugcd(GEN GCD, ulong x)
2854 : {
2855 283363493 : ulong N = lg(GCD)-1;
2856 283363493 : if (x >= N) x %= N;
2857 283363493 : return GCD[x+1];
2858 : }
2859 : /* 1_{gcd(x,N) = 1} * chi(x), return NULL if 0 */
2860 : static GEN
2861 402671825 : mychicgcd(GEN GCD, GEN VCHI, long x)
2862 : {
2863 402671825 : long N = lg(GCD)-1;
2864 402671825 : if (N == 1) return gen_1;
2865 328234296 : x = umodsu(x, N);
2866 328234296 : if (GCD[x+1] != 1) return NULL;
2867 272071847 : x %= vchip_FC(VCHI); if (!x) return gen_1;
2868 4678940 : return gel(gel(VCHI,1), x);
2869 : }
2870 :
2871 : /* contribution of scalar matrices to trace formula */
2872 : static GEN
2873 6403500 : TA1(long N, long k, GEN VCHI, GEN GCD, long n)
2874 : {
2875 : GEN S;
2876 : ulong m;
2877 6403500 : if (!uissquareall(n, &m)) return gen_0;
2878 387373 : if (m == 1) return A1(N,k); /* common */
2879 347410 : S = mychicgcd(GCD, VCHI, m);
2880 347410 : 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 127547 : mksqr(long N)
2886 : {
2887 127547 : pari_sp av = avma;
2888 127547 : long x, N2 = N << 1, N4 = N << 2;
2889 127547 : GEN v = const_vec(N2, cgetg(1, t_VECSMALL));
2890 127547 : gel(v, N2) = mkvecsmall(0); /* x = 0 */
2891 3497193 : for (x = 1; x <= N; x++)
2892 : {
2893 3369646 : long r = (((x*x - 1)%N4) >> 1) + 1;
2894 3369646 : gel(v,r) = vecsmall_append(gel(v,r), x);
2895 : }
2896 127547 : return gc_GEN(av, v);
2897 : }
2898 :
2899 : static GEN
2900 127547 : mkgcd(long N)
2901 : {
2902 : GEN GCD, d;
2903 : long i, N2;
2904 127547 : if (N == 1) return mkvecsmall(N);
2905 105105 : GCD = cgetg(N + 1, t_VECSMALL);
2906 105105 : d = GCD+1; /* GCD[i+1] = d[i] = gcd(i,N) = gcd(N-i,N), i = 0..N-1 */
2907 105105 : d[0] = N; d[1] = d[N-1] = 1; N2 = N>>1;
2908 1652791 : for (i = 2; i <= N2; i++) d[i] = d[N-i] = ugcd(N, i);
2909 105105 : 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 15230828 : mutglistall(long t, long N, long NF, GEN VCHI, long n, GEN MUP, GEN li, GEN GCD)
2916 : {
2917 15230828 : long i, lx = lg(li);
2918 15230828 : GEN DNF = mydivisorsu(NF), v = zerovec(NF);
2919 15230828 : long j, g, lDNF = lg(DNF);
2920 42489136 : for (i = 1; i < lx; i++)
2921 : {
2922 27258308 : long x = (li[i] + t) >> 1, y, lD;
2923 27258308 : GEN D, c = mychicgcd(GCD, VCHI, x);
2924 27258308 : if (li[i] && li[i] != N)
2925 : {
2926 18108670 : GEN c2 = mychicgcd(GCD, VCHI, t - x);
2927 18108670 : if (c2) c = c? gadd(c, c2): c2;
2928 : }
2929 27258308 : if (!c) continue;
2930 22108807 : y = (x*(x - t) + n) / N; /* exact division */
2931 22108807 : D = mydivisorsu(ugcd(labs(y), NF)); lD = lg(D);
2932 59562799 : 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 35176489 : for (j=2; j < lDNF; j++) { g = DNF[j]; gel(v,g) = gmulsg(MUP[g], gel(v,g)); }
2936 15230828 : return v;
2937 : }
2938 :
2939 : /* special case (N,F) = 1: easier */
2940 : static GEN
2941 161307502 : mutg1(long t, long N, GEN VCHI, GEN li, GEN GCD)
2942 : { /* (N,F) = 1 */
2943 161307502 : GEN S = NULL;
2944 161307502 : long i, lx = lg(li);
2945 338500041 : for (i = 1; i < lx; i++)
2946 : {
2947 177192539 : long x = (li[i] + t) >> 1;
2948 177192539 : GEN c = mychicgcd(GCD, VCHI, x);
2949 177192539 : if (c) S = S? gadd(S, c): c;
2950 177192539 : if (li[i] && li[i] != N)
2951 : {
2952 98008827 : c = mychicgcd(GCD, VCHI, t - x);
2953 98008827 : if (c) S = S? gadd(S, c): c;
2954 : }
2955 177192539 : if (S && !signe(S)) S = NULL; /* strive hard to add gen_0 */
2956 : }
2957 161307502 : 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 361569 : mfrhopol(long n)
2963 : {
2964 : #ifdef LONG_IS_64BIT
2965 309960 : const long M = 2642249;
2966 : #else
2967 51609 : const long M = 1629;
2968 : #endif
2969 361569 : long j, d = n >> 1; /* >= 1 */
2970 361569 : GEN P = cgetg(d + 3, t_POL);
2971 :
2972 361569 : if (n > M) pari_err_IMPL("mfrhopol for large weight"); /* avoid overflow */
2973 361569 : P[1] = evalvarn(0)|evalsigne(1);
2974 361569 : gel(P,2) = gen_1;
2975 361569 : gel(P,3) = utoineg(n-1); /* j = 1 */
2976 361569 : if (d > 1) gel(P,4) = utoipos(((n-3)*(n-2)) >> 1); /* j = 2 */
2977 361569 : if (d > 2) gel(P,5) = utoineg(((n-5)*(n-4)*(n-3)) / 6); /* j = 3 */
2978 1516474 : for (j = 4; j <= d; j++)
2979 1154905 : gel(P,j+2) = divis(mulis(gel(P,j+1), (n-2*j+1)*(n-2*j+2)), (n-j+1)*(-j));
2980 361569 : return P;
2981 : }
2982 :
2983 : /* polrecip(Q)(t2), assume Q(0) = 1 */
2984 : GEN
2985 3251109 : mfrhopol_u_eval(GEN Q, ulong t2)
2986 : {
2987 3251109 : GEN T = addiu(gel(Q,3), t2);
2988 3251108 : long l = lg(Q), j;
2989 37887745 : for (j = 4; j < l; j++) T = addii(gel(Q,j), mului(t2, T));
2990 3251112 : return T;
2991 : }
2992 : GEN
2993 56618 : mfrhopol_eval(GEN Q, GEN t2)
2994 : {
2995 : long l, j;
2996 : GEN T;
2997 56618 : 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 168101972 : mfrhopowsimp(GEN Q, GEN sh, long nu, long t, long t2, long n)
3007 : {
3008 : GEN T;
3009 168101972 : 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 51275 : case 3: return gmul(mulss(t, t2 - 2*n), sh);
3015 3240667 : default:
3016 3240667 : if (!t) return gmul(gmul2n(gel(Q, lg(Q) - 1), -1), sh);
3017 3194490 : T = mfrhopol_u_eval(Q, t2); if (odd(nu)) T = mulsi(t, T);
3018 3194490 : return gmul(T, sh);
3019 : }
3020 : }
3021 :
3022 : /* contribution of elliptic matrices to trace formula */
3023 : static GEN
3024 6403500 : TA2(long N, long k, GEN VCHI, long n, GEN SQRTS, GEN MUP, GEN GCD)
3025 : {
3026 6403500 : const long n4 = n << 2, N4 = N << 2, nu = k - 2;
3027 6403500 : const long st = (!odd(N) && odd(n)) ? 2 : 1;
3028 : long limt, t;
3029 : GEN S, Q;
3030 :
3031 6403500 : limt = usqrt(n4);
3032 6403500 : if (limt*limt == n4) limt--;
3033 6403500 : Q = nu > 3 ? ZX_z_unscale(mfrhopol(nu), n) : NULL;
3034 6403500 : S = gen_0;
3035 326044870 : for (t = odd(k)? st: 0; t <= limt; t += st) /* t^2 < 4n */
3036 : {
3037 319641370 : pari_sp av = avma;
3038 319641370 : long t2 = t*t, D = n4 - t2, F, D0, NF;
3039 : GEN sh, li;
3040 :
3041 319641370 : li = gel(SQRTS, (umodsu(-D - 1, N4) >> 1) + 1);
3042 328077728 : if (lg(li) == 1) continue;
3043 176538330 : D0 = mycoredisc2neg(D, &F);
3044 176538330 : NF = myugcd(GCD, F);
3045 176538330 : if (NF == 1)
3046 : { /* (N,F) = 1 => single value in mutglistall */
3047 161307502 : GEN mut = mutg1(t, N, VCHI, li, GCD);
3048 161307502 : if (!mut) { set_avma(av); continue; }
3049 156484775 : sh = gmul(uutoQ(hclassno6u_i(D,D0,F),6), mut);
3050 : }
3051 : else
3052 : {
3053 15230828 : GEN v = mutglistall(t, N, NF, VCHI, n, MUP, li, GCD);
3054 15230828 : GEN DF = mydivisorsu(F);
3055 15230828 : long i, lDF = lg(DF);
3056 15230828 : sh = gen_0;
3057 61388909 : for (i = 1; i < lDF; i++)
3058 : {
3059 46158081 : long Ff, f = DF[i], g = myugcd(GCD, f);
3060 46158081 : GEN mut = gel(v, g);
3061 46158081 : if (gequal0(mut)) continue;
3062 31212937 : Ff = DF[lDF-i]; /* F/f */
3063 31212937 : if (Ff == 1) sh = gadd(sh, mut);
3064 : else
3065 : {
3066 22367102 : GEN P = gel(myfactoru(Ff), 1);
3067 22367102 : long j, lP = lg(P);
3068 49329488 : for (j = 1; j < lP; j++) { long p = P[j]; Ff -= kross(D0, p)*Ff/p; }
3069 22367102 : sh = gadd(sh, gmulsg(Ff, mut));
3070 : }
3071 : }
3072 15230828 : if (gequal0(sh)) { set_avma(av); continue; }
3073 11617197 : if (D0 == -3) sh = gdivgu(sh, 3);
3074 11123882 : else if (D0 == -4) sh = gdivgu(sh, 2);
3075 10652089 : else sh = gmulgu(sh, myh(D0));
3076 : }
3077 168101972 : S = gc_upto(av, gadd(S, mfrhopowsimp(Q,sh,nu,t,t2,n)));
3078 : }
3079 6403500 : return S;
3080 : }
3081 :
3082 : /* compute global auxiliary data for TA3 */
3083 : static GEN
3084 127547 : mkbez(long N, long FC)
3085 : {
3086 127547 : long ct, i, NF = N/FC;
3087 127547 : GEN w, D = mydivisorsu(N);
3088 127547 : long l = lg(D);
3089 :
3090 127547 : w = cgetg(l, t_VEC);
3091 369824 : for (i = ct = 1; i < l; i++)
3092 : {
3093 347382 : long u, v, h, c = D[i], Nc = D[l-i];
3094 347382 : if (c > Nc) break;
3095 242277 : h = cbezout(c, Nc, &u, &v);
3096 242277 : if (h == 1) /* shortcut */
3097 174783 : gel(w, ct++) = mkvecsmall4(1,u*c,1,i);
3098 67494 : else if (!(NF%h))
3099 57582 : gel(w, ct++) = mkvecsmall4(h,u*(c/h),myeulerphiu(h),i);
3100 : }
3101 127547 : setlg(w,ct); stackdummy((pari_sp)(w+ct),(pari_sp)(w+l));
3102 127547 : return w;
3103 : }
3104 :
3105 : /* contribution of hyperbolic matrices to trace formula, d * nd = n,
3106 : * DN = divisorsu(N) */
3107 : static GEN
3108 33222755 : auxsum(GEN VCHI, GEN GCD, long d, long nd, GEN DN, GEN BEZ)
3109 : {
3110 33222755 : GEN S = gen_0;
3111 33222755 : long ct, g = nd - d, lDN = lg(DN), lBEZ = lg(BEZ);
3112 85332121 : for (ct = 1; ct < lBEZ; ct++)
3113 : {
3114 52109366 : GEN y, B = gel(BEZ, ct);
3115 52109366 : long ic, c, Nc, uch, h = B[1];
3116 52109366 : if (g%h) continue;
3117 50886480 : uch = B[2];
3118 50886480 : ic = B[4];
3119 50886480 : c = DN[ic];
3120 50886480 : Nc= DN[lDN - ic]; /* Nc = N/c */
3121 50886480 : if (ugcd(Nc, nd) == 1)
3122 43436149 : y = mychicgcd(GCD, VCHI, d + uch*g); /* 0 if (c,d) > 1 */
3123 : else
3124 7450331 : y = NULL;
3125 50886480 : if (c != Nc && ugcd(Nc, d) == 1)
3126 : {
3127 38319922 : GEN y2 = mychicgcd(GCD, VCHI, nd - uch*g); /* 0 if (c,nd) > 1 */
3128 38319922 : if (y2) y = y? gadd(y, y2): y2;
3129 : }
3130 50886480 : if (y) S = gadd(S, gmulsg(B[3], y));
3131 : }
3132 33222755 : return S;
3133 : }
3134 :
3135 : static GEN
3136 6403500 : TA3(long N, long k, GEN VCHI, GEN GCD, GEN Dn, GEN BEZ)
3137 : {
3138 6403500 : GEN S = gen_0, DN = mydivisorsu(N);
3139 6403500 : long i, l = lg(Dn);
3140 39626255 : for (i = 1; i < l; i++)
3141 : {
3142 39586292 : long d = Dn[i], nd = Dn[l-i]; /* = n/d */
3143 : GEN t, u;
3144 39586292 : if (d > nd) break;
3145 33222755 : t = auxsum(VCHI, GCD, d, nd, DN, BEZ);
3146 33222755 : if (isintzero(t)) continue;
3147 32094719 : u = powuu(d,k-1); if (d == nd) u = gmul2n(u,-1);
3148 32094719 : S = gadd(S, gmul(u,t));
3149 : }
3150 6403500 : return S;
3151 : }
3152 :
3153 : /* special contribution in weight 2 in trace formula */
3154 : static long
3155 6403500 : TA4(long k, GEN VCHIP, GEN Dn, GEN GCD)
3156 : {
3157 : long i, l, S;
3158 6403500 : 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 127547 : mkmup(long N)
3171 : {
3172 127547 : GEN fa = myfactoru(N), P = gel(fa,1), D = divisorsu_fact(fa);
3173 127547 : long i, lP = lg(P), lD = lg(D);
3174 127547 : GEN MUP = zero_zv(N);
3175 127547 : MUP[1] = 1;
3176 446831 : for (i = 2; i < lD; i++)
3177 : {
3178 319284 : long j, g = D[i], Ng = D[lD-i]; /* N/g */
3179 873985 : for (j = 1; j < lP; j++) { long p = P[j]; if (Ng%p) g += g/p; }
3180 319284 : MUP[D[i]] = g;
3181 : }
3182 127547 : 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 32942 : mfnewzerodata(long N, GEN CHIP)
3200 : {
3201 32942 : GEN V, M, L, faN = myfactoru(N), PN = gel(faN,1), EN = gel(faN,2);
3202 32942 : GEN G = gel(CHIP,1), chi = gel(CHIP,2);
3203 32942 : GEN fa = znstar_get_faN(G), P = ZV_to_zv(gel(fa,1)), E = gel(fa,2);
3204 32942 : long i, mod, j = 1, l = lg(PN);
3205 :
3206 32942 : M = cgetg(l, t_VECSMALL); M[1] = 0;
3207 32942 : V = cgetg(l, t_VEC);
3208 : /* Tr^new(n) = 0 if (n mod M[i]) in V[i] */
3209 32942 : 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 32837 : j = M[1]? 2: 1;
3253 70042 : for (i = odd(N)? 1: 2; i < l; i++) /* skip p=2, done above */
3254 : {
3255 37205 : long p = PN[i], e = EN[i];
3256 37205 : long z = zv_search(P, p), c = z? E[z]: 0; /* c = v_p(FC) */
3257 37205 : if ((e <= 2 && c == 1 && itos(gel(chi,z)) == (p>>1)) /* ord(CHI_p)=2 */
3258 35014 : || (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 34391 : else if (e >= 2 && c < e)
3266 : { /* sc: p */
3267 2660 : gel(V,j) = mkvecsmall(0);
3268 2660 : M[j] = p; j++;
3269 : }
3270 : }
3271 32837 : if (j == 1) return cgetg(1, t_VECSMALL);
3272 15379 : setlg(V,j); setlg(M,j); mod = zv_prod(M);
3273 15379 : L = zero_zv(mod);
3274 33677 : for (i = 1; i < j; i++)
3275 : {
3276 18298 : GEN v = gel(V,i);
3277 18298 : long s, m = M[i], lv = lg(v);
3278 47621 : for (s = 1; s < lv; s++)
3279 : {
3280 29323 : long a = v[s] + 1;
3281 56392 : do { L[a] = 1; a += m; } while (a <= mod);
3282 : }
3283 : }
3284 15379 : 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 2604859 : 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 127547 : inittrace(long N, GEN CHI, GEN VCHIP)
3295 : {
3296 : long FC;
3297 127547 : if (VCHIP)
3298 127540 : FC = mfcharmodulus(CHI);
3299 : else
3300 7 : VCHIP = mfcharinit(mfchartoprimitive(CHI, &FC));
3301 127547 : 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 32837 : inittrconj(long N, long FC)
3308 : {
3309 : GEN fa, P, E, v;
3310 : long i, k, l;
3311 :
3312 32837 : if (FC != 1) return cgetg(1,t_VECSMALL);
3313 :
3314 27230 : fa = myfactoru(N >> vals(N));
3315 27230 : P = gel(fa,1); l = lg(P);
3316 27230 : E = gel(fa,2);
3317 27230 : v = cgetg(l, t_VECSMALL);
3318 59346 : 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 27230 : setlg(v,k); return v;
3325 : }
3326 :
3327 : /* assume CHIP primitive, f(CHIP) | N; NZ = mfnewzerodata(N,CHIP) */
3328 : static GEN
3329 32837 : initnewtrace_i(long N, GEN CHIP, GEN NZ)
3330 : {
3331 32837 : GEN T = const_vec(N, cgetg(1,t_VEC)), D, VCHIP;
3332 32837 : long FC = mfcharmodulus(CHIP), N1, N2, i, l;
3333 :
3334 32837 : if (!NZ) NZ = mkvecsmall(1); /*Tr^new = 0; initialize data nevertheless*/
3335 32837 : VCHIP = mfcharinit(CHIP);
3336 32837 : N1 = N/FC; newd_params(N1, &N2);
3337 32837 : D = mydivisorsu(N1/N2); l = lg(D);
3338 32837 : N2 *= FC;
3339 160377 : for (i = 1; i < l; i++)
3340 : {
3341 127540 : long M = D[i]*N2;
3342 127540 : gel(T,M) = inittrace(M, CHIP, VCHIP);
3343 : }
3344 32837 : gel(T,N) = shallowconcat(gel(T,N), mkvec2(NZ, inittrconj(N,FC)));
3345 32837 : return T;
3346 : }
3347 : /* don't initialize if Tr^new = 0, return NULL */
3348 : static GEN
3349 32942 : initnewtrace(long N, GEN CHI)
3350 : {
3351 32942 : GEN CHIP = mfchartoprimitive(CHI, NULL), NZ = mfnewzerodata(N,CHIP);
3352 32942 : return NZ? initnewtrace_i(N, CHIP, NZ): NULL;
3353 : }
3354 :
3355 : /* (-1)^k */
3356 : static long
3357 8204 : m1pk(long k) { return odd(k)? -1 : 1; }
3358 : static long
3359 7847 : badchar(long N, long k, GEN CHI)
3360 7847 : { return mfcharparity(CHI) != m1pk(k) || (CHI && N % mfcharconductor(CHI)); }
3361 :
3362 :
3363 : static long
3364 43435 : mfcuspdim_i(long N, long k, GEN CHI, GEN vSP)
3365 : {
3366 43435 : pari_sp av = avma;
3367 : long FC;
3368 : GEN s;
3369 43435 : if (k <= 0) return 0;
3370 43435 : if (k == 1) return CHI? mf1cuspdim(N, CHI, vSP): 0;
3371 43176 : FC = CHI? mfcharconductor(CHI): 1;
3372 43176 : if (FC == 1) CHI = NULL;
3373 43176 : s = gsub(A1(N, k), gadd(A21(N, k, CHI), A22(N, k, CHI)));
3374 43176 : s = gadd(s, gsubsg(A4(k, FC), A3(N, FC)));
3375 43176 : 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 3409 : 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 861 : mffulldim(long N, long k, GEN CHI)
3386 : {
3387 861 : pari_sp av = avma;
3388 861 : long FC = CHI? mfcharconductor(CHI): 1;
3389 : GEN s;
3390 861 : if (k <= 0) return (k == 0 && FC == 1)? 1: 0;
3391 861 : if (k == 1) return gc_long(av, itos(A3(N, FC)) + mf1cuspdim(N, CHI, NULL));
3392 644 : if (FC == 1) CHI = NULL;
3393 644 : s = gsub(A1(N, k), gadd(A21(N, k, CHI), A22(N, k, CHI)));
3394 644 : s = gadd(s, A3(N, FC));
3395 644 : 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 6403500 : mfcusptrace_i(long N, long k, long n, GEN Dn, GEN S)
3415 : {
3416 6403500 : pari_sp av = avma;
3417 : GEN a, b, VCHIP, GCD;
3418 : long t;
3419 6403500 : if (!n) return gen_0;
3420 6403500 : VCHIP = gel(S,_VCHIP);
3421 6403500 : GCD = gel(S,_GCD);
3422 6403500 : t = TA4(k, VCHIP, Dn, GCD);
3423 6403500 : a = TA1(N, k, VCHIP, GCD, n); if (t) a = gaddgs(a,t);
3424 6403500 : b = TA2(N, k, VCHIP, n, gel(S,_SQRTS), gel(S,_MUP), GCD);
3425 6403500 : b = gadd(b, TA3(N, k, VCHIP, GCD, Dn, gel(S,_BEZ)));
3426 6403500 : b = gsub(a,b);
3427 6403500 : if (typ(b) != t_POL) return gc_upto(av, b);
3428 50834 : return gc_GEN(av, vchip_polmod(VCHIP, b));
3429 : }
3430 :
3431 : static GEN
3432 7667084 : mfcusptracecache(long N, long k, long n, GEN Dn, GEN S, cachenew_t *cache)
3433 : {
3434 7667084 : GEN C = NULL, T = gel(cache->vfull,N);
3435 7667084 : long lcache = lg(T);
3436 7667084 : if (n < lcache) C = gel(T, n);
3437 7667084 : if (C) cache->cuspHIT++; else C = mfcusptrace_i(N, k, n, Dn, S);
3438 7667084 : cache->cuspTOTAL++;
3439 7667084 : if (n < lcache) gel(T,n) = C;
3440 7667084 : return C;
3441 : }
3442 :
3443 : /* return the divisors of n, known to be among the elements of D */
3444 : static GEN
3445 324443 : div_restrict(GEN D, ulong n)
3446 : {
3447 : long i, j, l;
3448 324443 : GEN v, VDIV = caches[cache_DIV].cache;
3449 324443 : 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 208611 : trconj(GEN T, long N, long n)
3463 208611 : { return (lg(T) > 1 && N % n == 0 && zv_search(T, n)); }
3464 :
3465 : /* n > 0; trace formula on new space */
3466 : static GEN
3467 2604859 : mfnewtrace_i(long N, long k, long n, cachenew_t *cache)
3468 : {
3469 2604859 : GEN VCHIP, s, Dn, DN1, SN, S = cache->DATA;
3470 : long FC, N1, N2, N1N2, g, i, j, lDN1;
3471 :
3472 2604859 : if (!S) return gen_0;
3473 2604859 : SN = gel(S,N);
3474 2604859 : if (mfnewchkzero(gel(SN,_NEWLZ), n)) return gen_0;
3475 1879343 : if (k > 2 && trconj(gel(SN,_TRCONJ), N, n)) return gen_0;
3476 1879315 : VCHIP = gel(SN, _VCHIP); FC = vchip_FC(VCHIP);
3477 1879315 : N1 = N/FC; newt_params(N1, n, FC, &g, &N2);
3478 1879315 : N1N2 = N1/N2;
3479 1879315 : DN1 = mydivisorsu(N1N2); lDN1 = lg(DN1);
3480 1879315 : N2 *= FC;
3481 1879315 : Dn = mydivisorsu(n); /* this one is probably out of cache */
3482 1879315 : s = gmulsg(mubeta2(N1N2,n), mfcusptracecache(N2, k, n, Dn, gel(S,N2), cache));
3483 7342641 : for (i = 2; i < lDN1; i++)
3484 : { /* skip M1 = 1, done above */
3485 5463326 : long M1 = DN1[i], N1M1 = DN1[lDN1-i];
3486 5463326 : GEN Dg = mydivisorsu(ugcd(M1, g));
3487 5463326 : M1 *= N2;
3488 5463326 : s = gadd(s, gmulsg(mubeta2(N1M1,n),
3489 5463326 : mfcusptracecache(M1, k, n, Dn, gel(S,M1), cache)));
3490 5787769 : for (j = 2; j < lg(Dg); j++) /* skip d = 1, done above */
3491 : {
3492 324443 : long d = Dg[j], ndd = n/(d*d), M = M1/d;
3493 324443 : GEN z = mulsi(mubeta2(N1M1,ndd), powuu(d,k-1)), C = vchip_lift(VCHIP,d,z);
3494 324443 : GEN Dndd = div_restrict(Dn, ndd);
3495 324443 : s = gadd(s, gmul(C, mfcusptracecache(M, k, ndd, Dndd, gel(S,M), cache)));
3496 : }
3497 5463326 : s = vchip_mod(VCHIP, s);
3498 : }
3499 1879315 : 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 8400 : mfolddim_i(long N, long k, GEN CHIP, GEN vSP)
3531 : {
3532 8400 : long S, i, l, F = mfcharmodulus(CHIP), N1 = N / F, N2;
3533 : GEN D;
3534 8400 : newd_params(N1, &N2); /* will ensure mubeta != 0 */
3535 8400 : D = mydivisorsu(N1/N2); l = lg(D); S = 0;
3536 8400 : if (k == 1 && !vSP) vSP = get_vDIH(N, divisorsNF(N, F));
3537 32557 : for (i = 2; i < l; i++)
3538 : {
3539 24157 : long d = mfcuspdim_i(N / D[i], k, CHIP, vSP);
3540 24157 : if (d) S -= mubeta(D[i]) * d;
3541 : }
3542 8400 : 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 15869 : mfnewdim(long N, long k, GEN CHI)
3554 : {
3555 : pari_sp av;
3556 : long S, F;
3557 15869 : GEN vSP, CHIP = mfchartoprimitive(CHI, &F);
3558 15869 : vSP = (k == 1)? get_vDIH(N, divisorsNF(N, F)): NULL;
3559 15869 : S = mfcuspdim_i(N, k, CHIP, vSP); if (!S) return 0;
3560 7903 : 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 966 : mftraceform_new(long N, long k, GEN CHI)
3566 : {
3567 : GEN T;
3568 966 : if (k == 1) return initwt1newtrace(mfinit_Nkchi(N, 1, CHI, mf_CUSP, 0));
3569 945 : T = initnewtrace(N,CHI); if (!T) return mftrivial();
3570 945 : 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 gc_GEN(av, mftraceform_i(NK,space)); }
3596 :
3597 : static GEN
3598 17703 : hecke_data(long N, long n)
3599 17703 : { 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 24829 : mfhecke_i(long n, long N, GEN F)
3612 : {
3613 24829 : if (n == 1) return F;
3614 17332 : 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 gc_GEN(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 36022 : mfbd_i(GEN F, long d)
3645 : {
3646 : GEN D, NK, gk, CHI;
3647 36022 : if (d == 1) return F;
3648 13559 : if (d <= 0) pari_err_TYPE("mfbd [d <= 0]", stoi(d));
3649 13559 : 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 13559 : gk = mf_get_gk(F); CHI = mf_get_CHI(F);
3652 13559 : if (typ(gk) != t_INT) CHI = mfcharmul(CHI, get_mfchar(utoi(d << 2)));
3653 13559 : NK = mkgNK(muliu(mf_get_gN(F), d), gk, CHI, mf_get_field(F));
3654 13559 : return tag2(t_MF_BD, NK, F, D);
3655 : }
3656 : GEN
3657 266 : mfbd(GEN F, long d)
3658 : {
3659 266 : pari_sp av = avma;
3660 266 : if (!checkmf_i(F)) pari_err_TYPE("mfbd",F);
3661 266 : return gc_GEN(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 gc_GEN(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 7756 : mkMinv(GEN W, GEN a, GEN b, GEN P)
3759 : {
3760 7756 : GEN A = (b && typ(b) == t_POL)? Q_remove_denom(QXQ_inv(b,P), &b): NULL;
3761 7756 : if (a && b)
3762 : {
3763 1330 : a = Qdivii(a,b);
3764 1330 : if (typ(a) == t_INT) b = gen_1; else { b = gel(a,2); a = gel(a,1); }
3765 1330 : if (is_pm1(a)) a = NULL;
3766 : }
3767 7756 : if (a) A = A? ZX_Z_mul(A,a): a; else if (!A) A = gen_1;
3768 7756 : if (!b) b = gen_1;
3769 7756 : if (!P) P = gen_0;
3770 7756 : return mkvec4(W,b,A,P);
3771 : }
3772 : /* M square invertible QabM, return [M',d], M*M' = d*Id */
3773 : static GEN
3774 602 : QabM_Minv(GEN M, GEN P, long n)
3775 : {
3776 : GEN dW, W, dM;
3777 602 : M = Q_remove_denom(M, &dM);
3778 602 : W = P? ZabM_inv(liftpol_shallow(M), P, n, &dW): ZM_inv(M, &dW);
3779 602 : 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 5012 : mfclean(GEN M, GEN P, long n, int ratlift)
3797 : {
3798 5012 : GEN W, v, y, z, d, Minv, dM, MdM = Q_remove_denom(M, &dM);
3799 5012 : if (n <= 2)
3800 3906 : W = ZM_pseudoinv(MdM, &v, &d);
3801 : else
3802 1106 : W = ZabM_pseudoinv_i(liftpol_shallow(MdM), P, n, &v, &d, ratlift);
3803 5012 : y = gel(v,1);
3804 5012 : z = gel(v,2);
3805 5012 : if (lg(z) != lg(MdM)) M = vecpermute(M,z);
3806 5012 : M = rowslice(M, 1, y[lg(y)-1]);
3807 5012 : Minv = mkMinv(W, dM, d, P);
3808 5012 : return mkvec3(y, Minv, M);
3809 : }
3810 : /* call mfclean using only CHI */
3811 : static GEN
3812 4060 : mfcleanCHI(GEN M, GEN CHI, int ratlift)
3813 : {
3814 4060 : long n = mfcharorder(CHI);
3815 4060 : GEN P = (n <= 2)? NULL: mfcharpol(CHI);
3816 4060 : 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 33880 : newtrace_stripped(GEN DATA)
3822 33880 : { return DATA && (lg(DATA) == 5 && typ(gel(DATA,3)) == t_INT); }
3823 : /* f a t_MF_NEWTRACE */
3824 : static GEN
3825 33880 : newtrace_DATA(long N, GEN f)
3826 : {
3827 33880 : GEN DATA = gel(f,2);
3828 33880 : 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 33880 : reset_cachenew(cachenew_t *cache, long N, GEN tf)
3834 : {
3835 : long i, n, l;
3836 33880 : GEN v, DATA = newtrace_DATA(N,tf);
3837 33880 : cache->DATA = DATA;
3838 33880 : if (!DATA) return;
3839 33775 : n = cache->n;
3840 33775 : v = cache->vfull; l = N+1; /* = lg(DATA) */
3841 2203516 : for (i = 1; i < l; i++)
3842 2169741 : if (typ(gel(v,i)) == t_INT && lg(gel(DATA,i)) != 1)
3843 53648 : gel(v,i) = const_vec(n, NULL);
3844 33775 : 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 13265 : init_cachenew(cachenew_t *cache, long n, long N, GEN tf)
3850 : {
3851 13265 : long i, l = N+1; /* = lg(tf.DATA) when DATA != NULL */
3852 : GEN v;
3853 13265 : cache->n = n;
3854 13265 : cache->vnew = v = cgetg(l, t_VEC);
3855 945119 : for (i = 1; i < l; i++) gel(v,i) = (N % i)? gen_0: const_vec(n, NULL);
3856 13265 : cache->newHIT = cache->newTOTAL = cache->cuspHIT = cache->cuspTOTAL = 0;
3857 13265 : cache->vfull = v = zerovec(N);
3858 13265 : reset_cachenew(cache, N, tf);
3859 13265 : }
3860 : static void
3861 17318 : dbg_cachenew(cachenew_t *C)
3862 : {
3863 17318 : 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 17318 : }
3867 :
3868 : /* newtrace_{N,k}(d*i), i = n0, ..., n */
3869 : static GEN
3870 179977 : colnewtrace(long n0, long n, long d, long N, long k, cachenew_t *cache)
3871 : {
3872 179977 : GEN v = cgetg(n-n0+2, t_COL);
3873 : long i;
3874 4692947 : for (i = n0; i <= n; i++) gel(v, i-n0+1) = mfnewtracecache(N, k, i*d, cache);
3875 179977 : 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 89117 : heckenewtrace(long m0, long m, long l, long N, long NBIG, long k, long n, cachenew_t *cache)
3881 : {
3882 89117 : long lD, a, k1, nl = n*l;
3883 89117 : GEN D, V, v = colnewtrace(m0, m, nl, N, k, cache); /* d=1 */
3884 : GEN VCHIP;
3885 89117 : if (n == 1) return v;
3886 61187 : VCHIP = cache->VCHIP;
3887 61187 : D = mydivisorsu(u_ppo(n, NBIG)); lD = lg(D);
3888 61187 : k1 = k - 1;
3889 150150 : for (a = 2; a < lD; a++)
3890 : { /* d > 1, (d,NBIG) = 1 */
3891 88963 : long i, j, d = D[a], c = ugcd(l, d), dl = d/c, m0d = ceildivuu(m0, dl);
3892 88963 : GEN C = vchip_lift(VCHIP, d, powuu(d, k1));
3893 : /* m0=0: i = 1 => skip F(0) = 0 */
3894 88963 : if (!m0) { i = 1; j = dl; } else { i = 0; j = m0d*dl; }
3895 88963 : V = colnewtrace(m0d, m/dl, nl/(d*c), N, k, cache);
3896 : /* C = chi(d) d^(k-1) */
3897 1078273 : for (; j <= m; i++, j += dl)
3898 989310 : gel(v,j-m0+1) = gadd(gel(v,j-m0+1), vchip_mod(VCHIP, gmul(C,gel(V,i+1))));
3899 : }
3900 61187 : 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 12383 : mkmf(GEN x1, GEN x2, GEN x3, GEN x4, GEN x5)
3959 : {
3960 12383 : GEN MF = obj_init(5, MF_SPLITN);
3961 12383 : gel(MF,1) = x1;
3962 12383 : gel(MF,2) = x2;
3963 12383 : gel(MF,3) = x3;
3964 12383 : gel(MF,4) = x4;
3965 12383 : 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 7644 : get_badj(long N, long FC)
3971 : {
3972 7644 : GEN fa = myfactoru(N), P = gel(fa,1), E = gel(fa,2);
3973 7644 : long i, b = 1, l = lg(P);
3974 20321 : for (i = 1; i < l; i++)
3975 12677 : if (E[i] > 1 && u_lval(FC, P[i]) < E[i]) b *= P[i];
3976 7644 : return b;
3977 : }
3978 : /* in place, assume perm strictly increasing */
3979 : static void
3980 1358 : vecpermute_inplace(GEN v, GEN perm)
3981 : {
3982 1358 : long i, l = lg(perm);
3983 11690 : for (i = 1; i < l; i++) gel(v,i) = gel(v,perm[i]);
3984 1358 : }
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 15624 : 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 15624 : dim = mfnewdim(N, k, CHI);
3996 15624 : if (!dim && !init) return NULL;
3997 7644 : sb = mfsturmNk(N, k);
3998 7644 : CHIP = mfchartoprimitive(CHI, &FC);
3999 : /* remove newtrace data from S to save space in output: negligible slowdown */
4000 7644 : tf = tag(t_MF_NEWTRACE, mkNK(N,k,CHIP), CHIP);
4001 7644 : badj = get_badj(N, FC);
4002 : /* try sbsmall first: Sturm bound not sharp for new space */
4003 7644 : SB = ceilA1(N, k);
4004 7644 : listj = cgetg(2*sb + 3, t_VECSMALL);
4005 373667 : for (j = ctlj = 1; ctlj < 2*sb + 3; j++)
4006 366023 : if (ugcd(j, badj) == 1) listj[ctlj++] = j;
4007 7644 : if (init)
4008 : {
4009 4144 : init_cachenew(cache, (SB+1)*listj[dim+1], N, tf);
4010 4144 : if (init == -1 || !dim) return NULL; /* old space or dim = 0 */
4011 : }
4012 : else
4013 3500 : reset_cachenew(cache, N, tf);
4014 : /* cache.DATA is not NULL */
4015 7175 : ord = mfcharorder(CHIP);
4016 7175 : P = ord <= 2? NULL: mfcharpol(CHIP);
4017 7175 : vj = cgetg(dim+1, t_VECSMALL);
4018 7175 : M = cgetg(dim+1, t_MAT);
4019 7182 : for (two = 1, ct = 0, jin = 1; two <= 2; two++)
4020 : {
4021 7182 : long a, jlim = jin + sb;
4022 22078 : for (a = jin; a <= jlim; a++)
4023 : {
4024 : GEN z, vecz;
4025 22071 : ct++; vj[ct] = listj[a];
4026 22071 : gel(M, ct) = heckenewtrace(0, SB, 1, N, N, k, vj[ct], cache);
4027 22071 : if (ct < dim) continue;
4028 :
4029 7854 : z = QabM_indexrank(M, P, ord);
4030 7854 : vecz = gel(z, 2); ct = lg(vecz) - 1;
4031 7854 : if (ct == dim) { M = mkvec3(z, gen_0, M); break; } /*maximal rank, done*/
4032 679 : vecpermute_inplace(M, vecz);
4033 679 : vecpermute_inplace(vj, vecz);
4034 : }
4035 7182 : 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 7175 : S = cgetg(dim + 1, t_VEC);
4045 28525 : for (j = 1; j <= dim; j++) gel(S, j) = mfhecke_i(vj[j], N, tf);
4046 7175 : dbg_cachenew(cache);
4047 7175 : mf1 = mkvec4(utoipos(N), utoipos(k), CHI, utoi(mf_NEW));
4048 7175 : 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 49 : mfinittonew(GEN mf)
4053 : {
4054 49 : GEN CHI = MF_get_CHI(mf), S = MF_get_S(mf), vMjd = MFcusp_get_vMjd(mf);
4055 49 : GEN M = MF_get_M(mf), vj, mf1;
4056 49 : long i, j, l, l0 = lg(S), N0 = MF_get_N(mf);
4057 252 : for (i = l0-1; i > 0; i--)
4058 : {
4059 238 : long N = gel(vMjd,i)[1];
4060 238 : if (N != N0) break;
4061 : }
4062 49 : if (i == l0-1) return NULL;
4063 42 : S = vecslice(S, i+1, l0-1); /* forms of conductor N0 */
4064 42 : l = lg(S); vj = cgetg(l, t_VECSMALL);
4065 245 : for (j = 1; j < l; j++) vj[j] = gel(vMjd,j+i)[2];
4066 42 : M = vecslice(M, lg(M)-lg(S)+1, lg(M)-1); /* their coefficients */
4067 42 : M = mfcleanCHI(M, CHI, 0);
4068 42 : mf1 = mkvec4(utoipos(N0), MF_get_gk(mf), CHI, utoi(mf_NEW));
4069 42 : 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 82859 : RgC_Bd_expand(long m0, long m, GEN v, long d, long m0d)
4075 : {
4076 : long i, j;
4077 : GEN w;
4078 82859 : if (d == 1) return v;
4079 23744 : w = zerocol(m-m0+1);
4080 23744 : if (!m0) { i = 1; j = d; } else { i = 0; j = m0d*d; }
4081 470547 : for (; j <= m; i++, j += d) gel(w,j-m0+1) = gel(v,i+1);
4082 23744 : 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 9128 : bhnmat_extend(GEN M, long m, long r, GEN S, cachenew_t *cache)
4090 : {
4091 9128 : long i, mr, m0, m0r, Nold = 0, jold = 0, l = lg(S);
4092 9128 : GEN MAT = cgetg(l, t_MAT), v = NULL;
4093 9128 : if (M) { m0 = nbrows(M); m0r = m0 * r; } else m0 = m0r = 0;
4094 9128 : mr = m*r;
4095 91987 : for (i = 1; i < l; i++)
4096 : {
4097 : long d, j, md, N;
4098 82859 : GEN c, f = bhn_parse(gel(S,i), &d,&j); /* t_MF_NEWTRACE */
4099 82859 : N = mf_get_N(f);
4100 82859 : md = ceildivuu(m0r,d);
4101 82859 : if (N != Nold) { reset_cachenew(cache, N, f); Nold = N; jold = 0; }
4102 82859 : if (!cache->DATA) { gel(MAT,i) = zerocol(m+1); continue; }
4103 82859 : if (j != jold || md)
4104 66619 : { v = heckenewtrace(md, mr/d, 1, N, N, mf_get_k(f), j,cache); jold=j; }
4105 82859 : c = RgC_Bd_expand(m0r, mr, v, d, md);
4106 82859 : if (r > 1) c = c_deflate(m-m0, r, c);
4107 82859 : if (M) c = shallowconcat(gel(M,i), c);
4108 82859 : gel(MAT,i) = c;
4109 : }
4110 9128 : return MAT;
4111 : }
4112 :
4113 : /* k > 1 */
4114 : static GEN
4115 3227 : mfinitcusp(long N, long k, GEN CHI, cachenew_t *cache, long space)
4116 : {
4117 : long L, l, lDN1, FC, N1, d1, i, init;
4118 3227 : GEN vS, vMjd, DN1, vmf, CHIP = mfchartoprimitive(CHI, &FC);
4119 :
4120 3227 : d1 = (space == mf_OLD)? mfolddim_i(N, k, CHIP, NULL): mfcuspdim(N, k, CHIP);
4121 3227 : if (!d1) return NULL;
4122 2919 : N1 = N/FC; DN1 = mydivisorsu(N1); lDN1 = lg(DN1);
4123 2919 : init = (space == mf_OLD)? -1: 1;
4124 2919 : vmf = cgetg(lDN1, t_VEC);
4125 17318 : for (i = lDN1 - 1, l = 1; i; i--)
4126 : { /* by decreasing level to allow cache */
4127 14399 : GEN mf = mfnewinit(FC*DN1[i], k, CHIP, cache, init);
4128 14399 : if (mf) gel(vmf, l++) = mf;
4129 14399 : init = 0;
4130 : }
4131 2919 : setlg(vmf,l); vmf = vecreverse(vmf); /* reorder by increasing level */
4132 :
4133 2919 : L = mfsturmNk(N, k)+1;
4134 2919 : vS = vectrunc_init(L);
4135 2919 : vMjd = vectrunc_init(L);
4136 9240 : for (i = 1; i < l; i++)
4137 : {
4138 6321 : GEN DNM, mf = gel(vmf,i), S = MF_get_S(mf), vj = MFnew_get_vj(mf);
4139 6321 : long a, lDNM, lS = lg(S), M = MF_get_N(mf);
4140 6321 : DNM = mydivisorsu(N / M); lDNM = lg(DNM);
4141 25578 : for (a = 1; a < lS; a++)
4142 : {
4143 19257 : GEN tf = gel(S,a);
4144 19257 : long b, j = vj[a];
4145 47929 : for (b = 1; b < lDNM; b++)
4146 : {
4147 28672 : long d = DNM[b];
4148 28672 : vectrunc_append(vS, mfbd_i(tf, d));
4149 28672 : vectrunc_append(vMjd, mkvecsmall3(M, j, d));
4150 : }
4151 : }
4152 : }
4153 2919 : return mkmf(NULL, cgetg(1, t_VEC), vS, vMjd, NULL);
4154 : }
4155 :
4156 : long
4157 4557 : mfsturm_mf(GEN mf)
4158 : {
4159 4557 : GEN Mindex = MF_get_Mindex(mf);
4160 4557 : long n = lg(Mindex)-1;
4161 4557 : return n? Mindex[n]-1: 0;
4162 : }
4163 :
4164 : long
4165 812 : mfsturm(GEN T)
4166 : {
4167 : long N, nk, dk;
4168 812 : GEN CHI, mf = checkMF_i(T);
4169 812 : 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 350 : mfeigenbasis(GEN mf)
4193 : {
4194 350 : pari_sp ltop = avma;
4195 : GEN F, S, v, vP;
4196 : long i, l, k, dS;
4197 :
4198 350 : mf = checkMF(mf);
4199 350 : k = MF_get_k(mf);
4200 350 : S = MF_get_S(mf); dS = lg(S)-1;
4201 350 : if (!dS) return cgetg(1, t_VEC);
4202 343 : F = MF_get_newforms(mf);
4203 343 : vP = MF_get_fields(mf);
4204 343 : 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 133 : GEN (*L)(GEN, GEN) = (MF_get_space(mf) == mf_FULL)? mflinear: mflinear_bhn;
4217 133 : l = lg(F); v = cgetg(l, t_VEC);
4218 462 : for (i = 1; i < l; i++) gel(v,i) = L(mf, gel(F,i));
4219 : }
4220 896 : for (i = 1; i < l; i++) mf_setfield(gel(v,i), gel(vP,i));
4221 343 : return gc_GEN(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 7420 : Minv_RgC_mul(GEN Minv, GEN v)
4227 : {
4228 7420 : GEN M = gel(Minv,1), d = gel(Minv,2), A = gel(Minv,3);
4229 7420 : v = RgM_RgC_mul(M, v);
4230 7420 : if (!equali1(A))
4231 : {
4232 1869 : if (typ(A) == t_POL && degpol(A) > 0) A = mkpolmod(A, gel(Minv,4));
4233 1869 : v = RgC_Rg_mul(v, A);
4234 : }
4235 7420 : if (!equali1(d)) v = RgC_Rg_div(v, d);
4236 7420 : return v;
4237 : }
4238 : static GEN
4239 1295 : Minv_RgM_mul(GEN Minv, GEN B)
4240 : {
4241 1295 : long j, l = lg(B);
4242 1295 : GEN M = cgetg(l, t_MAT);
4243 6013 : for (j = 1; j < l; j++) gel(M,j) = Minv_RgC_mul(Minv, gel(B,j));
4244 1295 : 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 1386 : vecpermute_partial(GEN v, GEN perm, long *r)
4266 : {
4267 1386 : long i, n = lg(v)-1, l = lg(perm);
4268 : GEN w;
4269 1386 : 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 1414 : mftobasis_i(GEN mf, GEN F)
4282 : {
4283 : GEN v, Mindex, Minv;
4284 1414 : if (!MF_get_dim(mf)) return cgetg(1, t_COL);
4285 1414 : Mindex = MF_get_Mindex(mf);
4286 1414 : Minv = MF_get_Minv(mf);
4287 1414 : 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 1148 : GEN A = gel(Minv,1), d = gel(Minv,2);
4296 : long r;
4297 1148 : v = F;
4298 1148 : switch(typ(F))
4299 : {
4300 0 : case t_SER: v = sertocol(v);
4301 1148 : case t_VEC: case t_COL: break;
4302 0 : default: pari_err_TYPE("mftobasis", F);
4303 : }
4304 1148 : if (lg(v) == 1) pari_err_TYPE("mftobasis",v);
4305 1148 : v = vecpermute_partial(v, Mindex, &r);
4306 1148 : 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 910 : const_mat(long n, GEN x)
4316 : {
4317 910 : long j, l = n+1;
4318 910 : GEN A = cgetg(l,t_MAT);
4319 6902 : for (j = 1; j < l; j++) gel(A,j) = const_col(n, x);
4320 910 : 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 455 : mftonew_i(GEN mf, GEN L, long *plevel)
4326 : {
4327 : GEN S, listMjd, CHI, res, Aclos, Acoef, D, perm;
4328 455 : long N1, LC, lD, i, l, t, level, N = MF_get_N(mf);
4329 :
4330 455 : if (MF_get_k(mf) == 1) pari_err_IMPL("mftonew in weight 1");
4331 455 : listMjd = MFcusp_get_vMjd(mf);
4332 455 : CHI = MF_get_CHI(mf); LC = mfcharconductor(CHI);
4333 455 : S = MF_get_S(mf);
4334 :
4335 455 : N1 = N/LC;
4336 455 : D = mydivisorsu(N1); lD = lg(D);
4337 455 : perm = cgetg(N1+1, t_VECSMALL);
4338 3451 : for (i = 1; i < lD; i++) perm[D[i]] = i;
4339 455 : Aclos = const_mat(lD-1, cgetg(1,t_VEC));
4340 455 : Acoef = const_mat(lD-1, cgetg(1,t_VEC));
4341 455 : l = lg(listMjd);
4342 4669 : for (i = 1; i < l; i++)
4343 : {
4344 : long M, d;
4345 : GEN v;
4346 4214 : if (gequal0(gel(L,i))) continue;
4347 469 : v = gel(listMjd, i);
4348 469 : M = perm[ v[1]/LC ];
4349 469 : d = perm[ v[3] ];
4350 469 : gcoeff(Aclos,M,d) = vec_append(gcoeff(Aclos,M,d), gel(S,i));
4351 469 : gcoeff(Acoef,M,d) = shallowconcat(gcoeff(Acoef,M,d), gel(L,i));
4352 : }
4353 455 : res = cgetg(l, t_VEC); level = 1;
4354 3451 : for (i = t = 1; i < lD; i++)
4355 : {
4356 2996 : long j, M = D[i]*LC;
4357 2996 : GEN gM = utoipos(M);
4358 26530 : for (j = 1; j < lD; j++)
4359 : {
4360 23534 : GEN vf = gcoeff(Aclos,i,j), C, NK;
4361 : long d;
4362 23534 : if (lg(vf) == 1) continue;
4363 427 : d = D[j];
4364 427 : C = gcoeff(Acoef,i,j);
4365 427 : NK = mf_get_NK(gel(vf, 1));
4366 427 : if (d > 1)
4367 : { /* remove mfbd(, d) wrappers */
4368 175 : long h, lf = lg(vf);
4369 357 : for (h = 1; h < lf; h++)
4370 : {
4371 182 : GEN fd = gel(vf, h);
4372 182 : if (mf_get_type(fd) != t_MF_BD || !equaliu(gel(fd,3), d))
4373 0 : pari_err_BUG("mftonew [inconsistent multiplier]");
4374 182 : gel(vf, h) = gel(fd, 2);
4375 : }
4376 : }
4377 427 : level = ulcm(level, M*d);
4378 427 : gel(res,t++) = mkvec3(gM, utoipos(d), mflinear_i(NK,vf,C));
4379 : }
4380 : }
4381 455 : if (plevel) *plevel = level;
4382 455 : setlg(res, t); return res;
4383 : }
4384 : GEN
4385 217 : mftonew(GEN mf, GEN F)
4386 : {
4387 217 : pari_sp av = avma;
4388 : GEN ES;
4389 : long s;
4390 217 : mf = checkMF(mf);
4391 217 : s = MF_get_space(mf);
4392 217 : if (s != mf_FULL && s != mf_CUSP)
4393 7 : pari_err_TYPE("mftonew [not a full or cuspidal space]", mf);
4394 210 : ES = mftobasisES(mf,F);
4395 203 : if (!gequal0(gel(ES,1)))
4396 0 : pari_err_TYPE("mftonew [not a cuspidal form]", F);
4397 203 : F = gel(ES,2);
4398 203 : return gc_GEN(av, mftonew_i(mf,F, NULL));
4399 : }
4400 :
4401 : static GEN mfeisenstein_i(long k, GEN CHI1, GEN CHI2);
4402 :
4403 : /* mfinit(F * Theta) */
4404 : static GEN
4405 98 : mf2init(GEN mf)
4406 : {
4407 98 : GEN CHI = MF_get_CHI(mf), gk = gadd(MF_get_gk(mf), ghalf);
4408 98 : long N = MF_get_N(mf);
4409 98 : return mfinit_Nkchi(N, itou(gk), mfchiadjust(CHI, gk, N), mf_FULL, 0);
4410 : }
4411 :
4412 : static long
4413 623 : mfvec_first_cusp(GEN v)
4414 : {
4415 623 : long i, l = lg(v);
4416 1519 : for (i = 1; i < l; i++)
4417 : {
4418 1414 : GEN F = gel(v,i);
4419 1414 : long t = mf_get_type(F);
4420 1414 : if (t == t_MF_BD) { F = gel(F,2); t = mf_get_type(F); }
4421 1414 : if (t == t_MF_HECKE) { F = gel(F,3); t = mf_get_type(F); }
4422 1414 : if (t == t_MF_NEWTRACE) break;
4423 : }
4424 623 : return i;
4425 : }
4426 : /* vF a vector of mf F of type DIV(LINEAR(BAS,L), f) in (lcm) level N,
4427 : * F[2]=LINEAR(BAS,L), F[2][2]=BAS=fixed basis (Eisenstein or bhn type),
4428 : * F[2][3]=L, F[3]=f; mfvectomat(vF, n) */
4429 : static GEN
4430 630 : mflineardivtomat(long N, GEN vF, long n)
4431 : {
4432 630 : GEN F, M, f, fc, ME, dB, B, a0, V = NULL;
4433 630 : long lM, lF = lg(vF), j;
4434 :
4435 630 : if (lF == 1) return cgetg(1,t_MAT);
4436 623 : F = gel(vF,1);
4437 623 : if (lg(F) == 5)
4438 : { /* chicompat */
4439 273 : V = gmael(F,4,4);
4440 273 : if (typ(V) == t_INT) V = NULL;
4441 : }
4442 623 : M = gmael(F,2,2); /* BAS */
4443 623 : lM = lg(M);
4444 623 : j = mfvec_first_cusp(M);
4445 623 : if (j == 1) ME = NULL;
4446 : else
4447 : { /* BAS starts by Eisenstein */
4448 161 : ME = mfvectomat(vecslice(M,1,j-1), n, 1);
4449 161 : M = vecslice(M, j,lM-1);
4450 : }
4451 623 : M = bhnmat_extend_nocache(NULL, N, n, 1, M);
4452 623 : if (ME) M = shallowconcat(ME,M);
4453 : /* M = mfcoefs of BAS */
4454 623 : B = cgetg(lF, t_MAT);
4455 623 : dB= cgetg(lF, t_VEC);
4456 2947 : for (j = 1; j < lF; j++)
4457 : {
4458 2324 : GEN g = gel(vF, j); /* t_MF_DIV */
4459 2324 : gel(B,j) = RgM_RgC_mul(M, gmael(g,2,3));
4460 2324 : gel(dB,j)= gmael(g,2,4);
4461 : }
4462 623 : f = mfcoefsser(gel(F,3),n);
4463 623 : a0 = polcoef_i(f, 0, -1);
4464 623 : if (gequal0(a0) || gequal1(a0))
4465 322 : a0 = NULL;
4466 : else
4467 301 : f = gdiv(ser_unscale(f, a0), a0);
4468 623 : fc = ginv(f);
4469 2947 : for (j = 1; j < lF; j++)
4470 : {
4471 2324 : pari_sp av = avma;
4472 2324 : GEN LISer = RgV_to_ser_full(gel(B,j)), f;
4473 2324 : if (a0) LISer = gdiv(ser_unscale(LISer, a0), a0);
4474 2324 : f = gmul(LISer, fc);
4475 2324 : if (a0) f = ser_unscale(f, ginv(a0));
4476 2324 : f = sertocol(f); setlg(f, n+2);
4477 2324 : if (!gequal1(gel(dB,j))) f = RgC_Rg_div(f, gel(dB,j));
4478 2324 : gel(B,j) = gc_upto(av,f);
4479 : }
4480 623 : if (V) B = gmodulo(QabM_tracerel(V, 0, B), gel(V,1));
4481 623 : return B;
4482 : }
4483 :
4484 : static GEN
4485 350 : mfheckemat_mfcoefs(GEN mf, GEN B, GEN DATA)
4486 : {
4487 350 : GEN Mindex = MF_get_Mindex(mf), Minv = MF_get_Minv(mf);
4488 350 : long j, l = lg(B), sb = mfsturm_mf(mf);
4489 350 : GEN b = MF_get_basis(mf), Q = cgetg(l, t_VEC);
4490 1827 : for (j = 1; j < l; j++)
4491 : {
4492 1477 : GEN v = hecke_i(sb, 1, gel(B,j), gel(b,j), DATA); /* Tn b[j] */
4493 1477 : settyp(v,t_COL); gel(Q,j) = vecpermute(v, Mindex);
4494 : }
4495 350 : return Minv_RgM_mul(Minv,Q);
4496 : }
4497 : /* T_p^2, p prime, 1/2-integral weight; B = mfcoefs(mf,sb*p^2,1) or (mf,sb,p^2)
4498 : * if p|N */
4499 : static GEN
4500 7 : mfheckemat_mfcoefs_p2(GEN mf, long p, GEN B)
4501 : {
4502 7 : pari_sp av = avma;
4503 7 : GEN DATA = heckef2_data(MF_get_N(mf), p*p);
4504 7 : return gc_upto(av, mfheckemat_mfcoefs(mf, B, DATA));
4505 : }
4506 : /* convert Mindex from row-index to mfcoef indexation: a(n) is stored in
4507 : * mfcoefs()[n+1], so subtract 1 from all indices */
4508 : static GEN
4509 49 : Mindex_as_coef(GEN mf)
4510 : {
4511 49 : GEN v, Mindex = MF_get_Mindex(mf);
4512 49 : long i, l = lg(Mindex);
4513 49 : v = cgetg(l, t_VECSMALL);
4514 210 : for (i = 1; i < l; i++) v[i] = Mindex[i]-1;
4515 49 : return v;
4516 : }
4517 : /* T_p, p prime; B = mfcoefs(mf,sb*p,1) or (mf,sb,p) if p|N; integral weight */
4518 : static GEN
4519 35 : mfheckemat_mfcoefs_p(GEN mf, long p, GEN B)
4520 : {
4521 35 : pari_sp av = avma;
4522 35 : GEN vm, Q, C, Minv = MF_get_Minv(mf);
4523 35 : long lm, k, i, j, l = lg(B), N = MF_get_N(mf);
4524 :
4525 35 : if (N % p == 0) return Minv_RgM_mul(Minv, rowpermute(B, MF_get_Mindex(mf)));
4526 21 : k = MF_get_k(mf);
4527 21 : C = gmul(mfchareval(MF_get_CHI(mf), p), powuu(p, k-1));
4528 21 : vm = Mindex_as_coef(mf); lm = lg(vm);
4529 21 : Q = cgetg(l, t_MAT);
4530 147 : for (j = 1; j < l; j++) gel(Q,j) = cgetg(lm, t_COL);
4531 147 : for (i = 1; i < lm; i++)
4532 : {
4533 126 : long m = vm[i], mp = m*p;
4534 126 : GEN Cm = (m % p) == 0? C : NULL;
4535 1260 : for (j = 1; j < l; j++)
4536 : {
4537 1134 : GEN S = gel(B,j), s = gel(S, mp + 1);
4538 1134 : if (Cm) s = gadd(s, gmul(C, gel(S, m/p + 1)));
4539 1134 : gcoeff(Q, i, j) = s;
4540 : }
4541 : }
4542 21 : return gc_upto(av, Minv_RgM_mul(Minv,Q));
4543 : }
4544 : /* Matrix of T(p), p prime, dim(mf) > 0 and integral weight */
4545 : static GEN
4546 343 : mfheckemat_p(GEN mf, long p)
4547 : {
4548 343 : pari_sp av = avma;
4549 343 : long N = MF_get_N(mf), sb = mfsturm_mf(mf);
4550 343 : GEN B = (N % p)? mfcoefs_mf(mf, sb * p, 1): mfcoefs_mf(mf, sb, p);
4551 343 : return gc_upto(av, mfheckemat_mfcoefs(mf, B, hecke_data(N,p)));
4552 : }
4553 :
4554 : /* mf_NEW != (0), weight > 1, p prime. Use
4555 : * T(p) T(j) = T(j*p) + p^{k-1} \chi(p) 1_{p | j, p \nmid N} T(j/p) */
4556 : static GEN
4557 910 : mfnewmathecke_p(GEN mf, long p)
4558 : {
4559 910 : pari_sp av = avma;
4560 910 : GEN tf, vj = MFnew_get_vj(mf), CHI = MF_get_CHI(mf);
4561 910 : GEN Mindex = MF_get_Mindex(mf), Minv = MF_get_Minv(mf);
4562 910 : long N = MF_get_N(mf), k = MF_get_k(mf);
4563 910 : long i, j, lvj = lg(vj), lim = vj[lvj-1] * p;
4564 910 : GEN M, perm, V, need = zero_zv(lim);
4565 910 : GEN C = (N % p)? gmul(mfchareval(CHI,p), powuu(p,k-1)): NULL;
4566 910 : tf = mftraceform_new(N, k, CHI);
4567 3927 : for (i = 1; i < lvj; i++)
4568 : {
4569 3017 : j = vj[i]; need[j*p] = 1;
4570 3017 : if (N % p && j % p == 0) need[j/p] = 1;
4571 : }
4572 910 : perm = zero_zv(lim);
4573 910 : V = cgetg(lim+1, t_VEC);
4574 12614 : for (i = j = 1; i <= lim; i++)
4575 11704 : if (need[i]) { gel(V,j) = mfhecke_i(i, N, tf); perm[i] = j; j++; }
4576 910 : setlg(V, j);
4577 910 : V = bhnmat_extend_nocache(NULL, N, mfsturm_mf(mf), 1, V);
4578 910 : V = rowpermute(V, Mindex); /* V[perm[i]] = coeffs(T_i newtrace) */
4579 910 : M = cgetg(lvj, t_MAT);
4580 3927 : for (i = 1; i < lvj; i++)
4581 : {
4582 : GEN t;
4583 3017 : j = vj[i]; t = gel(V, perm[j*p]);
4584 3017 : if (C && j % p == 0) t = RgC_add(t, RgC_Rg_mul(gel(V, perm[j/p]),C));
4585 3017 : gel(M,i) = t;
4586 : }
4587 910 : return gc_upto(av, Minv_RgM_mul(Minv, M));
4588 : }
4589 :
4590 : GEN
4591 77 : mfheckemat(GEN mf, GEN vn)
4592 : {
4593 77 : pari_sp av = avma;
4594 77 : long lv, lvP, i, N, dim, nk, dk, p, sb, flint = (typ(vn)==t_INT);
4595 : GEN CHI, res, vT, FA, B, vP;
4596 :
4597 77 : mf = checkMF(mf);
4598 77 : if (typ(vn) != t_VECSMALL) vn = gtovecsmall(vn);
4599 77 : N = MF_get_N(mf); CHI = MF_get_CHI(mf); Qtoss(MF_get_gk(mf), &nk, &dk);
4600 77 : dim = MF_get_dim(mf);
4601 77 : lv = lg(vn);
4602 77 : res = cgetg(lv, t_VEC);
4603 77 : FA = cgetg(lv, t_VEC);
4604 77 : vP = cgetg(lv, t_VEC);
4605 77 : vT = const_vec(vecsmall_max(vn), NULL);
4606 182 : for (i = 1; i < lv; i++)
4607 : {
4608 105 : ulong n = (ulong)labs(vn[i]);
4609 : GEN fa;
4610 105 : if (!n) pari_err_TYPE("mfheckemat", vn);
4611 105 : if (dk == 1 || uissquareall(n, &n)) fa = myfactoru(n);
4612 0 : else { n = 0; fa = myfactoru(1); } /* dummy: T_{vn[i]} = 0 */
4613 105 : vn[i] = n;
4614 105 : gel(FA,i) = fa;
4615 105 : gel(vP,i) = gel(fa,1);
4616 : }
4617 77 : vP = shallowconcat1(vP); vecsmall_sort(vP);
4618 77 : vP = vecsmall_uniq_sorted(vP); /* all primes occurring in vn */
4619 77 : lvP = lg(vP); if (lvP == 1) goto END;
4620 56 : p = vP[lvP-1];
4621 56 : sb = mfsturm_mf(mf);
4622 56 : if (dk == 1 && nk != 1 && MF_get_space(mf) == mf_NEW)
4623 21 : B = NULL; /* special purpose mfnewmathecke_p is faster */
4624 35 : else if (lvP == 2 && N % p == 0)
4625 21 : B = mfcoefs_mf(mf, sb, dk==2? p*p: p); /* single prime | N, can optimize */
4626 : else
4627 14 : B = mfcoefs_mf(mf, sb * (dk==2? p*p: p), 1); /* general initialization */
4628 126 : for (i = 1; i < lvP; i++)
4629 : {
4630 70 : long j, l, q, e = 1;
4631 : GEN C, Tp, u1, u0;
4632 70 : p = vP[i];
4633 189 : for (j = 1; j < lv; j++) e = maxss(e, z_lval(vn[j], p));
4634 70 : if (!B)
4635 28 : Tp = mfnewmathecke_p(mf, p);
4636 42 : else if (dk == 2)
4637 7 : Tp = mfheckemat_mfcoefs_p2(mf,p, (lvP==2||N%p)? B: matdeflate(sb,p*p,B));
4638 : else
4639 35 : Tp = mfheckemat_mfcoefs_p(mf, p, (lvP==2||N%p)? B: matdeflate(sb,p,B));
4640 70 : gel(vT, p) = Tp;
4641 70 : if (e == 1) continue;
4642 14 : u0 = gen_1;
4643 14 : if (dk == 2)
4644 : {
4645 0 : C = N % p? gmul(mfchareval(CHI,p*p), powuu(p, nk-2)): NULL;
4646 0 : if (e == 2) u0 = uutoQ(p+1,p); /* special case T_{p^4} */
4647 : }
4648 : else
4649 14 : C = N % p? gmul(mfchareval(CHI,p), powuu(p, nk-1)): NULL;
4650 28 : for (u1=Tp, q=p, l=2; l <= e; l++)
4651 : { /* u0 = T_{p^{l-2}}, u1 = T_{p^{l-1}} for l > 2 */
4652 14 : GEN v = gmul(Tp, u1);
4653 14 : if (C) v = gsub(v, gmul(C, u0));
4654 : /* q = p^l, vT[q] = T_q for k integer else T_{q^2} */
4655 14 : q *= p; u0 = u1; gel(vT, q) = u1 = v;
4656 : }
4657 : }
4658 56 : END:
4659 : /* vT[p^e] = T_{p^e} for all p^e occurring below */
4660 182 : for (i = 1; i < lv; i++)
4661 : {
4662 105 : long n = vn[i], j, lP;
4663 : GEN fa, P, E, M;
4664 105 : if (n == 0) { gel(res,i) = zeromat(dim,dim); continue; }
4665 105 : if (n == 1) { gel(res,i) = matid(dim); continue; }
4666 77 : fa = gel(FA,i);
4667 77 : P = gel(fa,1); lP = lg(P);
4668 77 : E = gel(fa,2); M = gel(vT, upowuu(P[1], E[1]));
4669 84 : for (j = 2; j < lP; j++) M = RgM_mul(M, gel(vT, upowuu(P[j], E[j])));
4670 77 : gel(res,i) = M;
4671 : }
4672 77 : if (flint) res = gel(res,1);
4673 77 : return gc_GEN(av, res);
4674 : }
4675 :
4676 : /* f = \sum_i v[i] T_listj[i] (Trace Form) attached to v; replace by f/a_1(f) */
4677 : static GEN
4678 1505 : mf_normalize(GEN mf, GEN v)
4679 : {
4680 1505 : GEN c, dc = NULL, M = MF_get_M(mf), Mindex = MF_get_Mindex(mf);
4681 1505 : v = Q_primpart(v);
4682 1505 : c = RgMrow_RgC_mul(M, v, 2); /* a_1(f) */
4683 1505 : if (gequal1(c)) return v;
4684 910 : if (typ(c) == t_POL) c = gmodulo(c, mfcharpol(MF_get_CHI(mf)));
4685 910 : if (typ(c) == t_POLMOD && varn(gel(c,1)) == 1 && degpol(gel(c,1)) >= 40
4686 7 : && Mindex[1] == 2
4687 7 : && mfcharorder(MF_get_CHI(mf)) <= 2)
4688 7 : { /* normalize using expansion at infinity (small coefficients) */
4689 7 : GEN w, P = gel(c,1), a1 = gel(c,2);
4690 7 : long i, l = lg(Mindex);
4691 7 : w = cgetg(l, t_COL);
4692 7 : gel(w,1) = gen_1;
4693 280 : for (i = 2; i < l; i++)
4694 : {
4695 273 : c = liftpol_shallow(RgMrow_RgC_mul(M, v, Mindex[i]));
4696 273 : gel(w,i) = QXQ_div(c, a1, P);
4697 : }
4698 : /* w = expansion at oo of normalized form */
4699 7 : v = Minv_RgC_mul(MF_get_Minv(mf), Q_remove_denom(w, &dc));
4700 7 : v = gmodulo(v, P); /* back to mfbasis coefficients */
4701 : }
4702 : else
4703 : {
4704 903 : c = ginv(c);
4705 903 : if (typ(c) == t_POLMOD) c = Q_remove_denom(c, &dc);
4706 903 : v = RgC_Rg_mul(v, c);
4707 : }
4708 910 : if (dc) v = RgC_Rg_div(v, dc);
4709 910 : return v;
4710 : }
4711 : static void
4712 441 : pol_red(GEN NF, GEN *pP, GEN *pa, long flag)
4713 : {
4714 441 : GEN dP, a, P = *pP;
4715 441 : long d = degpol(P);
4716 :
4717 441 : *pa = a = pol_x(varn(P));
4718 441 : if (d * (NF ? nf_get_degree(NF): 1) > 30) return;
4719 :
4720 434 : dP = RgX_disc(P);
4721 434 : if (typ(dP) != t_INT)
4722 112 : { dP = gnorm(dP); if (typ(dP) != t_INT) pari_err_BUG("mfnewsplit"); }
4723 434 : if (d == 2 || expi(dP) < 62)
4724 : {
4725 399 : if (expi(dP) < 31)
4726 399 : P = NF? rnfpolredabs(NF, P,flag): polredabs0(P,flag);
4727 : else
4728 0 : P = NF? rnfpolredbest(NF,P,flag): polredbest(P,flag);
4729 399 : if (flag)
4730 : {
4731 371 : a = gel(P,2); if (typ(a) == t_POLMOD) a = gel(a,2);
4732 371 : P = gel(P,1);
4733 : }
4734 : }
4735 434 : *pP = P;
4736 434 : *pa = a;
4737 : }
4738 :
4739 : /* Diagonalize and normalize. See mfsplit for meaning of flag. */
4740 : static GEN
4741 1078 : mfspclean(GEN mf, GEN mf0, GEN NF, long ord, GEN simplesp, long flag)
4742 : {
4743 1078 : const long vz = 1;
4744 1078 : long i, l = lg(simplesp), dim = MF_get_dim(mf);
4745 1078 : GEN res = cgetg(l, t_MAT), pols = cgetg(l, t_VEC);
4746 1078 : GEN zeros = (mf == mf0)? NULL: zerocol(dim - MF_get_dim(mf0));
4747 2611 : for (i = 1; i < l; i++)
4748 : {
4749 1533 : GEN ATP = gel(simplesp, i), A = gel(ATP,1), P = gel(ATP,3);
4750 1533 : long d = degpol(P);
4751 1533 : GEN a, v = (flag && d > flag)? NULL: gel(A,1);
4752 1533 : if (d == 1) P = pol_x(vz);
4753 : else
4754 : {
4755 441 : pol_red(NF, &P, &a, !!v);
4756 441 : if (v)
4757 : { /* Mod(a,P) root of charpoly(T), K*gpowers(a) = eigenvector of T */
4758 413 : GEN K, den, M = cgetg(d+1, t_MAT), T = gel(ATP,2);
4759 : long j;
4760 413 : T = shallowtrans(T);
4761 413 : gel(M,1) = vec_ei(d,1); /* basis of cyclic vectors */
4762 1330 : for (j = 2; j <= d; j++) gel(M,j) = RgM_RgC_mul(T, gel(M,j-1));
4763 413 : M = Q_primpart(M);
4764 147 : K = NF? ZabM_inv(liftpol_shallow(M), nf_get_pol(NF), ord, &den)
4765 413 : : ZM_inv(M,&den);
4766 413 : K = shallowtrans(K);
4767 413 : v = gequalX(a)? pol_x_powers(d, vz): RgXQ_powers(a, d-1, P);
4768 413 : v = gmodulo(RgM_RgC_mul(A, RgM_RgC_mul(K,v)), P);
4769 : }
4770 : }
4771 1533 : if (v)
4772 : {
4773 1505 : v = mf_normalize(mf0, v); if (zeros) v = shallowconcat(zeros,v);
4774 1505 : gel(res,i) = v; if (flag) setlg(res,i+1);
4775 : }
4776 : else
4777 28 : gel(res,i) = zerocol(dim);
4778 1533 : gel(pols,i) = P;
4779 : }
4780 1078 : return mkvec2(res, pols);
4781 : }
4782 :
4783 : /* return v = v_{X-r}(P), and set Z = P / (X-r)^v */
4784 : static long
4785 70 : RgX_valrem_root(GEN P, GEN r, GEN *Z)
4786 : {
4787 : long v;
4788 140 : for (v = 0; degpol(P); v++)
4789 : {
4790 140 : GEN t, Q = RgX_div_by_X_x(P, r, &t);
4791 140 : if (!gequal0(t)) break;
4792 70 : P = Q;
4793 : }
4794 70 : *Z = P; return v;
4795 : }
4796 : static GEN
4797 1519 : mynffactor(GEN NF, GEN P, long dimlim)
4798 : {
4799 : long i, l, v;
4800 : GEN R, E;
4801 1519 : if (dimlim != 1)
4802 : {
4803 952 : R = NF? nffactor(NF, P): QX_factor(P);
4804 952 : if (!dimlim) return R;
4805 21 : E = gel(R,2);
4806 21 : R = gel(R,1); l = lg(R);
4807 98 : for (i = 1; i < l; i++)
4808 91 : if (degpol(gel(R,i)) > dimlim) break;
4809 21 : if (i == 1) return NULL;
4810 21 : setlg(E,i);
4811 21 : setlg(R,i); return mkmat2(R, E);
4812 : }
4813 : /* dimlim = 1 */
4814 567 : R = nfroots(NF, P); l = lg(R);
4815 567 : if (l == 1) return NULL;
4816 504 : v = varn(P);
4817 504 : settyp(R, t_COL);
4818 504 : if (degpol(P) == l-1)
4819 448 : E = const_col(l-1, gen_1);
4820 : else
4821 : {
4822 56 : E = cgetg(l, t_COL);
4823 126 : for (i = 1; i < l; i++) gel(E,i) = utoi(RgX_valrem_root(P, gel(R,i), &P));
4824 : }
4825 504 : R = deg1_from_roots(R, v);
4826 504 : return mkmat2(R, E);
4827 : }
4828 :
4829 : /* Let K be a number field attached to NF (Q if NF = NULL). A K-vector
4830 : * space of dimension d > 0 is given by a t_MAT A (n x d, full column rank)
4831 : * giving a K-basis, X a section (d x n: left pseudo-inverse of A). Return a
4832 : * pair (T, fa), where T is an element of the Hecke algebra (a sum of Tp taken
4833 : * from vector vTp) acting on A (a d x d t_MAT) and fa is the factorization of
4834 : * its characteristic polynomial, limited to factors of degree <= dimlim if
4835 : * dimlim != 0 (return NULL if there are no factors of degree <= dimlim) */
4836 : static GEN
4837 1344 : findbestsplit(GEN NF, GEN vTp, GEN A, GEN X, long dimlim, long vz)
4838 : {
4839 1344 : GEN T = NULL, Tkeep = NULL, fakeep = NULL;
4840 1344 : long lmax = 0, i, lT = lg(vTp);
4841 1771 : for (i = 1; i < lT; i++)
4842 : {
4843 1771 : GEN D, P, E, fa, TpA = gel(vTp,i);
4844 : long l;
4845 2800 : if (typ(TpA) == t_INT) break;
4846 1519 : if (lg(TpA) > lg(A)) TpA = RgM_mul(X, RgM_mul(TpA, A)); /* Tp | A */
4847 1519 : T = T ? RgM_add(T, TpA) : TpA;
4848 1519 : if (!NF) { P = QM_charpoly_ZX(T); setvarn(P, vz); }
4849 : else
4850 : {
4851 294 : P = charpoly(Q_remove_denom(T, &D), vz);
4852 294 : if (D) P = gdiv(RgX_unscale(P, D), powiu(D, degpol(P)));
4853 : }
4854 1519 : fa = mynffactor(NF, P, dimlim);
4855 1519 : if (!fa) return NULL;
4856 1456 : E = gel(fa, 2);
4857 : /* characteristic polynomial is separable ? */
4858 1456 : if (isint1(vecmax(E))) { Tkeep = T; fakeep = fa; break; }
4859 427 : l = lg(E);
4860 : /* characteristic polynomial has more factors than before ? */
4861 427 : if (l > lmax) { lmax = l; Tkeep = T; fakeep = fa; }
4862 : }
4863 1281 : return mkvec2(Tkeep, fakeep);
4864 : }
4865 :
4866 : static GEN
4867 294 : nfcontent(GEN nf, GEN v)
4868 : {
4869 294 : long i, l = lg(v);
4870 294 : GEN c = gel(v,1);
4871 1512 : for (i = 2; i < l; i++) c = idealadd(nf, c, gel(v,i));
4872 294 : if (typ(c) == t_MAT && gequal1(gcoeff(c,1,1))) c = gen_1;
4873 294 : return c;
4874 : }
4875 : static GEN
4876 455 : nf_primpart(GEN nf, GEN x)
4877 : {
4878 455 : switch(typ(x))
4879 : {
4880 294 : case t_COL:
4881 : {
4882 294 : GEN A = matalgtobasis(nf, x), c = nfcontent(nf, A);
4883 294 : if (typ(c) == t_INT) return x;
4884 35 : c = idealred_elt(nf,c);
4885 35 : A = Q_primpart( nfC_nf_mul(nf, A, Q_primpart(nfinv(nf,c))) );
4886 35 : A = liftpol_shallow( matbasistoalg(nf, A) );
4887 35 : if (gexpo(A) > gexpo(x)) A = x;
4888 35 : return A;
4889 : }
4890 455 : case t_MAT: pari_APPLY_same(nf_primpart(nf, gel(x,i)));
4891 0 : default:
4892 0 : pari_err_TYPE("nf_primpart", x);
4893 : return NULL; /*LCOV_EXCL_LINE*/
4894 : }
4895 : }
4896 :
4897 : /* rotate entries of v to accomodate new entry 'x' (push out oldest entry) */
4898 : static void
4899 1225 : vecpush(GEN v, GEN x)
4900 : {
4901 : long i;
4902 6125 : for (i = lg(v)-1; i > 1; i--) gel(v,i) = gel(v,i-1);
4903 1225 : gel(v,1) = x;
4904 1225 : }
4905 :
4906 : /* sort t_VEC of vector spaces by increasing dimension */
4907 : static GEN
4908 1078 : sort_by_dim(GEN v)
4909 : {
4910 1078 : long i, l = lg(v);
4911 1078 : GEN D = cgetg(l, t_VECSMALL);
4912 2611 : for (i = 1; i < l; i++) D[i] = lg(gmael(v,i,2));
4913 1078 : return vecpermute(v , vecsmall_indexsort(D));
4914 : }
4915 : static GEN
4916 1078 : split_starting_space(GEN mf)
4917 : {
4918 1078 : long d = MF_get_dim(mf), d2;
4919 1078 : GEN id = matid(d);
4920 1078 : switch(MF_get_space(mf))
4921 : {
4922 1071 : case mf_NEW:
4923 1071 : case mf_CUSP: return mkvec2(id, id);
4924 : }
4925 7 : d2 = lg(MF_get_S(mf))-1;
4926 7 : return mkvec2(vecslice(id, d-d2+1,d),
4927 : shallowconcat(zeromat(d2,d-d2),matid(d2)));
4928 : }
4929 : /* If dimlim > 0, keep only the dimension <= dimlim eigenspaces.
