Code coverage tests

This page documents the degree to which the PARI/GP source code is tested by our public test suite, distributed with the source distribution in directory src/test/. This is measured by the gcov utility; we then process gcov output using the lcov frond-end.

We test a few variants depending on Configure flags on the pari.math.u-bordeaux.fr machine (x86_64 architecture), and agregate them in the final report:

The target is to exceed 90% coverage for all mathematical modules (given that branches depending on DEBUGLEVEL or DEBUGMEM are not covered). This script is run to produce the results below.

LCOV - code coverage report
Current view: top level - basemath - modsym.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.14.0 lcov report (development 26712-590d837a1c) Lines: 2819 3025 93.2 %
Date: 2021-06-22 07:13:04 Functions: 294 299 98.3 %
Legend: Lines: hit not hit

          Line data    Source code
       1             : /* Copyright (C) 2011  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., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */
      14             : 
      15             : #include "pari.h"
      16             : #include "paripriv.h"
      17             : 
      18             : #define DEBUGLEVEL DEBUGLEVEL_ms
      19             : 
      20             : /* Adapted from shp_package/moments by Robert Pollack
      21             :  * http://www.math.mcgill.ca/darmon/programs/shp/shp.html */
      22             : static GEN mskinit(ulong N, long k, long sign);
      23             : static GEN mshecke_i(GEN W, ulong p);
      24             : static GEN ZSl2_star(GEN v);
      25             : static GEN getMorphism(GEN W1, GEN W2, GEN v);
      26             : static GEN voo_act_Gl2Q(GEN g, long k);
      27             : 
      28             : /* Input: P^1(Z/NZ) (formed by create_p1mod)
      29             :    Output: # P^1(Z/NZ) */
      30             : static long
      31        8316 : p1_size(GEN p1N) { return lg(gel(p1N,1)) - 1; }
      32             : static ulong
      33    53559009 : p1N_get_N(GEN p1N) { return gel(p1N,3)[2]; }
      34             : static GEN
      35    24978562 : p1N_get_hash(GEN p1N) { return gel(p1N,2); }
      36             : static GEN
      37        2219 : p1N_get_fa(GEN p1N) { return gel(p1N,4); }
      38             : static GEN
      39        2107 : p1N_get_div(GEN p1N) { return gel(p1N,5); }
      40             : static GEN
      41    22239623 : p1N_get_invsafe(GEN p1N) { return gel(p1N,6); }
      42             : static GEN
      43     7062188 : p1N_get_inverse(GEN p1N) { return gel(p1N,7); }
      44             : 
      45             : /* ms-specific accessors */
      46             : /* W = msinit, return the output of msinit_N */
      47             : static GEN
      48     5144244 : get_msN(GEN W) { return lg(W) == 4? gel(W,1): W; }
      49             : static GEN
      50     3044860 : msN_get_p1N(GEN W) { return gel(W,1); }
      51             : static GEN
      52      209062 : msN_get_genindex(GEN W) { return gel(W,5); }
      53             : static GEN
      54    39969097 : msN_get_E2fromE1(GEN W) { return gel(W,7); }
      55             : static GEN
      56        1393 : msN_get_annT2(GEN W) { return gel(W,8); }
      57             : static GEN
      58        1393 : msN_get_annT31(GEN W) { return gel(W,9); }
      59             : static GEN
      60        1358 : msN_get_singlerel(GEN W) { return gel(W,10); }
      61             : static GEN
      62      755426 : msN_get_section(GEN W) { return gel(W,12); }
      63             : 
      64             : static GEN
      65       87948 : ms_get_p1N(GEN W) { return msN_get_p1N(get_msN(W)); }
      66             : static long
      67       75362 : ms_get_N(GEN W) { return p1N_get_N(ms_get_p1N(W)); }
      68             : static GEN
      69        1680 : ms_get_hashcusps(GEN W) { W = get_msN(W); return gel(W,16); }
      70             : static GEN
      71       17059 : ms_get_section(GEN W) { return msN_get_section(get_msN(W)); }
      72             : static GEN
      73      204050 : ms_get_genindex(GEN W) { return msN_get_genindex(get_msN(W)); }
      74             : static long
      75      199444 : ms_get_nbgen(GEN W) { return lg(ms_get_genindex(W))-1; }
      76             : static long
      77     2428335 : ms_get_nbE1(GEN W)
      78             : {
      79             :   GEN W11;
      80     2428335 :   W = get_msN(W); W11 = gel(W,11);
      81     2428335 :   return W11[4] - W11[3];
      82             : }
      83             : 
      84             : /* msk-specific accessors */
      85             : static long
      86         126 : msk_get_dim(GEN W) { return gmael(W,3,2)[2]; }
      87             : static GEN
      88       82516 : msk_get_basis(GEN W) { return gmael(W,3,1); }
      89             : static long
      90       46984 : msk_get_weight(GEN W) { return gmael(W,3,2)[1]; }
      91             : static long
      92       21434 : msk_get_sign(GEN W)
      93             : {
      94       21434 :   GEN t = gel(W,2);
      95       21434 :   return typ(t)==t_INT? 0: itos(gel(t,1));
      96             : }
      97             : static GEN
      98         959 : msk_get_star(GEN W) { return gmael(W,2,2); }
      99             : static GEN
     100        3710 : msk_get_starproj(GEN W) { return gmael(W,2,3); }
     101             : 
     102             : static int
     103         434 : is_Qevproj(GEN x)
     104         434 : { return typ(x) == t_VEC && lg(x) == 5 && typ(gel(x,1)) == t_MAT; }
     105             : long
     106         224 : msdim(GEN W)
     107             : {
     108         224 :   if (is_Qevproj(W)) return lg(gel(W,1)) - 1;
     109         210 :   checkms(W);
     110         203 :   if (!msk_get_sign(W)) return msk_get_dim(W);
     111          91 :   return lg(gel(msk_get_starproj(W), 1)) - 1;
     112             : }
     113             : long
     114          14 : msgetlevel(GEN W) { checkms(W); return ms_get_N(W); }
     115             : long
     116          14 : msgetweight(GEN W) { checkms(W); return msk_get_weight(W); }
     117             : long
     118          28 : msgetsign(GEN W) { checkms(W); return msk_get_sign(W); }
     119             : 
     120             : void
     121       36547 : checkms(GEN W)
     122             : {
     123       36547 :   if (typ(W) != t_VEC || lg(W) != 4
     124       36547 :       || typ(gel(W,1)) != t_VEC || lg(gel(W,1)) != 17)
     125           7 :     pari_err_TYPE("checkms [please apply msinit]", W);
     126       36540 : }
     127             : 
     128             : /** MODULAR TO SYM **/
     129             : 
     130             : /* q a t_FRAC or t_INT */
     131             : static GEN
     132     2221884 : Q_log_init(ulong N, GEN q)
     133             : {
     134             :   long l, n;
     135             :   GEN Q;
     136             : 
     137     2221884 :   q = gboundcf(q, 0);
     138     2221884 :   l = lg(q);
     139     2221884 :   Q = cgetg(l, t_VECSMALL);
     140     2221884 :   Q[1] = 1;
     141    21320124 :   for (n=2; n <l; n++) Q[n] = umodiu(gel(q,n), N);
     142    19099094 :   for (n=3; n < l; n++)
     143    16877210 :     Q[n] = Fl_add(Fl_mul(Q[n], Q[n-1], N), Q[n-2], N);
     144     2221884 :   return Q;
     145             : }
     146             : 
     147             : /** INIT MODSYM STRUCTURE, WEIGHT 2 **/
     148             : 
     149             : /* num = [Gamma : Gamma_0(N)] = N * Prod_{p|N} (1+p^-1) */
     150             : static ulong
     151        2107 : count_Manin_symbols(ulong N, GEN P)
     152             : {
     153        2107 :   long i, l = lg(P);
     154        2107 :   ulong num = N;
     155        5467 :   for (i = 1; i < l; i++) { ulong p = P[i]; num *= p+1; num /= p; }
     156        2107 :   return num;
     157             : }
     158             : /* returns the list of "Manin symbols" (c,d) in (Z/NZ)^2, (c,d,N) = 1
     159             :  * generating H^1(X_0(N), Z) */
     160             : static GEN
     161        2107 : generatemsymbols(ulong N, ulong num, GEN divN)
     162             : {
     163        2107 :   GEN ret = cgetg(num+1, t_VEC);
     164        2107 :   ulong c, d, curn = 0;
     165             :   long i, l;
     166             :   /* generate Manin-symbols in two lists: */
     167             :   /* list 1: (c:1) for 0 <= c < N */
     168      231973 :   for (c = 0; c < N; c++) gel(ret, ++curn) = mkvecsmall2(c, 1);
     169        2107 :   if (N == 1) return ret;
     170             :   /* list 2: (c:d) with 1 <= c < N, c | N, 0 <= d < N, gcd(d,N) > 1, gcd(c,d)=1.
     171             :    * Furthermore, d != d0 (mod N/c) with c,d0 already in the list */
     172        2079 :   l = lg(divN) - 1;
     173             :   /* c = 1 first */
     174        2079 :   gel(ret, ++curn) = mkvecsmall2(1,0);
     175      227759 :   for (d = 2; d < N; d++)
     176      225680 :     if (ugcd(d,N) != 1UL)
     177       80871 :       gel(ret, ++curn) = mkvecsmall2(1,d);
     178             :   /* omit c = 1 (first) and c = N (last) */
     179        6370 :   for (i=2; i < l; i++)
     180             :   {
     181             :     ulong Novc, d0;
     182        4291 :     c = divN[i];
     183        4291 :     Novc = N / c;
     184      120575 :     for (d0 = 2; d0 <= Novc; d0++)
     185             :     {
     186      116284 :       ulong k, d = d0;
     187      116284 :       if (ugcd(d, Novc) == 1UL) continue;
     188      146902 :       for (k = 0; k < c; k++, d += Novc)
     189      134554 :         if (ugcd(c,d) == 1UL)
     190             :         {
     191       27356 :           gel(ret, ++curn) = mkvecsmall2(c,d);
     192       27356 :           break;
     193             :         }
     194             :     }
     195             :   }
     196        2079 :   if (curn != num) pari_err_BUG("generatemsymbols [wrong number of symbols]");
     197        2079 :   return ret;
     198             : }
     199             : 
     200             : static GEN
     201        2107 : inithashmsymbols(ulong N, GEN symbols)
     202             : {
     203        2107 :   GEN H = zerovec(N);
     204        2107 :   long k, l = lg(symbols);
     205             :   /* skip the (c:1), 0 <= c < N and (1:0) */
     206      110334 :   for (k=N+2; k < l; k++)
     207             :   {
     208      108227 :     GEN s = gel(symbols, k);
     209      108227 :     ulong c = s[1], d = s[2], Novc = N/c;
     210      108227 :     if (gel(H,c) == gen_0) gel(H,c) = const_vecsmall(Novc+1,0);
     211      108227 :     if (c != 1) { d %= Novc; if (!d) d = Novc; }
     212      108227 :     mael(H, c, d) = k;
     213             :   }
     214        2107 :   return H;
     215             : }
     216             : 
     217             : /** Helper functions for Sl2(Z) / Gamma_0(N) **/
     218             : /* M a 2x2 ZM in SL2(Z) */
     219             : GEN
     220     1237117 : SL2_inv_shallow(GEN M)
     221             : {
     222     1237117 :   GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
     223     1237117 :   GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
     224     1237117 :   return mkmat22(d,negi(b), negi(c),a);
     225             : }
     226             : /* SL2_inv(M)[2] */
     227             : static GEN
     228        3514 : SL2_inv2(GEN M)
     229             : {
     230        3514 :   GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
     231        3514 :   return mkcol2(negi(b),a);
     232             : }
     233             : /* M a 2x2 mat2 in SL2(Z) */
     234             : static GEN
     235      735553 : sl2_inv(GEN M)
     236             : {
     237      735553 :   long a=coeff(M,1,1), b=coeff(M,1,2), c=coeff(M,2,1), d=coeff(M,2,2);
     238      735553 :   return mkvec2(mkvecsmall2(d, -c), mkvecsmall2(-b, a));
     239             : }
     240             : /* Return the mat2 [a,b; c,d], not a zm to avoid GP problems */
     241             : static GEN
     242     1924573 : mat2(long a, long b, long c, long d)
     243     1924573 : { return mkvec2(mkvecsmall2(a,c), mkvecsmall2(b,d)); }
     244             : static GEN
     245      406882 : mat2_to_ZM(GEN M)
     246             : {
     247      406882 :   GEN A = gel(M,1), B = gel(M,2);
     248      406882 :   retmkmat2(mkcol2s(A[1],A[2]), mkcol2s(B[1],B[2]));
     249             : }
     250             : 
     251             : /* Input: a = 2-vector = path = {r/s,x/y}
     252             :  * Output: either [r,x;s,y] or [-r,x;-s,y], whichever has determinant > 0 */
     253             : static GEN
     254      118209 : path_to_ZM(GEN a)
     255             : {
     256      118209 :   GEN v = gel(a,1), w = gel(a,2);
     257      118209 :   long r = v[1], s = v[2], x = w[1], y = w[2];
     258      118209 :   if (cmpii(mulss(r,y), mulss(x,s)) < 0) { r = -r; s = -s; }
     259      118209 :   return mkmat22(stoi(r),stoi(x),stoi(s),stoi(y));
     260             : }
     261             : static GEN
     262     1196195 : path_to_zm(GEN a)
     263             : {
     264     1196195 :   GEN v = gel(a,1), w = gel(a,2);
     265     1196195 :   long r = v[1], s = v[2], x = w[1], y = w[2];
     266     1196195 :   if (cmpii(mulss(r,y), mulss(x,s)) < 0) { r = -r; s = -s; }
     267     1196195 :   return mat2(r,x,s,y);
     268             : }
     269             : /* path from c1 to c2 */
     270             : static GEN
     271      687715 : mkpath(GEN c1, GEN c2) { return mat2(c1[1], c2[1], c1[2], c2[2]); }
     272             : static long
     273      972104 : cc(GEN M) { GEN v = gel(M,1); return v[2]; }
     274             : static long
     275      972104 : dd(GEN M) { GEN v = gel(M,2); return v[2]; }
     276             : 
     277             : /*Input: a,b = 2 paths, N = integer
     278             :  *Output: 1 if the a,b are \Gamma_0(N)-equivalent; 0 otherwise */
     279             : static int
     280      112196 : gamma_equiv(GEN a, GEN b, ulong N)
     281             : {
     282      112196 :   pari_sp av = avma;
     283      112196 :   GEN m = path_to_zm(a);
     284      112196 :   GEN n = path_to_zm(b);
     285      112196 :   GEN d = subii(mulss(cc(m),dd(n)), mulss(dd(m),cc(n)));
     286      112196 :   return gc_bool(av, umodiu(d, N) == 0);
     287             : }
     288             : /* Input: a,b = 2 paths that are \Gamma_0(N)-equivalent, N = integer
     289             :  * Output: M in \Gamma_0(N) such that Mb=a */
     290             : static GEN
     291       58660 : gamma_equiv_matrix(GEN a, GEN b)
     292             : {
     293       58660 :   GEN m = path_to_ZM(a);
     294       58660 :   GEN n = path_to_ZM(b);
     295       58660 :   return ZM_mul(m, SL2_inv_shallow(n));
     296             : }
     297             : 
     298             : /*************/
     299             : /* P^1(Z/NZ) */
     300             : /*************/
     301             : /* a != 0 in Z/NZ. Return v in (Z/NZ)^* such that av = gcd(a, N) (mod N)*/
     302             : static ulong
     303      429520 : Fl_inverse(ulong a, ulong N) { ulong g; return Fl_invgen(a,N,&g); }
     304             : 
     305             : /* Input: N = integer
     306             :  * Output: creates P^1(Z/NZ) = [symbols, H, N]
     307             :  *   symbols: list of vectors [x,y] that give a set of representatives
     308             :  *            of P^1(Z/NZ)
     309             :  *   H: an M by M grid whose value at the r,c-th place is the index of the
     310             :  *      "standard representative" equivalent to [r,c] occurring in the first
     311             :  *      list. If gcd(r,c,N) > 1 the grid has value 0. */
     312             : static GEN
     313        2107 : create_p1mod(ulong N)
     314             : {
     315        2107 :   GEN fa = factoru(N), div = divisorsu_fact(fa);
     316        2107 :   ulong i, nsym = count_Manin_symbols(N, gel(fa,1));
     317        2107 :   GEN symbols = generatemsymbols(N, nsym, div);
     318        2107 :   GEN H = inithashmsymbols(N,symbols);
     319        2107 :   GEN invsafe = cgetg(N, t_VECSMALL), inverse = cgetg(N, t_VECSMALL);
     320      229866 :   for (i = 1; i < N; i++)
     321             :   {
     322      227759 :     invsafe[i] = Fl_invsafe(i,N);
     323      227759 :     inverse[i] = Fl_inverse(i,N);
     324             :   }
     325        2107 :   return mkvecn(7, symbols, H, utoipos(N), fa, div, invsafe, inverse);
     326             : }
     327             : 
     328             : /* Let (c : d) in P1(Z/NZ).
     329             :  * If c = 0 return (0:1). If d = 0 return (1:0).
     330             :  * Else replace by (cu : du), where u in (Z/NZ)^* such that C := cu = gcd(c,N).
     331             :  * In create_p1mod(), (c : d) is represented by (C:D) where D = du (mod N/c)
     332             :  * is smallest such that gcd(C,D) = 1. Return (C : du mod N/c), which need
     333             :  * not belong to P1(Z/NZ) ! A second component du mod N/c = 0 is replaced by
     334             :  * N/c in this case to avoid problems with array indices */
     335             : static void
     336    24978562 : p1_std_form(long *pc, long *pd, GEN p1N)
     337             : {
     338    24978562 :   ulong N = p1N_get_N(p1N);
     339             :   ulong u;
     340    24978562 :   *pc = umodsu(*pc, N); if (!*pc) { *pd = 1; return; }
     341    22503047 :   *pd = umodsu(*pd, N); if (!*pd) { *pc = 1; return; }
     342    22239623 :   u = p1N_get_invsafe(p1N)[*pd];
     343    22239623 :   if (u) { *pc = Fl_mul(*pc,u,N); *pd = 1; return; } /* (d,N) = 1 */
     344             : 
     345     7062188 :   u = p1N_get_inverse(p1N)[*pc];
     346     7062188 :   if (u > 1) { *pc = Fl_mul(*pc,u,N); *pd = Fl_mul(*pd,u,N); }
     347             :   /* c | N */
     348     7062188 :   if (*pc != 1) *pd %= (N / *pc);
     349     7062188 :   if (!*pd) *pd = N / *pc;
     350             : }
     351             : 
     352             : /* Input: v = [x,y] = elt of P^1(Z/NZ) = class in Gamma_0(N) \ PSL2(Z)
     353             :  * Output: returns the index of the standard rep equivalent to v */
     354             : static long
     355    24978562 : p1_index(long x, long y, GEN p1N)
     356             : {
     357    24978562 :   ulong N = p1N_get_N(p1N);
     358    24978562 :   GEN H = p1N_get_hash(p1N);
     359             : 
     360    24978562 :   p1_std_form(&x, &y, p1N);
     361    24978562 :   if (y == 1) return x+1;
     362     7325612 :   if (y == 0) return N+1;
     363     7062188 :   if (mael(H,x,y) == 0) pari_err_BUG("p1_index");
     364     7062188 :   return mael(H,x,y);
     365             : }
     366             : 
     367             : /* Cusps for \Gamma_0(N) */
     368             : 
     369             : /* \sum_{d | N} \phi(gcd(d, N/d)), using multiplicativity. fa = factor(N) */
     370             : ulong
     371        2191 : mfnumcuspsu_fact(GEN fa)
     372             : {
     373        2191 :   GEN P = gel(fa,1), E = gel(fa,2);
     374        2191 :   long i, l = lg(P);
     375        2191 :   ulong T = 1;
     376        5649 :   for (i = 1; i < l; i++)
     377             :   {
     378        3458 :     long e = E[i], e2 = e >> 1; /* floor(E[i] / 2) */
     379        3458 :     ulong p = P[i];
     380        3458 :     if (odd(e))
     381        3059 :       T *= 2 * upowuu(p, e2);
     382             :     else
     383         399 :       T *= (p+1) * upowuu(p, e2-1);
     384             :   }
     385        2191 :   return T;
     386             : }
     387             : ulong
     388           7 : mfnumcuspsu(ulong n)
     389           7 : { pari_sp av = avma; return gc_ulong(av, mfnumcuspsu_fact( factoru(n) )); }
     390             : /* \sum_{d | N} \phi(gcd(d, N/d)), using multiplicativity. fa = factor(N) */
     391             : GEN
     392          14 : mfnumcusps_fact(GEN fa)
     393             : {
     394          14 :   GEN P = gel(fa,1), E = gel(fa,2), T = gen_1;
     395          14 :   long i, l = lg(P);
     396          35 :   for (i = 1; i < l; i++)
     397             :   {
     398          21 :     GEN p = gel(P,i), c;
     399          21 :     long e = itos(gel(E,i)), e2 = e >> 1; /* floor(E[i] / 2) */
     400          21 :     if (odd(e))
     401           0 :       c = shifti(powiu(p, e2), 1);
     402             :     else
     403          21 :       c = mulii(addiu(p,1), powiu(p, e2-1));
     404          21 :     T = T? mulii(T, c): c;
     405             :   }
     406          14 :   return T? T: gen_1;
     407             : }
     408             : GEN
     409          21 : mfnumcusps(GEN n)
     410             : {
     411          21 :   pari_sp av = avma;
     412          21 :   GEN F = check_arith_pos(n,"mfnumcusps");
     413          21 :   if (!F)
     414             :   {
     415          14 :     if (lgefint(n) == 3) return utoi( mfnumcuspsu(n[2]) );
     416           7 :     F = absZ_factor(n);
     417             :   }
     418          14 :   return gerepileuptoint(av, mfnumcusps_fact(F));
     419             : }
     420             : 
     421             : /* to each cusp in \Gamma_0(N) P1(Q), represented by p/q, we associate a
     422             :  * unique index. Canonical representative: (1:0) or (p:q) with q | N, q < N,
     423             :  * p defined modulo d := gcd(N/q,q), (p,d) = 1.
     424             :  * Return [[N, nbcusps], H, cusps]*/
     425             : static GEN
     426        2107 : inithashcusps(GEN p1N)
     427             : {
     428        2107 :   ulong N = p1N_get_N(p1N);
     429        2107 :   GEN div = p1N_get_div(p1N), H = zerovec(N+1);
     430        2107 :   long k, ind, l = lg(div), ncusp = mfnumcuspsu_fact(p1N_get_fa(p1N));
     431        2107 :   GEN cusps = cgetg(ncusp+1, t_VEC);
     432             : 
     433        2107 :   gel(H,1) = mkvecsmall2(0/*empty*/, 1/* first cusp: (1:0) */);
     434        2107 :   gel(cusps, 1) = mkvecsmall2(1,0);
     435        2107 :   ind = 2;
     436        8477 :   for (k=1; k < l-1; k++) /* l-1: remove q = N */
     437             :   {
     438        6370 :     ulong p, q = div[k], d = ugcd(q, N/q);
     439        6370 :     GEN h = const_vecsmall(d+1,0);
     440        6370 :     gel(H,q+1) = h ;
     441       16044 :     for (p = 0; p < d; p++)
     442        9674 :       if (ugcd(p,d) == 1)
     443             :       {
     444        7784 :         h[p+1] = ind;
     445        7784 :         gel(cusps, ind) = mkvecsmall2(p,q);
     446        7784 :         ind++;
     447             :       }
     448             :   }
     449        2107 :   return mkvec3(mkvecsmall2(N,ind-1), H, cusps);
     450             : }
     451             : /* c = [p,q], (p,q) = 1, return a canonical representative for
     452             :  * \Gamma_0(N)(p/q) */
     453             : static GEN
     454      203469 : cusp_std_form(GEN c, GEN S)
     455             : {
     456      203469 :   long p, N = gel(S,1)[1], q = umodsu(c[2], N);
     457             :   ulong u, d;
     458      203469 :   if (q == 0) return mkvecsmall2(1, 0);
     459      201761 :   p = umodsu(c[1], N);
     460      201761 :   u = Fl_inverse(q, N);
     461      201761 :   q = Fl_mul(q,u, N);
     462      201761 :   d = ugcd(q, N/q);
     463      201761 :   return mkvecsmall2(Fl_div(p % d,u % d, d), q);
     464             : }
     465             : /* c = [p,q], (p,q) = 1, return the index of the corresponding cusp.
     466             :  * S from inithashcusps */
     467             : static ulong
     468      203469 : cusp_index(GEN c, GEN S)
     469             : {
     470             :   long p, q;
     471      203469 :   GEN H = gel(S,2);
     472      203469 :   c = cusp_std_form(c, S);
     473      203469 :   p = c[1]; q = c[2];
     474      203469 :   if (!mael(H,q+1,p+1)) pari_err_BUG("cusp_index");
     475      203469 :   return mael(H,q+1,p+1);
     476             : }
     477             : 
     478             : /* M a square invertible ZM, return a ZM iM such that iM M = M iM = d.Id */
     479             : static GEN
     480        3066 : ZM_inv_denom(GEN M)
     481             : {
     482        3066 :   GEN diM, iM = ZM_inv(M, &diM);
     483        3066 :   return mkvec2(iM, diM);
     484             : }
     485             : /* return M^(-1) v, dinv = ZM_inv_denom(M) OR Qevproj_init(M) */
     486             : static GEN
     487      744023 : ZC_apply_dinv(GEN dinv, GEN v)
     488             : {
     489             :   GEN x, c, iM;
     490      744023 :   if (lg(dinv) == 3)
     491             :   {
     492      665917 :     iM = gel(dinv,1);
     493      665917 :     c = gel(dinv,2);
     494             :   }
     495             :   else
     496             :   { /* Qevproj_init */
     497       78106 :     iM = gel(dinv,2);
     498       78106 :     c = gel(dinv,3);
     499       78106 :     v = typ(v) == t_MAT? rowpermute(v, gel(dinv,4))
     500       78106 :                        : vecpermute(v, gel(dinv,4));
     501             :   }
     502      744023 :   x = RgM_RgC_mul(iM, v);
     503      744023 :   if (!isint1(c)) x = RgC_Rg_div(x, c);
     504      744023 :   return x;
     505             : }
     506             : 
     507             : /* M an n x d ZM of rank d (basis of a Q-subspace), n >= d.
     508             :  * Initialize a projector on M */
     509             : GEN
     510        5537 : Qevproj_init(GEN M)
     511             : {
     512             :   GEN v, perm, MM, iM, diM;
     513        5537 :   v = ZM_indexrank(M); perm = gel(v,1);
     514        5537 :   MM = rowpermute(M, perm); /* square invertible */
     515        5537 :   iM = ZM_inv(MM, &diM);
     516        5537 :   return mkvec4(M, iM, diM, perm);
     517             : }
     518             : 
     519             : /* same with typechecks */
     520             : static GEN
     521         721 : Qevproj_init0(GEN M)
     522             : {
     523         721 :   switch(typ(M))
     524             :   {
     525         665 :     case t_VEC:
     526         665 :       if (lg(M) == 5) return M;
     527           0 :       break;
     528          49 :     case t_COL:
     529          49 :       M = mkmat(M);/*fall through*/
     530          56 :     case t_MAT:
     531          56 :       M = Q_primpart(M);
     532          56 :       RgM_check_ZM(M,"Qevproj_init");
     533          56 :       return Qevproj_init(M);
     534             :   }
     535           0 :   pari_err_TYPE("Qevproj_init",M);
     536             :   return NULL;/*LCOV_EXCL_LINE*/
     537             : }
     538             : 
     539             : /* T an n x n QM, pro = Qevproj_init(M), pro2 = Qevproj_init(M2); TM \subset M2.
     540             :  * Express these column vectors on M2's basis */
     541             : static GEN
     542        3661 : Qevproj_apply2(GEN T, GEN pro, GEN pro2)
     543             : {
     544        3661 :   GEN M = gel(pro,1), iM = gel(pro2,2), ciM = gel(pro2,3), perm = gel(pro2,4);
     545        3661 :   return RgM_Rg_div(RgM_mul(iM, RgM_mul(rowpermute(T,perm), M)), ciM);
     546             : }
     547             : /* T an n x n QM, stabilizing d-dimensional Q-vector space spanned by the
     548             :  * d columns of M, pro = Qevproj_init(M). Return dxd matrix of T acting on M */
     549             : GEN
     550        3031 : Qevproj_apply(GEN T, GEN pro) { return Qevproj_apply2(T, pro, pro); }
     551             : /* Qevproj_apply(T,pro)[,k] */
     552             : GEN
     553         819 : Qevproj_apply_vecei(GEN T, GEN pro, long k)
     554             : {
     555         819 :   GEN M = gel(pro,1), iM = gel(pro,2), ciM = gel(pro,3), perm = gel(pro,4);
     556         819 :   GEN v = RgM_RgC_mul(iM, RgM_RgC_mul(rowpermute(T,perm), gel(M,k)));
     557         819 :   return RgC_Rg_div(v, ciM);
     558             : }
     559             : 
     560             : static int
     561         434 : cmp_dim(void *E, GEN a, GEN b)
     562             : {
     563             :   long k;
     564             :   (void)E;
     565         434 :   a = gel(a,1);
     566         434 :   b = gel(b,1); k = lg(a)-lg(b);
     567         434 :   return k? ((k > 0)? 1: -1): 0;
     568             : }
     569             : 
     570             : /* FIXME: could use ZX_roots for deglim = 1 */
     571             : static GEN
     572         343 : ZX_factor_limit(GEN T, long deglim, long *pl)
     573             : {
     574         343 :   GEN fa = ZX_factor(T), P, E;
     575             :   long i, l;
     576         343 :   P = gel(fa,1); *pl = l = lg(P);
     577         343 :   if (deglim <= 0) return fa;
     578         224 :   E = gel(fa,2);
     579         567 :   for (i = 1; i < l; i++)
     580         406 :     if (degpol(gel(P,i)) > deglim) break;
     581         224 :   setlg(P,i);
     582         224 :   setlg(E,i); return fa;
     583             : }
     584             : 
     585             : /* Decompose the subspace H (Qevproj format) in simple subspaces.