4930 : * See mfsplit for the meaning of flag. */
4931 : static GEN
4932 1477 : split_ii(GEN mf, long dimlim, long flag, GEN vSP, long *pnewd)
4933 : {
4934 : forprime_t iter;
4935 1477 : GEN CHI = MF_get_CHI(mf), empty = cgetg(1, t_VEC), mf0 = mf;
4936 : GEN NF, POLCYC, todosp, Tpbigvec, simplesp;
4937 1477 : long N = MF_get_N(mf), k = MF_get_k(mf);
4938 1477 : long ord, FC, NEWT, dimsimple = 0, newd = -1;
4939 1477 : const long NBH = 5, vz = 1;
4940 : ulong p;
4941 :
4942 1477 : switch(MF_get_space(mf))
4943 : {
4944 1183 : case mf_NEW: break;
4945 287 : case mf_CUSP:
4946 : case mf_FULL:
4947 : {
4948 : GEN CHIP;
4949 287 : if (k > 1) { mf0 = mfinittonew(mf); break; }
4950 259 : CHIP = mfchartoprimitive(CHI, NULL);
4951 259 : newd = lg(MF_get_S(mf))-1 - mfolddim_i(N, k, CHIP, vSP);
4952 259 : break;
4953 : }
4954 7 : default: pari_err_TYPE("mfsplit [space does not contain newspace]", mf);
4955 : return NULL; /*LCOV_EXCL_LINE*/
4956 : }
4957 1470 : if (newd < 0) newd = mf0? MF_get_dim(mf0): 0;
4958 1470 : *pnewd = newd;
4959 1470 : if (!newd) return mkvec2(cgetg(1, t_MAT), empty);
4960 :
4961 1078 : NEWT = (k > 1 && MF_get_space(mf0) == mf_NEW);
4962 1078 : todosp = mkvec( split_starting_space(mf0) );
4963 1078 : simplesp = empty;
4964 1078 : FC = mfcharconductor(CHI);
4965 1078 : ord = mfcharorder(CHI);
4966 1078 : if (ord <= 2) NF = POLCYC = NULL;
4967 : else
4968 : {
4969 210 : POLCYC = mfcharpol(CHI);
4970 210 : NF = nfinit(POLCYC,DEFAULTPREC);
4971 : }
4972 1078 : Tpbigvec = zerovec(NBH);
4973 1078 : u_forprime_init(&iter, 2, ULONG_MAX);
4974 1512 : while (dimsimple < newd && (p = u_forprime_next(&iter)))
4975 : {
4976 : GEN nextsp;
4977 : long ind;
4978 1512 : if (N % (p*p) == 0 && N/p % FC == 0) continue; /* T_p = 0 in this case */
4979 1225 : vecpush(Tpbigvec, NEWT? mfnewmathecke_p(mf0,p): mfheckemat_p(mf0,p));
4980 1225 : nextsp = empty;
4981 1624 : for (ind = 1; ind < lg(todosp); ind++)
4982 : {
4983 1344 : GEN tmp = gel(todosp, ind), fa, P, E, D, Tp, DTp;
4984 1344 : GEN A = gel(tmp, 1);
4985 1344 : GEN X = gel(tmp, 2);
4986 : long lP, i;
4987 1344 : tmp = findbestsplit(NF, Tpbigvec, A, X, dimlim, vz);
4988 1463 : if (!tmp) continue; /* nothing there */
4989 1281 : Tp = gel(tmp, 1);
4990 1281 : fa = gel(tmp, 2);
4991 1281 : P = gel(fa, 1);
4992 1281 : E = gel(fa, 2); lP = lg(P);
4993 : /* lP > 1 */
4994 1281 : if (DEBUGLEVEL) err_printf("Exponents = %Ps\n", E);
4995 1281 : if (lP == 2)
4996 : {
4997 868 : GEN P1 = gel(P,1);
4998 868 : long e1 = itos(gel(E,1)), d1 = degpol(P1);
4999 868 : if (e1 * d1 == lg(Tp)-1)
5000 : {
5001 819 : if (e1 > 1) nextsp = vec_append(nextsp, mkvec2(A,X));
5002 : else
5003 : { /* simple module */
5004 721 : simplesp = vec_append(simplesp, mkvec3(A,Tp,P1));
5005 966 : if ((dimsimple += d1) == newd) goto END;
5006 : }
5007 119 : continue;
5008 : }
5009 : }
5010 : /* Found splitting */
5011 462 : DTp = Q_remove_denom(Tp, &D);
5012 1260 : for (i = 1; i < lP; i++)
5013 : {
5014 1043 : GEN Ai, Xi, dXi, AAi, v, y, Pi = gel(P,i);
5015 1043 : Ai = RgX_RgM_eval(D? RgX_rescale(Pi,D): Pi, DTp);
5016 1043 : Ai = QabM_ker(Ai, POLCYC, ord);
5017 1043 : if (NF) Ai = nf_primpart(NF, Ai);
5018 :
5019 1043 : AAi = RgM_mul(A, Ai);
5020 : /* gives section, works on nonsquare matrices */
5021 1043 : Xi = QabM_pseudoinv(Ai, POLCYC, ord, &v, &dXi);
5022 1043 : Xi = RgM_Rg_div(Xi, dXi);
5023 1043 : y = gel(v,1);
5024 1043 : if (isint1(gel(E,i)))
5025 : {
5026 812 : GEN Tpi = RgM_mul(Xi, RgM_mul(rowpermute(Tp,y), Ai));
5027 812 : simplesp = vec_append(simplesp, mkvec3(AAi, Tpi, Pi));
5028 812 : if ((dimsimple += degpol(Pi)) == newd) goto END;
5029 : }
5030 : else
5031 : {
5032 231 : Xi = RgM_mul(Xi, rowpermute(X,y));
5033 231 : nextsp = vec_append(nextsp, mkvec2(AAi, Xi));
5034 : }
5035 : }
5036 : }
5037 280 : todosp = nextsp; if (lg(todosp) == 1) break;
5038 : }
5039 0 : END:
5040 1078 : if (DEBUGLEVEL) err_printf("end split, need to clean\n");
5041 1078 : return mfspclean(mf, mf0, NF, ord, sort_by_dim(simplesp), flag);
5042 : }
5043 : static GEN
5044 28 : dim_filter(GEN v, long dim)
5045 : {
5046 28 : GEN P = gel(v,2);
5047 28 : long j, l = lg(P);
5048 140 : for (j = 1; j < l; j++)
5049 126 : if (degpol(gel(P,j)) > dim)
5050 : {
5051 14 : v = mkvec2(vecslice(gel(v,1),1,j-1), vecslice(P,1,j-1));
5052 14 : break;
5053 : }
5054 28 : return v;
5055 : }
5056 : static long
5057 287 : dim_sum(GEN v)
5058 : {
5059 287 : GEN P = gel(v,2);
5060 287 : long j, l = lg(P), d = 0;
5061 707 : for (j = 1; j < l; j++) d += degpol(gel(P,j));
5062 287 : return d;
5063 : }
5064 : static GEN
5065 1155 : split_i(GEN mf, long dimlim, long flag)
5066 1155 : { long junk; return split_ii(mf, dimlim, flag, NULL, &junk); }
5067 : /* mf is either already split or output by mfinit. Splitting is done only for
5068 : * newspace except in weight 1. If flag = 0 (default) split completely.
5069 : * If flag = d > 0, only give the Galois polynomials in degree > d
5070 : * Flag is ignored if dimlim = 1. */
5071 : GEN
5072 98 : mfsplit(GEN mf0, long dimlim, long flag)
5073 : {
5074 98 : pari_sp av = avma;
5075 98 : GEN v, mf = checkMF_i(mf0);
5076 98 : if (!mf) pari_err_TYPE("mfsplit", mf0);
5077 98 : if ((v = obj_check(mf, MF_SPLIT)))
5078 28 : { if (dimlim) v = dim_filter(v, dimlim); }
5079 70 : else if (dimlim && (v = obj_check(mf, MF_SPLITN)))
5080 21 : { v = (itos(gel(v,1)) >= dimlim)? dim_filter(gel(v,2), dimlim): NULL; }
5081 98 : if (!v)
5082 : {
5083 : long newd;
5084 70 : v = split_ii(mf, dimlim, flag, NULL, &newd);
5085 70 : if (lg(v) == 1) obj_insert(mf, MF_SPLITN, mkvec2(utoi(dimlim), v));
5086 70 : else if (!flag)
5087 : {
5088 49 : if (dim_sum(v) == newd) obj_insert(mf, MF_SPLIT,v);
5089 21 : else obj_insert(mf, MF_SPLITN, mkvec2(utoi(dimlim), v));
5090 : }
5091 : }
5092 98 : return gc_GEN(av, v);
5093 : }
5094 : static GEN
5095 238 : split(GEN mf) { return split_i(mf,0,0); }
5096 : GEN
5097 805 : MF_get_newforms(GEN mf) { return gel(obj_checkbuild(mf,MF_SPLIT,&split),1); }
5098 : GEN
5099 602 : MF_get_fields(GEN mf) { return gel(obj_checkbuild(mf,MF_SPLIT,&split),2); }
5100 :
5101 : /*************************************************************************/
5102 : /* Modular forms of Weight 1 */
5103 : /*************************************************************************/
5104 : /* S_1(G_0(N)), small N. Return 1 if definitely empty; return 0 if maybe
5105 : * nonempty */
5106 : static int
5107 16632 : wt1empty(long N)
5108 : {
5109 16632 : if (N <= 100) switch (N)
5110 : { /* nonempty [32/100] */
5111 5453 : case 23: case 31: case 39: case 44: case 46:
5112 : case 47: case 52: case 55: case 56: case 57:
5113 : case 59: case 62: case 63: case 68: case 69:
5114 : case 71: case 72: case 76: case 77: case 78:
5115 : case 79: case 80: case 83: case 84: case 87:
5116 : case 88: case 92: case 93: case 94: case 95:
5117 5453 : case 99: case 100: return 0;
5118 3549 : default: return 1;
5119 : }
5120 7630 : if (N <= 600) switch(N)
5121 : { /* empty [111/500] */
5122 336 : case 101: case 102: case 105: case 106: case 109:
5123 : case 113: case 121: case 122: case 123: case 125:
5124 : case 130: case 134: case 137: case 146: case 149:
5125 : case 150: case 153: case 157: case 162: case 163:
5126 : case 169: case 170: case 173: case 178: case 181:
5127 : case 182: case 185: case 187: case 193: case 194:
5128 : case 197: case 202: case 205: case 210: case 218:
5129 : case 221: case 226: case 233: case 241: case 242:
5130 : case 245: case 246: case 250: case 257: case 265:
5131 : case 267: case 269: case 274: case 277: case 281:
5132 : case 289: case 293: case 298: case 305: case 306:
5133 : case 313: case 314: case 317: case 326: case 337:
5134 : case 338: case 346: case 349: case 353: case 361:
5135 : case 362: case 365: case 369: case 370: case 373:
5136 : case 374: case 377: case 386: case 389: case 394:
5137 : case 397: case 401: case 409: case 410: case 421:
5138 : case 425: case 427: case 433: case 442: case 449:
5139 : case 457: case 461: case 466: case 481: case 482:
5140 : case 485: case 490: case 493: case 509: case 514:
5141 : case 521: case 530: case 533: case 534: case 538:
5142 : case 541: case 545: case 554: case 557: case 562:
5143 : case 565: case 569: case 577: case 578: case 586:
5144 336 : case 593: return 1;
5145 6979 : default: return 0;
5146 : }
5147 315 : return 0;
5148 : }
5149 :
5150 : static GEN
5151 28 : initwt1trace(GEN mf)
5152 : {
5153 28 : GEN S = MF_get_S(mf), v, H;
5154 : long l, i;
5155 28 : if (lg(S) == 1) return mftrivial();
5156 28 : H = mfheckemat(mf, Mindex_as_coef(mf));
5157 28 : l = lg(H); v = cgetg(l, t_VEC);
5158 63 : for (i = 1; i < l; i++) gel(v,i) = gtrace(gel(H,i));
5159 28 : v = Minv_RgC_mul(MF_get_Minv(mf), v);
5160 28 : return mflineardiv_linear(S, v, 1);
5161 : }
5162 : static GEN
5163 21 : initwt1newtrace(GEN mf)
5164 : {
5165 21 : GEN v, D, S, Mindex, CHI = MF_get_CHI(mf);
5166 21 : long FC, lD, i, sb, N1, N2, lM, N = MF_get_N(mf);
5167 21 : CHI = mfchartoprimitive(CHI, &FC);
5168 21 : if (N % FC || mfcharparity(CHI) == 1) return mftrivial();
5169 21 : D = mydivisorsu(N/FC); lD = lg(D);
5170 21 : S = MF_get_S(mf);
5171 21 : if (lg(S) == 1) return mftrivial();
5172 21 : N2 = newd_params2(N);
5173 21 : N1 = N / N2;
5174 21 : Mindex = MF_get_Mindex(mf);
5175 21 : lM = lg(Mindex);
5176 21 : sb = Mindex[lM-1];
5177 21 : v = zerovec(sb+1);
5178 42 : for (i = 1; i < lD; i++)
5179 : {
5180 21 : long M = FC*D[i], j;
5181 21 : GEN tf = initwt1trace(M == N? mf: mfinit_Nkchi(M, 1, CHI, mf_CUSP, 0));
5182 : GEN listd, w;
5183 21 : if (mf_get_type(tf) == t_MF_CONST) continue;
5184 21 : w = mfcoefs_i(tf, sb, 1);
5185 21 : if (M == N) { v = gadd(v, w); continue; }
5186 0 : listd = mydivisorsu(u_ppo(ugcd(N/M, N1), FC));
5187 0 : for (j = 1; j < lg(listd); j++)
5188 : {
5189 0 : long d = listd[j], d2 = d*d; /* coprime to FC */
5190 0 : GEN dk = mfchareval(CHI, d);
5191 0 : long NMd = N/(M*d), m;
5192 0 : for (m = 1; m <= sb/d2; m++)
5193 : {
5194 0 : long be = mubeta2(NMd, m);
5195 0 : if (be)
5196 : {
5197 0 : GEN c = gmul(dk, gmulsg(be, gel(w, m+1)));
5198 0 : long n = m*d2;
5199 0 : gel(v, n+1) = gadd(gel(v, n+1), c);
5200 : }
5201 : }
5202 : }
5203 : }
5204 21 : if (gequal0(gel(v,2))) return mftrivial();
5205 21 : v = vecpermute(v,Mindex);
5206 21 : v = Minv_RgC_mul(MF_get_Minv(mf), v);
5207 21 : return mflineardiv_linear(S, v, 1);
5208 : }
5209 :
5210 : /* i*p + 1, i*p < lim corresponding to a_p(f_j), a_{2p}(f_j)... */
5211 : static GEN
5212 1834 : pindices(long p, long lim)
5213 : {
5214 1834 : GEN v = cgetg(lim, t_VECSMALL);
5215 : long i, ip;
5216 22190 : for (i = 1, ip = p + 1; ip < lim; i++, ip += p) v[i] = ip;
5217 1834 : setlg(v, i); return v;
5218 : }
5219 :
5220 : /* assume !wt1empty(N), in particular N>25 */
5221 : /* Returns [[lim,p], mf (weight 2), p*lim x dim matrix] */
5222 : static GEN
5223 1834 : mf1_pre(long N)
5224 : {
5225 : pari_timer tt;
5226 : GEN mf, v, L, I, M, Minv, den;
5227 : long B, lim, LIM, p;
5228 :
5229 1834 : if (DEBUGLEVEL) timer_start(&tt);
5230 1834 : mf = mfinit_Nkchi(N, 2, mfchartrivial(), mf_CUSP, 0);
5231 1834 : if (DEBUGLEVEL)
5232 0 : timer_printf(&tt, "mf1basis [pre]: S_2(%ld), dim = %ld",
5233 : N, MF_get_dim(mf));
5234 1834 : M = MF_get_M(mf); Minv = MF_get_Minv(mf); den = gel(Minv,2);
5235 1834 : B = mfsturm_mf(mf);
5236 1834 : if (uisprime(N))
5237 : {
5238 392 : lim = 2 * MF_get_dim(mf); /* ensure mfstabiter's first kernel ~ square */
5239 392 : p = 2;
5240 : }
5241 : else
5242 : {
5243 : forprime_t S;
5244 1442 : u_forprime_init(&S, 2, N);
5245 2576 : while ((p = u_forprime_next(&S)))
5246 2576 : if (N % p) break;
5247 1442 : lim = B + 1;
5248 : }
5249 1834 : LIM = (N & (N - 1))? 2 * lim: 3 * lim; /* N power of 2 ? */
5250 1834 : L = mkvecsmall4(lim, LIM, mfsturmNk(N,1), p);
5251 1834 : M = bhnmat_extend_nocache(M, N, LIM-1, 1, MF_get_S(mf));
5252 1834 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis [pre]: bnfmat_extend");
5253 1834 : v = pindices(p, LIM);
5254 1834 : if (!LIM) return mkvec4(L, mf, M, v);
5255 1834 : I = RgM_Rg_div(ZM_mul(rowslice(M, B+2, LIM), gel(Minv,1)), den);
5256 1834 : I = Q_remove_denom(I, &den);
5257 1834 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis [prec]: Iden");
5258 1834 : return mkvec5(L, mf, M, v, mkvec2(I, den));
5259 : }
5260 :
5261 : /* lg(A) > 1, E a t_POL */
5262 : static GEN
5263 686 : mfmatsermul(GEN A, GEN E)
5264 : {
5265 686 : long j, l = lg(A), r = nbrows(A);
5266 686 : GEN M = cgetg(l, t_MAT);
5267 686 : E = RgXn_red_shallow(E, r+1);
5268 5866 : for (j = 1; j < l; j++)
5269 : {
5270 5180 : GEN c = RgV_to_RgX(gel(A,j), 0);
5271 5180 : gel(M, j) = RgX_to_RgC(RgXn_mul(c, E, r+1), r);
5272 : }
5273 686 : return M;
5274 : }
5275 : /* lg(Ap) > 1, Ep an Flxn */
5276 : static GEN
5277 1141 : mfmatsermul_Fl(GEN Ap, GEN Ep, ulong p)
5278 : {
5279 1141 : long j, l = lg(Ap), r = nbrows(Ap);
5280 1141 : GEN M = cgetg(l, t_MAT);
5281 42630 : for (j = 1; j < l; j++)
5282 : {
5283 41489 : GEN c = Flv_to_Flx(gel(Ap,j), 0);
5284 41489 : gel(M,j) = Flx_to_Flv(Flxn_mul(c, Ep, r+1, p), r);
5285 : }
5286 1141 : return M;
5287 : }
5288 :
5289 : /* CHI mod F | N, return mfchar of modulus N.
5290 : * FIXME: wasteful, G should be precomputed */
5291 : static GEN
5292 13048 : mfcharinduce(GEN CHI, long N)
5293 : {
5294 : GEN G, chi;
5295 13048 : if (mfcharmodulus(CHI) == N) return CHI;
5296 1463 : G = znstar0(utoipos(N), 1);
5297 1463 : chi = zncharinduce(gel(CHI,1), gel(CHI,2), G);
5298 1463 : CHI = leafcopy(CHI);
5299 1463 : gel(CHI,1) = G;
5300 1463 : gel(CHI,2) = chi; return CHI;
5301 : }
5302 :
5303 : static GEN
5304 3983 : gmfcharno(GEN CHI)
5305 : {
5306 3983 : GEN G = gel(CHI,1), chi = gel(CHI,2);
5307 3983 : return mkintmod(znconreyexp(G, chi), znstar_get_N(G));
5308 : }
5309 : static long
5310 13692 : mfcharno(GEN CHI)
5311 : {
5312 13692 : GEN n = znconreyexp(gel(CHI,1), gel(CHI,2));
5313 13692 : return itou(n);
5314 : }
5315 :
5316 : /* return k such that minimal mfcharacter in Galois orbit of CHI is CHI^k */
5317 : static long
5318 12138 : mfconreyminimize(GEN CHI)
5319 : {
5320 12138 : GEN G = gel(CHI,1), cyc, chi;
5321 12138 : cyc = ZV_to_zv(znstar_get_cyc(G));
5322 12138 : chi = ZV_to_zv(znconreychar(G, gel(CHI,2)));
5323 12138 : return zv_cyc_minimize(cyc, chi, coprimes_zv(mfcharorder(CHI)));
5324 : }
5325 :
5326 : /* find scalar c such that first nonzero entry of c*v is 1; return c*v */
5327 : static GEN
5328 2065 : RgV_normalize(GEN v, GEN *pc)
5329 : {
5330 2065 : long i, l = lg(v);
5331 5313 : for (i = 1; i < l; i++)
5332 : {
5333 5313 : GEN c = gel(v,i);
5334 5313 : if (!gequal0(c))
5335 : {
5336 2065 : if (gequal1(c)) break;
5337 679 : *pc = ginv(c); return RgV_Rg_mul(v, *pc);
5338 : }
5339 : }
5340 1386 : *pc = gen_1; return v;
5341 : }
5342 : /* pS != NULL; dim > 0 */
5343 : static GEN
5344 784 : mftreatdihedral(long N, GEN DIH, GEN POLCYC, long ordchi, GEN *pS)
5345 : {
5346 784 : long l = lg(DIH), lim = mfsturmNk(N, 1), i;
5347 784 : GEN Minv, C = cgetg(l, t_VEC), M = cgetg(l, t_MAT);
5348 2436 : for (i = 1; i < l; i++)
5349 : {
5350 1652 : GEN c, v = mfcoefs_i(gel(DIH,i), lim, 1);
5351 1652 : gel(M,i) = RgV_normalize(v, &c);
5352 1652 : gel(C,i) = Rg_col_ei(c, l-1, i);
5353 : }
5354 784 : Minv = gel(mfclean(M,POLCYC,ordchi,0),2);
5355 784 : M = RgM_Minv_mul(M, Minv);
5356 784 : C = RgM_Minv_mul(C, Minv);
5357 784 : *pS = vecmflinear(DIH, C); return M;
5358 : }
5359 :
5360 : /* same mode a maximal ideal above q */
5361 : static GEN
5362 2408 : Tpmod(GEN Ap, GEN A, ulong chip, long p, ulong q)
5363 : {
5364 2408 : GEN B = leafcopy(Ap);
5365 2408 : long i, ip, l = lg(B);
5366 86345 : for (i = 1, ip = p; ip < l; i++, ip += p)
5367 83937 : B[ip] = Fl_add(B[ip], Fl_mul(A[i], chip, q), q);
5368 2408 : return B;
5369 : }
5370 : /* Tp(f_1), ..., Tp(f_d) mod q */
5371 : static GEN
5372 301 : matTpmod(GEN xp, GEN x, ulong chip, long p, ulong q)
5373 2709 : { pari_APPLY_same(Tpmod(gel(xp,i), gel(x,i), chip, p, q)); }
5374 :
5375 : /* Ap[i] = a_{p*i}(F), A[i] = a_i(F), i = 1..lim
5376 : * Tp(f)[n] = a_{p*n}(f) + chi(p) a_{n/p}(f) * 1_{p | n} */
5377 : static GEN
5378 469 : Tp(GEN Ap, GEN A, GEN chip, long p)
5379 : {
5380 469 : GEN B = leafcopy(Ap);
5381 469 : long i, ip, l = lg(B);
5382 12915 : for (i = 1, ip = p; ip < l; i++, ip += p)
5383 12446 : gel(B,ip) = gadd(gel(B,ip), gmul(gel(A,i), chip));
5384 469 : return B;
5385 : }
5386 : /* Tp(f_1), ..., Tp(f_d) */
5387 : static GEN
5388 56 : matTp(GEN xp, GEN x, GEN chip, long p)
5389 525 : { pari_APPLY_same(Tp(gel(xp,i), gel(x,i), chip, p)); }
5390 :
5391 : static GEN
5392 378 : _RgXQM_mul(GEN x, GEN y, GEN T)
5393 378 : { return T? RgXQM_mul(x, y, T): RgM_mul(x, y); }
5394 : /* largest T-stable Q(CHI)-subspace of Q(CHI)-vector space spanned by columns
5395 : * of A */
5396 : static GEN
5397 28 : mfstabiter(GEN *pC, GEN A0, GEN chip, GEN TMP, GEN P, long ordchi)
5398 : {
5399 28 : GEN A, Ap, vp = gel(TMP,4), C = NULL;
5400 28 : long i, lA, lim1 = gel(TMP,1)[3], p = gel(TMP,1)[4];
5401 : pari_timer tt;
5402 :
5403 28 : Ap = rowpermute(A0, vp);
5404 28 : A = rowslice(A0, 2, nbrows(Ap)+1); /* remove a0 */
5405 : for(;;)
5406 28 : {
5407 56 : GEN R = shallowconcat(matTp(Ap, A, chip, p), A);
5408 56 : GEN B = QabM_ker(R, P, ordchi);
5409 56 : long lB = lg(B);
5410 56 : if (DEBUGLEVEL)
5411 0 : timer_printf(&tt, "mf1basis: Hecke intersection (dim %ld)", lB-1);
5412 56 : if (lB == 1) return NULL;
5413 56 : lA = lg(A); if (lB == lA) break;
5414 28 : B = rowslice(B, 1, lA-1);
5415 28 : Ap = _RgXQM_mul(Ap, B, P);
5416 28 : A = _RgXQM_mul(A, B, P);
5417 28 : C = C? _RgXQM_mul(C, B, P): B;
5418 : }
5419 28 : if (nbrows(A) < lim1)
5420 : {
5421 14 : A0 = rowslice(A0, 2, lim1);
5422 14 : A = C? _RgXQM_mul(A0, C, P): A0;
5423 : }
5424 : else /* all needed coefs computed */
5425 14 : A = rowslice(A, 1, lim1-1);
5426 28 : if (*pC) C = C? _RgXQM_mul(*pC, C, P): *pC;
5427 : /* put back a0 */
5428 119 : for (i = 1; i < lA; i++) gel(A,i) = vec_prepend(gel(A,i), gen_0);
5429 28 : *pC = C; return A;
5430 : }
5431 :
5432 : static long
5433 252 : mfstabitermod(GEN A, GEN vp, ulong chip, long p, ulong q)
5434 : {
5435 252 : GEN Ap, C = NULL;
5436 252 : Ap = rowpermute(A, vp);
5437 252 : A = rowslice(A, 2, nbrows(Ap)+1);
5438 : while (1)
5439 49 : {
5440 301 : GEN Rp = shallowconcat(matTpmod(Ap, A, chip, p, q), A);
5441 301 : GEN B = Flm_ker(Rp, q);
5442 301 : long lA = lg(A), lB = lg(B);
5443 301 : if (lB == 1) return 0;
5444 266 : if (lB == lA) return lA-1;
5445 49 : B = rowslice(B, 1, lA-1);
5446 49 : Ap = Flm_mul(Ap, B, q);
5447 49 : A = Flm_mul(A, B, q);
5448 49 : C = C? Flm_mul(C, B, q): B;
5449 : }
5450 : }
5451 :
5452 : static GEN
5453 595 : mfcharinv_i(GEN CHI)
5454 : {
5455 595 : GEN G = gel(CHI,1);
5456 595 : CHI = leafcopy(CHI); gel(CHI,2) = zncharconj(G, gel(CHI,2)); return CHI;
5457 : }
5458 :
5459 : /* upper bound dim S_1(Gamma_0(N),chi) performing the linear algebra mod p */
5460 : static long
5461 595 : mf1dimmod(GEN E1, GEN E, GEN chip, long ordchi, long dih, GEN TMP)
5462 : {
5463 595 : GEN E1i, A, vp, mf, C = NULL;
5464 595 : ulong q, r = QabM_init(ordchi, &q);
5465 : long lim, LIM, p;
5466 :
5467 595 : LIM = gel(TMP,1)[2]; lim = gel(TMP,1)[1];
5468 595 : mf= gel(TMP,2);
5469 595 : A = gel(TMP,3);
5470 595 : A = QabM_to_Flm(A, r, q);
5471 595 : E1 = QabX_to_Flx(E1, r, q);
5472 595 : E1i = Flxn_inv(E1, nbrows(A), q);
5473 595 : if (E)
5474 : {
5475 574 : GEN Iden = gel(TMP,5), I = gel(Iden,1), den = gel(Iden,2);
5476 574 : GEN Mindex = MF_get_Mindex(mf), F = rowslice(A, 1, LIM);
5477 574 : GEN E1ip = Flxn_red(E1i, LIM);
5478 574 : ulong d = den? umodiu(den, q): 1;
5479 574 : long i, nE = lg(E) - 1;
5480 : pari_sp av;
5481 :
5482 574 : I = ZM_to_Flm(I, q);
5483 574 : if (d != 1) I = Flm_Fl_mul(I, Fl_inv(d, q), q);
5484 574 : av = avma;
5485 1120 : for (i = 1; i <= nE; i++)
5486 : {
5487 889 : GEN e = Flxn_mul(E1ip, QabX_to_Flx(gel(E,i), r, q), LIM, q);
5488 889 : GEN B = mfmatsermul_Fl(F, e, q), z;
5489 889 : GEN B2 = Flm_mul(I, rowpermute(B, Mindex), q);
5490 889 : B = rowslice(B, lim+1,LIM);
5491 889 : z = Flm_ker(Flm_sub(B2, B, q), q);
5492 889 : if (lg(z)-1 == dih) return dih;
5493 546 : C = C? Flm_mul(C, z, q): z;
5494 546 : F = Flm_mul(F, z, q);
5495 546 : (void)gc_all(av, 2, &F,&C);
5496 : }
5497 231 : A = F;
5498 : }
5499 : /* use Schaeffer */
5500 252 : p = gel(TMP,1)[4]; vp = gel(TMP,4);
5501 252 : A = mfmatsermul_Fl(A, E1i, q);
5502 252 : return mfstabitermod(A, vp, Qab_to_Fl(chip, r, q), p, q);
5503 : }
5504 :
5505 : static GEN
5506 224 : mf1intermat(GEN A, GEN Mindex, GEN e, GEN Iden, long lim, GEN POLCYC)
5507 : {
5508 224 : long j, l = lg(A), LIM = nbrows(A);
5509 224 : GEN I = gel(Iden,1), den = gel(Iden,2), B = cgetg(l, t_MAT);
5510 :
5511 5257 : for (j = 1; j < l; j++)
5512 : {
5513 5033 : pari_sp av = avma;
5514 5033 : GEN c = RgV_to_RgX(gel(A,j), 0), c1, c2;
5515 5033 : c = RgX_to_RgC(RgXn_mul(c, e, LIM), LIM);
5516 5033 : if (POLCYC) c = liftpol_shallow(c);
5517 5033 : c1 = vecslice(c, lim+1, LIM);
5518 5033 : if (den) c1 = RgC_Rg_mul(c1, den);
5519 5033 : c2 = RgM_RgC_mul(I, vecpermute(c, Mindex));
5520 5033 : gel(B, j) = gc_upto(av, RgC_sub(c2, c1));
5521 : }
5522 224 : return B;
5523 : }
5524 : /* Compute the full S_1(\G_0(N),\chi); return NULL if space is empty; else
5525 : * if pS is NULL, return stoi(dim), where dim is the dimension; else *pS is
5526 : * set to a vector of forms making up a basis, and return the matrix of their
5527 : * Fourier expansions. pdih gives the dimension of the subspace generated by
5528 : * dihedral forms; TMP is from mf1_pre or NULL. */
5529 : static GEN
5530 11284 : mf1basis(long N, GEN CHI, GEN TMP, GEN vSP, GEN *pS, long *pdih)
5531 : {
5532 11284 : GEN E = NULL, EB, E1, E1i, dE1i, mf, A, C, POLCYC, DIH, Minv, chip;
5533 11284 : long nE = 0, p, LIM, lim, lim1, i, lA, dimp, ordchi, dih;
5534 : pari_timer tt;
5535 : pari_sp av;
5536 :
5537 11284 : if (pdih) *pdih = 0;
5538 11284 : if (pS) *pS = NULL;
5539 11284 : if (wt1empty(N) || mfcharparity(CHI) != -1) return NULL;
5540 10990 : ordchi = mfcharorder(CHI);
5541 10990 : if (uisprime(N) && ordchi > 4) return NULL;
5542 10962 : if (pS)
5543 : {
5544 3857 : DIH = mfdihedralcusp(N, CHI, vSP);
5545 3857 : dih = lg(DIH) - 1;
5546 : }
5547 : else
5548 : {
5549 7105 : DIH = NULL;
5550 7105 : dih = mfdihedralcuspdim(N, CHI, vSP);
5551 : }
5552 10962 : POLCYC = (ordchi <= 2)? NULL: mfcharpol(CHI);
5553 10962 : if (pdih) *pdih = dih;
5554 10962 : if (N <= 600) switch(N)
5555 : {
5556 : long m;
5557 126 : case 219: case 273: case 283: case 331: case 333: case 344: case 416:
5558 : case 438: case 468: case 491: case 504: case 546: case 553: case 563:
5559 : case 566: case 581: case 592:
5560 126 : break; /* one chi with both exotic and dihedral forms */
5561 9499 : default: /* only dihedral forms */
5562 9499 : if (!dih) return NULL;
5563 : /* fall through */
5564 : case 124: case 133: case 148: case 171: case 201: case 209: case 224:
5565 : case 229: case 248: case 261: case 266: case 288: case 296: case 301:
5566 : case 309: case 325: case 342: case 371: case 372: case 380: case 399:
5567 : case 402: case 403: case 404: case 408: case 418: case 432: case 444:
5568 : case 448: case 451: case 453: case 458: case 496: case 497: case 513:
5569 : case 522: case 527: case 532: case 576: case 579:
5570 : /* no chi with both exotic and dihedral; one chi with exotic forms */
5571 3248 : if (dih)
5572 : {
5573 2338 : if (!pS) return utoipos(dih);
5574 728 : return mftreatdihedral(N, DIH, POLCYC, ordchi, pS) ;
5575 : }
5576 910 : m = mfcharno(mfcharinduce(CHI,N));
5577 910 : if (N == 124 && (m != 67 && m != 87)) return NULL;
5578 784 : if (N == 133 && (m != 83 && m !=125)) return NULL;
5579 490 : if (N == 148 && (m !=105 && m !=117)) return NULL;
5580 364 : if (N == 171 && (m != 94 && m !=151)) return NULL;
5581 364 : if (N == 201 && (m != 29 && m !=104)) return NULL;
5582 364 : if (N == 209 && (m != 87 && m !=197)) return NULL;
5583 364 : if (N == 224 && (m != 95 && m !=191)) return NULL;
5584 364 : if (N == 229 && (m !=107 && m !=122)) return NULL;
5585 364 : if (N == 248 && (m != 87 && m !=191)) return NULL;
5586 273 : if (N == 261 && (m != 46 && m !=244)) return NULL;
5587 273 : if (N == 266 && (m != 83 && m !=125)) return NULL;
5588 273 : if (N == 288 && (m != 31 && m !=223)) return NULL;
5589 273 : if (N == 296 && (m !=105 && m !=265)) return NULL;
5590 : }
5591 595 : if (DEBUGLEVEL)
5592 0 : err_printf("mf1basis: start character %Ps, conductor = %ld, order = %ld\n",
5593 : gmfcharno(CHI), mfcharconductor(CHI), ordchi);
5594 595 : if (!TMP) TMP = mf1_pre(N);
5595 595 : lim = gel(TMP,1)[1]; LIM = gel(TMP,1)[2]; lim1 = gel(TMP,1)[3];
5596 595 : p = gel(TMP,1)[4];
5597 595 : mf = gel(TMP,2);
5598 595 : A = gel(TMP,3);
5599 595 : EB = mfeisensteinbasis(N, 1, mfcharinv_i(CHI));
5600 595 : nE = lg(EB) - 1;
5601 595 : E1 = RgV_to_RgX(mftocol(gel(EB,1), LIM-1, 1), 0); /* + O(x^LIM) */
5602 595 : if (--nE)
5603 574 : E = RgM_to_RgXV(mfvectomat(vecslice(EB, 2, nE+1), LIM-1, 1), 0);
5604 595 : chip = mfchareval(CHI, p); /* != 0 */
5605 595 : if (DEBUGLEVEL) timer_start(&tt);
5606 595 : av = avma; dimp = mf1dimmod(E1, E, chip, ordchi, dih, TMP);
5607 595 : set_avma(av);
5608 595 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: dim mod p is %ld", dimp);
5609 595 : if (!dimp) return NULL;
5610 280 : if (!pS) return utoi(dimp);
5611 224 : if (dimp == dih) return mftreatdihedral(N, DIH, POLCYC, ordchi, pS);
5612 168 : E1i = RgXn_inv(E1, LIM); /* E[1] does not vanish at oo */
5613 168 : if (POLCYC) E1i = liftpol_shallow(E1i);
5614 168 : E1i = Q_remove_denom(E1i, &dE1i);
5615 168 : if (DEBUGLEVEL)
5616 : {
5617 0 : GEN a0 = gel(E1,2);
5618 0 : if (typ(a0) == t_POLMOD) a0 = gnorm(a0);
5619 0 : a0 = Q_abs_shallow(a0);
5620 0 : timer_printf(&tt, "mf1basis: invert E; norm(a0(E)) = %Ps", a0);
5621 : }
5622 168 : C = NULL;
5623 168 : if (nE)
5624 : { /* mf attached to S2(N), fi = mfbasis(mf)
5625 : * M = coefs(f1,...,fd) up to LIM
5626 : * F = coefs(F1,...,FD) = M * C, for some matrix C over Q(chi),
5627 : * initially 1, eventually giving \cap_E S2 / E; D <= d.
5628 : * B = coefs(E/E1 F1, .., E/E1 FD); we want X in Q(CHI)^d and
5629 : * Y in Q(CHI)^D such that
5630 : * B * X = M * Y, i.e. Minv * rowpermute(B, Mindex * X) = Y
5631 : *(B - I * rowpermute(B, Mindex)) * X = 0.
5632 : * where I = M * Minv. Rows of (B - I * ...) are 0 up to lim so
5633 : * are not included */
5634 154 : GEN Mindex = MF_get_Mindex(mf), Iden = gel(TMP,5);
5635 : pari_timer TT;
5636 154 : pari_sp av = avma;
5637 154 : if (DEBUGLEVEL) timer_start(&TT);
5638 238 : for (i = 1; i <= nE; i++)
5639 : {
5640 224 : pari_sp av2 = avma;
5641 : GEN e, z, B;
5642 :
5643 224 : e = Q_primpart(RgXn_mul(E1i, gel(E,i), LIM));
5644 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: E[%ld] / E[1]", i+1);
5645 : /* the first time A is over Z and it is more efficient to lift than
5646 : * to let RgXn_mul use Kronecker's trick */
5647 224 : if (POLCYC && i == 1) e = liftpol_shallow(e);
5648 224 : B = mf1intermat(A, Mindex, e, Iden, lim, i == 1? NULL: POLCYC);
5649 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: ... intermat");
5650 224 : z = gc_upto(av2, QabM_ker(B, POLCYC, ordchi));
5651 224 : if (DEBUGLEVEL)
5652 0 : timer_printf(&TT, "mf1basis: ... kernel (dim %ld)",lg(z)-1);
5653 224 : if (lg(z) == 1) return NULL;
5654 224 : if (lg(z) == lg(A)) { set_avma(av2); continue; } /* no progress */
5655 224 : C = C? _RgXQM_mul(C, z, POLCYC): z;
5656 224 : A = _RgXQM_mul(A, z, POLCYC);
5657 224 : if (DEBUGLEVEL) timer_printf(&TT, "mf1basis: ... updates");
5658 224 : if (lg(z)-1 == dimp) break;
5659 84 : if (gc_needed(av, 1))
5660 : {
5661 0 : if (DEBUGMEM > 1) pari_warn(warnmem,"mf1basis i = %ld", i);
5662 0 : (void)gc_all(av, 2, &A, &C);
5663 : }
5664 : }
5665 154 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: intersection [total]");
5666 : }
5667 168 : lA = lg(A);
5668 168 : if (lA-1 == dimp)
5669 : {
5670 140 : A = mfmatsermul(rowslice(A, 1, lim1), E1i);
5671 140 : if (POLCYC) A = RgXQM_red(A, POLCYC);
5672 140 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: matsermul [1]");
5673 : }
5674 : else
5675 : {
5676 28 : A = mfmatsermul(A, E1i);
5677 28 : if (POLCYC) A = RgXQM_red(A, POLCYC);
5678 28 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: matsermul [2]");
5679 28 : A = mfstabiter(&C, A, chip, TMP, POLCYC, ordchi);
5680 28 : if (DEBUGLEVEL) timer_printf(&tt, "mf1basis: Hecke stability");
5681 28 : if (!A) return NULL;
5682 : }
5683 168 : if (dE1i) C = RgM_Rg_mul(C, dE1i);
5684 168 : if (POLCYC)
5685 : {
5686 147 : A = QXQM_to_mod_shallow(A, POLCYC);
5687 147 : C = QXQM_to_mod_shallow(C, POLCYC);
5688 : }
5689 168 : lA = lg(A);
5690 581 : for (i = 1; i < lA; i++)
5691 : {
5692 413 : GEN c, v = gel(A,i);
5693 413 : gel(A,i) = RgV_normalize(v, &c);
5694 413 : gel(C,i) = RgC_Rg_mul(gel(C,i), c);
5695 : }
5696 168 : Minv = gel(mfclean(A, POLCYC, ordchi, 0), 2);
5697 168 : A = RgM_Minv_mul(A, Minv);
5698 168 : C = RgM_Minv_mul(C, Minv);
5699 168 : *pS = vecmflineardiv0(MF_get_S(mf), C, gel(EB,1));
5700 168 : return A;
5701 : }
5702 :
5703 : static void
5704 413 : MF_set_space(GEN mf, long x) { gmael(mf,1,4) = utoi(x); }
5705 : static GEN
5706 252 : mf1_cusptonew(GEN mf, GEN vSP)
5707 : {
5708 252 : const long vy = 1;
5709 : long i, lP, dSnew, ct;
5710 252 : GEN vP, F, S, Snew, vF, v = split_ii(mf, 0, 0, vSP, &i);
5711 :
5712 252 : F = gel(v,1);
5713 252 : vP= gel(v,2); lP = lg(vP);
5714 252 : if (lP == 1) { obj_insert(mf, MF_SPLIT, v); return NULL; }
5715 238 : MF_set_space(mf, mf_NEW);
5716 238 : S = MF_get_S(mf);
5717 238 : dSnew = dim_sum(v);
5718 238 : Snew = cgetg(dSnew + 1, t_VEC); ct = 0;
5719 238 : vF = cgetg(lP, t_MAT);
5720 546 : for (i = 1; i < lP; i++)
5721 : {
5722 308 : GEN V, P = gel(vP,i), f = liftpol_shallow(gel(F,i));
5723 308 : long j, d = degpol(P);
5724 308 : gel(vF,i) = V = zerocol(dSnew);
5725 308 : if (d == 1)
5726 : {
5727 140 : gel(Snew, ct+1) = mflineardiv_linear(S, f, 0);
5728 140 : gel(V, ct+1) = gen_1;
5729 : }
5730 : else
5731 : {
5732 168 : f = RgXV_to_RgM(f,d);
5733 511 : for (j = 1; j <= d; j++)
5734 : {
5735 343 : gel(Snew, ct+j) = mflineardiv_linear(S, row(f,j), 0);
5736 343 : gel(V, ct+j) = mkpolmod(pol_xn(j-1,vy), P);
5737 : }
5738 : }
5739 308 : ct += d;
5740 : }
5741 238 : obj_insert(mf, MF_SPLIT, mkvec2(vF, vP));
5742 238 : gel(mf,3) = Snew; return mf;
5743 : }
5744 : static GEN
5745 3969 : mf1init(long N, GEN CHI, GEN TMP, GEN vSP, long space, long flraw)
5746 : {
5747 3969 : GEN mf, mf1, S, M = mf1basis(N, CHI, TMP, vSP, &S, NULL);
5748 3969 : if (!M) return NULL;
5749 952 : mf1 = mkvec4(stoi(N), gen_1, CHI, utoi(mf_CUSP));
5750 952 : mf = mkmf(mf1, cgetg(1,t_VEC), S, gen_0, NULL);
5751 952 : if (space == mf_NEW)
5752 : {
5753 252 : gel(mf,5) = mfcleanCHI(M,CHI, 0);
5754 252 : mf = mf1_cusptonew(mf, vSP); if (!mf) return NULL;
5755 238 : if (!flraw) M = mfcoefs_mf(mf, mfsturmNk(N,1)+1, 1);
5756 : }
5757 938 : gel(mf,5) = flraw? zerovec(3): mfcleanCHI(M, CHI, 0);
5758 938 : return mf;
5759 : }
5760 :
5761 : static GEN
5762 1029 : mfEMPTY(GEN mf1)
5763 : {
5764 1029 : GEN Minv = mkMinv(cgetg(1,t_MAT), NULL,NULL,NULL);
5765 1029 : GEN M = mkvec3(cgetg(1,t_VECSMALL), Minv, cgetg(1,t_MAT));
5766 1029 : return mkmf(mf1, cgetg(1,t_VEC), cgetg(1,t_VEC), cgetg(1,t_VEC), M);
5767 : }
5768 : static GEN
5769 616 : mfEMPTYall(long N, GEN gk, GEN vCHI, long space)
5770 : {
5771 : long i, l;
5772 : GEN v, gN, gs;
5773 616 : if (!vCHI) return cgetg(1, t_VEC);
5774 14 : gN = utoipos(N); gs = utoi(space);
5775 14 : l = lg(vCHI); v = cgetg(l, t_VEC);
5776 42 : for (i = 1; i < l; i++) gel(v,i) = mfEMPTY(mkvec4(gN,gk,gel(vCHI,i),gs));
5777 14 : return v;
5778 : }
5779 :
5780 : static GEN
5781 3983 : fmt_dim(GEN CHI, long d, long dih)
5782 3983 : { return mkvec4(gmfcharorder(CHI), gmfcharno(CHI), utoi(d), stoi(dih)); }
5783 : /* merge two vector of fmt_dim's for the same vector of characters. If CHI
5784 : * is not NULL, remove dim-0 spaces and add character from CHI */
5785 : static GEN
5786 7 : merge_dims(GEN V, GEN W, GEN CHI)
5787 : {
5788 7 : long i, j, id, l = lg(V);
5789 7 : GEN A = cgetg(l, t_VEC);
5790 7 : if (l == 1) return A;
5791 7 : id = CHI? 1: 3;
5792 21 : for (i = j = 1; i < l; i++)
5793 : {
5794 14 : GEN v = gel(V,i), w = gel(W,i);
5795 14 : long dv = itou(gel(v,id)), dvh = itou(gel(v,id+1)), d;
5796 14 : long dw = itou(gel(w,id)), dwh = itou(gel(w,id+1));
5797 14 : d = dv + dw;
5798 14 : if (d || CHI)
5799 14 : gel(A,j++) = CHI? fmt_dim(gel(CHI,i),d, dvh+dwh)
5800 14 : : mkvec2s(d,dvh+dwh);
5801 : }
5802 7 : setlg(A, j); return A;
5803 : }
5804 : static GEN
5805 3010 : mfdim0all(GEN w)
5806 : {
5807 3038 : if (w) retconst_vec(lg(w)-1, zerovec(2));
5808 3003 : return cgetg(1,t_VEC);
5809 : }
5810 : static long
5811 7315 : mf1cuspdim_i(long N, GEN CHI, GEN TMP, GEN vSP, long *dih)
5812 : {
5813 7315 : pari_sp av = avma;
5814 7315 : GEN b = mf1basis(N, CHI, TMP, vSP, NULL, dih);
5815 7315 : return gc_long(av, b? itou(b): 0);
5816 : }
5817 :
5818 : static long
5819 476 : mf1cuspdim(long N, GEN CHI, GEN vSP)
5820 : {
5821 476 : if (!vSP) vSP = get_vDIH(N, divisorsNF(N, mfcharconductor(CHI)));
5822 476 : return mf1cuspdim_i(N, CHI, NULL, vSP, NULL);
5823 : }
5824 : static GEN
5825 4144 : mf1cuspdimall(long N, GEN vCHI)
5826 : {
5827 : GEN z, TMP, w, vSP;
5828 : long i, j, l;
5829 4144 : if (wt1empty(N)) return mfdim0all(vCHI);
5830 1141 : w = mf1chars(N,vCHI);
5831 1141 : l = lg(w); if (l == 1) return cgetg(1,t_VEC);
5832 1141 : z = cgetg(l, t_VEC);
5833 1141 : TMP = mf1_pre(N); vSP = get_vDIH(N, NULL);
5834 7861 : for (i = j = 1; i < l; i++)
5835 : {
5836 6720 : GEN CHI = gel(w,i);
5837 6720 : long dih, d = mf1cuspdim_i(N, CHI, TMP, vSP, &dih);
5838 6720 : if (vCHI)
5839 42 : gel(z,j++) = mkvec2s(d, dih);
5840 6678 : else if (d)
5841 1428 : gel(z,j++) = fmt_dim(CHI, d, dih);
5842 : }
5843 1141 : setlg(z,j); return z;
5844 : }
5845 :
5846 : /* dimension of S_1(Gamma_1(N)) */
5847 : static long
5848 4123 : mf1cuspdimsum(long N)
5849 : {
5850 4123 : pari_sp av = avma;
5851 4123 : GEN v = mf1cuspdimall(N, NULL);
5852 4123 : long i, ct = 0, l = lg(v);
5853 5544 : for (i = 1; i < l; i++)
5854 : {
5855 1421 : GEN w = gel(v,i); /* [ord(CHI),*,dim,*] */
5856 1421 : ct += itou(gel(w,3))*myeulerphiu(itou(gel(w,1)));
5857 : }
5858 4123 : return gc_long(av,ct);
5859 : }
5860 :
5861 : static GEN
5862 56 : mf1newdimall(long N, GEN vCHI)
5863 : {
5864 : GEN z, w, vTMP, vSP, fa, P, E;
5865 : long i, c, l, lw, P1;
5866 56 : if (wt1empty(N)) return mfdim0all(vCHI);
5867 56 : w = mf1chars(N,vCHI);
5868 56 : lw = lg(w); if (lw == 1) return cgetg(1,t_VEC);
5869 56 : vTMP = const_vec(N, NULL);
5870 56 : vSP = get_vDIH(N, NULL);
5871 56 : gel(vTMP,N) = mf1_pre(N);
5872 : /* if p || N and p \nmid F(CHI), S_1^new(G0(N),chi) = 0 */
5873 56 : fa = znstar_get_faN(gmael(w,1,1));
5874 56 : P = gel(fa,1); l = lg(P);
5875 56 : E = gel(fa,2);
5876 154 : for (i = P1 = 1; i < l; i++)
5877 98 : if (E[i] == 1) P1 *= itou(gel(P,i));
5878 : /* P1 = \prod_{v_p(N) = 1} p */
5879 56 : z = cgetg(lw, t_VEC);
5880 182 : for (i = c = 1; i < lw; i++)
5881 : {
5882 : long S, j, l, F, dihnew;
5883 126 : GEN D, CHI = gel(w,i), CHIP = mfchartoprimitive(CHI,&F);
5884 :
5885 126 : S = F % P1? 0: mf1cuspdim_i(N, CHI, gel(vTMP,N), vSP, &dihnew);
5886 126 : if (!S)
5887 : {
5888 56 : if (vCHI) gel(z, c++) = zerovec(2);
5889 56 : continue;
5890 : }
5891 70 : D = mydivisorsu(N/F); l = lg(D);
5892 77 : for (j = l-2; j > 0; j--) /* skip last M = N */
5893 : {
5894 7 : long M = D[j]*F, m, s, dih;
5895 7 : GEN TMP = gel(vTMP,M);
5896 7 : if (wt1empty(M) || !(m = mubeta(D[l-j]))) continue; /*m = mubeta(N/M)*/
5897 7 : if (!TMP) gel(vTMP,M) = TMP = mf1_pre(M);
5898 7 : s = mf1cuspdim_i(M, CHIP, TMP, vSP, &dih);
5899 7 : if (s) { S += m * s; dihnew += m * dih; }
5900 : }
5901 70 : if (vCHI)
5902 63 : gel(z,c++) = mkvec2s(S, dihnew);
5903 7 : else if (S)
5904 7 : gel(z, c++) = fmt_dim(CHI, S, dihnew);
5905 : }
5906 56 : setlg(z,c); return z;
5907 : }
5908 :
5909 : static GEN
5910 28 : mf1olddimall(long N, GEN vCHI)
5911 : {
5912 : long i, j, l;
5913 : GEN z, w;
5914 28 : if (wt1empty(N)) return mfdim0all(vCHI);
5915 28 : w = mf1chars(N,vCHI);
5916 28 : l = lg(w); z = cgetg(l, t_VEC);
5917 84 : for (i = j = 1; i < l; i++)
5918 : {
5919 56 : GEN CHI = gel(w,i);
5920 56 : long d = mfolddim(N, 1, CHI);
5921 56 : if (vCHI)
5922 28 : gel(z,j++) = mkvec2s(d,d?-1:0);
5923 28 : else if (d)
5924 7 : gel(z, j++) = fmt_dim(CHI, d, -1);
5925 : }
5926 28 : setlg(z,j); return z;
5927 : }
5928 :
5929 : static long
5930 469 : mf1olddimsum(long N)
5931 : {
5932 : GEN D;
5933 469 : long N2, i, l, S = 0;
5934 469 : newd_params(N, &N2); /* will ensure mubeta != 0 */
5935 469 : D = mydivisorsu(N/N2); l = lg(D);
5936 2485 : for (i = 2; i < l; i++)
5937 : {
5938 2016 : long M = D[l-i]*N2, d = mf1cuspdimsum(M);
5939 2016 : if (d) S -= mubeta(D[i]) * d;
5940 : }
5941 469 : return S;
5942 : }
5943 : static long
5944 1050 : mf1newdimsum(long N)
5945 : {
5946 1050 : long S = mf1cuspdimsum(N);
5947 1050 : return S? S - mf1olddimsum(N): 0;
5948 : }
5949 :
5950 : /* return the automorphism of a degree-2 nf */
5951 : static GEN
5952 5768 : nf2_get_conj(GEN nf)
5953 : {
5954 5768 : GEN pol = nf_get_pol(nf);
5955 5768 : return deg1pol_shallow(gen_m1, negi(gel(pol,3)), varn(pol));
5956 : }
5957 : static int
5958 42 : foo_stable(GEN foo)
5959 42 : { return lg(foo) != 3 || equalii(gel(foo,1), gel(foo,2)); }
5960 :
5961 : static long
5962 224 : mfisdihedral(GEN vF, GEN DIH)
5963 : {
5964 224 : GEN vG = gel(DIH,1), M = gel(DIH,2), v, G, bnr, w, gen, D, f, nf, tau;
5965 224 : GEN bnr0 = NULL, f0, f0b, xin, foo;
5966 : long i, l, e, j, L, n;
5967 224 : if (lg(M) == 1) return 0;
5968 42 : v = RgM_RgC_invimage(M, vF);
5969 42 : if (!v) return 0;
5970 42 : l = lg(v);
5971 42 : for (i = 1; i < l; i++)
5972 42 : if (!gequal0(gel(v,i))) break;
5973 42 : if (i == l) return 0;
5974 42 : G = gel(vG,i);
5975 42 : bnr = gel(G,2); D = cyc_get_expo(bnr_get_cyc(bnr));
5976 42 : w = gel(G,3);
5977 42 : f = bnr_get_mod(bnr);
5978 42 : nf = bnr_get_nf(bnr);
5979 42 : tau = nf2_get_conj(nf);
5980 42 : f0 = gel(f,1); foo = gel(f,2);
5981 42 : f0b = galoisapply(nf, tau, f0);
5982 42 : xin = zv_to_ZV(gel(w,2)); /* xi(bnr.gen[i]) = e(xin[i] / D) */
5983 42 : if (!foo_stable(foo)) { foo = mkvec2(gen_1, gen_1); bnr0 = bnr; }
5984 42 : if (!gequal(f0, f0b))
5985 : {
5986 21 : f0 = idealmul(nf, f0, idealdivexact(nf, f0b, idealadd(nf, f0, f0b)));
5987 21 : bnr0 = bnr;
5988 : }
5989 42 : if (bnr0)
5990 : { /* conductor not ambiguous */
5991 : GEN S;
5992 28 : bnr = Buchray(bnr_get_bnf(bnr), mkvec2(f0, foo), nf_INIT | nf_GEN);
5993 28 : S = bnrsurjection(bnr, bnr0);
5994 28 : xin = FpV_red(RgV_RgM_mul(xin, gel(S,1)), D);
5995 : /* still xi(gen[i]) = e(xin[i] / D), for the new generators; D stays
5996 : * the same, not exponent(bnr.cyc) ! */
5997 : }
5998 42 : gen = bnr_get_gen(bnr); L = lg(gen);
5999 77 : for (j = 1, e = itou(D); j < L; j++)
6000 : {
6001 63 : GEN Ng = idealnorm(nf, gel(gen,j));
6002 63 : GEN a = shifti(gel(xin,j), 1); /* xi(g_j^2) = e(a/D) */
6003 63 : GEN b = FpV_dotproduct(xin, isprincipalray(bnr,Ng), D);
6004 63 : GEN m = Fp_sub(a, b, D); /* xi(g_j/g_j^\tau) = e(m/D) */
6005 63 : e = ugcd(e, itou(m)); if (e == 1) break;
6006 : }
6007 42 : n = itou(D) / e;
6008 42 : return n == 1? 4: 2*n;
6009 : }
6010 :
6011 : static ulong
6012 119 : myradicalu(ulong n) { return zv_prod(gel(myfactoru(n),1)); }
6013 :
6014 : /* list of fundamental discriminants unramified outside N, with sign s
6015 : * [s = 0 => no sign condition] */
6016 : static GEN
6017 119 : mfunram(long N, long s)
6018 : {
6019 119 : long cN = myradicalu(N >> vals(N)), p = 1, m = 1, l, c, i;
6020 119 : GEN D = mydivisorsu(cN), res;
6021 119 : l = lg(D);
6022 119 : if (s == 1) m = 0; else if (s == -1) p = 0;
6023 119 : res = cgetg(6*l - 5, t_VECSMALL);
6024 119 : c = 1;
6025 119 : if (!odd(N))
6026 : { /* d = 1 */
6027 56 : if (p) res[c++] = 8;
6028 56 : if (m) { res[c++] =-8; res[c++] =-4; }
6029 : }
6030 364 : for (i = 2; i < l; i++)
6031 : { /* skip d = 1, done above */
6032 245 : long d = D[i], d4 = d & 3L; /* d odd, squarefree, d4 = 1 or 3 */
6033 245 : if (d4 == 1) { if (p) res[c++] = d; }
6034 182 : else { if (m) res[c++] =-d; }
6035 245 : if (!odd(N))
6036 : {
6037 56 : if (p) { res[c++] = 8*d; if (d4 == 3) res[c++] = 4*d; }
6038 56 : if (m) { res[c++] =-8*d; if (d4 == 1) res[c++] =-4*d; }
6039 : }
6040 : }
6041 119 : setlg(res, c); return res;
6042 : }
6043 :
6044 : /* Return 1 if F is definitely not S4 type; return 0 on failure. */
6045 : static long
6046 105 : mfisnotS4(long N, GEN w)
6047 : {
6048 105 : GEN D = mfunram(N, 0);
6049 105 : long i, lD = lg(D), lw = lg(w);
6050 616 : for (i = 1; i < lD; i++)
6051 : {
6052 511 : long p, d = D[i], ok = 0;
6053 1442 : for (p = 2; p < lw; p++)
6054 1442 : if (w[p] && kross(d,p) == -1) { ok = 1; break; }
6055 511 : if (!ok) return 0;
6056 : }
6057 105 : return 1;
6058 : }
6059 :
6060 : /* Return 1 if Q(sqrt(5)) \not\subset Q(F), i.e. F is definitely not A5 type;
6061 : * return 0 on failure. */
6062 : static long
6063 105 : mfisnotA5(GEN F)
6064 : {
6065 105 : GEN CHI = mf_get_CHI(F), P = mfcharpol(CHI), T, Q;
6066 :
6067 105 : if (mfcharorder(CHI) % 5 == 0) return 0;
6068 105 : T = mf_get_field(F); if (degpol(T) == 1) return 1;
6069 105 : if (degpol(P) > 1) T = rnfequation(P,T);
6070 105 : Q = gsubgs(pol_xn(2,varn(T)), 5);
6071 105 : return (typ(nfisincl(Q, T)) == t_INT);
6072 : }
6073 :
6074 : /* v[p+1]^2 / chi(p) - 2 = z + 1/z with z primitive root of unity of order n,
6075 : * return n */
6076 : static long
6077 6741 : mffindrootof1(GEN v, long p, GEN CHI)
6078 : {
6079 6741 : GEN ap = gel(v,p+1), u0, u1, u1k, u2;
6080 6741 : long c = 1;
6081 6741 : if (gequal0(ap)) return 2;
6082 5033 : u0 = gen_2; u1k = u1 = gsubgs(gdiv(gsqr(ap), mfchareval(CHI, p)), 2);
6083 14812 : while (!gequalsg(2, liftpol_shallow(u1))) /* u1 = z^c + z^-c */
6084 : {
6085 9779 : u2 = gsub(gmul(u1k, u1), u0);
6086 9779 : u0 = u1; u1 = u2; c++;
6087 : }
6088 5033 : return c;
6089 : }
6090 :
6091 : /* we known that F is not dihedral */
6092 : static long
6093 182 : mfgaloistype_i(long N, GEN CHI, GEN F, GEN v)
6094 : {
6095 : forprime_t iter;
6096 182 : long lim = lg(v)-2;
6097 182 : GEN w = zero_zv(lim);
6098 : pari_sp av;
6099 : ulong p;
6100 182 : u_forprime_init(&iter, 2, lim);
6101 182 : av = avma;
6102 5292 : while((p = u_forprime_next(&iter))) if (N%p) switch(mffindrootof1(v, p, CHI))
6103 : {
6104 1400 : case 1: case 2: continue;
6105 3451 : case 3: w[p] = 1; break;
6106 70 : case 4: return -24; /* S4 */
6107 0 : case 5: return -60; /* A5 */
6108 7 : default: pari_err_DOMAIN("mfgaloistype", "form", "not a",
6109 : strtoGENstr("cuspidal eigenform"), F);
6110 0 : set_avma(av);
6111 : }
6112 105 : if (mfisnotS4(N,w) && mfisnotA5(F)) return -12; /* A4 */
6113 0 : return 0; /* FAILURE */
6114 : }
6115 :
6116 : static GEN
6117 224 : mfgaloistype0(long N, GEN CHI, GEN F, GEN DIH, long lim)
6118 : {
6119 224 : pari_sp av = avma;
6120 224 : GEN vF = mftocol(F, lim, 1);
6121 224 : long t = mfisdihedral(vF, DIH), bound;
6122 224 : if (t) return gc_stoi(av,t);
6123 182 : bound = maxss(200, 5*expu(N)*expu(N));
6124 : for(;;)
6125 : {
6126 182 : t = mfgaloistype_i(N, CHI, F, vF);
6127 175 : set_avma(av); if (t) return stoi(t);
6128 0 : if (lim > bound) return gen_0;
6129 0 : lim += lim >> 1;
6130 0 : vF = mfcoefs_i(F,lim,1);
6131 : }
6132 : }
6133 :
6134 : /* If f is NULL, give all the galoistypes, otherwise just for f */
6135 : /* Return 0 to indicate failure; in this case the type is either -12 or -60,
6136 : * most likely -12. FIXME using the Galois representation. */
6137 : GEN
6138 231 : mfgaloistype(GEN NK, GEN f)
6139 : {
6140 231 : pari_sp av = avma;
6141 231 : GEN CHI, T, F, DIH, SP, mf = checkMF_i(NK);
6142 : long N, k, lL, i, lim, SB;
6143 :
6144 231 : if (f && !checkmf_i(f)) pari_err_TYPE("mfgaloistype", f);
6145 224 : if (mf)
6146 : {
6147 189 : N = MF_get_N(mf);
6148 189 : k = MF_get_k(mf);
6149 189 : CHI = MF_get_CHI(mf);
6150 : }
6151 : else
6152 : {
6153 35 : checkNK(NK, &N, &k, &CHI, 0);
6154 35 : mf = f? NULL: mfinit_i(NK, mf_NEW);
6155 : }
6156 224 : if (k != 1) pari_err_DOMAIN("mfgaloistype", "k", "!=", gen_1, stoi(k));
6157 224 : SB = mf? mfsturm_mf(mf): mfsturmNk(N,1);
6158 224 : SP = get_DIH(N);
6159 224 : DIH = mfdihedralnew(N, CHI, SP);
6160 224 : lim = lg(DIH) == 1? 200: SB;
6161 224 : DIH = mkvec2(DIH, mfvectomat(DIH,SB,1));
6162 224 : if (f) return gc_INT(av, mfgaloistype0(N,CHI, f, DIH, lim));
6163 126 : F = mfeigenbasis(mf); lL = lg(F);
6164 126 : T = cgetg(lL, t_VEC);
6165 252 : for (i=1; i < lL; i++) gel(T,i) = mfgaloistype0(N, CHI, gel(F,i), DIH, lim);
6166 126 : return gc_upto(av, T);
6167 : }
6168 :
6169 : /******************************************************************/
6170 : /* Find all dihedral forms. */
6171 : /******************************************************************/
6172 : /* lim >= 2 */
6173 : static void
6174 14 : consttabdihedral(long lim) { cache_set(cache_DIH, mfdihedralall(lim)); }
6175 :
6176 : /* a ideal coprime to bnr modulus */
6177 : static long
6178 107611 : mfdiheval(GEN bnr, GEN w, GEN a)
6179 : {
6180 107611 : GEN L, cycn = gel(w,1), chin = gel(w,2);
6181 107611 : long ordmax = cycn[1];
6182 107611 : L = ZV_to_Flv(isprincipalray(bnr,a), ordmax);
6183 107611 : return Flv_dotproduct(chin, L, ordmax);
6184 : }
6185 :
6186 : /* x(t^k) mod T = polcyclo(m), 0 <= k < m */
6187 : static GEN
6188 30331 : Galois(GEN x, long k, GEN T, long m)
6189 : {
6190 : GEN B;
6191 : long i, ik, d;
6192 30331 : if (typ(x) != t_POL) return x;
6193 7455 : if (varn(x) != varn(T)) pari_APPLY_pol_normalized(Galois(gel(x,i), k, T, m));
6194 7420 : if ((d = degpol(x)) <= 0) return x;
6195 7063 : B = cgetg(m + 2, t_POL); B[1] = x[1]; gel(B,2) = gel(x,2);
6196 61565 : for (i = 1; i < m; i++) gel(B, i+2) = gen_0;
6197 23940 : for (i = 1, ik = k; i <= d; i++, ik = Fl_add(ik, k, m))
6198 16877 : gel(B, ik + 2) = gel(x, i+2);
6199 7063 : return QX_ZX_rem(normalizepol(B), T);
6200 : }
6201 : static GEN
6202 1022 : vecGalois(GEN x, long k, GEN T, long m)
6203 31332 : { pari_APPLY_same(Galois(gel(x,i), k, T, m)); }
6204 :
6205 : static GEN
6206 234178 : fix_pol(GEN S, GEN Pn, int *trace)
6207 : {
6208 234178 : if (typ(S) != t_POL) return S;
6209 118069 : S = RgX_rem(S, Pn);
6210 118069 : if (typ(S) == t_POL)
6211 : {
6212 118069 : switch(lg(S))
6213 : {
6214 45108 : case 2: return gen_0;
6215 20517 : case 3: return gel(S,2);
6216 : }
6217 52444 : *trace = 1;
6218 : }
6219 52444 : return S;
6220 : }
6221 :
6222 : static GEN
6223 13573 : dihan(GEN bnr, GEN w, GEN k0j, long m, ulong lim)
6224 : {
6225 13573 : GEN nf = bnr_get_nf(bnr), f = bid_get_ideal(bnr_get_bid(bnr));
6226 13573 : GEN v = zerovec(lim+1), cycn = gel(w,1), Tinit = gel(w,3);
6227 13573 : GEN Pn = gel(Tinit,lg(Tinit)==4? 2: 1);
6228 13573 : long j, ordmax = cycn[1];
6229 13573 : long D = itos(nf_get_disc(nf)), vt = varn(Pn);
6230 13573 : int trace = 0;
6231 : ulong p, n;
6232 : forprime_t T;
6233 :
6234 13573 : if (!lim) return v;
6235 13363 : gel(v,2) = gen_1;
6236 13363 : u_forprime_init(&T, 2, lim);
6237 : /* fill in prime powers first */
6238 116207 : while ((p = u_forprime_next(&T)))
6239 : {
6240 : GEN vP, vchiP, S;
6241 : long k, lP;
6242 : ulong q, qk;
6243 102844 : if (kross(D,p) >= 0) q = p;
6244 45192 : else if (!(q = umuluu_le(p,p,lim))) continue;
6245 : /* q = Norm P */
6246 65856 : vP = idealprimedec(nf, utoipos(p));
6247 65856 : lP = lg(vP);
6248 65856 : vchiP = cgetg(lP, t_VECSMALL);
6249 179081 : for (j = k = 1; j < lP; j++)
6250 : {
6251 113225 : GEN P = gel(vP,j);
6252 113225 : if (!idealval(nf, f, P)) vchiP[k++] = mfdiheval(bnr,w,P);
6253 : }
6254 65856 : if (k == 1) continue;
6255 62188 : setlg(vchiP, k); lP = k;
6256 62188 : if (lP == 2)
6257 : { /* one prime above p not dividing f */
6258 16765 : long s, s0 = vchiP[1];
6259 27069 : for (qk=q, s = s0;; s = Fl_add(s,s0,ordmax))
6260 : {
6261 27069 : S = Qab_zeta(s, ordmax, vt);
6262 27069 : gel(v, qk+1) = fix_pol(S, Pn, &trace);
6263 27069 : if (!(qk = umuluu_le(qk,q,lim))) break;
6264 : }
6265 : }
6266 : else /* two primes above p not dividing f */
6267 : {
6268 45423 : long s, s0 = vchiP[1], s1 = vchiP[2];
6269 45423 : for (qk=q, k = 1;; k++)
6270 18424 : { /* sum over a,b s.t. Norm( P1^a P2^b ) = q^k, i.e. a+b = k */
6271 : long a;
6272 63847 : GEN S = gen_0;
6273 220752 : for (a = 0; a <= k; a++)
6274 : {
6275 156905 : s = Fl_add(Fl_mul(a, s0, ordmax), Fl_mul(k-a, s1, ordmax), ordmax);
6276 156905 : S = gadd(S, Qab_zeta(s, ordmax, vt));
6277 : }
6278 63847 : gel(v, qk+1) = fix_pol(S, Pn, &trace);
6279 63847 : if (!(qk = umuluu_le(qk,q,lim))) break;
6280 : }
6281 : }
6282 : }
6283 : /* complete with nonprime powers */
6284 308098 : for (n = 2; n <= lim; n++)
6285 : {
6286 294735 : GEN S, fa = myfactoru(n), P = gel(fa, 1), E = gel(fa, 2);
6287 : long q;
6288 294735 : if (lg(P) == 2) continue;
6289 : /* not a prime power */
6290 143262 : q = upowuu(P[1],E[1]);
6291 143262 : S = gmul(gel(v, q + 1), gel(v, n/q + 1));
6292 143262 : gel(v, n+1) = fix_pol(S, Pn, &trace);
6293 : }
6294 13363 : if (trace)
6295 : {
6296 7154 : long k0 = k0j[1], jdeg = k0j[2];
6297 7154 : v = QabV_tracerel(Tinit, jdeg, v); /* Apply Galois Mod(k0, ordw) */
6298 7154 : if (k0 > 1) v = vecGalois(v, k0, gel(Tinit,1), m);
6299 : }
6300 13363 : return v;
6301 : }
6302 :
6303 : /* as cyc_normalize for t_VECSMALL cyc */
6304 : static GEN
6305 26810 : cyc_normalize_zv(GEN cyc)
6306 : {
6307 26810 : long i, o = cyc[1], l = lg(cyc); /* > 1 */
6308 26810 : GEN D = cgetg(l, t_VECSMALL);
6309 31185 : D[1] = o; for (i = 2; i < l; i++) D[i] = o / cyc[i];
6310 26810 : return D;
6311 : }
6312 : /* as char_normalize for t_VECSMALLs */
6313 : static GEN
6314 118517 : char_normalize_zv(GEN chi, GEN ncyc)
6315 : {
6316 118517 : long i, l = lg(chi);
6317 118517 : GEN c = cgetg(l, t_VECSMALL);
6318 118517 : if (l > 1) {
6319 118517 : c[1] = chi[1];
6320 160454 : for (i = 2; i < l; i++) c[i] = chi[i] * ncyc[i];
6321 : }
6322 118517 : return c;
6323 : }
6324 :
6325 : static GEN
6326 9331 : dihan_bnf(long D)
6327 : {
6328 9331 : GEN c = getrand(), bnf;
6329 9331 : setrand(gen_1);
6330 9331 : bnf = Buchall(quadpoly_i(stoi(D)), nf_FORCE, LOWDEFAULTPREC);
6331 9331 : setrand(c);
6332 9331 : return bnf;
6333 : }
6334 : static GEN
6335 37758 : dihan_bnr(GEN bnf, GEN A)
6336 : {
6337 37758 : GEN c = getrand(), bnr;
6338 37758 : setrand(gen_1);
6339 37758 : bnr = Buchray(bnf, A, nf_INIT|nf_GEN);
6340 37758 : setrand(c);
6341 37758 : return bnr;
6342 : }
6343 : /* Hecke xi * (D/.) = Dirichlet chi, return v in Q^r st chi(g_i) = e(v[i]).
6344 : * cycn = cyc_normalize_zv(bnr.cyc), chin = char_normalize_zv(chi,cyc) */
6345 : static GEN
6346 34489 : bnrchartwist2conrey(GEN chin, GEN cycn, GEN bnrconreyN, GEN kroconreyN)
6347 : {
6348 34489 : long l = lg(bnrconreyN), c1 = cycn[1], i;
6349 34489 : GEN v = cgetg(l, t_COL);
6350 125363 : for (i = 1; i < l; i++)
6351 : {
6352 90874 : GEN d = sstoQ(zv_dotproduct(chin, gel(bnrconreyN,i)), c1);
6353 90874 : if (kroconreyN[i] < 0) d = gadd(d, ghalf);
6354 90874 : gel(v,i) = d;
6355 : }
6356 34489 : return v;
6357 : }
6358 :
6359 : /* chi(g_i) = e(v[i]) denormalize wrt Conrey generators orders */
6360 : static GEN
6361 34489 : conreydenormalize(GEN znN, GEN v)
6362 : {
6363 34489 : GEN gcyc = znstar_get_conreycyc(znN), w;
6364 34489 : long l = lg(v), i;
6365 34489 : w = cgetg(l, t_COL);
6366 125363 : for (i = 1; i < l; i++)
6367 90874 : gel(w,i) = modii(gmul(gel(v,i), gel(gcyc,i)), gel(gcyc,i));
6368 34489 : return w;
6369 : }
6370 :
6371 : static long
6372 84028 : Miyake(GEN vchi, GEN gb, GEN cycn)
6373 : {
6374 84028 : long i, e = cycn[1], lb = lg(gb);
6375 84028 : GEN v = char_normalize_zv(vchi, cycn);
6376 124992 : for (i = 1; i < lb; i++)
6377 100268 : if ((zv_dotproduct(v, gel(gb,i)) - v[i]) % e) return 1;
6378 24724 : return 0;
6379 : }
6380 :
6381 : /* list of Hecke characters not induced by a Dirichlet character up to Galois
6382 : * conjugation, whose conductor is bnr.cond; cycn = cyc_normalize(bnr.cyc)*/
6383 : static GEN
6384 26810 : mklvchi(GEN bnr, GEN cycn, GEN gb)
6385 : {
6386 26810 : GEN cyc = bnr_get_cyc(bnr), cycsmall = ZV_to_zv(cyc);
6387 26810 : GEN vchi = cyc2elts(cycsmall);
6388 26810 : long ordmax = cycsmall[1], c, i, l;
6389 26810 : l = lg(vchi);
6390 304024 : for (i = c = 1; i < l; i++)
6391 : {
6392 277214 : GEN chi = gel(vchi,i);
6393 277214 : if (!gb || Miyake(chi, gb, cycn)) gel(vchi, c++) = Flv_to_ZV(chi);
6394 : }
6395 26810 : setlg(vchi, c); l = c;
6396 279300 : for (i = 1; i < l; i++)
6397 : {
6398 252490 : GEN chi = gel(vchi,i);
6399 : long n;
6400 252490 : if (!chi) continue;
6401 1055754 : for (n = 2; n < ordmax; n++)
6402 966476 : if (ugcd(n, ordmax) == 1)
6403 : {
6404 397670 : GEN tmp = ZV_ZV_mod(gmulsg(n, chi), cyc);
6405 : long j;
6406 7623539 : for (j = i+1; j < l; j++)
6407 7225869 : if (gel(vchi,j) && gequal(gel(vchi,j), tmp)) gel(vchi,j) = NULL;
6408 : }
6409 : }
6410 279300 : for (i = c = 1; i < l; i++)
6411 : {
6412 252490 : GEN chi = gel(vchi,i);
6413 252490 : if (chi && bnrisconductor(bnr, chi)) gel(vchi, c++) = chi;
6414 : }
6415 26810 : setlg(vchi, c); return vchi;
6416 : }
6417 :
6418 : static GEN
6419 7805 : get_gb(GEN bnr, GEN con)
6420 : {
6421 7805 : GEN gb, g = bnr_get_gen(bnr), nf = bnr_get_nf(bnr);
6422 7805 : long i, l = lg(g);
6423 7805 : gb = cgetg(l, t_VEC);
6424 18326 : for (i = 1; i < l; i++)
6425 10521 : gel(gb,i) = ZV_to_zv(isprincipalray(bnr, galoisapply(nf, con, gel(g,i))));
6426 7805 : return gb;
6427 : }
6428 : static GEN
6429 15862 : get_bnrconreyN(GEN bnr, GEN znN)
6430 : {
6431 15862 : GEN z, g = znstar_get_conreygen(znN);
6432 15862 : long i, l = lg(g);
6433 15862 : z = cgetg(l, t_VEC);
6434 57134 : for (i = 1; i < l; i++) gel(z,i) = ZV_to_zv(isprincipalray(bnr,gel(g,i)));
6435 15862 : return z;
6436 : }
6437 : /* con = NULL if D > 0 or if D < 0 and id != idcon. */
6438 : static GEN
6439 33698 : mfdihedralcommon(GEN bnf, GEN id, GEN znN, GEN kroconreyN, long vt,
6440 : long N, long D, GEN con)
6441 : {
6442 33698 : GEN bnr = dihan_bnr(bnf, id), cyc = ZV_to_zv( bnr_get_cyc(bnr) );
6443 : GEN bnrconreyN, cycn, cycN, Lvchi, res, P, vT;
6444 : long j, ordmax, l, lc, deghecke;
6445 :
6446 33698 : lc = lg(cyc); if (lc == 1) return NULL;
6447 26810 : cycn = cyc_normalize_zv(cyc);
6448 26810 : Lvchi = mklvchi(bnr, cycn, con? get_gb(bnr, con): NULL);
6449 26810 : l = lg(Lvchi);
6450 26810 : if (l == 1) return NULL;
6451 :
6452 15862 : bnrconreyN = get_bnrconreyN(bnr, znN);
6453 15862 : cycN = ZV_to_zv(znstar_get_cyc(znN));
6454 15862 : ordmax = cyc[1];
6455 15862 : vT = const_vec(odd(ordmax)? ordmax << 1: ordmax, NULL);
6456 15862 : P = polcyclo(ordmax, vt);
6457 15862 : gel(vT,ordmax) = Qab_trace_init(ordmax, ordmax, P, P);
6458 15862 : deghecke = myeulerphiu(ordmax);
6459 15862 : res = cgetg(l, t_VEC);
6460 50351 : for (j = 1; j < l; j++)
6461 : {
6462 34489 : GEN T, v, vchi = ZV_to_zv(gel(Lvchi,j));
6463 34489 : GEN chi, chin = char_normalize_zv(vchi, cycn);
6464 : long o, vnum, k0, degrel;
6465 34489 : v = bnrchartwist2conrey(chin, cycn, bnrconreyN, kroconreyN);
6466 34489 : o = itou(Q_denom(v));
6467 34489 : T = gel(vT, o);
6468 34489 : if (!T) gel(vT,o) = T = Qab_trace_init(ordmax, o, P, polcyclo(o,vt));
6469 34489 : chi = conreydenormalize(znN, v);
6470 34489 : vnum = itou(znconreyexp(znN, chi));
6471 34489 : chi = ZV_to_zv(znconreychar(znN,chi));
6472 34489 : degrel = deghecke / degpol(gel(T,1));
6473 34489 : k0 = zv_cyc_minimize(cycN, chi, coprimes_zv(o));
6474 34489 : vnum = Fl_powu(vnum, k0, N);
6475 : /* encodes degrel forms: jdeg = 0..degrel-1 */
6476 34489 : gel(res,j) = mkvec3(mkvecsmalln(5, N, k0 % o, vnum, D, degrel),
6477 : id, mkvec3(cycn,chin,T));
6478 : }
6479 15862 : return res;
6480 : }
6481 :
6482 : static long
6483 49364 : is_cond(long D, long n)
6484 : {
6485 49364 : if (D > 0) return n != 4 || (D&7L) == 1;
6486 30114 : return n != 2 && n != 3 && (n != 4 || (D&7L)!=1);
6487 : }
6488 : /* Append to v all dihedral weight 1 forms coming from D, if fundamental.
6489 : * level in [l1, l2] */
6490 : static void
6491 18718 : append_dihedral(GEN v, long D, long l1, long l2, long vt)
6492 : {
6493 18718 : long Da = labs(D), no, i, numi, ct, min, max;
6494 : GEN bnf, con, vI, resall, arch1, arch2;
6495 : pari_sp av;
6496 :
6497 : /* min <= Nf <= max */
6498 18718 : max = l2 / Da;
6499 18718 : if (l1 == l2)
6500 : { /* assume Da | l2 */
6501 140 : min = max;
6502 140 : if (D > 0 && min < 3) return;
6503 : }
6504 : else /* assume l1 < l2 */
6505 18578 : min = (l1 + Da-1)/Da;
6506 18718 : if (!sisfundamental(D)) return;
6507 :
6508 5726 : av = avma;
6509 5726 : bnf = dihan_bnf(D);
6510 5726 : con = nf2_get_conj(bnf_get_nf(bnf));
6511 5726 : vI = ideallist(bnf, max);
6512 55090 : numi = 0; for (i = min; i <= max; i++) numi += lg(gel(vI, i)) - 1;
6513 5726 : if (D > 0)
6514 : {
6515 1428 : numi <<= 1;
6516 1428 : arch1 = mkvec2(gen_1,gen_0);
6517 1428 : arch2 = mkvec2(gen_0,gen_1);
6518 : }
6519 : else
6520 4298 : arch1 = arch2 = NULL;
6521 5726 : resall = cgetg(numi+1, t_VEC); ct = 1;
6522 55090 : for (no = min; no <= max; no++) if (is_cond(D, no))
6523 : {
6524 44646 : long N = Da*no, lc, lI;
6525 44646 : GEN I = gel(vI, no), znN = znstar0(utoipos(N), 1), conreyN, kroconreyN;
6526 :
6527 44646 : conreyN = znstar_get_conreygen(znN); lc = lg(conreyN);
6528 44646 : kroconreyN = cgetg(lc, t_VECSMALL);
6529 166054 : for (i = 1; i < lc; i++) kroconreyN[i] = krosi(D, gel(conreyN, i));
6530 44646 : lI = lg(I);
6531 87822 : for (i = 1; i < lI; i++)
6532 : {
6533 43176 : GEN id = gel(I, i), idcon, z;
6534 : long j;
6535 43176 : if (typ(id) == t_INT) continue;
6536 28182 : idcon = galoisapply(bnf, con, id);
6537 51408 : for (j = i; j < lI; j++)
6538 51408 : if (gequal(idcon, gel(I, j))) { gel(I, j) = gen_0; break; }
6539 28182 : if (D < 0)
6540 : {
6541 17479 : GEN conk = i == j ? con : NULL;
6542 17479 : z = mfdihedralcommon(bnf, id, znN, kroconreyN, vt, N, D, conk);
6543 17479 : if (z) gel(resall, ct++) = z;
6544 : }
6545 : else
6546 : {
6547 : GEN ide;
6548 10703 : ide = mkvec2(id, arch1);
6549 10703 : z = mfdihedralcommon(bnf, ide, znN, kroconreyN, vt, N, D, NULL);
6550 10703 : if (z) gel(resall, ct++) = z;
6551 10703 : if (gequal(idcon,id)) continue;
6552 5516 : ide = mkvec2(id, arch2);
6553 5516 : z = mfdihedralcommon(bnf, ide, znN, kroconreyN, vt, N, D, NULL);
6554 5516 : if (z) gel(resall, ct++) = z;
6555 : }
6556 : }
6557 : }
6558 5726 : if (ct == 1) set_avma(av);
6559 : else
6560 : {
6561 4816 : setlg(resall, ct);
6562 4816 : vectrunc_append(v, gc_GEN(av, shallowconcat1(resall)));
6563 : }
6564 : }
6565 :
6566 : static long
6567 42042 : di_N(GEN a) { return gel(a,1)[1]; }
6568 : static GEN
6569 14 : mfdihedral(long N)
6570 : {
6571 14 : GEN D = mydivisorsu(N), res = vectrunc_init(2*N);
6572 14 : long j, l = lg(D), vt = fetch_user_var("t");
6573 105 : for (j = 2; j < l; j++)
6574 : { /* skip d = 1 */
6575 91 : long d = D[j];
6576 91 : if (d == 2) continue;
6577 84 : append_dihedral(res, -d, N,N, vt);
6578 84 : if (d >= 5 && D[l-j] >= 3) append_dihedral(res, d, N,N, vt);/* Nf >= 3 */
6579 : }
6580 14 : if (lg(res) > 1) res = shallowconcat1(res);
6581 14 : return res;
6582 : }
6583 : /* All primitive dihedral weight 1 forms of leven in [1, N], N > 1 */
6584 : static GEN
6585 14 : mfdihedralall(long N)
6586 : {
6587 14 : GEN res = vectrunc_init(2*N), z;
6588 14 : long D, ct, i, vt = fetch_user_var("t");
6589 :
6590 13986 : for (D = -3; D >= -N; D--) append_dihedral(res, D, 1,N, vt);
6591 : /* Nf >= 3 (GTM 193, prop 3.3.18) */
6592 4620 : for (D = N / 3; D >= 5; D--) append_dihedral(res, D, 1,N, vt);
6593 14 : ct = lg(res);
6594 14 : if (ct > 1)
6595 : { /* sort wrt N */
6596 14 : res = shallowconcat1(res);
6597 14 : res = vecpermute(res, indexvecsort(res, mkvecsmall(1)));
6598 14 : ct = lg(res);
6599 : }
6600 14 : z = const_vec(N, cgetg(1,t_VEC));
6601 7658 : for (i = 1; i < ct;)
6602 : { /* regroup result sharing the same N */
6603 7644 : long n = di_N(gel(res,i)), j = i+1, k;
6604 : GEN v;
6605 34412 : while (j < ct && di_N(gel(res,j)) == n) j++;
6606 7644 : gel(z, n) = v = cgetg(j-i+1, t_VEC);
6607 42056 : for (k = 1; i < j; k++,i++) gel(v,k) = gel(res,i);
6608 : }
6609 14 : return z;
6610 : }
6611 :
6612 : /* return [vF, index], where vecpermute(vF,index) generates dihedral forms
6613 : * for character CHI */
6614 : static GEN
6615 24969 : mfdihedralnew_i(long N, GEN CHI, GEN SP)
6616 : {
6617 : GEN bnf, Tinit, Pm, vf, M, V, NK;
6618 : long Dold, d, ordw, i, SB, c, l, k0, k1, chino, chinoorig, lv;
6619 :
6620 24969 : lv = lg(SP); if (lv == 1) return NULL;
6621 12138 : CHI = mfcharinduce(CHI,N);
6622 12138 : ordw = mfcharorder(CHI);
6623 12138 : chinoorig = mfcharno(CHI);
6624 12138 : k0 = mfconreyminimize(CHI);
6625 12138 : chino = Fl_powu(chinoorig, k0, N);
6626 12138 : k1 = Fl_inv(k0 % ordw, ordw);
6627 12138 : V = cgetg(lv, t_VEC);
6628 12138 : d = 0;
6629 39039 : for (i = l = 1; i < lv; i++)
6630 : {
6631 26901 : GEN sp = gel(SP,i), T = gel(sp,1);
6632 26901 : if (T[3] != chino) continue;
6633 4060 : d += T[5];
6634 4060 : if (k1 != 1)
6635 : {
6636 77 : GEN t = leafcopy(T);
6637 77 : t[3] = chinoorig;
6638 77 : t[2] = (t[2]*k1) % ordw;
6639 77 : sp = mkvec4(t, gel(sp,2), gel(sp,3), gel(sp,4));
6640 : }
6641 4060 : gel(V, l++) = sp;
6642 : }
6643 12138 : setlg(V, l); /* dihedral forms of level N and character CHI */
6644 12138 : if (l == 1) return NULL;
6645 :
6646 2555 : SB = mfsturmNk(N,1) + 1;
6647 2555 : M = cgetg(d+1, t_MAT);
6648 2555 : vf = cgetg(d+1, t_VEC);
6649 2555 : NK = mkNK(N, 1, CHI);
6650 2555 : bnf = NULL; Dold = 0;
6651 6615 : for (i = c = 1; i < l; i++)
6652 : { /* T = [N, k0, conreyno, D, degrel] */
6653 4060 : GEN bnr, Vi = gel(V,i), T = gel(Vi,1), id = gel(Vi,2), w = gel(Vi,3);
6654 4060 : long jdeg, k0i = T[2], D = T[4], degrel = T[5];
6655 :
6656 4060 : if (D != Dold) { Dold = D; bnf = dihan_bnf(D); }
6657 4060 : bnr = dihan_bnr(bnf, id);
6658 12054 : for (jdeg = 0; jdeg < degrel; jdeg++,c++)
6659 : {
6660 7994 : GEN k0j = mkvecsmall2(k0i, jdeg), an = dihan(bnr, w, k0j, ordw, SB);
6661 7994 : settyp(an, t_COL); gel(M,c) = an;
6662 7994 : gel(vf,c) = tag3(t_MF_DIHEDRAL, NK, bnr, w, k0j);
6663 : }
6664 : }
6665 2555 : Tinit = gmael3(V,1,3,3); Pm = gel(Tinit,1);
6666 2555 : V = QabM_indexrank(M, degpol(Pm)==1? NULL: Pm, ordw);
6667 2555 : return mkvec2(vf,gel(V,2));
6668 : }
6669 : static long
6670 16149 : mfdihedralnewdim(long N, GEN CHI, GEN SP)
6671 : {
6672 16149 : pari_sp av = avma;
6673 16149 : GEN S = mfdihedralnew_i(N, CHI, SP);
6674 16149 : return gc_long(av, S? lg(gel(S,2))-1: 0);
6675 : }
6676 : static GEN
6677 8820 : mfdihedralnew(long N, GEN CHI, GEN SP)
6678 : {
6679 8820 : pari_sp av = avma;
6680 8820 : GEN S = mfdihedralnew_i(N, CHI, SP);
6681 8820 : if (!S) retgc_const(av, cgetg(1, t_VEC));
6682 917 : return vecpermute(gel(S,1), gel(S,2));
6683 : }
6684 :
6685 : static long
6686 7105 : mfdihedralcuspdim(long N, GEN CHI, GEN vSP)
6687 : {
6688 7105 : pari_sp av = avma;
6689 : GEN D, CHIP;
6690 : long F, i, lD, dim;
6691 :
6692 7105 : CHIP = mfchartoprimitive(CHI, &F);
6693 7105 : D = mydivisorsu(N/F); lD = lg(D);
6694 7105 : dim = mfdihedralnewdim(N, CHI, gel(vSP,N)); /* d = 1 */
6695 16149 : for (i = 2; i < lD; i++)
6696 : {
6697 9044 : long d = D[i], a = mfdihedralnewdim(N/d, CHIP, gel(vSP, N/d));
6698 9044 : if (a) dim += a * mynumdivu(d);
6699 : }
6700 7105 : return gc_long(av,dim);
6701 : }
6702 :
6703 : static GEN
6704 7364 : mfbdall(GEN E, long N)
6705 : {
6706 7364 : GEN v, D = mydivisorsu(N);
6707 7364 : long i, j, nD = lg(D) - 1, nE = lg(E) - 1;
6708 7364 : v = cgetg(nD*nE + 1, t_VEC);
6709 10458 : for (j = 1; j <= nE; j++)
6710 : {
6711 3094 : GEN Ej = gel(E, j);
6712 9457 : for (i = 0; i < nD; i++) gel(v, i*nE + j) = mfbd_i(Ej, D[i+1]);
6713 : }
6714 7364 : return v;
6715 : }
6716 : static GEN
6717 3857 : mfdihedralcusp(long N, GEN CHI, GEN vSP)
6718 : {
6719 3857 : pari_sp av = avma;
6720 : GEN D, CHIP, z;
6721 : long F, i, lD;
6722 :
6723 3857 : CHIP = mfchartoprimitive(CHI, &F);
6724 3857 : D = mydivisorsu(N/F); lD = lg(D);
6725 3857 : z = cgetg(lD, t_VEC);
6726 3857 : gel(z,1) = mfdihedralnew(N, CHI, gel(vSP,N));
6727 8596 : for (i = 2; i < lD; i++) /* skip 1 */
6728 : {
6729 4739 : GEN LF = mfdihedralnew(N / D[i], CHIP, gel(vSP, N / D[i]));
6730 4739 : gel(z,i) = mfbdall(LF, D[i]);
6731 : }
6732 3857 : return gc_GEN(av, shallowconcat1(z));
6733 : }
6734 :
6735 : /* used to decide between ratlift and comatrix for ZM_inv; ratlift is better
6736 : * when N has many divisors */
6737 : static int
6738 2576 : abundant(ulong N) { return mynumdivu(N) >= 8; }
6739 :
6740 : /* CHI an mfchar */
6741 : static int
6742 371 : cmp_ord(void *E, GEN a, GEN b)
6743 : {
6744 371 : GEN chia = MF_get_CHI(a), chib = MF_get_CHI(b);
6745 371 : (void)E; return cmpii(gmfcharorder(chia), gmfcharorder(chib));
6746 : }
6747 : /* mfinit structure.
6748 : -- mf[1] contains [N,k,CHI,space],
6749 : -- mf[2] contains vector of closures of Eisenstein series, empty if not
6750 : full space.
6751 : -- mf[3] contains vector of closures, so #mf[3] = dimension of cusp/new space.
6752 : -- mf[4] contains the corresponding indices: either j for T(j)tf if newspace,
6753 : or [M,j,d] for B(d)T(j)tf_M if cuspspace or oldspace.
6754 : -- mf[5] contains the matrix M of first coefficients of basis, never cleaned.
6755 : * NK is either [N,k] or [N,k,CHI].