     586             :  * Eg for H = msnew */
     587             : static GEN
     588         266 : mssplit_i(GEN W, GEN H, long deglim)
     589             : {
     590         266 :   ulong p, N = ms_get_N(W);
     591             :   long first, dim;
     592             :   forprime_t S;
     593         266 :   GEN T1 = NULL, T2 = NULL, V;
     594         266 :   dim = lg(gel(H,1))-1;
     595         266 :   V = vectrunc_init(dim+1);
     596         266 :   if (!dim) return V;
     597         259 :   (void)u_forprime_init(&S, 2, ULONG_MAX);
     598         259 :   vectrunc_append(V, H);
     599         259 :   first = 1; /* V[1..first-1] contains simple subspaces */
     600         399 :   while ((p = u_forprime_next(&S)))
     601             :   {
     602             :     GEN T;
     603             :     long j, lV;
     604         399 :     if (N % p == 0) continue;
     605         336 :     if (T1 && T2) {
     606          21 :       T = RgM_add(T1,T2);
     607          21 :       T2 = NULL;
     608             :     } else {
     609         315 :       T2 = T1;
     610         315 :       T1 = T = mshecke(W, p, NULL);
     611             :     }
     612         336 :     lV = lg(V);
     613         679 :     for (j = first; j < lV; j++)
     614             :     {
     615         343 :       pari_sp av = avma;
     616             :       long lP;
     617         343 :       GEN Vj = gel(V,j), P = gel(Vj,1);
     618         343 :       GEN TVj = Qevproj_apply(T, Vj); /* c T | V_j */
     619         343 :       GEN ch = QM_charpoly_ZX(TVj), fa = ZX_factor_limit(ch,deglim, &lP);
     620         343 :       GEN F = gel(fa, 1), E = gel(fa, 2);
     621         343 :       long k, lF = lg(F);
     622         343 :       if (lF == 2 && lP == 2)
     623             :       {
     624         336 :         if (equali1(gel(E,1)))
     625             :         { /* simple subspace */
     626         168 :           swap(gel(V,first), gel(V,j));
     627         168 :           first++;
     628             :         }
     629             :         else
     630           0 :           set_avma(av);
     631             :       }
     632         175 :       else if (lF == 1) /* discard V[j] */
     633           7 :       { swap(gel(V,j), gel(V,lg(V)-1)); setlg(V, lg(V)-1); }
     634             :       else
     635             :       { /* can split Vj */
     636             :         GEN pows;
     637         168 :         long D = 1;
     638         658 :         for (k = 1; k < lF; k++)
     639             :         {
     640         490 :           long d = degpol(gel(F,k));
     641         490 :           if (d > D) D = d;
     642             :         }
     643             :         /* remove V[j] */
     644         168 :         gel(V,j) = gel(V,lg(V)-1); setlg(V, lg(V)-1);
     645         168 :         pows = RgM_powers(TVj, minss((long)2*sqrt((double)D), D));
     646         658 :         for (k = 1; k < lF; k++)
     647             :         {
     648         490 :           GEN f = gel(F,k);
     649         490 :           GEN K = QM_ker( RgX_RgMV_eval(f, pows)) ; /* Ker f(TVj) */
     650         490 :           GEN p = vec_Q_primpart( RgM_mul(P, K) );
     651         490 :           vectrunc_append(V, Qevproj_init(p));
     652         490 :           if (lg(K) == 2 || isint1(gel(E,k)))
     653             :           { /* simple subspace */
     654         406 :             swap(gel(V,first), gel(V, lg(V)-1));
     655         406 :             first++;
     656             :           }
     657             :         }
     658         168 :         if (j < first) j = first;
     659             :       }
     660             :     }
     661         336 :     if (first >= lg(V)) {
     662         259 :       gen_sort_inplace(V, NULL, cmp_dim, NULL);
     663         259 :       return V;
     664             :     }
     665             :   }
     666           0 :   pari_err_BUG("subspaces not found");
     667             :   return NULL;/*LCOV_EXCL_LINE*/
     668             : }
     669             : GEN
     670         266 : mssplit(GEN W, GEN H, long deglim)
     671             : {
     672         266 :   pari_sp av = avma;
     673         266 :   checkms(W);
     674         266 :   if (!msk_get_sign(W))
     675           0 :     pari_err_DOMAIN("mssplit","abs(sign)","!=",gen_1,gen_0);
     676         266 :   if (!H) H = msnew(W);
     677         266 :   H = Qevproj_init0(H);
     678         266 :   return gerepilecopy(av, mssplit_i(W,H,deglim));
     679             : }
     680             : 
     681             : /* proV = Qevproj_init of a Hecke simple subspace, return [ a_n, n <= B ] */
     682             : static GEN
     683         245 : msqexpansion_i(GEN W, GEN proV, ulong B)
     684             : {
     685         245 :   ulong p, N = ms_get_N(W), sqrtB;
     686         245 :   long i, d, k = msk_get_weight(W);
     687             :   forprime_t S;
     688         245 :   GEN T1=NULL, T2=NULL, TV=NULL, ch=NULL, v, dTiv, Tiv, diM, iM, L;
     689         245 :   switch(B)
     690             :   {
     691           0 :     case 0: return cgetg(1,t_VEC);
     692           0 :     case 1: return mkvec(gen_1);
     693             :   }
     694         245 :   (void)u_forprime_init(&S, 2, ULONG_MAX);
     695         357 :   while ((p = u_forprime_next(&S)))
     696             :   {
     697             :     GEN T;
     698         357 :     if (N % p == 0) continue;
     699         266 :     if (T1 && T2)
     700             :     {
     701           0 :       T = RgM_add(T1,T2);
     702           0 :       T2 = NULL;
     703             :     }
     704             :     else
     705             :     {
     706         266 :       T2 = T1;
     707         266 :       T1 = T = mshecke(W, p, NULL);
     708             :     }
     709         266 :     TV = Qevproj_apply(T, proV); /* T | V */
     710         266 :     ch = QM_charpoly_ZX(TV);
     711         266 :     if (ZX_is_irred(ch)) break;
     712          21 :     ch = NULL;
     713             :   }
     714         245 :   if (!ch) pari_err_BUG("q-Expansion not found");
     715             :   /* T generates the Hecke algebra (acting on V) */
     716         245 :   d = degpol(ch);
     717         245 :   v = vec_ei(d, 1); /* take v = e_1 */
     718         245 :   Tiv = cgetg(d+1, t_MAT); /* Tiv[i] = T^(i-1)v */
     719         245 :   gel(Tiv, 1) = v;
     720         343 :   for (i = 2; i <= d; i++) gel(Tiv, i) = RgM_RgC_mul(TV, gel(Tiv,i-1));
     721         245 :   Tiv = Q_remove_denom(Tiv, &dTiv);
     722         245 :   iM = ZM_inv(Tiv, &diM);
     723         245 :   if (dTiv) diM = gdiv(diM, dTiv);
     724         245 :   L = const_vec(B,NULL);
     725         245 :   sqrtB = usqrt(B);
     726         245 :   gel(L,1) = d > 1? mkpolmod(gen_1,ch): gen_1;
     727        2471 :   for (p = 2; p <= B; p++)
     728             :   {
     729        2226 :     pari_sp av = avma;
     730             :     GEN T, u, Tv, ap, P;
     731             :     ulong m;
     732        2226 :     if (gel(L,p)) continue;  /* p not prime */
     733         819 :     T = mshecke(W, p, NULL);
     734         819 :     Tv = Qevproj_apply_vecei(T, proV, 1); /* Tp.v */
     735             :     /* Write Tp.v = \sum u_i T^i v */
     736         819 :     u = RgC_Rg_div(RgM_RgC_mul(iM, Tv), diM);
     737         819 :     ap = gerepilecopy(av, RgV_to_RgX(u, 0));
     738         819 :     if (d > 1)
     739         399 :       ap = mkpolmod(ap,ch);
     740             :     else
     741         420 :       ap = simplify_shallow(ap);
     742         819 :     gel(L,p) = ap;
     743         819 :     if (!(N % p))
     744             :     { /* p divides the level */
     745         147 :       ulong C = B/p;
     746         546 :       for (m=1; m<=C; m++)
     747         399 :         if (gel(L,m)) gel(L,m*p) = gmul(gel(L,m), ap);
     748         147 :       continue;
     749             :     }
     750         672 :     P = powuu(p,k-1);
     751         672 :     if (p <= sqrtB) {
     752         119 :       ulong pj, oldpj = 1;
     753         546 :       for (pj = p; pj <= B; oldpj=pj, pj *= p)
     754             :       {
     755         427 :         GEN apj = (pj==p)? ap
     756         427 :                          : gsub(gmul(ap,gel(L,oldpj)), gmul(P,gel(L,oldpj/p)));
     757         427 :         gel(L,pj) = apj;
     758        3136 :         for (m = B/pj; m > 1; m--)
     759        2709 :           if (gel(L,m) && m%p) gel(L,m*pj) = gmul(gel(L,m), apj);
     760             :       }
     761             :     } else {
     762         553 :       gel(L,p) = ap;
     763        1092 :       for (m = B/p; m > 1; m--)
     764         539 :         if (gel(L,m)) gel(L,m*p) = gmul(gel(L,m), ap);
     765             :     }
     766             :   }
     767         245 :   return L;
     768             : }
     769             : GEN
     770         245 : msqexpansion(GEN W, GEN proV, ulong B)
     771             : {
     772         245 :   pari_sp av = avma;
     773         245 :   checkms(W);
     774         245 :   proV = Qevproj_init0(proV);
     775         245 :   return gerepilecopy(av, msqexpansion_i(W,proV,B));
     776             : }
     777             : 
     778             : static GEN
     779         217 : Qevproj_apply0(GEN T, GEN pro)
     780             : {
     781         217 :   GEN iM = gel(pro,2), perm = gel(pro,4);
     782         217 :   return vec_Q_primpart(ZM_mul(iM, rowpermute(T,perm)));
     783             : }
     784             : /* T a ZC or ZM */
     785             : GEN
     786        4186 : Qevproj_down(GEN T, GEN pro)
     787             : {
     788        4186 :   GEN iM = gel(pro,2), ciM = gel(pro,3), perm = gel(pro,4);
     789        4186 :   if (typ(T) == t_COL)
     790        4186 :     return RgC_Rg_div(ZM_ZC_mul(iM, vecpermute(T,perm)), ciM);
     791             :   else
     792           0 :     return RgM_Rg_div(ZM_mul(iM, rowpermute(T,perm)), ciM);
     793             : }
     794             : 
     795             : static GEN
     796         287 : Qevproj_star(GEN W, GEN H)
     797             : {
     798         287 :   long s = msk_get_sign(W);
     799         287 :   if (s)
     800             :   { /* project on +/- component */
     801         217 :     GEN A = RgM_mul(msk_get_star(W), H);
     802         217 :     A = (s > 0)? gadd(A, H): gsub(A, H);
     803             :     /* Im(star + sign) = Ker(star - sign) */
     804         217 :     H = QM_image_shallow(A);
     805         217 :     H = Qevproj_apply0(H, msk_get_starproj(W));
     806             :   }
     807         287 :   return H;
     808             : }
     809             : 
     810             : static GEN
     811        3059 : Tp_matrices(ulong p)
     812             : {
     813        3059 :   GEN v = cgetg(p+2, t_VEC);
     814             :   ulong i;
     815       28707 :   for (i = 1; i <= p; i++) gel(v,i) = mat2(1, i-1, 0, p);
     816        3059 :   gel(v,i) = mat2(p, 0, 0, 1);
     817        3059 :   return v;
     818             : }
     819             : static GEN
     820         987 : Up_matrices(ulong p)
     821             : {
     822         987 :   GEN v = cgetg(p+1, t_VEC);
     823             :   ulong i;
     824        6300 :   for (i = 1; i <= p; i++) gel(v,i) = mat2(1, i-1, 0, p);
     825         987 :   return v;
     826             : }
     827             : 
     828             : /* M = N/p. Classes of Gamma_0(M) / Gamma_O(N) when p | M */
     829             : static GEN
     830         182 : NP_matrices(ulong M, ulong p)
     831             : {
     832         182 :   GEN v = cgetg(p+1, t_VEC);
     833             :   ulong i;
     834        1365 :   for (i = 1; i <= p; i++) gel(v,i) = mat2(1, 0, (i-1)*M, 1);
     835         182 :   return v;
     836             : }
     837             : /* M = N/p. Extra class of Gamma_0(M) / Gamma_O(N) when p \nmid M */
     838             : static GEN
     839          98 : NP_matrix_extra(ulong M, ulong p)
     840             : {
     841          98 :   long w,z, d = cbezout(p, -M, &w, &z);
     842          98 :   if (d != 1) return NULL;
     843          98 :   return mat2(w,z,M,p);
     844             : }
     845             : static GEN
     846         112 : WQ_matrix(long N, long Q)
     847             : {
     848         112 :   long w,z, d = cbezout(Q, N/Q, &w, &z);
     849         112 :   if (d != 1) return NULL;
     850         112 :   return mat2(Q,1,-N*z,Q*w);
     851             : }
     852             : 
     853             : GEN
     854         287 : msnew(GEN W)
     855             : {
     856         287 :   pari_sp av = avma;
     857         287 :   GEN S = mscuspidal(W, 0);
     858         287 :   ulong N = ms_get_N(W);
     859         287 :   long s = msk_get_sign(W), k = msk_get_weight(W);
     860         287 :   if (N > 1 && (!uisprime(N) || (k == 12 || k > 14)))
     861             :   {
     862         112 :     GEN p1N = ms_get_p1N(W), P = gel(p1N_get_fa(p1N), 1);
     863         112 :     long i, nP = lg(P)-1;
     864         112 :     GEN v = cgetg(2*nP + 1, t_COL);
     865         112 :     S = gel(S,1); /* Q basis */
     866         294 :     for (i = 1; i <= nP; i++)
     867             :     {
     868         182 :       pari_sp av = avma, av2;
     869         182 :       long M = N/P[i];
     870         182 :       GEN T1,Td, Wi = mskinit(M, k, s);
     871         182 :       GEN v1 = NP_matrices(M, P[i]);
     872         182 :       GEN vd = Up_matrices(P[i]);
     873             :       /* p^2 \nmid N */
     874         182 :       if (M % P[i])
     875             :       {
     876          98 :         v1 = vec_append(v1, NP_matrix_extra(M,P[i]));
     877          98 :         vd = vec_append(vd, WQ_matrix(N,P[i]));
     878             :       }
     879         182 :       T1 = getMorphism(W, Wi, v1);
     880         182 :       Td = getMorphism(W, Wi, vd);
     881         182 :       if (s)
     882             :       {
     883         168 :         T1 = Qevproj_apply2(T1, msk_get_starproj(W), msk_get_starproj(Wi));
     884         168 :         Td = Qevproj_apply2(Td, msk_get_starproj(W), msk_get_starproj(Wi));
     885             :       }
     886         182 :       av2 = avma;
     887         182 :       T1 = RgM_mul(T1,S);
     888         182 :       Td = RgM_mul(Td,S);  /* multiply by S = restrict to mscusp */
     889         182 :       gerepileallsp(av, av2, 2, &T1, &Td);
     890         182 :       gel(v,2*i-1) = T1;
     891         182 :       gel(v,2*i)   = Td;
     892             :     }
     893         112 :     S = ZM_mul(S, QM_ker(matconcat(v))); /* Snew */
     894         112 :     S = Qevproj_init(vec_Q_primpart(S));
     895             :   }
     896         287 :   return gerepilecopy(av, S);
     897             : }
     898             : 
     899             : /* Solve the Manin relations for a congruence subgroup \Gamma by constructing
     900             :  * a well-formed fundamental domain for the action of \Gamma on upper half
     901             :  * space. See
     902             :  * Pollack and Stevens, Overconvergent modular symbols and p-adic L-functions
     903             :  * Annales scientifiques de l'ENS 44, fascicule 1 (2011), 1-42
     904             :  * http://math.bu.edu/people/rpollack/Papers/Overconvergent_modular_symbols_and_padic_Lfunctions.pdf
     905             :  *
     906             :  * FIXME: Implemented for \Gamma = \Gamma_0(N) only. */
     907             : 
     908             : /* linked lists */
     909             : typedef struct list_t { GEN data; struct list_t *next; } list_t;
     910             : static list_t *
     911      115164 : list_new(GEN x)
     912             : {
     913      115164 :   list_t *L = (list_t*)stack_malloc(sizeof(list_t));
     914      115164 :   L->data = x;
     915      115164 :   L->next = NULL; return L;
     916             : }
     917             : static void
     918      113085 : list_insert(list_t *L, GEN x)
     919             : {
     920      113085 :   list_t *l = list_new(x);
     921      113085 :   l->next = L->next;
     922      113085 :   L->next = l;
     923      113085 : }
     924             : 
     925             : /*Input: N > 1, p1N = P^1(Z/NZ)
     926             :  *Output: a connected fundamental domain for the action of \Gamma_0(N) on
     927             :  *  upper half space.  When \Gamma_0(N) is torsion free, the domain has the
     928             :  *  property that all of its vertices are cusps.  When \Gamma_0(N) has
     929             :  *  three-torsion, 2 extra triangles need to be added.
     930             :  *
     931             :  * The domain is constructed by beginning with the triangle with vertices 0,1
     932             :  * and oo.  Each adjacent triangle is successively tested to see if it contains
     933             :  * points not \Gamma_0(N) equivalent to some point in our region.  If a
     934             :  * triangle contains new points, it is added to the region.  This process is
     935             :  * continued until the region can no longer be extended (and still be a
     936             :  * fundamental domain) by added an adjacent triangle.  The list of cusps
     937             :  * between 0 and 1 are then returned
     938             :  *
     939             :  * Precisely, the function returns a list such that the elements of the list
     940             :  * with odd index are the cusps in increasing order.  The even elements of the
     941             :  * list are either an "x" or a "t".  A "t" represents that there is an element
     942             :  * of order three such that its fixed point is in the triangle directly
     943             :  * adjacent to the our region with vertices given by the cusp before and after
     944             :  * the "t".  The "x" represents that this is not the case. */
     945             : enum { type_X, type_DO /* ? */, type_T };
     946             : static GEN
     947        2079 : form_list_of_cusps(ulong N, GEN p1N)
     948             : {
     949        2079 :   pari_sp av = avma;
     950        2079 :   long i, position, nbC = 2;
     951             :   GEN v, L;
     952             :   list_t *C, *c;
     953             :   /* Let t be the index of a class in PSL2(Z) / \Gamma in our fixed enumeration
     954             :    * v[t] != 0 iff it is the class of z tau^r for z a previous alpha_i
     955             :    * or beta_i.
     956             :    * For \Gamma = \Gamma_0(N), the enumeration is given by p1_index.
     957             :    * We write cl(gamma) = the class of gamma mod \Gamma */
     958        2079 :   v = const_vecsmall(p1_size(p1N), 0);
     959        2079 :   i = p1_index( 0, 1, p1N); v[i] = 1;
     960        2079 :   i = p1_index( 1,-1, p1N); v[i] = 2;
     961        2079 :   i = p1_index(-1, 0, p1N); v[i] = 3;
     962             :   /* the value is unused [debugging]: what matters is whether it is != 0 */
     963        2079 :   position = 4;
     964             :   /* at this point, Fund = R, v contains the classes of Id, tau, tau^2 */
     965             : 
     966        2079 :   C  = list_new(mkvecsmall3(0,1, type_X));
     967        2079 :   list_insert(C, mkvecsmall3(1,1,type_DO));
     968             :   /* C is a list of triples[a,b,t], where c = a/b is a cusp, and t is the type
     969             :    * of the path between c and the PREVIOUS cusp in the list, coded as
     970             :    *   type_DO = "?", type_X = "x", type_T = "t"
     971             :    * Initially, C = [0/1,"?",1/1]; */
     972             : 
     973             :   /* loop through the current set of cusps C and check to see if more cusps
     974             :    * should be added */
     975             :   for (;;)
     976       10367 :   {
     977       12446 :     int done = 1;
     978      542836 :     for (c = C; c; c = c->next)
     979             :     {
     980             :       GEN cusp1, cusp2, gam;
     981             :       long pos, b1, b2, b;
     982             : 
     983      542836 :       if (!c->next) break;
     984      530390 :       cusp1 = c->data; /* = a1/b1 */
     985      530390 :       cusp2 = (c->next)->data; /* = a2/b2 */
     986      530390 :       if (cusp2[3] != type_DO) continue;
     987             : 
     988             :       /* gam (oo -> 0) = (cusp2 -> cusp1), gam in PSL2(Z) */
     989      224091 :       gam = path_to_zm(mkpath(cusp2, cusp1)); /* = [a2,a1;b2,b1] */
     990             :       /* we have normalized the cusp representation so that a1 b2 - a2 b1 = 1 */
     991      224091 :       b1 = coeff(gam,2,1); b2 = coeff(gam,2,2);
     992             :       /* gam.1  = (a1 + a2) / (b1 + b2) */
     993      224091 :       b = b1 + b2;
     994             :       /* Determine whether the adjacent triangle *below* (cusp1->cusp2)
     995             :        * should be added */
     996      224091 :       pos = p1_index(b1,b2, p1N); /* did we see cl(gam) before ? */
     997      224091 :       if (v[pos])
     998      112196 :         cusp2[3] = type_X; /* NO */
     999             :       else
    1000             :       { /* YES */
    1001             :         ulong B1, B2;
    1002      111895 :         v[pos] = position;
    1003      111895 :         i = p1_index(-(b1+b2), b1, p1N); v[i] = position+1;
    1004      111895 :         i = p1_index(b2, -(b1+b2), p1N); v[i] = position+2;
    1005             :         /* add cl(gam), cl(gam*TAU), cl(gam*TAU^2) to v */
    1006      111895 :         position += 3;
    1007             :         /* gam tau gam^(-1) in \Gamma ? */
    1008      111895 :         B1 = umodsu(b1, N);
    1009      111895 :         B2 = umodsu(b2, N);
    1010      111895 :         if ((Fl_sqr(B2,N) + Fl_sqr(B1,N) + Fl_mul(B1,B2,N)) % N == 0)
    1011         889 :           cusp2[3] = type_T;
    1012             :         else
    1013             :         {
    1014      111006 :           long a1 = coeff(gam, 1,1), a2 = coeff(gam, 1,2);
    1015      111006 :           long a = a1 + a2; /* gcd(a,b) = 1 */
    1016      111006 :           list_insert(c, mkvecsmall3(a,b,type_DO));
    1017      111006 :           c = c->next;
    1018      111006 :           nbC++;
    1019      111006 :           done = 0;
    1020             :         }
    1021             :       }
    1022             :     }
    1023       12446 :     if (done) break;
    1024             :   }
    1025        2079 :   L = cgetg(nbC+1, t_VEC); i = 1;
    1026      117243 :   for (c = C; c; c = c->next) gel(L,i++) = c->data;
    1027        2079 :   return gerepilecopy(av, L);
    1028             : }
    1029             : 
    1030             : /* W an msN. M in PSL2(Z). Return index of M in P1^(Z/NZ) = Gamma0(N) \ PSL2(Z),
    1031             :  * and M0 in Gamma_0(N) such that M = M0 * M', where M' = chosen
    1032             :  * section( PSL2(Z) -> P1^(Z/NZ) ). */
    1033             : static GEN
    1034      498463 : Gamma0N_decompose(GEN W, GEN M, long *index)
    1035             : {
    1036      498463 :   GEN p1N = msN_get_p1N(W), W3 = gel(W,3), section = msN_get_section(W);
    1037             :   GEN A;
    1038      498463 :   ulong N = p1N_get_N(p1N);
    1039      498463 :   ulong c = umodiu(gcoeff(M,2,1), N);
    1040      498463 :   ulong d = umodiu(gcoeff(M,2,2), N);
    1041      498463 :   long s, ind = p1_index(c, d, p1N); /* as an elt of P1(Z/NZ) */
    1042      498463 :   *index = W3[ind]; /* as an elt of F, E2, ... */
    1043      498463 :   M = ZM_zm_mul(M, sl2_inv(gel(section,ind)));
    1044             :   /* normalize mod +/-Id */
    1045      498463 :   A = gcoeff(M,1,1);
    1046      498463 :   s = signe(A);
    1047      498463 :   if (s < 0)
    1048      237125 :     M = ZM_neg(M);
    1049      261338 :   else if (!s)
    1050             :   {
    1051         378 :     GEN C = gcoeff(M,2,1);
    1052         378 :     if (signe(C) < 0) M = ZM_neg(M);
    1053             :   }
    1054      498463 :   return M;
    1055             : }
    1056             : /* W an msN; as above for a path. Return [[ind], M] */
    1057             : static GEN
    1058      234486 : path_Gamma0N_decompose(GEN W, GEN path)
    1059             : {
    1060      234486 :   GEN p1N = msN_get_p1N(W);
    1061      234486 :   GEN p1index_to_ind = gel(W,3);
    1062      234486 :   GEN section = msN_get_section(W);
    1063      234486 :   GEN M = path_to_zm(path);
    1064      234486 :   long p1index = p1_index(cc(M), dd(M), p1N);
    1065      234486 :   long ind = p1index_to_ind[p1index];
    1066      234486 :   GEN M0 = ZM_zm_mul(mat2_to_ZM(M), sl2_inv(gel(section,p1index)));
    1067      234486 :   return mkvec2(mkvecsmall(ind), M0);
    1068             : }
    1069             : 
    1070             : /*Form generators of H_1(X_0(N),{cusps},Z)
    1071             : *
    1072             : *Input: N = integer > 1, p1N = P^1(Z/NZ)
    1073             : *Output: [cusp_list,E,F,T2,T3,E1] where
    1074             : *  cusps_list = list of cusps describing fundamental domain of
    1075             : *    \Gamma_0(N).
    1076             : *  E = list of paths in the boundary of the fundamental domains and oriented
    1077             : *    clockwise such that they do not contain a point
    1078             : *    fixed by an element of order 2 and they are not an edge of a
    1079             : *    triangle containing a fixed point of an element of order 3
    1080             : *  F = list of paths in the interior of the domain with each
    1081             : *    orientation appearing separately
    1082             : * T2 = list of paths in the boundary of domain containing a point fixed
    1083             : *    by an element of order 2 (oriented clockwise)
    1084             : * T3 = list of paths in the boundard of domain which are the edges of
    1085             : *    some triangle containing a fixed point of a matrix of order 3 (both
    1086             : *    orientations appear)
    1087             : * E1 = a sublist of E such that every path in E is \Gamma_0(N)-equivalent to
    1088             : *    either an element of E1 or the flip (reversed orientation) of an element
    1089             : *    of E1.
    1090             : * (Elements of T2 are \Gamma_0(N)-equivalent to their own flip.)
    1091             : *
    1092             : * sec = a list from 1..#p1N of matrices describing a section of the map
    1093             : *   SL_2(Z) to P^1(Z/NZ) given by [a,b;c,d]-->[c,d].
    1094             : *   Given our fixed enumeration of P^1(Z/NZ), the j-th element of the list
    1095             : *   represents the image of the j-th element of P^1(Z/NZ) under the section. */
    1096             : 
    1097             : /* insert path in set T */
    1098             : static void
    1099      340144 : set_insert(hashtable *T, GEN path)
    1100      340144 : { hash_insert(T, path,  (void*)(T->nb + 1)); }
    1101             : 
    1102             : static GEN
    1103       18711 : hash_to_vec(hashtable *h)
    1104             : {
    1105       18711 :   GEN v = cgetg(h->nb + 1, t_VEC);
    1106             :   ulong i;
    1107     2696764 :   for (i = 0; i < h->len; i++)
    1108             :   {
    1109     2678053 :     hashentry *e = h->table[i];
    1110     3242064 :     while (e)
    1111             :     {
    1112      564011 :       GEN key = (GEN)e->key;
    1113      564011 :       long index = (long)e->val;
    1114      564011 :       gel(v, index) = key;
    1115      564011 :       e = e->next;
    1116             :     }
    1117             :   }
    1118       18711 :   return v;
    1119             : }
    1120             : 
    1121             : static long
    1122      173082 : path_to_p1_index(GEN path, GEN p1N)
    1123             : {
    1124      173082 :   GEN M = path_to_zm(path);
    1125      173082 :   return p1_index(cc(M), dd(M), p1N);
    1126             : }
    1127             : 
    1128             : /* Pollack-Stevens sets */
    1129             : typedef struct PS_sets_t {
    1130             :   hashtable *F, *T2, *T31, *T32, *E1, *E2;
    1131             :   GEN E2fromE1, stdE1;
    1132             : } PS_sets_t;
    1133             : 
    1134             : static hashtable *
    1135       16142 : set_init(long max)
    1136       16142 : { return hash_create(max, (ulong(*)(void*))&hash_GEN,
    1137             :                           (int(*)(void*,void*))&gidentical, 1); }
    1138             : /* T = E2fromE1[i] = [c, gamma] */
    1139             : static ulong
    1140    40032230 : E2fromE1_c(GEN T) { return itou(gel(T,1)); }
    1141             : static GEN
    1142      579957 : E2fromE1_Zgamma(GEN T) { return gel(T,2); }
    1143             : static GEN
    1144       57694 : E2fromE1_gamma(GEN T) { return gcoeff(gel(T,2),1,1); }
    1145             : 
    1146             : static void
    1147      115388 : insert_E(GEN path, PS_sets_t *S, GEN p1N)
    1148             : {
    1149      115388 :   GEN rev = vecreverse(path);
    1150      115388 :   long std = path_to_p1_index(rev, p1N);
    1151      115388 :   GEN v = gel(S->stdE1, std);
    1152      115388 :   if (v)
    1153             :   { /* [s, p1], where E1[s] is the path p1 = vecreverse(path) mod \Gamma */
    1154       57694 :     GEN gamma, p1 = gel(v,2);
    1155       57694 :     long r, s = itou(gel(v,1));
    1156             : 
    1157       57694 :     set_insert(S->E2, path);
    1158       57694 :     r = S->E2->nb;
    1159       57694 :     if (gel(S->E2fromE1, r) != gen_0) pari_err_BUG("insert_E");
    1160             : 
    1161       57694 :     gamma = gamma_equiv_matrix(rev, p1);
    1162             :     /* reverse(E2[r]) = gamma * E1[s] */
    1163       57694 :     gel(S->E2fromE1, r) = mkvec2(utoipos(s), to_famat_shallow(gamma,gen_m1));
    1164             :   }
    1165             :   else
    1166             :   {
    1167       57694 :     set_insert(S->E1, path);
    1168       57694 :     std = path_to_p1_index(path, p1N);
    1169       57694 :     gel(S->stdE1, std) = mkvec2(utoipos(S->E1->nb), path);
    1170             :   }
    1171      115388 : }
    1172             : 
    1173             : static GEN
    1174        8316 : cusp_infinity(void) { return mkvecsmall2(1,0); }
    1175             : 
    1176             : static void
    1177        2079 : form_E_F_T(ulong N, GEN p1N, GEN *pC, PS_sets_t *S)
    1178             : {
    1179        2079 :   GEN C, cusp_list = form_list_of_cusps(N, p1N);
    1180        2079 :   long nbgen = lg(cusp_list)-1, nbmanin = p1_size(p1N), r, s, i;
    1181             :   hashtable *F, *T2, *T31, *T32, *E1, *E2;
    1182             : 
    1183        2079 :   *pC = C = cgetg(nbgen+1, t_VEC);
    1184      117243 :   for (i = 1; i <= nbgen; i++)
    1185             :   {
    1186      115164 :     GEN c = gel(cusp_list,i);
    1187      115164 :     gel(C,i) = mkvecsmall2(c[1], c[2]);
    1188             :   }
    1189        2079 :   S->F  = F  = set_init(nbmanin);
    1190        2079 :   S->E1 = E1 = set_init(nbgen);
    1191        2079 :   S->E2 = E2 = set_init(nbgen);
    1192        2079 :   S->T2 = T2 = set_init(nbgen);
    1193        2079 :   S->T31 = T31 = set_init(nbgen);
    1194        2079 :   S->T32 = T32 = set_init(nbgen);
    1195             : 
    1196             :   /* T31 represents the three torsion paths going from left to right */
    1197             :   /* T32 represents the three torsion paths going from right to left */
    1198      115164 :   for (r = 1; r < nbgen; r++)
    1199             :   {
    1200      113085 :     GEN c2 = gel(cusp_list,r+1);
    1201      113085 :     if (c2[3] == type_T)
    1202             :     {
    1203         889 :       GEN c1 = gel(cusp_list,r), path = mkpath(c1,c2), path2 = vecreverse(path);
    1204         889 :       set_insert(T31, path);
    1205         889 :       set_insert(T32, path2);
    1206             :     }
    1207             :   }
    1208             : 
    1209             :   /* to record relations between E2 and E1 */
    1210        2079 :   S->E2fromE1 = zerovec(nbgen);
    1211        2079 :   S->stdE1 = const_vec(nbmanin, NULL);
    1212             : 
    1213             :   /* Assumption later: path [oo,0] is E1[1], path [1,oo] is E2[1] */
    1214             :   {
    1215        2079 :     GEN oo = cusp_infinity();
    1216        2079 :     GEN p1 = mkpath(oo, mkvecsmall2(0,1)); /* [oo, 0] */
    1217        2079 :     GEN p2 = mkpath(mkvecsmall2(1,1), oo); /* [1, oo] */
    1218        2079 :     insert_E(p1, S, p1N);
    1219        2079 :     insert_E(p2, S, p1N);
    1220             :   }
    1221             : 
    1222      115164 :   for (r = 1; r < nbgen; r++)
    1223             :   {
    1224      113085 :     GEN c1 = gel(cusp_list,r);
    1225    22298080 :     for (s = r+1; s <= nbgen; s++)
    1226             :     {
    1227    22184995 :       pari_sp av = avma;
    1228    22184995 :       GEN c2 = gel(cusp_list,s), path;
    1229    22184995 :       GEN d = subii(mulss(c1[1],c2[2]), mulss(c1[2],c2[1]));
    1230    22184995 :       set_avma(av);
    1231    22184995 :       if (!is_pm1(d)) continue;
    1232             : 
    1233      224091 :       path = mkpath(c1,c2);
    1234      224091 :       if (r+1 == s)
    1235             :       {
    1236      113085 :         GEN w = path;
    1237      113085 :         ulong hash = T31->hash(w); /* T31, T32 use the same hash function */
    1238      113085 :         if (!hash_search2(T31, w, hash) && !hash_search2(T32, w, hash))
    1239             :         {
    1240      112196 :           if (gamma_equiv(path, vecreverse(path), N))
    1241         966 :             set_insert(T2, path);
    1242             :           else
    1243      111230 :             insert_E(path, S, p1N);
    1244             :         }
    1245             :       } else {
    1246      111006 :         set_insert(F, mkvec2(path, mkvecsmall2(r,s)));
    1247      111006 :         set_insert(F, mkvec2(vecreverse(path), mkvecsmall2(s,r)));
    1248             :       }
    1249             :     }
    1250             :   }
    1251        2079 :   setlg(S->E2fromE1, E2->nb+1);
    1252        2079 : }
    1253             : 
    1254             : /* v = \sum n_i g_i, g_i in Sl(2,Z), return \sum n_i g_i^(-1) */
    1255             : static GEN
    1256      845705 : ZSl2_star(GEN v)
    1257             : {
    1258             :   long i, l;
    1259             :   GEN w, G;
    1260      845705 :   if (typ(v) == t_INT) return v;
    1261      845705 :   G = gel(v,1);
    1262      845705 :   w = cgetg_copy(G, &l);
    1263     2015363 :   for (i = 1; i < l; i++)
    1264             :   {
    1265     1169658 :     GEN g = gel(G,i);
    1266     1169658 :     if (typ(g) == t_MAT) g = SL2_inv_shallow(g);
    1267     1169658 :     gel(w,i) = g;
    1268             :   }
    1269      845705 :   return ZG_normalize(mkmat2(w, gel(v,2)));
    1270             : }
    1271             : 
    1272             : /* Input: h = set of unimodular paths, p1N = P^1(Z/NZ) = Gamma_0(N)\PSL2(Z)
    1273             :  * Output: Each path is converted to a matrix and then an element of P^1(Z/NZ)
    1274             :  * Append the matrix to W[12], append the index that represents
    1275             :  * these elements of P^1 (the classes mod Gamma_0(N) via our fixed
    1276             :  * enumeration to W[2]. */
    1277             : static void
    1278       12474 : paths_decompose(GEN W, hashtable *h, int flag)
    1279             : {
    1280       12474 :   GEN p1N = ms_get_p1N(W), section = ms_get_section(W);
    1281       12474 :   GEN v = hash_to_vec(h);
    1282       12474 :   long i, l = lg(v);
    1283      352618 :   for (i = 1; i < l; i++)
    1284             :   {
    1285      340144 :     GEN e = gel(v,i);
    1286      340144 :     GEN M = path_to_zm(flag? gel(e,1): e);
    1287      340144 :     long index = p1_index(cc(M), dd(M), p1N);
    1288      340144 :     vecsmalltrunc_append(gel(W,2), index);
    1289      340144 :     gel(section, index) = M;
    1290             :   }
    1291       12474 : }
    1292             : static void
    1293        2079 : fill_W2_W12(GEN W, PS_sets_t *S)
    1294             : {
    1295        2079 :   GEN p1N = msN_get_p1N(W);
    1296        2079 :   long n = p1_size(p1N);
    1297        2079 :   gel(W, 2) = vecsmalltrunc_init(n+1);
    1298        2079 :   gel(W,12) = cgetg(n+1, t_VEC);
    1299             :   /* F contains [path, [index cusp1, index cusp2]]. Others contain paths only */
    1300        2079 :   paths_decompose(W, S->F, 1);
    1301        2079 :   paths_decompose(W, S->E2, 0);
    1302        2079 :   paths_decompose(W, S->T32, 0);
    1303        2079 :   paths_decompose(W, S->E1, 0);
    1304        2079 :   paths_decompose(W, S->T2, 0);
    1305        2079 :   paths_decompose(W, S->T31, 0);
    1306        2079 : }
    1307             : 
    1308             : /* x t_VECSMALL, corresponds to a map x(i) = j, where 1 <= j <= max for all i
    1309             :  * Return y s.t. y[j] = i or 0 (not in image) */
    1310             : static GEN
    1311        4158 : reverse_list(GEN x, long max)
    1312             : {
    1313        4158 :   GEN y = const_vecsmall(max, 0);
    1314        4158 :   long r, lx = lg(x);
    1315      403851 :   for (r = 1; r < lx; r++) y[ x[r] ] = r;
    1316        4158 :   return y;
    1317             : }
    1318             : 
    1319             : /* go from C[a] to C[b]; return the indices of paths
    1320             :  * E.g. if a < b
    1321             :  *   (C[a]->C[a+1], C[a+1]->C[a+2], ... C[b-1]->C[b])
    1322             :  * (else reverse direction)
    1323             :  * = b - a paths */
    1324             : static GEN
    1325      216818 : F_indices(GEN W, long a, long b)
    1326             : {
    1327      216818 :   GEN v = cgetg(labs(b-a) + 1, t_VEC);
    1328      216818 :   long s, k = 1;
    1329      216818 :   if (a < b) {
    1330      108409 :     GEN index_forward = gel(W,13);
    1331      841127 :     for (s = a; s < b; s++) gel(v,k++) = gel(index_forward,s);
    1332             :   } else {
    1333      108409 :     GEN index_backward = gel(W,14);
    1334      841127 :     for (s = a; s > b; s--) gel(v,k++) = gel(index_backward,s);
    1335             :   }
    1336      216818 :   return v;
    1337             : }
    1338             : /* go from C[a] to C[b] via oo; return the indices of paths
    1339             :  * E.g. if a < b
    1340             :  *   (C[a]->C[a-1], ... C[2]->C[1],
    1341             :  *    C[1]->oo, oo-> C[end],
    1342             :  *    C[end]->C[end-1], ... C[b+1]->C[b])
    1343             :  *  a-1 + 2 + end-(b+1)+1 = end - b + a + 1 paths  */
    1344             : static GEN
    1345        5194 : F_indices_oo(GEN W, long end, long a, long b)
    1346             : {
    1347        5194 :   GEN index_oo = gel(W,15);
    1348        5194 :   GEN v = cgetg(end-labs(b-a)+1 + 1, t_VEC);
    1349        5194 :   long s, k = 1;
    1350             : 
    1351        5194 :   if (a < b) {
    1352        2597 :     GEN index_backward = gel(W,14);
    1353        2597 :     for (s = a; s > 1; s--) gel(v,k++) = gel(index_backward,s);
    1354        2597 :     gel(v,k++) = gel(index_backward,1); /* C[1] -> oo */
    1355        2597 :     gel(v,k++) = gel(index_oo,2); /* oo -> C[end] */
    1356       45542 :     for (s = end; s > b; s--) gel(v,k++) = gel(index_backward,s);
    1357             :   } else {
    1358        2597 :     GEN index_forward = gel(W,13);
    1359       45542 :     for (s = a; s < end; s++) gel(v,k++) = gel(index_forward,s);
    1360        2597 :     gel(v,k++) = gel(index_forward,end); /* C[end] -> oo */
    1361        2597 :     gel(v,k++) = gel(index_oo,1); /* oo -> C[1] */
    1362        2597 :     for (s = 1; s < b; s++) gel(v,k++) = gel(index_forward,s);
    1363             :   }
    1364        5194 :   return v;
    1365             : }
    1366             : /* index of oo -> C[1], oo -> C[end] */
    1367             : static GEN
    1368        2079 : indices_oo(GEN W, GEN C)
    1369             : {
    1370        2079 :   long end = lg(C)-1;
    1371        2079 :   GEN w, v = cgetg(2+1, t_VEC), oo = cusp_infinity();
    1372        2079 :   w = mkpath(oo, gel(C,1)); /* oo -> C[1]=0 */
    1373        2079 :   gel(v,1) = path_Gamma0N_decompose(W, w);
    1374        2079 :   w = mkpath(oo, gel(C,end)); /* oo -> C[end]=1 */
    1375        2079 :   gel(v,2) = path_Gamma0N_decompose(W, w);
    1376        2079 :   return v;
    1377             : }
    1378             : 
    1379             : /* index of C[1]->C[2], C[2]->C[3], ... C[end-1]->C[end], C[end]->oo
    1380             :  * Recall that C[1] = 0, C[end] = 1 */
    1381             : static GEN
    1382        2079 : indices_forward(GEN W, GEN C)
    1383             : {
    1384        2079 :   long s, k = 1, end = lg(C)-1;
    1385        2079 :   GEN v = cgetg(end+1, t_VEC);
    1386      117243 :   for (s = 1; s <= end; s++)
    1387             :   {
    1388      115164 :     GEN w = mkpath(gel(C,s), s == end? cusp_infinity(): gel(C,s+1));
    1389      115164 :     gel(v,k++) = path_Gamma0N_decompose(W, w);
    1390             :   }
    1391        2079 :   return v;
    1392             : }
    1393             : /* index of C[1]->oo, C[2]->C[1], ... C[end]->C[end-1] */
    1394             : static GEN
    1395        2079 : indices_backward(GEN W, GEN C)
    1396             : {
    1397        2079 :   long s, k = 1, end = lg(C)-1;
    1398        2079 :   GEN v = cgetg(end+1, t_VEC);
    1399      117243 :   for (s = 1; s <= end; s++)
    1400             :   {
    1401      115164 :     GEN w = mkpath(gel(C,s), s == 1? cusp_infinity(): gel(C,s-1));
    1402      115164 :     gel(v,k++) = path_Gamma0N_decompose(W, w);
    1403             :   }
    1404        2079 :   return v;
    1405             : }
    1406             : 
    1407             : /*[0,-1;1,-1]*/
    1408             : static GEN
    1409        2163 : mkTAU()
    1410        2163 : { return mkmat22(gen_0,gen_m1, gen_1,gen_m1); }
    1411             : /* S */
    1412             : static GEN
    1413          84 : mkS()
    1414          84 : { return mkmat22(gen_0,gen_1, gen_m1,gen_0); }
    1415             : /* N = integer > 1. Returns data describing Delta_0 = Z[P^1(Q)]_0 seen as
    1416             :  * a Gamma_0(N) - module. */
    1417             : static GEN
    1418        2107 : msinit_N(ulong N)
    1419             : {
    1420             :   GEN p1N, C, vecF, vecT2, vecT31, TAU, W, W2, singlerel, annT2, annT31;
    1421             :   GEN F_index;
    1422             :   ulong r, s, width;
    1423             :   long nball, nbgen, nbp1N;
    1424             :   hashtable *F, *T2, *T31, *T32, *E1, *E2;
    1425             :   PS_sets_t S;
    1426             : 
    1427        2107 :   W = zerovec(16);
    1428        2107 :   gel(W,1) = p1N = create_p1mod(N);
    1429        2107 :   gel(W,16)= inithashcusps(p1N);
    1430        2107 :   TAU = mkTAU();
    1431        2107 :   if (N == 1)
    1432             :   {
    1433          28 :     gel(W,5) = mkvecsmall(1);
    1434             :     /* cheat because sets are not disjoint if N=1 */
    1435          28 :     gel(W,11) = mkvecsmall5(0, 0, 1, 1, 2);
    1436          28 :     gel(W,12) = mkvec(mat2(1,0,0,1));
    1437          28 :     gel(W,8) = mkvec( mkmat22(gen_1,gen_1, mkS(),gen_1) );
    1438          28 :     gel(W,9) = mkvec( mkmat2(mkcol3(gen_1,TAU,ZM_sqr(TAU)),
    1439             :                              mkcol3(gen_1,gen_1,gen_1)) );
    1440          28 :     return W;
    1441             :   }
    1442        2079 :   nbp1N = p1_size(p1N);
    1443        2079 :   form_E_F_T(N,p1N, &C, &S);
    1444        2079 :   E1  = S.E1;
    1445        2079 :   E2  = S.E2;
    1446        2079 :   T31 = S.T31;
    1447        2079 :   T32 = S.T32;
    1448        2079 :   F   = S.F;
    1449        2079 :   T2  = S.T2;
    1450        2079 :   nbgen = lg(C)-1;
    1451             : 
    1452             :  /* Put our paths in the order: F,E2,T32,E1,T2,T31
    1453             :   * W2[j] associates to the j-th element of this list its index in P1. */
    1454        2079 :   fill_W2_W12(W, &S);
    1455        2079 :   W2 = gel(W, 2);
    1456        2079 :   nball = lg(W2)-1;
    1457        2079 :   gel(W,3) = reverse_list(W2, nbp1N);
    1458        2079 :   gel(W,5) = vecslice(gel(W,2), F->nb + E2->nb + T32->nb + 1, nball);
    1459        2079 :   gel(W,4) = reverse_list(gel(W,5), nbp1N);
    1460        2079 :   gel(W,13) = indices_forward(W, C);
    1461        2079 :   gel(W,14) = indices_backward(W, C);
    1462        2079 :   gel(W,15) = indices_oo(W, C);
    1463       10395 :   gel(W,11) = mkvecsmall5(F->nb,
    1464        2079 :                           F->nb + E2->nb,
    1465        2079 :                           F->nb + E2->nb + T32->nb,
    1466        2079 :                           F->nb + E2->nb + T32->nb + E1->nb,
    1467        2079 :                           F->nb + E2->nb + T32->nb + E1->nb + T2->nb);
    1468             :   /* relations between T32 and T31 [not stored!]