6756 : * mfinit does not do the splitting, only the basis generation. */
6757 :
6758 : /* Set flraw to 1 if do not need mf[5]: no mftobasis etc..., only the
6759 : expansions of the basis elements are needed. */
6760 :
6761 : static GEN
6762 5019 : mfinit_Nkchi(long N, long k, GEN CHI, long space, long flraw)
6763 : {
6764 5019 : GEN M = NULL, mf = NULL, mf1 = mkvec4(utoi(N), stoi(k), CHI, utoi(space));
6765 5019 : long sb = mfsturmNk(N, k);
6766 5019 : if (k < 0 || badchar(N, k, CHI)) return mfEMPTY(mf1);
6767 4984 : if (k == 0 || space == mf_EISEN) /*nothing*/;
6768 4823 : else if (k == 1)
6769 : {
6770 364 : switch (space)
6771 : {
6772 350 : case mf_NEW:
6773 : case mf_FULL:
6774 350 : case mf_CUSP: mf = mf1init(N, CHI, NULL, get_vDIH(N,NULL), space, flraw);
6775 350 : break;
6776 7 : case mf_OLD: pari_err_IMPL("mfinit in weight 1 for old space");
6777 7 : default: pari_err_FLAG("mfinit");
6778 : }
6779 : }
6780 : else /* k >= 2 */
6781 : {
6782 4459 : long ord = mfcharorder(CHI);
6783 4459 : GEN z = NULL, P = (ord <= 2)? NULL: mfcharpol(CHI);
6784 : cachenew_t cache;
6785 4459 : switch(space)
6786 : {
6787 1225 : case mf_NEW:
6788 1225 : mf = mfnewinit(N, k, CHI, &cache, 1);
6789 1225 : if (mf && !flraw) { M = MF_get_M(mf); z = MF_get_Mindex(mf); }
6790 1225 : break;
6791 3227 : case mf_OLD:
6792 : case mf_CUSP:
6793 : case mf_FULL:
6794 3227 : if (!(mf = mfinitcusp(N, k, CHI, &cache, space))) break;
6795 2919 : if (!flraw)
6796 : {
6797 2268 : M = bhnmat_extend(M, sb+1, 1, MF_get_S(mf), &cache);
6798 2268 : if (space != mf_FULL) gel(mf,5) = mfcleanCHI(M, CHI, abundant(N));
6799 : }
6800 2919 : dbg_cachenew(&cache); break;
6801 7 : default: pari_err_FLAG("mfinit");
6802 : }
6803 4452 : if (z) gel(mf,5) = mfclean2(M, z, P, ord);
6804 : }
6805 4963 : if (!mf) mf = mfEMPTY(mf1);
6806 : else
6807 : {
6808 3997 : gel(mf,1) = mf1;
6809 3997 : if (flraw) gel(mf,5) = zerovec(3);
6810 : }
6811 4963 : if (!space_is_cusp(space))
6812 : {
6813 840 : GEN E = mfeisensteinbasis(N, k, CHI);
6814 840 : gel(mf,2) = E;
6815 840 : if (!flraw)
6816 : {
6817 518 : if (M)
6818 210 : M = shallowconcat(mfvectomat(E, sb+1, 1), M);
6819 : else
6820 308 : M = mfcoefs_mf(mf, sb+1, 1);
6821 518 : gel(mf,5) = mfcleanCHI(M, CHI, abundant(N));
6822 : }
6823 : }
6824 4963 : return mf;
6825 : }
6826 :
6827 : /* mfinit for k = nk/dk */
6828 : static GEN
6829 2723 : mfinit_Nndkchi(long N, long nk, long dk, GEN CHI, long space, long flraw)
6830 266 : { return (dk == 2)? mf2init_Nkchi(N, nk >> 1, CHI, space, flraw)
6831 2989 : : mfinit_Nkchi(N, nk, CHI, space, flraw); }
6832 : static GEN
6833 3388 : mfinit_i(GEN NK, long space)
6834 : {
6835 : GEN CHI, mf;
6836 : long N, k, dk, joker;
6837 3388 : if (checkmf_i(NK))
6838 : {
6839 161 : N = mf_get_N(NK);
6840 161 : Qtoss(mf_get_gk(NK), &k, &dk);
6841 161 : CHI = mf_get_CHI(NK);
6842 : }
6843 3227 : else if ((mf = checkMF_i(NK)))
6844 : {
6845 21 : long s = MF_get_space(mf);
6846 21 : if (s == space) return mf;
6847 21 : Qtoss(MF_get_gk(mf), &k, &dk);
6848 21 : if (dk == 1 && k > 1 && space == mf_NEW && (s == mf_CUSP || s == mf_FULL))
6849 21 : return mfinittonew(mf);
6850 0 : N = MF_get_N(mf);
6851 0 : CHI = MF_get_CHI(mf);
6852 : }
6853 : else
6854 3206 : checkNK2(NK, &N, &k, &dk, &CHI, 1);
6855 3346 : joker = !CHI || typ(CHI) == t_COL;
6856 3346 : if (joker)
6857 : {
6858 1162 : GEN mf, vCHI = CHI;
6859 : long i, j, l;
6860 1162 : if (CHI && lg(CHI) == 1) return cgetg(1,t_VEC);
6861 1155 : if (k < 0) return mfEMPTYall(N, uutoQ(k,dk), CHI, space);
6862 1141 : if (k == 1 && dk == 1 && space != mf_EISEN)
6863 504 : {
6864 : GEN TMP, vSP, gN, gs;
6865 : pari_timer tt;
6866 1106 : if (space != mf_CUSP && space != mf_NEW)
6867 0 : pari_err_IMPL("mfinit([N,1,wildcard], space != cusp or new space)");
6868 1106 : if (wt1empty(N)) return mfEMPTYall(N, gen_1, CHI, space);
6869 504 : vCHI = mf1chars(N,vCHI);
6870 504 : l = lg(vCHI); mf = cgetg(l, t_VEC); if (l == 1) return mf;
6871 504 : TMP = mf1_pre(N); vSP = get_vDIH(N, NULL);
6872 504 : gN = utoipos(N); gs = utoi(space);
6873 504 : if (DEBUGLEVEL) timer_start(&tt);
6874 4123 : for (i = j = 1; i < l; i++)
6875 : {
6876 3619 : pari_sp av = avma;
6877 3619 : GEN c = gel(vCHI,i), z = mf1init(N, c, TMP, vSP, space, 0);
6878 3619 : if (z) z = gc_GEN(av, z);
6879 : else
6880 : {
6881 2905 : set_avma(av);
6882 2905 : if (CHI) z = mfEMPTY(mkvec4(gN,gen_1,c,gs));
6883 : }
6884 3619 : if (z) gel(mf, j++) = z;
6885 3619 : if (DEBUGLEVEL)
6886 0 : timer_printf(&tt, "mf1basis: character %ld / %ld (order = %ld)",
6887 : i, l-1, mfcharorder(c));
6888 : }
6889 : }
6890 : else
6891 : {
6892 35 : vCHI = mfchars(N,k,dk,vCHI);
6893 35 : l = lg(vCHI); mf = cgetg(l, t_VEC);
6894 119 : for (i = j = 1; i < l; i++)
6895 : {
6896 84 : pari_sp av = avma;
6897 84 : GEN v = mfinit_Nndkchi(N, k, dk, gel(vCHI,i), space, 0);
6898 84 : if (MF_get_dim(v) || CHI) gel(mf, j++) = v; else set_avma(av);
6899 : }
6900 : }
6901 539 : setlg(mf,j);
6902 539 : if (!CHI) gen_sort_inplace(mf, NULL, &cmp_ord, NULL);
6903 539 : return mf;
6904 : }
6905 2184 : return mfinit_Nndkchi(N, k, dk, CHI, space, 0);
6906 : }
6907 : GEN
6908 2408 : mfinit(GEN NK, long space)
6909 : {
6910 2408 : pari_sp av = avma;
6911 2408 : return gc_GEN(av, mfinit_i(NK, space));
6912 : }
6913 :
6914 : /* UTILITY FUNCTIONS */
6915 : static void
6916 364 : cusp_canon(GEN cusp, long N, long *pA, long *pC)
6917 : {
6918 364 : pari_sp av = avma;
6919 : long A, C, tc, cg;
6920 364 : if (N <= 0) pari_err_DOMAIN("mfcuspwidth","N","<=",gen_0,stoi(N));
6921 357 : if (!cusp || (tc = typ(cusp)) == t_INFINITY) { *pA = 1; *pC = N; return; }
6922 350 : if (tc != t_INT && tc != t_FRAC) pari_err_TYPE("checkcusp", cusp);
6923 350 : Qtoss(cusp, &A,&C);
6924 350 : if (N % C)
6925 : {
6926 : ulong uC;
6927 14 : long u = Fl_invgen((C-1)%N + 1, N, &uC);
6928 14 : A = Fl_mul(A, u, N);
6929 14 : C = (long)uC;
6930 : }
6931 350 : cg = ugcd(C, N/C);
6932 420 : while (ugcd(A, N) > 1) A += cg;
6933 350 : *pA = A % N; *pC = C; set_avma(av);
6934 : }
6935 : static long
6936 987 : mfcuspcanon_width(long N, long C)
6937 987 : { return (!C || C == N)? 1 : N / ugcd(N, Fl_sqr(umodsu(C,N),N)); }
6938 : /* v = [a,c] a ZC, width of cusp (a:c) */
6939 : static long
6940 9772 : mfZC_width(long N, GEN v)
6941 : {
6942 9772 : ulong C = umodiu(gel(v,2), N);
6943 9772 : return (C == 0)? 1: N / ugcd(N, Fl_sqr(C,N));
6944 : }
6945 : long
6946 161 : mfcuspwidth(GEN gN, GEN cusp)
6947 : {
6948 161 : long N = 0, A, C;
6949 : GEN mf;
6950 161 : if (typ(gN) == t_INT) N = itos(gN);
6951 42 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
6952 0 : else pari_err_TYPE("mfcuspwidth", gN);
6953 161 : cusp_canon(cusp, N, &A, &C);
6954 154 : return mfcuspcanon_width(N, C);
6955 : }
6956 :
6957 : /* Q a t_INT */
6958 : static GEN
6959 14 : findq(GEN al, GEN Q)
6960 : {
6961 : long n;
6962 14 : if (typ(al) == t_FRAC && cmpii(gel(al,2), Q) <= 0)
6963 0 : return mkvec(mkvec2(gel(al,1), gel(al,2)));
6964 14 : n = 1 + (long)ceil(2.0781*gtodouble(glog(Q, LOWDEFAULTPREC)));
6965 14 : return contfracpnqn(gboundcf(al,n), n);
6966 : }
6967 : static GEN
6968 91 : findqga(long N, GEN z)
6969 : {
6970 91 : GEN Q, LDC, CK = NULL, DK = NULL, ma, x, y = imag_i(z);
6971 : long j, l;
6972 91 : if (gcmpgs(gmulsg(2*N, y), 1) >= 0) return NULL;
6973 14 : x = real_i(z);
6974 14 : Q = ground(ginv(gsqrt(gmulsg(N, y), LOWDEFAULTPREC)));
6975 14 : LDC = findq(gmulsg(-N,x), Q);
6976 14 : ma = gen_1; l = lg(LDC);
6977 35 : for (j = 1; j < l; j++)
6978 : {
6979 21 : GEN D, DC = gel(LDC,j), C1 = gel(DC,2);
6980 21 : if (cmpii(C1,Q) > 0) break;
6981 21 : D = gel(DC,1);
6982 21 : if (ugcdiu(D,N) == 1)
6983 : {
6984 7 : GEN C = mului(N, C1), den;
6985 7 : den = gadd(gsqr(gmul(C,y)), gsqr(gadd(D, gmul(C,x))));
6986 7 : if (gcmp(den, ma) < 0) { ma = den; CK = C; DK = D; }
6987 : }
6988 : }
6989 14 : return DK? mkvec2(CK, DK): NULL;
6990 : }
6991 :
6992 : static long
6993 168 : valNC2(GEN P, GEN E, long e)
6994 : {
6995 168 : long i, d = 1, l = lg(P);
6996 504 : for (i = 1; i < l; i++)
6997 : {
6998 336 : long v = u_lval(e, P[i]) << 1;
6999 336 : if (v == E[i] + 1) v--;
7000 336 : d *= upowuu(P[i], v);
7001 : }
7002 168 : return d;
7003 : }
7004 :
7005 : static GEN
7006 49 : findqganew(long N, GEN z)
7007 : {
7008 49 : GEN MI, DI, x = real_i(z), y = imag_i(z), Ck = gen_0, Dk = gen_1, fa, P, E;
7009 : long i;
7010 49 : MI = uutoQ(1,N);
7011 49 : DI = mydivisorsu(mysqrtu(N));
7012 49 : fa = myfactoru(N); P = gel(fa,1); E = gel(fa,2);
7013 217 : for (i = 1; i < lg(DI); i++)
7014 : {
7015 168 : long e = DI[i], g;
7016 : GEN U, C, D, m;
7017 168 : (void)cxredsl2(gmulsg(e, z), &U);
7018 168 : C = gcoeff(U,2,1); if (!signe(C)) continue;
7019 168 : D = gcoeff(U,2,2);
7020 168 : g = ugcdiu(D,e);
7021 168 : if (g > 1) { C = muliu(C,e/g); D = diviuexact(D,g); } else C = muliu(C,e);
7022 168 : m = gadd(gsqr(gadd(gmul(C, x), D)), gsqr(gmul(C, y)));
7023 168 : m = gdivgu(m, valNC2(P, E, e));
7024 168 : if (gcmp(m, MI) < 0) { MI = m; Ck = C; Dk = D; }
7025 : }
7026 49 : return signe(Ck)? mkvec2(Ck, Dk): NULL;
7027 : }
7028 :
7029 : /* Return z' and U = [a,b;c,d] \in SL_2(Z), z' = U*z,
7030 : * Im(z')/width(U.oo) > sqrt(3)/(2N). Set *pczd = c*z+d */
7031 : static GEN
7032 182 : cxredga0N(long N, GEN z, GEN *pU, GEN *pczd, long flag)
7033 : {
7034 182 : GEN v = NULL, A, B, C, D;
7035 : long e;
7036 182 : if (N == 1) return cxredsl2_i(z, pU, pczd);
7037 140 : e = gexpo(gel(z,2));
7038 140 : if (e < 0) z = gprec_wensure(z, precision(z) + nbits2extraprec(-e));
7039 140 : v = flag? findqganew(N,z): findqga(N,z);
7040 140 : if (!v) { *pU = matid(2); *pczd = gen_1; return z; }
7041 56 : C = gel(v,1);
7042 56 : D = gel(v,2);
7043 56 : if (!is_pm1(bezout(C,D, &B,&A))) pari_err_BUG("cxredga0N [gcd > 1]");
7044 56 : B = negi(B);
7045 56 : *pU = mkmat2(mkcol2(A,C), mkcol2(B,D));
7046 56 : *pczd = gadd(gmul(C,z), D);
7047 56 : return gdiv(gadd(gmul(A,z), B), *pczd);
7048 : }
7049 :
7050 : static GEN
7051 161 : lfunthetaall(GEN b, GEN vL, GEN t, long bitprec)
7052 : {
7053 161 : long i, l = lg(vL);
7054 161 : GEN v = cgetg(l, t_VEC);
7055 350 : for (i = 1; i < l; i++)
7056 : {
7057 189 : GEN T, L = gel(vL,i), a0 = gel(L,1), ldata = gel(L,2);
7058 189 : GEN van = gel(ldata_get_an(ldata),2);
7059 189 : if (lg(van) == 1)
7060 : {
7061 0 : T = gmul(b, a0);
7062 0 : if (isexactzero(T)) { GEN z = real_0_bit(-bitprec); T = mkcomplex(z,z); }
7063 : }
7064 : else
7065 : {
7066 189 : T = gmul2n(lfuntheta(ldata, t, 0, bitprec), -1);
7067 189 : T = gmul(b, gadd(a0, T));
7068 : }
7069 189 : gel(v,i) = T;
7070 : }
7071 161 : return l == 2? gel(v,1): v;
7072 : }
7073 :
7074 : /* P in ZX, irreducible */
7075 : static GEN
7076 182 : ZX_roots(GEN P, long prec)
7077 : {
7078 182 : long d = degpol(P);
7079 182 : if (d == 1) return mkvec(gen_0);
7080 182 : if (d == 2 && isint1(gel(P,2)) && isintzero(gel(P,3)) && isint1(gel(P,4)))
7081 7 : return mkvec2(powIs(3), gen_I()); /* order as polroots */
7082 294 : return (ZX_sturm_irred(P) == d)? ZX_realroots_irred(P, prec)
7083 294 : : QX_complex_roots(P, prec);
7084 : }
7085 : /* initializations for RgX_RgV_eval / RgC_embed */
7086 : static GEN
7087 217 : rootspowers(GEN v)
7088 : {
7089 217 : long i, l = lg(v);
7090 217 : GEN w = cgetg(l, t_VEC);
7091 868 : for (i = 1; i < l; i++) gel(w,i) = gpowers(gel(v,i), l-2);
7092 217 : return w;
7093 : }
7094 : /* mf embeddings attached to Q(chi)/(T), chi attached to cyclotomic P */
7095 : static GEN
7096 924 : getembed(GEN P, GEN T, GEN zcyclo, long prec)
7097 : {
7098 : long i, l;
7099 : GEN v;
7100 924 : if (degpol(P) == 1) P = NULL; /* mfcharpol for quadratic char */
7101 924 : if (degpol(T) == 1) T = NULL; /* dim 1 orbit */
7102 924 : if (T && P)
7103 35 : { /* K(y) / (T(y)), K = Q(t)/(P) cyclotomic */
7104 35 : GEN vr = RgX_is_ZX(T)? ZX_roots(T,prec): roots(RgX_embed1(T,zcyclo), prec);
7105 35 : v = rootspowers(vr); l = lg(v);
7106 105 : for (i = 1; i < l; i++) gel(v,i) = mkcol3(P,zcyclo,gel(v,i));
7107 : }
7108 889 : else if (T)
7109 : { /* Q(y) / (T(y)), T noncyclotomic */
7110 182 : GEN vr = ZX_roots(T, prec);
7111 182 : v = rootspowers(vr); l = lg(v);
7112 763 : for (i = 1; i < l; i++) gel(v,i) = mkcol2(T, gel(v,i));
7113 : }
7114 : else /* cyclotomic or rational */
7115 707 : v = mkvec(P? mkvec2(P, zcyclo): cgetg(1,t_VEC));
7116 924 : return v;
7117 : }
7118 : static GEN
7119 777 : grootsof1_CHI(GEN CHI, long prec)
7120 777 : { return grootsof1(mfcharorder(CHI), prec); }
7121 : /* return the [Q(F):Q(chi)] embeddings of F */
7122 : static GEN
7123 609 : mfgetembed(GEN F, long prec)
7124 : {
7125 609 : GEN T = mf_get_field(F), CHI = mf_get_CHI(F), P = mfcharpol(CHI);
7126 609 : return getembed(P, T, grootsof1_CHI(CHI, prec), prec);
7127 : }
7128 : static GEN
7129 7 : mfchiembed(GEN mf, long prec)
7130 : {
7131 7 : GEN CHI = MF_get_CHI(mf), P = mfcharpol(CHI);
7132 7 : return getembed(P, pol_x(0), grootsof1_CHI(CHI, prec), prec);
7133 : }
7134 : /* mfgetembed for the successive eigenforms in MF_get_newforms */
7135 : static GEN
7136 161 : mfeigenembed(GEN mf, long prec)
7137 : {
7138 161 : GEN vP = MF_get_fields(mf), vF = MF_get_newforms(mf);
7139 161 : GEN zcyclo, vE, CHI = MF_get_CHI(mf), P = mfcharpol(CHI);
7140 161 : long i, l = lg(vP);
7141 161 : vF = Q_remove_denom(liftpol_shallow(vF), NULL);
7142 161 : prec += nbits2extraprec(gexpo(vF));
7143 161 : zcyclo = grootsof1_CHI(CHI, prec);
7144 161 : vE = cgetg(l, t_VEC);
7145 469 : for (i = 1; i < l; i++) gel(vE,i) = getembed(P, gel(vP,i), zcyclo, prec);
7146 161 : return vE;
7147 : }
7148 :
7149 : static int
7150 28 : checkPv(GEN P, GEN v)
7151 28 : { return typ(P) == t_POL && is_vec_t(typ(v)) && lg(v)-1 >= degpol(P); }
7152 : static int
7153 28 : checkemb_i(GEN E)
7154 : {
7155 28 : long t = typ(E), l = lg(E);
7156 28 : if (t == t_VEC) return l == 1 || (l == 3 && checkPv(gel(E,1), gel(E,2)));
7157 21 : if (t != t_COL) return 0;
7158 21 : if (l == 3) return checkPv(gel(E,1), gel(E,2));
7159 21 : return l == 4 && is_vec_t(typ(gel(E,2))) && checkPv(gel(E,1), gel(E,3));
7160 : }
7161 : static GEN
7162 28 : anyembed(GEN v, GEN E)
7163 : {
7164 28 : switch(typ(v))
7165 : {
7166 21 : case t_VEC: case t_COL: return mfvecembed(E, v);
7167 7 : case t_MAT: return mfmatembed(E, v);
7168 : }
7169 0 : return mfembed(E, v);
7170 : }
7171 : GEN
7172 49 : mfembed0(GEN E, GEN v, long prec)
7173 : {
7174 49 : pari_sp av = avma;
7175 49 : GEN mf, vE = NULL;
7176 49 : if (checkmf_i(E)) vE = mfgetembed(E, prec);
7177 35 : else if ((mf = checkMF_i(E))) vE = mfchiembed(mf, prec);
7178 49 : if (vE)
7179 : {
7180 21 : long i, l = lg(vE);
7181 : GEN w;
7182 21 : if (!v) return gc_GEN(av, l == 2? gel(vE,1): vE);
7183 0 : w = cgetg(l, t_VEC);
7184 0 : for (i = 1; i < l; i++) gel(w,i) = anyembed(v, gel(vE,i));
7185 0 : return gc_GEN(av, l == 2? gel(w,1): w);
7186 : }
7187 28 : if (!checkemb_i(E) || !v) pari_err_TYPE("mfembed", E);
7188 28 : return gc_GEN(av, anyembed(v,E));
7189 : }
7190 :
7191 : /* dummy lfun create for theta evaluation */
7192 : static GEN
7193 952 : mfthetaancreate(GEN van, GEN N, GEN k)
7194 : {
7195 952 : GEN L = zerovec(6);
7196 952 : gel(L,1) = lfuntag(t_LFUN_GENERIC, van);
7197 952 : gel(L,3) = mkvec2(gen_0, gen_1);
7198 952 : gel(L,4) = k;
7199 952 : gel(L,5) = N; return L;
7200 : }
7201 : /* destroy van and prepare to evaluate theta(sigma(van)), for all sigma in
7202 : * embeddings vector vE */
7203 : static GEN
7204 343 : van_embedall(GEN van, GEN vE, GEN gN, GEN gk)
7205 : {
7206 343 : GEN a0 = gel(van,1), vL;
7207 343 : long i, lE = lg(vE), l = lg(van);
7208 343 : van++; van[0] = evaltyp(t_VEC) | _evallg(l-1); /* remove a0 */
7209 343 : vL = cgetg(lE, t_VEC);
7210 917 : for (i = 1; i < lE; i++)
7211 : {
7212 574 : GEN E = gel(vE,i), v = mfvecembed(E, van);
7213 574 : gel(vL,i) = mkvec2(mfembed(E,a0), mfthetaancreate(v, gN, gk));
7214 : }
7215 343 : return vL;
7216 : }
7217 :
7218 : static int
7219 1134 : cusp_AC(GEN cusp, long *A, long *C)
7220 : {
7221 1134 : switch(typ(cusp))
7222 : {
7223 140 : case t_INFINITY: *A = 1; *C = 0; break;
7224 301 : case t_INT: *A = itos(cusp); *C = 1; break;
7225 448 : case t_FRAC: *A = itos(gel(cusp, 1)); *C = itos(gel(cusp, 2)); break;
7226 245 : case t_REAL: case t_COMPLEX:
7227 245 : *A = 0; *C = 0;
7228 245 : if (gsigne(imag_i(cusp)) <= 0)
7229 7 : pari_err_DOMAIN("mfeval","imag(tau)","<=",gen_0,cusp);
7230 238 : return 0;
7231 0 : default: pari_err_TYPE("cusp_AC", cusp);
7232 : }
7233 889 : return 1;
7234 : }
7235 : static GEN
7236 518 : cusp2mat(long A, long C)
7237 : { long B, D;
7238 518 : cbezout(A, C, &D, &B);
7239 518 : return mkmat22s(A, -B, C, D);
7240 : }
7241 : static GEN
7242 21 : mkS(void) { return mkmat22s(0,-1,1,0); }
7243 :
7244 : /* if t is a cusp, return F(t), else NULL */
7245 : static GEN
7246 364 : evalcusp(GEN mf, GEN F, GEN t, long prec)
7247 : {
7248 : long A, C;
7249 : GEN R;
7250 364 : if (!cusp_AC(t, &A,&C)) return NULL;
7251 196 : if (C % mf_get_N(F) == 0) return gel(mfcoefs_i(F, 0, 1), 1);
7252 175 : R = mfgaexpansion(mf, F, cusp2mat(A,C), 0, prec);
7253 175 : return gequal0(gel(R,1))? gmael(R,3,1): gen_0;
7254 : }
7255 : /* Evaluate an mf closure numerically, i.e., in the usual sense, either for a
7256 : * single tau or a vector of tau; for each, return a vector of results
7257 : * corresponding to all complex embeddings of F. If flag is nonzero, allow
7258 : * replacing F by F | gamma to increase imag(gamma^(-1).tau) [ expensive if
7259 : * MF_EISENSPACE not present ] */
7260 : static GEN
7261 168 : mfeval_i(GEN mf, GEN F, GEN vtau, long flag, long bitprec)
7262 : {
7263 : GEN L0, vL, vb, sqN, vczd, vTAU, vs, van, vE;
7264 168 : long N = MF_get_N(mf), N0, ta, lv, i, prec = nbits2prec(bitprec);
7265 168 : GEN gN = utoipos(N), gk = mf_get_gk(F), gk1 = gsubgs(gk,1), vgk;
7266 168 : long flscal = 0;
7267 :
7268 : /* gen_0 is ignored, second component assumes Ramanujan-Petersson in
7269 : * 1/2-integer weight */
7270 168 : vgk = mkvec2(gen_0, mfiscuspidal(mf,F)? gmul2n(gk1,-1): gk1);
7271 168 : ta = typ(vtau);
7272 168 : if (!is_vec_t(ta)) { flscal = 1; vtau = mkvec(vtau); ta = t_VEC; }
7273 168 : lv = lg(vtau);
7274 168 : sqN = sqrtr_abs(utor(N, prec));
7275 168 : vs = const_vec(lv-1, NULL);
7276 168 : vb = const_vec(lv-1, NULL);
7277 168 : vL = cgetg(lv, t_VEC);
7278 168 : vTAU = cgetg(lv, t_VEC);
7279 168 : vczd = cgetg(lv, t_VEC);
7280 168 : L0 = mfthetaancreate(NULL, gN, vgk); /* only for thetacost */
7281 168 : vE = mfgetembed(F, prec);
7282 168 : N0 = 0;
7283 357 : for (i = 1; i < lv; i++)
7284 : {
7285 196 : GEN z = gel(vtau,i), tau, U;
7286 : long w, n;
7287 :
7288 196 : gel(vs,i) = evalcusp(mf, F, z, prec);
7289 189 : if (gel(vs,i)) continue;
7290 161 : tau = cxredga0N(N, z, &U, &gel(vczd,i), flag);
7291 161 : if (!flag) w = 0; else { w = mfZC_width(N, gel(U,1)); tau = gdivgu(tau,w); }
7292 161 : gel(vTAU,i) = mulcxmI(gmul(tau, sqN));
7293 161 : n = lfunthetacost(L0, real_i(gel(vTAU,i)), 0, bitprec);
7294 161 : if (N0 < n) N0 = n;
7295 161 : if (flag)
7296 : {
7297 49 : GEN A, al, v = mfslashexpansion(mf, F, ZM_inv(U,NULL), n, 0, &A, prec);
7298 49 : gel(vL,i) = van_embedall(v, vE, gN, vgk);
7299 49 : al = gel(A,1);
7300 49 : if (!gequal0(al))
7301 7 : gel(vb,i) = gexp(gmul(gmul(gmulsg(w,al),PiI2(prec)), tau), prec);
7302 : }
7303 : }
7304 161 : if (!flag)
7305 : {
7306 112 : van = mfcoefs_i(F, N0, 1);
7307 112 : vL = const_vec(lv-1, van_embedall(van, vE, gN, vgk));
7308 : }
7309 350 : for (i = 1; i < lv; i++)
7310 : {
7311 : GEN T;
7312 189 : if (gel(vs,i)) continue;
7313 161 : T = gpow(gel(vczd,i), gneg(gk), prec);
7314 161 : if (flag && gel(vb,i)) T = gmul(T, gel(vb,i));
7315 161 : gel(vs,i) = lfunthetaall(T, gel(vL,i), gel(vTAU,i), bitprec);
7316 : }
7317 161 : return flscal? gel(vs,1): vs;
7318 : }
7319 :
7320 : static long
7321 1358 : mfistrivial(GEN F)
7322 : {
7323 1358 : switch(mf_get_type(F))
7324 : {
7325 7 : case t_MF_CONST: return lg(gel(F,2)) == 1;
7326 273 : case t_MF_LINEAR: case t_MF_LINEAR_BHN: return gequal0(gel(F,3));
7327 1078 : default: return 0;
7328 : }
7329 : }
7330 :
7331 : static long
7332 1176 : mf_same_k(GEN mf, GEN f) { return gequal(MF_get_gk(mf), mf_get_gk(f)); }
7333 : static long
7334 1134 : mf_same_CHI(GEN mf, GEN f)
7335 : {
7336 1134 : GEN F1, F2, chi1, chi2, CHI1 = MF_get_CHI(mf), CHI2 = mf_get_CHI(f);
7337 : /* are the primitive chars attached to CHI1 and CHI2 equal ? */
7338 1134 : F1 = znconreyconductor(gel(CHI1,1), gel(CHI1,2), &chi1);
7339 1134 : if (typ(F1) == t_VEC) F1 = gel(F1,1);
7340 1134 : F2 = znconreyconductor(gel(CHI2,1), gel(CHI2,2), &chi2);
7341 1134 : if (typ(F2) == t_VEC) F2 = gel(F2,1);
7342 1134 : return equalii(F1,F2) && ZV_equal(chi1,chi2);
7343 : }
7344 : /* check k and CHI rigorously, but not coefficients nor N */
7345 : static long
7346 245 : mfisinspace_i(GEN mf, GEN F)
7347 : {
7348 245 : return mfistrivial(F) || (mf_same_k(mf,F) && mf_same_CHI(mf,F));
7349 : }
7350 : static void
7351 7 : err_space(GEN F)
7352 7 : { pari_err_DOMAIN("mftobasis", "form", "does not belong to",
7353 0 : strtoGENstr("space"), F); }
7354 :
7355 : static long
7356 154 : mfcheapeisen(GEN mf)
7357 : {
7358 154 : long k, L, N = MF_get_N(mf);
7359 : GEN P;
7360 154 : if (N <= 70) return 1;
7361 84 : k = itos(gceil(MF_get_gk(mf)));
7362 84 : if (odd(k)) k--;
7363 84 : switch (k)
7364 : {
7365 0 : case 2: L = 190; break;
7366 14 : case 4: L = 162; break;
7367 70 : case 6:
7368 70 : case 8: L = 88; break;
7369 0 : case 10: L = 78; break;
7370 0 : default: L = 66; break;
7371 : }
7372 84 : P = gel(myfactoru(N), 1);
7373 84 : return P[lg(P)-1] <= L;
7374 : }
7375 :
7376 : static GEN
7377 189 : myimag_i(GEN x)
7378 : {
7379 189 : long tc = typ(x);
7380 189 : if (tc == t_INFINITY || tc == t_INT || tc == t_FRAC) return gen_1;
7381 196 : if (tc == t_VEC) pari_APPLY_same(myimag_i(gel(x,i)));
7382 154 : return imag_i(x);
7383 : }
7384 :
7385 : static GEN
7386 154 : mintau(GEN vtau)
7387 : {
7388 154 : if (!is_vec_t(typ(vtau))) return myimag_i(vtau);
7389 7 : return (lg(vtau) == 1)? gen_1: vecmin(myimag_i(vtau));
7390 : }
7391 :
7392 : /* initialization for mfgaexpansion: what does not depend on cusp */
7393 : static GEN
7394 1015 : mf_eisendec(GEN mf, GEN F, long prec)
7395 : {
7396 1015 : GEN B = liftpol_shallow(mfeisensteindec(mf, F)), v = variables_vecsmall(B);
7397 1015 : GEN Mvecj = obj_check(mf, MF_EISENSPACE);
7398 1015 : long l = lg(v), i, ord;
7399 1015 : if (lg(Mvecj) < 5) Mvecj = gel(Mvecj,1);
7400 1015 : ord = itou(gel(Mvecj,4));
7401 1071 : for (i = 1; i < l; i++)
7402 721 : if (v[i] != 1)
7403 : {
7404 : GEN d;
7405 : long e;
7406 665 : B = Q_remove_denom(B, &d);
7407 665 : e = gexpo(B);
7408 665 : if (e > 0) prec += nbits2prec(e);
7409 665 : B = gsubst(B, v[i], rootsof1u_cx(ord, prec));
7410 665 : if (d) B = gdiv(B, d);
7411 665 : break;
7412 : }
7413 1015 : return B;
7414 : }
7415 :
7416 : GEN
7417 168 : mfeval(GEN mf0, GEN F, GEN vtau, long bitprec)
7418 : {
7419 168 : pari_sp av = avma;
7420 168 : long flnew = 1;
7421 168 : GEN mf = checkMF_i(mf0);
7422 168 : if (!mf) pari_err_TYPE("mfeval", mf0);
7423 168 : if (!checkmf_i(F)) pari_err_TYPE("mfeval", F);
7424 168 : if (!mfisinspace_i(mf, F)) err_space(F);
7425 168 : if (!obj_check(mf, MF_EISENSPACE)) flnew = mfcheapeisen(mf);
7426 168 : if (flnew && gcmpgs(gmulsg(2*MF_get_N(mf), mintau(vtau)), 1) >= 0) flnew = 0;
7427 168 : return gc_GEN(av, mfeval_i(mf, F, vtau, flnew, bitprec));
7428 : }
7429 :
7430 : static long
7431 189 : val(GEN v, long bit)
7432 : {
7433 189 : long c, l = lg(v);
7434 392 : for (c = 1; c < l; c++)
7435 378 : if (gexpo(gel(v,c)) > -bit) return c-1;
7436 14 : return -1;
7437 : }
7438 : GEN
7439 203 : mfcuspval(GEN mf, GEN F, GEN cusp, long bitprec)
7440 : {
7441 203 : pari_sp av = avma;
7442 203 : long lvE, w, N, sb, n, A, C, prec = nbits2prec(bitprec);
7443 : GEN ga, gk, vE;
7444 203 : mf = checkMF(mf);
7445 203 : if (!checkmf_i(F)) pari_err_TYPE("mfcuspval",F);
7446 203 : N = MF_get_N(mf);
7447 203 : cusp_canon(cusp, N, &A, &C);
7448 203 : gk = mf_get_gk(F);
7449 203 : if (typ(gk) != t_INT)
7450 : {
7451 42 : GEN FT = mfmultheta(F), mf2 = obj_checkbuild(mf, MF_MF2INIT, &mf2init);
7452 42 : GEN r = mfcuspval(mf2, FT, cusp, bitprec);
7453 42 : if ((C & 3L) == 2)
7454 : {
7455 14 : GEN z = uutoQ(1,4);
7456 14 : r = gsub(r, typ(r) == t_VEC? const_vec(lg(r)-1, z): z);
7457 : }
7458 42 : return gc_upto(av, r);
7459 : }
7460 161 : vE = mfgetembed(F, prec);
7461 161 : lvE = lg(vE);
7462 161 : w = mfcuspcanon_width(N, C);
7463 161 : sb = w * mfsturmNk(N, itos(gk));
7464 161 : ga = cusp2mat(A,C);
7465 168 : for (n = 8;; n = minss(sb, n << 1))
7466 7 : {
7467 168 : GEN R = mfgaexpansion(mf, F, ga, n, prec), res = liftpol_shallow(gel(R,3));
7468 168 : GEN v = cgetg(lvE-1, t_VECSMALL);
7469 168 : long j, ok = 1;
7470 168 : res = RgC_embedall(res, vE);
7471 357 : for (j = 1; j < lvE; j++)
7472 : {
7473 189 : v[j] = val(gel(res,j), bitprec/2);
7474 189 : if (v[j] < 0) ok = 0;
7475 : }
7476 168 : if (ok)
7477 : {
7478 154 : res = cgetg(lvE, t_VEC);
7479 329 : for (j = 1; j < lvE; j++) gel(res,j) = gadd(gel(R,1), uutoQ(v[j], w));
7480 154 : return gc_GEN(av, lvE==2? gel(res,1): res);
7481 : }
7482 14 : if (n == sb) return lvE==2? mkoo(): const_vec(lvE-1, mkoo()); /* 0 */
7483 : }
7484 : }
7485 :
7486 : long
7487 231 : mfiscuspidal(GEN mf, GEN F)
7488 : {
7489 231 : pari_sp av = avma;
7490 : GEN mf2;
7491 231 : if (space_is_cusp(MF_get_space(mf))) return 1;
7492 105 : if (typ(mf_get_gk(F)) == t_INT)
7493 : {
7494 63 : GEN v = mftobasis(mf,F,0), vE = vecslice(v, 1, lg(MF_get_E(mf))-1);
7495 63 : return gc_long(av, gequal0(vE));
7496 : }
7497 42 : if (!gequal0(mfak_i(F, 0))) return 0;
7498 21 : mf2 = obj_checkbuild(mf, MF_MF2INIT, &mf2init);
7499 21 : return mfiscuspidal(mf2, mfmultheta(F));
7500 : }
7501 :
7502 : /* F = vector of newforms in mftobasis format */
7503 : static GEN
7504 105 : mffrickeeigen_i(GEN mf, GEN F, GEN vE, long prec)
7505 : {
7506 105 : GEN M, Z, L0, gN = MF_get_gN(mf), gk = MF_get_gk(mf);
7507 105 : long N0, i, lM, bit = prec2nbits(prec), k = itou(gk);
7508 105 : long LIM = 5; /* Sturm bound is enough */
7509 :
7510 105 : L0 = mfthetaancreate(NULL, gN, gk); /* only for thetacost */
7511 105 : START:
7512 105 : N0 = lfunthetacost(L0, gen_1, LIM, bit);
7513 105 : M = mfcoefs_mf(mf, N0, 1);
7514 105 : lM = lg(F);
7515 105 : Z = cgetg(lM, t_VEC);
7516 287 : for (i = 1; i < lM; i++)
7517 : { /* expansion of D * F[i] */
7518 182 : GEN D, z, van = RgM_RgC_mul(M, Q_remove_denom(gel(F,i), &D));
7519 182 : GEN L = van_embedall(van, gel(vE,i), gN, gk);
7520 182 : long l = lg(L), j, bit_add = D? expi(D): 0;
7521 182 : gel(Z,i) = z = cgetg(l, t_VEC);
7522 567 : for (j = 1; j < l; j++)
7523 : {
7524 : GEN v, C, C0;
7525 : long m, e;
7526 518 : for (m = 0; m <= LIM; m++)
7527 : {
7528 518 : v = lfuntheta(gmael(L,j,2), gen_1, m, bit);
7529 518 : if (gexpo(v) > bit_add - bit/2) break;
7530 : }
7531 385 : if (m > LIM) { LIM <<= 1; goto START; }
7532 385 : C = mulcxpowIs(gdiv(v,conj_i(v)), 2*m - k);
7533 385 : C0 = grndtoi(C, &e); if (e < 5-prec2nbits(precision(C))) C = C0;
7534 385 : gel(z,j) = C;
7535 : }
7536 : }
7537 105 : return Z;
7538 : }
7539 : static GEN
7540 84 : mffrickeeigen(GEN mf, GEN vE, long prec)
7541 : {
7542 84 : GEN D = obj_check(mf, MF_FRICKE);
7543 84 : if (D) { long p = gprecision(D); if (!p || p >= prec) return D; }
7544 77 : D = mffrickeeigen_i(mf, MF_get_newforms(mf), vE, prec);
7545 77 : return obj_insert(mf, MF_FRICKE, D);
7546 : }
7547 :
7548 : /* integral weight, new space for primitive quadratic character CHIP;
7549 : * MF = vector of embedded eigenforms coefs on mfbasis, by orbit.
7550 : * Assume N > Q > 1 and (Q,f(CHIP)) = 1 */
7551 : static GEN
7552 56 : mfatkineigenquad(GEN mf, GEN CHIP, long Q, GEN MF, long bitprec)
7553 : {
7554 : GEN L0, la2, S, F, vP, tau, wtau, Z, va, vb, den, coe, sqrtQ, sqrtN;
7555 56 : GEN M, gN, gk = MF_get_gk(mf);
7556 56 : long N0, x, yq, i, j, lF, dim, muQ, prec = nbits2prec(bitprec);
7557 56 : long N = MF_get_N(mf), k = itos(gk), NQ = N / Q;
7558 :
7559 : /* Q coprime to FC */
7560 56 : F = MF_get_newforms(mf);
7561 56 : vP = MF_get_fields(mf);
7562 56 : lF = lg(F);
7563 56 : Z = cgetg(lF, t_VEC);
7564 56 : S = MF_get_S(mf); dim = lg(S) - 1;
7565 56 : muQ = mymoebiusu(Q);
7566 56 : if (muQ)
7567 : {
7568 42 : GEN SQ = cgetg(dim+1,t_VEC), Qk = gpow(stoi(Q), sstoQ(k-2, 2), prec);
7569 42 : long i, bit2 = bitprec >> 1;
7570 154 : for (j = 1; j <= dim; j++) gel(SQ,j) = mfak_i(gel(S,j), Q);
7571 84 : for (i = 1; i < lF; i++)
7572 : {
7573 42 : GEN S = RgV_dotproduct(gel(F,i), SQ), T = gel(vP,i);
7574 : long e;
7575 42 : if (degpol(T) > 1 && typ(S) != t_POLMOD) S = gmodulo(S, T);
7576 42 : S = grndtoi(gdiv(conjvec(S, prec), Qk), &e);
7577 42 : if (e > -bit2) pari_err_PREC("mfatkineigenquad");
7578 42 : if (muQ == -1) S = gneg(S);
7579 42 : gel(Z,i) = S;
7580 : }
7581 42 : return Z;
7582 : }
7583 14 : la2 = mfchareval(CHIP, Q); /* 1 or -1 */
7584 14 : (void)cbezout(Q, NQ, &x, &yq);
7585 14 : sqrtQ = sqrtr_abs(utor(Q,prec));
7586 14 : tau = mkcomplex(gadd(sstoQ(-1, NQ), uutoQ(1, 1000)),
7587 : divru(sqrtQ, N));
7588 14 : den = gaddgs(gmulsg(NQ, tau), 1);
7589 14 : wtau = gdiv(gsub(gmulsg(x, tau), sstoQ(yq, Q)), den);
7590 14 : coe = gpowgs(gmul(sqrtQ, den), k);
7591 :
7592 14 : sqrtN = sqrtr_abs(utor(N,prec));
7593 14 : tau = mulcxmI(gmul(tau, sqrtN));
7594 14 : wtau = mulcxmI(gmul(wtau, sqrtN));
7595 14 : gN = utoipos(N);
7596 14 : L0 = mfthetaancreate(NULL, gN, gk); /* only for thetacost */
7597 14 : N0 = maxss(lfunthetacost(L0,real_i(tau), 0,bitprec),
7598 : lfunthetacost(L0,real_i(wtau),0,bitprec));
7599 14 : M = mfcoefs_mf(mf, N0, 1);
7600 14 : va = cgetg(dim+1, t_VEC);
7601 14 : vb = cgetg(dim+1, t_VEC);
7602 105 : for (j = 1; j <= dim; j++)
7603 : {
7604 91 : GEN L, v = vecslice(gel(M,j), 2, N0+1); /* remove a0 */
7605 91 : settyp(v, t_VEC); L = mfthetaancreate(v, gN, gk);
7606 91 : gel(va,j) = lfuntheta(L, tau,0,bitprec);
7607 91 : gel(vb,j) = lfuntheta(L,wtau,0,bitprec);
7608 : }
7609 84 : for (i = 1; i < lF; i++)
7610 : {
7611 70 : GEN z, FE = gel(MF,i);
7612 70 : long l = lg(FE);
7613 70 : z = cgetg(l, t_VEC);
7614 70 : for (j = 1; j < l; j++)
7615 : {
7616 70 : GEN f = gel(FE,j), a = RgV_dotproduct(va,f), b = RgV_dotproduct(vb,f);
7617 70 : GEN la = ground( gdiv(b, gmul(a,coe)) );
7618 70 : if (!gequal(gsqr(la), la2)) pari_err_PREC("mfatkineigenquad");
7619 70 : if (typ(la) == t_INT)
7620 : {
7621 70 : if (j != 1) pari_err_BUG("mfatkineigenquad");
7622 70 : z = const_vec(l-1, la); break;
7623 : }
7624 0 : gel(z,j) = la;
7625 : }
7626 70 : gel(Z,i) = z;
7627 : }
7628 14 : return Z;
7629 : }
7630 :
7631 : static GEN
7632 84 : myusqrt(ulong a, long prec)
7633 : {
7634 84 : if (a == 1UL) return gen_1;
7635 70 : if (uissquareall(a, &a)) return utoipos(a);
7636 49 : return sqrtr_abs(utor(a, prec));
7637 : }
7638 : /* Assume mf is a nontrivial new space, rational primitive character CHIP
7639 : * and (Q,FC) = 1 */
7640 : static GEN
7641 112 : mfatkinmatnewquad(GEN mf, GEN CHIP, long Q, long flag, long PREC)
7642 : {
7643 112 : GEN cM, M, D, MF, den, vE, F = MF_get_newforms(mf);
7644 112 : long i, c, e, prec, bitprec, lF = lg(F), N = MF_get_N(mf), k = MF_get_k(mf);
7645 :
7646 112 : if (Q == 1) return mkvec4(gen_0, matid(MF_get_dim(mf)), gen_1, mf);
7647 112 : den = gel(MF_get_Minv(mf), 2);
7648 112 : bitprec = expi(den) + 64;
7649 112 : if (!flag) bitprec = maxss(bitprec, prec2nbits(PREC));
7650 :
7651 35 : START:
7652 112 : prec = nbits2prec(bitprec);
7653 112 : vE = mfeigenembed(mf, prec);
7654 112 : M = cgetg(lF, t_VEC);
7655 294 : for (i = 1; i < lF; i++) gel(M,i) = RgC_embedall(gel(F,i), gel(vE,i));
7656 112 : if (Q != N)
7657 : {
7658 56 : D = mfatkineigenquad(mf, CHIP, Q, M, bitprec);
7659 56 : c = odd(k)? Q: 1;
7660 : }
7661 : else
7662 : {
7663 56 : D = mffrickeeigen(mf, vE, prec);
7664 56 : c = mfcharmodulus(CHIP); if (odd(k)) c = -Q/c;
7665 : }
7666 112 : D = shallowconcat1(D);
7667 112 : if (vec_isconst(D)) { MF = diagonal_shallow(D); flag = 0; }
7668 : else
7669 : {
7670 63 : M = shallowconcat1(M);
7671 63 : MF = RgM_mul(matmuldiagonal(M,D), ginv(M));
7672 : }
7673 112 : if (!flag) return mkvec4(gen_0, MF, gen_1, mf);
7674 :
7675 21 : if (c > 0)
7676 21 : cM = myusqrt(c, PREC);
7677 : else
7678 : {
7679 0 : MF = imag_i(MF); c = -c;
7680 0 : cM = mkcomplex(gen_0, myusqrt(c,PREC));
7681 : }
7682 21 : if (c != 1) MF = RgM_Rg_mul(MF, myusqrt(c,prec));
7683 21 : MF = grndtoi(RgM_Rg_mul(MF,den), &e);
7684 21 : if (e > -32) { bitprec <<= 1; goto START; }
7685 21 : MF = RgM_Rg_div(MF, den);
7686 21 : if (is_rational_t(typ(cM)) && !isint1(cM))
7687 0 : { MF = RgM_Rg_div(MF, cM); cM = gen_1; }
7688 21 : return mkvec4(gen_0, MF, cM, mf);
7689 : }
7690 :
7691 : /* let CHI mod N, Q || N, return \bar{CHI_Q} * CHI_{N/Q} */
7692 : static GEN
7693 105 : mfcharAL(GEN CHI, long Q)
7694 : {
7695 105 : GEN G = gel(CHI,1), c = gel(CHI,2), cycc, d, P, E, F;
7696 105 : long l = lg(c), N = mfcharmodulus(CHI), i;
7697 105 : if (N == Q) return mfcharconj(CHI);
7698 49 : if (N == 1) return CHI;
7699 42 : CHI = leafcopy(CHI);
7700 42 : gel(CHI,2) = d = leafcopy(c);
7701 42 : F = znstar_get_faN(G);
7702 42 : P = gel(F,1);
7703 42 : E = gel(F,2);
7704 42 : cycc = znstar_get_conreycyc(G);
7705 42 : if (!odd(Q) && equaliu(gel(P,1), 2) && E[1] >= 3)
7706 14 : gel(d,2) = Fp_neg(gel(d,2), gel(cycc,2));
7707 56 : else for (i = 1; i < l; i++)
7708 28 : if (!umodui(Q, gel(P,i))) gel(d,i) = Fp_neg(gel(d,i), gel(cycc,i));
7709 42 : return CHI;
7710 : }
7711 : static long
7712 238 : atkin_get_NQ(long N, long Q, const char *f)
7713 : {
7714 238 : long NQ = N / Q;
7715 238 : if (N % Q) pari_err_DOMAIN(f,"N % Q","!=",gen_0,utoi(Q));
7716 238 : if (ugcd(NQ, Q) > 1) pari_err_DOMAIN(f,"gcd(Q,N/Q)","!=",gen_1,utoi(Q));
7717 238 : return NQ;
7718 : }
7719 :
7720 : /* transform mf to new_NEW if possible */
7721 : static GEN
7722 1379 : MF_set_new(GEN mf)
7723 : {
7724 1379 : GEN vMjd, vj, gk = MF_get_gk(mf);
7725 : long l, j;
7726 1379 : if (MF_get_space(mf) != mf_CUSP
7727 1379 : || typ(gk) != t_INT || itou(gk) == 1) return mf;
7728 182 : vMjd = MFcusp_get_vMjd(mf); l = lg(vMjd);
7729 182 : if (l > 1 && gel(vMjd,1)[1] != MF_get_N(mf)) return mf; /* oldspace != 0 */
7730 175 : mf = shallowcopy(mf);
7731 175 : gel(mf,1) = shallowcopy(gel(mf,1));
7732 175 : MF_set_space(mf, mf_NEW);
7733 175 : vj = cgetg(l, t_VECSMALL);
7734 938 : for (j = 1; j < l; j++) vj[j] = gel(vMjd, j)[2];
7735 175 : gel(mf,4) = vj; return mf;
7736 : }
7737 :
7738 : /* if flag = 1, rationalize, else don't */
7739 : static GEN
7740 217 : mfatkininit_i(GEN mf, long Q, long flag, long prec)
7741 : {
7742 : GEN M, B, C, CHI, CHIAL, G, chi, P, z, g, mfB, s, Mindex, Minv;
7743 217 : long j, l, lim, ord, FC, NQ, cQ, nk, dk, N = MF_get_N(mf);
7744 :
7745 217 : B = MF_get_basis(mf); l = lg(B);
7746 217 : M = cgetg(l, t_MAT); if (l == 1) return mkvec4(gen_0,M,gen_1,mf);
7747 217 : Qtoss(MF_get_gk(mf), &nk,&dk);
7748 217 : Q = labs(Q);
7749 217 : NQ = atkin_get_NQ(N, Q, "mfatkininit");
7750 217 : CHI = MF_get_CHI(mf);
7751 217 : CHI = mfchartoprimitive(CHI, &FC);
7752 217 : ord = mfcharorder(CHI);
7753 217 : mf = MF_set_new(mf);
7754 217 : if (MF_get_space(mf) == mf_NEW && ord <= 2 && NQ % FC == 0 && dk == 1)
7755 112 : return mfatkinmatnewquad(mf, CHI, Q, flag, prec);
7756 : /* now flag != 0 */
7757 105 : G = gel(CHI,1);
7758 105 : chi = gel(CHI,2);
7759 105 : if (Q == N) { g = mkmat22s(0, -1, N, 0); cQ = NQ; } /* Fricke */
7760 : else
7761 : {
7762 28 : GEN F, gQP = utoi(ugcd(Q, FC));
7763 : long t, v;
7764 28 : chi = znchardecompose(G, chi, gQP);
7765 28 : F = znconreyconductor(G, chi, &chi);
7766 28 : G = znstar0(F,1);
7767 28 : (void)cbezout(Q, NQ, &t, &v);
7768 28 : g = mkmat22s(Q*t, 1, -N*v, Q);
7769 28 : cQ = -NQ*v;
7770 : }
7771 105 : C = s = gen_1;
7772 : /* N.B. G,chi are G_Q,chi_Q [primitive] at this point */
7773 105 : if (lg(chi) != 1) C = ginv( znchargauss(G, chi, gen_1, prec2nbits(prec)) );
7774 105 : if (dk == 1)
7775 84 : { if (odd(nk)) s = myusqrt(Q,prec); }
7776 : else
7777 : {
7778 21 : long r = nk >> 1; /* k-1/2 */
7779 21 : s = gpow(utoipos(Q), mkfracss(odd(r)? 1: 3, 4), prec);
7780 21 : if (odd(cQ))
7781 : {
7782 21 : long t = r + ((cQ-1) >> 1);
7783 21 : s = mkcomplex(s, odd(t)? gneg(s): s);
7784 : }
7785 : }
7786 105 : if (!isint1(s)) C = gmul(C, s);
7787 105 : CHIAL = mfcharAL(CHI, Q);
7788 105 : if (dk == 2)
7789 : {
7790 21 : ulong q = odd(Q)? Q << 2: Q, Nq = ulcm(q, mfcharmodulus(CHIAL));
7791 21 : CHIAL = induceN(Nq, CHIAL);
7792 21 : CHIAL = mfcharmul(CHIAL, induce(gel(CHIAL,1), utoipos(q)));
7793 : }
7794 105 : CHIAL = mfchartoprimitive(CHIAL,NULL);
7795 105 : mfB = gequal(CHIAL,CHI)? mf: mfinit_Nndkchi(N,nk,dk,CHIAL,MF_get_space(mf),0);
7796 105 : Mindex = MF_get_Mindex(mfB);
7797 105 : Minv = MF_get_Minv(mfB);
7798 105 : P = z = NULL;
7799 105 : if (ord > 2) { P = mfcharpol(CHI); z = rootsof1u_cx(ord, prec); }
7800 105 : lim = maxss(mfsturm(mfB), mfsturm(mf)) + 1;
7801 343 : for (j = 1; j < l; j++)
7802 : {
7803 238 : GEN v = mfslashexpansion(mf, gel(B,j), g, lim, 0, NULL, prec+EXTRAPREC64);
7804 : long junk;
7805 238 : if (!isint1(C)) v = RgV_Rg_mul(v, C);
7806 238 : v = bestapprnf(v, P, z, prec);
7807 238 : v = vecpermute_partial(v, Mindex, &junk);
7808 238 : v = Minv_RgC_mul(Minv, v); /* cf mftobasis_i */
7809 238 : gel(M, j) = v;
7810 : }
7811 105 : if (is_rational_t(typ(C)) && !gequal1(C)) { M = gdiv(M, C); C = gen_1; }
7812 105 : if (mfB == mf) mfB = gen_0;
7813 105 : return mkvec4(mfB, M, C, mf);
7814 : }
7815 : GEN
7816 91 : mfatkininit(GEN mf, long Q, long prec)
7817 : {
7818 91 : pari_sp av = avma;
7819 91 : mf = checkMF(mf); return gc_GEN(av, mfatkininit_i(mf, Q, 1, prec));
7820 : }
7821 : static void
7822 56 : checkmfa(GEN z)
7823 : {
7824 56 : if (typ(z) != t_VEC || lg(z) != 5 || typ(gel(z,2)) != t_MAT
7825 56 : || !checkMF_i(gel(z,4))
7826 56 : || (!isintzero(gel(z,1)) && !checkMF_i(gel(z,1))))
7827 0 : pari_err_TYPE("mfatkin [please apply mfatkininit()]",z);
7828 56 : }
7829 :
7830 : /* Apply atkin Q to closure F */
7831 : GEN
7832 56 : mfatkin(GEN mfa, GEN F)
7833 : {
7834 56 : pari_sp av = avma;
7835 : GEN z, mfB, MQ, mf;
7836 56 : checkmfa(mfa);
7837 56 : mfB= gel(mfa,1);
7838 56 : MQ = gel(mfa,2);
7839 56 : mf = gel(mfa,4);
7840 56 : if (typ(mfB) == t_INT) mfB = mf;
7841 56 : z = RgM_RgC_mul(MQ, mftobasis_i(mf,F));
7842 56 : return gc_upto(av, mflinear(mfB, z));
7843 : }
7844 :
7845 : GEN
7846 49 : mfatkineigenvalues(GEN mf, long Q, long prec)
7847 : {
7848 49 : pari_sp av = avma;
7849 : GEN vF, L, CHI, M, mfatk, C, MQ, vE, mfB;
7850 : long N, NQ, l, i;
7851 :
7852 49 : mf = checkMF(mf); N = MF_get_N(mf);
7853 49 : vF = MF_get_newforms(mf); l = lg(vF);
7854 : /* N.B. k is integral */
7855 49 : if (l == 1) retgc_const(av, cgetg(1, t_VEC));
7856 49 : L = cgetg(l, t_VEC);
7857 49 : if (Q == 1)
7858 : {
7859 7 : GEN vP = MF_get_fields(mf);
7860 21 : for (i = 1; i < l; i++) gel(L,i) = const_vec(degpol(gel(vP,i)), gen_1);
7861 7 : return L;
7862 : }
7863 42 : vE = mfeigenembed(mf,prec);
7864 42 : if (Q == N) return gc_upto(av, mffrickeeigen(mf, vE, prec));
7865 21 : Q = labs(Q);
7866 21 : NQ = atkin_get_NQ(N, Q, "mfatkineigenvalues"); /* != 1 */
7867 21 : mfatk = mfatkininit(mf, Q, prec);
7868 21 : mfB= gel(mfatk,1); if (typ(mfB) != t_VEC) mfB = mf;
7869 21 : MQ = gel(mfatk,2);
7870 21 : C = gel(mfatk,3);
7871 21 : M = row(mfcoefs_mf(mfB,1,1), 2); /* vec of a_1(b_i) for mfbasis functions */
7872 56 : for (i = 1; i < l; i++)
7873 : {
7874 35 : GEN c = RgV_dotproduct(RgM_RgC_mul(MQ,gel(vF,i)), M); /* C * eigen_i */
7875 35 : gel(L,i) = Rg_embedall_i(c, gel(vE,i));
7876 : }
7877 21 : if (!gequal1(C)) L = gdiv(L, C);
7878 21 : CHI = MF_get_CHI(mf);
7879 21 : if (mfcharorder(CHI) <= 2 && NQ % mfcharconductor(CHI) == 0) L = ground(L);
7880 21 : return gc_GEN(av, L);
7881 : }
7882 :
7883 : /* expand B_d V, keeping same length */
7884 : static GEN
7885 6370 : bdexpand(GEN V, long d)
7886 : {
7887 : GEN W;
7888 : long N, n;
7889 6370 : if (d == 1) return V;
7890 2352 : N = lg(V)-1; W = zerovec(N);
7891 46634 : for (n = 0; n <= (N-1)/d; n++) gel(W, n*d+1) = gel(V, n+1);
7892 2352 : return W;
7893 : }
7894 : /* expand B_d V, increasing length up to lim */
7895 : static GEN
7896 329 : bdexpandall(GEN V, long d, long lim)
7897 : {
7898 : GEN W;
7899 : long N, n;
7900 329 : if (d == 1) return V;
7901 35 : N = lg(V)-1; W = zerovec(lim);
7902 259 : for (n = 0; n <= N-1 && n*d <= lim; n++) gel(W, n*d+1) = gel(V, n+1);
7903 35 : return W;
7904 : }
7905 :
7906 : static void
7907 9394 : parse_vecj(GEN T, GEN *E1, GEN *E2)
7908 : {
7909 9394 : if (lg(T)==3) { *E1 = gel(T,1); *E2 = gel(T,2); }
7910 5194 : else { *E1 = T; *E2 = NULL; }
7911 9394 : }
7912 :
7913 : /* g in M_2(Z) ? */
7914 : static int
7915 3052 : check_M2Z(GEN g)
7916 3052 : { return typ(g) == t_MAT && lg(g) == 3 && lgcols(g) == 3 && RgM_is_ZM(g); }
7917 : /* g in SL_2(Z) ? */
7918 : static int
7919 1841 : check_SL2Z(GEN g) { return check_M2Z(g) && equali1(ZM_det(g)); }
7920 :
7921 : static GEN
7922 9499 : mfcharcxeval(GEN CHI, long n, long prec)
7923 : {
7924 9499 : ulong ord, N = mfcharmodulus(CHI);
7925 : GEN ordg;
7926 9499 : if (N == 1) return gen_1;
7927 3696 : if (ugcd(N, labs(n)) > 1) return gen_0;
7928 3696 : ordg = gmfcharorder(CHI);
7929 3696 : ord = itou(ordg);
7930 3696 : return rootsof1q_cx(znchareval_i(CHI,n,ordg), ord, prec);
7931 : }
7932 :
7933 : static GEN
7934 5159 : RgV_shift(GEN V, GEN gn)
7935 : {
7936 : long i, n, l;
7937 : GEN W;
7938 5159 : if (typ(gn) != t_INT) pari_err_BUG("RgV_shift [n not integral]");
7939 5159 : n = itos(gn);
7940 5159 : if (n < 0) pari_err_BUG("RgV_shift [n negative]");
7941 5159 : if (!n) return V;
7942 112 : W = cgetg_copy(V, &l); if (n > l-1) n = l-1;
7943 308 : for (i=1; i <= n; i++) gel(W,i) = gen_0;
7944 4900 : for ( ; i < l; i++) gel(W,i) = gel(V, i-n);
7945 112 : return W;
7946 : }
7947 : static GEN
7948 7868 : hash_eisengacx(hashtable *H, void *E, long w, GEN ga, long n, long prec)
7949 : {
7950 7868 : ulong h = H->hash(E);
7951 7868 : hashentry *e = hash_search2(H, E, h);
7952 : GEN v;
7953 7868 : if (e) v = (GEN)e->val;
7954 : else
7955 : {
7956 5390 : v = mfeisensteingacx((GEN)E, w, ga, n, prec);
7957 5390 : hash_insert2(H, E, (void*)v, h);
7958 : }
7959 7868 : return v;
7960 : }
7961 : static GEN
7962 5159 : vecj_expand(GEN B, hashtable *H, long w, GEN ga, long n, long prec)
7963 : {
7964 : GEN E1, E2, v;
7965 5159 : parse_vecj(B, &E1, &E2);
7966 5159 : v = hash_eisengacx(H, (void*)E1, w, ga, n, prec);
7967 5159 : if (E2)
7968 : {
7969 2653 : GEN u = hash_eisengacx(H, (void*)E2, w, ga, n, prec);
7970 2653 : GEN a = gadd(gel(v,1), gel(u,1));
7971 2653 : GEN b = RgV_mul_RgXn(gel(v,2), gel(u,2));
7972 2653 : v = mkvec2(a,b);
7973 : }
7974 5159 : return v;
7975 : }
7976 : static GEN
7977 1085 : shift_M(GEN M, GEN Valpha, long w)
7978 : {
7979 1085 : long i, l = lg(Valpha);
7980 1085 : GEN almin = vecmin(Valpha);
7981 6244 : for (i = 1; i < l; i++)
7982 : {
7983 5159 : GEN alpha = gel(Valpha, i), gsh = gmulsg(w, gsub(alpha,almin));
7984 5159 : gel(M,i) = RgV_shift(gel(M,i), gsh);
7985 : }
7986 1085 : return almin;
7987 : }
7988 : static GEN mfeisensteinspaceinit(GEN NK);
7989 : #if 0
7990 : /* ga in M_2^+(Z)), n >= 0 */
7991 : static GEN
7992 : mfgaexpansion_init(GEN mf, GEN ga, long n, long prec)
7993 : {
7994 : GEN M, Mvecj, vecj, almin, Valpha;
7995 : long i, w, l, N = MF_get_N(mf), c = itos(gcoeff(ga,2,1));
7996 : hashtable *H;
7997 :
7998 : if (c % N == 0)
7999 : { /* ga in G_0(N), trivial case; w = 1 */
8000 : GEN chid = mfcharcxeval(MF_get_CHI(mf), itos(gcoeff(ga,2,2)), prec);
8001 : return mkvec2(chid, utoi(n));
8002 : }
8003 :
8004 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
8005 : if (lg(Mvecj) < 5) pari_err_IMPL("mfgaexpansion_init in this case");
8006 : w = mfcuspcanon_width(N, c);
8007 : vecj = gel(Mvecj, 3);
8008 : l = lg(vecj);
8009 : M = cgetg(l, t_VEC);
8010 : Valpha = cgetg(l, t_VEC);
8011 : H = hash_create(l, (ulong(*)(void*))&hash_GEN,
8012 : (int(*)(void*,void*))&gidentical, 1);
8013 : for (i = 1; i < l; i++)
8014 : {
8015 : GEN v = vecj_expand(gel(vecj,i), H, w, ga, n, prec);
8016 : gel(Valpha,i) = gel(v,1);
8017 : gel(M,i) = gel(v,2);
8018 : }
8019 : almin = shift_M(M, Valpha, w);
8020 : return mkvec3(almin, utoi(w), M);
8021 : }
8022 : /* half-integer weight not supported; vF = [F,eisendec(F)].
8023 : * Minit = mfgaexpansion_init(mf, ga, n, prec) */
8024 : static GEN
8025 : mfgaexpansion_with_init(GEN Minit, GEN vF)
8026 : {
8027 : GEN v;
8028 : if (lg(Minit) == 3)
8029 : { /* ga in G_0(N) */
8030 : GEN chid = gel(Minit,1), gn = gel(Minit,2);
8031 : v = mfcoefs_i(gel(vF,1), itou(gn), 1);
8032 : v = mkvec3(gen_0, gen_1, RgV_Rg_mul(v,chid));
8033 : }
8034 : else
8035 : {
8036 : GEN V = RgM_RgC_mul(gel(Minit,3), gel(vF,2));
8037 : v = mkvec3(gel(Minit,1), gel(Minit,2), V);
8038 : }
8039 : return v;
8040 : }
8041 : #endif
8042 :
8043 : /* B = mfeisensteindec(F) already embedded, ga in M_2^+(Z)), n >= 0 */
8044 : static GEN
8045 1085 : mfgaexpansion_i(GEN mf, GEN B0, GEN ga, long n, long prec)
8046 : {
8047 1085 : GEN M, Mvecj, vecj, almin, Valpha, B, E = NULL;
8048 1085 : long i, j, w, nw, l, N = MF_get_N(mf), bit = prec2nbits(prec) / 2;
8049 : hashtable *H;
8050 :
8051 1085 : Mvecj = obj_check(mf, MF_EISENSPACE);
8052 1085 : if (lg(Mvecj) < 5) { E = gel(Mvecj, 2); Mvecj = gel(Mvecj, 1); }
8053 1085 : vecj = gel(Mvecj, 3);
8054 1085 : l = lg(vecj);
8055 1085 : B = cgetg(l, t_COL);
8056 1085 : M = cgetg(l, t_VEC);
8057 1085 : Valpha = cgetg(l, t_VEC);
8058 1085 : w = mfZC_width(N, gel(ga,1));
8059 1085 : nw = E ? n + w : n;
8060 1085 : H = hash_create(l, (ulong(*)(void*))&hash_GEN,
8061 : (int(*)(void*,void*))&gidentical, 1);
8062 9177 : for (i = j = 1; i < l; i++)
8063 : {
8064 : GEN v;
8065 8092 : if (gequal0(gel(B0,i))) continue;
8066 5159 : v = vecj_expand(gel(vecj,i), H, w, ga, nw, prec);
8067 5159 : gel(B,j) = gel(B0,i);
8068 5159 : gel(Valpha,j) = gel(v,1);
8069 5159 : gel(M,j) = gel(v,2); j++;
8070 : }
8071 1085 : setlg(Valpha, j);
8072 1085 : setlg(B, j);
8073 1085 : setlg(M, j); l = j;
8074 1085 : if (l == 1) return mkvec3(gen_0, utoi(w), zerovec(n+1));
8075 1085 : almin = shift_M(M, Valpha, w);
8076 1085 : B = RgM_RgC_mul(M, B); l = lg(B);
8077 151445 : for (i = 1; i < l; i++)
8078 150360 : if (gexpo(gel(B,i)) < -bit) gel(B,i) = gen_0;
8079 1085 : settyp(B, t_VEC);
8080 1085 : if (E)
8081 : {
8082 : GEN v, e;
8083 56 : long ell = 0, vB, ve;
8084 126 : for (i = 1; i < l; i++)
8085 126 : if (!gequal0(gel(B,i))) break;
8086 56 : vB = i-1;
8087 56 : v = hash_eisengacx(H, (void*)E, w, ga, n + vB, prec);
8088 56 : e = gel(v,2); l = lg(e);
8089 56 : for (i = 1; i < l; i++)
8090 56 : if (!gequal0(gel(e,i))) break;
8091 56 : ve = i-1;
8092 56 : almin = gsub(almin, gel(v,1));
8093 56 : if (gsigne(almin) < 0)
8094 : {
8095 0 : GEN gell = gceil(gmulsg(-w, almin));
8096 0 : ell = itos(gell);
8097 0 : almin = gadd(almin, gdivgu(gell, w));
8098 0 : if (nw < ell) pari_err_IMPL("alpha < 0 in mfgaexpansion");
8099 : }
8100 56 : if (ve) { ell += ve; e = vecslice(e, ve+1, l-1); }
8101 56 : B = vecslice(B, ell + 1, minss(n + ell + 1, lg(B)-1));
8102 56 : B = RgV_div_RgXn(B, e);
8103 : }
8104 1085 : return mkvec3(almin, utoi(w), B);
8105 : }
8106 :
8107 : /* Theta multiplier: assume 4 | C, (C,D)=1 */
8108 : static GEN
8109 343 : mfthetamultiplier(GEN C, GEN D)
8110 : {
8111 343 : long s = kronecker(C, D);
8112 343 : if (Mod4(D) == 1) return s > 0 ? gen_1: gen_m1;
8113 84 : return s > 0? powIs(3): gen_I();
8114 : }
8115 : /* theta | [*,*;C,D] defined over Q(i) [else over Q] */
8116 : static int
8117 56 : mfthetaI(long C, long D) { return odd(C) || (D & 3) == 3; }
8118 : /* (theta | M) [0..n], assume (C,D) = 1 */
8119 : static GEN
8120 343 : mfthetaexpansion(GEN M, long n)
8121 : {
8122 343 : GEN w, s, al, sla, E, V = zerovec(n+1), C = gcoeff(M,2,1), D = gcoeff(M,2,2);
8123 343 : long lim, la, f, C4 = Mod4(C);
8124 343 : switch (C4)
8125 : {
8126 70 : case 0: al = gen_0; w = gen_1;
8127 70 : s = mfthetamultiplier(C,D);
8128 70 : lim = usqrt(n); gel(V, 1) = s;
8129 70 : s = gmul2n(s, 1);
8130 756 : for (f = 1; f <= lim; f++) gel(V, f*f + 1) = s;
8131 70 : break;
8132 105 : case 2: al = uutoQ(1,4); w = gen_1;
8133 105 : E = subii(C, shifti(D,1)); /* (E, D) = 1 */
8134 105 : s = gmul2n(mfthetamultiplier(E, D), 1);
8135 105 : if ((!signe(E) && equalim1(D)) || (signe(E) > 0 && signe(C) < 0))
8136 14 : s = gneg(s);
8137 105 : lim = (usqrt(n << 2) - 1) >> 1;
8138 966 : for (f = 0; f <= lim; f++) gel(V, f*(f+1) + 1) = s;
8139 105 : break;
8140 168 : default: al = gen_0; w = utoipos(4);
8141 168 : la = (-Mod4(D)*C4) & 3L;
8142 168 : E = negi(addii(D, mului(la, C)));
8143 168 : s = mfthetamultiplier(E, C); /* (E,C) = 1 */
8144 168 : if (signe(C) < 0 && signe(E) >= 0) s = gneg(s);
8145 168 : s = gsub(s, mulcxI(s));
8146 168 : sla = gmul(s, powIs(-la));
8147 168 : lim = usqrt(n); gel(V, 1) = gmul2n(s, -1);
8148 1708 : for (f = 1; f <= lim; f++) gel(V, f*f + 1) = odd(f) ? sla : s;
8149 168 : break;
8150 : }
8151 343 : return mkvec3(al, w, V);
8152 : }
8153 :
8154 : /* F 1/2 integral weight */
8155 : static GEN
8156 343 : mf2gaexpansion(GEN mf2, GEN F, GEN ga, long n, long prec)
8157 : {
8158 343 : GEN FT = mfmultheta(F), mf = obj_checkbuild(mf2, MF_MF2INIT, &mf2init);
8159 343 : GEN res, V1, Tres, V2, al, V, gsh, C = gcoeff(ga,2,1);
8160 343 : long w2, N = MF_get_N(mf), w = mfcuspcanon_width(N, umodiu(C,N));
8161 343 : long ext = (Mod4(C) != 2)? 0: (w+3) >> 2;
8162 343 : long prec2 = prec + nbits2extraprec((long)M_PI/(2*M_LN2)*sqrt(n + ext));
8163 343 : res = mfgaexpansion(mf, FT, ga, n + ext, prec2);
8164 343 : Tres = mfthetaexpansion(ga, n + ext);
8165 343 : V1 = gel(res,3);
8166 343 : V2 = gel(Tres,3);
8167 343 : al = gsub(gel(res,1), gel(Tres,1));
8168 343 : w2 = itos(gel(Tres,2));
8169 343 : if (w != itos(gel(res,2)) || w % w2)
8170 0 : pari_err_BUG("mf2gaexpansion [incorrect w2 or w]");
8171 343 : if (w2 != w) V2 = bdexpand(V2, w/w2);
8172 343 : V = RgV_div_RgXn(V1, V2);
8173 343 : gsh = gfloor(gmulsg(w, al));
8174 343 : if (!gequal0(gsh))
8175 : {
8176 35 : al = gsub(al, gdivgu(gsh, w));
8177 35 : if (gsigne(gsh) > 0)
8178 : {
8179 0 : V = RgV_shift(V, gsh);
8180 0 : V = vecslice(V, 1, n + 1);
8181 : }
8182 : else
8183 : {
8184 35 : long sh = -itos(gsh), i;
8185 35 : if (sh > ext) pari_err_BUG("mf2gaexpansion [incorrect sh]");
8186 154 : for (i = 1; i <= sh; i++)
8187 119 : if (!gequal0(gel(V,i))) pari_err_BUG("mf2gaexpansion [sh too large]");
8188 35 : V = vecslice(V, sh+1, n + sh+1);
8189 : }
8190 : }
8191 343 : obj_free(mf); return mkvec3(al, stoi(w), gprec_wtrunc(V, prec));
8192 : }
8193 :
8194 : static GEN
8195 77 : mfgaexpansionatkin(GEN mf, GEN F, GEN C, GEN D, long Q, long n, long prec)
8196 : {
8197 77 : GEN mfa = mfatkininit_i(mf, Q, 0, prec), MQ = gel(mfa,2);
8198 77 : long i, FC, k = MF_get_k(mf);
8199 77 : GEN x, v, V, z, s, CHI = mfchartoprimitive(MF_get_CHI(mf), &FC);
8200 :
8201 : /* V = mfcoefs(F | w_Q, n), can't use mfatkin because MQ nonrational */
8202 77 : V = RgM_RgC_mul(mfcoefs_mf(mf,n,1), RgM_RgC_mul(MQ, mftobasis_i(mf,F)));
8203 77 : (void)bezout(utoipos(Q), C, &x, &v);
8204 77 : s = mfchareval(CHI, (umodiu(x, FC) * umodiu(D, FC)) % FC);
8205 77 : s = gdiv(s, gpow(utoipos(Q), uutoQ(k,2), prec));
8206 77 : V = RgV_Rg_mul(V, s);
8207 77 : z = rootsof1powinit(umodiu(D,Q)*umodiu(v,Q) % Q, Q, prec);
8208 11613 : for (i = 1; i <= n+1; i++) gel(V,i) = gmul(gel(V,i), rootsof1pow(z, i-1));
8209 77 : return mkvec3(gen_0, utoipos(Q), V);
8210 : }
8211 :
8212 : static long
8213 70 : inveis_extraprec(long N, GEN ga, GEN Mvecj, long n)
8214 : {
8215 70 : long e, w = mfZC_width(N, gel(ga,1));
8216 70 : GEN f, E = gel(Mvecj,2), v = mfeisensteingacx(E, w, ga, n, DEFAULTPREC);
8217 70 : v = gel(v,2);
8218 70 : f = RgV_to_RgX(v,0); n -= RgX_valrem(f, &f);
8219 70 : e = gexpo(RgXn_inv(f, n+1));
8220 70 : return (e > 0)? nbits2extraprec(e): 0;
8221 : }
8222 : /* allow F of the form [F, mf_eisendec(F)]~ */
8223 : static GEN
8224 1834 : mfgaexpansion(GEN mf, GEN F, GEN ga, long n, long prec)
8225 : {
8226 1834 : GEN v, EF = NULL, res, Mvecj, c, d;
8227 : long precnew, N;
8228 :
8229 1834 : if (n < 0) pari_err_DOMAIN("mfgaexpansion", "n", "<", gen_0, stoi(n));
8230 1834 : if (typ(F) == t_COL && lg(F) == 3) { EF = gel(F,2); F = gel(F,1); }
8231 1834 : if (!checkmf_i(F)) pari_err_TYPE("mfgaexpansion", F);
8232 1834 : if (!check_SL2Z(ga)) pari_err_TYPE("mfgaexpansion",ga);
8233 1834 : if (typ(mf_get_gk(F)) != t_INT) return mf2gaexpansion(mf, F, ga, n, prec);
8234 1491 : c = gcoeff(ga,2,1);
8235 1491 : d = gcoeff(ga,2,2);
8236 1491 : N = MF_get_N(mf);
8237 1491 : if (!umodiu(c, mf_get_N(F)))
8238 : { /* trivial case: ga in Gamma_0(N) */
8239 329 : long w = mfcuspcanon_width(N, umodiu(c,N));
8240 329 : GEN CHI = mf_get_CHI(F);
8241 329 : GEN chid = mfcharcxeval(CHI, umodiu(d,mfcharmodulus(CHI)), prec);
8242 329 : v = mfcoefs_i(F, n/w, 1); if (!isint1(chid)) v = RgV_Rg_mul(v,chid);
8243 329 : return mkvec3(gen_0, stoi(w), bdexpandall(v,w,n+1));
8244 : }
8245 1162 : mf = MF_set_new(mf);
8246 1162 : if (MF_get_space(mf) == mf_NEW)
8247 : {
8248 483 : long cN = umodiu(c,N), g = ugcd(cN,N), Q = N/g;
8249 483 : GEN CHI = MF_get_CHI(mf);
8250 483 : if (ugcd(cN, Q)==1 && mfcharorder(CHI) <= 2
8251 231 : && g % mfcharconductor(CHI) == 0
8252 119 : && degpol(mf_get_field(F)) == 1)
8253 77 : return mfgaexpansionatkin(mf, F, c, d, Q, n, prec);
8254 : }
8255 1085 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
8256 1085 : precnew = prec;
8257 1085 : if (lg(Mvecj) < 5) precnew += inveis_extraprec(N, ga, Mvecj, n);
8258 1085 : if (!EF) EF = mf_eisendec(mf, F, precnew);
8259 1085 : res = mfgaexpansion_i(mf, EF, ga, n, precnew);
8260 1085 : return precnew == prec ? res : gprec_wtrunc(res, prec);
8261 : }
8262 :
8263 : /* parity = -1 or +1 */
8264 : static GEN
8265 217 : findd(long N, long parity)
8266 : {
8267 217 : GEN L, D = mydivisorsu(N);
8268 217 : long i, j, l = lg(D);
8269 217 : L = cgetg(l, t_VEC);
8270 1218 : for (i = j = 1; i < l; i++)
8271 : {
8272 1001 : long d = D[i];
8273 1001 : if (parity == -1) d = -d;
8274 1001 : if (sisfundamental(d)) gel(L,j++) = stoi(d);
8275 : }
8276 217 : setlg(L,j); return L;
8277 : }
8278 : /* does ND contain a divisor of N ? */
8279 : static int
8280 413 : seenD(long N, GEN ND)
8281 : {
8282 413 : long j, l = lg(ND);
8283 427 : for (j = 1; j < l; j++)
8284 14 : if (N % ND[j] == 0) return 1;
8285 413 : return 0;
8286 : }
8287 : static GEN
8288 63 : search_levels(GEN vN, const char *f)
8289 : {
8290 63 : switch(typ(vN))
8291 : {
8292 28 : case t_INT: vN = mkvecsmall(itos(vN)); break;
8293 35 : case t_VEC: case t_COL: vN = ZV_to_zv(vN); break;
8294 0 : case t_VECSMALL: vN = leafcopy(vN); break;
8295 0 : default: pari_err_TYPE(f, vN);
8296 : }
8297 63 : vecsmall_sort(vN); return vN;
8298 : }
8299 : GEN
8300 28 : mfsearch(GEN NK, GEN V, long space)
8301 : {
8302 28 : pari_sp av = avma;
8303 : GEN F, gk, NbyD, vN;
8304 : long n, nk, dk, parity, nV, i, lvN;
8305 :
8306 28 : if (typ(NK) != t_VEC || lg(NK) != 3) pari_err_TYPE("mfsearch", NK);
8307 28 : gk = gel(NK,2);
8308 28 : if (typ(gmul2n(gk, 1)) != t_INT) pari_err_TYPE("mfsearch [k]", gk);
8309 28 : switch(typ(V))
8310 : {
8311 28 : case t_VEC: V = shallowtrans(V);
8312 28 : case t_COL: break;
8313 0 : default: pari_err_TYPE("mfsearch [V]", V);
8314 : }
8315 28 : vN = search_levels(gel(NK,1), "mfsearch [N]");
8316 28 : if (gequal0(V)) { set_avma(av); retmkvec(mftrivial()); }
8317 14 : lvN = lg(vN);
8318 :
8319 14 : Qtoss(gk, &nk,&dk);
8320 14 : parity = (dk == 1 && odd(nk)) ? -1 : 1;
8321 14 : nV = lg(V)-2;
8322 14 : F = cgetg(1, t_VEC);
8323 14 : NbyD = const_vec(vN[lvN-1], cgetg(1,t_VECSMALL));
8324 231 : for (n = 1; n < lvN; n++)
8325 : {
8326 217 : long N = vN[n];
8327 : GEN L;
8328 217 : if (N <= 0 || (dk == 2 && (N & 3))) continue;
8329 217 : L = findd(N, parity);
8330 630 : for (i = 1; i < lg(L); i++)
8331 : {
8332 413 : GEN mf, M, CO, gD = gel(L,i);
8333 413 : GEN *ND = (GEN*)NbyD + itou(gD); /* points to NbyD[|D|] */
8334 :
8335 413 : if (seenD(N, *ND)) continue;
8336 413 : mf = mfinit_Nndkchi(N, nk, dk, get_mfchar(gD), space, 1);
8337 413 : M = mfcoefs_mf(mf, nV, 1);
8338 413 : CO = inverseimage(M, V); if (lg(CO) == 1) continue;
8339 :
8340 42 : F = vec_append(F, mflinear(mf,CO));
8341 42 : *ND = vecsmall_append(*ND, N); /* add to NbyD[|D|] */
8342 : }
8343 : }
8344 14 : return gc_GEN(av, F);
8345 : }
8346 :
8347 : static GEN
8348 889 : search_from_split(GEN mf, GEN vap, GEN vlp)
8349 : {
8350 889 : pari_sp av = avma;
8351 889 : long lvlp = lg(vlp), j, jv, l1;
8352 889 : GEN v, NK, S1, S, M = NULL;
8353 :
8354 889 : S1 = gel(split_i(mf, 1, 0), 1); /* rational newforms */
8355 889 : l1 = lg(S1);
8356 889 : if (l1 == 1) return gc_NULL(av);
8357 455 : v = cgetg(l1, t_VEC);
8358 455 : S = MF_get_S(mf);
8359 455 : NK = mf_get_NK(gel(S,1));
8360 455 : if (lvlp > 1) M = rowpermute(mfcoefs_mf(mf, vlp[lvlp-1], 1), vlp);
8361 980 : for (j = jv = 1; j < l1; j++)
8362 : {
8363 525 : GEN vF = gel(S1,j);
8364 : long t;
8365 658 : for (t = lvlp-1; t > 0; t--)
8366 : { /* lhs = vlp[j]-th coefficient of eigenform */
8367 595 : GEN rhs = gel(vap,t), lhs = RgMrow_RgC_mul(M, vF, t);
8368 595 : if (!gequal(lhs, rhs)) break;
8369 : }
8370 525 : if (!t) gel(v,jv++) = mflinear_i(NK,S,vF);
8371 : }
8372 455 : if (jv == 1) return gc_NULL(av);
8373 63 : setlg(v,jv); return v;
8374 : }
8375 : GEN
8376 35 : mfeigensearch(GEN NK, GEN AP)
8377 : {
8378 35 : pari_sp av = avma;
8379 35 : GEN k, vN, vap, vlp, vres = cgetg(1, t_VEC), D;
8380 : long n, lvN, i, l, even;
8381 :
8382 35 : if (!AP) l = 1;
8383 : else
8384 : {
8385 28 : l = lg(AP);
8386 28 : if (typ(AP) != t_VEC) pari_err_TYPE("mfeigensearch",AP);
8387 : }
8388 35 : vap = cgetg(l, t_VEC);
8389 35 : vlp = cgetg(l, t_VECSMALL);
8390 35 : if (l > 1)
8391 : {
8392 28 : GEN perm = indexvecsort(AP, mkvecsmall(1));
8393 77 : for (i = 1; i < l; i++)
8394 : {
8395 49 : GEN v = gel(AP,perm[i]), gp, ap;
8396 49 : if (typ(v) != t_VEC || lg(v) != 3) pari_err_TYPE("mfeigensearch", AP);
8397 49 : gp = gel(v,1);
8398 49 : ap = gel(v,2);
8399 49 : if (typ(gp) != t_INT || (typ(ap) != t_INT && typ(ap) != t_INTMOD))
8400 0 : pari_err_TYPE("mfeigensearch", AP);
8401 49 : gel(vap,i) = ap;
8402 49 : vlp[i] = itos(gp)+1; if (vlp[i] < 0) pari_err_TYPE("mfeigensearch", AP);
8403 : }
8404 : }
8405 35 : l = lg(NK);
8406 35 : if (typ(NK) != t_VEC || l != 3) pari_err_TYPE("mfeigensearch",NK);
8407 35 : k = gel(NK,2);
8408 35 : vN = search_levels(gel(NK,1), "mfeigensearch [N]");
8409 35 : lvN = lg(vN);
8410 35 : vecsmall_sort(vlp);
8411 35 : even = !mpodd(k);
8412 980 : for (n = 1; n < lvN; n++)
8413 : {
8414 945 : pari_sp av2 = avma;
8415 : GEN mf, L;
8416 945 : long N = vN[n];
8417 945 : if (even) D = gen_1;
8418 : else
8419 : {
8420 112 : long r = (N&3L);
8421 112 : if (r == 1 || r == 2) continue;
8422 56 : D = stoi( corediscs(-N, NULL) ); /* < 0 */
8423 : }
8424 889 : mf = mfinit_i(mkvec3(utoipos(N), k, D), mf_NEW);
8425 889 : L = search_from_split(mf, vap, vlp);
8426 889 : if (L) vres = shallowconcat(vres, L); else set_avma(av2);
8427 : }
8428 35 : return gc_GEN(av, vres);
8429 : }
8430 :
8431 : /* tf_{N,k}(n) */
8432 : static GEN
8433 4512970 : mfnewtracecache(long N, long k, long n, cachenew_t *cache)
8434 : {
8435 4512970 : GEN C = NULL, S;
8436 : long lcache;
8437 4512970 : if (!n) return gen_0;
8438 4374202 : S = gel(cache->vnew,N);
8439 4374202 : lcache = lg(S);
8440 4374202 : if (n < lcache) C = gel(S, n);
8441 4374202 : if (C) cache->newHIT++;
8442 2604859 : else C = mfnewtrace_i(N,k,n,cache);
8443 4374202 : cache->newTOTAL++;
8444 4374202 : if (n < lcache) gel(S,n) = C;
8445 4374202 : return C;
8446 : }
8447 :
8448 : static long
8449 1393 : mfdim_Nkchi(long N, long k, GEN CHI, long space)
8450 : {
8451 1393 : if (k < 0 || badchar(N,k,CHI)) return 0;
8452 1092 : if (k == 0)
8453 35 : return mfcharistrivial(CHI) && !space_is_cusp(space)? 1: 0;
8454 1057 : switch(space)
8455 : {
8456 245 : case mf_NEW: return mfnewdim(N,k,CHI);
8457 196 : case mf_CUSP:return mfcuspdim(N,k,CHI);
8458 168 : case mf_OLD: return mfolddim(N,k,CHI);
8459 217 : case mf_FULL:return mffulldim(N,k,CHI);
8460 231 : case mf_EISEN: return mfeisensteindim(N,k,CHI);
8461 0 : default: pari_err_FLAG("mfdim");
8462 : }
8463 : return 0;/*LCOV_EXCL_LINE*/
8464 : }
8465 : static long
8466 2114 : mf1dimsum(long N, long space)
8467 : {
8468 2114 : switch(space)
8469 : {
8470 1050 : case mf_NEW: return mf1newdimsum(N);
8471 1057 : case mf_CUSP: return mf1cuspdimsum(N);
8472 7 : case mf_OLD: return mf1olddimsum(N);
8473 : }
8474 0 : pari_err_FLAG("mfdim");
8475 : return 0; /*LCOV_EXCL_LINE*/
8476 : }
8477 : /* mfdim for k = nk/dk */
8478 : static long
8479 44744 : mfdim_Nndkchi(long N, long nk, long dk, GEN CHI, long space)
8480 43463 : { return (dk == 2)? mf2dim_Nkchi(N, nk >> 1, CHI, space)
8481 88186 : : mfdim_Nkchi(N, nk, CHI, space); }
8482 : /* FIXME: use direct dim Gamma1(N) formula, don't compute individual spaces */
8483 : static long
8484 252 : mfkdimsum(long N, long k, long dk, long space)
8485 : {
8486 252 : GEN w = mfchars(N, k, dk, NULL);
8487 252 : long i, j, D = 0, l = lg(w);
8488 1239 : for (i = j = 1; i < l; i++)
8489 : {
8490 987 : GEN CHI = gel(w,i);
8491 987 : long d = mfdim_Nndkchi(N,k,dk,CHI,space);
8492 987 : if (d) D += d * myeulerphiu(mfcharorder(CHI));
8493 : }
8494 252 : return D;
8495 : }
8496 : static GEN
8497 105 : mf1dims(long N, GEN vCHI, long space)
8498 : {
8499 105 : GEN D = NULL;
8500 105 : switch(space)
8501 : {
8502 56 : case mf_NEW: D = mf1newdimall(N, vCHI); break;
8503 21 : case mf_CUSP:D = mf1cuspdimall(N, vCHI); break;
8504 28 : case mf_OLD: D = mf1olddimall(N, vCHI); break;
8505 0 : default: pari_err_FLAG("mfdim");
8506 : }
8507 105 : return D;
8508 : }
8509 : static GEN
8510 2961 : mfkdims(long N, long k, long dk, GEN vCHI, long space)
8511 : {
8512 2961 : GEN D, w = mfchars(N, k, dk, vCHI);
8513 2961 : long i, j, l = lg(w);
8514 2961 : D = cgetg(l, t_VEC);
8515 46592 : for (i = j = 1; i < l; i++)
8516 : {
8517 43631 : GEN CHI = gel(w,i);
8518 43631 : long d = mfdim_Nndkchi(N,k,dk,CHI,space);
8519 43631 : if (vCHI)
8520 574 : gel(D, j++) = mkvec2s(d, 0);
8521 43057 : else if (d)
8522 2520 : gel(D, j++) = fmt_dim(CHI, d, 0);
8523 : }
8524 2961 : setlg(D,j); return D;
8525 : }
8526 : GEN
8527 5719 : mfdim(GEN NK, long space)
8528 : {
8529 5719 : pari_sp av = avma;
8530 : long N, k, dk, joker;
8531 : GEN CHI, mf;
8532 5719 : if ((mf = checkMF_i(NK))) return utoi(MF_get_dim(mf));
8533 5586 : checkNK2(NK, &N, &k, &dk, &CHI, 2);
8534 5586 : if (!CHI) joker = 1;
8535 : else
8536 2611 : switch(typ(CHI))
8537 : {
8538 2373 : case t_INT: joker = 2; break;
8539 112 : case t_COL: joker = 3; break;
8540 126 : default: joker = 0; break;
8541 : }
8542 5586 : if (joker)
8543 : {
8544 : long d;
8545 : GEN D;
8546 5460 : if (k < 0) switch(joker)
8547 : {
8548 0 : case 1: return cgetg(1,t_VEC);
8549 7 : case 2: return gen_0;
8550 0 : case 3: return mfdim0all(CHI);
8551 : }
8552 5453 : if (k == 0)
8553 : {
8554 28 : if (space_is_cusp(space)) switch(joker)
8555 : {
8556 7 : case 1: return cgetg(1,t_VEC);
8557 0 : case 2: return gen_0;
8558 7 : case 3: return mfdim0all(CHI);
8559 : }
8560 14 : switch(joker)
8561 : {
8562 : long i, l;
8563 7 : case 1: retmkvec(fmt_dim(mfchartrivial(),0,0));
8564 0 : case 2: return gen_1;
8565 7 : case 3: l = lg(CHI); D = cgetg(l,t_VEC);
8566 35 : for (i = 1; i < l; i++)
8567 : {
8568 28 : long t = mfcharistrivial(gel(CHI,i));
8569 28 : gel(D,i) = mkvec2(t? gen_1: gen_0, gen_0);
8570 : }
8571 7 : return D;
8572 : }
8573 : }
8574 5425 : if (dk == 1 && k == 1 && space != mf_EISEN)
8575 105 : {
8576 2219 : long fix = 0, space0 = space;
8577 2219 : if (space == mf_FULL) space = mf_CUSP; /* remove Eisenstein part */
8578 2219 : if (joker == 2)
8579 : {
8580 2114 : d = mf1dimsum(N, space);
8581 2114 : if (space0 == mf_FULL) d += mfkdimsum(N,k,dk,mf_EISEN);/*add it back*/
8582 2114 : return gc_utoi(av, d);
8583 : }
8584 : /* must initialize explicitly: trivial spaces for E_k/S_k differ */
8585 105 : if (space0 == mf_FULL)
8586 : {
8587 7 : if (!CHI) fix = 1; /* must remove 0 spaces */
8588 7 : CHI = mfchars(N, k, dk, CHI);
8589 : }
8590 105 : D = mf1dims(N, CHI, space);
8591 105 : if (space0 == mf_FULL)
8592 : {
8593 7 : GEN D2 = mfkdims(N, k, dk, CHI, mf_EISEN);
8594 7 : D = merge_dims(D, D2, fix? CHI: NULL);
8595 : }
8596 : }
8597 : else
8598 : {
8599 3206 : if (joker==2) { d = mfkdimsum(N,k,dk,space); return gc_utoi(av,d); }
8600 2954 : D = mfkdims(N, k, dk, CHI, space);
8601 : }
8602 3059 : if (!CHI) return gc_upto(av, vecsort(D, mkvecsmall(1)));
8603 105 : return gc_GEN(av, D);
8604 : }
8605 126 : return utoi( mfdim_Nndkchi(N, k, dk, CHI, space) );
8606 : }
8607 :
8608 : GEN
8609 357 : mfbasis(GEN NK, long space)
8610 : {
8611 357 : pari_sp av = avma;
8612 : long N, k, dk;
8613 : GEN mf, CHI;
8614 357 : if ((mf = checkMF_i(NK))) return gconcat(gel(mf,2), gel(mf,3));
8615 14 : checkNK2(NK, &N, &k, &dk, &CHI, 0);
8616 14 : if (dk == 2) return gc_GEN(av, mf2basis(N, k>>1, CHI, NULL, space));
8617 14 : mf = mfinit_Nkchi(N, k, CHI, space, 1);
8618 14 : return gc_GEN(av, MF_get_basis(mf));
8619 : }
8620 :
8621 : static GEN
8622 49 : deg1ser_shallow(GEN a1, GEN a0, long v, long e)
8623 49 : { return RgX_to_ser(deg1pol_shallow(a1, a0, v), e+2); }
8624 : /* r / x + O(1) */
8625 : static GEN
8626 49 : simple_pole(GEN r)
8627 : {
8628 49 : GEN S = deg1ser_shallow(gen_0, r, 0, 1);
8629 49 : setvalser(S, -1); return S;
8630 : }
8631 :
8632 : /* F form, E embedding; mfa = mfatkininit or root number (eigenform case) */
8633 : static GEN
8634 161 : mflfuncreate(GEN mfa, GEN F, GEN E, GEN N, GEN gk)
8635 : {
8636 161 : GEN LF = cgetg(8,t_VEC), polar = cgetg(1,t_COL), eps;
8637 161 : long k = itou(gk);
8638 161 : gel(LF,1) = lfuntag(t_LFUN_MFCLOS, mkvec3(F,E,gen_1));
8639 161 : if (typ(mfa) != t_VEC)
8640 98 : eps = mfa; /* cuspidal eigenform: root number; no poles */
8641 : else
8642 : { /* mfatkininit */
8643 63 : GEN a0, b0, vF, vG, G = NULL;
8644 63 : GEN M = gel(mfa,2), C = gel(mfa,3), mf = gel(mfa,4);
8645 63 : M = gdiv(mfmatembed(E, M), C);
8646 63 : vF = mfvecembed(E, mftobasis_i(mf, F));
8647 63 : vG = RgM_RgC_mul(M, vF);
8648 63 : if (gequal(vF,vG)) eps = gen_1;
8649 49 : else if (gequal(vF,gneg(vG))) eps = gen_m1;
8650 : else
8651 : { /* not self-dual */
8652 42 : eps = NULL;
8653 42 : G = mfatkin(mfa, F);
8654 42 : gel(LF,2) = lfuntag(t_LFUN_MFCLOS, mkvec3(G,E,ginv(C)));
8655 42 : gel(LF,6) = powIs(k);
8656 : }
8657 : /* polar part */
8658 63 : a0 = mfembed(E, mfcoef(F,0));
8659 63 : b0 = eps? gmul(eps,a0): gdiv(mfembed(E, mfcoef(G,0)), C);
8660 63 : if (!gequal0(b0))
8661 : {
8662 28 : b0 = mulcxpowIs(gmul2n(b0,1), k);
8663 28 : polar = vec_append(polar, mkvec2(gk, simple_pole(b0)));
8664 : }
8665 63 : if (!gequal0(a0))
8666 : {
8667 21 : a0 = gneg(gmul2n(a0,1));
8668 21 : polar = vec_append(polar, mkvec2(gen_0, simple_pole(a0)));
8669 : }
8670 : }
8671 161 : if (eps) /* self-dual */
8672 : {
8673 119 : gel(LF,2) = mfcharorder(mf_get_CHI(F)) <= 2? gen_0: gen_1;
8674 119 : gel(LF,6) = mulcxpowIs(eps,k);
8675 : }
8676 161 : gel(LF,3) = mkvec2(gen_0, gen_1);
8677 161 : gel(LF,4) = gk;
8678 161 : gel(LF,5) = N;
8679 161 : if (lg(polar) == 1) setlg(LF,7); else gel(LF,7) = polar;
8680 161 : return LF;
8681 : }
8682 : static GEN
8683 133 : mflfuncreateall(long sd, GEN mfa, GEN F, GEN vE, GEN gN, GEN gk)
8684 : {
8685 133 : long i, l = lg(vE);
8686 133 : GEN L = cgetg(l, t_VEC);
8687 294 : for (i = 1; i < l; i++)
8688 161 : gel(L,i) = mflfuncreate(sd? gel(mfa,i): mfa, F, gel(vE,i), gN, gk);
8689 133 : return L;
8690 : }
8691 : GEN
8692 84 : lfunmf(GEN mf, GEN F, long bitprec)
8693 : {
8694 84 : pari_sp av = avma;
8695 84 : long i, l, prec = nbits2prec(bitprec);
8696 : GEN L, gk, gN;
8697 84 : mf = checkMF(mf);
8698 84 : gk = MF_get_gk(mf);
8699 84 : gN = MF_get_gN(mf);
8700 84 : if (typ(gk)!=t_INT) pari_err_IMPL("half-integral weight");
8701 84 : if (F)
8702 : {
8703 : GEN v;
8704 77 : long s = MF_get_space(mf);
8705 77 : if (!checkmf_i(F)) pari_err_TYPE("lfunmf", F);
8706 77 : if (!mfisinspace_i(mf, F)) err_space(F);
8707 77 : L = NULL;
8708 77 : if ((s == mf_NEW || s == mf_CUSP || s == mf_FULL)
8709 63 : && gequal(mfcoefs_i(F,1,1), mkvec2(gen_0,gen_1)))
8710 : { /* check if eigenform */
8711 35 : GEN vP, vF, b = mftobasis_i(mf, F);
8712 35 : long lF, d = degpol(mf_get_field(F));
8713 35 : v = mfsplit(mf, d, 0);
8714 35 : vF = gel(v,1);
8715 35 : vP = gel(v,2); lF = lg(vF);
8716 35 : for (i = 1; i < lF; i++)
8717 28 : if (degpol(gel(vP,i)) == d && gequal(gel(vF,i), b))
8718 : {
8719 28 : GEN vE = mfgetembed(F, prec);
8720 28 : GEN Z = mffrickeeigen_i(mf, mkvec(b), mkvec(vE), prec);
8721 28 : L = mflfuncreateall(1, gel(Z,1), F, vE, gN, gk);
8722 28 : break;
8723 : }
8724 : }
8725 77 : if (!L)
8726 : { /* not an eigenform: costly general case */
8727 49 : GEN mfa = mfatkininit_i(mf, itou(gN), 1, prec);
8728 49 : L = mflfuncreateall(0,mfa, F, mfgetembed(F,prec), gN, gk);
8729 : }
8730 77 : if (lg(L) == 2) L = gel(L,1);
8731 : }
8732 : else
8733 : {
8734 7 : GEN M = mfeigenbasis(mf), vE = mfeigenembed(mf, prec);
8735 7 : GEN v = mffrickeeigen(mf, vE, prec);
8736 7 : l = lg(vE); L = cgetg(l, t_VEC);
8737 63 : for (i = 1; i < l; i++)
8738 56 : gel(L,i) = mflfuncreateall(1,gel(v,i), gel(M,i), gel(vE,i), gN, gk);
8739 : }
8740 84 : return gc_GEN(av, L);
8741 : }
8742 :
8743 : GEN
8744 28 : mffromell(GEN E)
8745 : {
8746 28 : pari_sp av = avma;
8747 : GEN mf, F, z, v, S;
8748 : long N, i, l;
8749 :
8750 28 : checkell(E);
8751 28 : if (ell_get_type(E) != t_ELL_Q) pari_err_TYPE("mfffromell [E not over Q]", E);
8752 28 : N = itos(ellQ_get_N(E));
8753 28 : mf = mfinit_i(mkvec2(utoi(N), gen_2), mf_NEW);
8754 28 : v = split_i(mf, 1, 0);
8755 28 : S = gel(v,1); l = lg(S); /* rational newforms */
8756 28 : F = tag(t_MF_ELL, mkNK(N,2,mfchartrivial()), E);
8757 28 : z = mftobasis_i(mf, F);
8758 28 : for(i = 1; i < l; i++)
8759 28 : if (gequal(z, gel(S,i))) break;
8760 28 : if (i == l) pari_err_BUG("mffromell [E is not modular]");
8761 28 : return gc_GEN(av, mkvec3(mf, F, z));
8762 : }
8763 :
8764 : /* returns -1 if not, degree otherwise */
8765 : long
8766 140 : polishomogeneous(GEN P)
8767 : {
8768 : long i, D, l;
8769 140 : if (typ(P) != t_POL) return 0;
8770 77 : D = -1; l = lg(P);
8771 322 : for (i = 2; i < l; i++)
8772 : {
8773 245 : GEN c = gel(P,i);
8774 : long d;
8775 245 : if (gequal0(c)) continue;
8776 112 : d = polishomogeneous(c);
8777 112 : if (d < 0) return -1;
8778 112 : if (D < 0) D = d + i-2; else if (D != d + i-2) return -1;
8779 : }
8780 77 : return D;
8781 : }
8782 :
8783 : /* M a pp((Gram q)^(-1)) ZM; P a homogeneous t_POL, is P spherical ? */
8784 : static int
8785 28 : RgX_isspherical(GEN M, GEN P)
8786 : {
8787 28 : pari_sp av = avma;
8788 28 : GEN S, v = variables_vecsmall(P);
8789 28 : long i, j, l = lg(v);
8790 28 : if (l > lg(M)) pari_err(e_MISC, "too many variables in mffromqf");
8791 21 : S = gen_0;
8792 63 : for (j = 1; j < l; j++)
8793 : {
8794 42 : GEN Mj = gel(M, j), Pj = deriv(P, v[j]);
8795 105 : for (i = 1; i <= j; i++)
8796 : {
8797 63 : GEN c = gel(Mj, i);
8798 63 : if (!signe(c)) continue;
8799 42 : if (i != j) c = shifti(c, 1);
8800 42 : S = gadd(S, gmul(c, deriv(Pj, v[i])));
8801 : }
8802 : }
8803 21 : return gc_bool(av, gequal0(S));
8804 : }
8805 :
8806 : static GEN
8807 49 : c_QFsimple_i(long n, GEN Q, GEN P)
8808 : {
8809 49 : GEN V, v = qfrep0(Q, utoi(n), 1);
8810 49 : long i, l = lg(v);
8811 49 : V = cgetg(l+1, t_VEC);
8812 49 : if (!P || equali1(P))
8813 : {
8814 42 : gel(V,1) = gen_1;
8815 420 : for (i = 2; i <= l; i++) gel(V,i) = utoi(v[i-1] << 1);
8816 : }
8817 : else
8818 : {
8819 7 : gel(V,1) = gcopy(P);
8820 7 : for (i = 2; i <= l; i++) gel(V,i) = gmulgu(P, v[i-1] << 1);
8821 : }
8822 49 : return V;
8823 : }
8824 :
8825 : /* v a t_VECSMALL of variable numbers, lg(r) >= lg(v), r is a vector of
8826 : * scalars [not involving any variable in v] */
8827 : static GEN
8828 14 : gsubstvec_i(GEN e, GEN v, GEN r)
8829 : {
8830 14 : long i, l = lg(v);
8831 42 : for(i = 1; i < l; i++) e = gsubst(e, v[i], gel(r,i));
8832 14 : return e;
8833 : }
8834 : static GEN
8835 56 : c_QF_i(long n, GEN Q, GEN P)
8836 : {
8837 56 : pari_sp av = avma;
8838 : GEN V, v, va;
8839 : long i, l;
8840 56 : if (!P || typ(P) != t_POL) return gc_upto(av, c_QFsimple_i(n, Q, P));
8841 7 : v = gel(minim(Q, utoi(2*n), NULL), 3);
8842 7 : va = variables_vecsmall(P);
8843 7 : V = zerovec(n + 1); l = lg(v);
8844 21 : for (i = 1; i < l; i++)
8845 : {
8846 14 : pari_sp av = avma;
8847 14 : GEN X = gel(v,i);
8848 14 : long c = (itos(qfeval(Q, X)) >> 1) + 1;
8849 14 : gel(V, c) = gc_upto(av, gadd(gel(V, c), gsubstvec_i(P, va, X)));
8850 : }
8851 7 : return gmul2n(V, 1);
8852 : }
8853 :
8854 : GEN
8855 77 : mffromqf(GEN Q, GEN P)
8856 : {
8857 77 : pari_sp av = avma;
8858 : GEN G, Qi, F, D, N, mf, v, gk, chi;
8859 : long m, d, space;
8860 77 : if (typ(Q) != t_MAT) pari_err_TYPE("mffromqf", Q);
8861 77 : if (!RgM_is_ZM(Q) || !qfiseven(Q))
8862 0 : pari_err_TYPE("mffromqf [not integral or even]", Q);
8863 77 : m = lg(Q)-1;
8864 77 : Qi = ZM_inv(Q, &N);
8865 77 : if (!qfiseven(Qi)) N = shifti(N, 1);
8866 77 : d = 0;
8867 77 : if (!P || gequal1(P)) P = NULL;
8868 : else
8869 : {
8870 35 : P = simplify_shallow(P);
8871 35 : if (typ(P) == t_POL)
8872 : {
8873 28 : d = polishomogeneous(P);
8874 28 : if (d < 0) pari_err_TYPE("mffromqf [not homogeneous t_POL]", P);
8875 28 : if (!RgX_isspherical(Qi, P))
8876 7 : pari_err_TYPE("mffromqf [not a spherical t_POL]", P);
8877 : }
8878 : }
8879 63 : gk = uutoQ(m + 2*d, 2);
8880 63 : D = ZM_det(Q);
8881 63 : if (!odd(m)) { if ((m & 3) == 2) D = negi(D); } else D = shifti(D, 1);
8882 63 : space = d > 0 ? mf_CUSP : mf_FULL;
8883 63 : G = znstar0(N,1);
8884 63 : chi = mkvec2(G, znchar_quad(G,D));
8885 63 : mf = mfinit(mkvec3(N, gk, chi), space);
8886 63 : if (odd(d))
8887 : {
8888 7 : F = mftrivial();
8889 7 : v = zerocol(MF_get_dim(mf));
8890 : }
8891 : else
8892 : {
8893 56 : F = c_QF_i(mfsturm(mf), Q, P);
8894 56 : v = mftobasis_i(mf, F);
8895 56 : F = mflinear(mf, v);
8896 : }
8897 63 : return gc_GEN(av, mkvec3(mf, F, v));
8898 : }
8899 :
8900 : /***********************************************************************/
8901 : /* Eisenstein Series */
8902 : /***********************************************************************/
8903 : /* \sigma_{k-1}(\chi,n) */
8904 : static GEN
8905 24192 : sigchi(long k, GEN CHI, long n)
8906 : {
8907 24192 : pari_sp av = avma;
8908 24192 : GEN S = gen_1, D = mydivisorsu(u_ppo(n,mfcharmodulus(CHI)));
8909 24192 : long i, l = lg(D), ord = mfcharorder(CHI), vt = varn(mfcharpol(CHI));
8910 83671 : for (i = 2; i < l; i++) /* skip D[1] = 1 */
8911 : {
8912 59479 : long d = D[i], a = mfcharevalord(CHI, d, ord);
8913 59479 : S = gadd(S, Qab_Czeta(a, ord, powuu(d, k-1), vt));
8914 : }
8915 24192 : return gc_upto(av,S);
8916 : }
8917 :
8918 : /* write n = n0*n1*n2, (n0,N1*N2) = 1, n1 | N1^oo, n2 | N2^oo;
8919 : * return NULL if (n,N1,N2) > 1, else return factoru(n0) */
8920 : static GEN
8921 649418 : sigchi2_dec(long n, long N1, long N2, long *pn1, long *pn2)
8922 : {
8923 649418 : GEN P0, E0, P, E, fa = myfactoru(n);
8924 : long i, j, l;
8925 649418 : *pn1 = 1;
8926 649418 : *pn2 = 1;
8927 649418 : if (N1 == 1 && N2 == 1) return fa;
8928 635593 : P = gel(fa,1); l = lg(P);
8929 635593 : E = gel(fa,2);
8930 635593 : P0 = cgetg(l, t_VECSMALL);
8931 635593 : E0 = cgetg(l, t_VECSMALL);
8932 1472373 : for (i = j = 1; i < l; i++)
8933 : {
8934 942088 : long p = P[i], e = E[i];
8935 942088 : if (N1 % p == 0)
8936 : {
8937 142520 : if (N2 % p == 0) return NULL;
8938 37212 : *pn1 *= upowuu(p,e);
8939 : }
8940 799568 : else if (N2 % p == 0)
8941 129059 : *pn2 *= upowuu(p,e);
8942 670509 : else { P0[j] = p; E0[j] = e; j++; }
8943 : }
8944 530285 : setlg(P0, j);
8945 530285 : setlg(E0, j); return mkvec2(P0,E0);
8946 : }
8947 :
8948 : /* sigma_{k-1}(\chi_1,\chi_2,n), ord multiple of lcm(ord(CHI1),ord(CHI2)) */
8949 : static GEN
8950 593803 : sigchi2(long k, GEN CHI1, GEN CHI2, long n, long ord)
8951 : {
8952 593803 : pari_sp av = avma;
8953 : GEN S, D;
8954 593803 : long i, l, n1, n2, vt, N1 = mfcharmodulus(CHI1), N2 = mfcharmodulus(CHI2);
8955 593803 : D = sigchi2_dec(n, N1, N2, &n1, &n2); if (!D) return gc_const(av, gen_0);
8956 493178 : D = divisorsu_fact(D); l = lg(D);
8957 493178 : vt = varn(mfcharpol(CHI1));
8958 2123142 : for (i = 1, S = gen_0; i < l; i++)
8959 : { /* S += d^(k-1)*chi1(d)*chi2(n/d) */
8960 1629964 : long a, d = n2*D[i], nd = n1*D[l-i]; /* (d,N1)=1; (n/d,N2) = 1 */
8961 1629964 : a = mfcharevalord(CHI1, d, ord) + mfcharevalord(CHI2, nd, ord);
8962 1629964 : if (a >= ord) a -= ord;
8963 1629964 : S = gadd(S, Qab_Czeta(a, ord, powuu(d, k-1), vt));
8964 : }
8965 493178 : return gc_upto(av, S);
8966 : }
8967 :
8968 : /**************************************************************************/
8969 : /** Dirichlet characters with precomputed values **/
8970 : /**************************************************************************/
8971 : /* CHI mfchar */
8972 : static GEN
8973 14266 : mfcharcxinit(GEN CHI, long prec)
8974 : {
8975 14266 : GEN G = gel(CHI,1), chi = gel(CHI,2), z, V;
8976 14266 : GEN v = ncharvecexpo(G, znconrey_normalized(G,chi));
8977 14266 : long n, l = lg(v), o = mfcharorder(CHI);
8978 14266 : V = cgetg(l, t_VEC);
8979 14266 : z = grootsof1(o, prec); /* Mod(t, Phi_o(t)) -> e(1/o) */
8980 123893 : for (n = 1; n < l; n++) gel(V,n) = v[n] < 0? gen_0: gel(z, v[n]+1);
8981 14266 : return mkvecn(6, G, chi, gmfcharorder(CHI), v, V, mfcharpol(CHI));
8982 : }
8983 : /* v a "CHIvec" */
8984 : static long
8985 24695398 : CHIvec_N(GEN v) { return itou(znstar_get_N(gel(v,1))); }
8986 : static GEN
8987 15456 : CHIvec_CHI(GEN v)
8988 15456 : { return mkvec4(gel(v,1), gel(v,2), gel(v,3), gel(v,6)); }
8989 : /* character order */
8990 : static long
8991 35777 : CHIvec_ord(GEN v) { return itou(gel(v,3)); }
8992 : /* character exponents, i.e. t such that chi(n) = e(t) */
8993 : static GEN
8994 422457 : CHIvec_expo(GEN v) { return gel(v,4); }
8995 : /* character values chi(n) */
8996 : static GEN
8997 24048745 : CHIvec_val(GEN v) { return gel(v,5); }
8998 : /* CHI(n) */
8999 : static GEN
9000 24038714 : mychareval(GEN v, long n)
9001 : {
9002 24038714 : long N = CHIvec_N(v), ind = n%N;
9003 24038714 : if (ind <= 0) ind += N;
9004 24038714 : return gel(CHIvec_val(v), ind);
9005 : }
9006 : /* return c such that CHI(n) = e(c / ordz) or -1 if (n,N) > 1 */
9007 : static long
9008 422457 : mycharexpo(GEN v, long n)
9009 : {
9010 422457 : long N = CHIvec_N(v), ind = n%N;
9011 422457 : if (ind <= 0) ind += N;
9012 422457 : return CHIvec_expo(v)[ind];
9013 : }
9014 : /* faster than mfcharparity */
9015 : static long
9016 54082 : CHIvec_parity(GEN v) { return mycharexpo(v,-1) ? -1: 1; }
9017 : /**************************************************************************/
9018 :
9019 : static ulong
9020 55615 : sigchi2_Fl(long k, GEN CHI1vec, GEN CHI2vec, long n, GEN vz, ulong p)
9021 : {
9022 55615 : pari_sp av = avma;
9023 55615 : long ordz = lg(vz)-2, i, l, n1, n2;
9024 55615 : ulong S = 0;
9025 55615 : GEN D = sigchi2_dec(n, CHIvec_N(CHI1vec), CHIvec_N(CHI2vec), &n1, &n2);
9026 55615 : if (!D) return gc_ulong(av,S);
9027 50932 : D = divisorsu_fact(D);
9028 50932 : l = lg(D);
9029 172263 : for (i = 1; i < l; i++)
9030 : { /* S += d^(k-1)*chi1(d)*chi2(n/d) */
9031 121331 : long a, d = n2*D[i], nd = n1*D[l-i]; /* (d,N1)=1, (n/d,N2)=1 */
9032 121331 : a = mycharexpo(CHI2vec, nd) + mycharexpo(CHI1vec, d);
9033 121331 : if (a >= ordz) a -= ordz;
9034 121331 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, Fl_powu(d,k-1,p), p), p);
9035 : }
9036 50932 : return gc_ulong(av,S);
9037 : }
9038 :
9039 : /**********************************************************************/
9040 : /* Fourier expansions of Eisenstein series */
9041 : /**********************************************************************/
9042 : /* L(CHI_t,0) / 2, CHI_t(n) = CHI(n)(t/n) as a character modulo N*t,
9043 : * order(CHI) | ord != 0 */
9044 : static GEN
9045 2618 : charLFwt1(long N, GEN CHI, long ord, long t)
9046 : {
9047 : GEN S;
9048 : long r, vt;
9049 :
9050 2618 : if (N == 1 && t == 1) return mkfrac(gen_m1,stoi(4));
9051 2618 : S = gen_0; vt = varn(mfcharpol(CHI));
9052 295435 : for (r = 1; r < N; r++)
9053 : { /* S += r*chi(r) */
9054 : long a, c;
9055 292817 : if (ugcd(N,r) != 1) continue;
9056 233310 : a = mfcharevalord(CHI,r,ord);
9057 233310 : c = (t != 1 && kross(t, r) < 0)? -r: r;
9058 233310 : S = gadd(S, Qab_Czeta(a, ord, stoi(c), vt));
9059 : }
9060 2618 : return gdivgs(S, -2*N);
9061 : }
9062 : /* L(CHI,0) / 2, mod p */
9063 : static ulong
9064 2002 : charLFwt1_Fl(GEN CHIvec, GEN vz, ulong p)
9065 : {
9066 2002 : long r, m = CHIvec_N(CHIvec);
9067 : ulong S;
9068 2002 : if (m == 1) return Rg_to_Fl(mkfrac(gen_m1,stoi(4)), p);
9069 2002 : S = 0;
9070 95977 : for (r = 1; r < m; r++)
9071 : { /* S += r*chi(r) */
9072 93975 : long a = mycharexpo(CHIvec,r);
9073 93975 : if (a < 0) continue;
9074 91616 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, r, p), p);
9075 : }
9076 2002 : return Fl_div(Fl_neg(S,p), 2*m, p);
9077 : }
9078 : /* L(CHI_t,1-k) / 2, CHI_t(n) = CHI(n) * (t/n), order(CHI) | ord != 0;
9079 : * assume conductor of CHI_t divides N */
9080 : static GEN
9081 4361 : charLFwtk(long N, long k, GEN CHI, long ord, long t)
9082 : {
9083 : GEN S, P, dS;
9084 : long r, vt;
9085 :
9086 4361 : if (k == 1) return charLFwt1(N, CHI, ord, t);
9087 1743 : if (N == 1 && t == 1) return gdivgs(bernfrac(k),-2*k);
9088 1057 : vt = varn(mfcharpol(CHI));
9089 1057 : P = bern_init(N, k, &dS);
9090 1057 : dS = mul_denom(dS, stoi(-2*N*k));
9091 14126 : for (r = 1, S = gen_0; r < N; r++)
9092 : { /* S += P(r)*chi(r) */
9093 : long a;
9094 : GEN C;
9095 13069 : if (ugcd(r,N) != 1) continue;
9096 10472 : a = mfcharevalord(CHI,r,ord);
9097 10472 : C = ZX_Z_eval(P, utoi(r));
9098 10472 : if (t != 1 && kross(t, r) < 0) C = gneg(C);
9099 10472 : S = gadd(S, Qab_Czeta(a, ord, C, vt));
9100 : }
9101 1057 : return gdiv(S, dS);
9102 : }
9103 : /* L(CHI,1-k) / 2, mod p */
9104 : static ulong
9105 2891 : charLFwtk_Fl(long k, GEN CHIvec, GEN vz, ulong p)
9106 : {
9107 : GEN P, dS;
9108 : long r, m;
9109 : ulong S, d;
9110 2891 : if (k == 1) return charLFwt1_Fl(CHIvec, vz, p);
9111 889 : m = CHIvec_N(CHIvec);
9112 889 : if (m == 1) return Rg_to_Fl(gdivgs(bernfrac(k),-2*k), p);
9113 504 : P = ZX_to_Flx(bern_init(m, k, &dS), p);
9114 10402 : for (r = 1, S = 0; r < m; r++)
9115 : { /* S += P(r)*chi(r) */
9116 9898 : long a = mycharexpo(CHIvec,r);
9117 9898 : if (a < 0) continue;
9118 8638 : S = Fl_add(S, Qab_Czeta_Fl(a, vz, Flx_eval(P,r,p), p), p);
9119 : }
9120 504 : d = (2 * k * m) % p; if (dS) d = Fl_mul(d, umodiu(dS, p), p);
9121 504 : return Fl_div(Fl_neg(S,p), d, p);
9122 : }
9123 :
9124 : static GEN
9125 7917 : mfeisenstein2_0(long k, GEN CHI1, GEN CHI2, long ord)
9126 : {
9127 7917 : long N1 = mfcharmodulus(CHI1), N2 = mfcharmodulus(CHI2);
9128 7917 : if (k == 1 && N1 == 1) return charLFwtk(N2, 1, CHI2, ord, 1);
9129 5306 : if (N2 == 1) return charLFwtk(N1, k, CHI1, ord, 1);
9130 3773 : return gen_0;
9131 : }
9132 : static ulong
9133 4382 : mfeisenstein2_0_Fl(long k, GEN CHI1vec, GEN CHI2vec, GEN vz, ulong p)
9134 : {
9135 4382 : if (k == 1 && CHIvec_N(CHI1vec) == 1)
9136 2002 : return charLFwtk_Fl(k, CHI2vec, vz, p);
9137 2380 : else if (CHIvec_N(CHI2vec) == 1)
9138 889 : return charLFwtk_Fl(k, CHI1vec, vz, p);
9139 1491 : else return 0;
9140 : }
9141 : static GEN
9142 133 : NK_eisen2(long k, GEN CHI1, GEN CHI2, long ord)
9143 : {
9144 133 : long o, N = mfcharmodulus(CHI1)*mfcharmodulus(CHI2);
9145 133 : GEN CHI = mfcharmul(CHI1, CHI2);
9146 133 : o = mfcharorder(CHI);
9147 133 : if ((ord & 3) == 2) ord >>= 1;
9148 133 : if ((o & 3) == 2) o >>= 1;
9149 133 : if (ord != o) pari_err_IMPL("mfeisenstein for these characters");
9150 126 : return mkNK(N, k, CHI);
9151 : }
9152 : static GEN
9153 371 : mfeisenstein_i(long k, GEN CHI1, GEN CHI2)
9154 : {
9155 371 : long s = 1, ord, vt;
9156 : GEN E0, NK, vchi, T;
9157 371 : if (CHI2) { CHI2 = get_mfchar(CHI2); if (mfcharparity(CHI2) < 0) s = -s; }
9158 371 : if (CHI1) { CHI1 = get_mfchar(CHI1); if (mfcharparity(CHI1) < 0) s = -s; }
9159 357 : if (s != m1pk(k)) return mftrivial();
9160 336 : if (!CHI1) CHI1 = mfchartrivial();
9161 336 : if (!CHI2)
9162 : { /* E_k(chi1) */
9163 203 : vt = varn(mfcharpol(CHI1));
9164 203 : ord = mfcharorder(CHI1);
9165 203 : NK = mkNK(mfcharmodulus(CHI1), k, CHI1);
9166 203 : E0 = charLFwtk(mfcharmodulus(CHI1), k, CHI1, ord, 1);
9167 203 : vchi = mkvec3(E0, mkvec(mfcharpol(CHI1)), CHI1);
9168 203 : return tag(t_MF_EISEN, NK, vchi);
9169 : }
9170 : /* E_k(chi1,chi2) */
9171 133 : vt = varn(mfcharpol(CHI1));
9172 133 : ord = ulcm(mfcharorder(CHI1), mfcharorder(CHI2));
9173 133 : NK = NK_eisen2(k, CHI1, CHI2, ord);
9174 126 : E0 = mfeisenstein2_0(k, CHI1, CHI2, ord);
9175 126 : T = mkvec(polcyclo(ord, vt));
9176 126 : vchi = mkvec4(E0, T, CHI1, CHI2);
9177 126 : return tag2(t_MF_EISEN, NK, vchi, mkvecsmall2(ord,0));
9178 : }
9179 : GEN
9180 371 : mfeisenstein(long k, GEN CHI1, GEN CHI2)
9181 : {
9182 371 : pari_sp av = avma;
9183 371 : if (k < 1) pari_err_DOMAIN("mfeisenstein", "k", "<", gen_1, stoi(k));
9184 371 : return gc_GEN(av, mfeisenstein_i(k, CHI1, CHI2));
9185 : }
9186 :
9187 : static GEN
9188 2625 : mfeisenstein2all(long N0, GEN NK, long k, GEN CHI1, GEN CHI2, GEN T, long o)
9189 : {
9190 2625 : GEN E, E0 = mfeisenstein2_0(k, CHI1,CHI2, o), vchi = mkvec4(E0, T, CHI1,CHI2);
9191 2625 : long j, d = (lg(T)==4)? itou(gmael(T,3,1)): 1;
9192 2625 : E = cgetg(d+1, t_VEC);
9193 5369 : for (j=1; j<=d; j++) gel(E,j) = tag2(t_MF_EISEN, NK,vchi,mkvecsmall2(o,j-1));
9194 2625 : return mfbdall(E, N0 / mf_get_N(gel(E,1)));
9195 : }
9196 :
9197 : /* list of characters on G = (Z/NZ)^*, v[i] = NULL if (i,N) > 1, else
9198 : * the conductor of Conrey label i, [conductor, primitive char].
9199 : * Trivial chi (label 1) comes first */
9200 : static GEN
9201 1169 : zncharsG(GEN G)
9202 : {
9203 1169 : long i, l, N = itou(znstar_get_N(G));
9204 : GEN vCHI, V;
9205 1169 : if (N == 1) return mkvec2(gen_1,cgetg(1,t_COL));
9206 1169 : vCHI = const_vec(N,NULL);
9207 1169 : V = cyc2elts(znstar_get_conreycyc(G));
9208 1169 : l = lg(V);
9209 207739 : for (i = 1; i < l; i++)
9210 : {
9211 206570 : GEN chi0, chi = zc_to_ZC(gel(V,i)), n, F;
9212 206570 : F = znconreyconductor(G, chi, &chi0);
9213 206570 : if (typ(F) != t_INT) F = gel(F,1);
9214 206570 : n = znconreyexp(G, chi);
9215 206570 : gel(vCHI, itos(n)) = mkvec2(chi0, F);
9216 : }
9217 1169 : return vCHI;
9218 : }
9219 :
9220 : /* CHI primitive, f(CHI) | N. Return pairs (CHI1,CHI2) both primitive
9221 : * such that f(CHI1)*f(CHI2) | N and CHI1 * CHI2 = CHI;
9222 : * if k = 1, CHI1 is even; if k = 2, omit (1,1) if CHI = 1 */
9223 : static GEN
9224 1407 : mfeisensteinbasis_i(long N0, long k, GEN CHI)
9225 : {
9226 1407 : GEN G = gel(CHI,1), chi = gel(CHI,2), vT = const_vec(myeulerphiu(N0), NULL);
9227 1407 : GEN CHI0, GN, chiN, Lchi, LG, V, RES, NK, T, C = mfcharpol(CHI);
9228 1407 : long i, j, l, n, n1, N, ord = mfcharorder(CHI);
9229 1407 : long F = mfcharmodulus(CHI), vt = varn(mfcharpol(CHI));
9230 :
9231 1407 : CHI0 = (F == 1)? CHI: mfchartrivial();
9232 1407 : j = 1; RES = cgetg(N0+1, t_VEC);
9233 1407 : T = gel(vT,ord) = Qab_trace_init(ord, ord, C, C);
9234 1407 : if (F != 1 || k != 2)
9235 : { /* N1 = 1 */
9236 1253 : NK = mkNK(F, k, CHI);
9237 1253 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI0, CHI, T, ord);
9238 1253 : if (F != 1 && k != 1)
9239 329 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI, CHI0, T, ord);
9240 : }
9241 1407 : if (N0 == 1) { setlg(RES,j); return RES; }
9242 1316 : GN = G; chiN = chi;
9243 1316 : if (F == N0) N = N0;
9244 : else
9245 : {
9246 714 : GEN faN = myfactoru(N0), P = gel(faN,1), E = gel(faN,2);
9247 714 : long lP = lg(P);
9248 1848 : for (i = N = 1; i < lP; i++)
9249 : {
9250 1134 : long p = P[i];
9251 1134 : N *= upowuu(p, maxuu(E[i]/2, z_lval(F,p)));
9252 : }
9253 714 : if ((N & 3) == 2) N >>= 1;
9254 714 : if (N == 1) { setlg(RES,j); return RES; }
9255 567 : if (F != N)
9256 : {
9257 133 : GN = znstar0(utoipos(N),1);
9258 133 : chiN = zncharinduce(G, chi, GN);
9259 : }
9260 : }
9261 1169 : LG = const_vec(N, NULL); /* LG[d] = znstar(d,1) or NULL */
9262 1169 : gel(LG,1) = gel(CHI0,1);
9263 1169 : gel(LG,F) = G;
9264 1169 : gel(LG,N) = GN;
9265 1169 : Lchi = coprimes_zv(N);
9266 1169 : n = itou(znconreyexp(GN,chiN));
9267 1169 : V = zncharsG(GN); l = lg(V);
9268 263305 : for (n1 = 2; n1 < l; n1++) /* skip 1 (trivial char) */
9269 : {
9270 262136 : GEN v = gel(V,n1), w, chi1, chi2, G1, G2, CHI1, CHI2;
9271 : long N12, N1, N2, no, o12, t, m;
9272 262136 : if (!Lchi[n1] || n1 == n) continue; /* skip trivial chi2 */
9273 204197 : chi1 = gel(v,1); N1 = itou(gel(v,2)); /* conductor of chi1 */
9274 204197 : w = gel(V, Fl_div(n,n1,N));
9275 204197 : chi2 = gel(w,1); N2 = itou(gel(w,2)); /* conductor of chi2 */
9276 204197 : N12 = N1 * N2;
9277 204197 : if (N0 % N12) continue;
9278 :
9279 1771 : G1 = gel(LG,N1); if (!G1) gel(LG,N1) = G1 = znstar0(utoipos(N1), 1);
9280 1771 : if (k == 1 && zncharisodd(G1,chi1)) continue;
9281 1043 : G2 = gel(LG,N2); if (!G2) gel(LG,N2) = G2 = znstar0(utoipos(N2), 1);
9282 1043 : CHI1 = mfcharGL(G1, chi1);
9283 1043 : CHI2 = mfcharGL(G2, chi2);
9284 1043 : o12 = ulcm(mfcharorder(CHI1), mfcharorder(CHI2));
9285 : /* remove Galois orbit: same trace */
9286 1043 : no = Fl_powu(n1, ord, N);
9287 1414 : for (t = 1+ord, m = n1; t <= o12; t += ord)
9288 : { /* m <-> CHI1^t, if t in Gal(Q(chi1,chi2)/Q), omit (CHI1^t,CHI2^t) */
9289 371 : m = Fl_mul(m, no, N); if (!m) break;
9290 371 : if (ugcd(t, o12) == 1) Lchi[m] = 0;
9291 : }
9292 1043 : T = gel(vT,o12);
9293 1043 : if (!T) T = gel(vT,o12) = Qab_trace_init(o12, ord, polcyclo(o12,vt), C);
9294 1043 : NK = mkNK(N12, k, CHI);
9295 1043 : gel(RES, j++) = mfeisenstein2all(N0, NK, k, CHI1, CHI2, T, o12);
9296 : }
9297 1169 : setlg(RES,j); return RES;
9298 : }
9299 :
9300 : static GEN
9301 721 : mfbd_E2(GEN E2, long d, GEN CHI)
9302 : {
9303 721 : GEN E2d = mfbd_i(E2, d);
9304 721 : GEN F = mkvec2(E2, E2d), L = mkvec2(gen_1, utoineg(d));
9305 : /* cannot use mflinear_i: E2 and E2d do not have the same level */
9306 721 : return tag3(t_MF_LINEAR, mkNK(d,2,CHI), F, L, gen_1);
9307 : }
9308 : /* C-basis of E_k(Gamma_0(N),chi). If k = 1, the first basis element must not
9309 : * vanish at oo [used in mf1basis]. Here E_1(CHI), whose q^0 coefficient
9310 : * does not vanish (since L(CHI,0) does not) *if* CHI is not trivial; which
9311 : * must be the case in weight 1.
9312 : *
9313 : * (k>=3): In weight k >= 3, basis is B(d) E(CHI1,(CHI/CHI1)_prim), where
9314 : * CHI1 is primitive modulo N1, and if N2 is the conductor of CHI/CHI1
9315 : * then d*N1*N2 | N.
9316 : * (k=2): In weight k=2, same if CHI is nontrivial. If CHI is trivial, must
9317 : * not take CHI1 trivial, and must add E_2(tau)-dE_2(d tau)), where
9318 : * d|N, d > 1.