    1469             :    * T32[i] = - T31[i] */
    1470             : 
    1471             :   /* relations of F */
    1472        2079 :   width = E1->nb + T2->nb + T31->nb;
    1473             :   /* F_index[r] = [index_1, ..., index_k], where index_i is the p1_index()
    1474             :    * of the elementary unimodular path between 2 consecutive cusps
    1475             :    * [in E1,E2,T2,T31 or T32] */
    1476        2079 :   F_index = cgetg(F->nb+1, t_VEC);
    1477        2079 :   vecF = hash_to_vec(F);
    1478      224091 :   for (r = 1; r <= F->nb; r++)
    1479             :   {
    1480      222012 :     GEN w = gel(gel(vecF,r), 2);
    1481      222012 :     long a = w[1], b = w[2], d = labs(b - a);
    1482             :     /* c1 = cusp_list[a],  c2 = cusp_list[b], ci != oo */
    1483      222012 :     gel(F_index,r) = (nbgen-d >= d-1)? F_indices(W, a,b)
    1484      222012 :                                      : F_indices_oo(W, lg(C)-1,a,b);
    1485             :   }
    1486             : 
    1487        2079 :   singlerel = cgetg(width+1, t_VEC);
    1488             :   /* form the single boundary relation */
    1489       59773 :   for (s = 1; s <= E2->nb; s++)
    1490             :   { /* reverse(E2[s]) = gamma * E1[c] */
    1491       57694 :     GEN T = gel(S.E2fromE1,s), gamma = E2fromE1_gamma(T);
    1492       57694 :     gel(singlerel, E2fromE1_c(T)) = mkmat22(gen_1,gen_1, gamma,gen_m1);
    1493             :   }
    1494        3934 :   for (r = E1->nb + 1; r <= width; r++) gel(singlerel, r) = gen_1;
    1495             : 
    1496             :   /* form the 2-torsion relations */
    1497        2079 :   annT2 = cgetg(T2->nb+1, t_VEC);
    1498        2079 :   vecT2 = hash_to_vec(T2);
    1499        3045 :   for (r = 1; r <= T2->nb; r++)
    1500             :   {
    1501         966 :     GEN w = gel(vecT2,r);
    1502         966 :     GEN gamma = gamma_equiv_matrix(vecreverse(w), w);
    1503         966 :     gel(annT2, r) = mkmat22(gen_1,gen_1, gamma,gen_1);
    1504             :   }
    1505             : 
    1506             :   /* form the 3-torsion relations */
    1507        2079 :   annT31 = cgetg(T31->nb+1, t_VEC);
    1508        2079 :   vecT31 = hash_to_vec(T31);
    1509        2968 :   for (r = 1; r <= T31->nb; r++)
    1510             :   {
    1511         889 :     GEN M = path_to_ZM( vecreverse(gel(vecT31,r)) );
    1512         889 :     GEN gamma = ZM_mul(ZM_mul(M, TAU), SL2_inv_shallow(M));
    1513         889 :     gel(annT31, r) = mkmat2(mkcol3(gen_1,gamma,ZM_sqr(gamma)),
    1514             :                             mkcol3(gen_1,gen_1,gen_1));
    1515             :   }
    1516        2079 :   gel(W,6) = F_index;
    1517        2079 :   gel(W,7) = S.E2fromE1;
    1518        2079 :   gel(W,8) = annT2;
    1519        2079 :   gel(W,9) = annT31;
    1520        2079 :   gel(W,10)= singlerel;
    1521        2079 :   return W;
    1522             : }
    1523             : static GEN
    1524         112 : cusp_to_P1Q(GEN c) { return c[2]? sstoQ(c[1], c[2]): mkoo(); }
    1525             : static GEN
    1526          21 : mspathgens_i(GEN W)
    1527             : {
    1528             :   GEN R, r, g, section, gen, annT2, annT31;
    1529             :   long i, l;
    1530          21 :   checkms(W); W = get_msN(W);
    1531          21 :   section = msN_get_section(W);
    1532          21 :   gen = ms_get_genindex(W);
    1533          21 :   l = lg(gen);
    1534          21 :   g = cgetg(l,t_VEC);
    1535          77 :   for (i = 1; i < l; i++)
    1536             :   {
    1537          56 :     GEN p = gel(section,gen[i]);
    1538          56 :     gel(g,i) = mkvec2(cusp_to_P1Q(gel(p,1)), cusp_to_P1Q(gel(p,2)));
    1539             :   }
    1540          21 :   annT2 = msN_get_annT2(W);
    1541          21 :   annT31= msN_get_annT31(W);
    1542          21 :   if (ms_get_N(W) == 1)
    1543             :   {
    1544           7 :     R = cgetg(3, t_VEC);
    1545           7 :     gel(R,1) = mkvec( mkvec2(gel(annT2,1), gen_1) );
    1546           7 :     gel(R,2) = mkvec( mkvec2(gel(annT31,1), gen_1) );
    1547             :   }
    1548             :   else
    1549             :   {
    1550          14 :     GEN singlerel = msN_get_singlerel(W);
    1551          14 :     long j, nbT2 = lg(annT2)-1, nbT31 = lg(annT31)-1, nbE1 = ms_get_nbE1(W);
    1552          14 :     R = cgetg(nbT2+nbT31+2, t_VEC);
    1553          14 :     l = lg(singlerel);
    1554          14 :     r = cgetg(l, t_VEC);
    1555          42 :     for (i = 1; i <= nbE1; i++)
    1556          28 :       gel(r,i) = mkvec2(gel(singlerel, i), utoi(i));
    1557          35 :     for (; i < l; i++)
    1558          21 :       gel(r,i) = mkvec2(gen_1, utoi(i));
    1559          14 :     gel(R,1) = r; j = 2;
    1560          35 :     for (i = 1; i <= nbT2; i++,j++)
    1561          21 :       gel(R,j) = mkvec( mkvec2(gel(annT2,i), utoi(i + nbE1)) );
    1562          14 :     for (i = 1; i <= nbT31; i++,j++)
    1563           0 :       gel(R,j) = mkvec( mkvec2(gel(annT31,i), utoi(i + nbE1 + nbT2)) );
    1564             :   }
    1565          21 :   return mkvec2(g,R);
    1566             : }
    1567             : GEN
    1568          21 : mspathgens(GEN W)
    1569             : {
    1570          21 :   pari_sp av = avma;
    1571          21 :   return gerepilecopy(av, mspathgens_i(W));
    1572             : }
    1573             : /* Modular symbols in weight k: Hom_Gamma(Delta, Q[x,y]_{k-2}) */
    1574             : /* A symbol phi is represented by the {phi(g_i)}, {phi(g'_i)}, {phi(g''_i)}
    1575             :  * where the {g_i, g'_i, g''_i} are the Z[\Gamma]-generators of Delta,
    1576             :  * g_i corresponds to E1, g'_i to T2, g''_i to T31.
    1577             :  */
    1578             : 
    1579             : /* FIXME: export. T^1, ..., T^n */
    1580             : static GEN
    1581      701834 : RgX_powers(GEN T, long n)
    1582             : {
    1583      701834 :   GEN v = cgetg(n+1, t_VEC);
    1584             :   long i;
    1585      701834 :   gel(v, 1) = T;
    1586     1643600 :   for (i = 1; i < n; i++) gel(v,i+1) = RgX_mul(gel(v,i), T);
    1587      701834 :   return v;
    1588             : }
    1589             : 
    1590             : /* g = [a,b;c,d] a mat2. Return (X^{k-2} | g)(X,Y)[X = 1]. */
    1591             : static GEN
    1592        2604 : voo_act_Gl2Q(GEN g, long k)
    1593             : {
    1594        2604 :   GEN mc = stoi(-coeff(g,2,1)), d = stoi(coeff(g,2,2));
    1595        2604 :   return RgX_to_RgC(gpowgs(deg1pol_shallow(mc, d, 0), k-2), k-1);
    1596             : }
    1597             : 
    1598             : struct m_act {
    1599             :   long dim, k, p;
    1600             :   GEN q;
    1601             :   GEN(*act)(struct m_act *,GEN);
    1602             : };
    1603             : 
    1604             : /* g = [a,b;c,d]. Return (P | g)(X,Y)[X = 1] = P(dX - cY, -b X + aY)[X = 1],
    1605             :  * for P = X^{k-2}, X^{k-3}Y, ..., Y^{k-2} */
    1606             : GEN
    1607      350749 : RgX_act_Gl2Q(GEN g, long k)
    1608             : {
    1609             :   GEN a,b,c,d, V1,V2,V;
    1610             :   long i;
    1611      350749 :   if (k == 2) return matid(1);
    1612      350749 :   a = gcoeff(g,1,1); b = gcoeff(g,1,2);
    1613      350749 :   c = gcoeff(g,2,1); d = gcoeff(g,2,2);
    1614      350749 :   V1 = RgX_powers(deg1pol_shallow(gneg(c), d, 0), k-2); /* d - c Y */
    1615      350749 :   V2 = RgX_powers(deg1pol_shallow(a, gneg(b), 0), k-2); /*-b + a Y */
    1616      350749 :   V = cgetg(k, t_MAT);
    1617      350749 :   gel(V,1)   = RgX_to_RgC(gel(V1, k-2), k-1);
    1618      819280 :   for (i = 1; i < k-2; i++)
    1619             :   {
    1620      468531 :     GEN v1 = gel(V1, k-2-i); /* (d-cY)^(k-2-i) */
    1621      468531 :     GEN v2 = gel(V2, i); /* (-b+aY)^i */
    1622      468531 :     gel(V,i+1) = RgX_to_RgC(RgX_mul(v1,v2), k-1);
    1623             :   }
    1624      350749 :   gel(V,k-1) = RgX_to_RgC(gel(V2, k-2), k-1);
    1625      350749 :   return V; /* V[i+1] = X^i | g */
    1626             : }
    1627             : /* z in Z[Gl2(Q)], return the matrix of z acting on V */
    1628             : static GEN
    1629      600649 : act_ZGl2Q(GEN z, struct m_act *T, hashtable *H)
    1630             : {
    1631      600649 :   GEN S = NULL, G, E;
    1632             :   pari_sp av;
    1633             :   long l, j;
    1634             :   /* paranoia: should not occur */
    1635      600649 :   if (typ(z) == t_INT) return scalarmat_shallow(z, T->dim);
    1636      600649 :   G = gel(z,1); l = lg(G);
    1637      600649 :   E = gel(z,2); av = avma;
    1638     1770307 :   for (j = 1; j < l; j++)
    1639             :   {
    1640     1169658 :     GEN M, g = gel(G,j), n = gel(E,j);
    1641     1169658 :     if (typ(g) == t_INT) /* = 1 */
    1642        3948 :       M = n; /* n*Id_dim */
    1643             :     else
    1644             :     { /*search in H succeeds because of preload*/
    1645     1165710 :       M = H? (GEN)hash_search(H,g)->val: T->act(T,g);
    1646     1165710 :       if (is_pm1(n))
    1647     1158185 :       { if (signe(n) < 0) M = RgM_neg(M); }
    1648             :       else
    1649        7525 :         M = RgM_Rg_mul(M, n);
    1650             :     }
    1651     1169658 :     if (!S) { S = M; continue; }
    1652      569009 :     S = gadd(S, M);
    1653      569009 :     if (gc_needed(av,1))
    1654             :     {
    1655           0 :       if(DEBUGMEM>1) pari_warn(warnmem,"act_ZGl2Q, j = %ld",j);
    1656           0 :       S = gerepileupto(av, S);
    1657             :     }
    1658             :   }
    1659      600649 :   return gerepilecopy(av, S);
    1660             : }
    1661             : static GEN
    1662      350602 : _RgX_act_Gl2Q(struct m_act *S, GEN z) { return RgX_act_Gl2Q(z, S->k); }
    1663             : /* acting on (X^{k-2},...,Y^{k-2}) */
    1664             : GEN
    1665       60907 : RgX_act_ZGl2Q(GEN z, long k)
    1666             : {
    1667             :   struct m_act T;
    1668       60907 :   T.k = k;
    1669       60907 :   T.dim = k-1;
    1670       60907 :   T.act=&_RgX_act_Gl2Q;
    1671       60907 :   return act_ZGl2Q(z, &T, NULL);
    1672             : }
    1673             : 
    1674             : /* First pass, identify matrices in Sl_2 to convert to operators;
    1675             :  * insert operators in hashtable. This allows GC in act_ZGl2Q */
    1676             : static void
    1677     1069894 : hash_preload(GEN M, struct m_act *S, hashtable *H)
    1678             : {
    1679     1069894 :   if (typ(M) != t_INT)
    1680             :   {
    1681     1069894 :     ulong h = H->hash(M);
    1682     1069894 :     hashentry *e = hash_search2(H, M, h);
    1683     1069894 :     if (!e) hash_insert2(H, M, S->act(S,M), h);
    1684             :   }
    1685     1069894 : }
    1686             : /* z a sparse operator */
    1687             : static void
    1688      539728 : hash_vecpreload(GEN z, struct m_act *S, hashtable *H)
    1689             : {
    1690      539728 :   GEN G = gel(z,1);
    1691      539728 :   long i, l = lg(G);
    1692     1609622 :   for (i = 1; i < l; i++) hash_preload(gel(G,i), S, H);
    1693      539728 : }
    1694             : static void
    1695       40677 : ZGl2QC_preload(struct m_act *S, GEN v, hashtable *H)
    1696             : {
    1697       40677 :   GEN val = gel(v,2);
    1698       40677 :   long i, l = lg(val);
    1699      580405 :   for (i = 1; i < l; i++) hash_vecpreload(gel(val,i), S, H);
    1700       40677 : }
    1701             : /* Given a sparse vector of elements in Z[G], convert it to a (sparse) vector
    1702             :  * of operators on V (given by t_MAT) */
    1703             : static void
    1704       40691 : ZGl2QC_to_act(struct m_act *S, GEN v, hashtable *H)
    1705             : {
    1706       40691 :   GEN val = gel(v,2);
    1707       40691 :   long i, l = lg(val);
    1708      580433 :   for (i = 1; i < l; i++) gel(val,i) = act_ZGl2Q(gel(val,i), S, H);
    1709       40691 : }
    1710             : 
    1711             : /* For all V[i] in Z[\Gamma], find the P such that  P . V[i]^* = 0;
    1712             :  * write P in basis X^{k-2}, ..., Y^{k-2} */
    1713             : static GEN
    1714        1260 : ZGV_tors(GEN V, long k)
    1715             : {
    1716        1260 :   long i, l = lg(V);
    1717        1260 :   GEN v = cgetg(l, t_VEC);
    1718        1764 :   for (i = 1; i < l; i++)
    1719             :   {
    1720         504 :     GEN a = ZSl2_star(gel(V,i));
    1721         504 :     gel(v,i) = ZM_ker(RgX_act_ZGl2Q(a,k));
    1722             :   }
    1723        1260 :   return v;
    1724             : }
    1725             : 
    1726             : static long
    1727   110641321 : set_from_index(GEN W11, long i)
    1728             : {
    1729   110641321 :   if (i <= W11[1]) return 1;
    1730    96328113 :   if (i <= W11[2]) return 2;
    1731    56359394 :   if (i <= W11[3]) return 3;
    1732    56171556 :   if (i <= W11[4]) return 4;
    1733     2309825 :   if (i <= W11[5]) return 5;
    1734      255626 :   return 6;
    1735             : }
    1736             : 
    1737             : /* det M = 1 */
    1738             : static void
    1739     1535667 : treat_index(GEN W, GEN M, long index, GEN v)
    1740             : {
    1741     1535667 :   GEN W11 = gel(W,11);
    1742     1535667 :   long shift = W11[3]; /* #F + #E2 + T32 */
    1743     1535667 :   switch(set_from_index(W11, index))
    1744             :   {
    1745      251167 :     case 1: /*F*/
    1746             :     {
    1747      251167 :       GEN F_index = gel(W,6), ind = gel(F_index, index);
    1748      251167 :       long j, l = lg(ind);
    1749     1288371 :       for (j = 1; j < l; j++)
    1750             :       {
    1751     1037204 :         GEN IND = gel(ind,j), M0 = gel(IND,2);
    1752     1037204 :         long index = mael(IND,1,1);
    1753     1037204 :         treat_index(W, ZM_mul(M,M0), index, v);
    1754             :       }
    1755      251167 :       break;
    1756             :     }
    1757             : 
    1758      579957 :     case 2: /*E2, E2[r] + gamma * E1[s] = 0 */
    1759             :     {
    1760      579957 :       long r = index - W11[1];
    1761      579957 :       GEN z = gel(msN_get_E2fromE1(W), r);
    1762             : 
    1763      579957 :       index = E2fromE1_c(z);
    1764      579957 :       M = G_ZG_mul(M, E2fromE1_Zgamma(z)); /* M * (-gamma) */
    1765      579957 :       gel(v, index) = ZG_add(gel(v, index), M);
    1766      579957 :       break;
    1767             :     }
    1768             : 
    1769        5922 :     case 3: /*T32, T32[i] = -T31[i] */
    1770             :     {
    1771        5922 :       long T3shift = W11[5] - W11[2]; /* #T32 + #E1 + #T2 */
    1772        5922 :       index += T3shift;
    1773        5922 :       index -= shift;
    1774        5922 :       gel(v, index) = ZG_add(gel(v, index), to_famat_shallow(M,gen_m1));
    1775        5922 :       break;
    1776             :     }
    1777      698621 :     default: /*E1,T2,T31*/
    1778      698621 :       index -= shift;
    1779      698621 :       gel(v, index) = ZG_add(gel(v, index), to_famat_shallow(M,gen_1));
    1780      698621 :       break;
    1781             :   }
    1782     1535667 : }
    1783             : static void
    1784   109105654 : treat_index_trivial(GEN v, GEN W, long index)
    1785             : {
    1786   109105654 :   GEN W11 = gel(W,11);
    1787   109105654 :   long shift = W11[3]; /* #F + #E2 + T32 */
    1788   109105654 :   switch(set_from_index(W11, index))
    1789             :   {
    1790    14062041 :     case 1: /*F*/
    1791             :     {
    1792    14062041 :       GEN F_index = gel(W,6), ind = gel(F_index, index);
    1793    14062041 :       long j, l = lg(ind);
    1794    99889426 :       for (j = 1; j < l; j++)
    1795             :       {
    1796    85827385 :         GEN IND = gel(ind,j);
    1797    85827385 :         treat_index_trivial(v, W, mael(IND,1,1));
    1798             :       }
    1799    14062041 :       break;
    1800             :     }
    1801             : 
    1802    39388762 :     case 2: /*E2, E2[r] + gamma * E1[s] = 0 */
    1803             :     {
    1804    39388762 :       long r = index - W11[1];
    1805    39388762 :       long s = E2fromE1_c(gel(msN_get_E2fromE1(W), r));
    1806    39388762 :       v[s]--;
    1807    39388762 :       break;
    1808             :     }
    1809             : 
    1810     2473205 :     case 3: case 5: case 6: /*T32,T2,T31*/
    1811     2473205 :       break;
    1812             : 
    1813    53181646 :     case 4: /*E1*/
    1814    53181646 :       v[index-shift]++;
    1815    53181646 :       break;
    1816             :   }
    1817   109105654 : }
    1818             : 
    1819             : static GEN
    1820      178213 : M2_log(GEN W, GEN M)
    1821             : {
    1822      178213 :   GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
    1823      178213 :   GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
    1824             :   GEN  u, v, D, V;
    1825             :   long index, s;
    1826             : 
    1827      178213 :   W = get_msN(W);
    1828      178213 :   V = zerovec(ms_get_nbgen(W));
    1829             : 
    1830      178213 :   D = subii(mulii(a,d), mulii(b,c));
    1831      178213 :   s = signe(D);
    1832      178213 :   if (!s) return V;
    1833      176869 :   if (is_pm1(D))
    1834             :   { /* shortcut, no need to apply Manin's trick */
    1835       63399 :     if (s < 0) { b = negi(b); d = negi(d); }
    1836       63399 :     M = Gamma0N_decompose(W, mkmat22(a,b, c,d), &index);
    1837       63399 :     treat_index(W, M, index, V);
    1838             :   }
    1839             :   else
    1840             :   {
    1841             :     GEN U, B, P, Q, PQ, C1,C2;
    1842             :     long i, l;
    1843      113470 :     (void)bezout(a,c,&u,&v);
    1844      113470 :     B = addii(mulii(b,u), mulii(d,v));
    1845             :     /* [u,v;-c,a] [a,b; c,d] = [1,B; 0,D], i.e. M = U [1,B;0,D] */
    1846      113470 :     U = mkmat22(a,negi(v), c,u);
    1847             : 
    1848             :     /* {1/0 -> B/D} as \sum g_i, g_i unimodular paths */
    1849      113470 :     PQ = ZV_allpnqn( gboundcf(gdiv(B,D), 0) );
    1850      113470 :     P = gel(PQ,1); l = lg(P);
    1851      113470 :     Q = gel(PQ,2);
    1852      113470 :     C1 = gel(U,1);
    1853      548534 :     for (i = 1; i < l; i++, C1 = C2)
    1854             :     {
    1855             :       GEN M;
    1856      435064 :       C2 = ZM_ZC_mul(U, mkcol2(gel(P,i), gel(Q,i)));
    1857      435064 :       if (!odd(i)) C1 = ZC_neg(C1);
    1858      435064 :       M = Gamma0N_decompose(W, mkmat2(C1,C2), &index);
    1859      435064 :       treat_index(W, M, index, V);
    1860             :     }
    1861             :   }
    1862      176869 :   return V;
    1863             : }
    1864             : 
    1865             : /* express +oo->q=a/b in terms of the Z[G]-generators, trivial action */
    1866             : static void
    1867     2221884 : Q_log_trivial(GEN v, GEN W, GEN q)
    1868             : {
    1869     2221884 :   GEN Q, W3 = gel(W,3), p1N = msN_get_p1N(W);
    1870     2221884 :   ulong c,d, N = p1N_get_N(p1N);
    1871             :   long i, lx;
    1872             : 
    1873     2221884 :   Q = Q_log_init(N, q);
    1874     2221884 :   lx = lg(Q);
    1875     2221884 :   c = 0;
    1876    23542008 :   for (i = 1; i < lx; i++, c = d)
    1877             :   {
    1878             :     long index;
    1879    21320124 :     d = Q[i];
    1880    21320124 :     if (c && !odd(i)) c = N - c;
    1881    21320124 :     index = W3[ p1_index(c,d,p1N) ];
    1882    21320124 :     treat_index_trivial(v, W, index);
    1883             :   }
    1884     2221884 : }
    1885             : static void
    1886      804069 : M2_log_trivial(GEN V, GEN W, GEN M)
    1887             : {
    1888      804069 :   GEN p1N = gel(W,1), W3 = gel(W,3);
    1889      804069 :   ulong N = p1N_get_N(p1N);
    1890      804069 :   GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
    1891      804069 :   GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
    1892             :   GEN  u, v, D;
    1893             :   long index, s;
    1894             : 
    1895      804069 :   D = subii(mulii(a,d), mulii(b,c));
    1896      804069 :   s = signe(D);
    1897      811293 :   if (!s) return;
    1898      804062 :   if (is_pm1(D))
    1899             :   { /* shortcut, not need to apply Manin's trick */
    1900      301000 :     if (s < 0) d = negi(d);
    1901      301000 :     index = W3[ p1_index(umodiu(c,N),umodiu(d,N),p1N) ];
    1902      301000 :     treat_index_trivial(V, W, index);
    1903             :   }
    1904             :   else
    1905             :   {
    1906             :     GEN U, B, P, Q, PQ;
    1907             :     long i, l;
    1908      503062 :     if (!signe(c)) { Q_log_trivial(V,W,gdiv(b,d)); return; }
    1909      495838 :     (void)bezout(a,c,&u,&v);
    1910      495838 :     B = addii(mulii(b,u), mulii(d,v));
    1911             :     /* [u,v;-c,a] [a,b; c,d] = [1,B; 0,D], i.e. M = U [1,B;0,D] */
    1912      495838 :     U = mkvec2(c, u);
    1913             : 
    1914             :     /* {1/0 -> B/D} as \sum g_i, g_i unimodular paths */
    1915      495838 :     PQ = ZV_allpnqn( gboundcf(gdiv(B,D), 0) );
    1916      495838 :     P = gel(PQ,1); l = lg(P);
    1917      495838 :     Q = gel(PQ,2);
    1918     2152983 :     for (i = 1; i < l; i++, c = d)
    1919             :     {
    1920     1657145 :       d = addii(mulii(gel(U,1),gel(P,i)), mulii(gel(U,2),gel(Q,i)));
    1921     1657145 :       if (!odd(i)) c = negi(c);
    1922     1657145 :       index = W3[ p1_index(umodiu(c,N),umodiu(d,N),p1N) ];
    1923     1657145 :       treat_index_trivial(V, W, index);
    1924             :     }
    1925             :   }
    1926             : }
    1927             : 
    1928             : static GEN
    1929       16772 : cusp_to_ZC(GEN c)
    1930             : {
    1931       16772 :   switch(typ(c))
    1932             :   {
    1933          35 :     case t_INFINITY:
    1934          35 :       return mkcol2(gen_1,gen_0);
    1935          91 :     case t_INT:
    1936          91 :       return mkcol2(c,gen_1);
    1937         140 :     case t_FRAC:
    1938         140 :       return mkcol2(gel(c,1),gel(c,2));
    1939       16506 :     case t_VECSMALL:
    1940       16506 :       return mkcol2(stoi(c[1]), stoi(c[2]));
    1941           0 :     default:
    1942           0 :       pari_err_TYPE("mspathlog",c);
    1943             :       return NULL;/*LCOV_EXCL_LINE*/
    1944             :   }
    1945             : }
    1946             : static GEN
    1947        8386 : path2_to_M2(GEN p)
    1948        8386 : { return mkmat2(cusp_to_ZC(gel(p,1)), cusp_to_ZC(gel(p,2))); }
    1949             : static GEN
    1950       16149 : path_to_M2(GEN p)
    1951             : {
    1952       16149 :   if (lg(p) != 3) pari_err_TYPE("mspathlog",p);
    1953       16142 :   switch(typ(p))
    1954             :   {
    1955        9870 :     case t_MAT:
    1956        9870 :       RgM_check_ZM(p,"mspathlog");
    1957        9870 :       break;
    1958        6272 :     case t_VEC:
    1959        6272 :       p = path2_to_M2(p);
    1960        6272 :       break;
    1961           0 :     default: pari_err_TYPE("mspathlog",p);
    1962             :   }
    1963       16142 :   return p;
    1964             : }
    1965             : /* Expresses path p as \sum x_i g_i, where the g_i are our distinguished
    1966             :  * generators and x_i \in Z[\Gamma]. Returns [x_1,...,x_n] */
    1967             : GEN
    1968       12691 : mspathlog(GEN W, GEN p)
    1969             : {
    1970       12691 :   pari_sp av = avma;
    1971       12691 :   checkms(W);
    1972       12691 :   return gerepilecopy(av, M2_log(W, path_to_M2(p)));
    1973             : }
    1974             : 
    1975             : /** HECKE OPERATORS **/
    1976             : /* [a,b;c,d] * cusp */
    1977             : static GEN
    1978     1922172 : cusp_mul(long a, long b, long c, long d, GEN cusp)
    1979             : {
    1980     1922172 :   long x = cusp[1], y = cusp[2];
    1981     1922172 :   long A = a*x+b*y, B = c*x+d*y, u = cgcd(A,B);
    1982     1922172 :   if (u != 1) { A /= u; B /= u; }
    1983     1922172 :   return mkcol2s(A, B);
    1984             : }
    1985             : /* f in Gl2(Q), act on path (zm), return path_to_M2(f.path) */
    1986             : static GEN
    1987      961086 : Gl2Q_act_path(GEN f, GEN path)
    1988             : {
    1989      961086 :   long a = coeff(f,1,1), b = coeff(f,1,2);
    1990      961086 :   long c = coeff(f,2,1), d = coeff(f,2,2);
    1991      961086 :   GEN c1 = cusp_mul(a,b,c,d, gel(path,1));
    1992      961086 :   GEN c2 = cusp_mul(a,b,c,d, gel(path,2));
    1993      961086 :   return mkmat2(c1,c2);
    1994             : }
    1995             : 
    1996             : static GEN
    1997     2417037 : init_act_trivial(GEN W) { return const_vecsmall(ms_get_nbE1(W), 0); }
    1998             : static GEN
    1999        3444 : mspathlog_trivial(GEN W, GEN p)
    2000             : {
    2001             :   GEN v;
    2002        3444 :   W = get_msN(W);
    2003        3444 :   v = init_act_trivial(W);
    2004        3444 :   M2_log_trivial(v, W, path_to_M2(p));
    2005        3437 :   return v;
    2006             : }
    2007             : 
    2008             : /* map from W1=Hom(Delta_0(N1),Q) -> W2=Hom(Delta_0(N2),Q), weight 2,
    2009             :  * trivial action. v a t_VEC of Gl2_Q (\sum v[i] in Z[Gl2(Q)]).
    2010             :  * Return the matrix attached to the action of v. */
    2011             : static GEN
    2012        3668 : getMorphism_trivial(GEN WW1, GEN WW2, GEN v)
    2013             : {
    2014        3668 :   GEN T, section, gen, W1 = get_msN(WW1), W2 = get_msN(WW2);
    2015             :   long j, lv, d2;
    2016        3668 :   if (ms_get_N(W1) == 1) return cgetg(1,t_MAT);
    2017        3668 :   if (ms_get_N(W2) == 1) return zeromat(0, ms_get_nbE1(W1));
    2018        3668 :   section = msN_get_section(W2);
    2019        3668 :   gen = msN_get_genindex(W2);
    2020        3668 :   d2 = ms_get_nbE1(W2);
    2021        3668 :   T = cgetg(d2+1, t_MAT);
    2022        3668 :   lv = lg(v);
    2023      202601 :   for (j = 1; j <= d2; j++)
    2024             :   {
    2025      198933 :     GEN w = gel(section, gen[j]);
    2026      198933 :     GEN t = init_act_trivial(W1);
    2027      198933 :     pari_sp av = avma;
    2028             :     long l;
    2029      999565 :     for (l = 1; l < lv; l++) M2_log_trivial(t, W1, Gl2Q_act_path(gel(v,l), w));
    2030      198933 :     gel(T,j) = t; set_avma(av);
    2031             :   }
    2032        3668 :   return shallowtrans(zm_to_ZM(T));
    2033             : }
    2034             : 
    2035             : static GEN
    2036      165522 : RgV_sparse(GEN v, GEN *pind)
    2037             : {
    2038             :   long i, l, k;
    2039      165522 :   GEN w = cgetg_copy(v,&l), ind = cgetg(l, t_VECSMALL);
    2040    17143112 :   for (i = k = 1; i < l; i++)
    2041             :   {
    2042    16977590 :     GEN c = gel(v,i);
    2043    16977590 :     if (typ(c) == t_INT) continue;
    2044      784798 :     gel(w,k) = c; ind[k] = i; k++;
    2045             :   }
    2046      165522 :   setlg(w,k); setlg(ind,k);
    2047      165522 :   *pind = ind; return w;
    2048             : }
    2049             : 
    2050             : static int
    2051      162568 : mat2_isidentity(GEN M)
    2052             : {
    2053      162568 :   GEN A = gel(M,1), B = gel(M,2);
    2054      162568 :   return A[1] == 1 && A[2] == 0 && B[1] == 0 && B[2] == 1;
    2055             : }
    2056             : /* path a mat22/mat22s, return log(f.path)^* . f in sparse form */
    2057             : static GEN
    2058      165522 : M2_logf(GEN Wp, GEN path, GEN f)
    2059             : {
    2060      165522 :   pari_sp av = avma;
    2061             :   GEN ind, L;
    2062             :   long i, l;
    2063      165522 :   if (f)
    2064      160454 :     path = Gl2Q_act_path(f, path);
    2065        5068 :   else if (typ(gel(path,1)) == t_VECSMALL)
    2066        2114 :     path = path2_to_M2(path);
    2067      165522 :   L = M2_log(Wp, path);
    2068      165522 :   L = RgV_sparse(L,&ind); l = lg(L);
    2069      950320 :   for (i = 1; i < l; i++) gel(L,i) = ZSl2_star(gel(L,i));
    2070      165522 :   if (f) ZGC_G_mul_inplace(L, mat2_to_ZM(f));
    2071      165522 :   return gerepilecopy(av, mkvec2(ind,L));
    2072             : }
    2073             : 
    2074             : static hashtable *
    2075        3668 : Gl2act_cache(long dim) { return set_init(dim*10); }
    2076             : 
    2077             : /* f zm/ZM in Gl_2(Q), acts from the left on Delta, which is generated by
    2078             :  * (g_i) as Z[Gamma1]-module, and by (G_i) as Z[Gamma2]-module.
    2079             :  * We have f.G_j = \sum_i \lambda_{i,j} g_i,   \lambda_{i,j} in Z[Gamma1]
    2080             :  * For phi in Hom_Gamma1(D,V), g in D, phi | f is in Hom_Gamma2(D,V) and
    2081             :  *  (phi | f)(G_j) = phi(f.G_j) | f
    2082             :  *                 = phi( \sum_i \lambda_{i,j} g_i ) | f
    2083             :  *                 = \sum_i phi(g_i) | (\lambda_{i,j}^* f)
    2084             :  *                 = \sum_i phi(g_i) | \mu_{i,j}(f)
    2085             :  * More generally
    2086             :  *  (\sum_k (phi |v_k))(G_j) = \sum_i phi(g_i) | \Mu_{i,j}
    2087             :  * with \Mu_{i,j} = \sum_k \mu{i,j}(v_k)
    2088             :  * Return the \Mu_{i,j} matrix as vector of sparse columns of operators on V */
    2089             : static GEN
    2090        3192 : init_dual_act(GEN v, GEN W1, GEN W2, struct m_act *S)
    2091             : {
    2092        3192 :   GEN section = ms_get_section(W2), gen = ms_get_genindex(W2);
    2093             :   /* HACK: the actions we consider in dimension 1 are trivial and in
    2094             :    * characteristic != 2, 3 => torsion generators are 0
    2095             :    * [satisfy e.g. (1+gamma).g = 0 => \phi(g) | 1+gamma  = 0 => \phi(g) = 0 */
    2096        3192 :   long j, lv = lg(v), dim = S->dim == 1? ms_get_nbE1(W2): lg(gen)-1;
    2097        3192 :   GEN T = cgetg(dim+1, t_VEC);
    2098        3192 :   hashtable *H = Gl2act_cache(dim);
    2099       40929 :   for (j = 1; j <= dim; j++)
    2100             :   {
    2101       37737 :     pari_sp av = avma;
    2102       37737 :     GEN w = gel(section, gen[j]); /* path_to_zm( E1/T2/T3 element ) */
    2103       37737 :     GEN t = NULL;
    2104             :     long k;
    2105      200305 :     for (k = 1; k < lv; k++)
    2106             :     {
    2107      162568 :       GEN tk, f = gel(v,k);
    2108      162568 :       if (typ(gel(f,1)) != t_VECSMALL) f = ZM_to_zm(f);
    2109      162568 :       if (mat2_isidentity(f)) f = NULL;
    2110      162568 :       tk = M2_logf(W1, w, f); /* mu_{.,j}(v[k]) as sparse vector */
    2111      162568 :       t = t? ZGCs_add(t, tk): tk;
    2112             :     }
    2113       37737 :     gel(T,j) = gerepilecopy(av, t);
    2114             :   }
    2115       40929 :   for (j = 1; j <= dim; j++)
    2116             :   {
    2117       37737 :     ZGl2QC_preload(S, gel(T,j), H);
    2118       37737 :     ZGl2QC_to_act(S, gel(T,j), H);
    2119             :   }
    2120        3192 :   return T;
    2121             : }
    2122             : 
    2123             : /* modular symbol given by phi[j] = \phi(G_j)
    2124             :  * \sum L[i]*phi[i], L a sparse column of operators */
    2125             : static GEN
    2126      354354 : dense_act_col(GEN col, GEN phi)
    2127             : {
    2128      354354 :   GEN s = NULL, colind = gel(col,1), colval = gel(col,2);
    2129      354354 :   long i, l = lg(colind), lphi = lg(phi);
    2130     5630121 :   for (i = 1; i < l; i++)
    2131             :   {
    2132     5278490 :     long a = colind[i];
    2133             :     GEN t;
    2134     5278490 :     if (a >= lphi) break; /* happens if k=2: torsion generator t omitted */
    2135     5275767 :     t = gel(phi, a); /* phi(G_a) */
    2136     5275767 :     t = RgM_RgC_mul(gel(colval,i), t);
    2137     5275767 :     s = s? RgC_add(s, t): t;
    2138             :   }
    2139      354354 :   return s;
    2140             : }
    2141             : /* modular symbol given by \phi( G[ind[j]] ) = val[j]
    2142             :  * \sum L[i]*phi[i], L a sparse column of operators */
    2143             : static GEN
    2144      779093 : sparse_act_col(GEN col, GEN phi)
    2145             : {
    2146      779093 :   GEN s = NULL, colind = gel(col,1), colval = gel(col,2);
    2147      779093 :   GEN ind = gel(phi,2), val = gel(phi,3);
    2148      779093 :   long a, l = lg(ind);
    2149      779093 :   if (lg(gel(phi,1)) == 1) return RgM_RgC_mul(gel(colval,1), gel(val,1));
    2150     3033205 :   for (a = 1; a < l; a++)
    2151             :   {
    2152     2254413 :     GEN t = gel(val, a); /* phi(G_i) */
    2153     2254413 :     long i = zv_search(colind, ind[a]);
    2154     2254413 :     if (!i) continue;
    2155      540603 :     t = RgM_RgC_mul(gel(colval,i), t);
    2156      540603 :     s = s? RgC_add(s, t): t;
    2157             :   }
    2158      778792 :   return s;
    2159             : }
    2160             : static int
    2161       69139 : phi_sparse(GEN phi) { return typ(gel(phi,1)) == t_VECSMALL; }
    2162             : /* phi in Hom_Gamma1(Delta, V), return the matrix whose colums are the
    2163             :  *   \sum_i phi(g_i) | \mu_{i,j} = (phi|f)(G_j),
    2164             :  * see init_dual_act. */
    2165             : static GEN
    2166       69139 : dual_act(long dimV, GEN act, GEN phi)
    2167             : {
    2168       69139 :   long l = lg(act), j;
    2169       69139 :   GEN v = cgetg(l, t_MAT);
    2170       69139 :   GEN (*ACT)(GEN,GEN) = phi_sparse(phi)? sparse_act_col: dense_act_col;
    2171     1199254 :   for (j = 1; j < l; j++)
    2172             :   {
    2173     1130115 :     pari_sp av = avma;
    2174     1130115 :     GEN s = ACT(gel(act,j), phi);
    2175     1130115 :     gel(v,j) = s? gerepileupto(av,s): zerocol(dimV);
    2176             :   }
    2177       69139 :   return v;
    2178             : }
    2179             : 
    2180             : /* in level N > 1 */
    2181             : static void
    2182       59087 : msk_get_st(GEN W, long *s, long *t)
    2183       59087 : { GEN st = gmael(W,3,3); *s = st[1]; *t = st[2]; }
    2184             : static GEN
    2185       59087 : msk_get_link(GEN W) { return gmael(W,3,4); }
    2186             : static GEN
    2187       59402 : msk_get_inv(GEN W) { return gmael(W,3,5); }
    2188             : /* \phi in Hom(Delta, V), \phi(G_k) = phi[k]. Write \phi as
    2189             :  *   \sum_{i,j} mu_{i,j} phi_{i,j}, mu_{i,j} in Q */
    2190             : static GEN
    2191       59402 : getMorphism_basis(GEN W, GEN phi)
    2192             : {
    2193       59402 :   GEN R, Q, Ls, T0, T1, Ts, link, basis, inv = msk_get_inv(W);
    2194             :   long i, j, r, s, t, dim, lvecT;
    2195             : 
    2196       59402 :   if (ms_get_N(W) == 1) return ZC_apply_dinv(inv, gel(phi,1));
    2197       59087 :   lvecT = lg(phi);
    2198       59087 :   basis = msk_get_basis(W);
    2199       59087 :   dim = lg(basis)-1;
    2200       59087 :   R = zerocol(dim);
    2201       59087 :   msk_get_st(W, &s, &t);
    2202       59087 :   link = msk_get_link(W);
    2203      789922 :   for (r = 2; r < lvecT; r++)
    2204             :   {
    2205             :     GEN Tr, L;
    2206      730835 :     if (r == s) continue;
    2207      671748 :     Tr = gel(phi,r); /* Phi(G_r), r != 1,s */
    2208      671748 :     L = gel(link, r);
    2209      671748 :     Q = ZC_apply_dinv(gel(inv,r), Tr);
    2210             :     /* write Phi(G_r) as sum_{a,b} mu_{a,b} Phi_{a,b}(G_r) */
    2211     3510668 :     for (j = 1; j < lg(L); j++) gel(R, L[j]) = gel(Q,j);
    2212             :   }
    2213       59087 :   Ls = gel(link, s);
    2214       59087 :   T1 = gel(phi,1); /* Phi(G_1) */
    2215       59087 :   gel(R, Ls[t]) = gel(T1, 1);
    2216             : 
    2217       59087 :   T0 = NULL;
    2218      789922 :   for (i = 2; i < lg(link); i++)
    2219             :   {
    2220             :     GEN L;
    2221      730835 :     if (i == s) continue;
    2222      671748 :     L = gel(link,i);
    2223     3510668 :     for (j =1 ; j < lg(L); j++)
    2224             :     {
    2225     2838920 :       long n = L[j]; /* phi_{i,j} = basis[n] */
    2226     2838920 :       GEN mu_ij = gel(R, n);
    2227     2838920 :       GEN phi_ij = gel(basis, n), pols = gel(phi_ij,3);
    2228     2838920 :       GEN z = RgC_Rg_mul(gel(pols, 3), mu_ij);
    2229     2838920 :       T0 = T0? RgC_add(T0, z): z; /* += mu_{i,j} Phi_{i,j} (G_s) */
    2230             :     }
    2231             :   }
    2232       59087 :   Ts = gel(phi,s); /* Phi(G_s) */
    2233       59087 :   if (T0) Ts = RgC_sub(Ts, T0);
    2234             :   /* solve \sum_{j!=t} mu_{s,j} Phi_{s,j}(G_s) = Ts */
    2235       59087 :   Q = ZC_apply_dinv(gel(inv,s), Ts);
    2236      234080 :   for (j = 1; j < t; j++) gel(R, Ls[j]) = gel(Q,j);
    2237             :   /* avoid mu_{s,t} */
    2238       59906 :   for (j = t; j < lg(Q); j++) gel(R, Ls[j+1]) = gel(Q,j);
    2239       59087 :   return R;
    2240             : }
    2241             : 
    2242             : /* a = s(g_i) for some modular symbol s; b in Z[G]
    2243             :  * return s(b.g_i) = b^* . s(g_i) */
    2244             : static GEN
    2245      115626 : ZGl2Q_act_s(GEN b, GEN a, long k)
    2246             : {
    2247      115626 :   if (typ(b) == t_INT)
    2248             :   {
    2249       58604 :     if (!signe(b)) return gen_0;
    2250          14 :     switch(typ(a))
    2251             :     {
    2252          14 :       case t_POL:
    2253          14 :         a = RgX_to_RgC(a, k-1); /*fall through*/
    2254          14 :       case t_COL:
    2255          14 :         a = RgC_Rg_mul(a,b);
    2256          14 :         break;
    2257           0 :       default: a = scalarcol_shallow(b,k-1);
    2258             :     }
    2259             :   }
    2260             :   else
    2261             :   {
    2262       57022 :     b = RgX_act_ZGl2Q(ZSl2_star(b), k);
    2263       57022 :     switch(typ(a))
    2264             :     {
    2265          63 :       case t_POL:
    2266          63 :         a = RgX_to_RgC(a, k-1); /*fall through*/
    2267       45262 :       case t_COL:
    2268       45262 :         a = RgM_RgC_mul(b,a);
    2269       45262 :         break;
    2270       11760 :       default: a = RgC_Rg_mul(gel(b,1),a);
    2271             :     }
    2272             :   }
    2273       57036 :   return a;
    2274             : }
    2275             : 
    2276             : static int
    2277          21 : checksymbol(GEN W, GEN s)
    2278             : {
    2279             :   GEN t, annT2, annT31, singlerel;
    2280             :   long i, k, l, nbE1, nbT2, nbT31;
    2281          21 :   k = msk_get_weight(W);
    2282          21 :   W = get_msN(W);
    2283          21 :   nbE1 = ms_get_nbE1(W);
    2284          21 :   singlerel = gel(W,10);
    2285          21 :   l = lg(singlerel);
    2286          21 :   if (k == 2)
    2287             :   {
    2288           0 :     for (i = nbE1+1; i < l; i++)
    2289           0 :       if (!gequal0(gel(s,i))) return 0;
    2290           0 :     return 1;
    2291             :   }
    2292          21 :   annT2 = msN_get_annT2(W); nbT2 = lg(annT2)-1;
    2293          21 :   annT31 = msN_get_annT31(W);nbT31 = lg(annT31)-1;
    2294          21 :   t = NULL;
    2295          84 :   for (i = 1; i < l; i++)
    2296             :   {
    2297          63 :     GEN a = gel(s,i);
    2298          63 :     a = ZGl2Q_act_s(gel(singlerel,i), a, k);
    2299          63 :     t = t? gadd(t, a): a;
    2300             :   }
    2301          21 :   if (!gequal0(t)) return 0;
    2302          14 :   for (i = 1; i <= nbT2; i++)
    2303             :   {
    2304           0 :     GEN a = gel(s,i + nbE1);
    2305           0 :     a = ZGl2Q_act_s(gel(annT2,i), a, k);
    2306           0 :     if (!gequal0(a)) return 0;
    2307             :   }
    2308          28 :   for (i = 1; i <= nbT31; i++)
    2309             :   {
    2310          14 :     GEN a = gel(s,i + nbE1 + nbT2);
    2311          14 :     a = ZGl2Q_act_s(gel(annT31,i), a, k);
    2312          14 :     if (!gequal0(a)) return 0;
    2313             :   }
    2314          14 :   return 1;
    2315             : }
    2316             : GEN
    2317          56 : msissymbol(GEN W, GEN s)
    2318             : {
    2319             :   long k, nbgen;
    2320          56 :   checkms(W);
    2321          56 :   k = msk_get_weight(W);
    2322          56 :   nbgen = ms_get_nbgen(W);
    2323          56 :   switch(typ(s))
    2324             :   {
    2325          21 :     case t_VEC: /* values s(g_i) */
    2326          21 :       if (lg(s)-1 != nbgen) return gen_0;
    2327          21 :       break;
    2328          28 :     case t_COL:
    2329          28 :       if (msk_get_sign(W))
    2330             :       {
    2331           0 :         GEN star = gel(msk_get_starproj(W), 1);
    2332           0 :         if (lg(star) == lg(s)) return gen_1;
    2333             :       }
    2334          28 :       if (k == 2) /* on the dual basis of (g_i) */
    2335             :       {
    2336           0 :         if (lg(s)-1 != nbgen) return gen_0;
    2337             :       }
    2338             :       else
    2339             :       {
    2340          28 :         GEN basis = msk_get_basis(W);
    2341          28 :         return (lg(s) == lg(basis))? gen_1: gen_0;
    2342             :       }
    2343           0 :       break;
    2344           7 :     case t_MAT:
    2345             :     {
    2346           7 :       long i, l = lg(s);
    2347           7 :       GEN v = cgetg(l, t_VEC);
    2348          21 :       for (i = 1; i < l; i++) gel(v,i) = msissymbol(W,gel(s,i))? gen_1: gen_0;
    2349           7 :       return v;
    2350             :     }
    2351           0 :     default: return gen_0;
    2352             :   }
    2353          21 :   return checksymbol(W,s)? gen_1: gen_0;
    2354             : }
    2355             : 
    2356             : /* map op: W1 = Hom(Delta_0(N1),V) -> W2 = Hom(Delta_0(N2),V), given by
    2357             :  * \sum v[i], v[i] in Gl2(Q) */
    2358             : static GEN
    2359        6377 : getMorphism(GEN W1, GEN W2, GEN v)
    2360             : {
    2361             :   struct m_act S;
    2362             :   GEN B1, M, act;
    2363        6377 :   long a, l, k = msk_get_weight(W1);
    2364        6377 :   if (k == 2) return getMorphism_trivial(W1,W2,v);
    2365        2709 :   S.k = k;
    2366        2709 :   S.dim = k-1;
    2367        2709 :   S.act = &_RgX_act_Gl2Q;
    2368        2709 :   act = init_dual_act(v,W1,W2,&S);
    2369        2709 :   B1 = msk_get_basis(W1);
    2370        2709 :   l = lg(B1); M = cgetg(l, t_MAT);
    2371       61180 :   for (a = 1; a < l; a++)
    2372             :   {
    2373       58471 :     pari_sp av = avma;
    2374       58471 :     GEN phi = dual_act(S.dim, act, gel(B1,a));
    2375       58471 :     GEN D = getMorphism_basis(W2, phi);
    2376       58471 :     gel(M,a) = gerepilecopy(av, D);
    2377             :   }
    2378        2709 :   return M;
    2379             : }
    2380             : static GEN
    2381        5215 : msendo(GEN W, GEN v) { return getMorphism(W, W, v); }
    2382             : 
    2383             : static GEN
    2384        2527 : endo_project(GEN W, GEN e, GEN H)
    2385             : {
    2386        2527 :   if (msk_get_sign(W)) e = Qevproj_apply(e, msk_get_starproj(W));
    2387        2527 :   if (H) e = Qevproj_apply(e, Qevproj_init0(H));
    2388        2527 :   return e;
    2389             : }
    2390             : static GEN
    2391        3381 : mshecke_i(GEN W, ulong p)
    2392             : {
    2393        3381 :   GEN v = ms_get_N(W) % p? Tp_matrices(p): Up_matrices(p);
    2394        3381 :   return msendo(W,v);
    2395             : }
    2396             : GEN
    2397        2478 : mshecke(GEN W, long p, GEN H)
    2398             : {
    2399        2478 :   pari_sp av = avma;
    2400             :   GEN T;
    2401        2478 :   checkms(W);
    2402        2478 :   if (p <= 1) pari_err_PRIME("mshecke",stoi(p));
    2403        2478 :   T = mshecke_i(W,p);
    2404        2478 :   T = endo_project(W,T,H);
    2405        2478 :   return gerepilecopy(av, T);
    2406             : }
    2407             : 
    2408             : static GEN
    2409          42 : msatkinlehner_i(GEN W, long Q)
    2410             : {
    2411          42 :   long N = ms_get_N(W);
    2412             :   GEN v;
    2413          42 :   if (Q == 1) return matid(msk_get_dim(W));
    2414          28 :   if (Q == N) return msendo(W, mkvec(mat2(0,1,-N,0)));
    2415          21 :   if (N % Q) pari_err_DOMAIN("msatkinlehner","N % Q","!=",gen_0,stoi(Q));
    2416          14 :   v = WQ_matrix(N, Q);
    2417          14 :   if (!v) pari_err_DOMAIN("msatkinlehner","gcd(Q,N/Q)","!=",gen_1,stoi(Q));
    2418          14 :   return msendo(W,mkvec(v));
    2419             : }
    2420             : GEN
    2421          42 : msatkinlehner(GEN W, long Q, GEN H)
    2422             : {
    2423          42 :   pari_sp av = avma;
    2424             :   GEN w;
    2425             :   long k;
    2426          42 :   checkms(W);
    2427          42 :   k = msk_get_weight(W);
    2428          42 :   if (Q <= 0) pari_err_DOMAIN("msatkinlehner","Q","<=",gen_0,stoi(Q));
    2429          42 :   w = msatkinlehner_i(W,Q);
    2430          35 :   w = endo_project(W,w,H);
    2431          35 :   if (k > 2 && Q != 1) w = RgM_Rg_div(w, powuu(Q,(k-2)>>1));
    2432          35 :   return gerepilecopy(av, w);
    2433             : }
    2434             : 
    2435             : static GEN
    2436        1813 : msstar_i(GEN W) { return msendo(W, mkvec(mat2(-1,0,0,1))); }
    2437             : GEN
    2438          14 : msstar(GEN W, GEN H)
    2439             : {
    2440          14 :   pari_sp av = avma;
    2441             :   GEN s;
    2442          14 :   checkms(W);
    2443          14 :   s = msstar_i(W);
    2444          14 :   s = endo_project(W,s,H);
    2445          14 :   return gerepilecopy(av, s);
    2446             : }
    2447             : 
    2448             : #if 0
    2449             : /* is \Gamma_0(N) cusp1 = \Gamma_0(N) cusp2 ? */
    2450             : static int
    2451             : iscuspeq(ulong N, GEN cusp1, GEN cusp2)
    2452             : {
    2453             :   long p1, q1, p2, q2, s1, s2, d;
    2454             :   p1 = cusp1[1]; p2 = cusp2[1];
    2455             :   q1 = cusp1[2]; q2 = cusp2[2];
    2456             :   d = Fl_mul(umodsu(q1,N),umodsu(q2,N), N);
    2457             :   d = ugcd(d, N);
    2458             : 
    2459             :   s1 = q1 > 2? Fl_inv(umodsu(p1,q1), q1): 1;
    2460             :   s2 = q2 > 2? Fl_inv(umodsu(p2,q2), q2): 1;
    2461             :   return Fl_mul(s1,q2,d) == Fl_mul(s2,q1,d);
    2462             : }
    2463             : #endif
    2464             : 
    2465             : /* return E_c(r) */
    2466             : static GEN
    2467        2604 : get_Ec_r(GEN c, long k)
    2468             : {
    2469        2604 :   long p = c[1], q = c[2], u, v;
    2470             :   GEN gr;
    2471        2604 :   (void)cbezout(p, q, &u, &v);
    2472        2604 :   gr = mat2(p, -v, q, u); /* g . (1:0) = (p:q) */
    2473        2604 :   return voo_act_Gl2Q(sl2_inv(gr), k);
    2474             : }
    2475             : /* N > 1; returns the modular symbol attached to the cusp c := p/q via the rule
    2476             :  * E_c(path from a to b in Delta_0) := E_c(b) - E_c(a), where
    2477             :  * E_c(r) := 0 if r != c mod Gamma
    2478             :  *           v_oo | gamma_r^(-1)
    2479             :  * where v_oo is stable by T = [1,1;0,1] (i.e x^(k-2)) and
    2480             :  * gamma_r . (1:0) = r, for some gamma_r in SL_2(Z) * */
    2481             : static GEN
    2482         462 : msfromcusp_trivial(GEN W, GEN c)
    2483             : {
    2484         462 :   GEN section = ms_get_section(W), gen = ms_get_genindex(W);
    2485         462 :   GEN S = ms_get_hashcusps(W);
    2486         462 :   long j, ic = cusp_index(c, S), l = ms_get_nbE1(W)+1;
    2487         462 :   GEN phi = cgetg(l, t_COL);
    2488       90356 :   for (j = 1; j < l; j++)
    2489             :   {
    2490       89894 :     GEN vj, g = gel(section, gen[j]); /* path_to_zm(generator) */
    2491       89894 :     GEN c1 = gel(g,1), c2 = gel(g,2);
    2492       89894 :     long i1 = cusp_index(c1, S);
    2493       89894 :     long i2 = cusp_index(c2, S);
    2494       89894 :     if (i1 == ic)
    2495        3290 :       vj = (i2 == ic)?  gen_0: gen_1;
    2496             :     else
    2497       86604 :       vj = (i2 == ic)? gen_m1: gen_0;
    2498       89894 :     gel(phi, j) = vj;
    2499             :   }
    2500         462 :   return phi;
    2501             : }
    2502             : static GEN
    2503        1393 : msfromcusp_i(GEN W, GEN c)
    2504             : {
    2505             :   GEN section, gen, S, phi;
    2506        1393 :   long j, ic, l, k = msk_get_weight(W);
    2507        1393 :   if (k == 2)
    2508             :   {
    2509         462 :     long N = ms_get_N(W);
    2510         462 :     return N == 1? cgetg(1,t_COL): msfromcusp_trivial(W, c);
    2511             :   }
    2512         931 :   k = msk_get_weight(W);
    2513         931 :   section = ms_get_section(W);
    2514         931 :   gen = ms_get_genindex(W);
    2515         931 :   S = ms_get_hashcusps(W);
    2516         931 :   ic = cusp_index(c, S);
    2517         931 :   l = lg(gen);
    2518         931 :   phi = cgetg(l, t_COL);
    2519       12075 :   for (j = 1; j < l; j++)
    2520             :   {
    2521       11144 :     GEN vj = NULL, g = gel(section, gen[j]); /* path_to_zm(generator) */
    2522       11144 :     GEN c1 = gel(g,1), c2 = gel(g,2);
    2523       11144 :     long i1 = cusp_index(c1, S);
    2524       11144 :     long i2 = cusp_index(c2, S);
    2525       11144 :     if (i1 == ic) vj = get_Ec_r(c1, k);
    2526       11144 :     if (i2 == ic)
    2527             :     {
    2528        1302 :       GEN s = get_Ec_r(c2, k);
    2529        1302 :       vj = vj? gsub(vj, s): gneg(s);
    2530             :     }
    2531       11144 :     if (!vj) vj = zerocol(k-1);
    2532       11144 :     gel(phi, j) = vj;
    2533             :   }
    2534         931 :   return getMorphism_basis(W, phi);
    2535             : }
    2536             : GEN
    2537          28 : msfromcusp(GEN W, GEN c)
    2538             : {
    2539          28 :   pari_sp av = avma;
    2540             :   long N;
    2541          28 :   checkms(W);
    2542          28 :   N = ms_get_N(W);
    2543          28 :   switch(typ(c))
    2544             :   {
    2545           7 :     case t_INFINITY:
    2546           7 :       c = mkvecsmall2(1,0);
    2547           7 :       break;
    2548          14 :     case t_INT:
    2549          14 :       c = mkvecsmall2(smodis(c,N), 1);
    2550          14 :       break;
    2551           7 :     case t_FRAC:
    2552           7 :       c = mkvecsmall2(smodis(gel(c,1),N), smodis(gel(c,2),N));
    2553           7 :       break;
    2554           0 :     default:
    2555           0 :       pari_err_TYPE("msfromcusp",c);
    2556             :   }
    2557          28 :   return gerepilecopy(av, msfromcusp_i(W,c));
    2558             : }
    2559             : 
    2560             : static GEN
    2561         287 : mseisenstein_i(GEN W)
    2562             : {
    2563         287 :   GEN M, S = ms_get_hashcusps(W), cusps = gel(S,3);
    2564         287 :   long i, l = lg(cusps);
    2565         287 :   if (msk_get_weight(W)==2) l--;
    2566         287 :   M = cgetg(l, t_MAT);
    2567        1652 :   for (i = 1; i < l; i++) gel(M,i) = msfromcusp_i(W, gel(cusps,i));
    2568         287 :   return Qevproj_init(Qevproj_star(W, QM_image_shallow(M)));
    2569             : }
    2570             : GEN
    2571          21 : mseisenstein(GEN W)
    2572             : {
    2573          21 :   pari_sp av = avma;
    2574          21 :   checkms(W); return gerepilecopy(av, mseisenstein_i(W));
    2575             : }
    2576             : 
    2577             : /* upper bound for log_2 |charpoly(T_p|S)|, where S is a cuspidal subspace of
    2578             :  * dimension d, k is the weight */
    2579             : #if 0
    2580             : static long
    2581             : TpS_char_bound(ulong p, long k, long d)
    2582             : { /* |eigenvalue| <= 2 p^(k-1)/2 */
    2583             :   return d * (2 + (log2((double)p)*(k-1))/2);
    2584             : }
    2585             : #endif
    2586             : static long
    2587         266 : TpE_char_bound(ulong p, long k, long d)
    2588             : { /* |eigenvalue| <= 2 p^(k-1) */
    2589         266 :   return d * (2 + log2((double)p)*(k-1));
    2590             : }
    2591             : 
    2592             : static GEN eisker(GEN M);
    2593             : static int
    2594         294 : use_Petersson(long N, long k, long s)
    2595             : {
    2596         294 :   if (!s)
    2597             :   {
    2598          70 :     if (N == 1)  return 1;
    2599          49 :     if (N <= 3)  return k >= 42;
    2600          42 :     if (N == 4)  return k >= 30;
    2601          42 :     if (N == 5)  return k >= 20;
    2602          42 :     if (N <= 10) return k >= 14;
    2603          35 :     if (N <= 16) return k >= 10;
    2604           7 :     if (N <= 28) return k >= 8;
    2605           7 :     if (N <= 136 || N == 180 || N == 200 || N == 225) return k >= 6;
    2606           0 :     return k >= 4;
    2607             :   }
    2608         224 :   if (s < 0)
    2609             :   {
    2610           0 :     if (N <= 64 || N == 100 || N == 128 || N == 144 || N == 225
    2611           0 :         || N == 351 || N == 375) return k >= 8;
    2612           0 :     return k >= 6;
    2613             :   }
    2614         224 :   if (N == 1) return 1;
    2615         217 :   if (N == 2) return k >= 56;
    2616         217 :   if (N == 3) return k >= 68;
    2617         182 :   if (N == 4) return k >= 78;
    2618         175 :   if (N == 5) return k >= 38;
    2619         147 :   if (N == 6) return k >= 24;
    2620         147 :   if (N == 7) return k >= 44;
    2621         140 :   if (N <= 9) return k >= 28;
    2622         133 :   if (N <= 13) return k >= 20;
    2623          98 :   if (N <= 21 || N == 50) return k >= 14;
    2624          70 :   if (N == 24 || N == 25) return k >= 16;
    2625          70 :   if (N <= 58 || N == 63 || N == 72 || N == 84 || N == 208 || N == 224) return k >= 10;
    2626          42 :   if (N <= 128 || N == 144 || N == 145 || N == 160 || N == 168 || N == 175 ||
    2627          21 :       N == 180 || N == 252 || N == 253 || N == 273 || N == 320 || N == 335 ||
    2628          42 :       N == 336 || N == 345 || N == 360) return k >= 8;
    2629          21 :   return k >= 6;
    2630             : }
    2631             : /* eisspace^-(N) = 0 */
    2632             : static int
    2633          49 : isminustriv(GEN F)
    2634             : {
    2635          49 :   GEN P = gel(F,1), E = gel(F,2);
    2636          49 :   long i = 1, l = lg(P);
    2637          49 :   if (l == 1) return 1;
    2638          49 :   if (P[1] == 2)
    2639             :   {
    2640           7 :     if (E[1] >= 4) return 0;
    2641           7 :     i++;
    2642             :   }
    2643          98 :   for (; i < l; i++)
    2644          49 :     if (E[i] > 1) return 0;
    2645          49 :   return 1;
    2646             : }
    2647             : 
    2648             : GEN
    2649         343 : mscuspidal(GEN W, long flag)
    2650             : {
    2651         343 :   pari_sp av = avma;
    2652             :   GEN M, E, S;
    2653             :   ulong p, N;
    2654             :   long k, s;
    2655             : 
    2656         343 :   checkms(W);
    2657         343 :   N = ms_get_N(W);
    2658         343 :   k = msk_get_weight(W);
    2659         343 :   s = msk_get_sign(W);
    2660         343 :   E = flag? mseisenstein_i(W): NULL;
    2661         343 :   if (s < 0 && isminustriv(factoru(N))) M = matid(msdim(W));
    2662         294 :   else if (use_Petersson(N, k, s)) M = eisker(W);
    2663             :   else
    2664             :   {
    2665             :     GEN T, TE, chE;
    2666             :     forprime_t F;
    2667             :     long bit;
    2668             :     pari_timer ti;
    2669             : 
    2670         266 :     if (!E) E = mseisenstein_i(W);
    2671         266 :     (void)u_forprime_init(&F, 2, ULONG_MAX);
    2672         392 :     while ((p = u_forprime_next(&F)))
    2673         392 :       if (N % p) break;
    2674         266 :     if (DEBUGLEVEL) timer_start(&ti);
    2675         266 :     T = mshecke(W, p, NULL);
    2676         266 :     if (DEBUGLEVEL) timer_printf(&ti,"Tp, p = %ld", p);
    2677         266 :     TE = Qevproj_apply(T, E); /* T_p | E */
    2678         266 :     if (DEBUGLEVEL) timer_printf(&ti,"Qevproj_init(E)");
    2679         266 :     bit = TpE_char_bound(p, k, lg(TE)-1);
    2680         266 :     chE = QM_charpoly_ZX_bound(TE, bit);
    2681         266 :     chE = ZX_radical(chE);
    2682         266 :     M = RgX_RgM_eval(chE, T);
    2683         266 :     M = QM_image_shallow(M);
    2684             :   }
    2685         343 :   S = Qevproj_init(M);
    2686         343 :   return gerepilecopy(av, flag? mkvec2(S,E): S);
    2687             : }
    2688             : 
    2689             : /** INIT ELLSYM STRUCTURE **/
    2690             : /* V a vector of ZM. If all of them have 0 last row, return NULL.