9319 : * (k=1): In weight k=1, same as k >= 3 except that we restrict to CHI1 even */
9320 : static GEN
9321 1435 : mfeisensteinbasis(long N, long k, GEN CHI)
9322 : {
9323 : long i, F;
9324 : GEN L;
9325 1435 : if (badchar(N, k, CHI)) return cgetg(1, t_VEC);
9326 1435 : if (k == 0) return mfcharistrivial(CHI)? mkvec(mf1()): cgetg(1, t_VEC);
9327 1407 : CHI = mfchartoprimitive(CHI, &F);
9328 1407 : L = mfeisensteinbasis_i(N, k, CHI);
9329 1407 : if (F == 1 && k == 2)
9330 : {
9331 154 : GEN v, E2 = mfeisenstein(2, NULL, NULL), D = mydivisorsu(N);
9332 154 : long nD = lg(D)-1;
9333 154 : v = cgetg(nD, t_VEC); L = vec_append(L,v);
9334 868 : for (i = 1; i < nD; i++) gel(v,i) = mfbd_E2(E2, D[i+1], CHI);
9335 : }
9336 1407 : return lg(L) == 1? L: shallowconcat1(L);
9337 : }
9338 :
9339 : static GEN
9340 77 : not_in_space(GEN F, long flag)
9341 : {
9342 77 : if (!flag) err_space(F);
9343 70 : return cgetg(1, t_COL);
9344 : }
9345 : /* when flag set, no error */
9346 : GEN
9347 1029 : mftobasis(GEN mf, GEN F, long flag)
9348 : {
9349 1029 : pari_sp av2, av = avma;
9350 : GEN G, v, y, gk;
9351 1029 : long N, B, ismf = checkmf_i(F);
9352 :
9353 1029 : mf = checkMF(mf);
9354 1029 : if (ismf)
9355 : {
9356 938 : if (mfistrivial(F)) return zerocol(MF_get_dim(mf));
9357 931 : if (!mf_same_k(mf, F) || !mf_same_CHI(mf, F)) return not_in_space(F, flag);
9358 : }
9359 980 : N = MF_get_N(mf);
9360 980 : gk = MF_get_gk(mf);
9361 980 : if (ismf)
9362 : {
9363 889 : long NF = mf_get_N(F);
9364 889 : B = maxuu(mfsturmNgk(NF,gk), mfsturmNgk(N,gk)) + 1;
9365 889 : v = mfcoefs_i(F,B,1);
9366 : }
9367 : else
9368 : {
9369 91 : B = mfsturmNgk(N, gk) + 1;
9370 91 : switch(typ(F))
9371 : { /* F(0),...,F(lg(v)-2) */
9372 63 : case t_SER: v = sertocol(F); settyp(v,t_VEC); break;
9373 14 : case t_VEC: v = F; break;
9374 7 : case t_COL: v = shallowtrans(F); break;
9375 7 : default: pari_err_TYPE("mftobasis",F);
9376 : v = NULL;/*LCOV_EXCL_LINE*/
9377 : }
9378 84 : if (flag) B = minss(B, lg(v)-2);
9379 : }
9380 973 : y = mftobasis_i(mf, v);
9381 973 : if (typ(y) == t_VEC)
9382 : {
9383 21 : if (flag) return gc_GEN(av, y);
9384 0 : pari_err(e_MISC, "not enough coefficients in mftobasis");
9385 : }
9386 952 : av2 = avma;
9387 952 : if (MF_get_space(mf) == mf_FULL || mfsturm(mf)+1 == B) return y;
9388 476 : G = mflinear(mf, y);
9389 476 : if (!gequal(v, mfcoefs_i(G, lg(v)-2,1))) y = NULL;
9390 476 : if (!y) { set_avma(av); return not_in_space(F, flag); }
9391 441 : set_avma(av2); return gc_upto(av, y);
9392 : }
9393 :
9394 : /* assume N > 0; first cusp is always 0 */
9395 : static GEN
9396 49 : mfcusps_i(long N)
9397 : {
9398 : long i, c, l;
9399 : GEN D, v;
9400 :
9401 49 : if (N == 1) return mkvec(gen_0);
9402 49 : D = mydivisorsu(N); l = lg(D); /* left on stack */
9403 49 : c = mfnumcuspsu_fact(myfactoru(N));
9404 49 : v = cgetg(c + 1, t_VEC);
9405 350 : for (i = c = 1; i < l; i++)
9406 : {
9407 301 : long C = D[i], NC = D[l-i], lima = ugcd(C, NC), A0, A;
9408 889 : for (A0 = 0; A0 < lima; A0++)
9409 588 : if (ugcd(A0, lima) == 1)
9410 : {
9411 539 : A = A0; while (ugcd(A,C) > 1) A += lima;
9412 392 : gel(v, c++) = uutoQ(A, C);
9413 : }
9414 : }
9415 49 : return v;
9416 : }
9417 : /* List of cusps of Gamma_0(N) */
9418 : GEN
9419 28 : mfcusps(GEN gN)
9420 : {
9421 : long N;
9422 : GEN mf;
9423 28 : if (typ(gN) == t_INT) N = itos(gN);
9424 14 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
9425 0 : else { pari_err_TYPE("mfcusps", gN); N = 0; }
9426 28 : if (N <= 0) pari_err_DOMAIN("mfcusps", "N", "<=", gen_0, stoi(N));
9427 28 : return mfcusps_i(N);
9428 : }
9429 :
9430 : long
9431 315 : mfcuspisregular(GEN NK, GEN cusp)
9432 : {
9433 : long v, N, dk, nk, t, o;
9434 : GEN mf, CHI, go, A, C, g, c, d;
9435 315 : if ((mf = checkMF_i(NK)))
9436 : {
9437 49 : GEN gk = MF_get_gk(mf);
9438 49 : N = MF_get_N(mf);
9439 49 : CHI = MF_get_CHI(mf);
9440 49 : Qtoss(gk, &nk, &dk);
9441 : }
9442 : else
9443 266 : checkNK2(NK, &N, &nk, &dk, &CHI, 0);
9444 315 : if (typ(cusp) == t_INFINITY) return 1;
9445 315 : if (typ(cusp) == t_FRAC) { A = gel(cusp,1); C = gel(cusp,2); }
9446 28 : else { A = cusp; C = gen_1; }
9447 315 : g = diviuexact(mului(N,C), ugcd(N, Fl_sqr(umodiu(C,N), N)));
9448 315 : c = mulii(negi(C),g);
9449 315 : d = addiu(mulii(A,g), 1);
9450 315 : if (!CHI) return 1;
9451 315 : go = gmfcharorder(CHI);
9452 315 : v = vali(go); if (v < 2) go = shifti(go, 2-v);
9453 315 : t = itou( znchareval(gel(CHI,1), gel(CHI,2), d, go) );
9454 315 : if (dk == 1) return t == 0;
9455 154 : o = itou(go);
9456 154 : if (kronecker(c,d) < 0) t = Fl_add(t, o/2, o);
9457 154 : if (Mod4(d) == 1) return t == 0;
9458 14 : t = Fl_sub(t, Fl_mul(o/4, nk, o), o);
9459 14 : return t == 0;
9460 : }
9461 :
9462 : /* Some useful closures */
9463 :
9464 : /* sum_{d|n} d^k */
9465 : static GEN
9466 48020 : mysumdivku(ulong n, ulong k)
9467 : {
9468 48020 : GEN fa = myfactoru(n);
9469 48020 : return k == 1? usumdiv_fact(fa): usumdivk_fact(fa,k);
9470 : }
9471 : static GEN
9472 882 : c_Ek(long n, long d, GEN F)
9473 : {
9474 882 : GEN E = cgetg(n + 2, t_VEC), C = gel(F,2);
9475 882 : long i, k = mf_get_k(F);
9476 882 : gel (E, 1) = gen_1;
9477 26264 : for (i = 1; i <= n; i++)
9478 : {
9479 25382 : pari_sp av = avma;
9480 25382 : gel(E, i+1) = gc_upto(av, gmul(C, mysumdivku(i*d, k-1)));
9481 : }
9482 882 : return E;
9483 : }
9484 :
9485 : GEN
9486 406 : mfEk(long k)
9487 : {
9488 406 : pari_sp av = avma;
9489 : GEN E0, NK;
9490 406 : if (k < 0 || odd(k)) pari_err_TYPE("mfEk [incorrect k]", stoi(k));
9491 406 : if (!k) return mf1();
9492 399 : E0 = gdivsg(-2*k, bernfrac(k));
9493 399 : NK = mkNK(1,k,mfchartrivial());
9494 399 : return gc_GEN(av, tag(t_MF_Ek, NK, E0));
9495 : }
9496 :
9497 : GEN
9498 56 : mfDelta(void)
9499 : {
9500 56 : pari_sp av = avma;
9501 56 : return gc_GEN(av, tag0(t_MF_DELTA, mkNK(1,12,mfchartrivial())));
9502 : }
9503 :
9504 : GEN
9505 798 : mfTheta(GEN psi)
9506 : {
9507 798 : pari_sp av = avma;
9508 : GEN N, gk, psi2;
9509 : long par;
9510 798 : if (!psi) { psi = mfchartrivial(); N = utoipos(4); par = 1; }
9511 : else
9512 : {
9513 : long FC;
9514 21 : psi = get_mfchar(psi);
9515 21 : FC = mfcharconductor(psi);
9516 21 : if (mfcharmodulus(psi) != FC)
9517 0 : pari_err_TYPE("mfTheta [nonprimitive character]", psi);
9518 21 : par = mfcharparity(psi);
9519 21 : N = shifti(sqru(FC),2);
9520 : }
9521 798 : if (par > 0) { gk = ghalf; psi2 = psi; }
9522 7 : else { gk = gsubsg(2, ghalf); psi2 = mfcharmul(psi, get_mfchar(stoi(-4))); }
9523 798 : return gc_GEN(av, tag(t_MF_THETA, mkgNK(N, gk, psi2, pol_x(1)), psi));
9524 : }
9525 :
9526 : /* Output 0 if not desired eta product: if flag=0 (default) require
9527 : * holomorphic at cusps. If flag set, accept meromorphic, but sill in some
9528 : * modular function space */
9529 : GEN
9530 210 : mffrometaquo(GEN eta, long flag)
9531 : {
9532 210 : pari_sp av = avma;
9533 : GEN NK, N, k, BR, P;
9534 210 : long v, cusp = 0;
9535 210 : if (!etaquotype(&eta, &N,&k,&P, &v, NULL, flag? NULL: &cusp) || cusp < 0)
9536 14 : return gc_const(av, gen_0);
9537 196 : if (lg(gel(eta,1)) == 1) { set_avma(av); return mf1(); }
9538 189 : BR = mkvec2(ZV_to_zv(gel(eta,1)), ZV_to_zv(gel(eta,2)));
9539 189 : if (v < 0) v = 0;
9540 189 : NK = mkgNK(N, k, get_mfchar(P), pol_x(1));
9541 189 : return gc_GEN(av, tag2(t_MF_ETAQUO, NK, BR, utoi(v)));
9542 : }
9543 :
9544 : /* Q^(-r) */
9545 : static GEN
9546 375 : RgXn_negpow(GEN Q, long r, long L)
9547 : {
9548 375 : if (r < 0) r = -r; else Q = RgXn_inv_i(Q, L);
9549 375 : if (r != 1) Q = RgXn_powu_i(Q, r, L);
9550 375 : return Q;
9551 : }
9552 : /* flag same as in mffrometaquo: if set, accept meromorphic. */
9553 : static GEN
9554 49 : mfisetaquo_i(GEN F, long flag)
9555 : {
9556 : GEN gk, P, E, M, S, G, CHI, v, w;
9557 : long b, l, L, N, vS, m, j;
9558 49 : const long bextra = 10;
9559 :
9560 49 : if (!checkmf_i(F)) pari_err_TYPE("mfisetaquo",F);
9561 49 : CHI = mf_get_CHI(F); if (mfcharorder(CHI) > 2) return NULL;
9562 49 : N = mf_get_N(F);
9563 49 : gk = mf_get_gk(F);
9564 49 : b = mfsturmNgk(N, gk);
9565 49 : L = maxss(N, b) + bextra;
9566 49 : S = mfcoefs_i(F, L, 1);
9567 49 : if (!RgV_is_ZV(S)) return NULL;
9568 889 : for (vS = 1; vS <= L+1; vS++)
9569 889 : if (signe(gel(S,vS))) break;
9570 49 : vS--;
9571 49 : if (vS >= bextra - 1) { L += vS; S = mfcoefs_i(F, L, 1); }
9572 49 : if (vS) { S = vecslice(S, vS+1, L+1); L -= vS; }
9573 49 : S = RgV_to_RgX(S, 0); l = lg(S)-2;
9574 49 : P = cgetg(l, t_COL);
9575 49 : E = cgetg(l, t_COL); w = v = gen_0; /* w = weight, v = valuation */
9576 1908 : for (m = j = 1; m+2 < lg(S); m++)
9577 : {
9578 1866 : GEN c = gel(S,m+2);
9579 : long r;
9580 1866 : if (is_bigint(c)) return NULL;
9581 1859 : r = -itos(c);
9582 1859 : if (r)
9583 : {
9584 375 : S = ZXn_mul(S, RgXn_negpow(eta_ZXn(m, L), r, L), L);
9585 375 : gel(P,j) = utoipos(m);
9586 375 : gel(E,j) = stoi(r);
9587 375 : v = addmuliu(v, gel(E,j), m);
9588 375 : w = addis(w, r);
9589 375 : j++;
9590 : }
9591 : }
9592 42 : if (!equalii(w, gmul2n(gk, 1)) || (!flag && !equalii(v, muluu(24,vS))))
9593 7 : return NULL;
9594 35 : setlg(P, j);
9595 35 : setlg(E, j); M = mkmat2(P, E); G = mffrometaquo(M, flag);
9596 35 : return (typ(G) != t_INT
9597 35 : && (mfsturmmf(G) <= b + bextra || mfisequal(F, G, b)))? M: NULL;
9598 : }
9599 : GEN
9600 49 : mfisetaquo(GEN F, long flag)
9601 : {
9602 49 : pari_sp av = avma;
9603 49 : GEN M = mfisetaquo_i(F, flag);
9604 49 : return M? gc_GEN(av, M): gc_const(av, gen_0);
9605 : }
9606 :
9607 : #if 0
9608 : /* number of primitive characters modulo N */
9609 : static ulong
9610 : numprimchars(ulong N)
9611 : {
9612 : GEN fa, P, E;
9613 : long i, l;
9614 : ulong n;
9615 : if ((N & 3) == 2) return 0;
9616 : fa = myfactoru(N);
9617 : P = gel(fa,1); l = lg(P);
9618 : E = gel(fa,2);
9619 : for (i = n = 1; i < l; i++)
9620 : {
9621 : ulong p = P[i], e = E[i];
9622 : if (e == 2) n *= p-2; else n *= (p-1)*(p-1)*upowuu(p,e-2);
9623 : }
9624 : return n;
9625 : }
9626 : #endif
9627 :
9628 : /* Space generated by products of two Eisenstein series */
9629 :
9630 : static int
9631 73759 : cmp_small_priority(void *E, GEN a, GEN b)
9632 : {
9633 73759 : GEN prio = (GEN)E;
9634 73759 : return cmpss(prio[(long)a], prio[(long)b]);
9635 : }
9636 : static long
9637 1288 : znstar_get_expo(GEN G) { return itou(cyc_get_expo(znstar_get_cyc(G))); }
9638 :
9639 : /* Return [vchi, bymod, vG]:
9640 : * vG[f] = znstar(f,1) for f a conductor of (at least) a char mod N; else NULL
9641 : * bymod[f] = vecsmall of conrey indexes of chars modulo f | N; else NULL
9642 : * vchi[n] = a list of CHIvec [G0,chi0,o,ncharvecexpo(G0,nchi0),...]:
9643 : * chi0 = primitive char attached to Conrey Mod(n,N)
9644 : * (resp. NULL if (n,N) > 1) */
9645 : static GEN
9646 644 : charsmodN(long N)
9647 : {
9648 644 : GEN D, G, prio, phio, dummy = cgetg(1,t_VEC);
9649 644 : GEN vP, vG = const_vec(N,NULL), vCHI = const_vec(N,NULL);
9650 644 : GEN bymod = const_vec(N,NULL);
9651 644 : long pn, i, l, vt = fetch_user_var("t");
9652 644 : D = mydivisorsu(N); l = lg(D);
9653 3920 : for (i = 1; i < l; i++)
9654 3276 : gel(bymod, D[i]) = vecsmalltrunc_init(myeulerphiu(D[i])+1);
9655 644 : gel(vG,N) = G = znstar0(utoipos(N),1);
9656 644 : pn = znstar_get_expo(G); /* exponent(Z/NZ)^* */
9657 644 : vP = const_vec(pn,NULL);
9658 26845 : for (i = 1; i <= N; i++)
9659 : {
9660 : GEN P, gF, G0, chi0, nchi0, chi, v, go;
9661 : long j, F, o;
9662 26201 : if (ugcd(i,N) != 1) continue;
9663 13937 : chi = znconreylog(G, utoipos(i));
9664 13937 : gF = znconreyconductor(G, chi, &chi0);
9665 13937 : F = (typ(gF) == t_INT)? itou(gF): itou(gel(gF,1));
9666 13937 : G0 = gel(vG, F); if (!G0) G0 = gel(vG,F) = znstar0(gF, 1);
9667 13937 : nchi0 = znconreylog_normalize(G0,chi0);
9668 13937 : go = gel(nchi0,1); o = itou(go); /* order(chi0) */
9669 13937 : v = ncharvecexpo(G0, nchi0);
9670 13937 : if (!equaliu(go, pn)) v = zv_z_mul(v, pn / o);
9671 13937 : P = gel(vP, o); if (!P) P = gel(vP,o) = polcyclo(o,vt);
9672 : /* mfcharcxinit with dummy complex powers */
9673 13937 : gel(vCHI,i) = mkvecn(6, G0, chi0, go, v, dummy, P);
9674 13937 : D = mydivisorsu(N / F); l = lg(D);
9675 40138 : for (j = 1; j < l; j++) vecsmalltrunc_append(gel(bymod, F*D[j]), i);
9676 : }
9677 644 : phio = zero_zv(pn); l = lg(vCHI); prio = cgetg(l, t_VEC);
9678 26845 : for (i = 1; i < l; i++)
9679 : {
9680 26201 : GEN CHI = gel(vCHI,i);
9681 : long o;
9682 26201 : if (!CHI) continue;
9683 13937 : o = CHIvec_ord(CHI);
9684 13937 : if (!phio[o]) phio[o] = myeulerphiu(o);
9685 13937 : prio[i] = phio[o];
9686 : }
9687 644 : l = lg(bymod);
9688 : /* sort characters by increasing value of phi(order) */
9689 26845 : for (i = 1; i < l; i++)
9690 : {
9691 26201 : GEN z = gel(bymod,i);
9692 26201 : if (z) gen_sort_inplace(z, (void*)prio, &cmp_small_priority, NULL);
9693 : }
9694 644 : return mkvec3(vCHI, bymod, vG);
9695 : }
9696 :
9697 : static GEN
9698 5166 : mfeisenstein2pure(long k, GEN CHI1, GEN CHI2, long ord, GEN P, long lim)
9699 : {
9700 5166 : GEN c, V = cgetg(lim+2, t_COL);
9701 : long n;
9702 5166 : c = mfeisenstein2_0(k, CHI1, CHI2, ord);
9703 5166 : if (P) c = grem(c, P);
9704 5166 : gel(V,1) = c;
9705 99120 : for (n=1; n <= lim; n++)
9706 : {
9707 93954 : c = sigchi2(k, CHI1, CHI2, n, ord);
9708 93954 : if (P) c = grem(c, P);
9709 93954 : gel(V,n+1) = c;
9710 : }
9711 5166 : return V;
9712 : }
9713 : static GEN
9714 4382 : mfeisenstein2pure_Fl(long k, GEN CHI1vec, GEN CHI2vec, GEN vz, ulong p, long lim)
9715 : {
9716 4382 : GEN V = cgetg(lim+2, t_VECSMALL);
9717 : long n;
9718 4382 : V[1] = mfeisenstein2_0_Fl(k, CHI1vec, CHI2vec, vz, p);
9719 59997 : for (n=1; n <= lim; n++) V[n+1] = sigchi2_Fl(k, CHI1vec, CHI2vec, n, vz, p);
9720 4382 : return V;
9721 : }
9722 :
9723 : static GEN
9724 252 : getcolswt2(GEN M, GEN D, ulong p)
9725 : {
9726 252 : GEN R, v = gel(M,1);
9727 252 : long i, l = lg(M) - 1;
9728 252 : R = cgetg(l, t_MAT); /* skip D[1] = 1 */
9729 1008 : for (i = 1; i < l; i++)
9730 : {
9731 756 : GEN w = Flv_Fl_mul(gel(M,i+1), D[i+1], p);
9732 756 : gel(R,i) = Flv_sub(v, w, p);
9733 : }
9734 252 : return R;
9735 : }
9736 : static GEN
9737 5425 : expandbd(GEN V, long d)
9738 : {
9739 : long L, n, nd;
9740 : GEN W;
9741 5425 : if (d == 1) return V;
9742 2086 : L = lg(V)-1; W = zerocol(L); /* nd = n/d */
9743 17871 : for (n = nd = 0; n < L; n += d, nd++) gel(W, n+1) = gel(V, nd+1);
9744 2086 : return W;
9745 : }
9746 : static GEN
9747 7021 : expandbd_Fl(GEN V, long d)
9748 : {
9749 : long L, n, nd;
9750 : GEN W;
9751 7021 : if (d == 1) return V;
9752 2639 : L = lg(V)-1; W = zero_Flv(L); /* nd = n/d */
9753 16366 : for (n = nd = 0; n < L; n += d, nd++) W[n+1] = V[nd+1];
9754 2639 : return W;
9755 : }
9756 : static void
9757 4382 : getcols_i(GEN *pM, GEN *pvj, GEN gk, GEN CHI1vec, GEN CHI2vec, long NN1, GEN vz,
9758 : ulong p, long lim)
9759 : {
9760 4382 : GEN CHI1 = CHIvec_CHI(CHI1vec), CHI2 = CHIvec_CHI(CHI2vec);
9761 4382 : long N2 = CHIvec_N(CHI2vec);
9762 4382 : GEN vj, M, D = mydivisorsu(NN1/N2);
9763 4382 : long i, l = lg(D), k = gk[2];
9764 4382 : GEN V = mfeisenstein2pure_Fl(k, CHI1vec, CHI2vec, vz, p, lim);
9765 4382 : M = cgetg(l, t_MAT);
9766 11403 : for (i = 1; i < l; i++) gel(M,i) = expandbd_Fl(V, D[i]);
9767 4382 : if (k == 2 && N2 == 1 && CHIvec_N(CHI1vec) == 1)
9768 : {
9769 252 : M = getcolswt2(M, D, p); l--;
9770 252 : D = vecslice(D, 2, l);
9771 : }
9772 4382 : *pM = M;
9773 4382 : *pvj = vj = cgetg(l, t_VEC);
9774 11151 : for (i = 1; i < l; i++) gel(vj,i) = mkvec4(gk, CHI1, CHI2, utoipos(D[i]));
9775 4382 : }
9776 :
9777 : /* find all CHI1, CHI2 mod N such that CHI1*CHI2 = CHI, f(CHI1)*f(CHI2) | N.
9778 : * set M = mfcoefs(B_e E(CHI1,CHI2), lim), vj = [e,i1,i2] */
9779 : static void
9780 1806 : getcols(GEN *pM, GEN *pv, long k, long nCHI, GEN allN, GEN vz, ulong p,
9781 : long lim)
9782 : {
9783 1806 : GEN vCHI = gel(allN,1), gk = utoi(k);
9784 1806 : GEN M = cgetg(1,t_MAT), v = cgetg(1,t_VEC);
9785 1806 : long i1, N = lg(vCHI)-1;
9786 86135 : for (i1 = 1; i1 <= N; i1++)
9787 : {
9788 84329 : GEN CHI1vec = gel(vCHI, i1), CHI2vec, M1, v1;
9789 : long NN1, i2;
9790 147329 : if (!CHI1vec) continue;
9791 66220 : if (k == 1 && CHIvec_parity(CHI1vec) == -1) continue;
9792 41461 : NN1 = N/CHIvec_N(CHI1vec); /* N/f(chi1) */;
9793 41461 : i2 = Fl_div(nCHI,i1, N);
9794 41461 : if (!i2) i2 = 1;
9795 41461 : CHI2vec = gel(vCHI,i2);
9796 41461 : if (NN1 % CHIvec_N(CHI2vec)) continue; /* f(chi1)f(chi2) | N ? */
9797 3220 : getcols_i(&M1, &v1, gk, CHI1vec, CHI2vec, NN1, vz, p, lim);
9798 3220 : M = shallowconcat(M, M1);
9799 3220 : v = shallowconcat(v, v1);
9800 : }
9801 1806 : *pM = M;
9802 1806 : *pv = v;
9803 1806 : }
9804 :
9805 : static void
9806 1190 : update_Mj(GEN *M, GEN *vecj, GEN *pz, ulong p)
9807 : {
9808 : GEN perm;
9809 1190 : *pz = Flm_indexrank(*M, p); perm = gel(*pz,2);
9810 1190 : *M = vecpermute(*M, perm);
9811 1190 : *vecj = vecpermute(*vecj, perm);
9812 1190 : }
9813 : static int
9814 427 : getcolsgen(long dim, GEN *pM, GEN *pvj, GEN *pz, long k, long ell, long nCHI,
9815 : GEN allN, GEN vz, ulong p, long lim)
9816 : {
9817 427 : GEN vCHI = gel(allN,1), bymod = gel(allN,2), gell = utoi(ell);
9818 427 : long i1, N = lg(vCHI)-1;
9819 427 : long L = lim+1;
9820 427 : if (lg(*pvj)-1 >= dim) update_Mj(pM, pvj, pz, p);
9821 427 : if (lg(*pvj)-1 == dim) return 1;
9822 1575 : for (i1 = 1; i1 <= N; i1++)
9823 : {
9824 1554 : GEN CHI1vec = gel(vCHI, i1), T;
9825 : long par1, j, l, N1, NN1;
9826 :
9827 1554 : if (!CHI1vec) continue;
9828 1533 : par1 = CHIvec_parity(CHI1vec);
9829 1533 : if (ell == 1 && par1 == -1) continue;
9830 952 : if (odd(ell)) par1 = -par1;
9831 952 : N1 = CHIvec_N(CHI1vec);
9832 952 : NN1 = N/N1;
9833 952 : T = gel(bymod, NN1); l = lg(T);
9834 3612 : for (j = 1; j < l; j++)
9835 : {
9836 3031 : long i2 = T[j], l1, l2, j1, s, nC;
9837 3031 : GEN M, M1, M2, vj, vj1, vj2, CHI2vec = gel(vCHI, i2);
9838 3031 : if (CHIvec_parity(CHI2vec) != par1) continue;
9839 1162 : nC = Fl_div(nCHI, Fl_mul(i1,i2,N), N);
9840 1162 : getcols(&M2, &vj2, k-ell, nC, allN, vz, p, lim);
9841 1162 : l2 = lg(M2); if (l2 == 1) continue;
9842 1162 : getcols_i(&M1, &vj1, gell, CHI1vec, CHI2vec, NN1, vz, p, lim);
9843 1162 : l1 = lg(M1);
9844 1162 : M1 = Flm_to_FlxV(M1, 0);
9845 1162 : M2 = Flm_to_FlxV(M2, 0);
9846 1162 : M = cgetg((l1-1)*(l2-1) + 1, t_MAT);
9847 1162 : vj = cgetg((l1-1)*(l2-1) + 1, t_VEC);
9848 2863 : for (j1 = s = 1; j1 < l1; j1++)
9849 : {
9850 1701 : GEN E = gel(M1,j1), v = gel(vj1,j1);
9851 : long j2;
9852 7112 : for (j2 = 1; j2 < l2; j2++, s++)
9853 : {
9854 5411 : GEN c = Flx_to_Flv(Flxn_mul(E, gel(M2,j2), L, p), L);
9855 5411 : gel(M,s) = c;
9856 5411 : gel(vj,s) = mkvec2(v, gel(vj2,j2));
9857 : }
9858 : }
9859 1162 : *pM = shallowconcat(*pM, M);
9860 1162 : *pvj = shallowconcat(*pvj, vj);
9861 1162 : if (lg(*pvj)-1 >= dim) update_Mj(pM, pvj, pz, p);
9862 1162 : if (lg(*pvj)-1 == dim) return 1;
9863 : }
9864 : }
9865 21 : if (ell == 1)
9866 : {
9867 21 : update_Mj(pM, pvj, pz, p);
9868 21 : return (lg(*pvj)-1 == dim);
9869 : }
9870 0 : return 0;
9871 : }
9872 :
9873 : static GEN
9874 1645 : mkF2bd(long d, long lim)
9875 : {
9876 1645 : GEN V = zerovec(lim + 1);
9877 : long n;
9878 1645 : gel(V, 1) = sstoQ(-1, 24);
9879 24248 : for (n = 1; n <= lim/d; n++) gel(V, n*d + 1) = mysumdivku(n, 1);
9880 1645 : return V;
9881 : }
9882 :
9883 : static GEN
9884 5782 : mkeisen(GEN E, long ord, GEN P, long lim)
9885 : {
9886 5782 : long k = itou(gel(E,1)), e = itou(gel(E,4));
9887 5782 : GEN CHI1 = gel(E,2), CHI2 = gel(E,3);
9888 5782 : if (k == 2 && mfcharistrivial(CHI1) && mfcharistrivial(CHI2))
9889 616 : return gsub(mkF2bd(1,lim), gmulgu(mkF2bd(e,lim), e));
9890 : else
9891 : {
9892 5166 : GEN V = mfeisenstein2pure(k, CHI1, CHI2, ord, P, lim);
9893 5166 : return expandbd(V, e);
9894 : }
9895 : }
9896 : static GEN
9897 602 : mkM(GEN vj, long pn, GEN P, long lim)
9898 : {
9899 602 : long j, l = lg(vj), L = lim+1;
9900 602 : GEN M = cgetg(l, t_MAT);
9901 4837 : for (j = 1; j < l; j++)
9902 : {
9903 : GEN E1, E2;
9904 4235 : parse_vecj(gel(vj,j), &E1,&E2);
9905 4235 : E1 = RgV_to_RgX(mkeisen(E1, pn, P, lim), 0);
9906 4235 : if (E2)
9907 : {
9908 1547 : E2 = RgV_to_RgX(mkeisen(E2, pn, P, lim), 0);
9909 1547 : E1 = RgXn_mul(E1, E2, L);
9910 : }
9911 4235 : E1 = RgX_to_RgC(E1, L);
9912 4235 : if (P && E2) E1 = RgXQV_red(E1, P);
9913 4235 : gel(M,j) = E1;
9914 : }
9915 602 : return M;
9916 : }
9917 :
9918 : /* assume N > 2 */
9919 : static GEN
9920 35 : mffindeisen1(long N)
9921 : {
9922 35 : GEN G = znstar0(utoipos(N), 1), L = chargalois(G, NULL), chi0 = NULL;
9923 35 : long j, m = N, l = lg(L);
9924 259 : for (j = 1; j < l; j++)
9925 : {
9926 245 : GEN chi = gel(L,j);
9927 245 : long r = myeulerphiu(itou(zncharorder(G,chi)));
9928 245 : if (r >= m) continue;
9929 182 : chi = znconreyfromchar(G, chi);
9930 182 : if (zncharisodd(G,chi)) { m = r; chi0 = chi; if (r == 1) break; }
9931 : }
9932 35 : if (!chi0) pari_err_BUG("mffindeisen1 [no Eisenstein series found]");
9933 35 : chi0 = znchartoprimitive(G,chi0);
9934 35 : return mfcharGL(gel(chi0,1), gel(chi0,2));
9935 : }
9936 :
9937 : static GEN
9938 644 : mfeisensteinspaceinit_i(long N, long k, GEN CHI)
9939 : {
9940 644 : GEN M, Minv, vj, vG, GN, allN, P, vz, z = NULL;
9941 644 : long nCHI, lim, ell, ord, dim = mffulldim(N, k, CHI);
9942 : ulong r, p;
9943 :
9944 644 : if (!dim) retmkvec3(cgetg(1,t_VECSMALL),
9945 : mkvec2(cgetg(1,t_MAT),gen_1),cgetg(1,t_VEC));
9946 644 : lim = mfsturmNk(N, k) + 1;
9947 644 : allN = charsmodN(N);
9948 644 : vG = gel(allN,3);
9949 644 : GN = gel(vG,N);
9950 644 : ord = znstar_get_expo(GN);
9951 644 : P = ord <= 2? NULL: polcyclo(ord, varn(mfcharpol(CHI)));
9952 644 : CHI = induce(GN, CHI); /* lift CHI mod N before mfcharno*/
9953 644 : nCHI = mfcharno(CHI);
9954 644 : r = QabM_init(ord, &p);
9955 644 : vz = Fl_powers(r, ord, p);
9956 644 : getcols(&M, &vj, k, nCHI, allN, vz, p, lim);
9957 665 : for (ell = k>>1; ell >= 1; ell--)
9958 427 : if (getcolsgen(dim, &M, &vj, &z, k, ell, nCHI, allN, vz, p, lim)) break;
9959 644 : if (!z) update_Mj(&M, &vj, &z, p);
9960 644 : if (lg(vj) - 1 < dim) return NULL;
9961 602 : M = mkM(vj, ord, P, lim);
9962 602 : Minv = QabM_Minv(rowpermute(M, gel(z,1)), P, ord);
9963 602 : return mkvec4(gel(z,1), Minv, vj, utoi(ord));
9964 : }
9965 : /* true mf */
9966 : static GEN
9967 602 : mfeisensteinspaceinit(GEN mf)
9968 : {
9969 602 : pari_sp av = avma;
9970 602 : GEN z, CHI = MF_get_CHI(mf);
9971 602 : long N = MF_get_N(mf), k = MF_get_k(mf);
9972 602 : if (!CHI) CHI = mfchartrivial();
9973 602 : z = mfeisensteinspaceinit_i(N, k, CHI);
9974 602 : if (!z)
9975 : {
9976 35 : GEN E, CHIN = mffindeisen1(N), CHI0 = mfchartrivial();
9977 35 : z = mfeisensteinspaceinit_i(N, k+1, mfcharmul(CHI, CHIN));
9978 35 : if (z) E = mkvec4(gen_1, CHI0, CHIN, gen_1);
9979 : else
9980 : {
9981 7 : z = mfeisensteinspaceinit_i(N, k+2, CHI);
9982 7 : E = mkvec4(gen_2, CHI0, CHI0, utoipos(N));
9983 : }
9984 35 : z = mkvec2(z, E);
9985 : }
9986 602 : return gc_GEN(av, z);
9987 : }
9988 :
9989 : /* decomposition of modular form on eisenspace */
9990 : static GEN
9991 1015 : mfeisensteindec(GEN mf, GEN F)
9992 : {
9993 1015 : pari_sp av = avma;
9994 : GEN M, Mindex, Mvecj, V, B, CHI;
9995 : long o, ord;
9996 :
9997 1015 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
9998 1015 : if (lg(Mvecj) < 5)
9999 : {
10000 56 : GEN E, e = gel(Mvecj,2), gkE = gel(e,1);
10001 56 : long dE = itou(gel(e,4));
10002 56 : Mvecj = gel(Mvecj,1);
10003 56 : E = mfeisenstein(itou(gkE), NULL, gel(e,3));
10004 56 : if (dE != 1) E = mfbd_E2(E, dE, gel(e,2)); /* here k = 2 */
10005 56 : F = mfmul(F, E);
10006 : }
10007 1015 : M = gel(Mvecj, 2);
10008 1015 : if (lg(M) == 1) return cgetg(1, t_VEC);
10009 1015 : Mindex = gel(Mvecj, 1);
10010 1015 : ord = itou(gel(Mvecj,4));
10011 1015 : V = mfcoefs(F, Mindex[lg(Mindex)-1]-1, 1); settyp(V, t_COL);
10012 1015 : CHI = mf_get_CHI(F);
10013 1015 : o = mfcharorder(CHI);
10014 1015 : if (o > 2 && o != ord)
10015 : { /* convert Mod(.,polcyclo(o)) to Mod(., polcyclo(N)) for o | N,
10016 : * o and N both != 2 (mod 4) */
10017 84 : GEN z, P = gel(M,4); /* polcyclo(ord) */
10018 84 : long vt = varn(P);
10019 84 : z = gmodulo(pol_xn(ord/o, vt), P);
10020 84 : if (ord % o) pari_err_TYPE("mfeisensteindec", V);
10021 84 : V = gsubst(liftpol_shallow(V), vt, z);
10022 : }
10023 1015 : B = Minv_RgC_mul(M, vecpermute(V, Mindex));
10024 1015 : return gc_upto(av, B);
10025 : }
10026 :
10027 : /*********************************************************************/
10028 : /* END EISENSPACE */
10029 : /*********************************************************************/
10030 :
10031 : static GEN
10032 70 : sertocol2(GEN S, long l)
10033 : {
10034 70 : GEN C = cgetg(l + 2, t_COL);
10035 : long i;
10036 420 : for (i = 0; i <= l; i++) gel(C, i+1) = polcoef_i(S, i, -1);
10037 70 : return C;
10038 : }
10039 :
10040 : /* Compute polynomial P0 such that F=E4^(k/4)P0(E6/E4^(3/2)). */
10041 : static GEN
10042 14 : mfcanfindp0(GEN F, long k)
10043 : {
10044 14 : pari_sp ltop = avma;
10045 : GEN E4, E6, V, V1, Q, W, res, M, B;
10046 : long l, j;
10047 14 : l = k/6 + 2;
10048 14 : V = mfcoefsser(F,l);
10049 14 : E4 = mfcoefsser(mfEk(4),l);
10050 14 : E6 = mfcoefsser(mfEk(6),l);
10051 14 : V1 = gdiv(V, gpow(E4, uutoQ(k,4), 0));
10052 14 : Q = gdiv(E6, gpow(E4, uutoQ(3,2), 0));
10053 14 : W = gpowers(Q, l - 1);
10054 14 : M = cgetg(l + 1, t_MAT);
10055 70 : for (j = 1; j <= l; j++) gel(M,j) = sertocol2(gel(W,j), l);
10056 14 : B = sertocol2(V1, l);
10057 14 : res = inverseimage(M, B);
10058 14 : if (lg(res) == 1) err_space(F);
10059 14 : return gc_GEN(ltop, gtopolyrev(res, 0));
10060 : }
10061 :
10062 : /* Compute the first n+1 Taylor coeffs at tau=I of a modular form
10063 : * on SL_2(Z). */
10064 : GEN
10065 14 : mftaylor(GEN F, long n, long flreal, long prec)
10066 : {
10067 14 : pari_sp ltop = avma;
10068 14 : GEN P0, Pm1 = gen_0, v;
10069 14 : GEN X2 = mkpoln(3, ghalf,gen_0,gneg(ghalf)); /* (x^2-1) / 2 */
10070 : long k, m;
10071 14 : if (!checkmf_i(F)) pari_err_TYPE("mftaylor",F);
10072 14 : k = mf_get_k(F);
10073 14 : if (mf_get_N(F) != 1 || k < 0) pari_err_IMPL("mftaylor for this form");
10074 14 : P0 = mfcanfindp0(F, k);
10075 14 : v = cgetg(n+2, t_VEC); gel(v, 1) = RgX_coeff(P0,0);
10076 154 : for (m = 0; m < n; m++)
10077 : {
10078 140 : GEN P1 = gdivgu(gmulsg(-(k + 2*m), RgX_shift(P0,1)), 12);
10079 140 : P1 = gadd(P1, gmul(X2, RgX_deriv(P0)));
10080 140 : if (m) P1 = gsub(P1, gdivgu(gmulsg(m*(m+k-1), Pm1), 144));
10081 140 : Pm1 = P0; P0 = P1;
10082 140 : gel(v, m+2) = RgX_coeff(P0, 0);
10083 : }
10084 14 : if (flreal)
10085 : {
10086 7 : GEN pi2 = Pi2n(1, prec), pim4 = gmulsg(-2, pi2), VPC;
10087 7 : GEN C = gmulsg(3, gdiv(gpowgs(ggamma(uutoQ(1,4), prec), 8), gpowgs(pi2, 6)));
10088 : /* E_4(i): */
10089 7 : GEN facn = gen_1;
10090 7 : VPC = gpowers(gmul(pim4, gsqrt(C, prec)), n);
10091 7 : C = gpow(C, uutoQ(k,4), prec);
10092 84 : for (m = 0; m <= n; m++)
10093 : {
10094 77 : gel(v, m+1) = gdiv(gmul(C, gmul(gel(v, m+1), gel(VPC, m+1))), facn);
10095 77 : facn = gmulgu(facn, m+1);
10096 : }
10097 : }
10098 14 : return gc_GEN(ltop, v);
10099 : }
10100 :
10101 : #if 0
10102 : /* To be used in mfeigensearch() */
10103 : GEN
10104 : mfreadratfile()
10105 : {
10106 : GEN eqn;
10107 : pariFILE *F = pari_fopengz("rateigen300.gp");
10108 : eqn = gp_readvec_stream(F->file);
10109 : pari_fclose(F);
10110 : return eqn;
10111 : }
10112 : #endif
10113 : /*****************************************************************/
10114 : /* EISENSTEIN CUSPS: COMPLEX DIRECTLY: one F_k */
10115 : /*****************************************************************/
10116 :
10117 : /* CHIvec = charinit(CHI); data = [N1g/g1,N2g/g2,g1/g,g2/g,C/g1,C/g2,
10118 : * (N1g/g1)^{-1},(N2g/g2)^{-1}] */
10119 :
10120 : /* nm = n/m;
10121 : * z1 = powers of \z_{C/g}^{(Ae/g)^{-1}},
10122 : * z2 = powers of \z_N^{A^{-1}(g1g2/C)}]
10123 : * N.B. : we compute value and conjugate at the end, so it is (Ae/g)^{-1}
10124 : * and not -(Ae/g)^{-1} */
10125 : static GEN
10126 7830900 : eiscnm(long nm, long m, GEN CHI1vec, GEN CHI2vec, GEN data, GEN z1)
10127 : {
10128 7830900 : long Cg1 = data[5], s10 = (nm*data[7]) % Cg1, r10 = (nm - data[1]*s10) / Cg1;
10129 7830900 : long Cg2 = data[6], s20 = (m *data[8]) % Cg2, r20 = (m - data[2]*s20) / Cg2;
10130 : long j1, r1, s1;
10131 7830900 : GEN T = gen_0;
10132 18857104 : for (j1 = 0, r1 = r10, s1 = s10; j1 < data[3]; j1++, r1 -= data[1], s1 += Cg1)
10133 : {
10134 11026204 : GEN c1 = mychareval(CHI1vec, r1);
10135 11026204 : if (!gequal0(c1))
10136 : {
10137 : long j2, r2, s2;
10138 8127560 : GEN S = gen_0;
10139 21136570 : for (j2 = 0, r2 = r20, s2 = s20; j2 < data[4]; j2++, r2 -= data[2], s2 += Cg2)
10140 : {
10141 13009010 : GEN c2 = mychareval(CHI2vec, r2);
10142 13009010 : if (!gequal0(c2)) S = gadd(S, gmul(c2, rootsof1pow(z1, s1*s2)));
10143 : }
10144 8127560 : T = gadd(T, gmul(c1, S));
10145 : }
10146 : }
10147 7830900 : return conj_i(T);
10148 : }
10149 :
10150 : static GEN
10151 612675 : fg1g2n(long n, long k, GEN CHI1vec, GEN CHI2vec, GEN data, GEN z1, GEN z2)
10152 : {
10153 612675 : pari_sp av = avma;
10154 612675 : GEN S = gen_0, D = mydivisorsu(n);
10155 612675 : long i, l = lg(D);
10156 4528125 : for (i = 1; i < l; i++)
10157 : {
10158 3915450 : long m = D[i], nm = D[l-i]; /* n/m */
10159 3915450 : GEN u = eiscnm( nm, m, CHI1vec, CHI2vec, data, z1);
10160 3915450 : GEN v = eiscnm(-nm, -m, CHI1vec, CHI2vec, data, z1);
10161 3915450 : GEN w = odd(k) ? gsub(u, v) : gadd(u, v);
10162 3915450 : S = gadd(S, gmul(powuu(m, k-1), w));
10163 : }
10164 612675 : return gc_upto(av, gmul(S, rootsof1pow(z2, n)));
10165 : }
10166 :
10167 : static GEN
10168 14266 : gausssumcx(GEN CHIvec, long prec)
10169 : {
10170 : GEN z, S, V;
10171 14266 : long m, N = CHIvec_N(CHIvec);
10172 14266 : if (N == 1) return gen_1;
10173 7693 : V = CHIvec_val(CHIvec);
10174 7693 : z = rootsof1u_cx(N, prec);
10175 7693 : S = gmul(z, gel(V, N));
10176 103054 : for (m = N-1; m >= 1; m--) S = gmul(z, gadd(gel(V, m), S));
10177 7693 : return S;
10178 : }
10179 :
10180 : /* Computation of Q_k(\z_N^s) as a polynomial in \z_N^s. FIXME: explicit
10181 : * formula ? */
10182 : static GEN
10183 2338 : mfqk(long k, long N)
10184 : {
10185 : GEN X, P, ZI, Q, Xm1, invden;
10186 : long i;
10187 2338 : ZI = gdivgu(RgX_shift_shallow(RgV_to_RgX(identity_ZV(N-1), 0), 1), N);
10188 2338 : if (k == 1) return ZI;
10189 1176 : P = gsubgs(pol_xn(N,0), 1);
10190 1176 : invden = RgXQ_powu(ZI, k, P);
10191 1176 : X = pol_x(0); Q = gneg(X); Xm1 = gsubgs(X, 1);
10192 2884 : for (i = 2; i < k; i++)
10193 1708 : Q = RgX_shift_shallow(ZX_add(gmul(Xm1, ZX_deriv(Q)), gmulsg(-i, Q)), 1);
10194 1176 : return RgXQ_mul(Q, invden, P);
10195 : }
10196 :
10197 : /* CHI mfchar; M is a multiple of the conductor of CHI, but is NOT
10198 : * necessarily its modulus */
10199 : static GEN
10200 3346 : mfskcx(long k, GEN CHI, long M, long prec)
10201 : {
10202 : GEN S, CHIvec, P;
10203 : long F, m, i, l;
10204 3346 : CHI = mfchartoprimitive(CHI, &F);
10205 3346 : CHIvec = mfcharcxinit(CHI, prec);
10206 3346 : if (F == 1) S = gdivgu(bernfrac(k), k);
10207 : else
10208 : {
10209 2338 : GEN Q = mfqk(k, F), V = CHIvec_val(CHIvec);
10210 2338 : S = gmul(gel(V, F), RgX_coeff(Q, 0));
10211 39431 : for (m = 1; m < F; m++) S = gadd(S, gmul(gel(V, m), RgX_coeff(Q, m)));
10212 2338 : S = conj_i(S);
10213 : }
10214 : /* prime divisors of M not dividing f(chi) */
10215 3346 : P = gel(myfactoru(u_ppo(M/F,F)), 1); l = lg(P);
10216 3500 : for (i = 1; i < l; i++)
10217 : {
10218 154 : long p = P[i];
10219 154 : S = gmul(S, gsubsg(1, gdiv(mychareval(CHIvec, p), powuu(p, k))));
10220 : }
10221 3346 : return gmul(gmul(gausssumcx(CHIvec, prec), S), powuu(M/F, k));
10222 : }
10223 :
10224 : static GEN
10225 6146 : f00_i(long k, GEN CHI1vec, GEN CHI2vec, GEN G2, GEN S, long prec)
10226 : {
10227 : GEN c, a;
10228 6146 : long N1 = CHIvec_N(CHI1vec), N2 = CHIvec_N(CHI2vec);
10229 6146 : if (S[2] != N1) return gen_0;
10230 3346 : c = mychareval(CHI1vec, S[3]);
10231 3346 : if (isintzero(c)) return gen_0;
10232 3346 : a = mfskcx(k, mfchardiv(CHIvec_CHI(CHI2vec), CHIvec_CHI(CHI1vec)), N1*N2, prec);
10233 3346 : a = gmul(a, conj_i(gmul(c,G2)));
10234 3346 : return gdiv(a, mulsi(-N2, powuu(S[1], k-1)));
10235 : }
10236 :
10237 : static GEN
10238 4865 : f00(long k, GEN CHI1vec,GEN CHI2vec, GEN G1,GEN G2, GEN data, long prec)
10239 : {
10240 : GEN T1, T2;
10241 4865 : T2 = f00_i(k, CHI1vec, CHI2vec, G2, data, prec);
10242 4865 : if (k > 1) return T2;
10243 1281 : T1 = f00_i(k, CHI2vec, CHI1vec, G1, data, prec);
10244 1281 : return gadd(T1, T2);
10245 : }
10246 :
10247 : /* ga in SL_2(Z), find beta [a,b;c,d] in Gamma_0(N) and mu in Z such that
10248 : * beta * ga * T^u = [A',B';C',D'] with C' | N and N | B', C' > 0 */
10249 : static void
10250 5460 : mfgatogap(GEN ga, long N, long *pA, long *pC, long *pD, long *pd, long *pmu)
10251 : {
10252 5460 : GEN A = gcoeff(ga,1,1), B = gcoeff(ga,1,2);
10253 5460 : GEN C = gcoeff(ga,2,1), D = gcoeff(ga,2,2), a, b, c, d;
10254 : long t, Ap, Cp, B1, D1, mu;
10255 5460 : Cp = itou(bezout(muliu(A,N), C, &c, &d)); /* divides N */
10256 5460 : t = 0;
10257 5460 : if (Cp > 1)
10258 : { /* (d, N/Cp) = 1, find t such that (d - t*(A*N/Cp), N) = 1 */
10259 2604 : long dN = umodiu(d,Cp), Q = (N/Cp * umodiu(A,Cp)) % Cp;
10260 2989 : while (ugcd(dN, Cp) > 1) { t++; dN = Fl_sub(dN, Q, Cp); }
10261 : }
10262 5460 : if (t)
10263 : {
10264 385 : c = addii(c, mului(t, diviuexact(C,Cp)));
10265 385 : d = subii(d, mului(t, muliu(A, N/Cp))); /* (d,N) = 1 */
10266 : }
10267 5460 : D1 = umodiu(mulii(d,D), N);
10268 5460 : (void)bezout(d, mulis(c,-N), &a, &b); /* = 1 */
10269 5460 : t = 0; Ap = umodiu(addii(mulii(a,A), mulii(b,C)), N); /* (Ap,Cp) = 1 */
10270 7483 : while (ugcd(Ap, N) > 1) { t++; Ap = Fl_add(Ap, Cp, N); }
10271 5460 : B1 = umodiu(a,N)*umodiu(B,N) + umodiu(b,N)*umodiu(D,N) + t*D1;
10272 5460 : B1 %= N;
10273 5460 : *pmu = mu = Fl_neg(Fl_div(B1, Ap, N), N);
10274 : /* A', D' and d only needed modulo N */
10275 5460 : *pd = umodiu(d, N);
10276 5460 : *pA = Ap;
10277 5460 : *pC = Cp; *pD = (D1 + Cp*mu) % N;
10278 5460 : }
10279 :
10280 : #if 0
10281 : /* CHI is a mfchar, return alpha(CHI) */
10282 : static long
10283 : mfalchi(GEN CHI, long AN, long cg)
10284 : {
10285 : GEN G = gel(CHI,1), chi = gel(CHI,2), go = gmfcharorder(CHI);
10286 : long o = itou(go), a = itos( znchareval(G, chi, stoi(1 + AN/cg), go) );
10287 : if (a < 0 || (cg * a) % o) pari_err_BUG("mfalchi");
10288 : return (cg * a) / o;
10289 : }
10290 : #endif
10291 : /* return A such that CHI1(t) * CHI2(t) = e(A) or NULL if (t,N1*N2) > 1 */
10292 : static GEN
10293 10920 : mfcharmuleval(GEN CHI1vec, GEN CHI2vec, long t)
10294 : {
10295 10920 : long a1 = mycharexpo(CHI1vec, t), o1 = CHIvec_ord(CHI1vec);
10296 10920 : long a2 = mycharexpo(CHI2vec, t), o2 = CHIvec_ord(CHI2vec);;
10297 10920 : if (a1 < 0 || a2 < 0) return NULL;
10298 10920 : return sstoQ(a1*o2 + a2*o1, o1*o2);
10299 : }
10300 : static GEN
10301 5460 : mfcharmulcxeval(GEN CHI1vec, GEN CHI2vec, long t, long prec)
10302 : {
10303 5460 : GEN A = mfcharmuleval(CHI1vec, CHI2vec, t);
10304 : long n, d;
10305 5460 : if (!A) return gen_0;
10306 5460 : Qtoss(A, &n,&d); return rootsof1q_cx(n, d, prec);
10307 : }
10308 : /* alpha(CHI1 * CHI2) */
10309 : static long
10310 5460 : mfalchi2(GEN CHI1vec, GEN CHI2vec, long AN, long cg)
10311 : {
10312 5460 : GEN A = mfcharmuleval(CHI1vec, CHI2vec, 1 + AN/cg);
10313 : long a;
10314 5460 : if (!A) pari_err_BUG("mfalchi2");
10315 5460 : A = gmulsg(cg, A);
10316 5460 : if (typ(A) != t_INT) pari_err_BUG("mfalchi2");
10317 5460 : a = itos(A) % cg; if (a < 0) a += cg;
10318 5460 : return a;
10319 : }
10320 :
10321 : /* return g = (a,b), set u >= 0 s.t. g = a * u (mod b) */
10322 : static long
10323 21840 : mybezout(long a, long b, long *pu)
10324 : {
10325 21840 : long junk, g = cbezout(a, b, pu, &junk);
10326 21840 : if (*pu < 0) *pu += b/g;
10327 21840 : return g;
10328 : }
10329 :
10330 : /* E = [k, CHI1,CHI2, e], CHI1 and CHI2 primitive mfchars such that,
10331 : * CHI1(-1)*CHI2(-1) = (-1)^k; expansion of (B_e (E_k(CHI1,CHI2))) | ga.
10332 : * w is the width for the space of the calling function. */
10333 : static GEN
10334 5460 : mfeisensteingacx(GEN E, long w, GEN ga, long lim, long prec)
10335 : {
10336 5460 : GEN CHI1vec, CHI2vec, CHI1 = gel(E,2), CHI2 = gel(E,3), v, S, ALPHA;
10337 : GEN G1, G2, z1, z2, data;
10338 5460 : long k = itou(gel(E,1)), e = itou(gel(E,4));
10339 5460 : long N1 = mfcharmodulus(CHI1);
10340 5460 : long N2 = mfcharmodulus(CHI2), N = e * N1 * N2;
10341 : long NsurC, cg, wN, A, C, Ai, d, mu, alchi, na, da;
10342 : long eg, g, gH, U, u0, u1, u2, Aig, H, m, n, t, Cg, NC1, NC2;
10343 :
10344 5460 : mfgatogap(ga, N, &A, &C, &Ai, &d, &mu);
10345 5460 : CHI1vec = mfcharcxinit(CHI1, prec);
10346 5460 : CHI2vec = mfcharcxinit(CHI2, prec);
10347 5460 : NsurC = N/C; cg = ugcd(C, NsurC); wN = NsurC / cg;
10348 5460 : if (w%wN) pari_err_BUG("mfeisensteingacx [wN does not divide w]");
10349 5460 : alchi = mfalchi2(CHI1vec, CHI2vec, A*N, cg);
10350 5460 : ALPHA = sstoQ(alchi, NsurC);
10351 :
10352 5460 : g = mybezout(A*e, C, &u0); Cg = C/g; eg = e/g;
10353 5460 : NC1 = mybezout(N1, Cg, &u1);
10354 5460 : NC2 = mybezout(N2, Cg, &u2);
10355 5460 : H = (NC1*NC2*g)/Cg;
10356 5460 : Aig = (Ai*H)%N; if (Aig < 0) Aig += N;
10357 5460 : z1 = rootsof1powinit(u0, Cg, prec);
10358 5460 : z2 = rootsof1powinit(Aig, N, prec);
10359 5460 : data = mkvecsmalln(8, N1/NC1, N2/NC2, NC1, NC2, Cg/NC1, Cg/NC2, u1, u2);
10360 5460 : v = zerovec(lim + 1);
10361 : /* need n*H = alchi (mod cg) */
10362 5460 : gH = mybezout(H, cg, &U);
10363 5460 : if (gH > 1)
10364 : {
10365 511 : if (alchi % gH) return mkvec2(gen_0, v);
10366 511 : alchi /= gH; cg /= gH; H /= gH;
10367 : }
10368 5460 : G1 = gausssumcx(CHI1vec, prec);
10369 5460 : G2 = gausssumcx(CHI2vec, prec);
10370 5460 : if (!alchi)
10371 4865 : gel(v,1) = f00(k, CHI1vec,CHI2vec,G1,G2, mkvecsmall3(NC2,Cg,A*eg), prec);
10372 5460 : n = Fl_mul(alchi,U,cg); if (!n) n = cg;
10373 5460 : m = (n*H - alchi) / cg; /* positive, exact division */
10374 618135 : for (; m <= lim; n+=cg, m+=H)
10375 612675 : gel(v, m+1) = fg1g2n(n, k, CHI1vec, CHI2vec, data, z1,z2);
10376 5460 : t = (2*e)/g; if (odd(k)) t = -t;
10377 5460 : v = gdiv(v, gmul(conj_i(gmul(G1,G2)), mulsi(t, powuu(eg*N2/NC2, k-1))));
10378 5460 : if (k == 2 && N1 == 1 && N2 == 1) v = gsub(mkF2bd(wN,lim), gmulsg(e,v));
10379 :
10380 5460 : Qtoss(ALPHA, &na,&da);
10381 5460 : S = conj_i( mfcharmulcxeval(CHI1vec,CHI2vec,d,prec) ); /* CHI(1/d) */
10382 5460 : if (wN > 1)
10383 : {
10384 4151 : GEN z = rootsof1powinit(-mu, wN, prec);
10385 4151 : long i, l = lg(v);
10386 578837 : for (i = 1; i < l; i++) gel(v,i) = gmul(gel(v,i), rootsof1pow(z,i-1));
10387 : }
10388 5460 : v = RgV_Rg_mul(v, gmul(S, rootsof1q_cx(-mu*na, da, prec)));
10389 5460 : return mkvec2(ALPHA, bdexpand(v, w/wN));
10390 : }
10391 :
10392 : /*****************************************************************/
10393 : /* END EISENSTEIN CUSPS */
10394 : /*****************************************************************/
10395 :
10396 : static GEN
10397 1596 : mfchisimpl(GEN CHI)
10398 : {
10399 : GEN G, chi;
10400 1596 : if (typ(CHI) == t_INT) return CHI;
10401 1596 : G = gel(CHI, 1); chi = gel(CHI, 2);
10402 1596 : switch(mfcharorder(CHI))
10403 : {
10404 1148 : case 1: chi = gen_1; break;
10405 427 : case 2: chi = znchartokronecker(G,chi,1); break;
10406 21 : default:chi = mkintmod(znconreyexp(G,chi), znstar_get_N(G)); break;
10407 : }
10408 1596 : return chi;
10409 : }
10410 :
10411 : GEN
10412 700 : mfparams(GEN F)
10413 : {
10414 700 : pari_sp av = avma;
10415 : GEN z, mf, CHI;
10416 700 : if ((mf = checkMF_i(F)))
10417 : {
10418 14 : long N = MF_get_N(mf);
10419 14 : GEN gk = MF_get_gk(mf);
10420 14 : CHI = MF_get_CHI(mf);
10421 14 : z = mkvec5(utoi(N), gk, CHI, utoi(MF_get_space(mf)), mfcharpol(CHI));
10422 : }
10423 : else
10424 : {
10425 686 : if (!checkmf_i(F)) pari_err_TYPE("mfparams", F);
10426 686 : z = vec_append(mf_get_NK(F), mfcharpol(mf_get_CHI(F)));
10427 : }
10428 700 : gel(z,3) = mfchisimpl(gel(z,3));
10429 700 : return gc_GEN(av, z);
10430 : }
10431 :
10432 : GEN
10433 14 : mfisCM(GEN F)
10434 : {
10435 14 : pari_sp av = avma;
10436 : forprime_t S;
10437 : GEN D, v;
10438 : long N, k, lD, sb, p, i;
10439 14 : if (!checkmf_i(F)) pari_err_TYPE("mfisCM", F);
10440 14 : N = mf_get_N(F);
10441 14 : k = mf_get_k(F); if (N < 0 || k < 0) pari_err_IMPL("mfisCM for this F");
10442 14 : D = mfunram(N, -1);
10443 14 : lD = lg(D);
10444 14 : sb = maxss(mfsturmNk(N, k), 4*N);
10445 14 : v = mfcoefs_i(F, sb, 1);
10446 14 : u_forprime_init(&S, 2, sb);
10447 504 : while ((p = u_forprime_next(&S)))
10448 : {
10449 490 : GEN ap = gel(v, p+1);
10450 490 : if (!gequal0(ap))
10451 406 : for (i = 1; i < lD; i++)
10452 245 : if (kross(D[i], p) == -1) { D = vecsplice(D, i); lD--; }
10453 : }
10454 14 : if (lD == 1) return gc_const(av, gen_0);
10455 14 : if (lD == 2) return gc_stoi(av, D[1]);
10456 7 : if (k > 1) pari_err_BUG("mfisCM");
10457 7 : return gc_upto(av, zv_to_ZV(D));
10458 : }
10459 :
10460 : static long
10461 287 : mfspace_i(GEN mf, GEN F)
10462 : {
10463 : GEN v, vF, gk;
10464 : long n, nE, i, l, s, N;
10465 :
10466 287 : mf = checkMF(mf); s = MF_get_space(mf);
10467 287 : if (!F) return s;
10468 287 : if (!checkmf_i(F)) pari_err_TYPE("mfspace",F);
10469 287 : v = mftobasis(mf, F, 1);
10470 287 : n = lg(v)-1; if (!n) return -1;
10471 224 : nE = lg(MF_get_E(mf))-1;
10472 224 : switch(s)
10473 : {
10474 56 : case mf_NEW: case mf_OLD: case mf_EISEN: return s;
10475 140 : case mf_FULL:
10476 140 : if (mf_get_type(F) == t_MF_THETA) return mf_EISEN;
10477 133 : if (!gequal0(vecslice(v,1,nE)))
10478 63 : return gequal0(vecslice(v,nE+1,n))? mf_EISEN: mf_FULL;
10479 : }
10480 : /* mf is mf_CUSP or mf_FULL, F a cusp form */
10481 98 : gk = mf_get_gk(F);
10482 98 : if (typ(gk) == t_FRAC || equali1(gk)) return mf_CUSP;
10483 84 : vF = mftonew_i(mf, vecslice(v, nE+1, n), &N);
10484 84 : if (N != MF_get_N(mf)) return mf_OLD;
10485 56 : l = lg(vF);
10486 91 : for (i = 1; i < l; i++)
10487 56 : if (itos(gmael(vF,i,1)) != N) return mf_CUSP;
10488 35 : return mf_NEW;
10489 : }
10490 : long
10491 287 : mfspace(GEN mf, GEN F)
10492 287 : { pari_sp av = avma; return gc_long(av, mfspace_i(mf,F)); }
10493 : static GEN
10494 21 : lfunfindchi(GEN ldata, GEN van, long prec)
10495 : {
10496 21 : GEN gN = ldata_get_conductor(ldata), gk = ldata_get_k(ldata);
10497 21 : GEN G = znstar0(gN,1), cyc = znstar_get_conreycyc(G), L, go, vz;
10498 21 : long N = itou(gN), odd = typ(gk) == t_INT && mpodd(gk);
10499 21 : long i, j, o, l, B0 = 2, B = lg(van)-1, bit = 10 - prec2nbits(prec);
10500 :
10501 : /* if van is integral, chi must be trivial */
10502 21 : if (typ(van) == t_VECSMALL) return mfcharGL(G, zerocol(lg(cyc)-1));
10503 14 : L = cyc2elts(cyc); l = lg(L);
10504 42 : for (i = j = 1; i < l; i++)
10505 : {
10506 28 : GEN chi = zc_to_ZC(gel(L,i));
10507 28 : if (zncharisodd(G,chi) == odd) gel(L,j++) = mfcharGL(G,chi);
10508 : }
10509 14 : setlg(L,j); l = j;
10510 14 : if (l <= 2) return gel(L,1);
10511 0 : o = znstar_get_expo(G); go = utoi(o);
10512 0 : vz = grootsof1(o, prec);
10513 : for (;;)
10514 0 : {
10515 : long n;
10516 0 : for (n = B0; n <= B; n++)
10517 : {
10518 : GEN an, r;
10519 : long j;
10520 0 : if (ugcd(n, N) != 1) continue;
10521 0 : an = gel(van,n); if (gexpo(an) < bit) continue;
10522 0 : r = gdiv(an, conj_i(an));
10523 0 : for (i = 1; i < l; i++)
10524 : {
10525 0 : GEN CHI = gel(L,i);
10526 0 : if (gexpo(gsub(r, gel(vz, znchareval_i(CHI,n,go)+1))) > bit)
10527 0 : gel(L,i) = NULL;
10528 : }
10529 0 : for (i = j = 1; i < l; i++)
10530 0 : if (gel(L,i)) gel(L,j++) = gel(L,i);
10531 0 : l = j; setlg(L,l);
10532 0 : if (l == 2) return gel(L,1);
10533 : }
10534 0 : B0 = B+1; B <<= 1;
10535 0 : van = ldata_vecan(ldata_get_an(ldata), B, prec);
10536 : }
10537 : }
10538 :
10539 : GEN
10540 21 : mffromlfun(GEN L, long prec)
10541 : {
10542 21 : pari_sp av = avma;
10543 21 : GEN ldata = lfunmisc_to_ldata_shallow(L), Vga = ldata_get_gammavec(ldata);
10544 21 : GEN van, a0, CHI, NK, gk = ldata_get_k(ldata);
10545 : long N, space;
10546 21 : if (!gequal(Vga, mkvec2(gen_0, gen_1))) pari_err_TYPE("mffromlfun", L);
10547 21 : N = itou(ldata_get_conductor(ldata));
10548 21 : van = ldata_vecan(ldata_get_an(ldata), mfsturmNgk(N,gk) + 2, prec);
10549 21 : CHI = lfunfindchi(ldata, van, prec);
10550 21 : if (typ(van) != t_VEC) van = vecsmall_to_vec_inplace(van);
10551 21 : space = (lg(ldata) == 7)? mf_CUSP: mf_FULL;
10552 21 : a0 = (space == mf_CUSP)? gen_0: gneg(lfun(L, gen_0, prec2nbits(prec)));
10553 21 : NK = mkvec3(utoi(N), gk, mfchisimpl(CHI));
10554 21 : return gc_GEN(av, mkvec3(NK, utoi(space), shallowconcat(a0, van)));
10555 : }
10556 : /*******************************************************************/
10557 : /* */
10558 : /* HALF-INTEGRAL WEIGHT */
10559 : /* */
10560 : /*******************************************************************/
10561 : /* We use the prefix mf2; k represents the weight -1/2, so e.g.