    2691             :  * Otherwise return [m,i,j], where m = V[i][last,j] contains the value
    2692             :  * of smallest absolute value */
    2693             : static GEN
    2694         945 : RgMV_find_non_zero_last_row(long offset, GEN V)
    2695             : {
    2696         945 :   long i, lasti = 0, lastj = 0, lV = lg(V);
    2697         945 :   GEN m = NULL;
    2698        4109 :   for (i = 1; i < lV; i++)
    2699             :   {
    2700        3164 :     GEN M = gel(V,i);
    2701        3164 :     long j, n, l = lg(M);
    2702        3164 :     if (l == 1) continue;
    2703        2849 :     n = nbrows(M);
    2704       13860 :     for (j = 1; j < l; j++)
    2705             :     {
    2706       11011 :       GEN a = gcoeff(M, n, j);
    2707       11011 :       if (!gequal0(a) && (!m || abscmpii(a, m) < 0))
    2708             :       {
    2709        1596 :         m = a; lasti = i; lastj = j;
    2710        1596 :         if (is_pm1(m)) goto END;
    2711             :       }
    2712             :     }
    2713             :   }
    2714         945 : END:
    2715         945 :   if (!m) return NULL;
    2716         630 :   return mkvec2(m, mkvecsmall2(lasti+offset, lastj));
    2717             : }
    2718             : /* invert the d_oo := (\gamma_oo - 1) operator, acting on
    2719             :  * [x^(k-2), ..., y^(k-2)] */
    2720             : static GEN
    2721         630 : Delta_inv(GEN doo, long k)
    2722             : {
    2723         630 :   GEN M = RgX_act_ZGl2Q(doo, k);
    2724         630 :   M = RgM_minor(M, k-1, 1); /* 1st column and last row are 0 */
    2725         630 :   return ZM_inv_denom(M);
    2726             : }
    2727             : /* The ZX P = \sum a_i x^i y^{k-2-i} is given by the ZV [a_0, ..., a_k-2]~,
    2728             :  * return Q and d such that P = doo Q + d y^k-2, where d in Z and Q */
    2729             : static GEN
    2730       12873 : doo_decompose(GEN dinv, GEN P, GEN *pd)
    2731             : {
    2732       12873 :   long l = lg(P); *pd = gel(P, l-1);
    2733       12873 :   P = vecslice(P, 1, l-2);
    2734       12873 :   return vec_prepend(ZC_apply_dinv(dinv, P), gen_0);
    2735             : }
    2736             : 
    2737             : static GEN
    2738       12873 : get_phi_ij(long i,long j,long n, long s,long t,GEN P_st,GEN Q_st,GEN d_st,
    2739             :            GEN P_ij, GEN lP_ij, GEN dinv)
    2740             : {
    2741             :   GEN ind, pols;
    2742       12873 :   if (i == s && j == t)
    2743             :   {
    2744         630 :     ind = mkvecsmall(1);
    2745         630 :     pols = mkvec(scalarcol_shallow(gen_1, lg(P_st)-1)); /* x^{k-2} */
    2746             :   }
    2747             :   else
    2748             :   {
    2749       12243 :     GEN d_ij, Q_ij = doo_decompose(dinv, lP_ij, &d_ij);
    2750       12243 :     GEN a = ZC_Z_mul(P_ij, d_st);
    2751       12243 :     GEN b = ZC_Z_mul(P_st, negi(d_ij));
    2752       12243 :     GEN c = RgC_sub(RgC_Rg_mul(Q_ij, d_st), RgC_Rg_mul(Q_st, d_ij));
    2753       12243 :     if (i == s) { /* j != t */
    2754        1659 :       ind = mkvecsmall2(1, s);
    2755        1659 :       pols = mkvec2(c, ZC_add(a, b));
    2756             :     } else {
    2757       10584 :       ind = mkvecsmall3(1, i, s);
    2758       10584 :       pols = mkvec3(c, a, b); /* image of g_1, g_i, g_s */
    2759             :     }
    2760       12243 :     pols = Q_primpart(pols);
    2761             :   }
    2762       12873 :   return mkvec3(mkvecsmall3(i,j,n), ind, pols);
    2763             : }
    2764             : 
    2765             : static GEN
    2766        1155 : mskinit_trivial(GEN WN)
    2767             : {
    2768        1155 :   long dim = ms_get_nbE1(WN);
    2769        1155 :   return mkvec3(WN, gen_0, mkvec2(gen_0,mkvecsmall2(2, dim)));
    2770             : }
    2771             : /* sum of #cols of the matrices contained in V */
    2772             : static long
    2773        1260 : RgMV_dim(GEN V)
    2774             : {
    2775        1260 :   long l = lg(V), d = 0, i;
    2776        1764 :   for (i = 1; i < l; i++) d += lg(gel(V,i)) - 1;
    2777        1260 :   return d;
    2778             : }
    2779             : static GEN
    2780         630 : mskinit_nontrivial(GEN WN, long k)
    2781             : {
    2782         630 :   GEN annT2 = gel(WN,8), annT31 = gel(WN,9), singlerel = gel(WN,10);
    2783             :   GEN link, basis, monomials, Inv;
    2784         630 :   long nbE1 = ms_get_nbE1(WN);
    2785         630 :   GEN dinv = Delta_inv(ZG_neg( ZSl2_star(gel(singlerel,1)) ), k);
    2786         630 :   GEN p1 = cgetg(nbE1+1, t_VEC), remove;
    2787         630 :   GEN p2 = ZGV_tors(annT2, k);
    2788         630 :   GEN p3 = ZGV_tors(annT31, k);
    2789         630 :   GEN gentor = shallowconcat(p2, p3);
    2790             :   GEN P_st, lP_st, Q_st, d_st;
    2791             :   long n, i, dim, s, t, u;
    2792         630 :   gel(p1, 1) = cgetg(1,t_MAT); /* dummy */
    2793        3381 :   for (i = 2; i <= nbE1; i++) /* skip 1st element = (\gamma_oo-1)g_oo */
    2794             :   {
    2795        2751 :     GEN z = gel(singlerel, i);
    2796        2751 :     gel(p1, i) = RgX_act_ZGl2Q(ZSl2_star(z), k);
    2797             :   }
    2798         630 :   remove = RgMV_find_non_zero_last_row(nbE1, gentor);
    2799         630 :   if (!remove) remove = RgMV_find_non_zero_last_row(0, p1);
    2800         630 :   if (!remove) pari_err_BUG("msinit [no y^k-2]");
    2801         630 :   remove = gel(remove,2); /* [s,t] */
    2802         630 :   s = remove[1];
    2803         630 :   t = remove[2];
    2804             :   /* +1 because of = x^(k-2), but -1 because of Manin relation */
    2805         630 :   dim = (k-1)*(nbE1-1) + RgMV_dim(p2) + RgMV_dim(p3);
    2806             :   /* Let (g_1,...,g_d) be the Gamma-generators of Delta, g_1 = g_oo.
    2807             :    * We describe modular symbols by the collection phi(g_1), ..., phi(g_d)
    2808             :    * \in V := Q[x,y]_{k-2}, with right Gamma action.
    2809             :    * For each i = 1, .., d, let V_i \subset V be the Q-vector space of
    2810             :    * allowed values for phi(g_i): with basis (P^{i,j}) given by the monomials
    2811             :    * x^(j-1) y^{k-2-(j-1)}, j = 1 .. k-1
    2812             :    * (g_i in E_1) or the solution of the torsion equations (1 + gamma)P = 0
    2813             :    * (g_i in T2) or (1 + gamma + gamma^2)P = 0 (g_i in T31). All such P
    2814             :    * are chosen in Z[x,y] with Q_content 1.
    2815             :    *
    2816             :    * The Manin relation (singlerel) is of the form \sum_i \lambda_i g_i = 0,
    2817             :    * where \lambda_i = 1 if g_i in T2 or T31, and \lambda_i = (1 - \gamma_i)
    2818             :    * for g_i in E1.
    2819             :    *
    2820             :    * If phi \in Hom_Gamma(Delta, V), it is defined by phi(g_i) := P_i in V
    2821             :    * with \sum_i P_i . \lambda_i^* = 0, where (\sum n_i g_i)^* :=
    2822             :    * \sum n_i \gamma_i^(-1).
    2823             :    *
    2824             :    * We single out gamma_1 / g_1 (g_oo in Pollack-Stevens paper) and
    2825             :    * write P_{i,j} \lambda_i^* =  Q_{i,j} (\gamma_1 - 1)^* + d_{i,j} y^{k-2}
    2826             :    * where d_{i,j} is a scalar and Q_{i,j} in V; we normalize Q_{i,j} to
    2827             :    * that the coefficient of x^{k-2} is 0.
    2828             :    *
    2829             :    * There exist (s,t) such that d_{s,t} != 0.
    2830             :    * A Q-basis of the (dual) space of modular symbols is given by the
    2831             :    * functions phi_{i,j}, 2 <= i <= d, 1 <= j <= k-1, mapping
    2832             :    *  g_1 -> d_{s,t} Q_{i,j} - d_{i,j} Q_{s,t} + [(i,j)=(s,t)] x^{k-2}
    2833             :    * If i != s
    2834             :    *   g_i -> d_{s,t} P_{i,j}
    2835             :    *   g_s -> - d_{i,j} P_{s,t}
    2836             :    * If i = s, j != t
    2837             :    *   g_i -> d_{s,t} P_{i,j} - d_{i,j} P_{s,t}
    2838             :    * And everything else to 0. Again we normalize the phi_{i,j} such that
    2839             :    * their image has content 1. */
    2840         630 :   monomials = matid(k-1); /* represent the monomials x^{k-2}, ... , y^{k-2} */
    2841         630 :   if (s <= nbE1) /* in E1 */
    2842             :   {
    2843         315 :     P_st = gel(monomials, t);
    2844         315 :     lP_st = gmael(p1, s, t); /* P_{s,t} lambda_s^* */
    2845             :   }
    2846             :   else /* in T2, T31 */
    2847             :   {
    2848         315 :     P_st = gmael(gentor, s - nbE1, t);
    2849         315 :     lP_st = P_st;
    2850             :   }
    2851         630 :   Q_st = doo_decompose(dinv, lP_st, &d_st);
    2852         630 :   basis = cgetg(dim+1, t_VEC);
    2853         630 :   link = cgetg(nbE1 + lg(gentor), t_VEC);
    2854         630 :   gel(link,1) = cgetg(1,t_VECSMALL); /* dummy */
    2855         630 :   n = 1;
    2856        3381 :   for (i = 2; i <= nbE1; i++)
    2857             :   {
    2858        2751 :     GEN L = cgetg(k, t_VECSMALL);
    2859             :     long j;
    2860             :     /* link[i][j] = n gives correspondance between phi_{i,j} and basis[n] */
    2861        2751 :     gel(link,i) = L;
    2862       14056 :     for (j = 1; j < k; j++)
    2863             :     {
    2864       11305 :       GEN lP_ij = gmael(p1, i, j); /* P_{i,j} lambda_i^* */
    2865       11305 :       GEN P_ij = gel(monomials,j);
    2866       11305 :       L[j] = n;
    2867       11305 :       gel(basis, n) = get_phi_ij(i,j,n, s,t, P_st, Q_st, d_st, P_ij, lP_ij, dinv);
    2868       11305 :       n++;
    2869             :     }
    2870             :   }
    2871        1134 :   for (u = 1; u < lg(gentor); u++,i++)
    2872             :   {
    2873         504 :     GEN V = gel(gentor,u);
    2874         504 :     long j, lV = lg(V);
    2875         504 :     GEN L = cgetg(lV, t_VECSMALL);
    2876         504 :     gel(link,i) = L;
    2877        2072 :     for (j = 1; j < lV; j++)
    2878             :     {
    2879        1568 :       GEN lP_ij = gel(V, j); /* P_{i,j} lambda_i^* = P_{i,j} */
    2880        1568 :       GEN P_ij = lP_ij;
    2881        1568 :       L[j] = n;
    2882        1568 :       gel(basis, n) = get_phi_ij(i,j,n, s,t, P_st, Q_st, d_st, P_ij, lP_ij, dinv);
    2883        1568 :       n++;
    2884             :     }
    2885             :   }
    2886         630 :   Inv = cgetg(lg(link), t_VEC);
    2887         630 :   gel(Inv,1) = cgetg(1, t_MAT); /* dummy */
    2888        3885 :   for (i = 2; i < lg(link); i++)
    2889             :   {
    2890        3255 :     GEN M, inv, B = gel(link,i);
    2891        3255 :     long j, lB = lg(B);
    2892        3255 :     if (i == s) { B = vecsplice(B, t); lB--; } /* remove phi_st */
    2893        3255 :     M = cgetg(lB, t_MAT);
    2894       15498 :     for (j = 1; j < lB; j++)
    2895             :     {
    2896       12243 :       GEN phi_ij = gel(basis, B[j]), pols = gel(phi_ij,3);
    2897       12243 :       gel(M, j) = gel(pols, 2); /* phi_ij(g_i) */
    2898             :     }
    2899        3255 :     if (i <= nbE1 && i != s) /* maximal rank k-1 */
    2900        2436 :       inv = ZM_inv_denom(M);
    2901             :     else /* i = s (rank k-2) or from torsion: rank k/3 or k/2 */
    2902         819 :       inv = Qevproj_init(M);
    2903        3255 :     gel(Inv,i) = inv;
    2904             :   }
    2905         630 :   return mkvec3(WN, gen_0, mkvec5(basis, mkvecsmall2(k, dim), mkvecsmall2(s,t),
    2906             :                                   link, Inv));
    2907             : }
    2908             : static GEN
    2909        1799 : add_star(GEN W, long sign)
    2910             : {
    2911        1799 :   GEN s = msstar_i(W);
    2912        1799 :   GEN K = sign? QM_ker(gsubgs(s, sign)): cgetg(1,t_MAT);
    2913        1799 :   gel(W,2) = mkvec3(stoi(sign), s, Qevproj_init(K));
    2914        1799 :   return W;
    2915             : }
    2916             : /* WN = msinit_N(N) */
    2917             : static GEN
    2918        1799 : mskinit(ulong N, long k, long sign)
    2919             : {
    2920        1799 :   GEN W, WN = msinit_N(N);
    2921        1799 :   if (N == 1)
    2922             :   {
    2923          14 :     GEN basis, M = RgXV_to_RgM(mfperiodpolbasis(k, 0), k-1);
    2924          14 :     GEN T = cgetg(1, t_VECSMALL), ind = mkvecsmall(1);
    2925          14 :     long i, l = lg(M);
    2926          14 :     basis = cgetg(l, t_VEC);
    2927          70 :     for (i = 1; i < l; i++) gel(basis,i) = mkvec3(T, ind, mkvec(gel(M,i)));
    2928          14 :     W = mkvec3(WN, gen_0, mkvec5(basis, mkvecsmall2(k, l-1), mkvecsmall2(0,0),
    2929             :                                  gen_0, Qevproj_init(M)));
    2930             :   }
    2931             :   else
    2932        1785 :     W = k == 2? mskinit_trivial(WN)
    2933        1785 :               : mskinit_nontrivial(WN, k);
    2934        1799 :   return add_star(W, sign);
    2935             : }
    2936             : GEN
    2937         476 : msinit(GEN N, GEN K, long sign)
    2938             : {
    2939         476 :   pari_sp av = avma;
    2940             :   GEN W;
    2941             :   long k;
    2942         476 :   if (typ(N) != t_INT) pari_err_TYPE("msinit", N);
    2943         476 :   if (typ(K) != t_INT) pari_err_TYPE("msinit", K);
    2944         476 :   k = itos(K);
    2945         476 :   if (k < 2) pari_err_DOMAIN("msinit","k", "<", gen_2,K);
    2946         476 :   if (odd(k)) pari_err_IMPL("msinit [odd weight]");
    2947         476 :   if (signe(N) <= 0) pari_err_DOMAIN("msinit","N", "<=", gen_0,N);
    2948         476 :   W = mskinit(itou(N), k, sign);
    2949         476 :   return gerepilecopy(av, W);
    2950             : }
    2951             : 
    2952             : /* W = msinit, xpm integral modular symbol of weight 2, c t_FRAC
    2953             :  * Return image of <oo->c> */
    2954             : GEN
    2955     2214660 : mseval2_ooQ(GEN W, GEN xpm, GEN c)
    2956             : {
    2957     2214660 :   pari_sp av = avma;
    2958             :   GEN v;
    2959     2214660 :   W = get_msN(W);
    2960     2214660 :   v = init_act_trivial(W);
    2961     2214660 :   Q_log_trivial(v, W, c); /* oo -> (a:b), c = a/b */
    2962     2214660 :   return gerepileuptoint(av, ZV_zc_mul(xpm, v));
    2963             : }
    2964             : 
    2965             : static GEN
    2966       20314 : eval_single(GEN s, long k, GEN B, long v)
    2967             : {
    2968             :   long i, l;
    2969       20314 :   GEN A = cgetg_copy(s,&l);
    2970      135863 :   for (i=1; i<l; i++) gel(A,i) = ZGl2Q_act_s(gel(B,i), gel(s,i), k);
    2971       20314 :   A = RgV_sum(A);
    2972       20314 :   if (is_vec_t(typ(A))) A = RgV_to_RgX(A, v);
    2973       20314 :   return A;
    2974             : }
    2975             : /* Evaluate symbol s on mspathlog B (= sum p_i g_i, p_i in Z[G]). Allow
    2976             :  * s = t_MAT [ collection of symbols, return a vector ]*/
    2977             : static GEN
    2978       16072 : mseval_by_values(GEN W, GEN s, GEN p, long v)
    2979             : {
    2980       16072 :   long i, l, k = msk_get_weight(W);
    2981             :   GEN A;
    2982       16072 :   if (k == 2)
    2983             :   { /* trivial represention: don't bother with Z[G] */
    2984        3444 :     GEN B = mspathlog_trivial(W,p);
    2985        3437 :     if (typ(s) != t_MAT) return RgV_zc_mul(s,B);
    2986        3367 :     l = lg(s); A = cgetg(l, t_VEC);
    2987       10101 :     for (i = 1; i < l; i++) gel(A,i) = RgV_zc_mul(gel(s,i), B);
    2988             :   }
    2989             :   else
    2990             :   {
    2991       12628 :     GEN B = mspathlog(W,p);
    2992       12628 :     if (typ(s) != t_MAT) return eval_single(s, k, B, v);
    2993         812 :     l = lg(s); A = cgetg(l, t_VEC);
    2994        9310 :     for (i = 1; i < l; i++) gel(A,i) = eval_single(gel(s,i), k, B, v);
    2995             :   }
    2996        4179 :   return A;
    2997             : }
    2998             : 
    2999             : /* express symbol on the basis phi_{i,j} */
    3000             : static GEN
    3001       20692 : symtophi(GEN W, GEN s)
    3002             : {
    3003       20692 :   GEN e, basis = msk_get_basis(W);
    3004       20692 :   long i, l = lg(basis);
    3005       20692 :   if (lg(s) != l) pari_err_TYPE("mseval",s);
    3006       20692 :   e = zerovec(ms_get_nbgen(W));
    3007      313670 :   for (i=1; i<l; i++)
    3008             :   {
    3009      292978 :     GEN phi, ind, pols, c = gel(s,i);
    3010             :     long j, m;
    3011      292978 :     if (gequal0(c)) continue;
    3012      122696 :     phi = gel(basis,i);
    3013      122696 :     ind = gel(phi,2); m = lg(ind);
    3014      122696 :     pols = gel(phi,3);
    3015      470806 :     for (j=1; j<m; j++)
    3016             :     {
    3017      348110 :       long t = ind[j];
    3018      348110 :       gel(e,t) = gadd(gel(e,t), gmul(c, gel(pols,j)));
    3019             :     }
    3020             :   }
    3021       20692 :   return e;
    3022             : }
    3023             : /* evaluate symbol s on path p */
    3024             : GEN
    3025       17031 : mseval(GEN W, GEN s, GEN p)
    3026             : {
    3027       17031 :   pari_sp av = avma;
    3028       17031 :   long i, k, l, v = 0;
    3029       17031 :   checkms(W);
    3030       17031 :   k = msk_get_weight(W);
    3031       17031 :   switch(typ(s))
    3032             :   {
    3033           7 :     case t_VEC: /* values s(g_i) */
    3034           7 :       if (lg(s)-1 != ms_get_nbgen(W)) pari_err_TYPE("mseval",s);
    3035           7 :       if (!p) return gcopy(s);
    3036           0 :       v = gvar(s);
    3037           0 :       break;
    3038       12831 :     case t_COL:
    3039       12831 :       if (msk_get_sign(W))
    3040             :       {
    3041         399 :         GEN star = gel(msk_get_starproj(W), 1);
    3042         399 :         if (lg(star) == lg(s)) s = RgM_RgC_mul(star, s);
    3043             :       }
    3044       12831 :       if (k == 2) /* on the dual basis of (g_i) */
    3045             :       {
    3046         637 :         if (lg(s)-1 != ms_get_nbE1(W)) pari_err_TYPE("mseval",s);
    3047         637 :         if (!p) return gtrans(s);
    3048             :       }
    3049             :       else
    3050       12194 :         s = symtophi(W,s);
    3051       12271 :       break;
    3052        4193 :     case t_MAT:
    3053        4193 :       l = lg(s);
    3054        4193 :       if (!p)
    3055             :       {
    3056           7 :         GEN v = cgetg(l, t_VEC);
    3057          28 :         for (i = 1; i < l; i++) gel(v,i) = mseval(W, gel(s,i), NULL);
    3058           7 :         return v;
    3059             :       }
    3060        4186 :       if (l == 1) return cgetg(1, t_VEC);
    3061        4179 :       if (msk_get_sign(W))
    3062             :       {
    3063          84 :         GEN star = gel(msk_get_starproj(W), 1);
    3064          84 :         if (lg(star) == lgcols(s)) s = RgM_mul(star, s);
    3065             :       }
    3066        4179 :       if (k == 2)
    3067        3367 :       { if (nbrows(s) != ms_get_nbE1(W)) pari_err_TYPE("mseval",s); }
    3068             :       else
    3069             :       {
    3070         812 :         GEN t = cgetg(l, t_MAT);
    3071        9310 :         for (i = 1; i < l; i++) gel(t,i) = symtophi(W,gel(s,i));
    3072         812 :         s = t;
    3073             :       }
    3074        4179 :       break;
    3075           0 :     default: pari_err_TYPE("mseval",s);
    3076             :   }
    3077       16450 :   if (p)
    3078       16072 :     s = mseval_by_values(W, s, p, v);
    3079             :   else
    3080             :   {
    3081         378 :     l = lg(s);
    3082        3675 :     for (i = 1; i < l; i++)
    3083             :     {
    3084        3297 :       GEN c = gel(s,i);
    3085        3297 :       if (!isintzero(c)) gel(s,i) = RgV_to_RgX(gel(s,i), v);
    3086             :     }
    3087             :   }
    3088       16443 :   return gerepilecopy(av, s);
    3089             : }
    3090             : 
    3091             : static GEN
    3092        1792 : allxpm(GEN W, GEN xpm, long f)
    3093             : {
    3094        1792 :   GEN v, L = coprimes_zv(f);
    3095        1792 :   long a, nonzero = 0;
    3096        1792 :   v = const_vec(f, NULL);
    3097        6797 :   for (a = 1; a <= f; a++)
    3098             :   {
    3099             :     GEN c;
    3100        5005 :     if (!L[a]) continue;
    3101        3696 :     c = mseval2_ooQ(W, xpm, sstoQ(a, f));
    3102        3696 :     if (!gequal0(c)) { gel(v,a) = c; nonzero = 1; }
    3103             :   }
    3104        1792 :   return nonzero? v: NULL;
    3105             : }
    3106             : /* \sum_{a mod f} chi(a) x(a/f) */
    3107             : static GEN
    3108         854 : seval(GEN G, GEN chi, GEN vx)
    3109             : {
    3110         854 :   GEN vZ, T, s = gen_0, go = zncharorder(G,chi);
    3111         854 :   long i, l = lg(vx), o = itou(go);
    3112         854 :   T = polcyclo(o,0);
    3113         854 :   vZ = mkvec2(RgXQ_powers(RgX_rem(pol_x(0), T), o-1, T), go);
    3114        3458 :   for (i = 1; i < l; i++)
    3115             :   {
    3116        2604 :     GEN x = gel(vx,i);
    3117        2604 :     if (x) s = gadd(s, gmul(x, znchareval(G, chi, utoi(i), vZ)));
    3118             :   }
    3119         854 :   return gequal0(s)? NULL: poleval(s, rootsof1u_cx(o, DEFAULTPREC));
    3120             : }
    3121             : 
    3122             : /* Let W = msinit(conductor(E), 2), xpm an integral modular symbol with the same
    3123             :  * eigenvalues as L_E. There exist a unique C such that
    3124             :  *   C*L(E,(D/.),1)_{xpm} = L(E,(D/.),1) / w1(E_D) != 0,
    3125             :  * for all D fundamental, sign(D) = s, and such that E_D has rank 0.
    3126             :  * Return C * ellQtwist_bsdperiod(E,s) */
    3127             : static GEN
    3128         854 : ell_get_Cw(GEN LE, GEN W, GEN xpm, long s)
    3129             : {
    3130         854 :   long f, NE = ms_get_N(W);
    3131         854 :   const long bit = 64;
    3132             : 
    3133         854 :   for (f = 1;; f++)
    3134        1750 :   { /* look for chi with conductor f coprime to N(E) and parity s
    3135             :      * such that L(E,chi,1) != 0 */
    3136        2604 :     pari_sp av = avma;
    3137             :     GEN vchi, vx, G;
    3138             :     long l, i;
    3139        2604 :     if ((f & 3) == 2 || ugcd(NE,f) != 1) continue;
    3140        1792 :     vx = allxpm(W, xpm, f); if (!vx) continue;
    3141         854 :     G = znstar0(utoipos(f),1);
    3142         854 :     vchi = chargalois(G,NULL); l = lg(vchi);
    3143        1470 :     for (i = 1; i < l; i++)
    3144             :     {
    3145        1470 :       pari_sp av2 = avma;
    3146        1470 :       GEN tau, z, S, L, chi = gel(vchi,i);
    3147        1470 :       long o = zncharisodd(G,chi);
    3148        1470 :       if ((s > 0 && o) || (s < 0 && !o)
    3149        1470 :           || itos(zncharconductor(G, chi)) != f) continue;
    3150         854 :       S = seval(G, chi, vx);
    3151         854 :       if (!S) { set_avma(av2); continue; }
    3152             : 
    3153         854 :       L = lfuntwist(LE, mkvec2(G, zncharconj(G,chi)), bit);
    3154         854 :       z = lfun(L, gen_1, bit);
    3155         854 :       tau = znchargauss(G, chi, gen_1, bit);
    3156         854 :       return gdiv(gmul(z, tau), S); /* C * w */
    3157             :     }
    3158           0 :     set_avma(av);
    3159             :   }
    3160             : }
    3161             : static GEN
    3162         742 : ell_get_scale(GEN LE, GEN W, long sign, GEN x)
    3163             : {
    3164         742 :   if (sign)
    3165         630 :     return ell_get_Cw(LE, W, gel(x,1), sign);
    3166             :   else
    3167             :   {
    3168         112 :     GEN Cwp = ell_get_Cw(LE, W, gel(x,1), 1);
    3169         112 :     GEN Cwm = ell_get_Cw(LE, W, gel(x,2),-1);
    3170         112 :     return mkvec2(Cwp, Cwm);
    3171             :   }
    3172             : }
    3173             : /* E minimal */
    3174             : static GEN
    3175         931 : msfromell_scale(GEN x, GEN Cw, GEN E, long s)
    3176             : {
    3177         931 :   GEN B = int2n(32);
    3178         931 :   if (s)
    3179             :   {
    3180         630 :     GEN C = gdiv(Cw, ellQtwist_bsdperiod(E,s));
    3181         630 :     return ZC_Q_mul(gel(x,1), bestappr(C,B));
    3182             :   }
    3183             :   else
    3184             :   {
    3185         301 :     GEN xp = gel(x,1), Cp = gdiv(gel(Cw,1), ellQtwist_bsdperiod(E, 1)), L;
    3186         301 :     GEN xm = gel(x,2), Cm = gdiv(gel(Cw,2), ellQtwist_bsdperiod(E,-1));
    3187         301 :     xp = ZC_Q_mul(xp, bestappr(Cp,B));
    3188         301 :     xm = ZC_Q_mul(xm, bestappr(Cm,B));
    3189         301 :     if (signe(ell_get_disc(E)) > 0)
    3190         133 :       L = mkmat2(xp, xm); /* E(R) has 2 connected components */
    3191             :     else
    3192         168 :       L = mkmat2(gsub(xp,xm), gmul2n(xm,1));
    3193         301 :     return mkvec3(xp, xm, L);
    3194             :   }
    3195             : }
    3196             : /* v != 0 */
    3197             : static GEN
    3198         854 : Flc_normalize(GEN v, ulong p)
    3199             : {
    3200         854 :   long i, l = lg(v);
    3201        1512 :   for (i = 1; i < l; i++)
    3202        1512 :     if (v[i])
    3203             :     {
    3204         854 :       if (v[i] != 1) v = Flv_Fl_div(v, v[i], p);
    3205         854 :       return v;
    3206             :     }
    3207           0 :   return NULL;
    3208             : }
    3209             : /* K \cap Ker M  [F_l vector spaces]. K = NULL means full space */
    3210             : static GEN
    3211         903 : msfromell_ker(GEN K, GEN M, ulong l)
    3212             : {
    3213         903 :   GEN B, Ml = ZM_to_Flm(M, l);
    3214         903 :   if (K) Ml = Flm_mul(Ml, K, l);
    3215         903 :   B = Flm_ker(Ml, l);
    3216         903 :   if (!K) K = B;
    3217         161 :   else if (lg(B) < lg(K))
    3218         147 :     K = Flm_mul(K, B, l);
    3219         903 :   return K;
    3220             : }
    3221             : /* K = \cap_p Ker(T_p - a_p), 2-dimensional. Set *xl to the 1-dimensional
    3222             :  * Fl-basis  such that star . xl = sign . xl if sign != 0 and
    3223             :  * star * xl[1] = xl[1]; star * xl[2] = -xl[2] if sign = 0 */
    3224             : static void
    3225         742 : msfromell_l(GEN *pxl, GEN K, GEN star, long sign, ulong l)
    3226             : {
    3227         742 :   GEN s = ZM_to_Flm(star, l);
    3228         742 :   GEN a = gel(K,1), Sa = Flm_Flc_mul(s,a,l);
    3229         742 :   GEN b = gel(K,2);
    3230         742 :   GEN t = Flv_add(a,Sa,l), xp, xm;
    3231         742 :   if (zv_equal0(t))
    3232             :   {
    3233         140 :     xm = a;
    3234         140 :     xp = Flv_add(b,Flm_Flc_mul(s,b,l), l);
    3235             :   }
    3236             :   else
    3237             :   {
    3238         602 :     xp = t; t = Flv_sub(a, Sa, l);
    3239         602 :     xm = zv_equal0(t)? Flv_sub(b, Flm_Flc_mul(s,b,l), l): t;
    3240             :   }
    3241             :   /* xp = 0 on Im(S - 1), xm = 0 on Im(S + 1) */
    3242         742 :   if (sign > 0)
    3243         518 :     *pxl = mkmat(Flc_normalize(xp, l));
    3244         224 :   else if (sign < 0)
    3245         112 :     *pxl = mkmat(Flc_normalize(xm, l));
    3246             :   else
    3247         112 :     *pxl = mkmat2(Flc_normalize(xp, l), Flc_normalize(xm, l));
    3248         742 : }
    3249             : /* return a primitive symbol */
    3250             : static GEN
    3251         742 : msfromell_ratlift(GEN x, GEN q)
    3252             : {
    3253         742 :   GEN B = sqrti(shifti(q,-1));
    3254         742 :   GEN r = FpM_ratlift(x, q, B, B, NULL);
    3255         742 :   if (r) r = Q_primpart(r);
    3256         742 :   return r;
    3257             : }
    3258             : static int
    3259         742 : msfromell_check(GEN x, GEN vT, GEN star, long sign)
    3260             : {
    3261             :   long i, l;
    3262             :   GEN sx;
    3263         742 :   if (!x) return 0;
    3264         742 :   l = lg(vT);
    3265        1645 :   for (i = 1; i < l; i++)
    3266             :   {
    3267         903 :     GEN T = gel(vT,i);
    3268         903 :     if (!gequal0(ZM_mul(T, x))) return 0; /* fail */
    3269             :   }
    3270         742 :   sx = ZM_mul(star,x);
    3271         742 :   if (sign)
    3272         630 :     return ZV_equal(gel(sx,1), sign > 0? gel(x,1): ZC_neg(gel(x,1)));
    3273             :   else
    3274         112 :     return ZV_equal(gel(sx,1),gel(x,1)) && ZV_equal(gel(sx,2),ZC_neg(gel(x,2)));
    3275             : }
    3276             : GEN
    3277         742 : msfromell(GEN E0, long sign)
    3278             : {
    3279         742 :   pari_sp av = avma, av2;
    3280         742 :   GEN T, Cw, E, NE, star, q, vT, xl, xr, W, x = NULL, K = NULL;
    3281             :   long lE, single;
    3282             :   ulong p, l, N;
    3283             :   forprime_t S, Sl;
    3284             : 
    3285         742 :   if (typ(E0) != t_VEC) pari_err_TYPE("msfromell",E0);
    3286         742 :   lE = lg(E0);
    3287         742 :   if (lE == 1) return cgetg(1,t_VEC);
    3288         742 :   single = (typ(gel(E0,1)) != t_VEC);
    3289         742 :   E = single ? E0: gel(E0,1);
    3290         742 :   NE = ellQ_get_N(E);
    3291             :   /* must make it integral for ellap; we have minimal model at hand */
    3292         742 :   T = obj_check(E, Q_MINIMALMODEL); if (lg(T) != 2) E = gel(T,3);
    3293         742 :   N = itou(NE); av2 = avma;
    3294         742 :   W = gerepilecopy(av2, mskinit(N,2,0));
    3295         742 :   star = msk_get_star(W);
    3296         742 :   (void)u_forprime_init(&Sl, 1UL<<29, ULONG_MAX);
    3297             :   /* loop for p <= count_Manin_symbols(N) / 6 would be enough */
    3298         742 :   (void)u_forprime_init(&S, 2, ULONG_MAX);
    3299         742 :   vT = cgetg(1, t_VEC);
    3300         742 :   l = u_forprime_next(&Sl);
    3301        1267 :   while( (p = u_forprime_next(&S)) )
    3302             :   {
    3303             :     GEN M;
    3304        1267 :     if (N % p == 0) continue;
    3305         903 :     av2 = avma;
    3306         903 :     M = RgM_Rg_sub_shallow(mshecke_i(W, p), ellap(E, utoipos(p)));
    3307         903 :     M = gerepilecopy(av2, M);
    3308         903 :     vT = vec_append(vT, M); /* for certification at the end */
    3309         903 :     K = msfromell_ker(K, M, l);
    3310         903 :     if (lg(K) == 3) break;
    3311             :   }
    3312         742 :   if (!p) pari_err_BUG("msfromell: ran out of primes");
    3313             : 
    3314             :   /* mod one l should be enough */
    3315         742 :   msfromell_l(&xl, K, star, sign, l);
    3316         742 :   x = ZM_init_CRT(xl, l);
    3317         742 :   q = utoipos(l);
    3318         742 :   xr = msfromell_ratlift(x, q);
    3319             :   /* paranoia */
    3320         742 :   while (!msfromell_check(xr, vT, star, sign) && (l = u_forprime_next(&Sl)) )
    3321             :   {
    3322           0 :     GEN K = NULL;
    3323           0 :     long i, lvT = lg(vT);
    3324           0 :     for (i = 1; i < lvT; i++)
    3325             :     {
    3326           0 :       K = msfromell_ker(K, gel(vT,i), l);
    3327           0 :       if (lg(K) == 3) break;
    3328             :     }
    3329           0 :     if (i >= lvT) { x = NULL; continue; }
    3330           0 :     msfromell_l(&xl, K, star, sign, l);
    3331           0 :     ZM_incremental_CRT(&x, xl, &q, l);
    3332           0 :     xr = msfromell_ratlift(x, q);
    3333             :   }
    3334             :   /* linear form = 0 on all Im(Tp - ap) and Im(S - sign) if sign != 0 */
    3335         742 :   Cw = ell_get_scale(lfuncreate(E), W, sign, xr);
    3336         742 :   if (single)
    3337         693 :     x = msfromell_scale(xr, Cw, E, sign);
    3338             :   else
    3339             :   { /* assume all E0[i] isogenous, given by minimal models */
    3340          49 :     GEN v = cgetg(lE, t_VEC);
    3341             :     long i;
    3342         287 :     for (i=1; i<lE; i++) gel(v,i) = msfromell_scale(xr, Cw, gel(E0,i), sign);
    3343          49 :     x = v;
    3344             :   }
    3345         742 :   return gerepilecopy(av, mkvec2(W, x));
    3346             : }
    3347             : 
    3348             : GEN
    3349          21 : msfromhecke(GEN W, GEN v, GEN H)
    3350             : {
    3351          21 :   pari_sp av = avma;
    3352          21 :   long i, l = lg(v);
    3353          21 :   GEN K = NULL;
    3354          21 :   checkms(W);
    3355          21 :   if (typ(v) != t_VEC) pari_err_TYPE("msfromhecke",v);
    3356          49 :   for (i = 1; i < l; i++)
    3357             :   {
    3358          28 :     GEN K2, T, p, P, c = gel(v,i);
    3359          28 :     if (typ(c) != t_VEC || lg(c) != 3) pari_err_TYPE("msfromhecke",v);
    3360          28 :     p = gel(c,1);
    3361          28 :     if (typ(p) != t_INT) pari_err_TYPE("msfromhecke",v);
    3362          28 :     P = gel(c,2);
    3363          28 :     switch(typ(P))
    3364             :     {
    3365          21 :       case t_INT:
    3366          21 :         P = deg1pol_shallow(gen_1, negi(P), 0);
    3367          21 :         break;
    3368           7 :       case t_POL:
    3369           7 :         if (RgX_is_ZX(P)) break;
    3370             :       default:
    3371           0 :         pari_err_TYPE("msfromhecke",v);
    3372             :     };
    3373          28 :     T = mshecke(W, itos(p), H);
    3374          28 :     T = Q_primpart(RgX_RgM_eval(P, T));
    3375          28 :     if (K) T = ZM_mul(T,K);
    3376          28 :     K2 = ZM_ker(T);
    3377          28 :     if (!K) K = K2;
    3378           7 :     else if (lg(K2) < lg(K)) K = ZM_mul(K,K2);
    3379             :   }
    3380          21 :   return gerepilecopy(av, K);
    3381             : }
    3382             : 
    3383             : /* OVERCONVERGENT MODULAR SYMBOLS */
    3384             : 
    3385             : static GEN
    3386        2933 : mspadic_get_Wp(GEN W) { return gel(W,1); }
    3387             : static GEN
    3388         483 : mspadic_get_Tp(GEN W) { return gel(W,2); }
    3389             : static GEN
    3390         483 : mspadic_get_bin(GEN W) { return gel(W,3); }
    3391             : static GEN
    3392         476 : mspadic_get_actUp(GEN W) { return gel(W,4); }
    3393             : static GEN
    3394         476 : mspadic_get_q(GEN W) { return gel(W,5); }
    3395             : static long
    3396        1456 : mspadic_get_p(GEN W) { return gel(W,6)[1]; }
    3397             : static long
    3398        1211 : mspadic_get_n(GEN W) { return gel(W,6)[2]; }
    3399             : static long
    3400         161 : mspadic_get_flag(GEN W) { return gel(W,6)[3]; }
    3401             : static GEN
    3402         483 : mspadic_get_M(GEN W) { return gel(W,7); }
    3403             : static GEN
    3404         483 : mspadic_get_C(GEN W) { return gel(W,8); }
    3405             : static long
    3406         973 : mspadic_get_weight(GEN W) { return msk_get_weight(mspadic_get_Wp(W)); }
    3407             : 
    3408             : void
    3409         980 : checkmspadic(GEN W)
    3410             : {
    3411         980 :   if (typ(W) != t_VEC || lg(W) != 9) pari_err_TYPE("checkmspadic",W);
    3412         980 :   checkms(mspadic_get_Wp(W));
    3413         980 : }
    3414             : 
    3415             : /* f in M_2(Z) \cap GL_2(Q), p \nmid a [ and for the result to mean anything
    3416             :  * p | c, but not needed here]. Return the matrix M in M_D(Z), D = M+k-1
    3417             :  * such that, if v = \int x^i d mu, i < D, is a vector of D moments of mu,
    3418             :  * then M * v is the vector of moments of mu | f  mod p^D */
    3419             : static GEN
    3420      276073 : moments_act_i(struct m_act *S, GEN f)
    3421             : {
    3422      276073 :   long j, k = S->k, D = S->dim;
    3423      276073 :   GEN a = gcoeff(f,1,1), b = gcoeff(f,1,2);
    3424      276073 :   GEN c = gcoeff(f,2,1), d = gcoeff(f,2,2);
    3425      276073 :   GEN u, z, q = S->q, mat = cgetg(D+1, t_MAT);
    3426             : 
    3427      276073 :   a = modii(a,q);
    3428      276073 :   c = modii(c,q);
    3429      276073 :   z = FpX_powu(deg1pol(c,a,0), k-2, q); /* (a+cx)^(k-2) */
    3430             :   /* u := (b+dx) / (a+cx) mod (q,x^D) = (b/a +d/a*x) / (1 - (-c/a)*x) */
    3431      276073 :   if (!equali1(a))
    3432             :   {
    3433      271229 :     GEN ai = Fp_inv(a,q);
    3434      271229 :     b = Fp_mul(b,ai,q);
    3435      271229 :     c = Fp_mul(c,ai,q);
    3436      271229 :     d = Fp_mul(d,ai,q);
    3437             :   }
    3438      276073 :   u = deg1pol_shallow(d, b, 0);
    3439             :   /* multiply by 1 / (1 - (-c/a)*x) */
    3440      276073 :   if (signe(c))
    3441             :   {
    3442      269640 :     GEN C = Fp_neg(c,q), v = cgetg(D+2,t_POL);
    3443      269640 :     v[1] = evalsigne(1)|evalvarn(0);
    3444      269640 :     gel(v, 2) = gen_1; gel(v, 3) = C;
    3445     1405138 :     for (j = 4; j < D+2; j++)
    3446             :     {
    3447     1329027 :       GEN t = Fp_mul(gel(v,j-1), C, q);
    3448     1329027 :       if (!signe(t)) { setlg(v,j); break; }
    3449     1135498 :       gel(v,j) = t;
    3450             :     }
    3451      269640 :     u = FpXn_mul(u, v, D, q);
    3452             :   }
    3453     2369024 :   for (j = 1; j <= D; j++)
    3454             :   {
    3455     2092951 :     gel(mat,j) = RgX_to_RgC(z, D); /* (a+cx)^(k-2) * ((b+dx)/(a+cx))^(j-1) */
    3456     2092951 :     if (j != D) z = FpXn_mul(z, u, D, q);
    3457             :   }
    3458      276073 :   return shallowtrans(mat);
    3459             : }
    3460             : static GEN
    3461      275611 : moments_act(struct m_act *S, GEN f)
    3462      275611 : { pari_sp av = avma; return gerepilecopy(av, moments_act_i(S,f)); }
    3463             : static GEN
    3464         483 : init_moments_act(GEN W, long p, long n, GEN q, GEN v)
    3465             : {
    3466             :   struct m_act S;
    3467         483 :   long k = msk_get_weight(W);
    3468         483 :   S.p = p;
    3469         483 :   S.k = k;
    3470         483 :   S.q = q;
    3471         483 :   S.dim = n+k-1;
    3472         483 :   S.act = &moments_act; return init_dual_act(v,W,W,&S);
    3473             : }
    3474             : 
    3475             : static void
    3476        6762 : clean_tail(GEN phi, long c, GEN q)
    3477             : {
    3478        6762 :   long a, l = lg(phi);
    3479      214438 :   for (a = 1; a < l; a++)
    3480             :   {
    3481      207676 :     GEN P = FpC_red(gel(phi,a), q); /* phi(G_a) = vector of moments */
    3482      207676 :     long j, lP = lg(P);
    3483     1007825 :     for (j = c; j < lP; j++) gel(P,j) = gen_0; /* reset garbage to 0 */
    3484      207676 :     gel(phi,a) = P;
    3485             :   }
    3486        6762 : }
    3487             : /* concat z to all phi[i] */
    3488             : static GEN
    3489         630 : concat2(GEN phi, GEN z)
    3490             : {
    3491             :   long i, l;
    3492         630 :   GEN v = cgetg_copy(phi,&l);
    3493       29022 :   for (i = 1; i < l; i++) gel(v,i) = shallowconcat(gel(phi,i), z);
    3494         630 :   return v;
    3495             : }
    3496             : static GEN
    3497         630 : red_mod_FilM(GEN phi, ulong p, long k, long flag)
    3498             : {
    3499             :   long a, l;
    3500         630 :   GEN den = gen_1, v = cgetg_copy(phi, &l);
    3501         630 :   if (flag)
    3502             :   {
    3503         343 :     phi = Q_remove_denom(phi, &den);
    3504         343 :     if (!den) { den = gen_1; flag = 0; }
    3505             :   }
    3506       29386 :   for (a = 1; a < l; a++)
    3507             :   {
    3508       28756 :     GEN P = gel(phi,a), q = den;
    3509             :     long j;
    3510      207676 :     for (j = lg(P)-1; j >= k+1; j--)
    3511             :     {
    3512      178920 :       q = muliu(q,p);
    3513      178920 :       gel(P,j) = modii(gel(P,j),q);
    3514             :     }
    3515       28756 :     q = muliu(q,p);
    3516       93380 :     for (     ; j >= 1; j--)
    3517       64624 :       gel(P,j) = modii(gel(P,j),q);
    3518       28756 :     gel(v,a) = P;
    3519             :   }
    3520         630 :   if (flag) v = gdiv(v, den);
    3521         630 :   return v;
    3522             : }
    3523             : 
    3524             : /* denom(C) | p^(2(k-1) - v_p(ap)) */
    3525             : static GEN
    3526         154 : oms_dim2(GEN W, GEN phi, GEN C, GEN ap)
    3527             : {
    3528         154 :   long t, i, k = mspadic_get_weight(W);
    3529         154 :   long p = mspadic_get_p(W), n = mspadic_get_n(W);
    3530         154 :   GEN phi1 = gel(phi,1), phi2 = gel(phi,2);
    3531         154 :   GEN v, q = mspadic_get_q(W);
    3532         154 :   GEN act = mspadic_get_actUp(W);
    3533             : 
    3534         154 :   t = signe(ap)? Z_lval(ap,p) : k-1;
    3535         154 :   phi1 = concat2(phi1, zerovec(n));
    3536         154 :   phi2 = concat2(phi2, zerovec(n));
    3537        2107 :   for (i = 1; i <= n; i++)
    3538             :   {
    3539        1953 :     phi1 = dual_act(k-1, act, phi1);
    3540        1953 :     phi1 = dual_act(k-1, act, phi1);
    3541        1953 :     clean_tail(phi1, k + i*t, q);
    3542             : 
    3543        1953 :     phi2 = dual_act(k-1, act, phi2);
    3544        1953 :     phi2 = dual_act(k-1, act, phi2);
    3545        1953 :     clean_tail(phi2, k + i*t, q);
    3546             :   }
    3547         154 :   C = gpowgs(C,n);
    3548         154 :   v = RgM_RgC_mul(C, mkcol2(phi1,phi2));
    3549         154 :   phi1 = red_mod_FilM(gel(v,1), p, k, 1);
    3550         154 :   phi2 = red_mod_FilM(gel(v,2), p, k, 1);
    3551         154 :   return mkvec2(phi1,phi2);
    3552             : }
    3553             : 
    3554             : /* flag = 0 iff alpha is a p-unit */
    3555             : static GEN
    3556         322 : oms_dim1(GEN W, GEN phi, GEN alpha, long flag)
    3557             : {
    3558         322 :   long i, k = mspadic_get_weight(W);
    3559         322 :   long p = mspadic_get_p(W), n = mspadic_get_n(W);
    3560         322 :   GEN q = mspadic_get_q(W);
    3561         322 :   GEN act = mspadic_get_actUp(W);
    3562         322 :   phi = concat2(phi, zerovec(n));
    3563        3178 :   for (i = 1; i <= n; i++)
    3564             :   {
    3565        2856 :     phi = dual_act(k-1, act, phi);
    3566        2856 :     clean_tail(phi, k + i, q);
    3567             :   }
    3568         322 :   phi = gmul(lift_shallow(gpowgs(alpha,n)), phi);
    3569         322 :   phi = red_mod_FilM(phi, p, k, flag);
    3570         322 :   return mkvec(phi);
    3571             : }
    3572             : 
    3573             : /* lift polynomial P in RgX[X,Y]_{k-2} to a distribution \mu such that
    3574             :  * \int (Y - X z)^(k-2) d\mu(z) = P(X,Y)
    3575             :  * Return the t_VEC of k-1 first moments of \mu: \int z^i d\mu(z), 0<= i < k-1.
    3576             :  *   \sum_j (-1)^(k-2-j) binomial(k-2,j) Y^j \int z^(k-2-j) d\mu(z) = P(1,Y)
    3577             :  * Input is P(1,Y), bin = vecbinomial(k-2): bin[j] = binomial(k-2,j-1) */
    3578             : static GEN
    3579       38626 : RgX_to_moments(GEN P, GEN bin)
    3580             : {
    3581       38626 :   long j, k = lg(bin);
    3582             :   GEN Pd, Bd;
    3583       38626 :   if (typ(P) != t_POL) P = scalarpol(P,0);
    3584       38626 :   P = RgX_to_RgC(P, k-1); /* deg <= k-2 */
    3585       38626 :   settyp(P, t_VEC);
    3586       38626 :   Pd = P+1;  /* Pd[i] = coeff(P,i) */
    3587       38626 :   Bd = bin+1;/* Bd[i] = binomial(k-2,i) */
    3588       46249 :   for (j = 1; j < k-2; j++)
    3589             :   {
    3590        7623 :     GEN c = gel(Pd,j);
    3591        7623 :     if (odd(j)) c = gneg(c);
    3592        7623 :     gel(Pd,j) = gdiv(c, gel(Bd,j));
    3593             :   }
    3594       38626 :   return vecreverse(P);
    3595             : }
    3596             : static GEN
    3597         882 : RgXC_to_moments(GEN v, GEN bin)
    3598             : {
    3599             :   long i, l;
    3600         882 :   GEN w = cgetg_copy(v,&l);
    3601       39508 :   for (i=1; i<l; i++) gel(w,i) = RgX_to_moments(gel(v,i),bin);
    3602         882 :   return w;
    3603             : }
    3604             : 
    3605             : /* W an mspadic, assume O[2] is integral, den is the cancelled denominator
    3606             :  * or NULL, L = log(path)^* in sparse form */
    3607             : static GEN
    3608        2954 : omseval_int(struct m_act *S, GEN PHI, GEN L, hashtable *H)
    3609             : {
    3610             :   long i, l;
    3611        2954 :   GEN v = cgetg_copy(PHI, &l);
    3612        2954 :   ZGl2QC_to_act(S, L, H); /* as operators on V */
    3613        6286 :   for (i = 1; i < l; i++)
    3614             :   {
    3615        3332 :     GEN T = dense_act_col(L, gel(PHI,i));
    3616        3332 :     gel(v,i) = T? FpC_red(T,S->q): zerocol(S->dim);
    3617             :   }
    3618        2954 :   return v;
    3619             : }
    3620             : 
    3621             : GEN
    3622          14 : msomseval(GEN W, GEN phi, GEN path)
    3623             : {
    3624             :   struct m_act S;
    3625          14 :   pari_sp av = avma;
    3626             :   GEN v, Wp;
    3627             :   long n, vden;
    3628          14 :   checkmspadic(W);
    3629          14 :   if (typ(phi) != t_COL || lg(phi) != 4)  pari_err_TYPE("msomseval",phi);
    3630          14 :   vden = itos(gel(phi,2));
    3631          14 :   phi = gel(phi,1);
    3632          14 :   n = mspadic_get_n(W);
    3633          14 :   Wp= mspadic_get_Wp(W);
    3634          14 :   S.k = mspadic_get_weight(W);
    3635          14 :   S.p = mspadic_get_p(W);
    3636          14 :   S.q = powuu(S.p, n+vden);
    3637          14 :   S.dim = n + S.k - 1;
    3638          14 :   S.act = &moments_act;
    3639          14 :   path = path_to_M2(path);
    3640          14 :   v = omseval_int(&S, phi, M2_logf(Wp,path,NULL), NULL);
    3641          14 :   return gerepilecopy(av, v);
    3642             : }
    3643             : /* W = msinit(N,k,...); if flag < 0 or flag >= k-1, allow all symbols;
    3644             :  * else commit to v_p(a_p) <= flag (ordinary if flag = 0)*/
    3645             : GEN
    3646         490 : mspadicinit(GEN W, long p, long n, long flag)
    3647             : {
    3648         490 :   pari_sp av = avma;
    3649             :   long a, N, k;
    3650             :   GEN P, C, M, bin, Wp, Tp, q, pn, actUp, teich, pas;
    3651             : 
    3652         490 :   checkms(W);
    3653         490 :   N = ms_get_N(W);
    3654         490 :   k = msk_get_weight(W);
    3655         490 :   if (flag < 0) flag = 1; /* worst case */
    3656         357 :   else if (flag >= k) flag = k-1;
    3657             : 
    3658         490 :   bin = vecbinomial(k-2);
    3659         490 :   Tp = mshecke(W, p, NULL);
    3660         490 :   if (N % p == 0)
    3661             :   {
    3662          91 :     if ((N/p) % p == 0) pari_err_IMPL("mspadicinit when p^2 | N");
    3663             :     /* a_p != 0 */
    3664          84 :     Wp = W;
    3665          84 :     M = gen_0;
    3666          84 :     flag = (k-2) / 2; /* exact valuation */
    3667             :     /* will multiply by matrix with denominator p^(k-2)/2 in mspadicint.