10562 : k = 2 is weight 5/2. N is the level, so 4\mid N, and CHI is the
10563 : character, always even. */
10564 :
10565 : static long
10566 3360 : lamCO(long r, long s, long p)
10567 : {
10568 3360 : if ((s << 1) <= r)
10569 : {
10570 1232 : long rp = r >> 1;
10571 1232 : if (odd(r)) return upowuu(p, rp) << 1;
10572 336 : else return (p + 1)*upowuu(p, rp - 1);
10573 : }
10574 2128 : else return upowuu(p, r - s) << 1;
10575 : }
10576 :
10577 : static int
10578 1568 : condC(GEN faN, GEN valF)
10579 : {
10580 1568 : GEN P = gel(faN, 1), E = gel(faN, 2);
10581 1568 : long l = lg(P), i;
10582 3696 : for (i = 1; i < l; i++)
10583 3024 : if ((P[i] & 3L) == 3)
10584 : {
10585 1120 : long r = E[i];
10586 1120 : if (odd(r) || r < (valF[i] << 1)) return 1;
10587 : }
10588 672 : return 0;
10589 : }
10590 :
10591 : /* returns 2*zetaCO; weight is k + 1/2 */
10592 : static long
10593 3696 : zeta2CO(GEN faN, GEN valF, long r2, long s2, long k)
10594 : {
10595 3696 : if (r2 >= 4) return lamCO(r2, s2, 2) << 1;
10596 2912 : if (r2 == 3) return 6;
10597 1568 : if (condC(faN, valF)) return 4;
10598 672 : if (odd(k)) return s2 ? 3 : 5; else return s2 ? 5: 3;
10599 : }
10600 :
10601 : /* returns 4 times last term in formula */
10602 : static long
10603 3696 : dim22(long N, long F, long k)
10604 : {
10605 3696 : pari_sp av = avma;
10606 3696 : GEN vF, faN = myfactoru(N), P = gel(faN, 1), E = gel(faN, 2);
10607 3696 : long i, D, l = lg(P);
10608 3696 : vF = cgetg(l, t_VECSMALL);
10609 9968 : for (i = 1; i < l; i++) vF[i] = u_lval(F, P[i]);
10610 3696 : D = zeta2CO(faN, vF, E[1], vF[1], k);
10611 6272 : for (i = 2; i < l; i++) D *= lamCO(E[i], vF[i], P[i]);
10612 3696 : return gc_long(av,D);
10613 : }
10614 :
10615 : /* PSI not necessarily primitive, of conductor F */
10616 : static int
10617 13846 : charistotallyeven(GEN PSI, long F)
10618 : {
10619 13846 : pari_sp av = avma;
10620 13846 : GEN P = gel(myfactoru(F), 1);
10621 13846 : GEN G = gel(PSI,1), psi = gel(PSI,2);
10622 : long i;
10623 14350 : for (i = 1; i < lg(P); i++)
10624 : {
10625 532 : GEN psip = znchardecompose(G, psi, utoipos(P[i]));
10626 532 : if (zncharisodd(G, psip)) return gc_bool(av,0);
10627 : }
10628 13818 : return gc_bool(av,1);
10629 : }
10630 :
10631 : static GEN
10632 299775 : get_PSI(GEN CHI, long t)
10633 : {
10634 299775 : long r = t & 3L, t2 = (r == 2 || r == 3) ? t << 2 : t;
10635 299775 : return mfcharmul_i(CHI, induce(gel(CHI,1), utoipos(t2)));
10636 : }
10637 : /* space = mf_CUSP, mf_EISEN or mf_FULL, weight k + 1/2 */
10638 : static long
10639 41363 : mf2dimwt12(long N, GEN CHI, long space)
10640 : {
10641 41363 : pari_sp av = avma;
10642 41363 : GEN D = mydivisorsu(N >> 2);
10643 41363 : long i, l = lg(D), dim3 = 0, dim4 = 0;
10644 :
10645 41363 : CHI = induceN(N, CHI);
10646 341138 : for (i = 1; i < l; i++)
10647 : {
10648 299775 : long rp, t = D[i], Mt = D[l-i];
10649 299775 : GEN PSI = get_PSI(CHI,t);
10650 299775 : rp = mfcharconductor(PSI);
10651 299775 : if (Mt % (rp*rp) == 0) { dim4++; if (charistotallyeven(PSI,rp)) dim3++; }
10652 : }
10653 41363 : set_avma(av);
10654 41363 : switch (space)
10655 : {
10656 40439 : case mf_CUSP: return dim4 - dim3;
10657 462 : case mf_EISEN:return dim3;
10658 462 : case mf_FULL: return dim4;
10659 : }
10660 : return 0; /*LCOV_EXCL_LINE*/
10661 : }
10662 :
10663 : static long
10664 693 : mf2dimwt32(long N, GEN CHI, long F, long space)
10665 : {
10666 : long D;
10667 693 : switch(space)
10668 : {
10669 231 : case mf_CUSP: D = mypsiu(N) - 6*dim22(N, F, 1);
10670 231 : if (D%24) pari_err_BUG("mfdim");
10671 231 : return D/24 + mf2dimwt12(N, CHI, 4);
10672 231 : case mf_FULL: D = mypsiu(N) + 6*dim22(N, F, 0);
10673 231 : if (D%24) pari_err_BUG("mfdim");
10674 231 : return D/24 + mf2dimwt12(N, CHI, 1);
10675 231 : case mf_EISEN: D = dim22(N, F, 0) + dim22(N, F, 1);
10676 231 : if (D & 3L) pari_err_BUG("mfdim");
10677 231 : return (D >> 2) - mf2dimwt12(N, CHI, 3);
10678 : }
10679 : return 0; /*LCOV_EXCL_LINE*/
10680 : }
10681 :
10682 : /* F = conductor(CHI), weight k = r+1/2 */
10683 : static long
10684 43729 : checkmf2(long N, long r, GEN CHI, long F, long space)
10685 : {
10686 43729 : switch(space)
10687 : {
10688 43708 : case mf_FULL: case mf_CUSP: case mf_EISEN: break;
10689 14 : case mf_NEW: case mf_OLD:
10690 14 : pari_err_TYPE("half-integral weight [new/old spaces]", utoi(space));
10691 7 : default:
10692 7 : pari_err_TYPE("half-integral weight [incorrect space]",utoi(space));
10693 : }
10694 43708 : if (N & 3L)
10695 0 : pari_err_DOMAIN("half-integral weight", "N % 4", "!=", gen_0, stoi(N));
10696 43708 : return r >= 0 && mfcharparity(CHI) == 1 && N % F == 0;
10697 : }
10698 :
10699 : /* weight k = r + 1/2 */
10700 : static long
10701 43463 : mf2dim_Nkchi(long N, long r, GEN CHI, ulong space)
10702 : {
10703 43463 : long D, D2, F = mfcharconductor(CHI);
10704 43463 : if (!checkmf2(N, r, CHI, F, space)) return 0;
10705 43442 : if (r == 0) return mf2dimwt12(N, CHI, space);
10706 2772 : if (r == 1) return mf2dimwt32(N, CHI, F, space);
10707 2079 : if (space == mf_EISEN)
10708 : {
10709 693 : D = dim22(N, F, r) + dim22(N, F, 1-r);
10710 693 : if (D & 3L) pari_err_BUG("mfdim");
10711 693 : return D >> 2;
10712 : }
10713 1386 : D2 = space == mf_FULL? dim22(N, F, 1-r): -dim22(N, F, r);
10714 1386 : D = (2*r-1)*mypsiu(N) + 6*D2;
10715 1386 : if (D%24) pari_err_BUG("mfdim");
10716 1386 : return D/24;
10717 : }
10718 :
10719 : /* weight k=r+1/2 */
10720 : static GEN
10721 266 : mf2init_Nkchi(long N, long r, GEN CHI, long space, long flraw)
10722 : {
10723 266 : GEN CHI1, Minv, Minvmat, B, M, gk = gaddsg(r,ghalf);
10724 266 : GEN mf1 = mkvec4(utoi(N),gk,CHI,utoi(space));
10725 : long L;
10726 266 : if (!checkmf2(N, r, CHI, mfcharconductor(CHI), space)) return mfEMPTY(mf1);
10727 266 : if (space==mf_EISEN) pari_err_IMPL("half-integral weight Eisenstein space");
10728 266 : L = mfsturmNgk(N, gk) + 1;
10729 266 : B = mf2basis(N, r, CHI, &CHI1, space);
10730 266 : M = mflineardivtomat(N,B,L); /* defined modulo T = charpol(CHI) */
10731 266 : if (flraw) M = mkvec3(gen_0,gen_0,M);
10732 : else
10733 : {
10734 266 : long o1 = mfcharorder(CHI1), o = mfcharorder(CHI);
10735 266 : M = mfcleanCHI(M, CHI, 0);
10736 266 : Minv = gel(M,2);
10737 266 : Minvmat = RgM_Minv_mul(NULL, Minv); /* mod T */
10738 266 : if (o1 != o)
10739 : {
10740 133 : GEN tr = Qab_trace_init(o, o1, mfcharpol(CHI), mfcharpol(CHI1));
10741 133 : Minvmat = QabM_tracerel(tr, 0, Minvmat);
10742 : }
10743 : /* Minvmat mod T1 = charpol(CHI1) */
10744 266 : B = vecmflineardiv_linear(B, Minvmat);
10745 266 : gel(M,3) = RgM_Minv_mul(gel(M,3), Minv);
10746 266 : gel(M,2) = mkMinv(matid(lg(B)-1), NULL,NULL,NULL);
10747 : }
10748 266 : return mkmf(mf1, cgetg(1,t_VEC), B, gen_0, M);
10749 : }
10750 :
10751 : /**************************************************************************/
10752 : /* Kohnen + space */
10753 : /**************************************************************************/
10754 :
10755 : static GEN
10756 21 : mfkohnenbasis_i(GEN mf, GEN CHI, long eps, long sb)
10757 : {
10758 21 : GEN M = mfcoefs_mf(mf, sb, 1), p, P;
10759 21 : long c, i, n = mfcharorder(CHI), l = sb + 2;
10760 21 : p = cgetg(l, t_VECSMALL);
10761 : /* keep the a_n, n = (2 or 2+eps) mod 4 */
10762 210 : for (i = 3, c = 1; i < l; i+=4) p[c++] = i;
10763 203 : for (i = 3+eps; i < l; i+=4) p[c++] = i;
10764 21 : P = n <= 2? NULL: mfcharpol(CHI);
10765 21 : setlg(p, c);
10766 21 : return QabM_ker(rowpermute(M, p), P, n);
10767 : }
10768 : GEN
10769 21 : mfkohnenbasis(GEN mf)
10770 : {
10771 21 : pari_sp av = avma;
10772 : GEN gk, CHI, CHIP, K;
10773 : long N4, r, eps, sb;
10774 21 : mf = checkMF(mf);
10775 21 : if (MF_get_space(mf) != mf_CUSP)
10776 0 : pari_err_TYPE("mfkohnenbasis [not a cuspidal space", mf);
10777 21 : if (!MF_get_dim(mf)) return cgetg(1, t_MAT);
10778 21 : N4 = MF_get_N(mf) >> 2; gk = MF_get_gk(mf); CHI = MF_get_CHI(mf);
10779 21 : if (typ(gk) == t_INT) pari_err_TYPE("mfkohnenbasis", gk);
10780 21 : r = MF_get_r(mf);
10781 21 : CHIP = mfcharchiliftprim(CHI, N4);
10782 21 : eps = CHIP==CHI? 1: -1;
10783 21 : if (odd(r)) eps = -eps;
10784 21 : if (uissquarefree(N4))
10785 : {
10786 14 : long d = mfdim_Nkchi(N4, 2*r, mfcharpow(CHI, gen_2), mf_CUSP);
10787 14 : sb = mfsturmNgk(N4 << 2, gk) + 1;
10788 14 : K = mfkohnenbasis_i(mf, CHIP, eps, sb);
10789 14 : if (lg(K) - 1 == d) return gc_GEN(av, K);
10790 : }
10791 7 : sb = mfsturmNgk(N4 << 4, gk) + 1;
10792 7 : K = mfkohnenbasis_i(mf, CHIP, eps, sb);
10793 7 : return gc_GEN(av, K);
10794 : }
10795 :
10796 : static GEN
10797 21 : get_Shimura(GEN mf, GEN CHI, GEN vB, long t)
10798 : {
10799 21 : long N = MF_get_N(mf), r = MF_get_k(mf) >> 1;
10800 21 : long i, d = MF_get_dim(mf), sb = mfsturm_mf(mf);
10801 21 : GEN a = cgetg(d+1, t_MAT);
10802 84 : for (i = 1; i <= d; i++)
10803 : {
10804 63 : pari_sp av = avma;
10805 63 : GEN f = c_deflate(sb*sb, t, gel(vB,i));
10806 63 : f = mftobasis_i(mf, RgV_shimura(f, sb, t, N, r, CHI));
10807 63 : gel(a,i) = gc_upto(av, f);
10808 : }
10809 21 : return a;
10810 : }
10811 : static long
10812 35 : QabM_rank(GEN M, GEN P, long n)
10813 : {
10814 35 : GEN z = QabM_indexrank(M, P, n);
10815 35 : return lg(gel(z,2))-1;
10816 : }
10817 : /* discard T[*i] */
10818 : static void
10819 0 : discard_Ti(GEN T, long *i, long *lt)
10820 : {
10821 0 : long j, l = *lt-1;
10822 0 : for (j = *i; j < l; j++) T[j] = T[j+1];
10823 0 : (*i)--; *lt = l;
10824 0 : }
10825 : /* return [mf3, bijection, mfkohnenbasis, codeshi] */
10826 : static GEN
10827 14 : mfkohnenbijection_i(GEN mf)
10828 : {
10829 14 : GEN CHI = MF_get_CHI(mf), K = mfkohnenbasis(mf);
10830 : GEN mres, dMi, Mi, M, C, vB, mf3, SHI, T, P;
10831 14 : long N4 = MF_get_N(mf)>>2, r = MF_get_r(mf), dK = lg(K) - 1;
10832 : long i, c, n, oldr, lt, ltold, sb3, t, limt;
10833 14 : const long MAXlt = 100;
10834 :
10835 14 : mf3 = mfinit_Nkchi(N4, r<<1, mfcharpow(CHI,gen_2), mf_CUSP, 0);
10836 14 : if (MF_get_dim(mf3) != dK)
10837 0 : pari_err_BUG("mfkohnenbijection [different dimensions]");
10838 14 : if (!dK) return mkvec4(mf3, cgetg(1, t_MAT), K, cgetg(1, t_VEC));
10839 14 : CHI = mfcharchiliftprim(CHI, N4);
10840 14 : n = mfcharorder(CHI);
10841 14 : P = n<=2? NULL: mfcharpol(CHI);
10842 14 : SHI = cgetg(MAXlt, t_COL);
10843 14 : T = cgetg(MAXlt, t_VECSMALL);
10844 14 : sb3 = mfsturm_mf(mf3);
10845 14 : limt = 6; oldr = 0; vB = C = M = NULL;
10846 98 : for (t = lt = ltold = 1; lt < MAXlt; t++)
10847 : {
10848 : pari_sp av;
10849 98 : if (!uissquarefree(t)) continue;
10850 84 : T[lt++] = t; if (t <= limt) continue;
10851 14 : av = avma;
10852 14 : if (vB) gunclone(vB);
10853 : /* could improve the rest but 99% of running time is spent here */
10854 14 : vB = gclone( RgM_mul(mfcoefs_mf(mf, t*sb3*sb3, 1), K) );
10855 14 : set_avma(av);
10856 21 : for (i = ltold; i < lt; i++)
10857 : {
10858 : pari_sp av;
10859 : long r;
10860 21 : M = get_Shimura(mf3, CHI, vB, T[i]);
10861 21 : r = QabM_rank(M, P, n); if (!r) { discard_Ti(T, &i, <); continue; }
10862 21 : gel(SHI, i) = M; setlg(SHI, i+1);
10863 21 : if (r >= dK) { C = vecsmall_ei(dK, i); goto DONE; }
10864 14 : if (i == 1) { oldr = r; continue; }
10865 7 : av = avma; M = shallowmatconcat(SHI);
10866 7 : r = QabM_rank(M, P, n); /* >= rank(sum C[j] SHI[j]), probably sharp */
10867 7 : if (r >= dK)
10868 : {
10869 7 : M = RgV_sum(SHI);
10870 7 : if (QabM_rank(M, P, n) >= dK) { C = const_vecsmall(dK, 1); goto DONE; }
10871 0 : C = random_Flv(dK, 16);
10872 0 : M = RgV_zc_mul(SHI, C);
10873 0 : if (QabM_rank(M, P, n) >= dK) goto DONE;
10874 : }
10875 0 : else if (r == oldr) discard_Ti(T, &i, <);
10876 0 : oldr = r; set_avma(av);
10877 : }
10878 0 : limt *= 2; ltold = lt;
10879 : }
10880 0 : pari_err_BUG("mfkohnenbijection");
10881 14 : DONE:
10882 14 : gunclone(vB); lt = lg(SHI);
10883 14 : Mi = QabM_pseudoinv(M,P,n, NULL,&dMi); Mi = RgM_Rg_div(Mi,dMi);
10884 14 : mres = cgetg(lt, t_VEC);
10885 35 : for (i = c = 1; i < lt; i++)
10886 21 : if (C[i]) gel(mres,c++) = mkvec2s(T[i], C[i]);
10887 14 : setlg(mres,c); return mkvec4(mf3, Mi, K, mres);
10888 : }
10889 : GEN
10890 14 : mfkohnenbijection(GEN mf)
10891 : {
10892 14 : pari_sp av = avma;
10893 : long N;
10894 14 : mf = checkMF(mf); N = MF_get_N(mf);
10895 14 : if (!uissquarefree(N >> 2))
10896 0 : pari_err_TYPE("mfkohnenbijection [N/4 not squarefree]", utoi(N));
10897 14 : if (MF_get_space(mf) != mf_CUSP || MF_get_r(mf) == 0 || !mfshimura_space_cusp(mf))
10898 0 : pari_err_TYPE("mfkohnenbijection [incorrect mf for Kohnen]", mf);
10899 14 : return gc_GEN(av, mfkohnenbijection_i(mf));
10900 : }
10901 :
10902 : static int
10903 7 : checkbij_i(GEN b)
10904 : {
10905 7 : return typ(b) == t_VEC && lg(b) == 5 && checkMF_i(gel(b,1))
10906 7 : && typ(gel(b,2)) == t_MAT
10907 7 : && typ(gel(b,3)) == t_MAT
10908 14 : && typ(gel(b,4)) == t_VEC;
10909 : }
10910 :
10911 : /* bij is the output of mfkohnenbijection */
10912 : GEN
10913 7 : mfkohneneigenbasis(GEN mf, GEN bij)
10914 : {
10915 7 : pari_sp av = avma;
10916 : GEN mf3, mf30, B, KM, M, k;
10917 : long r, i, l, N4;
10918 7 : mf = checkMF(mf);
10919 7 : if (!checkbij_i(bij))
10920 0 : pari_err_TYPE("mfkohneneigenbasis [bijection]", bij);
10921 7 : if (MF_get_space(mf) != mf_CUSP)
10922 0 : pari_err_TYPE("mfkohneneigenbasis [not a cuspidal space]", mf);
10923 7 : if (!MF_get_dim(mf))
10924 0 : retmkvec3(cgetg(1, t_VEC), cgetg(1, t_VEC), cgetg(1, t_VEC));
10925 7 : N4 = MF_get_N(mf) >> 2; k = MF_get_gk(mf);
10926 7 : if (typ(k) == t_INT) pari_err_TYPE("mfkohneneigenbasis", k);
10927 7 : if (!uissquarefree(N4))
10928 0 : pari_err_TYPE("mfkohneneigenbasis [N not squarefree]", utoipos(N4));
10929 7 : r = MF_get_r(mf);
10930 7 : KM = RgM_mul(gel(bij,3), gel(bij,2));
10931 7 : mf3 = gel(bij,1);
10932 7 : mf30 = mfinit_Nkchi(N4, 2*r, MF_get_CHI(mf3), mf_NEW, 0);
10933 7 : B = mfcoefs_mf(mf30, mfsturm_mf(mf3), 1); l = lg(B);
10934 7 : M = cgetg(l, t_MAT);
10935 21 : for (i=1; i<l; i++) gel(M,i) = RgM_RgC_mul(KM, mftobasis_i(mf3, gel(B,i)));
10936 7 : return gc_GEN(av, mkvec3(mf30, M, RgM_mul(M, MF_get_newforms(mf30))));
10937 : }
10938 : /*************************** End Kohnen ************************************/
10939 : /***************************************************************************/
10940 :
10941 : static GEN desc(GEN F);
10942 : static GEN
10943 504 : desc_mfeisen(GEN F)
10944 : {
10945 504 : GEN R, gk = mf_get_gk(F);
10946 504 : if (typ(gk) == t_FRAC)
10947 7 : R = gsprintf("H_{%Ps}", gk);
10948 : else
10949 : {
10950 497 : GEN vchi = gel(F, 2), CHI = mfchisimpl(gel(vchi, 3));
10951 497 : long k = itou(gk);
10952 497 : if (lg(vchi) < 5) R = gsprintf("F_%ld(%Ps)", k, CHI);
10953 : else
10954 : {
10955 294 : GEN CHI2 = mfchisimpl(gel(vchi, 4));
10956 294 : R = gsprintf("F_%ld(%Ps, %Ps)", k, CHI, CHI2);
10957 : }
10958 : }
10959 504 : return R;
10960 : }
10961 : static GEN
10962 35 : desc_hecke(GEN F)
10963 : {
10964 : long n, N;
10965 35 : GEN D = gel(F,2);
10966 35 : if (typ(D) == t_VECSMALL) { N = D[3]; n = D[1]; }
10967 14 : else { GEN nN = gel(D,2); n = nN[1]; N = nN[2]; } /* half integer */
10968 35 : return gsprintf("T_%ld(%ld)(%Ps)", N, n, desc(gel(F,3)));
10969 : }
10970 : static GEN
10971 98 : desc_linear(GEN FLD, GEN dL)
10972 : {
10973 98 : GEN F = gel(FLD,2), L = gel(FLD,3), R = strtoGENstr("LIN([");
10974 98 : long n = lg(F) - 1, i;
10975 168 : for (i = 1; i <= n; i++)
10976 : {
10977 168 : R = shallowconcat(R, desc(gel(F,i))); if (i == n) break;
10978 70 : R = shallowconcat(R, strtoGENstr(", "));
10979 : }
10980 98 : return shallowconcat(R, gsprintf("], %Ps)", gdiv(L, dL)));
10981 : }
10982 : static GEN
10983 21 : desc_dihedral(GEN F)
10984 : {
10985 21 : GEN bnr = gel(F,2), D = nf_get_disc(bnr_get_nf(bnr)), f = bnr_get_mod(bnr);
10986 21 : GEN cyc = bnr_get_cyc(bnr);
10987 21 : GEN w = gel(F,3), chin = zv_to_ZV(gel(w,2)), o = utoi(gel(w,1)[1]);
10988 21 : GEN chi = char_denormalize(cyc, o, chin);
10989 21 : if (lg(gel(f,2)) == 1) f = gel(f,1);
10990 21 : return gsprintf("DIH(%Ps, %Ps, %Ps, %Ps)",D,f,cyc,chi);
10991 : }
10992 :
10993 : static void
10994 1043 : unpack0(GEN *U)
10995 1043 : { if (U) *U = mkvec2(cgetg(1, t_VEC), cgetg(1, t_VEC)); }
10996 : static void
10997 42 : unpack2(GEN F, GEN *U)
10998 42 : { if (U) *U = mkvec2(mkvec2(gel(F,2), gel(F,3)), cgetg(1, t_VEC)); }
10999 : static void
11000 308 : unpack23(GEN F, GEN *U)
11001 308 : { if (U) *U = mkvec2(mkvec(gel(F,2)), mkvec(gel(F,3))); }
11002 : static GEN
11003 1540 : desc_i(GEN F, GEN *U)
11004 : {
11005 1540 : switch(mf_get_type(F))
11006 : {
11007 7 : case t_MF_CONST: unpack0(U); return gsprintf("CONST(%Ps)", gel(F,2));
11008 504 : case t_MF_EISEN: unpack0(U); return desc_mfeisen(F);
11009 154 : case t_MF_Ek: unpack0(U); return gsprintf("E_%ld", mf_get_k(F));
11010 63 : case t_MF_DELTA: unpack0(U); return gsprintf("DELTA");
11011 35 : case t_MF_THETA: unpack0(U);
11012 35 : return gsprintf("THETA(%Ps)", mfchisimpl(gel(F,2)));
11013 56 : case t_MF_ETAQUO: unpack0(U);
11014 56 : return gsprintf("ETAQUO(%Ps, %Ps)", gel(F,2), gel(F,3));
11015 56 : case t_MF_ELL: unpack0(U);
11016 56 : return gsprintf("ELL(%Ps)", vecslice(gel(F,2), 1, 5));
11017 7 : case t_MF_TRACE: unpack0(U); return gsprintf("TR(%Ps)", mfparams(F));
11018 140 : case t_MF_NEWTRACE: unpack0(U); return gsprintf("TR^new(%Ps)", mfparams(F));
11019 21 : case t_MF_DIHEDRAL: unpack0(U); return desc_dihedral(F);
11020 28 : case t_MF_MUL: unpack2(F, U);
11021 28 : return gsprintf("MUL(%Ps, %Ps)", desc(gel(F,2)), desc(gel(F,3)));
11022 14 : case t_MF_DIV: unpack2(F, U);
11023 14 : return gsprintf("DIV(%Ps, %Ps)", desc(gel(F,2)), desc(gel(F,3)));
11024 14 : case t_MF_POW: unpack23(F, U);
11025 14 : return gsprintf("POW(%Ps, %ld)", desc(gel(F,2)), itos(gel(F,3)));
11026 14 : case t_MF_SHIFT: unpack23(F, U);
11027 14 : return gsprintf("SHIFT(%Ps, %ld)", desc(gel(F,2)), itos(gel(F,3)));
11028 14 : case t_MF_DERIV: unpack23(F, U);
11029 14 : return gsprintf("DER^%ld(%Ps)", itos(gel(F,3)), desc(gel(F,2)));
11030 21 : case t_MF_DERIVE2: unpack23(F, U);
11031 21 : return gsprintf("DERE2^%ld(%Ps)", itos(gel(F,3)), desc(gel(F,2)));
11032 14 : case t_MF_TWIST: unpack23(F, U);
11033 14 : return gsprintf("TWIST(%Ps, %Ps)", desc(gel(F,2)), gel(F,3));
11034 231 : case t_MF_BD: unpack23(F, U);
11035 231 : return gsprintf("B(%ld)(%Ps)", itou(gel(F,3)), desc(gel(F,2)));
11036 14 : case t_MF_BRACKET:
11037 14 : if (U) *U = mkvec2(mkvec2(gel(F,2), gel(F,3)), mkvec(gel(F,4)));
11038 14 : return gsprintf("MULRC_%ld(%Ps, %Ps)", itos(gel(F,4)), desc(gel(F,2)), desc(gel(F,3)));
11039 98 : case t_MF_LINEAR_BHN:
11040 : case t_MF_LINEAR:
11041 98 : if (U) *U = mkvec2(gel(F,2), mkvec(gdiv(gel(F,3), gel(F,4))));
11042 98 : return desc_linear(F,gel(F,4));
11043 35 : case t_MF_HECKE:
11044 35 : if (U) *U = mkvec2(mkvec(gel(F,3)), mkvec(stoi(gel(F,2)[1])));
11045 35 : return desc_hecke(F);
11046 0 : default: pari_err_TYPE("mfdescribe",F);
11047 : return NULL;/*LCOV_EXCL_LINE*/
11048 : }
11049 : }
11050 : static GEN
11051 623 : desc(GEN F) { return desc_i(F, NULL); }
11052 : GEN
11053 966 : mfdescribe(GEN F, GEN *U)
11054 : {
11055 966 : pari_sp av = avma;
11056 : GEN mf;
11057 966 : if ((mf = checkMF_i(F)))
11058 : {
11059 49 : const char *f = NULL;
11060 49 : switch (MF_get_space(mf))
11061 : {
11062 7 : case mf_NEW: f = "S_%Ps^new(G_0(%ld, %Ps))"; break;
11063 14 : case mf_CUSP: f = "S_%Ps(G_0(%ld, %Ps))"; break;
11064 7 : case mf_OLD: f = "S_%Ps^old(G_0(%ld, %Ps))"; break;
11065 7 : case mf_EISEN:f = "E_%Ps(G_0(%ld, %Ps))"; break;
11066 14 : case mf_FULL: f = "M_%Ps(G_0(%ld, %Ps))"; break;
11067 : }
11068 49 : if (U) *U = cgetg(1, t_VEC);
11069 49 : return gsprintf(f, MF_get_gk(mf), MF_get_N(mf), mfchisimpl(MF_get_CHI(mf)));
11070 : }
11071 917 : if (!checkmf_i(F)) pari_err_TYPE("mfdescribe", F);
11072 917 : F = desc_i(F, U); return gc_all(av, U ? 2: 1, &F, U);
11073 : }
11074 :
11075 : /***********************************************************************/
11076 : /* Eisenstein series H_r of weight r+1/2 */
11077 : /***********************************************************************/
11078 : /* radical(u_ppo(g,q)) */
11079 : static long
11080 28 : u_pporad(long g, long q)
11081 : {
11082 28 : GEN F = myfactoru(g), P = gel(F,1);
11083 : long i, l, n;
11084 28 : if (q == 1) return zv_prod(P);
11085 28 : l = lg(P);
11086 35 : for (i = n = 1; i < l; i++)
11087 : {
11088 7 : long p = P[i];
11089 7 : if (q % p) n *= p;
11090 : }
11091 28 : return n;
11092 : }
11093 : static void
11094 266 : c_F2TH4(long n, GEN *pF2, GEN *pTH4)
11095 : {
11096 266 : GEN v = mfcoefs_i(mfEk(2), n, 1), v2 = bdexpand(v,2), v4 = bdexpand(v,4);
11097 266 : GEN F2 = gdivgs(ZC_add(ZC_sub(v, ZC_z_mul(v2,3)), ZC_z_mul(v4,2)), -24);
11098 266 : GEN TH4 = gdivgs(ZC_sub(v, ZC_z_mul(v4,4)), -3);
11099 266 : settyp(F2,t_VEC); *pF2 = F2;
11100 266 : settyp(TH4,t_VEC);*pTH4= TH4;
11101 266 : }
11102 : /* r > 0, N >= 0 */
11103 : static GEN
11104 77 : mfEHcoef(long r, long N)
11105 : {
11106 : long D0, f, i, l, s;
11107 : GEN S, Df;
11108 :
11109 77 : if (r == 1) return hclassno(utoi(N));
11110 77 : if (N == 0) return gdivgs(bernfrac(2*r), -2*r);
11111 56 : s = N & 3L;
11112 56 : if (odd(r))
11113 : {
11114 42 : if (s == 2 || s == 1) return gen_0;
11115 14 : D0 = mycoredisc2neg(N,&f);
11116 : }
11117 : else
11118 : {
11119 14 : if (s == 2 || s == 3) return gen_0;
11120 14 : D0 = mycoredisc2pos(N,&f);
11121 : }
11122 28 : Df = mydivisorsu(u_pporad(f, D0)); l = lg(Df);
11123 28 : S = gen_0;
11124 63 : for (i = 1; i < l; i++)
11125 : {
11126 35 : long d = Df[i], s = mymoebiusu(d)*kross(D0, d); /* != 0 */
11127 35 : GEN c = gmul(powuu(d, r-1), mysumdivku(f/d, 2*r-1));
11128 35 : S = s > 0? addii(S, c): subii(S, c);
11129 : }
11130 28 : return gmul(lfunquadneg_naive(D0, r), S);
11131 : }
11132 : static GEN
11133 266 : mfEHmat(long lim, long r)
11134 : {
11135 266 : long j, l, d = r/2;
11136 : GEN f2, th4, th3, v, vth4, vf2;
11137 266 : c_F2TH4(lim, &f2, &th4);
11138 266 : f2 = RgV_to_ser(f2, 0, lim+3);
11139 266 : th4 = RgV_to_ser(th4, 0, lim+3);
11140 266 : th3 = RgV_to_ser(c_theta(lim, 1, mfchartrivial()), 0, lim+3);
11141 266 : if (odd(r)) th3 = gpowgs(th3, 3);
11142 266 : vth4 = gpowers(th4, d);
11143 266 : vf2 = gpowers0(f2, d, th3); /* th3 f2^j */
11144 266 : l = d+2; v = cgetg(l, t_VEC);
11145 924 : for (j = 1; j < l; j++)
11146 658 : gel(v, j) = ser2rfrac_i(gmul(gel(vth4, l-j), gel(vf2, j)));
11147 266 : return RgXV_to_RgM(v, lim);
11148 : }
11149 : static GEN
11150 7 : Hfind(long r, GEN *pden)
11151 : {
11152 7 : long lim = (r/2)+3, i;
11153 : GEN res, M, B;
11154 :
11155 7 : if (r <= 0) pari_err_DOMAIN("mfEH", "r", "<=", gen_0, stoi(r));
11156 7 : M = mfEHmat(lim, r);
11157 7 : B = cgetg(lim+1, t_COL);
11158 56 : for (i = 1; i <= lim; i++) gel(B, i) = mfEHcoef(r, i-1);
11159 7 : res = QM_gauss(M, B);
11160 7 : if (lg(res) == 1) pari_err_BUG("mfEH");
11161 7 : return Q_remove_denom(res,pden);
11162 : }
11163 : GEN
11164 266 : mfEH(GEN gk)
11165 : {
11166 266 : pari_sp av = avma;
11167 266 : GEN v, d, NK, gr = gsub(gk, ghalf);
11168 : long r;
11169 266 : if (typ(gr) != t_INT) pari_err_TYPE("mfEH", gk);
11170 266 : r = itos(gr);
11171 266 : switch (r)
11172 : {
11173 7 : case 1: v=cgetg(1,t_VEC); d=gen_1; break;
11174 133 : case 2: v=mkvec2s(1,-20); d=utoipos(120); break;
11175 56 : case 3: v=mkvec2s(-1,14); d=utoipos(252); break;
11176 35 : case 4: v=mkvec3s(1,-16,16); d=utoipos(240); break;
11177 7 : case 5: v=mkvec3s(-1,22,-88); d=utoipos(132); break;
11178 14 : case 6: v=mkvec4s(691,-18096,110136,-4160); d=utoipos(32760); break;
11179 7 : case 7: v=mkvec4s(-1,30,-240,224); d=utoipos(12); break;
11180 7 : default: v = Hfind(r, &d); break;
11181 : }
11182 266 : NK = mkgNK(utoipos(4), gaddgs(ghalf,r), mfchartrivial(), pol_x(1));
11183 266 : return gc_GEN(av, tag(t_MF_EISEN, NK, mkvec2(v,d)));
11184 : }
11185 :
11186 : /**********************************************************/
11187 : /* T(f^2) for half-integral weight */
11188 : /**********************************************************/
11189 :
11190 : /* T_p^2 V, p2 = p^2, c1 = chi(p) (-1/p)^r p^(r-1), c2 = chi(p^2)*p^(2r-1) */
11191 : static GEN
11192 70 : tp2apply(GEN V, long p, long p2, GEN c1, GEN c2)
11193 : {
11194 70 : long lw = (lg(V) - 2)/p2 + 1, m, n;
11195 70 : GEN a0 = gel(V,1), W = cgetg(lw + 1, t_VEC);
11196 :
11197 70 : gel(W,1) = gequal0(a0)? gen_0: gmul(a0, gaddsg(1, c2));
11198 11109 : for (n = 1; n < lw; n++)
11199 : {
11200 11039 : GEN c = gel(V, p2*n + 1);
11201 11039 : if (n%p) c = gadd(c, gmulsg(kross(n,p), gmul(gel(V,n+1), c1)));
11202 11039 : gel(W, n+1) = c; /* a(p^2*n) + c1 * (n/p) a(n) */
11203 : }
11204 1253 : for (m = 1, n = p2; n < lw; m++, n += p2)
11205 1183 : gel(W, n+1) = gadd(gel(W,n+1), gmul(gel(V,m+1), c2));
11206 70 : return W;
11207 : }
11208 :
11209 : /* T_{p^{2e}} V; can derecursify [Purkait, Hecke operators in half-integral
11210 : * weight, Prop 4.3], not worth it */
11211 : static GEN
11212 70 : tp2eapply(GEN V, long p, long p2, long e, GEN q, GEN c1, GEN c2)
11213 : {
11214 70 : GEN V4 = NULL;
11215 70 : if (e > 1)
11216 : {
11217 21 : V4 = vecslice(V, 1, (lg(V) - 2)/(p2*p2) + 1);
11218 21 : V = tp2eapply(V, p, p2, e-1, q, c1, c2);
11219 : }
11220 70 : V = tp2apply(V, p, p2, c1, c2);
11221 70 : if (e > 1)
11222 28 : V = gsub(V, (e == 2)? gmul(q, V4)
11223 7 : : gmul(c2, tp2eapply(V4, p, p2, e-2, q, c1, c2)));
11224 70 : return V;
11225 : }
11226 : /* weight k = r+1/2 */
11227 : static GEN
11228 98 : RgV_heckef2(long n, long d, GEN V, GEN F, GEN DATA)
11229 : {
11230 98 : GEN CHI = mf_get_CHI(F), fa = gel(DATA,1), P = gel(fa,1), E = gel(fa,2);
11231 98 : long i, l = lg(P), r = mf_get_r(F), s4 = odd(r)? -4: 4, k2m2 = (r<<1)-1;
11232 98 : if (typ(V) == t_COL) V = shallowtrans(V);
11233 140 : for (i = 1; i < l; i++)
11234 : { /* p does not divide N */
11235 42 : long p = P[i], e = E[i], p2 = p*p;
11236 42 : GEN c1, c2, a, b, q = NULL, C = mfchareval(CHI,p), C2 = gsqr(C);
11237 42 : a = r? powuu(p,r-1): mkfrac(gen_1,utoipos(p)); /* p^(r-1) = p^(k-3/2) */
11238 42 : b = r? mulii(powuu(p,r), a): a; /* p^(2r-1) = p^(2k-2) */
11239 42 : c1 = gmul(C, gmulsg(kross(s4,p),a));
11240 42 : c2 = gmul(C2, b);
11241 42 : if (e > 1)
11242 : {
11243 14 : q = r? powuu(p,k2m2): a;
11244 14 : if (e == 2) q = gmul(q, uutoQ(p+1,p)); /* special case T_{p^4} */
11245 14 : q = gmul(C2, q); /* chi(p^2) [ p^(2k-2) or (p+1)p^(2k-3) ] */
11246 : }
11247 42 : V = tp2eapply(V, p, p2, e, q, c1, c2);
11248 : }
11249 98 : return c_deflate(n, d, V);
11250 : }
11251 :
11252 : static GEN
11253 1211 : GL2toSL2(GEN g, GEN *abd)
11254 : {
11255 : GEN A, B, C, D, u, v, a, b, d, q;
11256 1211 : g = Q_primpart(g);
11257 1211 : if (!check_M2Z(g)) pari_err_TYPE("GL2toSL2", g);
11258 1211 : A = gcoeff(g,1,1); B = gcoeff(g,1,2);
11259 1211 : C = gcoeff(g,2,1); D = gcoeff(g,2,2);
11260 1211 : a = bezout(A, C, &u, &v);
11261 1211 : if (!equali1(a)) { A = diviiexact(A,a); C = diviiexact(C,a); }
11262 1211 : d = subii(mulii(A,D), mulii(B,C));
11263 1211 : if (signe(d) <= 0) pari_err_TYPE("GL2toSL2",g);
11264 1204 : q = dvmdii(addii(mulii(u,B), mulii(v,D)), d, &b);
11265 1204 : *abd = (equali1(a) && equali1(d))? NULL: mkvec3(a, b, d);
11266 1204 : return mkmat22(A, subii(mulii(q,A), v), C, addii(mulii(q,C), u));
11267 : }
11268 :
11269 : static GEN
11270 8582 : Rg_approx(GEN t, long bit)
11271 : {
11272 8582 : GEN a = real_i(t), b = imag_i(t);
11273 8582 : long e1 = gexpo(a), e2 = gexpo(b);
11274 8582 : if (e2 < -bit) { t = e1 < -bit? gen_0: a; }
11275 6510 : else if (e1 < -bit) t = gmul(b, gen_I());
11276 8582 : return t;
11277 : }
11278 : static GEN
11279 126 : RgV_approx(GEN x, long bit)
11280 840 : { pari_APPLY_same(Rg_approx(gel(x,i), bit)); }
11281 : /* m != 2 (mod 4), D t_INT; V has "denominator" D, recognize in Q(zeta_m) */
11282 : static GEN
11283 126 : bestapprnf2(GEN V, long m, GEN D, long prec)
11284 : {
11285 126 : long i, j, f, vt = fetch_user_var("t"), bit = prec2nbits_mul(prec, 0.8);
11286 126 : GEN Tinit, Vl, H, Pf, P = polcyclo(m, vt);
11287 :
11288 126 : V = liftpol_shallow(V);
11289 126 : V = gmul(RgV_approx(V, bit), D);
11290 126 : V = bestapprnf(V, P, NULL, prec);
11291 126 : Vl = liftpol_shallow(V);
11292 126 : H = coprimes_zv(m);
11293 672 : for (i = 2; i < m; i++)
11294 : {
11295 546 : if (H[i] != 1) continue;
11296 280 : if (!gequal(Vl, vecGalois(Vl, i, P, m))) H[i] = 0;
11297 14 : else for (j = i; j < m; j *= i) H[i] = 3;
11298 : }
11299 126 : f = znstar_conductor_bits(Flv_to_F2v(H));
11300 126 : if (f == 1) return gdiv(V, D);
11301 98 : if (f == m) return gmodulo(gdiv(V, D), P);
11302 7 : Pf = polcyclo(f, vt);
11303 7 : Tinit = Qab_trace_init(m, f, P, Pf);
11304 7 : return gmodulo(gdiv(QabV_tracerel(Tinit, 0, Vl), D), Pf);
11305 : }
11306 :
11307 : /* f | ga expansion; [f, mf_eisendec(f)]~ allowed */
11308 : GEN
11309 1148 : mfslashexpansion(GEN mf, GEN f, GEN ga, long n, long flrat, GEN *params, long prec)
11310 : {
11311 1148 : pari_sp av = avma;
11312 1148 : GEN a, b, d, res, al, V, M, ad, abd, gk, A, awd = NULL;
11313 : long i, w;
11314 :
11315 1148 : mf = checkMF(mf);
11316 1148 : gk = MF_get_gk(mf);
11317 1148 : M = GL2toSL2(ga, &abd);
11318 1141 : if (abd) { a = gel(abd,1); b = gel(abd,2); d = gel(abd,3); }
11319 903 : else { a = d = gen_1; b = gen_0; }
11320 1141 : ad = gdiv(a,d);
11321 1141 : res = mfgaexpansion(mf, f, M, n, prec);
11322 1141 : al = gel(res,1);
11323 1141 : w = itou(gel(res,2));
11324 1141 : V = gel(res,3);
11325 1141 : if (flrat)
11326 : {
11327 126 : GEN CHI = MF_get_CHI(mf);
11328 126 : long N = MF_get_N(mf), F = mfcharconductor(CHI);
11329 126 : long ord = mfcharorder(CHI), k, deg;
11330 126 : long B = umodiu(gcoeff(M,1,2), N);
11331 126 : long C = umodiu(gcoeff(M,2,1), N);
11332 126 : long D = umodiu(gcoeff(M,2,2), N);
11333 126 : long CD = (C * D) % N, BC = (B * C) % F;
11334 : GEN CV, t;
11335 : /* weight of f * Theta in 1/2-integral weight */
11336 126 : k = typ(gk) == t_INT? (long) itou(gk): MF_get_r(mf)+1;
11337 126 : CV = odd(k) ? powuu(N, k - 1) : powuu(N, k >> 1);
11338 126 : deg = ulcm(ulcm(ord, N/ugcd(N,CD)), F/ugcd(F,BC));
11339 126 : if ((deg & 3) == 2) deg >>= 1;
11340 126 : if (typ(gk) != t_INT && odd(deg) && mfthetaI(C,D)) deg <<= 2;
11341 126 : V = bestapprnf2(V, deg, CV, prec);
11342 126 : if (abd && !signe(b))
11343 : { /* can [a,0; 0,d] be simplified to id ? */
11344 7 : long nk, dk; Qtoss(gk, &nk, &dk);
11345 7 : if (ispower(ad, utoipos(2*dk), &t)) /* t^(2*dk) = a/d or t = NULL */
11346 : {
11347 7 : V = RgV_Rg_mul(V, powiu(t,nk));
11348 7 : awd = gdiv(a, muliu(d,w));
11349 : }
11350 : }
11351 : }
11352 1015 : else if (abd)
11353 : { /* ga = M * [a,b;0,d] * rational, F := f | M = q^al * \sum V[j] q^(j/w) */
11354 231 : GEN u, t = NULL, wd = muliu(d,w);
11355 : /* a > 0, 0 <= b < d; f | ga = (a/d)^(k/2) * F(tau + b/d) */
11356 231 : if (signe(b))
11357 : {
11358 : long ns, ds;
11359 : GEN z;
11360 0 : Qtoss(gdiv(b, wd), &ns, &ds); z = rootsof1powinit(ns, ds, prec);
11361 0 : for (i = 1; i <= n+1; i++) gel(V,i) = gmul(gel(V,i), rootsof1pow(z, i-1));
11362 0 : if (!gequal0(al)) t = gexp(gmul(PiI2(prec), gmul(al, gdiv(b,d))), prec);
11363 : }
11364 231 : awd = gdiv(a, wd);
11365 231 : u = gpow(ad, gmul2n(gk,-1), prec);
11366 231 : t = t? gmul(t, u): u;
11367 231 : V = RgV_Rg_mul(V, t);
11368 : }
11369 1141 : if (!awd) A = mkmat22(a, b, gen_0, d);
11370 : else
11371 : { /* rescale and update w from [a,0; 0,d] */
11372 : long ns;
11373 238 : Qtoss(awd, &ns, &w); /* update w */
11374 238 : V = bdexpand(V, ns);
11375 238 : if (!gequal0(al))
11376 : {
11377 0 : GEN adal = gmul(ad, al), sh = gfloor(adal);
11378 0 : al = gsub(adal, sh);
11379 0 : V = RgV_shift(V, sh);
11380 : }
11381 238 : A = matid(2);
11382 : }
11383 1141 : if (params) *params = mkvec3(al, utoipos(w), A);
11384 1141 : return gc_all(av,params?2:1,&V,params);
11385 : }
11386 :
11387 : /**************************************************************/
11388 : /* Alternative method for 1/2-integral weight */
11389 : /**************************************************************/
11390 : static GEN
11391 266 : mf2basis(long N, long r, GEN CHI, GEN *pCHI1, long space)
11392 : {
11393 : GEN CHI1, CHI2, mf1, mf2, B1, B2, BT, M1, M2, M, M2i, T, Th, v, den;
11394 266 : long sb, N2, o1, o2, k1 = r + 1;
11395 :
11396 266 : if (odd(k1))
11397 : {
11398 154 : CHI1 = mfcharmul(CHI, get_mfchar(stoi(-4)));
11399 154 : CHI2 = mfcharmul(CHI, get_mfchar(stoi(-8)));
11400 : }
11401 : else
11402 : {
11403 112 : CHI1 = CHI;
11404 112 : CHI2 = mfcharmul(CHI, get_mfchar(utoi(8)));
11405 : }
11406 266 : mf1 = mfinit_Nkchi(N, k1, CHI1, space, 1);
11407 266 : if (pCHI1) *pCHI1 = CHI1;
11408 266 : B1 = MF_get_basis(mf1); if (lg(B1) == 1) return cgetg(1,t_VEC);
11409 259 : N2 = ulcm(8, N);
11410 259 : mf2 = mfinit_Nkchi(N2, k1, CHI2, space, 1);
11411 259 : B2 = MF_get_basis(mf2); if (lg(B2) == 1) return cgetg(1,t_VEC);
11412 259 : sb = mfsturmNgk(N2, gaddsg(k1, ghalf));
11413 259 : M1 = mfcoefs_mf(mf1, sb, 1);
11414 259 : M2 = mfcoefs_mf(mf2, sb, 1);
11415 259 : Th = mfTheta(NULL);
11416 259 : BT = mfcoefs_i(Th, sb, 1);
11417 259 : M1 = mfmatsermul(M1, RgV_to_RgX(expandbd(BT,2),0));
11418 259 : M2 = mfmatsermul(M2, RgV_to_RgX(BT,0));
11419 259 : o1= mfcharorder(CHI1);
11420 259 : T = (o1 <= 2)? NULL: mfcharpol(CHI1);
11421 259 : if (o1 > 2) M1 = liftpol_shallow(M1);
11422 259 : o2= mfcharorder(CHI2);
11423 259 : if (T)
11424 : {
11425 14 : if (o2 == o1) M2 = liftpol_shallow(M2);
11426 : else
11427 : {
11428 0 : GEN tr = Qab_trace_init(o2, o1, mfcharpol(CHI2), mfcharpol(CHI1));
11429 0 : M2 = QabM_tracerel(tr, 0, M2);
11430 : }
11431 : }
11432 : /* now everything is defined mod T = mfcharpol(CHI1) */
11433 259 : M2i = QabM_pseudoinv_i(M2, T, o1, &v, &den);
11434 259 : M = RgM_mul(M2i, rowpermute(M1, gel(v,1)));
11435 259 : M = RgM_mul(M2, M);
11436 259 : M1 = RgM_Rg_mul(M1, den);
11437 259 : M = RgM_sub(M1, M); if (T) M = RgXQM_red(M, T);
11438 259 : return vecmflineardiv0(B1, QabM_ker(M, T, o1), Th);
11439 : }
11440 :
11441 : /*******************************************************************/
11442 : /* Integration */
11443 : /*******************************************************************/
11444 : static GEN
11445 490 : vanembed(GEN F, GEN v, long prec)
11446 : {
11447 490 : GEN CHI = mf_get_CHI(F);
11448 490 : long o = mfcharorder(CHI);
11449 490 : if (o > 2 || degpol(mf_get_field(F)) > 1) v = liftpol_shallow(v);
11450 490 : if (o > 2) v = gsubst(v, varn(mfcharpol(CHI)), rootsof1u_cx(o, prec));
11451 490 : return v;
11452 : }
11453 :
11454 : static long
11455 1253 : mfperiod_prelim_double(double t0, long k, long bitprec)
11456 : {
11457 1253 : double nlim, c = 2*M_PI*t0;
11458 1253 : nlim = ceil(bitprec * M_LN2 / c);
11459 1253 : c -= (k - 1)/(2*nlim); if (c < 1) c = 1.;
11460 1253 : nlim += ceil((0.7 + (k-1)/2*log(nlim))/c);
11461 1253 : return (long)nlim;
11462 : }
11463 : static long
11464 301 : mfperiod_prelim(GEN t0, long k, long bitprec)
11465 301 : { return mfperiod_prelim_double(gtodouble(t0), k, bitprec); }
11466 :
11467 : /* (-X)^(k-2) * P(-1/X) = (-1)^{k-2} P|_{k-2} S */
11468 : static GEN
11469 1288 : RgX_act_S(GEN P, long k)
11470 : {
11471 1288 : P = RgX_unscale(RgX_recipspec_shallow(P+2, lgpol(P), k-1), gen_m1);
11472 1288 : setvarn(P,0); return P;
11473 : }
11474 : static int
11475 2842 : RgX_act_typ(GEN P, long k)
11476 : {
11477 2842 : switch(typ(P))
11478 : {
11479 35 : case t_RFRAC: return t_RFRAC;
11480 2807 : case t_POL:
11481 2807 : if (varn(P) == 0)
11482 : {
11483 2807 : long d = degpol(P);
11484 2807 : if (d > k-2) return t_RFRAC;
11485 2653 : if (d) return t_POL;
11486 : }
11487 : }
11488 1211 : return 0;
11489 : }
11490 : static GEN
11491 2576 : act_S(GEN P, long k)
11492 : {
11493 : GEN X;
11494 2576 : switch(RgX_act_typ(P, k))
11495 : {
11496 140 : case t_RFRAC:
11497 140 : X = gneg(pol_x(0));
11498 140 : return gmul(gsubst(P, 0, ginv(X)), gpowgs(X, k - 2));
11499 1288 : case t_POL: return RgX_act_S(P, k);
11500 : }
11501 1148 : return P;
11502 : }
11503 :
11504 : static GEN
11505 203 : AX_B(GEN M)
11506 203 : { GEN A = gcoeff(M,1,1), B = gcoeff(M,1,2); return deg1pol_shallow(A,B,0); }
11507 : static GEN
11508 203 : CX_D(GEN M)
11509 203 : { GEN C = gcoeff(M,2,1), D = gcoeff(M,2,2); return deg1pol_shallow(C,D,0); }
11510 :
11511 : /* P|_{2-k}M = (CX+D)^{k-2}P((AX+B)/(CX+D)) */
11512 : static GEN
11513 154 : RgX_act_gen(GEN P, GEN M, long k)
11514 : {
11515 154 : GEN S = gen_0, PCD, PAB;
11516 : long i;
11517 154 : PCD = gpowers(CX_D(M), k-2);
11518 154 : PAB = gpowers(AX_B(M), k-2);
11519 833 : for (i = 0; i <= k-2; i++)
11520 : {
11521 679 : GEN t = RgX_coeff(P, i);
11522 679 : if (!gequal0(t)) S = gadd(S, gmul(t, gmul(gel(PCD, k-i-1), gel(PAB, i+1))));
11523 : }
11524 154 : return S;
11525 : }
11526 : static GEN
11527 266 : act_GL2(GEN P, GEN M, long k)
11528 : {
11529 266 : switch(RgX_act_typ(P, k))
11530 : {
11531 49 : case t_RFRAC:
11532 : {
11533 49 : GEN AB = AX_B(M), CD = CX_D(M);
11534 49 : return gmul(gsubst(P, 0, gdiv(AB, CD)), gpowgs(CD, k - 2));
11535 : }
11536 154 : case t_POL: return RgX_act_gen(P, M, k);
11537 : }
11538 63 : return P;
11539 : }
11540 : static GEN
11541 7 : vecact_GL2(GEN x, GEN M, long k)
11542 21 : { pari_APPLY_same(act_GL2(gel(x,i), M, k)); }
11543 :
11544 : static GEN
11545 2863 : RgX_approx(GEN x, long bit)
11546 10731 : { pari_APPLY_pol_normalized(Rg_approx(gel(x,i),bit)); }
11547 :
11548 : static GEN normalizeapprox(GEN x, long bit);
11549 : static GEN
11550 2898 : normalizeapprox_i(GEN x, long bit)
11551 : {
11552 2898 : GEN D = gen_1;
11553 2954 : if (is_vec_t(typ(x))) pari_APPLY_same(normalizeapprox(gel(x,i), bit));
11554 2870 : if (typ(x) == t_RFRAC && varn(gel(x,2)) == 0) { D = gel(x,2); x = gel(x,1); }
11555 2870 : if (typ(x) != t_POL || varn(x) != 0) return gdiv(x, D);
11556 2863 : return gdiv(RgX_approx(x, bit), D);
11557 : }
11558 : static GEN
11559 56 : normalizeapprox(GEN x, long bit)
11560 : {
11561 56 : pari_sp av = avma;
11562 56 : return gc_upto(av, normalizeapprox_i(x, bit));
11563 : }
11564 :
11565 : /* make sure T is a t_POL in variable 0 */
11566 : static GEN
11567 2863 : toRgX0(GEN T)
11568 2863 : { return typ(T) == t_POL && varn(T) == 0? T: scalarpol_shallow(T,0); }
11569 :
11570 : /* integrate by summing nlim+1 terms of van [may be < lg(van)]
11571 : * van can be an expansion with vector coefficients
11572 : * \int_A^oo \sum_n van[n] * q^(n/w + al) * P(z-A) dz, q = e(z) */
11573 : static GEN
11574 945 : intAoo(GEN van, long nlim, GEN al, long w, GEN P, GEN A, long k, long prec)
11575 : {
11576 : GEN alw, P1, piI2A, q, S, van0;
11577 945 : long n, vz = varn(gel(P,2));
11578 :
11579 945 : if (nlim < 1) nlim = 1;
11580 945 : alw = gmulsg(w, al);
11581 945 : P1 = RgX_translate(P, gneg(A));
11582 945 : piI2A = gmul(PiI2n(1, prec), A);
11583 945 : q = gexp(gdivgu(piI2A, w), prec);
11584 945 : S = gen_0;
11585 121674 : for (n = nlim; n >= 1; n--)
11586 : {
11587 120729 : GEN t = gsubst(P1, vz, gdivsg(w, gaddsg(n, alw)));
11588 120729 : S = gadd(gmul(gel(van, n+1), t), gmul(q, S));
11589 : }
11590 945 : S = gmul(q, S);
11591 945 : van0 = gel(van, 1);
11592 945 : if (!gequal0(al))
11593 : {
11594 42 : S = gadd(S, gmul(gsubst(P1, vz, ginv(al)), van0));
11595 42 : S = gmul(S, gexp(gmul(piI2A, al), prec));
11596 : }
11597 903 : else if (!gequal0(van0))
11598 231 : S = gsub(S, gdivgu(gmul(van0, gpowgs(gsub(pol_x(0), A), k-1)), k-1));
11599 945 : if (is_vec_t(typ(S)))
11600 : {
11601 637 : long j, l = lg(S);
11602 3192 : for (j = 1; j < l; j++) gel(S,j) = toRgX0(gel(S,j));
11603 : }
11604 : else
11605 308 : S = toRgX0(S);
11606 945 : return gneg(S);
11607 : }
11608 :
11609 : /* \sum_{j <= k} X^j * (Y / (2I\pi))^{k+1-j} k! / j! */
11610 : static GEN
11611 259 : get_P(long k, long v, long prec)
11612 : {
11613 259 : GEN a, S = cgetg(k + 1, t_POL), u = invr(Pi2n(1, prec+EXTRAPREC64));
11614 259 : long j, K = k-2;
11615 259 : S[1] = evalsigne(1)|evalvarn(0); a = u;
11616 259 : gel(S,K+2) = monomial(mulcxpowIs(a,3), 1, v); /* j = K */
11617 1176 : for(j = K-1; j >= 0; j--)
11618 : {
11619 917 : a = mulrr(mulru(a,j+1), u);
11620 917 : gel(S,j+2) = monomial(mulcxpowIs(a,3*(K+1-j)), K+1-j, v);
11621 : }
11622 259 : return S;
11623 : }
11624 :
11625 : static GEN
11626 2555 : getw1w2(long N, GEN ga)
11627 2555 : { return mkvecsmall2(mfZC_width(N, gel(ga,1)),
11628 2555 : mfZC_width(N, gel(ga,2))); }
11629 :
11630 : static GEN
11631 147 : intAoowithvanall(GEN mf, GEN vanall, GEN P, GEN cosets, long bitprec)
11632 : {
11633 147 : GEN vvan = gel(vanall,1), vaw = gel(vanall,2), W1W2, resall;
11634 147 : long prec = nbits2prec(bitprec), N, k, lco, j;
11635 :
11636 147 : N = MF_get_N(mf); k = MF_get_k(mf);
11637 147 : lco = lg(cosets);
11638 147 : W1W2 = cgetg(lco, t_VEC); resall = cgetg(lco, t_VEC);
11639 2702 : for (j = 1; j < lco; j++) gel(W1W2,j) = getw1w2(N, gel(cosets, j));
11640 2702 : for (j = 1; j < lco; j++)
11641 : {
11642 2555 : GEN w1w2j = gel(W1W2,j), alj, M, VAN, RES, AR, Q;
11643 : long jq, c, w1, w2, w;
11644 2555 : if (!w1w2j) continue;
11645 637 : alj = gel(vaw,j);
11646 637 : w1 = w1w2j[1]; Q = cgetg(lco, t_VECSMALL);
11647 637 : w2 = w1w2j[2]; M = cgetg(lco, t_COL);
11648 8267 : for (c = 1, jq = j; jq < lco; jq++)
11649 : {
11650 7630 : GEN W = gel(W1W2, jq);
11651 7630 : if (jq == j || (W && gequal(W, w1w2j) && gequal(gel(vaw, jq), alj)))
11652 : {
11653 2555 : Q[c] = jq; gel(W1W2, jq) = NULL;
11654 2555 : gel(M, c) = gel(vvan, jq); c++;
11655 : }
11656 : }
11657 637 : setlg(M,c); VAN = shallowmatconcat(M);
11658 637 : AR = mkcomplex(gen_0, sqrtr_abs(divru(utor(w1, prec+EXTRAPREC64), w2)));
11659 637 : w = itos(gel(alj,2));
11660 637 : RES = intAoo(VAN, lg(VAN)-2, gel(alj,1),w, P, AR, k, prec);
11661 3192 : for (jq = 1; jq < c; jq++) gel(resall, Q[jq]) = gel(RES, jq);
11662 : }
11663 147 : return resall;
11664 : }
11665 :
11666 : GEN
11667 539 : mftobasisES(GEN mf, GEN F)
11668 : {
11669 539 : GEN v = mftobasis(mf, F, 0);
11670 532 : long nE = lg(MF_get_E(mf))-1;
11671 532 : return mkvec2(vecslice(v,1,nE), vecslice(v,nE+1,lg(v)-1));
11672 : }
11673 :
11674 : static long
11675 0 : wt1mulcond(GEN F, long D, long space)
11676 : {
11677 0 : GEN E = mfeisenstein_i(1, mfchartrivial(), get_mfchar(stoi(D))), mf;
11678 0 : F = mfmul(F, E);
11679 0 : mf = mfinit_Nkchi(mf_get_N(F), mf_get_k(F), mf_get_CHI(F), space, 0);
11680 0 : return mfconductor(mf, F);
11681 : }
11682 : static int
11683 7 : wt1newlevel(long N)
11684 : {
11685 7 : GEN P = gel(myfactoru(N),1);
11686 7 : long l = lg(P), i;
11687 14 : for (i = 1; i < l; i++)
11688 7 : if (!wt1empty(N/P[i])) return 0;
11689 7 : return 1;
11690 : }
11691 : long
11692 175 : mfconductor(GEN mf, GEN F)
11693 : {
11694 175 : pari_sp av = avma;
11695 : GEN gk;
11696 : long space, N, M;
11697 :
11698 175 : mf = checkMF(mf);
11699 175 : if (!checkmf_i(F)) pari_err_TYPE("mfconductor",F);
11700 175 : if (mfistrivial(F)) return 1;
11701 175 : space = MF_get_space(mf);
11702 175 : if (space == mf_NEW) return mf_get_N(F);
11703 175 : gk = MF_get_gk(mf);
11704 175 : if (isint1(gk))
11705 : {
11706 7 : N = mf_get_N(F);
11707 7 : if (!wt1newlevel(N))
11708 : {
11709 0 : long s = space_is_cusp(space)? mf_CUSP: mf_FULL;
11710 0 : N = ugcd(N, wt1mulcond(F,-3,s));
11711 0 : if (!wt1newlevel(N)) N = ugcd(N, wt1mulcond(F,-4,s));
11712 : }
11713 7 : return gc_long(av,N);
11714 : }
11715 168 : if (typ(gk) != t_INT)
11716 : {
11717 42 : F = mfmultheta(F);
11718 42 : mf = obj_checkbuild(mf, MF_MF2INIT, &mf2init); /* mf_FULL */
11719 : }
11720 168 : N = 1;
11721 168 : if (space_is_cusp(space))
11722 : {
11723 7 : F = mftobasis_i(mf, F);
11724 7 : if (typ(gk) != t_INT) F = vecslice(F, lg(MF_get_E(mf)), lg(F) - 1);
11725 : }
11726 : else
11727 : {
11728 161 : GEN EF = mftobasisES(mf, F), vE = gel(EF,1), B = MF_get_E(mf);
11729 161 : long i, l = lg(B);
11730 1267 : for (i = 1; i < l; i++)
11731 1106 : if (!gequal0(gel(vE,i))) N = ulcm(N, mf_get_N(gel(B, i)));
11732 161 : F = gel(EF,2);
11733 : }
11734 168 : (void)mftonew_i(mf, F, &M); /* M = conductor of cuspidal part */
11735 168 : return gc_long(av, ulcm(M, N));
11736 : }
11737 :
11738 : static GEN
11739 1463 : fs_get_MF(GEN fs) { return gel(fs,1); }
11740 : static GEN
11741 847 : fs_get_vES(GEN fs) { return gel(fs,2); }
11742 : static GEN
11743 1596 : fs_get_pols(GEN fs) { return gel(fs,3); }
11744 : static GEN
11745 2191 : fs_get_cosets(GEN fs) { return gel(fs,4); }
11746 : static long
11747 630 : fs_get_bitprec(GEN fs) { return itou(gel(fs,5)); }
11748 : static GEN
11749 1246 : fs_get_vE(GEN fs) { return gel(fs,6); }
11750 : static GEN
11751 70 : fs_get_EF(GEN fs) { return gel(fs,7); }
11752 : static GEN
11753 1890 : fs_get_expan(GEN fs) { return gel(fs,8); }
11754 : static GEN
11755 28 : fs_set_expan(GEN fs, GEN vanall)
11756 28 : { GEN f = shallowcopy(fs); gel(f,8) = vanall; return f; }
11757 : static int
11758 49 : mfs_checkmf(GEN fs, GEN mf)
11759 49 : { GEN mfF = fs_get_MF(fs); return gequal(gel(mfF,1), gel(mf,1)); }
11760 : static long
11761 798 : checkfs_i(GEN v)
11762 798 : { return typ(v) == t_VEC && lg(v) == 9 && checkMF_i(fs_get_MF(v))
11763 567 : && typ(fs_get_vES(v)) == t_VEC
11764 567 : && typ(fs_get_pols(v)) == t_VEC
11765 567 : && typ(fs_get_cosets(v)) == t_VEC
11766 567 : && typ(fs_get_vE(v)) == t_VEC
11767 567 : && lg(fs_get_pols(v)) == lg(fs_get_cosets(v))
11768 567 : && typ(fs_get_expan(v)) == t_VEC
11769 567 : && lg(fs_get_expan(v)) == 3
11770 567 : && lg(gel(fs_get_expan(v), 1)) == lg(fs_get_cosets(v))
11771 1596 : && typ(gel(v,5)) == t_INT; }
11772 : GEN
11773 18858 : checkMF_i(GEN mf)
11774 : {
11775 18858 : long l = lg(mf);
11776 : GEN v;
11777 18858 : if (typ(mf) != t_VEC) return NULL;
11778 18830 : if (l == 9) return checkMF_i(fs_get_MF(mf));
11779 18830 : if (l != 7) return NULL;
11780 7623 : v = gel(mf,1);
11781 7623 : if (typ(v) != t_VEC || lg(v) != 5) return NULL;
11782 7623 : return (typ(gel(v,1)) == t_INT
11783 7623 : && typ(gmul2n(gel(v,2), 1)) == t_INT
11784 7623 : && typ(gel(v,3)) == t_VEC
11785 15246 : && typ(gel(v,4)) == t_INT)? mf: NULL; }
11786 : GEN
11787 3941 : checkMF(GEN T)
11788 : {
11789 3941 : GEN mf = checkMF_i(T);
11790 3941 : if (!mf) pari_err_TYPE("checkMF [please use mfinit]", T);
11791 3941 : return mf;
11792 : }
11793 :
11794 : /* c,d >= 0; c * Nc = N, find coset whose image in P1(Z/NZ) ~ (c, d + k(N/c)) */
11795 : static GEN
11796 11963 : coset_complete(long c, long d, long Nc)
11797 : {
11798 : long a, b;
11799 13307 : while (ugcd(c, d) > 1) d += Nc;
11800 11963 : (void)cbezout(c, d, &b, &a);
11801 11963 : return mkmat22s(a, -b, c, d);
11802 : }
11803 : /* right cosets of $\G_0(N)$: $\G=\bigsqcup_j \G_0(N)\ga_j$. */
11804 : /* We choose them with c\mid N and d mod N/c, not the reverse */
11805 : GEN
11806 168 : mfcosets(GEN gN)
11807 : {
11808 168 : pari_sp av = avma;
11809 : GEN V, D, mf;
11810 168 : long l, i, ct, N = 0;
11811 168 : if (typ(gN) == t_INT) N = itos(gN);
11812 14 : else if ((mf = checkMF_i(gN))) N = MF_get_N(mf);
11813 7 : else pari_err_TYPE("mfcosets", gN);
11814 161 : if (N <= 0) pari_err_DOMAIN("mfcosets", "N", "<=", gen_0, stoi(N));
11815 161 : V = cgetg(mypsiu(N) + 1, t_VEC);