    3668             :      * Except if p = 2 (multiply by alpha^2) */
    3669          84 :     if (p == 2) n += k-2; else n += (k-2)/2;
    3670          84 :     pn = powuu(p,n);
    3671             :     /* For accuracy mod p^n, oms_dim1 require p^(k/2*n) */
    3672          84 :     q = powiu(pn, k/2);
    3673             :   }
    3674             :   else
    3675             :   { /* p-stabilize */
    3676         399 :     long s = msk_get_sign(W);
    3677             :     GEN M1, M2;
    3678             : 
    3679         399 :     Wp = mskinit(N*p, k, s);
    3680         399 :     M1 = getMorphism(W, Wp, mkvec(mat2(1,0,0,1)));
    3681         399 :     M2 = getMorphism(W, Wp, mkvec(mat2(p,0,0,1)));
    3682         399 :     if (s)
    3683             :     {
    3684         147 :       GEN SW = msk_get_starproj(W), SWp = msk_get_starproj(Wp);
    3685         147 :       M1 = Qevproj_apply2(M1, SW, SWp);
    3686         147 :       M2 = Qevproj_apply2(M2, SW, SWp);
    3687             :     }
    3688         399 :     M = mkvec2(M1,M2);
    3689         399 :     n += Z_lval(Q_denom(M), p); /*den. introduced by p-stabilization*/
    3690             :     /* in supersingular case: will multiply by matrix with denominator p^k
    3691             :      * in mspadicint. Except if p = 2 (multiply by alpha^2) */
    3692         399 :     if (flag) { if (p == 2) n += 2*k-2; else n += k; }
    3693         399 :     pn = powuu(p,n);
    3694             :     /* For accuracy mod p^n, supersingular require p^((2k-1-v_p(a_p))*n) */
    3695         399 :     if (flag) /* k-1 also takes care of a_p = 0. Worst case v_p(a_p) = flag */
    3696         231 :       q = powiu(pn, 2*k-1 - flag);
    3697             :     else
    3698         168 :       q = pn;
    3699             :   }
    3700         483 :   actUp = init_moments_act(Wp, p, n, q, Up_matrices(p));
    3701             : 
    3702         483 :   if (p == 2) C = gen_0;
    3703             :   else
    3704             :   {
    3705         427 :     pas = matpascal(n);
    3706         427 :     teich = teichmullerinit(p, n+1);
    3707         427 :     P = gpowers(utoipos(p), n);
    3708         427 :     C = cgetg(p, t_VEC);
    3709        2317 :     for (a = 1; a < p; a++)
    3710             :     { /* powb[j+1] = ((a - w(a)) / p)^j mod p^n */
    3711        1890 :       GEN powb = Fp_powers(diviuexact(subui(a, gel(teich,a)), p), n, pn);
    3712        1890 :       GEN Ca = cgetg(n+2, t_VEC);
    3713        1890 :       long j, r, ai = Fl_inv(a, p); /* a^(-1) */
    3714        1890 :       gel(C,a) = Ca;
    3715       22134 :       for (j = 0; j <= n; j++)
    3716             :       {
    3717       20244 :         GEN Caj = cgetg(j+2, t_VEC);
    3718       20244 :         GEN atij = gel(teich, Fl_powu(ai,j,p));/* w(a)^(-j) = w(a^(-j) mod p) */
    3719       20244 :         gel(Ca,j+1) = Caj;
    3720      158200 :         for (r = 0; r <= j; r++)
    3721             :         {
    3722      137956 :           GEN c = Fp_mul(gcoeff(pas,j+1,r+1), gel(powb, j-r+1), pn);
    3723      137956 :           c = Fp_mul(c,atij,pn); /* binomial(j,r)*b^(j-r)*w(a)^(-j) mod p^n */
    3724      137956 :           gel(Caj,r+1) = mulii(c, gel(P,j+1)); /* p^j * c mod p^(n+j) */
    3725             :         }
    3726             :       }
    3727             :     }
    3728             :   }
    3729         483 :   return gerepilecopy(av, mkvecn(8, Wp,Tp, bin, actUp, q,
    3730             :                                  mkvecsmall3(p,n,flag), M, C));
    3731             : }
    3732             : 
    3733             : #if 0
    3734             : /* assume phi an ordinary OMS */
    3735             : static GEN
    3736             : omsactgl2(GEN W, GEN phi, GEN M)
    3737             : {
    3738             :   GEN q, Wp, act;
    3739             :   long p, k, n;
    3740             :   checkmspadic(W);
    3741             :   Wp = mspadic_get_Wp(W);
    3742             :   p = mspadic_get_p(W);
    3743             :   k = mspadic_get_weight(W);
    3744             :   n = mspadic_get_n(W);
    3745             :   q = mspadic_get_q(W);
    3746             :   act = init_moments_act(Wp, p, n, q, M);
    3747             :   phi = gel(phi,1);
    3748             :   return dual_act(k-1, act, gel(phi,1));
    3749             : }
    3750             : #endif
    3751             : 
    3752             : static GEN
    3753         483 : eigenvalue(GEN T, GEN x)
    3754             : {
    3755         483 :   long i, l = lg(x);
    3756         637 :   for (i = 1; i < l; i++)
    3757         637 :     if (!isintzero(gel(x,i))) break;
    3758         483 :   if (i == l) pari_err_DOMAIN("mstooms", "phi", "=", gen_0, x);
    3759         483 :   return gdiv(RgMrow_RgC_mul(T,x,i), gel(x,i));
    3760             : }
    3761             : 
    3762             : /* p coprime to ap, return unit root of x^2 - ap*x + p^(k-1), accuracy p^n */
    3763             : GEN
    3764         532 : mspadic_unit_eigenvalue(GEN ap, long k, GEN p, long n)
    3765             : {
    3766         532 :   GEN sqrtD, D = subii(sqri(ap), shifti(powiu(p,k-1),2));
    3767         532 :   if (absequaliu(p,2))
    3768             :   {
    3769          35 :     n++; sqrtD = Zp_sqrt(D, p, n);
    3770          35 :     if (mod4(sqrtD) != mod4(ap)) sqrtD = negi(sqrtD);
    3771             :   }
    3772             :   else
    3773         497 :     sqrtD = Zp_sqrtlift(D, ap, p, n);
    3774             :   /* sqrtD = ap (mod p) */
    3775         532 :   return gmul2n(gadd(ap, cvtop(sqrtD,p,n)), -1);
    3776             : }
    3777             : 
    3778             : /* W = msinit(N,k,...); phi = T_p/U_p - eigensymbol */
    3779             : GEN
    3780         483 : mstooms(GEN W, GEN phi)
    3781             : {
    3782         483 :   pari_sp av = avma;
    3783             :   GEN Wp, bin, Tp, c, alpha, ap, phi0, M;
    3784             :   long k, p, vden;
    3785             : 
    3786         483 :   checkmspadic(W);
    3787         483 :   if (typ(phi) != t_COL)
    3788             :   {
    3789         161 :     if (!is_Qevproj(phi)) pari_err_TYPE("mstooms",phi);
    3790         161 :     phi = gel(phi,1);
    3791         161 :     if (lg(phi) != 2) pari_err_TYPE("mstooms [dim_Q (eigenspace) > 1]",phi);
    3792         161 :     phi = gel(phi,1);
    3793             :   }
    3794             : 
    3795         483 :   Wp = mspadic_get_Wp(W);
    3796         483 :   Tp = mspadic_get_Tp(W);
    3797         483 :   bin = mspadic_get_bin(W);
    3798         483 :   k = msk_get_weight(Wp);
    3799         483 :   p = mspadic_get_p(W);
    3800         483 :   M = mspadic_get_M(W);
    3801             : 
    3802         483 :   phi = Q_remove_denom(phi, &c);
    3803         483 :   ap = eigenvalue(Tp, phi);
    3804         483 :   vden = c? Z_lvalrem(c, p, &c): 0;
    3805             : 
    3806         483 :   if (typ(M) == t_INT)
    3807             :   { /* p | N */
    3808             :     GEN c1;
    3809          84 :     alpha = ap;
    3810          84 :     alpha = ginv(alpha);
    3811          84 :     phi0 = mseval(Wp, phi, NULL);
    3812          84 :     phi0 = RgXC_to_moments(phi0, bin);
    3813          84 :     phi0 = Q_remove_denom(phi0, &c1);
    3814          84 :     if (c1) { vden += Z_lvalrem(c1, p, &c1); c = mul_denom(c,c1); }
    3815          84 :     if (umodiu(ap,p)) /* p \nmid a_p */
    3816          49 :       phi = oms_dim1(W, phi0, alpha, 0);
    3817             :     else
    3818             :     {
    3819          35 :       phi = oms_dim1(W, phi0, alpha, 1);
    3820          35 :       phi = Q_remove_denom(phi, &c1);
    3821          35 :       if (c1) { vden += Z_lvalrem(c1, p, &c1); c = mul_denom(c,c1); }
    3822             :     }
    3823             :   }
    3824             :   else
    3825             :   { /* p-stabilize */
    3826             :     GEN M1, M2, phi1, phi2, c1;
    3827         399 :     if (typ(M) != t_VEC || lg(M) != 3) pari_err_TYPE("mstooms",W);
    3828         399 :     M1 = gel(M,1);
    3829         399 :     M2 = gel(M,2);
    3830             : 
    3831         399 :     phi1 = RgM_RgC_mul(M1, phi);
    3832         399 :     phi2 = RgM_RgC_mul(M2, phi);
    3833         399 :     phi1 = mseval(Wp, phi1, NULL);
    3834         399 :     phi2 = mseval(Wp, phi2, NULL);
    3835             : 
    3836         399 :     phi1 = RgXC_to_moments(phi1, bin);
    3837         399 :     phi2 = RgXC_to_moments(phi2, bin);
    3838         399 :     phi = Q_remove_denom(mkvec2(phi1,phi2), &c1);
    3839         399 :     phi1 = gel(phi,1);
    3840         399 :     phi2 = gel(phi,2);
    3841         399 :     if (c1) { vden += Z_lvalrem(c1, p, &c1); c = mul_denom(c,c1); }
    3842             :     /* all polynomials multiplied by c p^vden */
    3843         399 :     if (umodiu(ap, p))
    3844             :     {
    3845         238 :       alpha = mspadic_unit_eigenvalue(ap, k, utoipos(p), mspadic_get_n(W));
    3846         238 :       alpha = ginv(alpha);
    3847         238 :       phi0 = gsub(phi1, gmul(lift_shallow(alpha),phi2));
    3848         238 :       phi = oms_dim1(W, phi0, alpha, 0);
    3849             :     }
    3850             :     else
    3851             :     { /* p | ap, alpha = [a_p, -1; p^(k-1), 0] */
    3852         161 :       long flag = mspadic_get_flag(W);
    3853         161 :       if (!flag || (signe(ap) && Z_lval(ap,p) < flag))
    3854           7 :         pari_err_TYPE("mstooms [v_p(ap) > mspadicinit flag]", phi);
    3855         154 :       alpha = mkmat22(ap,gen_m1, powuu(p, k-1),gen_0);
    3856         154 :       alpha = ginv(alpha);
    3857         154 :       phi = oms_dim2(W, mkvec2(phi1,phi2), gsqr(alpha), ap);
    3858         154 :       phi = Q_remove_denom(phi, &c1);
    3859         154 :       if (c1) { vden += Z_lvalrem(c1, p, &c1); c = mul_denom(c,c1); }
    3860             :     }
    3861             :   }
    3862         476 :   if (vden) c = mul_denom(c, powuu(p,vden));
    3863         476 :   if (p == 2) alpha = gsqr(alpha);
    3864         476 :   if (c) alpha = gdiv(alpha,c);
    3865         476 :   if (typ(alpha) == t_MAT)
    3866             :   { /* express in basis (omega,-p phi(omega)) */
    3867         154 :     gcoeff(alpha,2,1) = gdivgs(gcoeff(alpha,2,1), -p);
    3868         154 :     gcoeff(alpha,2,2) = gdivgs(gcoeff(alpha,2,2), -p);
    3869             :     /* at the end of mspadicint we shall multiply result by [1,0;0,-1/p]*alpha
    3870             :      * vden + k is the denominator of this matrix */
    3871             :   }
    3872             :   /* phi is integral-valued */
    3873         476 :   return gerepilecopy(av, mkcol3(phi, stoi(vden), alpha));
    3874             : }
    3875             : 
    3876             : /* HACK: the v[j] have different lengths */
    3877             : static GEN
    3878        2156 : FpVV_dotproduct(GEN v, GEN w, GEN p)
    3879             : {
    3880        2156 :   long j, l = lg(v);
    3881        2156 :   GEN T = cgetg(l, t_VEC);
    3882       26026 :   for (j = 1; j < l; j++) gel(T,j) = FpV_dotproduct(gel(v,j),w,p);
    3883        2156 :   return T;
    3884             : }
    3885             : 
    3886             : /* \int (-4z)^j given \int z^j */
    3887             : static GEN
    3888          98 : twistmoment_m4(GEN v)
    3889             : {
    3890             :   long i, l;
    3891          98 :   GEN w = cgetg_copy(v, &l);
    3892        2009 :   for (i = 1; i < l; i++)
    3893             :   {
    3894        1911 :     GEN c = gel(v,i);
    3895        1911 :     if (i > 1) c = gmul2n(c, (i-1)<<1);
    3896        1911 :     gel(w,i) = odd(i)? c: gneg(c);
    3897             :   }
    3898          98 :   return w;
    3899             : }
    3900             : /* \int (4z)^j given \int z^j */
    3901             : static GEN
    3902          98 : twistmoment_4(GEN v)
    3903             : {
    3904             :   long i, l;
    3905          98 :   GEN w = cgetg_copy(v, &l);
    3906        2009 :   for (i = 1; i < l; i++)
    3907             :   {
    3908        1911 :     GEN c = gel(v,i);
    3909        1911 :     if (i > 1) c = gmul2n(c, (i-1)<<1);
    3910        1911 :     gel(w,i) = c;
    3911             :   }
    3912          98 :   return w;
    3913             : }
    3914             : /* W an mspadic, phi eigensymbol, p \nmid D. Return C(x) mod FilM */
    3915             : GEN
    3916         483 : mspadicmoments(GEN W, GEN PHI, long D)
    3917             : {
    3918         483 :   pari_sp av = avma;
    3919         483 :   long na, ia, b, lphi, aD = labs(D), pp, p, k, n, vden;
    3920             :   GEN Wp, Dact, vL, v, C, pn, phi;
    3921             :   struct m_act S;
    3922             :   hashtable *H;
    3923             : 
    3924         483 :   checkmspadic(W);
    3925         483 :   Wp = mspadic_get_Wp(W);
    3926         483 :   p = mspadic_get_p(W);
    3927         483 :   k = mspadic_get_weight(W);
    3928         483 :   n = mspadic_get_n(W);
    3929         483 :   C = mspadic_get_C(W);
    3930         483 :   if (typ(PHI) != t_COL || lg(PHI) != 4 || typ(gel(PHI,1)) != t_VEC)
    3931         476 :     PHI = mstooms(W, PHI);
    3932         476 :   vden = itos( gel(PHI,2) );
    3933         476 :   phi = gel(PHI,1); lphi = lg(phi);
    3934         476 :   if (p == 2) { na = 2; pp = 4; }
    3935         420 :   else        { na = p-1; pp = p; }
    3936         476 :   pn = powuu(p, n + vden);
    3937             : 
    3938         476 :   S.p = p;
    3939         476 :   S.k = k;
    3940         476 :   S.q = pn;
    3941         476 :   S.dim = n+k-1;
    3942         476 :   S.act = &moments_act;
    3943         476 :   H = Gl2act_cache(ms_get_nbgen(Wp));
    3944         476 :   if (D == 1) Dact = NULL;
    3945             :   else
    3946             :   {
    3947          63 :     GEN gaD = utoi(aD), Dk = Fp_pows(stoi(D), 2-k, pn);
    3948          63 :     if (!sisfundamental(D)) pari_err_TYPE("mspadicmoments", stoi(D));
    3949          63 :     if (D % p == 0) pari_err_DOMAIN("mspadicmoments","p","|", stoi(D), utoi(p));
    3950          63 :     Dact = cgetg(aD, t_VEC);
    3951         532 :     for (b = 1; b < aD; b++)
    3952             :     {
    3953         469 :       GEN z = NULL;
    3954         469 :       long s = kross(D, b);
    3955         469 :       if (s)
    3956             :       {
    3957         462 :         pari_sp av2 = avma;
    3958             :         GEN d;
    3959         462 :         z = moments_act_i(&S, mkmat22(gaD,utoipos(b), gen_0,gaD));
    3960         462 :         d = s > 0? Dk: Fp_neg(Dk, pn);
    3961         924 :         z = equali1(d)? gerepilecopy(av2, z)
    3962         462 :                       : gerepileupto(av2, FpM_Fp_mul(z, d, pn));
    3963             :       }
    3964         469 :       gel(Dact,b) = z;
    3965             :     }
    3966             :   }
    3967         476 :   vL = cgetg(na+1,t_VEC);
    3968             :   /* first pass to precompute log(paths), preload matrices and allow GC later */
    3969        2464 :   for (ia = 1; ia <= na; ia++)
    3970             :   {
    3971             :     GEN path, La;
    3972        1988 :     long a = (p == 2 && ia == 2)? -1: ia;
    3973        1988 :     if (Dact)
    3974             :     { /* twist by D */
    3975         224 :       La = cgetg(aD, t_VEC);
    3976        1442 :       for (b = 1; b < aD; b++)
    3977             :       {
    3978        1218 :         GEN Actb = gel(Dact,b);
    3979        1218 :         if (!Actb) continue;
    3980             :         /* oo -> a/pp + b/|D|*/
    3981        1176 :         path = mkmat22(gen_1, addii(mulss(a, aD), muluu(pp, b)),
    3982             :                        gen_0, muluu(pp, aD));
    3983        1176 :         gel(La,b) = M2_logf(Wp,path,NULL);
    3984        1176 :         ZGl2QC_preload(&S, gel(La,b), H);
    3985             :       }
    3986             :     }
    3987             :     else
    3988             :     {
    3989        1764 :       path = mkmat22(gen_1,stoi(a), gen_0, utoipos(pp));
    3990        1764 :       La = M2_logf(Wp,path,NULL);
    3991        1764 :       ZGl2QC_preload(&S, La, H);
    3992             :     }
    3993        1988 :     gel(vL,ia) = La;
    3994             :   }
    3995         476 :   v = cgetg(na+1,t_VEC);
    3996             :   /* second pass, with GC */
    3997        2464 :   for (ia = 1; ia <= na; ia++)
    3998             :   {
    3999        1988 :     pari_sp av2 = avma;
    4000        1988 :     GEN vca, Ca = gel(C,ia), La = gel(vL,ia), va = cgetg(lphi, t_VEC);
    4001             :     long i;
    4002        1988 :     if (!Dact) vca = omseval_int(&S, phi, La, H);
    4003             :     else
    4004             :     { /* twist by D */
    4005         224 :       vca = cgetg(lphi,t_VEC);
    4006        1442 :       for (b = 1; b < aD; b++)
    4007             :       {
    4008        1218 :         GEN T, Actb = gel(Dact,b);
    4009        1218 :         if (!Actb) continue;
    4010        1176 :         T = omseval_int(&S, phi, gel(La,b), H);
    4011        2352 :         for (i = 1; i < lphi; i++)
    4012             :         {
    4013        1176 :           GEN z = FpM_FpC_mul(Actb, gel(T,i), pn);
    4014        1176 :           gel(vca,i) = b==1? z: ZC_add(gel(vca,i), z);
    4015             :         }
    4016             :       }
    4017             :     }
    4018        1988 :     if (p != 2)
    4019        4032 :     { for (i=1; i<lphi; i++) gel(va,i) = FpVV_dotproduct(Ca,gel(vca,i),pn); }
    4020         112 :     else if (ia == 1) /* \tilde{a} = 1 */
    4021         154 :     { for (i=1; i<lphi; i++) gel(va,i) = twistmoment_4(gel(vca,i)); }
    4022             :     else /* \tilde{a} = -1 */
    4023         154 :     { for (i=1; i<lphi; i++) gel(va,i) = twistmoment_m4(gel(vca,i)); }
    4024        1988 :     gel(v,ia) = gerepilecopy(av2, va);
    4025             :   }
    4026         476 :   return gerepilecopy(av, mkvec3(v, gel(PHI,3), mkvecsmall4(p,n+vden,n,D)));
    4027             : }
    4028             : static void
    4029        1918 : checkoms(GEN v)
    4030             : {
    4031        1918 :   if (typ(v) != t_VEC || lg(v) != 4 || typ(gel(v,1)) != t_VEC
    4032        1918 :       || typ(gel(v,3))!=t_VECSMALL)
    4033           0 :     pari_err_TYPE("checkoms [apply mspadicmoments]", v);
    4034        1918 : }
    4035             : static long
    4036        4284 : oms_get_p(GEN oms) { return gel(oms,3)[1]; }
    4037             : static long
    4038        4186 : oms_get_n(GEN oms) { return gel(oms,3)[2]; }
    4039             : static long
    4040        2464 : oms_get_n0(GEN oms) { return gel(oms,3)[3]; }
    4041             : static long
    4042        1918 : oms_get_D(GEN oms) { return gel(oms,3)[4]; }
    4043             : static int
    4044          98 : oms_is_supersingular(GEN oms) { GEN v = gel(oms,1); return lg(gel(v,1)) == 3; }
    4045             : 
    4046             : /* sum(j = 1, n, (-1)^(j+1)/j * x^j) */
    4047             : static GEN
    4048         784 : log1x(long n)
    4049             : {
    4050         784 :   long i, l = n+3;
    4051         784 :   GEN v = cgetg(l, t_POL);
    4052         784 :   v[1] = evalvarn(0)|evalsigne(1); gel(v,2) = gen_0;
    4053        8904 :   for (i = 3; i < l; i++) gel(v,i) = ginv(stoi(odd(i)? i-2: 2-i));
    4054         784 :   return v;
    4055             : }
    4056             : 
    4057             : /* S = (1+x)^zk log(1+x)^logj (mod x^(n+1)) */
    4058             : static GEN
    4059        1820 : xlog1x(long n, long zk, long logj, long *pteich)
    4060             : {
    4061        1820 :   GEN S = logj? RgXn_powu_i(log1x(n), logj, n+1): NULL;
    4062        1820 :   if (zk)
    4063             :   {
    4064        1183 :     GEN L = deg1pol_shallow(gen_1, gen_1, 0); /* x+1 */
    4065        1183 :     *pteich += zk;
    4066        1183 :     if (zk < 0) { L = RgXn_inv(L,n+1); zk = -zk; }
    4067        1183 :     if (zk != 1) L = RgXn_powu_i(L, zk, n+1);
    4068        1183 :     S = S? RgXn_mul(S, L, n+1): L;
    4069             :   }
    4070        1820 :   return S;
    4071             : }
    4072             : 
    4073             : /* oms from mspadicmoments; integrate teichmuller^i * S(x) [S = NULL: 1]*/
    4074             : static GEN
    4075        2366 : mspadicint(GEN oms, long teichi, GEN S)
    4076             : {
    4077        2366 :   pari_sp av = avma;
    4078        2366 :   long p = oms_get_p(oms), n = oms_get_n(oms), n0 = oms_get_n0(oms);
    4079        2366 :   GEN vT = gel(oms,1), alpha = gel(oms,2), gp = utoipos(p);
    4080        2366 :   long loss = S? Z_lval(Q_denom(S), p): 0;
    4081        2366 :   long nfinal = minss(n-loss, n0);
    4082        2366 :   long i, la, l = lg(gel(vT,1));
    4083        2366 :   GEN res = cgetg(l, t_COL), teich = NULL;
    4084             : 
    4085        2366 :   if (S) S = RgX_to_RgC(S,lg(gmael(vT,1,1))-1);
    4086        2366 :   if (p == 2)
    4087             :   {
    4088         448 :     la = 3; /* corresponds to [1,-1] */
    4089         448 :     teichi &= 1;
    4090             :   }
    4091             :   else
    4092             :   {
    4093        1918 :     la = p; /* corresponds to [1,2,...,p-1] */
    4094        1918 :     teichi = umodsu(teichi, p-1);
    4095        1918 :     if (teichi) teich = teichmullerinit(p, n);
    4096             :   }
    4097        5446 :   for (i=1; i<l; i++)
    4098             :   {
    4099        3080 :     pari_sp av2 = avma;
    4100        3080 :     GEN s = gen_0;
    4101             :     long ia;
    4102       14756 :     for (ia = 1; ia < la; ia++)
    4103             :     { /* Ta[j+1] correct mod p^n */
    4104       11676 :       GEN Ta = gmael(vT,ia,i), v = S? RgV_dotproduct(Ta, S): gel(Ta,1);
    4105       11676 :       if (teichi && ia != 1)
    4106             :       {
    4107        3843 :         if (p != 2)
    4108        3626 :           v = gmul(v, gel(teich, Fl_powu(ia,teichi,p)));
    4109             :         else
    4110         217 :           if (teichi) v = gneg(v);
    4111             :       }
    4112       11676 :       s = gadd(s, v);
    4113             :     }
    4114        3080 :     s = gadd(s, zeropadic(gp,nfinal));
    4115        3080 :     gel(res,i) = gerepileupto(av2, s);
    4116             :   }
    4117        2366 :   return gerepileupto(av, gmul(alpha, res));
    4118             : }
    4119             : /* integrate P = polynomial in log(x); vlog[j+1] = mspadicint(0,log(1+x)^j) */
    4120             : static GEN
    4121         539 : mspadicint_RgXlog(GEN P, GEN vlog)
    4122             : {
    4123         539 :   long i, d = degpol(P);
    4124         539 :   GEN s = gmul(gel(P,2), gel(vlog,1));
    4125        1848 :   for (i = 1; i <= d; i++) s = gadd(s, gmul(gel(P,i+2), gel(vlog,i+1)));
    4126         539 :   return s;
    4127             : };
    4128             : 
    4129             : /* oms from mspadicmoments */
    4130             : GEN
    4131          98 : mspadicseries(GEN oms, long teichi)
    4132             : {
    4133          98 :   pari_sp av = avma;
    4134             :   GEN S, L, X, vlog, s, s2, u, logu, bin;
    4135             :   long j, p, m, n, step, stop;
    4136          98 :   checkoms(oms);
    4137          98 :   n = oms_get_n0(oms);
    4138          98 :   if (n < 1)
    4139             :   {
    4140           0 :     s = zeroser(0,0);
    4141           0 :     if (oms_is_supersingular(oms)) s = mkvec2(s,s);
    4142           0 :     return gerepilecopy(av, s);
    4143             :   }
    4144          98 :   p = oms_get_p(oms);
    4145          98 :   vlog = cgetg(n+1, t_VEC);
    4146          98 :   step = p == 2? 2: 1;
    4147          98 :   stop = 0;
    4148          98 :   S = NULL;
    4149          98 :   L = log1x(n);
    4150         644 :   for (j = 0; j < n; j++)
    4151             :   {
    4152         616 :     if (j) stop += step + u_lval(j,p); /* = step*j + v_p(j!) */
    4153         616 :     if (stop >= n) break;
    4154             :     /* S = log(1+x)^j */
    4155         546 :     gel(vlog,j+1) = mspadicint(oms,teichi,S);
    4156         546 :     S = S? RgXn_mul(S, L, n+1): L;
    4157             :   }
    4158          98 :   m = j;
    4159          98 :   u = utoipos(p == 2? 5: 1+p);
    4160          98 :   logu = glog(cvtop(u, utoipos(p), 4*m), 0);
    4161          98 :   X = gdiv(pol_x(0), logu);
    4162          98 :   s = cgetg(m+1, t_VEC);
    4163          98 :   s2 = oms_is_supersingular(oms)? cgetg(m+1, t_VEC): NULL;
    4164          98 :   bin = pol_1(0);
    4165         539 :   for (j = 0; j < m; j++)
    4166             :   { /* bin = binomial(x/log(1+p+O(p^(4*n))), j) mod x^m */
    4167         539 :     GEN a, v = mspadicint_RgXlog(bin, vlog);
    4168         539 :     int done = 1;
    4169         539 :     gel(s,j+1) = a = gel(v,1);
    4170         539 :     if (!gequal0(a) || valp(a) > 0) done = 0; else setlg(s,j+1);
    4171         539 :     if (s2)
    4172             :     {
    4173         119 :       gel(s2,j+1) = a = gel(v,2);
    4174         119 :       if (!gequal0(a) || valp(a) > 0) done = 0; else setlg(s2,j+1);
    4175             :     }
    4176         539 :     if (done || j == m-1) break;
    4177         441 :     bin = RgXn_mul(bin, gdivgs(gsubgs(X, j), j+1), m);
    4178             :   }
    4179          98 :   s = RgV_to_ser(s,0,lg(s)+1);
    4180          98 :   if (s2) { s2 = RgV_to_ser(s2,0,lg(s2)+1); s = mkvec2(s, s2); }
    4181          98 :   if (kross(oms_get_D(oms), p) >= 0) return gerepilecopy(av, s);
    4182           7 :   return gerepileupto(av, gneg(s));
    4183             : }
    4184             : void
    4185        1911 : mspadic_parse_chi(GEN s, GEN *s1, GEN *s2)
    4186             : {
    4187        1911 :   if (!s) *s1 = *s2 = gen_0;
    4188        1778 :   else switch(typ(s))
    4189             :   {
    4190        1274 :     case t_INT: *s1 = *s2 = s; break;
    4191         504 :     case t_VEC:
    4192         504 :       if (lg(s) == 3)
    4193             :       {
    4194         504 :         *s1 = gel(s,1);
    4195         504 :         *s2 = gel(s,2);
    4196         504 :         if (typ(*s1) == t_INT && typ(*s2) == t_INT) break;
    4197             :       }
    4198           0 :     default: pari_err_TYPE("mspadicL",s);
    4199           0 :              *s1 = *s2 = NULL;
    4200             :   }
    4201        1911 : }
    4202             : /* oms from mspadicmoments
    4203             :  * r-th derivative of L(f,chi^s,psi) in direction <chi>
    4204             :    - s \in Z_p \times \Z/(p-1)\Z, s-> chi^s=<\chi>^s_1 omega^s_2)
    4205             :    - Z -> Z_p \times \Z/(p-1)\Z par s-> (s, s mod p-1).
    4206             :  */
    4207             : GEN
    4208        1820 : mspadicL(GEN oms, GEN s, long r)
    4209             : {
    4210        1820 :   pari_sp av = avma;
    4211             :   GEN s1, s2, z, S;
    4212             :   long p, n, teich;
    4213        1820 :   checkoms(oms);
    4214        1820 :   p = oms_get_p(oms);
    4215        1820 :   n = oms_get_n(oms);
    4216        1820 :   mspadic_parse_chi(s, &s1,&s2);
    4217        1820 :   teich = umodiu(subii(s2,s1), p==2? 2: p-1);
    4218        1820 :   S = xlog1x(n, itos(s1), r, &teich);
    4219        1820 :   z = mspadicint(oms, teich, S);
    4220        1820 :   if (lg(z) == 2) z = gel(z,1);
    4221        1820 :   if (kross(oms_get_D(oms), p) < 0) z = gneg(z);
    4222        1820 :   return gerepilecopy(av, z);
    4223             : }
    4224             : 
    4225             : /****************************************************************************/
    4226             : 
    4227             : struct siegel
    4228             : {
    4229             :   GEN V, Ast;
    4230             :   long N; /* level */
    4231             :   long oo; /* index of the [oo,0] path */
    4232             :   long k1, k2; /* two distinguished indices */
    4233             :   long n; /* #W, W = initial segment [in siegelstepC] already normalized */
    4234             : };
    4235             : 
    4236             : static void
    4237         378 : siegel_init(struct siegel *C, GEN M)
    4238             : {
    4239             :   GEN CPI, CP, MM, V, W, Ast;
    4240         378 :   GEN m = gel(M,11), M2 = gel(M,2), S = msN_get_section(M);
    4241         378 :   GEN E2fromE1 = msN_get_E2fromE1(M);
    4242         378 :   long m0 = lg(M2)-1;
    4243         378 :   GEN E2  = vecslice(M2, m[1]+1, m[2]);/* E2 */
    4244         378 :   GEN E1T = vecslice(M2, m[3]+1, m0); /* E1,T2,T31 */
    4245         378 :   GEN L = shallowconcat(E1T, E2);
    4246         378 :   long i, l = lg(L), n = lg(E1T)-1, lE = lg(E2);
    4247             : 
    4248         378 :   Ast = cgetg(l, t_VECSMALL);
    4249        6195 :   for (i = 1; i < lE; ++i)
    4250             :   {
    4251        5817 :     long j = E2fromE1_c(gel(E2fromE1,i));
    4252        5817 :     Ast[n+i] = j;
    4253        5817 :     Ast[j] = n+i;
    4254             :   }
    4255         686 :   for (; i<=n; ++i) Ast[i] = i;
    4256         378 :   MM = cgetg (l,t_VEC);
    4257             : 
    4258       12320 :   for (i = 1; i < l; i++)
    4259             :   {
    4260       11942 :     GEN c = gel(S, L[i]);
    4261       11942 :     long c12, c22, c21 = ucoeff(c,2,1);
    4262       11942 :     if (!c21) { gel(MM,i) = gen_0; continue; }
    4263       11564 :     c22 = ucoeff(c,2,2);
    4264       11564 :     if (!c22) { gel(MM,i) = gen_m1; continue; }
    4265       11186 :     c12 = ucoeff(c,1,2);
    4266       11186 :     gel(MM,i) = sstoQ(c12, c22); /* right extremity > 0 */
    4267             :   }
    4268         378 :   CP = indexsort(MM);
    4269         378 :   CPI = cgetg(l, t_VECSMALL);
    4270         378 :   V = cgetg(l, t_VEC);
    4271         378 :   W = cgetg(l, t_VECSMALL);
    4272       12320 :   for (i = 1; i < l; ++i)
    4273             :   {
    4274       11942 :     gel(V,i) = mat2_to_ZM(gel(S, L[CP[i]]));
    4275       11942 :     CPI[CP[i]] = i;
    4276             :   }
    4277       12320 :   for (i = 1; i < l; ++i) W[CPI[i]] = CPI[Ast[i]];
    4278         378 :   C->V = V;
    4279         378 :   C->Ast = W;
    4280         378 :   C->n = 0;
    4281         378 :   C->oo = 2;
    4282         378 :   C->N = ms_get_N(M);
    4283         378 : }
    4284             : 
    4285             : static double
    4286           0 : ZMV_size(GEN v)
    4287             : {
    4288           0 :   long i, l = lg(v);
    4289           0 :   GEN z = cgetg(l, t_VECSMALL);
    4290           0 :   for (i = 1; i < l; i++) z[i] = gexpo(gel(v,i));
    4291           0 :   return ((double)zv_sum(z)) / (4*(l-1));
    4292             : }
    4293             : 
    4294             : /* apply permutation perm to struct S. Don't follow k1,k2 */
    4295             : static void
    4296        5558 : siegel_perm0(struct siegel *S, GEN perm)
    4297             : {
    4298        5558 :   pari_sp av = avma;
    4299        5558 :   long i, l = lg(S->V);
    4300        5558 :   GEN V2 = cgetg(l, t_VEC), Ast2 = cgetg(l, t_VECSMALL);
    4301        5558 :   GEN V = S->V, Ast = S->Ast;
    4302             : 
    4303      267078 :   for (i = 1; i < l; i++) gel(V2,perm[i]) = gel(V,i);
    4304      267078 :   for (i = 1; i < l; i++) Ast2[perm[i]] = perm[Ast[i]];
    4305      267078 :   for (i = 1; i < l; i++) { S->Ast[i] = Ast2[i]; gel(V,i) = gel(V2,i); }
    4306        5558 :   set_avma(av); S->oo = perm[S->oo];
    4307        5558 : }
    4308             : /* apply permutation perm to full struct S */
    4309             : static void
    4310        5194 : siegel_perm(struct siegel *S, GEN perm)
    4311             : {
    4312        5194 :   siegel_perm0(S, perm);
    4313        5194 :   S->k1 = perm[S->k1];
    4314        5194 :   S->k2 = perm[S->k2];
    4315        5194 : }
    4316             : /* cyclic permutation of lg = l-1 moving a -> 1, a+1 -> 2, etc.  */
    4317             : static GEN
    4318        2884 : rotate_perm(long l, long a)
    4319             : {
    4320        2884 :   GEN p = cgetg(l, t_VECSMALL);
    4321        2884 :   long i, j = 1;
    4322       86905 :   for (i = a; i < l; i++) p[i] = j++;
    4323       49329 :   for (i = 1; i < a; i++) p[i] = j++;
    4324        2884 :   return p;
    4325             : }
    4326             : 
    4327             : /* a1 < c1 <= a2 < c2*/
    4328             : static GEN
    4329        2520 : basic_op_perm(long l, long a1, long a2, long c1, long c2)
    4330             : {
    4331        2520 :   GEN p = cgetg(l, t_VECSMALL);
    4332        2520 :   long i, j = 1;
    4333        2520 :   p[a1] = j++;
    4334       22568 :   for (i = c1; i < a2; i++)   p[i] = j++;
    4335       32284 :   for (i = a1+1; i < c1; i++) p[i] = j++;
    4336        2520 :   p[a2] = j++;
    4337       44891 :   for (i = c2; i < l; i++)    p[i] = j++;
    4338        2520 :   for (i = 1; i < a1; i++)    p[i] = j++;
    4339       29855 :   for (i = a2+1; i < c2; i++) p[i] = j++;
    4340        2520 :   return p;
    4341             : }
    4342             : static GEN
    4343         154 : basic_op_perm_elliptic(long l, long a1)
    4344             : {
    4345         154 :   GEN p = cgetg(l, t_VECSMALL);
    4346         154 :   long i, j = 1;
    4347         154 :   p[a1] = j++;
    4348        2660 :   for (i = 1; i < a1; i++)   p[i] = j++;
    4349        3990 :   for (i = a1+1; i < l; i++) p[i] = j++;
    4350         154 :   return p;
    4351             : }
    4352             : static GEN
    4353       14616 : ZM2_rev(GEN T) { return mkmat2(gel(T,2), ZC_neg(gel(T,1))); }
    4354             : 
    4355             : /* In place, V = vector of consecutive paths, between x <= y.
    4356             :  * V[x..y-1] <- g*V[x..y-1] */
    4357             : static void
    4358        5733 : path_vec_mul(GEN V, long x, long y, GEN g)
    4359             : {
    4360             :   long j;
    4361             :   GEN M;
    4362        5733 :   if (x == y) return;
    4363        3360 :   M = gel(V,x); gel(V,x) = ZM_mul(g,M);
    4364       37709 :   for (j = x+1; j < y; j++) /* V[j] <- g*V[j], optimized */
    4365             :   {
    4366       34349 :     GEN Mnext = gel(V,j); /* Mnext[,1] = M[,2] */
    4367       34349 :     GEN gM = gel(V,j-1), u = gel(gM,2);
    4368       34349 :     if (!ZV_equal(gel(M,2), gel(Mnext,1))) u = ZC_neg(u);
    4369       34349 :     gel(V,j) = mkmat2(u, ZM_ZC_mul(g,gel(Mnext,2)));
    4370       34349 :     M = Mnext;
    4371             :   }
    4372             : }
    4373             : 
    4374        4830 : static long prev(GEN V, long i) { return (i == 1)? lg(V)-1: i-1; }
    4375        4830 : static long next(GEN V, long i) { return (i == lg(V)-1)? 1: i+1; }
    4376             : static GEN
    4377       19810 : ZM_det2(GEN u, GEN v)
    4378             : {
    4379       19810 :   GEN a = gel(u,1), c = gel(u,2);
    4380       19810 :   GEN b = gel(v,1), d = gel(v,2); return subii(mulii(a,d), mulii(b,c));
    4381             : }
    4382             : static GEN
    4383       14616 : ZM2_det(GEN T) { return ZM_det2(gel(T,1),gel(T,2)); }
    4384             : static long
    4385        5194 : ZM_det2_sign(GEN u, GEN v)
    4386             : {
    4387        5194 :   pari_sp av = avma;
    4388        5194 :   long s = signe(ZM_det2(u, v));
    4389        5194 :   return gc_long(av, s);
    4390             : }
    4391             : static void
    4392        4466 : fill1(GEN V, long a)
    4393             : {
    4394        4466 :   long p = prev(V,a), n = next(V,a);
    4395        4466 :   GEN u = gmael(V,p,2), v = gmael(V,n,1);
    4396        4466 :   if (ZM_det2_sign(u,v) < 0) v = ZC_neg(v);
    4397        4466 :   gel(V,a) = mkmat2(u, v);
    4398        4466 : }
    4399             : /* a1 < a2 */
    4400             : static void
    4401        2520 : fill2(GEN V, long a1, long a2)
    4402             : {
    4403        2520 :   if (a2 != a1+1) { fill1(V,a1); fill1(V,a2); } /* non adjacent, reconnect */
    4404             :   else
    4405             :   { /* parabolic */
    4406         364 :     long p = prev(V,a1), n = next(V,a2);
    4407         364 :     GEN u, v, C = gmael(V,a1,2), mC = NULL; /* = \pm V[a2][1] */
    4408         364 :     u = gmael(V,p,2); v = C;
    4409         364 :     if (ZM_det2_sign(u,v) < 0) v = mC = ZC_neg(C);
    4410         364 :     gel(V,a1) = mkmat2(u,v);
    4411         364 :     v = gmael(V,n,1); u = C;
    4412         364 :     if (ZM_det2_sign(u,v) < 0) u = mC? mC: ZC_neg(C);
    4413         364 :     gel(V,a2) = mkmat2(u,v);
    4414             :   }
    4415        2520 : }
    4416             : 
    4417             : /* DU = det(U), return g = T*U^(-1) or NULL if not in Gamma0(N); if N = 0,
    4418             :  * only test whether g is integral */
    4419             : static GEN
    4420       14903 : ZM2_div(GEN T, GEN U, GEN DU, long N)
    4421             : {
    4422       14903 :   GEN a=gcoeff(U,1,1), b=gcoeff(U,1,2), c=gcoeff(U,2,1), d=gcoeff(U,2,2);
    4423       14903 :   GEN e=gcoeff(T,1,1), f=gcoeff(T,1,2), g=gcoeff(T,2,1), h=gcoeff(T,2,2);
    4424             :   GEN A, B, C, D, r;
    4425             : 
    4426       14903 :   C = dvmdii(subii(mulii(d,g), mulii(c,h)), DU, &r);
    4427       14903 :   if (r != gen_0 || (N && smodis(C,N))) return NULL;
    4428       14616 :   A = dvmdii(subii(mulii(d,e), mulii(c,f)), DU, &r);
    4429       14616 :   if (r != gen_0) return NULL;
    4430       14616 :   B = dvmdii(subii(mulii(a,f), mulii(b,e)), DU, &r);
    4431       14616 :   if (r != gen_0) return NULL;
    4432       14616 :   D = dvmdii(subii(mulii(a,h), mulii(g,b)), DU, &r);
    4433       14616 :   if (r != gen_0) return NULL;
    4434       14616 :   return mkmat22(A,B,C,D);
    4435             : }
    4436             : 
    4437             : static GEN
    4438       14616 : get_g(struct siegel *S, long a1)
    4439             : {
    4440       14616 :   pari_sp av = avma;
    4441       14616 :   long a2 = S->Ast[a1];
    4442       14616 :   GEN a = gel(S->V,a1), ar = ZM2_rev(gel(S->V,a2)), Dar = ZM2_det(ar);
    4443       14616 :   GEN g = ZM2_div(a, ar, Dar, S->N);
    4444       14616 :   if (!g)
    4445             :   {
    4446         287 :     GEN tau = mkmat22(gen_0,gen_m1, gen_1,gen_m1); /*[0,-1;1,-1]*/
    4447         287 :     g = ZM2_div(ZM_mul(ar, tau), ar, Dar, 0);
    4448             :   }
    4449       14616 :   return gerepilecopy(av, g);
    4450             : }
    4451             : /* input V = (X1 a X2 | X3 a^* X4) + Ast
    4452             :  * a1 = index of a
    4453             :  * a2 = index of a^*, inferred from a1. We must have a != a^*
    4454             :  * c1 = first cut [ index of first path in X3 ]
    4455             :  * c2 = second cut [ either in X4 or X1, index of first path ]
    4456             :  * Assume a < a^* (cf Paranoia below): c1 or c2 must be in
    4457             :  *    ]a,a^*], and the other in the "complement" ]a^*,a] */
    4458             : static void
    4459        2520 : basic_op(struct siegel *S, long a1, long c1, long c2)
    4460             : {
    4461             :   pari_sp av;
    4462        2520 :   long l = lg(S->V), a2 = S->Ast[a1];
    4463             :   GEN g;
    4464             : 
    4465        2520 :   if (a1 == a2)
    4466             :   { /* a = a^* */
    4467           0 :     g = get_g(S, a1);
    4468           0 :     path_vec_mul(S->V, a1+1, l, g);
    4469           0 :     av = avma;
    4470           0 :     siegel_perm(S, basic_op_perm_elliptic(l, a1));
    4471             :     /* fill the hole left at a1, reconnect the path */
    4472           0 :     set_avma(av); fill1(S->V, a1); return;
    4473             :   }
    4474             : 
    4475             :   /* Paranoia: (a,a^*) conjugate, call 'a' the first one */
    4476        2520 :   if (a2 < a1) lswap(a1,a2);
    4477             :   /* Now a1 < a2 */
    4478        2520 :   if (c1 <= a1 || c1 > a2) lswap(c1,c2); /* ensure a1 < c1 <= a2 */
    4479        2520 :   if (c2 < a1)
    4480             :   { /* if cut c2 is in X1 = X11|X12, rotate to obtain
    4481             :        (a X2 | X3 a^* X4 X11|X12): then a1 = 1 */
    4482             :     GEN p;
    4483        2520 :     av = avma; p = rotate_perm(l, a1);
    4484        2520 :     siegel_perm(S, p);
    4485        2520 :     a1 = 1; /* = p[a1] */
    4486        2520 :     a2 = S->Ast[1]; /* > a1 */
    4487        2520 :     c1 = p[c1];
    4488        2520 :     c2 = p[c2]; set_avma(av);
    4489             :   }
    4490             :   /* Now a1 < c1 <= a2 < c2; a != a^* */
    4491        2520 :   g = get_g(S, a1);
    4492        2520 :   if (S->oo >= c1 && S->oo < c2) /* W inside [c1..c2[ */
    4493         539 :   { /* c2 -> c1 excluding a1 */
    4494         539 :     GEN gi = SL2_inv_shallow(g); /* g a^* = a; gi a = a^* */
    4495         539 :     path_vec_mul(S->V, 1, a1, gi);
    4496         539 :     path_vec_mul(S->V, a1+1, c1, gi);
    4497         539 :     path_vec_mul(S->V, c2, l, gi);
    4498             :   }
    4499             :   else
    4500             :   { /* c1 -> c2 excluding a2 */
    4501        1981 :     path_vec_mul(S->V, c1, a2, g);
    4502        1981 :     path_vec_mul(S->V, a2+1, c2, g);
    4503             :   }
    4504        2520 :   av = avma;
    4505        2520 :   siegel_perm(S, basic_op_perm(l, a1,a2, c1,c2));
    4506        2520 :   set_avma(av);
    4507             :   /* fill the holes left at a1,a2, reconnect the path */
    4508        2520 :   fill2(S->V, a1, a2);
    4509             : }
    4510             : /* a = a^* (elliptic case) */
    4511             : static void
    4512         154 : basic_op_elliptic(struct siegel *S, long a1)
    4513             : {
    4514             :   pari_sp av;
    4515         154 :   long l = lg(S->V);
    4516         154 :   GEN g = get_g(S, a1);
    4517         154 :   path_vec_mul(S->V, a1+1, l, g);
    4518         154 :   av = avma; siegel_perm(S, basic_op_perm_elliptic(l, a1));
    4519             :   /* fill the hole left at a1 (now at 1), reconnect the path */
    4520         154 :   set_avma(av); fill1(S->V, 1);
    4521         154 : }
    4522             : 
    4523             : /* input V = W X a b Y a^* Z b^* T, W already normalized
    4524             :  * X = [n+1, k1-1], Y = [k2+1, Ast[k1]-1],
    4525             :  * Z = [Ast[k1]+1, Ast[k2]-1], T = [Ast[k2]+1, oo].