11816 161 : D = mydivisorsu(N); l = lg(D);
11817 588 : for (i = ct = 1; i < l; i++)
11818 : {
11819 427 : long d, c = D[i], Nc = D[l-i], e = ugcd(Nc, c);
11820 3332 : for (d = 0; d < Nc; d++)
11821 2905 : if (ugcd(d,e) == 1) gel(V, ct++) = coset_complete(c, d, Nc);
11822 : }
11823 161 : return gc_GEN(av, V);
11824 : }
11825 : static int
11826 35469 : cmp_coset(void *E, GEN A, GEN B)
11827 : {
11828 35469 : ulong N = (ulong)E, Nc, c = itou(gcoeff(A,2,1));
11829 35469 : int r = cmpuu(c, itou(gcoeff(B,2,1)));
11830 35469 : if (r) return r;
11831 30660 : Nc = N / c;
11832 30660 : return cmpuu(umodiu(gcoeff(A,2,2), Nc), umodiu(gcoeff(B,2,2), Nc));
11833 : }
11834 : /* M in SL_2(Z) */
11835 : static long
11836 9198 : mftocoset_i(ulong N, GEN M, GEN cosets)
11837 : {
11838 9198 : pari_sp av = avma;
11839 9198 : long A = itos(gcoeff(M,1,1)), c, u, v, Nc, i;
11840 9198 : long C = itos(gcoeff(M,2,1)), D = itos(gcoeff(M,2,2));
11841 : GEN ga;
11842 9198 : c = cbezout(N*A, C, &u, &v); Nc = N/c;
11843 9198 : ga = coset_complete(c, umodsu(v*D, Nc), Nc);
11844 9198 : i = gen_search(cosets, ga, (void*)N, &cmp_coset);
11845 9198 : if (i < 0) pari_err_BUG("mftocoset [no coset found]");
11846 9198 : return gc_long(av,i);
11847 : }
11848 : /* (U * V^(-1))[2,2] mod N, assuming V in SL2(Z) */
11849 : static long
11850 9177 : SL2_div_D(ulong N, GEN U, GEN V)
11851 : {
11852 9177 : long c = umodiu(gcoeff(U,2,1), N), d = umodiu(gcoeff(U,2,2), N);
11853 9177 : long a2 = umodiu(gcoeff(V,1,1), N), b2 = umodiu(gcoeff(V,1,2), N);
11854 9177 : return (a2*d - b2*c) % (long)N;
11855 : }
11856 : static long
11857 9177 : mftocoset_iD(ulong N, GEN M, GEN cosets, long *D)
11858 : {
11859 9177 : long i = mftocoset_i(N, M, cosets);
11860 9177 : *D = SL2_div_D(N, M, gel(cosets,i)); return i;
11861 : }
11862 : GEN
11863 7 : mftocoset(ulong N, GEN M, GEN cosets)
11864 : {
11865 : long i;
11866 7 : if (!check_SL2Z(M)) pari_err_TYPE("mftocoset",M);
11867 7 : i = mftocoset_i(N, M, cosets);
11868 7 : retmkvec2(gdiv(M,gel(cosets,i)), utoipos(i));
11869 : }
11870 :
11871 : static long
11872 2555 : getnlim2(long N, long w1, long w2, long nlim, long k, long bitprec)
11873 : {
11874 2555 : if (w2 == N) return nlim;
11875 483 : return mfperiod_prelim_double(1./sqrt((double)w1*w2), k, bitprec + 32);
11876 : }
11877 :
11878 : /* g * S, g 2x2 */
11879 : static GEN
11880 1337 : ZM_mulS(GEN g)
11881 1337 : { return mkmat2(gel(g,2), ZC_neg(gel(g,1))); }
11882 : /* g * T, g 2x2 */
11883 : static GEN
11884 4634 : ZM_mulT(GEN g)
11885 4634 : { return mkmat2(gel(g,1), ZC_add(gel(g,2), gel(g,1))); }
11886 : /* g * T^(-1), g 2x2 */
11887 : static GEN
11888 2352 : ZM_mulTi(GEN g)
11889 2352 : { return mkmat2(gel(g,1), ZC_sub(gel(g,2), gel(g,1))); }
11890 :
11891 : /* Compute all slashexpansions for all cosets */
11892 : static GEN
11893 175 : mfgaexpansionall(GEN mf, GEN FE, GEN cosets, double height, long prec)
11894 : {
11895 175 : GEN CHI = MF_get_CHI(mf), vres, vresaw;
11896 175 : long lco, j, k = MF_get_k(mf), N = MF_get_N(mf), bitprec = prec2nbits(prec);
11897 :
11898 175 : lco = lg(cosets);
11899 175 : vres = const_vec(lco-1, NULL);
11900 175 : vresaw = cgetg(lco, t_VEC);
11901 2912 : for (j = 1; j < lco; j++) if (!gel(vres,j))
11902 : {
11903 455 : GEN ga = gel(cosets, j), van, aw, al, z, gai;
11904 455 : long w1 = mfZC_width(N, gel(ga,1));
11905 455 : long w2 = mfZC_width(N, gel(ga,2));
11906 : long nlim, nlim2, daw, da, na, i;
11907 455 : double sqNinvdbl = height ? height/w1 : 1./sqrt((double)w1*N);
11908 455 : nlim = mfperiod_prelim_double(sqNinvdbl, k, bitprec + 32);
11909 455 : van = mfslashexpansion(mf, FE, ga, nlim, 0, &aw, prec + EXTRAPREC64);
11910 455 : van = vanembed(gel(FE, 1), van, prec + EXTRAPREC64);
11911 455 : al = gel(aw, 1);
11912 455 : nlim2 = height? nlim: getnlim2(N, w1, w2, nlim, k, bitprec);
11913 455 : gel(vres, j) = vecslice(van, 1, nlim2 + 1);
11914 455 : gel(vresaw, j) = aw;
11915 455 : Qtoss(al, &na, &da); daw = da*w1;
11916 455 : z = rootsof1powinit(1, daw, prec + EXTRAPREC64);
11917 455 : gai = ga;
11918 2737 : for (i = 1; i < w1; i++)
11919 : {
11920 : GEN V, coe;
11921 2282 : long Di, n, ind, w2, s = ((i*na) % da) * w1, t = i*da;
11922 2282 : gai = ZM_mulT(gai);
11923 2282 : ind = mftocoset_iD(N, gai, cosets, &Di);
11924 2282 : w2 = mfZC_width(N, gel(gel(cosets,ind), 2));
11925 2282 : nlim2 = height? nlim: getnlim2(N, w1, w2, nlim, k, bitprec);
11926 2282 : gel(vresaw, ind) = aw;
11927 2282 : V = cgetg(nlim2 + 2, t_VEC);
11928 909034 : for (n = 0; n <= nlim2; n++, s = Fl_add(s, t, daw))
11929 906752 : gel(V, n+1) = gmul(gel(van, n+1), rootsof1pow(z, s));
11930 2282 : coe = mfcharcxeval(CHI, Di, prec + EXTRAPREC64);
11931 2282 : if (!gequal1(coe)) V = RgV_Rg_mul(V, conj_i(coe));
11932 2282 : gel(vres, ind) = V;
11933 : }
11934 : }
11935 175 : return mkvec2(vres, vresaw);
11936 : }
11937 :
11938 : /* Compute all period pols of F|_k\ga_j, vF = mftobasis(F_S) */
11939 : static GEN
11940 168 : mfperiodpols_i(GEN mf, GEN FE, GEN cosets, GEN *pvan, long bit)
11941 : {
11942 168 : long N, i, prec = nbits2prec(bit), k = MF_get_k(mf);
11943 168 : GEN vP, P, CHI, intall = gen_0;
11944 :
11945 168 : *pvan = gen_0;
11946 168 : if (k == 0 && gequal0(gel(FE,2)))
11947 0 : return cosets? const_vec(lg(cosets)-1, pol_0(0)): pol_0(0);
11948 168 : N = MF_get_N(mf);
11949 168 : CHI = MF_get_CHI(mf);
11950 168 : P = get_P(k, fetch_var(), prec);
11951 168 : if (!cosets)
11952 : { /* ga = id */
11953 21 : long nlim, PREC = prec + EXTRAPREC64;
11954 21 : GEN F = gel(FE,1), sqNinv = invr(sqrtr_abs(utor(N, PREC))); /* A/w */
11955 : GEN AR, v, van, T1, T2;
11956 :
11957 21 : nlim = mfperiod_prelim(sqNinv, k, bit + 32);
11958 : /* F|id: al = 0, w = 1 */
11959 21 : v = mfcoefs_i(F, nlim, 1);
11960 21 : van = vanembed(F, v, PREC);
11961 21 : AR = mkcomplex(gen_0, sqNinv);
11962 21 : T1 = intAoo(van, nlim, gen_0,1, P, AR, k, prec);
11963 21 : if (N == 1) T2 = T1;
11964 : else
11965 : { /* F|S: al = 0, w = N */
11966 7 : v = mfgaexpansion(mf, FE, mkS(), nlim, PREC);
11967 7 : van = vanembed(F, gel(v,3), PREC);
11968 7 : AR = mkcomplex(gen_0, mulur(N,sqNinv));
11969 7 : T2 = intAoo(van, nlim, gen_0,N, P, AR, k, prec);
11970 : }
11971 21 : T1 = gsub(T1, act_S(T2, k));
11972 21 : T1 = normalizeapprox_i(T1, bit-20);
11973 21 : vP = gprec_wtrunc(T1, prec);
11974 : }
11975 : else
11976 : {
11977 147 : long lco = lg(cosets);
11978 147 : GEN vanall = mfgaexpansionall(mf, FE, cosets, 0, prec);
11979 147 : *pvan = vanall;
11980 147 : intall = intAoowithvanall(mf, vanall, P, cosets, bit);
11981 147 : vP = const_vec(lco-1, NULL);
11982 2702 : for (i = 1; i < lco; i++)
11983 : {
11984 2555 : GEN P, P1, P2, c, ga = gel(cosets, i);
11985 : long iS, DS;
11986 2646 : if (gel(vP,i)) continue;
11987 1323 : P1 = gel(intall, i);
11988 1323 : iS = mftocoset_iD(N, ZM_mulS(ga), cosets, &DS);
11989 1323 : c = mfcharcxeval(CHI, DS, prec + EXTRAPREC64);
11990 1323 : P2 = gel(intall, iS);
11991 :
11992 1323 : P = act_S(isint1(c)? P2: gmul(c, P2), k);
11993 1323 : P = normalizeapprox_i(gsub(P1, P), bit-20);
11994 1323 : gel(vP,i) = gprec_wtrunc(P, prec);
11995 1323 : if (iS == i) continue;
11996 :
11997 1232 : P = act_S(isint1(c)? P1: gmul(conj_i(c), P1), k);
11998 1232 : if (!odd(k)) P = gneg(P);
11999 1232 : P = normalizeapprox_i(gadd(P, P2), bit-20);
12000 1232 : gel(vP,iS) = gprec_wtrunc(P, prec);
12001 : }
12002 : }
12003 168 : delete_var(); return vP;
12004 : }
12005 :
12006 : /* when cosets = NULL, return a "fake" symbol containing only fs(oo->0) */
12007 : static GEN
12008 168 : mfsymbol_i(GEN mf, GEN F, GEN cosets, long bit)
12009 : {
12010 168 : GEN FE, van, vP, vE, Mvecj, vES = mftobasisES(mf,F);
12011 168 : long precnew, prec = nbits2prec(bit), k = MF_get_k(mf);
12012 168 : vE = mfgetembed(F, prec);
12013 168 : Mvecj = obj_checkbuild(mf, MF_EISENSPACE, &mfeisensteinspaceinit);
12014 168 : if (lg(Mvecj) >= 5) precnew = prec;
12015 : else
12016 : {
12017 14 : long N = MF_get_N(mf), n = mfperiod_prelim_double(1/(double)N, k, bit + 32);
12018 14 : precnew = prec + inveis_extraprec(N, mkS(), Mvecj, n);
12019 : }
12020 168 : FE = mkcol2(F, mf_eisendec(mf,F,precnew));
12021 168 : vP = mfperiodpols_i(mf, FE, cosets, &van, bit);
12022 168 : return mkvecn(8, mf, vES, vP, cosets, utoi(bit), vE, FE, van);
12023 : }
12024 :
12025 : static GEN
12026 56 : fs2_get_cusps(GEN f) { return gel(f,3); }
12027 : static GEN
12028 56 : fs2_get_MF(GEN f) { return gel(f,1); }
12029 : static GEN
12030 56 : fs2_get_W(GEN f) { return gel(f,2); }
12031 : static GEN
12032 56 : fs2_get_F(GEN f) { return gel(f,4); }
12033 : static long
12034 0 : fs2_get_bitprec(GEN f) { return itou(gel(f,5)); }
12035 : static GEN
12036 56 : fs2_get_al0(GEN f) { return gel(f,6); }
12037 : static GEN
12038 21 : fs2_get_den(GEN f) { return gel(f,7); }
12039 : static int
12040 210 : checkfs2_i(GEN f)
12041 : {
12042 : GEN W, C, F, al0;
12043 : long l;
12044 210 : if (typ(f) != t_VEC || lg(f) != 8 || typ(gel(f,5)) != t_INT) return 0;
12045 35 : C = fs2_get_cusps(f); l = lg(C);
12046 35 : W = fs2_get_W(f);
12047 35 : F = fs2_get_F(f);
12048 35 : al0 = fs2_get_al0(f);
12049 35 : return checkMF_i(fs2_get_MF(f))
12050 35 : && typ(W) == t_VEC && typ(F) == t_VEC && typ(al0) == t_VECSMALL
12051 70 : && lg(W) == l && lg(F) == l && lg(al0) == l;
12052 : }
12053 : static GEN fs2_init(GEN mf, GEN F, long bit);
12054 : GEN
12055 175 : mfsymbol(GEN mf, GEN F, long bit)
12056 : {
12057 175 : pari_sp av = avma;
12058 175 : GEN cosets = NULL;
12059 175 : if (!F)
12060 : {
12061 35 : F = mf;
12062 35 : if (!checkmf_i(F)) pari_err_TYPE("mfsymbol", F);
12063 35 : mf = mfinit_i(F, mf_FULL);
12064 : }
12065 140 : else if (!checkmf_i(F)) pari_err_TYPE("mfsymbol", F);
12066 175 : if (checkfs2_i(mf)) return fs2_init(mf, F, bit);
12067 175 : if (checkfs_i(mf))
12068 : {
12069 0 : cosets = fs_get_cosets(mf);
12070 0 : mf = fs_get_MF(mf);
12071 : }
12072 175 : else if (checkMF_i(mf))
12073 : {
12074 175 : GEN gk = MF_get_gk(mf);
12075 175 : if (typ(gk) != t_INT || equali1(gk)) return fs2_init(mf, F, bit);
12076 154 : if (signe(gk) <= 0) pari_err_TYPE("mfsymbol [k <= 0]", mf);
12077 147 : cosets = mfcosets(MF_get_gN(mf));
12078 : }
12079 0 : else pari_err_TYPE("mfsymbol",mf);
12080 147 : return gc_GEN(av, mfsymbol_i(mf, F, cosets, bit));
12081 : }
12082 :
12083 : static GEN
12084 14 : RgX_by_parity(GEN P, long odd)
12085 : {
12086 14 : long i, l = lg(P);
12087 : GEN Q;
12088 14 : if (l < 4) return odd ? pol_x(0): P;
12089 14 : Q = cgetg(l, t_POL); Q[1] = P[1];
12090 91 : for (i = odd? 2: 3; i < l; i += 2) gel(Q,i) = gen_0;
12091 91 : for (i = odd? 3: 2; i < l; i += 2) gel(Q,i) = gel(P,i);
12092 14 : return normalizepol_lg(Q, l);
12093 : }
12094 : /* flag 0: period polynomial of F, >0 or <0 with corresponding parity */
12095 : GEN
12096 35 : mfperiodpol(GEN mf0, GEN F, long flag, long bit)
12097 : {
12098 35 : pari_sp av = avma;
12099 35 : GEN pol, mf = checkMF_i(mf0);
12100 35 : if (!mf) pari_err_TYPE("mfperiodpol",mf0);
12101 35 : if (checkfs_i(F))
12102 : {
12103 14 : GEN mfpols = fs_get_pols(F);
12104 14 : if (!mfs_checkmf(F, mf)) pari_err_TYPE("mfperiodpol [different mf]",F);
12105 14 : pol = veclast(mfpols); /* trivial coset is last */
12106 : }
12107 : else
12108 : {
12109 21 : GEN gk = MF_get_gk(mf);
12110 21 : if (typ(gk) != t_INT) pari_err_TYPE("mfperiodpol [half-integral k]", mf);
12111 21 : if (equali1(gk)) pari_err_TYPE("mfperiodpol [k = 1]", mf);
12112 21 : F = mfsymbol_i(mf, F, NULL, bit);
12113 21 : pol = fs_get_pols(F);
12114 : }
12115 35 : if (flag) pol = RgX_by_parity(pol, flag < 0);
12116 35 : return gc_GEN(av, RgX_embedall(pol, fs_get_vE(F)));
12117 : }
12118 :
12119 : static int
12120 35 : mfs_iscusp(GEN mfs) { return gequal0(gmael(mfs,2,1)); }
12121 : /* given cusps s1 and s2 (rationals or oo)
12122 : * compute $\int_{s1}^{s2}(X-\tau)^{k-2}F|_k\ga_j(\tau)\,d\tau$ */
12123 : /* If flag = 1, do not give an error message if divergent, but
12124 : give the rational function as result. */
12125 :
12126 : static GEN
12127 126 : col2cusp(GEN v)
12128 : {
12129 : GEN A, C;
12130 126 : if (lg(v) != 3 || !RgV_is_ZV(v)) pari_err_TYPE("col2cusp",v);
12131 126 : A = gel(v,1);
12132 126 : C = gel(v,2);
12133 126 : if (gequal0(C))
12134 : {
12135 0 : if (gequal0(A)) pari_err_TYPE("mfsymboleval", mkvec2(A, C));
12136 0 : return mkoo();
12137 : }
12138 126 : return gdiv(A, C);
12139 : }
12140 : /* g.oo */
12141 : static GEN
12142 112 : mat2cusp(GEN g) { return col2cusp(gel(g,1)); }
12143 :
12144 : static GEN
12145 7 : pathmattovec(GEN path)
12146 7 : { return mkvec2(col2cusp(gel(path,1)), col2cusp(gel(path,2))); }
12147 :
12148 : static void
12149 546 : get_mf_F(GEN fs, GEN *mf, GEN *F)
12150 : {
12151 546 : if (lg(fs) == 3) { *mf = gel(fs,1); *F = gel(fs,2); }
12152 546 : else { *mf = fs_get_MF(fs); *F = NULL; }
12153 546 : }
12154 : static GEN
12155 189 : mfgetvan(GEN fs, GEN ga, GEN *pal, long nlim, long prec)
12156 : {
12157 : GEN van, mf, F, W;
12158 : long PREC;
12159 189 : get_mf_F(fs, &mf, &F);
12160 189 : if (!F)
12161 : {
12162 189 : GEN vanall = fs_get_expan(fs), cosets = fs_get_cosets(fs);
12163 189 : long D, jga = mftocoset_iD(MF_get_N(mf), ga, cosets, &D);
12164 189 : van = gmael(vanall, 1, jga);
12165 189 : W = gmael(vanall, 2, jga);
12166 189 : if (lg(van) >= nlim + 2)
12167 : {
12168 182 : GEN z = mfcharcxeval(MF_get_CHI(mf), D, prec);
12169 182 : if (!gequal1(z)) van = RgV_Rg_mul(van, z);
12170 182 : *pal = gel(W,1); return van;
12171 : }
12172 7 : F = gel(fs_get_EF(fs), 1);
12173 : }
12174 7 : PREC = prec + EXTRAPREC64;
12175 7 : van = mfslashexpansion(mf, F, ga, nlim, 0, &W, PREC);
12176 7 : van = vanembed(F, van, PREC);
12177 7 : *pal = gel(W,1); return van;
12178 : }
12179 : /* Computation of int_A^oo (f | ga)(t)(X-t)^{k-2} dt, assuming convergence;
12180 : * fs is either a symbol or a triple [mf,F,bitprec]. A != oo and im(A) > 0 */
12181 : static GEN
12182 77 : intAoo0(GEN fs, GEN A, GEN ga, GEN P, long bit)
12183 : {
12184 77 : long nlim, N, k, w, prec = nbits2prec(bit);
12185 : GEN van, mf, F, al;
12186 77 : get_mf_F(fs, &mf,&F); N = MF_get_N(mf); k = MF_get_k(mf);
12187 77 : w = mfZC_width(N, gel(ga,1));
12188 77 : nlim = mfperiod_prelim(gdivgu(imag_i(A), w), k, bit + 32);
12189 77 : van = mfgetvan(fs, ga, &al, nlim, prec);
12190 77 : return intAoo(van, nlim, al,w, P, A, k, prec);
12191 : }
12192 :
12193 : /* fs symbol, naive summation, A != oo, im(A) > 0 and B = oo or im(B) > 0 */
12194 : static GEN
12195 112 : mfsymboleval_direct(GEN fs, GEN path, GEN ga, GEN P)
12196 : {
12197 112 : GEN A, B, van, S, al, mf = fs_get_MF(fs);
12198 112 : long w, nlimA, nlimB = 0, N = MF_get_N(mf), k = MF_get_k(mf);
12199 112 : long bit = fs_get_bitprec(fs), prec = nbits2prec(bit);
12200 :
12201 112 : A = gel(path, 1);
12202 112 : B = gel(path, 2); if (typ(B) == t_INFINITY) B = NULL;
12203 112 : w = mfZC_width(N, gel(ga,1));
12204 112 : nlimA = mfperiod_prelim(gdivgu(imag_i(A),w), k, bit + 32);
12205 112 : if (B) nlimB = mfperiod_prelim(gdivgu(imag_i(B),w), k, bit + 32);
12206 112 : van = mfgetvan(fs, ga, &al, maxss(nlimA,nlimB), prec);
12207 112 : S = intAoo(van, nlimA, al,w, P, A, k, prec);
12208 112 : if (B) S = gsub(S, intAoo(van, nlimB, al,w, P, B, k, prec));
12209 112 : return RgX_embedall(S, fs_get_vE(fs));
12210 : }
12211 :
12212 : /* Computation of int_A^oo (f | ga)(t)(X-t)^{k-2} dt, assuming convergence;
12213 : * fs is either a symbol or a pair [mf,F]. */
12214 : static GEN
12215 77 : mfsymbolevalpartial(GEN fs, GEN A, GEN ga, long bit)
12216 : {
12217 : GEN Y, F, S, P, mf;
12218 77 : long N, k, w, prec = nbits2prec(bit);
12219 :
12220 77 : get_mf_F(fs, &mf, &F);
12221 77 : N = MF_get_N(mf); w = mfZC_width(N, gel(ga,1));
12222 77 : k = MF_get_k(mf);
12223 77 : Y = gdivgu(imag_i(A), w);
12224 77 : P = get_P(k, fetch_var(), prec);
12225 77 : if (lg(fs) != 3 && gtodouble(Y)*(2*N) < 1)
12226 21 : { /* true symbol + low imaginary part: use GL_2 action to improve */
12227 21 : GEN U, ga2, czd, A2 = cxredga0N(N, A, &U, &czd, 1);
12228 21 : GEN vE = fs_get_vE(fs);
12229 21 : ga2 = ZM_mul(ga, ZM_inv(U, NULL));
12230 21 : S = RgX_embedall(intAoo0(fs, A2, ga2, P, bit), vE);
12231 21 : S = gsub(S, mfsymboleval(fs, mkvec2(mat2cusp(U), mkoo()), ga2, bit));
12232 21 : S = typ(S) == t_VEC? vecact_GL2(S, U, k): act_GL2(S, U, k);
12233 : }
12234 : else
12235 : {
12236 56 : S = intAoo0(fs, A, ga, P, bit);
12237 56 : S = RgX_embedall(S, F? mfgetembed(F,prec): fs_get_vE(fs));
12238 : }
12239 77 : delete_var(); return normalizeapprox_i(S, bit-20);
12240 : }
12241 :
12242 : static GEN
12243 42 : actal(GEN x, GEN vabd)
12244 : {
12245 42 : if (typ(x) == t_INFINITY) return x;
12246 35 : return gdiv(gadd(gmul(gel(vabd,1), x), gel(vabd,2)), gel(vabd,3));
12247 : }
12248 :
12249 : static GEN
12250 14 : unact(GEN z, GEN vabd, long k, long prec)
12251 : {
12252 14 : GEN res = gsubst(z, 0, actal(pol_x(0), vabd));
12253 14 : GEN CO = gpow(gdiv(gel(vabd,3), gel(vabd,1)), sstoQ(k-2, 2), prec);
12254 14 : return gmul(CO, res);
12255 : }
12256 :
12257 : GEN
12258 210 : mfsymboleval(GEN fs, GEN path, GEN ga, long bitprec)
12259 : {
12260 210 : pari_sp av = avma;
12261 210 : GEN tau, V, LM, S, CHI, mfpols, cosets, al, be, mf, F, vabd = NULL;
12262 : long D, B, m, u, v, a, b, c, d, j, k, N, prec, tsc1, tsc2;
12263 :
12264 210 : if (checkfs_i(fs))
12265 : {
12266 203 : get_mf_F(fs, &mf, &F);
12267 203 : bitprec = minss(bitprec, fs_get_bitprec(fs));
12268 : }
12269 : else
12270 : {
12271 7 : if (checkfs2_i(fs)) pari_err_TYPE("mfsymboleval [need integral k > 1]",fs);
12272 0 : if (typ(fs) != t_VEC || lg(fs) != 3) pari_err_TYPE("mfsymboleval",fs);
12273 0 : get_mf_F(fs, &mf, &F);
12274 0 : mf = checkMF_i(mf);
12275 0 : if (!mf ||!checkmf_i(F)) pari_err_TYPE("mfsymboleval",fs);
12276 : }
12277 203 : if (lg(path) != 3) pari_err_TYPE("mfsymboleval",path);
12278 203 : if (typ(path) == t_MAT) path = pathmattovec(path);
12279 203 : if (typ(path) != t_VEC) pari_err_TYPE("mfsymboleval",path);
12280 203 : al = gel(path,1);
12281 203 : be = gel(path,2);
12282 203 : ga = ga? GL2toSL2(ga, &vabd): matid(2);
12283 203 : if (vabd)
12284 : {
12285 14 : al = actal(al, vabd);
12286 14 : be = actal(be, vabd); path = mkvec2(al, be);
12287 : }
12288 203 : tsc1 = cusp_AC(al, &a, &c);
12289 203 : tsc2 = cusp_AC(be, &b, &d);
12290 203 : prec = nbits2prec(bitprec);
12291 203 : k = MF_get_k(mf);
12292 203 : if (!tsc1)
12293 : {
12294 42 : GEN z2, z = mfsymbolevalpartial(fs, al, ga, bitprec);
12295 42 : if (tsc2)
12296 28 : z2 = d? mfsymboleval(fs, mkvec2(be, mkoo()), ga, bitprec): gen_0;
12297 : else
12298 14 : z2 = mfsymbolevalpartial(fs, be, ga, bitprec);
12299 42 : z = gsub(z, z2);
12300 42 : if (vabd) z = unact(z, vabd, k, prec);
12301 42 : return gc_upto(av, normalizeapprox_i(z, bitprec-20));
12302 : }
12303 161 : else if (!tsc2)
12304 : {
12305 21 : GEN z = mfsymbolevalpartial(fs, be, ga, bitprec);
12306 21 : if (c) z = gsub(mfsymboleval(fs, mkvec2(al, mkoo()), ga, bitprec), z);
12307 7 : else z = gneg(z);
12308 21 : if (vabd) z = unact(z, vabd, k, prec);
12309 21 : return gc_upto(av, normalizeapprox_i(z, bitprec-20));
12310 : }
12311 140 : if (F) pari_err_TYPE("mfsymboleval", fs);
12312 140 : D = a*d-b*c;
12313 140 : if (!D) { set_avma(av); return RgX_embedall(gen_0, fs_get_vE(fs)); }
12314 126 : mfpols = fs_get_pols(fs);
12315 126 : cosets = fs_get_cosets(fs);
12316 126 : CHI = MF_get_CHI(mf); N = MF_get_N(mf);
12317 126 : cbezout(a, c, &u, &v); B = u*b + v*d; tau = mkmat22s(a, -v, c, u);
12318 126 : V = gcf(sstoQ(B, D));
12319 126 : LM = shallowconcat(mkcol2(gen_1, gen_0), contfracpnqn(V, lg(V)));
12320 126 : S = gen_0; m = lg(LM) - 2;
12321 364 : for (j = 0; j < m; j++)
12322 : {
12323 : GEN M, P;
12324 : long D, iN;
12325 238 : M = mkmat2(gel(LM, j+2), gel(LM, j+1));
12326 238 : if (!odd(j)) gel(M,1) = ZC_neg(gel(M,1));
12327 238 : M = ZM_mul(tau, M);
12328 238 : iN = mftocoset_iD(N, ZM_mul(ga, M), cosets, &D);
12329 238 : P = gmul(gel(mfpols,iN), mfcharcxeval(CHI,D,prec));
12330 238 : S = gadd(S, act_GL2(P, ZM_inv(M, NULL), k));
12331 : }
12332 126 : if (typ(S) == t_RFRAC)
12333 : {
12334 : GEN R, S1, co;
12335 21 : gel(S,2) = primitive_part(gel(S,2), &co);
12336 21 : if (co) gel(S,1) = gdiv(gel(S,1), gtofp(co,prec));
12337 21 : S1 = poldivrem(gel(S,1), gel(S,2), &R);
12338 21 : if (gexpo(R) < -bitprec + 20) S = S1;
12339 : }
12340 126 : if (vabd) S = unact(S, vabd, k, prec);
12341 126 : S = RgX_embedall(S, fs_get_vE(fs));
12342 126 : return gc_upto(av, normalizeapprox_i(S, bitprec-20));
12343 : }
12344 :
12345 : /* v a scalar or t_POL; set *pw = a if expo(a) > E for some coefficient;
12346 : * take the 'a' with largest exponent */
12347 : static void
12348 5740 : improve(GEN v, GEN *pw, long *E)
12349 : {
12350 5740 : if (typ(v) != t_POL)
12351 : {
12352 4270 : long e = gexpo(v);
12353 4270 : if (e > *E) { *E = e; *pw = v; }
12354 : }
12355 : else
12356 : {
12357 1470 : long j, l = lg(v);
12358 5740 : for (j = 2; j < l; j++) improve(gel(v,j), pw, E);
12359 : }
12360 5740 : }
12361 : static GEN
12362 518 : polabstorel(GEN rnfeq, GEN x)
12363 : {
12364 518 : if (typ(x) != t_POL) return x;
12365 3500 : pari_APPLY_pol_normalized(eltabstorel(rnfeq, gel(x,i)));
12366 : }
12367 : static GEN
12368 1519 : bestapprnfrel(GEN x, GEN polabs, GEN roabs, GEN rnfeq, long prec)
12369 : {
12370 1519 : x = bestapprnf(x, polabs, roabs, prec);
12371 1519 : if (rnfeq) x = polabstorel(rnfeq, liftpol_shallow(x));
12372 1519 : return x;
12373 : }
12374 : /* v vector of polynomials polynomial in C[X] (possibly scalar).
12375 : * Set *w = coeff with largest exponent and return T / *w, rationalized */
12376 : static GEN
12377 98 : normal(GEN v, GEN polabs, GEN roabs, GEN rnfeq, GEN *w, long prec)
12378 : {
12379 98 : long i, l = lg(v), E = -(long)HIGHEXPOBIT;
12380 : GEN dv;
12381 1568 : for (i = 1; i < l; i++) improve(gel(v,i), w, &E);
12382 98 : v = RgV_Rg_mul(v, ginv(*w));
12383 1568 : for (i = 1; i < l; i++)
12384 1470 : gel(v,i) = bestapprnfrel(gel(v,i), polabs,roabs,rnfeq,prec);
12385 98 : v = Q_primitive_part(v,&dv);
12386 98 : if (dv) *w = gmul(*w,dv);
12387 98 : return v;
12388 : }
12389 :
12390 : static GEN mfpetersson_i(GEN FS, GEN GS);
12391 :
12392 : GEN
12393 42 : mfmanin(GEN FS, long bitprec)
12394 : {
12395 42 : pari_sp av = avma;
12396 : GEN mf, M, vp, vm, cosets, CHI, vpp, vmm, f, T, P, vE, polabs, roabs, rnfeq;
12397 : GEN pet;
12398 : long N, k, lco, i, prec, lvE;
12399 :
12400 42 : if (!checkfs_i(FS))
12401 : {
12402 7 : if (checkfs2_i(FS)) pari_err_TYPE("mfmanin [need integral k > 1]",FS);
12403 0 : pari_err_TYPE("mfmanin",FS);
12404 : }
12405 35 : if (!mfs_iscusp(FS)) pari_err_TYPE("mfmanin [noncuspidal]",FS);
12406 35 : mf = fs_get_MF(FS);
12407 35 : vp = fs_get_pols(FS);
12408 35 : cosets = fs_get_cosets(FS);
12409 35 : bitprec = fs_get_bitprec(FS);
12410 35 : N = MF_get_N(mf); k = MF_get_k(mf); CHI = MF_get_CHI(mf);
12411 35 : lco = lg(cosets); vm = cgetg(lco, t_VEC);
12412 35 : prec = nbits2prec(bitprec);
12413 476 : for (i = 1; i < lco; i++)
12414 : {
12415 441 : GEN g = gel(cosets, i), c;
12416 441 : long A = itos(gcoeff(g,1,1)), B = itos(gcoeff(g,1,2));
12417 441 : long C = itos(gcoeff(g,2,1)), D = itos(gcoeff(g,2,2));
12418 441 : long Dbar, ibar = mftocoset_iD(N, mkmat22s(-B,-A,D,C), cosets, &Dbar);
12419 :
12420 441 : c = mfcharcxeval(CHI, Dbar, prec); if (odd(k)) c = gneg(c);
12421 441 : T = RgX_Rg_mul(gel(vp,ibar), c);
12422 441 : if (typ(T) == t_POL && varn(T) == 0) T = RgX_recip(T);
12423 441 : gel(vm,i) = T;
12424 : }
12425 35 : vpp = gadd(vp,vm);
12426 35 : vmm = gsub(vp,vm);
12427 :
12428 35 : vE = fs_get_vE(FS); lvE = lg(vE);
12429 35 : f = gel(fs_get_EF(FS), 1);
12430 35 : P = mf_get_field(f); if (degpol(P) == 1) P = NULL;
12431 35 : T = mfcharpol(CHI); if (degpol(T) == 1) T = NULL;
12432 35 : if (T && P)
12433 : {
12434 7 : rnfeq = nf_rnfeqsimple(T, P);
12435 7 : polabs = gel(rnfeq,1);
12436 7 : roabs = gel(QX_complex_roots(polabs,prec), 1);
12437 : }
12438 : else
12439 : {
12440 28 : rnfeq = roabs = NULL;
12441 28 : polabs = P? P: T;
12442 : }
12443 35 : pet = mfpetersson_i(FS, NULL);
12444 35 : M = cgetg(lvE, t_VEC);
12445 84 : for (i = 1; i < lvE; i++)
12446 : {
12447 49 : GEN p, m, wp, wm, petdiag, r, E = gel(vE,i);
12448 49 : p = normal(RgXV_embed(vpp, E), polabs, roabs, rnfeq, &wp, prec);
12449 49 : m = normal(RgXV_embed(vmm, E), polabs, roabs, rnfeq, &wm, prec);
12450 49 : petdiag = typ(pet)==t_MAT? gcoeff(pet,i,i): pet;
12451 49 : r = gdiv(imag_i(gmul(wp, conj_i(wm))), petdiag);
12452 49 : r = bestapprnfrel(r, polabs, roabs, rnfeq, prec);
12453 49 : gel(M,i) = mkvec2(mkvec2(p,m), mkvec3(wp,wm,r));
12454 : }
12455 35 : return gc_GEN(av, lvE == 2? gel(M,1): M);
12456 : }
12457 :
12458 : /* flag = 0: full, flag = +1 or -1, odd/even */
12459 : /* Basis of period polynomials in level 1. */
12460 : GEN
12461 49 : mfperiodpolbasis(long k, long flag)
12462 : {
12463 49 : pari_sp av = avma;
12464 49 : long i, j, n = k - 2;
12465 : GEN M, C, v;
12466 49 : if (k <= 4) return cgetg(1,t_VEC);
12467 35 : M = cgetg(k, t_MAT);
12468 35 : C = matpascal(n);
12469 35 : if (!flag)
12470 392 : for (j = 0; j <= n; j++)
12471 : {
12472 371 : gel(M, j+1) = v = cgetg(k, t_COL);
12473 4767 : for (i = 0; i <= j; i++) gel(v, i+1) = gcoeff(C, j+1, i+1);
12474 4396 : for (; i <= n; i++) gel(v, i+1) = gcoeff(C, n-j+1, i-j+1);
12475 : }
12476 : else
12477 168 : for (j = 0; j <= n; j++)
12478 : {
12479 154 : gel(M, j+1) = v = cgetg(k, t_COL);
12480 1848 : for (i = 0; i <= n; i++)
12481 : {
12482 1694 : GEN a = i < j ? gcoeff(C, j+1, i+1) : gen_0;
12483 1694 : if (i + j >= n)
12484 : {
12485 924 : GEN b = gcoeff(C, j+1, i+j-n+1);
12486 924 : a = flag < 0 ? addii(a,b) : subii(a,b);
12487 : }
12488 1694 : gel(v, i+1) = a;
12489 : }
12490 : }
12491 35 : return gc_GEN(av, RgM_to_RgXV(ZM_ker(M), 0));
12492 : }
12493 :
12494 : static int
12495 168 : zero_at_cusp(GEN mf, GEN F, GEN c)
12496 : {
12497 168 : GEN v = evalcusp(mf, F, c, LOWDEFAULTPREC);
12498 168 : return (gequal0(v) || gexpo(v) <= -62);
12499 : }
12500 : /* Compute list E of j such that F|_k g_j vanishes at oo: return [E, s(E)] */
12501 : static void
12502 14 : mffvanish(GEN mf, GEN F, GEN G, GEN cosets, GEN *pres, GEN *press)
12503 : {
12504 14 : long j, lc = lg(cosets), N = MF_get_N(mf);
12505 : GEN v, vs;
12506 14 : *pres = v = zero_zv(lc-1);
12507 14 : *press= vs = zero_zv(lc-1);
12508 105 : for (j = 1; j < lc; j++)
12509 : {
12510 91 : GEN ga = gel(cosets,j), c = mat2cusp(ga);
12511 91 : if (zero_at_cusp(mf, F, c))
12512 14 : v[j] = vs[ mftocoset_i(N, ZM_mulS(ga), cosets) ] = 1;
12513 77 : else if (!zero_at_cusp(mf, G, c))
12514 0 : pari_err_IMPL("divergent Petersson product");
12515 : }
12516 14 : }
12517 : static GEN
12518 140 : Haberland(GEN PF, GEN PG, GEN vEF, GEN vEG, long k)
12519 : {
12520 140 : GEN S = gen_0, vC = vecbinomial(k-2); /* vC[n+1] = (-1)^n binom(k-2,n) */
12521 140 : long n, j, l = lg(PG);
12522 406 : for (n = 2; n < k; n+=2) gel(vC,n) = negi(gel(vC,n));
12523 2583 : for (j = 1; j < l; j++)
12524 : {
12525 2443 : GEN PFj = gel(PF,j), PGj = gel(PG,j);
12526 10038 : for (n = 0; n <= k-2; n++)
12527 : {
12528 7595 : GEN a = RgX_coeff(PGj, k-2-n), b = RgX_coeff(PFj, n);
12529 7595 : a = Rg_embedall(a, vEG);
12530 7595 : b = Rg_embedall(b, vEF);
12531 7595 : a = conj_i(a); if (typ(a) == t_VEC) settyp(a, t_COL);
12532 : /* a*b = scalar or t_VEC or t_COL or t_MAT */
12533 7595 : S = gadd(S, gdiv(gmul(a,b), gel(vC,n+1)));
12534 : }
12535 : }
12536 140 : S = mulcxpowIs(gmul2n(S, 1-k), 1+k);
12537 140 : return vEF==vEG? real_i(S): S;
12538 : }
12539 : /* F1S, F2S both symbols, same mf */
12540 : static GEN
12541 14 : mfpeterssonnoncusp(GEN F1S, GEN F2S)
12542 : {
12543 14 : pari_sp av = avma;
12544 : GEN mf, F1, F2, GF1, GF2, P2, cosets, vE1, vE2, FE1, FE2, P;
12545 : GEN I, IP1, RHO, RHOP1, INF, res, ress;
12546 14 : const double height = sqrt(3.)/2;
12547 : long k, r, j, bitprec, prec;
12548 :
12549 14 : mf = fs_get_MF(F1S);
12550 14 : FE1 = fs_get_EF(F1S); F1 = gel(FE1, 1);
12551 14 : FE2 = fs_get_EF(F2S); F2 = gel(FE2, 1);
12552 14 : cosets = fs_get_cosets(F1S);
12553 14 : bitprec = minuu(fs_get_bitprec(F1S), fs_get_bitprec(F2S));
12554 14 : prec = nbits2prec(bitprec);
12555 14 : F1S = fs_set_expan(F1S, mfgaexpansionall(mf, FE1, cosets, height, prec));
12556 14 : if (F2S != F1S)
12557 14 : F2S = fs_set_expan(F2S, mfgaexpansionall(mf, FE2, cosets, height, prec));
12558 14 : k = MF_get_k(mf); r = lg(cosets) - 1;
12559 14 : vE1 = fs_get_vE(F1S);
12560 14 : vE2 = fs_get_vE(F2S);
12561 14 : I = gen_I();
12562 14 : IP1 = mkcomplex(gen_1,gen_1);
12563 14 : RHO = rootsof1u_cx(3, prec+EXTRAPREC64);
12564 14 : RHOP1 = gaddsg(1, RHO);
12565 14 : INF = mkoo();
12566 14 : mffvanish(mf, F1, F2, cosets, &res, &ress);
12567 14 : P2 = fs_get_pols(F2S);
12568 14 : GF1 = cgetg(r+1, t_VEC);
12569 14 : GF2 = cgetg(r+1, t_VEC); P = get_P(k, fetch_var(), prec);
12570 105 : for (j = 1; j <= r; j++)
12571 : {
12572 91 : GEN g = gel(cosets,j);
12573 91 : if (res[j]) {
12574 14 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHOP1,INF), g, P);
12575 14 : gel(GF2,j) = mfsymboleval_direct(F2S, mkvec2(I,IP1), g, P);
12576 77 : } else if (ress[j]) {
12577 7 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHOP1,RHO), g, P);
12578 7 : gel(GF2,j) = mfsymboleval_direct(F2S, mkvec2(I,INF), g, P);
12579 : } else {
12580 70 : gel(GF1,j) = mfsymboleval_direct(F1S, mkvec2(RHO,I), g, P);
12581 70 : gel(GF2,j) = gneg(gel(P2,j)); /* - symboleval(F2S, [0,oo] */
12582 : }
12583 : }
12584 14 : delete_var();
12585 14 : return gc_upto(av, gdivgu(Haberland(GF1,GF2, vE1,vE2, k), r));
12586 : }
12587 :
12588 : /* Petersson product of F and G, given by mfsymbol's [k > 1 integral] */
12589 : static GEN
12590 140 : mfpetersson_i(GEN FS, GEN GS)
12591 : {
12592 140 : pari_sp av = avma;
12593 : GEN mf, ESF, ESG, PF, PG, PH, CHI, cosets, vEF, vEG;
12594 : long k, r, j, N, bitprec, prec;
12595 :
12596 140 : if (!checkfs_i(FS)) pari_err_TYPE("mfpetersson",FS);
12597 140 : mf = fs_get_MF(FS);
12598 140 : ESF = fs_get_vES(FS);
12599 140 : if (!GS) GS = FS;
12600 : else
12601 : {
12602 35 : if (!checkfs_i(GS)) pari_err_TYPE("mfpetersson",GS);
12603 35 : if (!mfs_checkmf(GS, mf))
12604 0 : pari_err_TYPE("mfpetersson [different mf]", mkvec2(FS,GS));
12605 : }
12606 140 : ESG = fs_get_vES(GS);
12607 140 : if (!gequal0(gel(ESF,1)) || !gequal0(gel(ESG,1)))
12608 14 : return mfpeterssonnoncusp(FS, GS);
12609 126 : if (gequal0(gel(ESF,2)) || gequal0(gel(ESG,2))) return gc_const(av, gen_0);
12610 126 : N = MF_get_N(mf);
12611 126 : k = MF_get_k(mf);
12612 126 : CHI = MF_get_CHI(mf);
12613 126 : PF = fs_get_pols(FS); vEF = fs_get_vE(FS);
12614 126 : PG = fs_get_pols(GS); vEG = fs_get_vE(GS);
12615 126 : cosets = fs_get_cosets(FS);
12616 126 : bitprec = minuu(fs_get_bitprec(FS), fs_get_bitprec(GS));
12617 126 : prec = nbits2prec(bitprec);
12618 126 : r = lg(PG)-1;
12619 126 : PH = cgetg(r+1, t_VEC);
12620 2478 : for (j = 1; j <= r; j++)
12621 : {
12622 2352 : GEN ga = gel(cosets,j), PGj1, PGjm1;
12623 : long iT, D;
12624 2352 : iT = mftocoset_iD(N, ZM_mulTi(ga), cosets, &D);
12625 2352 : PGj1 = RgX_translate(gel(PG, iT), gen_1);
12626 2352 : PGj1 = RgX_Rg_mul(PGj1, mfcharcxeval(CHI, D, prec));
12627 2352 : iT = mftocoset_iD(N, ZM_mulT(ga), cosets, &D);
12628 2352 : PGjm1 = RgX_translate(gel(PG,iT), gen_m1);
12629 2352 : PGjm1 = RgX_Rg_mul(PGjm1, mfcharcxeval(CHI, D, prec));
12630 2352 : gel(PH,j) = gsub(PGj1, PGjm1);
12631 : }
12632 126 : return gc_upto(av, gdivgu(Haberland(PF, PH, vEF, vEG, k), 6*r));
12633 : }
12634 :
12635 : /****************************************************************/
12636 : /* Petersson products using Nelson-Collins */
12637 : /****************************************************************/
12638 : /* Compute W(k,z) = sum_{m >= 1} (mz)^{k-1}(mzK_{k-2}(mz)-K_{k-1}(mz))
12639 : * for z>0 and absolute accuracy < 2^{-B}.
12640 : * K_k(x) ~ (Pi/(2x))^{1/2} e^{-x} */
12641 :
12642 : static void
12643 10304 : Wparams(GEN *ph, long *pN, long k, double x, long prec)
12644 : {
12645 10304 : double B = prec2nbits(prec) + 10;
12646 10304 : double C = B + k*log(x)/M_LN2 + 1, D = C*M_LN2 + 2.065;
12647 10304 : double F = 2 * ((C - 1) * M_LN2 + log(gtodouble(mpfact(k)))) / x;
12648 10304 : double T = log(F) * (1 + 2*k/x/F), PI2 = M_PI*M_PI;
12649 10304 : *pN = (long)ceil((T/PI2) * (D + log(D/PI2)));
12650 10304 : *ph = gprec_w(dbltor(T / *pN), prec);
12651 10304 : }
12652 :
12653 : static void
12654 10304 : Wcoshall(GEN *pCH, GEN *pCHK, GEN *pCHK1, GEN h, long N, long k, long prec)
12655 : {
12656 10304 : GEN CH, CHK, CHK1, z = gexp(h, prec);
12657 10304 : GEN PO = gpowers(z, N), POK1 = gpowers(gpowgs(z, k-1), N);
12658 10304 : GEN E = ginv(gel(PO, N + 1)); /* exp(-hN) */
12659 10304 : GEN E1 = ginv(gel(POK1, N + 1)); /* exp(-(k-1)h) */
12660 : long j;
12661 10304 : *pCH = CH = cgetg(N+1, t_VEC);
12662 10304 : *pCHK = CHK = cgetg(N+1, t_VEC);
12663 10304 : *pCHK1 = CHK1 = cgetg(N+1, t_VEC);
12664 146048 : for (j = 1; j <= N; j++)
12665 : {
12666 135744 : GEN eh = gel(PO, j+1), emh = gmul(gel(PO, N-j+1), E); /* e^{jh}, e^{-jh} */
12667 135744 : GEN ek1h = gel(POK1, j+1), ek1mh = gmul(gel(POK1, N-j+1), E1);
12668 135744 : gel(CH, j) = gmul2n(gadd(eh, emh), -1); /* cosh(jh) */
12669 135744 : gel(CHK1,j) = gmul2n(gadd(ek1h, ek1mh), -1); /* cosh((k-1)jh) */
12670 135744 : gel(CHK, j) = gmul2n(gadd(gmul(eh, ek1h), gmul(emh, ek1mh)), -1);
12671 : }
12672 10304 : }
12673 :
12674 : /* computing W(k,x) via integral */
12675 : static GEN
12676 10304 : Wint(long k, GEN vP, GEN x, long prec)
12677 : {
12678 : GEN P, P1, S1, S, h, CH, CHK, CHK1;
12679 : long N, j;
12680 10304 : Wparams(&h, &N, k, gtodouble(x), prec);
12681 10304 : Wcoshall(&CH, &CHK, &CHK1, h, N, k, prec);
12682 10304 : P = gel(vP, k+1); P1 = gel(vP, k); S = S1 = NULL;
12683 156352 : for (j = 0; j <= N; j++)
12684 : {
12685 146048 : GEN eh = gexp(j? gmul(x, gel(CH, j)): x, prec);
12686 146048 : GEN eh1 = gsubgs(eh, 1), eh1k = gpowgs(eh1, k), t1, t;
12687 146048 : t = gdiv(poleval(P, eh), gmul(eh1, eh1k));
12688 146048 : t1 = gdiv(poleval(P1, eh), eh1k);
12689 146048 : if (j)
12690 : {
12691 135744 : S = gadd(S, gmul(t, gel(CHK, j)));
12692 135744 : S1 = gadd(S1, gmul(t1, gel(CHK1, j)));
12693 : }
12694 : else
12695 : {
12696 10304 : S = gmul2n(t, -1);
12697 10304 : S1 = gmul2n(t1, -1);
12698 : }
12699 : }
12700 10304 : return gmul(gmul(h, gpowgs(x, k-1)), gsub(gmul(x, S), gmulsg(2*k-1, S1)));
12701 : }
12702 :
12703 : static GEN
12704 21 : get_vP(long k)
12705 : {
12706 21 : GEN P, v = cgetg(k+2, t_VEC), Q = deg1pol_shallow(gen_1,gen_m1,0);
12707 : long j;
12708 21 : gel(v,1) = gen_1;
12709 21 : gel(v,2) = P = pol_x(0);
12710 28 : for (j = 2; j <= k; j++)
12711 7 : gel(v,j+1) = P = RgX_shift_shallow(gsub(gmulsg(j, P),
12712 : gmul(Q, ZX_deriv(P))), 1);
12713 21 : return v;
12714 : }
12715 : /* vector of (-1)^j(1/(exp(x)-1))^(j) [x = z] * z^j for 0<=j<=r */
12716 : static GEN
12717 63742 : VS(long r, GEN z, GEN V, long prec)
12718 : {
12719 63742 : GEN e = gexp(z, prec), c = ginv(gsubgs(e,1));
12720 63742 : GEN T = gpowers0(gmul(c, z), r, c);
12721 : long j;
12722 63742 : V = gsubst(V, 0, e);
12723 143864 : for (j = 1; j <= r + 1; j++) gel(V,j) = gmul(gel(V,j), gel(T,j));
12724 63742 : return V;
12725 : }
12726 :
12727 : /* U(r,x)=sum_{m >= 1} (mx)^k K_k(mx), k = r+1/2 */
12728 : static GEN
12729 71932 : Unelson(long r, GEN V)
12730 : {
12731 71932 : GEN S = gel(V,r+1), C = gen_1; /* (r+j)! / j! / (r-j)! */
12732 : long j;
12733 71932 : if (!r) return S;
12734 40950 : for (j = 1; j <= r; j++)
12735 : {
12736 24570 : C = gdivgu(gmulgu(C, (r+j)*(r-j+1)), j);
12737 24570 : S = gadd(S, gmul2n(gmul(C, gel(V, r-j+1)), -j));
12738 : }
12739 16380 : return S;
12740 : }
12741 : /* W(r+1/2,z) / sqrt(Pi/2) */
12742 : static GEN
12743 63742 : Wint2(long r, GEN vP, GEN z, long prec)
12744 : {
12745 63742 : GEN R, V = VS(r, z, vP, prec);
12746 63742 : R = Unelson(r, V);
12747 63742 : if (r) R = gsub(R, gmulsg(2*r, Unelson(r-1, V)));
12748 63742 : return R;
12749 : }
12750 : typedef GEN(*Wfun_t)(long, GEN, GEN, long);
12751 : static GEN
12752 74046 : WfromZ(GEN Z, GEN vP, GEN gkm1, Wfun_t W, long k, GEN pi4, long prec)
12753 : {
12754 74046 : pari_sp av = avma;
12755 74046 : GEN Zk = gpow(Z, gkm1, prec), z = gmul(pi4, gsqrt(Z,prec));
12756 74046 : return gc_upto(av, gdiv(W(k, vP, z, prec), Zk));
12757 : }
12758 : /* mf a true mf or an fs2 */
12759 : static GEN
12760 21 : fs2_init(GEN mf, GEN F, long bit)
12761 : {
12762 21 : pari_sp av = avma;
12763 21 : long i, l, lim, N, k, k2, prec = nbits2prec(bit);
12764 : GEN DEN, cusps, tab, gk, gkm1, W0, vW, vVW, vVF, vP, al0;
12765 21 : GEN vE = mfgetembed(F, prec), pi4 = Pi2n(2, prec);
12766 : Wfun_t Wf;
12767 :
12768 21 : if (lg(mf) == 7)
12769 : {
12770 21 : vW = cusps = NULL; /* true mf */
12771 21 : DEN = tab = NULL; /* -Wall */
12772 : }
12773 : else
12774 : { /* mf already an fs2, reset if its precision is too low */
12775 0 : vW = (fs2_get_bitprec(mf) < bit)? NULL: fs2_get_W(mf);
12776 0 : cusps = fs2_get_cusps(mf);
12777 0 : DEN = fs2_get_den(mf);
12778 0 : mf = fs2_get_MF(mf);
12779 : }
12780 21 : N = MF_get_N(mf);
12781 21 : gk = MF_get_gk(mf); gkm1 = gsubgs(gk, 1);
12782 21 : k2 = itos(gmul2n(gk,1));
12783 21 : Wf = odd(k2)? Wint2: Wint;
12784 21 : k = k2 >> 1; vP = get_vP(k);
12785 21 : if (vW) lim = (lg(gel(vW,1)) - 2) / N; /* vW[1] attached to cusp 0, width N */
12786 : else
12787 : { /* true mf */
12788 21 : double B = (bit + 10)*M_LN2;
12789 21 : double L = (B + k2*log(B)/2 + k2*k2*log(B)/(4*B)) / (4*M_PI);
12790 : long n, Lw;
12791 21 : lim = ((long)ceil(L*L));
12792 21 : Lw = N*lim;
12793 21 : tab = cgetg(Lw+1,t_VEC);
12794 59157 : for (n = 1; n <= Lw; n++)
12795 59136 : gel(tab,n) = WfromZ(uutoQ(n,N), vP, gkm1, Wf, k, pi4, prec);
12796 21 : if (!cusps) cusps = mfcusps_i(N);
12797 21 : DEN = gmul2n(gmulgu(gpow(Pi2n(3, prec), gkm1, prec), mypsiu(N)), -2);
12798 21 : if (odd(k2)) DEN = gdiv(DEN, sqrtr_abs(Pi2n(-1,prec)));
12799 : }
12800 21 : l = lg(cusps);
12801 21 : vVF = cgetg(l, t_VEC);
12802 21 : vVW = cgetg(l, t_VEC);
12803 21 : al0 = cgetg(l, t_VECSMALL);
12804 21 : W0 = k2==1? ginv(pi4): gen_0;
12805 203 : for (i = 1; i < l; i++)
12806 : {
12807 : long A, C, w, wi, Lw, n;
12808 : GEN VF, W, paramsF, al;
12809 182 : (void)cusp_AC(gel(cusps,i), &A,&C);
12810 182 : wi = ugcd(N, C*C); w = N / wi; Lw = w * lim;
12811 182 : VF = mfslashexpansion(mf, F, cusp2mat(A,C), Lw, 0, ¶msF, prec);
12812 : /* paramsF[2] = w */
12813 182 : al = gel(paramsF, 1); if (gequal0(al)) al = NULL;
12814 100240 : for (n = 0; n <= Lw; n++)
12815 : {
12816 100058 : GEN a = gel(VF,n+1);
12817 100058 : gel(VF,n+1) = gequal0(a)? gen_0: Rg_embedall(a, vE);
12818 : }
12819 182 : if (vW)
12820 0 : W = gel(vW, i);
12821 : else
12822 : {
12823 182 : W = cgetg(Lw+2, t_VEC);
12824 100240 : for (n = 0; n <= Lw; n++)
12825 100058 : gel(W, n+1) = al? WfromZ(gadd(al,uutoQ(n,w)),vP,gkm1,Wf,k,pi4, prec)
12826 100058 : : (n? gel(tab, n * wi): W0);
12827 : }
12828 182 : al0[i] = !al;
12829 182 : gel(vVF, i) = VF;
12830 182 : gel(vVW, i) = W;
12831 : }
12832 21 : if (k2 <= 1) al0 = zero_zv(l-1); /* no need to test for convergence */
12833 21 : return gc_GEN(av, mkvecn(7, mf,vVW,cusps,vVF,utoipos(bit),al0,DEN));
12834 : }
12835 :
12836 : static GEN
12837 21 : mfpetersson2(GEN Fs, GEN Gs)
12838 : {
12839 21 : pari_sp av = avma;
12840 21 : GEN VC, RES, vF, vG, vW = fs2_get_W(Fs), al0 = fs2_get_al0(Fs);
12841 21 : long N = MF_get_N(fs2_get_MF(Fs)), j, lC;
12842 :
12843 21 : VC = fs2_get_cusps(Fs); lC = lg(VC);
12844 21 : vF = fs2_get_F(Fs);
12845 21 : vG = Gs? fs2_get_F(Gs): vF;
12846 21 : RES = gen_0;
12847 203 : for (j = 1; j < lC; j++)
12848 : {
12849 182 : GEN W = gel(vW,j), VF = gel(vF,j), VG = gel(vG,j), T = gen_0;
12850 182 : long A, C, w, n, L = lg(W);
12851 182 : pari_sp av = avma;
12852 182 : (void)cusp_AC(gel(VC,j), &A,&C); w = N/ugcd(N, C*C);
12853 182 : if (al0[j] && !isintzero(gel(VF,1)) && !isintzero(gel(VG,1)))
12854 0 : pari_err_IMPL("divergent Petersson product");
12855 100240 : for (n = 1; n < L; n++)
12856 : {
12857 100058 : GEN b = gel(VF,n), a = gel(VG,n);
12858 100058 : if (!isintzero(a) && !isintzero(b))
12859 : {
12860 79964 : T = gadd(T, gmul(gel(W,n), gmul(conj_i(a),b)));
12861 79964 : if (gc_needed(av,2)) T = gc_upto(av,T);
12862 : }
12863 : }
12864 182 : if (w != 1) T = gmulgu(T,w);
12865 182 : RES = gc_upto(av, gadd(RES, T));
12866 : }
12867 21 : if (!Gs) RES = real_i(RES);
12868 21 : return gc_upto(av, gdiv(RES, fs2_get_den(Fs)));
12869 : }
12870 :
12871 : static long
12872 161 : symbol_type(GEN F)
12873 : {
12874 161 : if (checkfs_i(F)) return 1;
12875 21 : if (checkfs2_i(F)) return 2;
12876 0 : return 0;
12877 : }
12878 : static int
12879 35 : symbol_same_mf(GEN F, GEN G) { return gequal(gmael(F,1,1), gmael(G,1,1)); }
12880 : GEN
12881 126 : mfpetersson(GEN F, GEN G)
12882 : {
12883 126 : long tF = symbol_type(F);
12884 126 : if (!tF) pari_err_TYPE("mfpetersson",F);
12885 126 : if (G)
12886 : {
12887 35 : long tG = symbol_type(G);
12888 35 : if (!tG) pari_err_TYPE("mfpetersson",F);
12889 35 : if (tF != tG || !symbol_same_mf(F,G))
12890 0 : pari_err_TYPE("mfpetersson [incompatible symbols]", mkvec2(F,G));
12891 : }
12892 126 : return (tF == 1)? mfpetersson_i(F, G): mfpetersson2(F, G);
12893 : }
12894 :
12895 : /****************************************************************/
12896 : /* projective Galois representation, weight 1 */
12897 : /****************************************************************/
12898 : static void
12899 392 : moreorders(long N, GEN CHI, GEN F, GEN *pP, GEN *pO, ulong *bound)
12900 : {
12901 392 : pari_sp av = avma;
12902 : forprime_t iter;
12903 392 : ulong a = *bound+1, b = 2*(*bound), p;
12904 392 : long i = 1;
12905 392 : GEN P, O, V = mfcoefs_i(F, b, 1);
12906 392 : *bound = b;
12907 392 : P = cgetg(b-a+2, t_VECSMALL);
12908 392 : O = cgetg(b-a+2, t_VECSMALL);
12909 392 : u_forprime_init(&iter, a, b);
12910 2310 : while((p = u_forprime_next(&iter))) if (N % p)
12911 : {
12912 1813 : O[i] = mffindrootof1(V, p, CHI);
12913 1813 : P[i++] = p;
12914 : }
12915 392 : setlg(P, i); *pP = shallowconcat(*pP, P);
12916 392 : setlg(O, i); *pO = shallowconcat(*pO, O);
12917 392 : (void)gc_all(av, 2, pP, pO);
12918 392 : }
12919 :
12920 : static GEN
12921 182 : search_abelian(GEN nf, long n, long k, GEN N, GEN CHI, GEN F,
12922 : GEN *pP, GEN *pO, ulong *bound, long prec)
12923 : {
12924 182 : pari_sp av = avma;
12925 : GEN bnr, cond, H, cyc, gn, T, Bquo, P, E;
12926 182 : long sN = itos(N), r1 = nf_get_r1(nf), i, j, d;
12927 :
12928 182 : cond = idealfactor(nf, N);
12929 182 : P = gel(cond,1);
12930 182 : E = gel(cond,2);
12931 679 : for (i = j = 1; i < lg(P); i++)
12932 : {
12933 497 : GEN pr = gel(P,i), Ej = gen_1;
12934 497 : long p = itos(pr_get_p(pr));
12935 497 : if (p == n)
12936 : {
12937 98 : long e = pr_get_e(pr); /* 1 + [e*p/(p-1)] */
12938 98 : Ej = utoipos(1 + (e*p) / (p-1));
12939 : }
12940 : else
12941 : {
12942 399 : long f = pr_get_f(pr);
12943 399 : if (Fl_powu(p % n, f, n) != 1) continue;
12944 : }
12945 462 : gel(P,j) = pr;
12946 462 : gel(E,j) = Ej; j++;
12947 : }
12948 182 : setlg(P,j);
12949 182 : setlg(E,j);
12950 182 : cond = mkvec2(cond, const_vec(r1, gen_1));
12951 182 : bnr = Buchraymod(Buchall(nf, nf_FORCE, prec), cond, nf_INIT, utoipos(n));
12952 182 : cyc = bnr_get_cyc(bnr);
12953 182 : d = lg(cyc)-1;
12954 182 : H = zv_diagonal(ZV_to_Flv(cyc, n));
12955 182 : gn = utoi(n);
12956 182 : for (i = 1;;)
12957 : {
12958 2646 : for(j = 2; i < lg(*pO); i++)
12959 : {
12960 2072 : long o, q = (*pP)[i];
12961 2072 : GEN pr = idealprimedec_galois(nf, stoi(q));
12962 2072 : o = ((*pO)[i] / pr_get_f(pr)) % n;
12963 2072 : if (o)
12964 : {
12965 1442 : GEN v = ZV_to_Flv(isprincipalray(bnr, pr), n);
12966 1442 : H = vec_append(H, Flv_Fl_mul(v, o, n));
12967 : }
12968 : }
12969 574 : H = Flm_image(H, n); if (lg(cyc)-lg(H) <= k) break;
12970 392 : moreorders(sN, CHI, F, pP, pO, bound);
12971 : }
12972 182 : H = hnfmodid(shallowconcat(zm_to_ZM(H), diagonal_shallow(cyc)), gn);
12973 :
12974 182 : Bquo = cgetg(k+1, t_MAT);
12975 812 : for (i = j = 1; i <= d; i++)
12976 630 : if (!equali1(gcoeff(H,i,i))) gel(Bquo,j++) = col_ei(d,i);
12977 :
12978 441 : for (i = 1, T = NULL; i<=k; i++)
12979 : {
12980 259 : GEN Hi = hnfmodid(shallowconcat(H, vecsplice(Bquo,i)), gn);
12981 259 : GEN pol = rnfkummer(bnr, Hi, prec);
12982 259 : T = T? nfcompositum(nf, T, pol, 2): pol;
12983 : }
12984 182 : T = rnfequation(nf, T); return gc_all(av, 3, &T, pP, pO);
12985 : }
12986 :
12987 : static GEN
12988 77 : search_solvable(GEN LG, GEN mf, GEN F, long prec)
12989 : {
12990 77 : GEN N = MF_get_gN(mf), CHI = MF_get_CHI(mf), pol, O, P, nf, Nfa;
12991 77 : long i, l = lg(LG), v = fetch_var();
12992 77 : ulong bound = 1;
12993 77 : O = cgetg(1, t_VECSMALL); /* projective order of rho(Frob_p) */
12994 77 : P = cgetg(1, t_VECSMALL);
12995 77 : Nfa = Z_factor(N);
12996 77 : pol = pol_x(v);
12997 259 : for (i = 1; i < l; i++)
12998 : { /* n prime, find a (Z/nZ)^k - extension */
12999 182 : GEN G = gel(LG,i);
13000 182 : long n = G[1], k = G[2];
13001 182 : nf = nfinitred(mkvec2(pol,Nfa), prec);
13002 182 : pol = search_abelian(nf, n, k, N, CHI, F, &P, &O, &bound, prec);
13003 182 : setvarn(pol,v);
13004 : }
13005 77 : delete_var(); setvarn(pol,0); return pol;
13006 : }
13007 :
13008 : static GEN
13009 0 : search_A5(GEN mf, GEN F)
13010 : {
13011 0 : GEN CHI = MF_get_CHI(mf), O, P, L;
13012 0 : long N = MF_get_N(mf), i, j, lL, nd, r;
13013 0 : ulong bound = 1;
13014 0 : r = radicalu(N);
13015 0 : L = veccond_to_A5(zv_z_mul(divisorsu(N/r),r), 2); lL = lg(L); nd = lL-1;
13016 0 : if (nd == 1) return gmael(L,1,1);
13017 0 : O = cgetg(1, t_VECSMALL); /* projective order of rho(Frob_p) */
13018 0 : P = cgetg(1, t_VECSMALL);
13019 0 : for(i = 1; nd > 1; )
13020 : {
13021 : long l;
13022 0 : moreorders(N, CHI, F, &P, &O, &bound);
13023 0 : l = lg(P);
13024 0 : for ( ; i < l; i++)
13025 : {
13026 0 : ulong p = P[i], f = O[i];
13027 0 : for (j = 1; j < lL; j++)
13028 0 : if (gel(L,j))
13029 : {
13030 0 : GEN FE = ZpX_primedec(gmael(L,j,1), utoi(p)), F = gel(FE,1);
13031 0 : long nF = lg(F)-1;
13032 0 : if (!equaliu(gel(F, nF), f)) { gel(L,j) = NULL; nd--; }
13033 : }
13034 0 : if (nd <= 1) break;
13035 : }
13036 : }
13037 0 : for (j = 1; j < lL; j++)
13038 0 : if (gel(L,j)) return gmael(L,j,1);
13039 0 : return NULL;
13040 : }
13041 :
13042 : GEN
13043 77 : mfgaloisprojrep(GEN mf, GEN F, long prec)
13044 : {
13045 77 : pari_sp av = avma;
13046 77 : GEN LG = NULL;
13047 77 : if (!checkMF_i(mf) && !checkmf_i(F)) pari_err_TYPE("mfgaloisrep", F);
13048 77 : switch( itos(mfgaloistype(mf,F)) )
13049 : {
13050 49 : case 0: case -12:
13051 49 : LG = mkvec2(mkvecsmall2(3,1), mkvecsmall2(2,2)); break;
13052 28 : case -24:
13053 28 : LG = mkvec3(mkvecsmall2(2,1), mkvecsmall2(3,1), mkvecsmall2(2,2)); break;
13054 0 : case -60: return gc_GEN(av, search_A5(mf, F));
13055 0 : default: pari_err_IMPL("mfgaloisprojrep for types D_n");
13056 : }
13057 77 : return gc_GEN(av, search_solvable(LG, mf, F, prec));
13058 : }
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