    4526             :  * Assume that X doesn't start by c c^* or a b a^* b^*. */
    4527             : static void
    4528        1057 : siegelstep(struct siegel *S)
    4529             : {
    4530        1057 :   if (S->Ast[S->k1] == S->k1)
    4531             :   {
    4532         154 :     basic_op_elliptic(S, S->k1);
    4533         154 :     S->n++;
    4534             :   }
    4535         903 :   else if (S->Ast[S->k1] == S->k1+1)
    4536             :   {
    4537         364 :     basic_op(S, S->k1, S->Ast[S->k1], 1); /* 1: W starts there */
    4538         364 :     S->n += 2;
    4539             :   }
    4540             :   else
    4541             :   {
    4542         539 :     basic_op(S, S->k2, S->Ast[S->k1], 1); /* 1: W starts there */
    4543         539 :     basic_op(S, S->k1, S->k2, S->Ast[S->k2]);
    4544         539 :     basic_op(S, S->Ast[S->k2], S->k2, S->Ast[S->k1]);
    4545         539 :     basic_op(S, S->k1, S->Ast[S->k1], S->Ast[S->k2]);
    4546         539 :     S->n += 4;
    4547             :   }
    4548        1057 : }
    4549             : 
    4550             : /* normalize hyperbolic polygon */
    4551             : static void
    4552         301 : mssiegel(struct siegel *S)
    4553             : {
    4554         301 :   pari_sp av = avma;
    4555             :   long k, t, nv;
    4556             : #ifdef COUNT
    4557             :   long countset[16];
    4558             :   for (k = 0; k < 16; k++) countset[k] = 0;
    4559             : #endif
    4560             : 
    4561         301 :   nv = lg(S->V)-1;
    4562         301 :   if (DEBUGLEVEL>1) err_printf("nv = %ld, expo = %.2f\n", nv,ZMV_size(S->V));
    4563         301 :   t = 0;
    4564        2205 :   while (S->n < nv)
    4565             :   {
    4566        1904 :     if (S->Ast[S->n+1] == S->n+1) { S->n++; continue; }
    4567        1778 :     if (S->Ast[S->n+1] == S->n+2) { S->n += 2; continue; }
    4568        1134 :     if (S->Ast[S->n+1] == S->n+3 && S->Ast[S->n+2] == S->n+4) { S->n += 4; continue; }
    4569        1057 :     k = nv;
    4570        1127 :     while (k > S->n)
    4571             :     {
    4572        1127 :       if (S->Ast[k] == k) { k--; continue; }
    4573        1099 :       if (S->Ast[k] == k-1) { k -= 2; continue; }
    4574        1057 :       if (S->Ast[k] == k-2 && S->Ast[k-1] == k-3) { k -= 4; continue; }
    4575        1057 :       break;
    4576             :     }
    4577        1057 :     if (k != nv)
    4578             :     {
    4579          63 :       pari_sp av2 = avma;
    4580          63 :       siegel_perm0(S, rotate_perm(nv+1, k+1));
    4581          63 :       set_avma(av2); S->n += nv-k;
    4582             :     }
    4583             : 
    4584        6223 :     for (k = S->n+1; k <= nv; k++)
    4585        6223 :       if (S->Ast[k] <= k) { t = S->Ast[k]; break; }
    4586        1057 :     S->k1 = t;
    4587        1057 :     S->k2 = t+1;
    4588             : #ifdef COUNT
    4589             :     countset[ ((S->k1-1 == S->n)
    4590             :               | ((S->k2 == S->Ast[S->k1]-1) << 1)
    4591             :               | ((S->Ast[S->k1] == S->Ast[S->k2]-1) << 2)
    4592             :               | ((S->Ast[S->k2] == nv) << 3)) ]++;
    4593             : #endif
    4594        1057 :     siegelstep(S);
    4595        1057 :     if (gc_needed(av,2))
    4596             :     {
    4597           0 :       if(DEBUGMEM>1) pari_warn(warnmem,"mspolygon, n = %ld",S->n);
    4598           0 :       gerepileall(av, 2, &S->V, &S->Ast);
    4599             :     }
    4600             :   }
    4601         301 :   if (DEBUGLEVEL>1) err_printf("expo = %.2f\n", ZMV_size(S->V));
    4602             : #ifdef COUNT
    4603             :   for (k = 0; k < 16; k++)
    4604             :     err_printf("%3ld: %6ld\n", k, countset[k]);
    4605             : #endif
    4606         301 : }
    4607             : 
    4608             : /* return a vector of char* */
    4609             : static GEN
    4610           0 : Ast2v(GEN Ast)
    4611             : {
    4612           0 :   long j = 0, k, l = lg(Ast);
    4613           0 :   GEN v = const_vec(l-1, NULL);
    4614           0 :   for (k=1; k < l; k++)
    4615             :   {
    4616             :     char *sj;
    4617           0 :     if (gel(v,k)) continue;
    4618           0 :     j++;
    4619           0 :     sj = stack_sprintf("$%ld$", j);
    4620           0 :     gel(v,k) = (GEN)sj;
    4621           0 :     if (Ast[k] != k) gel(v,Ast[k]) = (GEN)stack_sprintf("$%ld^*$", j);
    4622             :   }
    4623           0 :   return v;
    4624             : };
    4625             : 
    4626             : static void
    4627           0 : decorate(pari_str *s, GEN g, GEN arc, double high)
    4628             : {
    4629           0 :   double a = gtodouble(gcoeff(g,1,1)), c = gtodouble(gcoeff(g,2,1));
    4630           0 :   double d = gtodouble(gcoeff(g,2,2));
    4631           0 :   if (a + d)
    4632             :   {
    4633           0 :     double t, u, C = 360/(2*M_PI), x = (a-d) / (2*c), y = 0.8660254/fabs(c);
    4634           0 :     long D1 = itos(gcoeff(arc,2,1));
    4635           0 :     long D2 = itos(gcoeff(arc,2,2));
    4636           0 :     str_printf(s, "\\coordinate (ellpt) at (%.4f,%.4f);\n\\draw (ellpt) node {$\\bullet$}\n", x, y);
    4637           0 :     if (D1)
    4638             :     {
    4639           0 :       t = gtodouble(gcoeff(arc,1,1)) / D1;
    4640           0 :       u = (x*x + y*y - t*t)/(x-t)/2;
    4641           0 :       str_printf(s, "arc (%.4f:180:%.4f)\n", C*atan2(y,x-u), fabs(t-u));
    4642             :     }
    4643             :     else
    4644           0 :       str_printf(s, "-- (%.4f,%.4f)\n", x, high);
    4645           0 :     if (D2)
    4646             :     {
    4647           0 :       t = gtodouble(gcoeff(arc,1,2)) / D2;
    4648           0 :       u = (x*x + y*y - t*t)/(x-t)/2;
    4649           0 :       str_printf(s, "(ellpt) arc (%.4f:0:%.4f);\n", C*atan2(y,x-u), fabs(t-u));
    4650             :     }
    4651             :     else
    4652           0 :       str_printf(s, "(ellpt) -- (%.4f,%.4f);\n", x, high);
    4653             :   }
    4654             :   else
    4655           0 :     str_printf(s, "\\draw (%.4f,%.4f) node {$\\circ$};\n",a/c,fabs(1/c));
    4656           0 : }
    4657             : 
    4658             : static GEN
    4659           0 : polygon2tex(GEN V, GEN Ast, GEN G)
    4660             : {
    4661           0 :   pari_sp av = avma;
    4662             :   pari_str s;
    4663           0 :   long j, l = lg(V), flag = (l <= 16);
    4664           0 :   double d, high = (l < 4)? 1.2: 0.5;
    4665           0 :   GEN v = Ast2v(Ast), r1 = NULL, r2 = NULL;
    4666             : 
    4667           0 :   for (j = 1; j < l; j++)
    4668             :   {
    4669           0 :     GEN arc = gel(V,j);
    4670           0 :     if (!signe(gcoeff(arc,2,1)))
    4671           0 :       r1 = gdiv(gcoeff(arc,1,2), gcoeff(arc,2,2));
    4672           0 :     else if (!signe(gcoeff(arc,2,2)))
    4673           0 :       r2 = gdiv(gcoeff(arc,1,1), gcoeff(arc,2,1));
    4674             :   }
    4675           0 :   if (!r1 || !r2) pari_err_BUG("polgon2tex");
    4676           0 :   str_init(&s, 1); d = fabs(gtodouble(gsub(r1,r2)));
    4677           0 :   str_printf(&s, "\n\\begin{tikzpicture}[scale=%.2f]\n",
    4678             :                  d? (10 / d): 10);
    4679           0 :   for (j = 1; j < l; j++)
    4680             :   {
    4681           0 :     GEN arc = gel(V,j);
    4682           0 :     if (itos(gcoeff(arc,2,1)))
    4683             :     {
    4684           0 :       GEN a = gdiv(gcoeff(arc,1,1), gcoeff(arc,2,1));
    4685           0 :       double aa = gtodouble(a);
    4686           0 :       str_printf(&s, "\\draw (%.4f,0) ", aa);
    4687           0 :       if (flag || j == 2 || j == l-1)
    4688             :       {
    4689             :         long n, d;
    4690           0 :         Qtoss(a, &n, &d);
    4691           0 :         if (d == 1)
    4692           0 :           str_printf(&s, "node [below] {$%ld$}\n", n);
    4693             :         else
    4694           0 :           str_printf(&s, "node [below] {$\\frac{%ld}{%ld}$}\n", n, d);
    4695             :       }
    4696           0 :       if (itos(gcoeff(arc,2,2)))
    4697             :       {
    4698           0 :         GEN b = gdiv(gcoeff(arc,1,2),gcoeff(arc,2,2));
    4699           0 :         str_printf(&s, "arc (%s:%.4f) ", (gcmp(a,b)<0)?"180:0":"0:180",
    4700           0 :                    fabs((gtodouble(b)-aa)/2));
    4701           0 :         if (flag)
    4702           0 :           str_printf(&s, "node [midway, above] {%s} ", (char*)gel(v,j));
    4703             :       }
    4704             :       else
    4705             :       {
    4706           0 :         str_printf(&s, "-- (%.4f,%.4f) ", aa, high);
    4707           0 :         if (flag)
    4708           0 :           str_printf(&s, "node [very near end, right] {%s}",(char*)gel(v,j));
    4709             :       }
    4710             :     }
    4711             :     else
    4712             :     {
    4713           0 :       GEN b = gdiv(gcoeff(arc,1,2), gcoeff(arc,2,2));
    4714           0 :       double bb = gtodouble(b);
    4715           0 :       str_printf(&s, "\\draw (%.4f,%.4f)--(%.4f,0)\n", bb, high, bb);
    4716           0 :       if (flag)
    4717           0 :         str_printf(&s,"node [very near start, left] {%s}\n", (char*)gel(v,j));
    4718             :     }
    4719           0 :     str_printf(&s,";\n");
    4720           0 :     if (Ast[j] == j) decorate(&s, gel(G,j), arc, high);
    4721             :   }
    4722           0 :   str_printf(&s, "\n\\end{tikzpicture}");
    4723           0 :   return gerepileuptoleaf(av, strtoGENstr(s.string));
    4724             : }
    4725             : 
    4726             : static GEN
    4727           0 : circle2tex(GEN Ast, GEN G)
    4728             : {
    4729           0 :   pari_sp av = avma;
    4730           0 :   GEN v = Ast2v(Ast);
    4731             :   pari_str s;
    4732           0 :   long u, n = lg(Ast)-1;
    4733           0 :   const double ang = 360./n;
    4734             : 
    4735           0 :   if (n > 30)
    4736             :   {
    4737           0 :     v = const_vec(n, (GEN)"");
    4738           0 :     gel(v,1) = (GEN)"$(1,\\infty)$";
    4739             :   }
    4740           0 :   str_init(&s, 1);
    4741           0 :   str_puts(&s, "\n\\begingroup\n\
    4742             :   \\def\\geo#1#2{(#2:1) arc (90+#2:270+#1:{tan((#2-#1)/2)})}\n\
    4743             :   \\def\\sgeo#1#2{(#2:1) -- (#1:1)}\n\
    4744             :   \\def\\unarc#1#2#3{({#1 * #3}:1.2) node {#2}}\n\
    4745             :   \\def\\cell#1#2{({#1 * #2}:0.95) circle(0.05)}\n\
    4746             :   \\def\\link#1#2#3#4#5{\\unarc{#1}{#2}{#5}\\geo{#1*#5}{#3*#5}\\unarc{#3}{#4}{#5}}\n\
    4747             :   \\def\\slink#1#2#3#4#5{\\unarc{#1}{#2}{#5}\\sgeo{#1*#5}{#3*#5}\\unarc{#3}{#4}{#5}}");
    4748             : 
    4749           0 :   str_puts(&s, "\n\\begin{tikzpicture}[scale=4]\n");
    4750           0 :   str_puts(&s, "\\draw (0, 0) circle(1);\n");
    4751           0 :   for (u=1; u <= n; u++)
    4752             :   {
    4753           0 :     if (Ast[u] == u)
    4754             :     {
    4755           0 :       str_printf(&s,"\\draw\\unarc{%ld}{%s}{%.4f}; \\draw\\cell{%ld}{%.4f};\n",
    4756           0 :                  u, v[u], ang, u, ang);
    4757           0 :       if (ZM_isscalar(gpowgs(gel(G,u),3), NULL))
    4758           0 :         str_printf(&s,"\\fill \\cell{%ld}{%.4f};\n", u, ang);
    4759             :     }
    4760           0 :     else if(Ast[u] > u)
    4761           0 :       str_printf(&s, "\\draw \\%slink {%ld}{%s}{%ld}{%s}{%.4f};\n",
    4762           0 :                      (Ast[u] - u)*ang > 179? "s": "", u, v[u], Ast[u], v[Ast[u]], ang);
    4763             :   }
    4764           0 :   str_printf(&s, "\\end{tikzpicture}\\endgroup");
    4765           0 :   return gerepileuptoleaf(av, strtoGENstr(s.string));
    4766             : }
    4767             : 
    4768             : GEN
    4769         434 : mspolygon(GEN M, long flag)
    4770             : {
    4771         434 :   pari_sp av = avma;
    4772             :   struct siegel T;
    4773         434 :   GEN v, msN = NULL, G = NULL;
    4774         434 :   if (typ(M) == t_INT)
    4775             :   {
    4776         308 :     long N = itos(M);
    4777         308 :     if (N <= 0) pari_err_DOMAIN("msinit","N", "<=", gen_0,M);
    4778         308 :     msN = msinit_N(N);
    4779             :   }
    4780         126 :   else if (checkfarey_i(M))
    4781             :   {
    4782           0 :     T.V = gel(M,1);
    4783           0 :     T.Ast = gel(M,2);
    4784           0 :     G = gel(M,3);
    4785             :   }
    4786             :   else
    4787         126 :   { checkms(M); msN = get_msN(M); }
    4788         434 :   if (flag < 0 || flag > 3) pari_err_FLAG("mspolygon");
    4789         434 :   if (!G)
    4790             :   {
    4791         434 :     if (ms_get_N(msN) == 1)
    4792             :     {
    4793          56 :       GEN S = mkS();
    4794          56 :       T.V = mkvec2(matid(2), S);
    4795          56 :       T.Ast = mkvecsmall2(1,2);
    4796          56 :       G = mkvec2(S, mkTAU());
    4797             :     }
    4798             :     else
    4799             :     {
    4800             :       long i, l;
    4801         378 :       siegel_init(&T, msN);
    4802         378 :       l = lg(T.V);
    4803         378 :       if (flag & 1)
    4804             :       {
    4805         301 :         long oo2 = 0;
    4806             :         pari_sp av;
    4807         301 :         mssiegel(&T);
    4808        3451 :         for (i = 1; i < l; i++)
    4809             :         {
    4810        3451 :           GEN c = gel(T.V, i);
    4811        3451 :           GEN c22 = gcoeff(c,2,2); if (!signe(c22)) { oo2 = i; break; }
    4812             :         }
    4813         301 :         if (!oo2) pari_err_BUG("mspolygon");
    4814         301 :         av = avma; siegel_perm0(&T, rotate_perm(l, oo2));
    4815         301 :         set_avma(av);
    4816             :       }
    4817         378 :       G = cgetg(l, t_VEC);
    4818       12320 :       for (i = 1; i < l; i++) gel(G,i) = get_g(&T, i);
    4819             :     }
    4820             :   }
    4821         434 :   if (flag & 2)
    4822           0 :     v = mkvec5(T.V, T.Ast, G, polygon2tex(T.V,T.Ast,G), circle2tex(T.Ast,G));
    4823             :   else
    4824         434 :     v = mkvec3(T.V, T.Ast, G);
    4825         434 :   return gerepilecopy(av, v);
    4826             : }
    4827             : 
    4828             : #if 0
    4829             : static int
    4830             : iselliptic(GEN Ast, long i) { return i == Ast[i]; }
    4831             : static int
    4832             : isparabolic(GEN Ast, long i)
    4833             : { long i2 = Ast[i]; return (i2 == i+1 || i2 == i-1); }
    4834             : #endif
    4835             : 
    4836             : /* M from msinit, F QM maximal rank */
    4837             : GEN
    4838          77 : mslattice(GEN M, GEN F)
    4839             : {
    4840          77 :   pari_sp av = avma;
    4841             :   long i, ivB, j, k, l, lF;
    4842             :   GEN D, U, G, A, vB, m, d;
    4843             : 
    4844          77 :   checkms(M);
    4845          77 :   if (!F) F = gel(mscuspidal(M, 0), 1);
    4846             :   else
    4847             :   {
    4848          49 :     if (is_Qevproj(F)) F = gel(F,1);
    4849          49 :     if (typ(F) != t_MAT) pari_err_TYPE("mslattice",F);
    4850             :   }
    4851          77 :   lF = lg(F); if (lF == 1) return cgetg(1, t_MAT);
    4852          77 :   D = mspolygon(M,0);
    4853          77 :   k = msk_get_weight(M);
    4854          77 :   F = vec_Q_primpart(F);
    4855          77 :   if (typ(F)!=t_MAT || !RgM_is_ZM(F)) pari_err_TYPE("mslattice",F);
    4856          77 :   G = gel(D,3); l = lg(G);
    4857          77 :   A = gel(D,2);
    4858          77 :   vB = cgetg(l, t_COL);
    4859          77 :   d = mkcol2(gen_0,gen_1);
    4860          77 :   m = mkmat2(d, d);
    4861        7091 :   for (i = ivB = 1; i < l; i++)
    4862             :   {
    4863        7014 :     GEN B, vb, g = gel(G,i);
    4864        7014 :     if (A[i] < i) continue;
    4865        3514 :     gel(m,2) = SL2_inv2(g);
    4866        3514 :     vb = mseval(M, F, m);
    4867        3514 :     if (k == 2) B = vb;
    4868             :     else
    4869             :     {
    4870             :       long lB;
    4871         147 :       B = RgXV_to_RgM(vb, k-1);
    4872             :       /* add coboundaries */
    4873         147 :       B = shallowconcat(B, RgM_Rg_sub(RgX_act_Gl2Q(g, k), gen_1));
    4874             :       /* beware: the basis for RgX_act_Gl2Q is (X^(k-2),...,Y^(k-2)) */
    4875         147 :       lB = lg(B);
    4876        3444 :       for (j = 1; j < lB; j++) gel(B,j) = vecreverse(gel(B,j));
    4877             :     }
    4878        3514 :     gel(vB, ivB++) = B;
    4879             :   }
    4880          77 :   setlg(vB, ivB);
    4881          77 :   vB = shallowmatconcat(vB);
    4882          77 :   if (ZM_equal0(vB)) return gerepilecopy(av, F);
    4883             : 
    4884          77 :   (void)QM_ImQ_hnfall(vB, &U, 0);
    4885          77 :   if (k > 2) U = rowslice(U, 1, lgcols(U)-k); /* remove coboundary part */
    4886          77 :   U = Q_remove_denom(U, &d);
    4887          77 :   F = ZM_hnf(ZM_mul(F, U));
    4888          77 :   if (d) F = RgM_Rg_div(F, d);
    4889          77 :   return gerepileupto(av, F);
    4890             : }
    4891             : 
    4892             : /**** Petersson scalar product ****/
    4893             : /* TODO:
    4894             :  * Eisspace: represent functions by coordinates of nonzero entries in matrix */
    4895             : 
    4896             : /* oo -> g^(-1) oo */
    4897             : static GEN
    4898        6181 : cocycle(GEN g)
    4899        6181 : { return mkmat22(gen_1, gcoeff(g,2,2), gen_0, negi(gcoeff(g,2,1))); }
    4900             : 
    4901             : /* CD = binomial_init(k-2); return <P,Q> * D (integral) */
    4902             : static GEN
    4903       18151 : bil(GEN P, GEN Q, GEN CD)
    4904             : {
    4905       18151 :   GEN s, C = gel(CD,1);
    4906       18151 :   long i, n = lg(C)-2; /* k - 2 */
    4907       18151 :   if (!n) return gmul(P,Q);
    4908       18130 :   if (typ(P) != t_POL) P = scalarpol_shallow(P,0);
    4909       18130 :   if (typ(Q) != t_POL) Q = scalarpol_shallow(Q,0);
    4910       18130 :   s = gen_0;
    4911       37282 :   for (i = n - degpol(Q); i <= degpol(P); i++)
    4912             :   {
    4913       19152 :     GEN t = gmul(gmul(RgX_coeff(P,i), RgX_coeff(Q, n-i)), gel(C,i+1));
    4914       19152 :     s = odd(i)? gsub(s, t): gadd(s, t);
    4915             :   }
    4916       18130 :   return s;
    4917             : }
    4918             : 
    4919             : /* Let D = lcm {binomial(n,k), k = 0..n} = lcm([1..n+1]) / (n+1)
    4920             :  * Return [C, D] where C[i] = D / binomial(n,i+1), i = 0..n */
    4921             : static GEN
    4922        1379 : binomial_init(long n, GEN vC)
    4923             : {
    4924        1379 :   GEN C = vC? shallowcopy(vC): vecbinomial(n), c = C + 1;
    4925        1379 :   GEN D = diviuexact(ZV_lcm(identity_ZV(n+1)), n+1);
    4926        1379 :   long k, d = (n + 1) >> 1;
    4927             : 
    4928        1379 :   gel(c,0) = D;
    4929        2961 :   for (k = 1; k <= d; k++) gel(c, k) = diviiexact(D, gel(c, k));
    4930        2961 :   for (     ; k <= n;  k++) gel(c, k) = gel(c, n-k);
    4931        1379 :   return mkvec2(C, D);
    4932             : }
    4933             : 
    4934             : static void
    4935        1351 : mspetersson_i(GEN W, GEN F, GEN G, GEN *pvf, GEN *pvg, GEN *pC)
    4936             : {
    4937        1351 :   GEN WN = get_msN(W), annT2, annT31, section, c, vf, vg;
    4938             :   long i, n1, n2, n3;
    4939             : 
    4940        1351 :   annT2 = msN_get_annT2(WN);
    4941        1351 :   annT31 = msN_get_annT31(WN);
    4942        1351 :   section = msN_get_section(WN);
    4943             : 
    4944        1351 :   if (ms_get_N(WN) == 1)
    4945             :   {
    4946           7 :     vf = cgetg(3, t_VEC);
    4947           7 :     vg = cgetg(3, t_VEC);
    4948           7 :     gel(vf,1) = mseval(W, F, gel(section,1));
    4949           7 :     gel(vf,2) = gneg(gel(vf,1));
    4950           7 :     n1 = 0;
    4951             :   }
    4952             :   else
    4953             :   {
    4954        1344 :     GEN singlerel = msN_get_singlerel(WN);
    4955        1344 :     GEN gen = msN_get_genindex(WN);
    4956        1344 :     long l = lg(gen);
    4957        1344 :     vf = cgetg(l, t_VEC);
    4958        1344 :     vg = cgetg(l, t_VEC); /* generators of Delta ordered as E1,T2,T31 */
    4959        7476 :     for (i = 1; i < l; i++) gel(vf, i) = mseval(W, F, gel(section,gen[i]));
    4960        1344 :     n1 = ms_get_nbE1(WN); /* E1 */
    4961        7420 :     for (i = 1; i <= n1; i++)
    4962             :     {
    4963        6076 :       c = cocycle(gcoeff(gel(singlerel,i),2,1));
    4964        6076 :       gel(vg, i) = mseval(W, G, c);
    4965             :     }
    4966             :   }
    4967        1351 :   n2 = lg(annT2)-1; /* T2 */
    4968        1386 :   for (i = 1; i <= n2; i++)
    4969             :   {
    4970          35 :     c = cocycle(gcoeff(gel(annT2,i), 2,1));
    4971          35 :     gel(vg, i+n1) = gmul2n(mseval(W, G, c), -1);
    4972             :   }
    4973        1351 :   n3 = lg(annT31)-1; /* T31 */
    4974        1386 :   for (i = 1; i <= n3; i++)
    4975             :   {
    4976             :     GEN f;
    4977          35 :     c = cocycle(gcoeff(gel(annT31,i), 2,1));
    4978          35 :     f = mseval(W, G, c);
    4979          35 :     c = cocycle(gcoeff(gel(annT31,i), 3,1));
    4980          35 :     gel(vg, i+n1+n2) = gdivgs(gadd(f, mseval(W, G, c)), 3);
    4981             :   }
    4982        1351 :   *pC = binomial_init(msk_get_weight(W) - 2, NULL);
    4983        1351 :   *pvf = vf;
    4984        1351 :   *pvg = vg;
    4985        1351 : }
    4986             : 
    4987             : /* Petersson product on Hom_G(Delta_0, V_k) */
    4988             : GEN
    4989        1351 : mspetersson(GEN W, GEN F, GEN G)
    4990             : {
    4991        1351 :   pari_sp av = avma;
    4992             :   GEN vf, vg, CD, cf, cg, A;
    4993             :   long k, l, tG, tF;
    4994        1351 :   checkms(W);
    4995        1351 :   if (!F) F = matid(msdim(W));
    4996        1351 :   if (!G) G = F;
    4997        1351 :   tF = typ(F);
    4998        1351 :   tG = typ(G);
    4999        1351 :   if (tF == t_MAT && tG != t_MAT) pari_err_TYPE("mspetersson",G);
    5000        1351 :   if (tG == t_MAT && tF != t_MAT) pari_err_TYPE("mspetersson",F);
    5001        1351 :   mspetersson_i(W, F, G, &vf, &vg, &CD);
    5002        1351 :   vf = Q_primitive_part(vf, &cf);
    5003        1351 :   vg = Q_primitive_part(vg, &cg);
    5004        1351 :   A = div_content(mul_content(cf, cg), gel(CD,2));
    5005        1351 :   l = lg(vf);
    5006        1351 :   if (tF != t_MAT)
    5007             :   { /* <F,G>, two symbols */
    5008        1274 :     GEN s = gen_0;
    5009        7105 :     for (k = 1; k < l; k++) s = gadd(s, bil(gel(vf,k), gel(vg,k), CD));
    5010        1274 :     return gerepileupto(av, gmul(s, A));
    5011             :   }
    5012          77 :   else if (F != G)
    5013             :   { /* <(f_1,...,f_m), (g_1,...,g_n)> */
    5014           0 :     long iF, iG, lF = lg(F), lG = lg(G);
    5015           0 :     GEN M = cgetg(lG, t_MAT);
    5016           0 :     for (iG = 1; iG < lG; iG++)
    5017             :     {
    5018           0 :       GEN c = cgetg(lF, t_COL);
    5019           0 :       gel(M,iG) = c;
    5020           0 :       for (iF = 1; iF < lF; iF++)
    5021             :       {
    5022           0 :         GEN s = gen_0;
    5023           0 :         for (k = 1; k < l; k++)
    5024           0 :           s = gadd(s, bil(gmael(vf,k,iF), gmael(vg,k,iG), CD));
    5025           0 :         gel(c,iF) = s; /* M[iF,iG] = <F[iF], G[iG] > */
    5026             :       }
    5027             :     }
    5028           0 :     return gerepileupto(av, RgM_Rg_mul(M, A));
    5029             :   }
    5030             :   else
    5031             :   { /* <(f_1,...,f_n), (f_1,...,f_n)> */
    5032          77 :     long iF, iG, n = lg(F)-1;
    5033          77 :     GEN M = zeromatcopy(n,n);
    5034         693 :     for (iG = 1; iG <= n; iG++)
    5035        3192 :       for (iF = iG+1; iF <= n; iF++)
    5036             :       {
    5037        2576 :         GEN s = gen_0;
    5038       14728 :         for (k = 1; k < l; k++)
    5039       12152 :           s = gadd(s, bil(gmael(vf,k,iF), gmael(vg,k,iG), CD));
    5040        2576 :         gcoeff(M,iF,iG) = s; /* <F[iF], F[iG] > */
    5041        2576 :         gcoeff(M,iG,iF) = gneg(s);
    5042             :       }
    5043          77 :     return gerepileupto(av, RgM_Rg_mul(M, A));
    5044             :   }
    5045             : }
    5046             : 
    5047             : /* action of g in SL_2(Z/NZ) on functions f: (Z/NZ)^2 -> Q given by sparse
    5048             :  * matrix M. */
    5049             : static GEN
    5050         168 : actf(long N, GEN M, GEN g)
    5051             : {
    5052             :   long n, a, b, c, d, l;
    5053             :   GEN m;
    5054         168 :   c = umodiu(gcoeff(g,2,1), N); if (!c) return M;
    5055           0 :   d = umodiu(gcoeff(g,2,2), N);
    5056           0 :   a = umodiu(gcoeff(g,1,1), N);
    5057           0 :   b = umodiu(gcoeff(g,1,2), N);
    5058           0 :   m = cgetg_copy(M, &l);
    5059           0 :   for (n = 1; n < l; n++)
    5060             :   {
    5061           0 :     GEN v = gel(M,n);
    5062           0 :     long i = v[1], j = v[2];
    5063           0 :     long I = Fl_add(Fl_mul(a,i,N), Fl_mul(c,j,N), N);
    5064           0 :     long J = Fl_add(Fl_mul(b,i,N), Fl_mul(d,j,N), N);
    5065           0 :     if (!I) I = N;
    5066           0 :     if (!J) J = N;
    5067           0 :     gel(m,n) = mkvecsmall2(I,J);
    5068             :   }
    5069           0 :   return m;
    5070             : }
    5071             : 
    5072             : /* q1 = N/a, q2 = q1/d, (u,a) = 1. Gamma_0(N)-orbit attached to [q1,q2,u]
    5073             :  * in (Z/N)^2; set of [q1 v, q2 w], v in (Z/a)^*, w in Z/a*d,
    5074             :  * w mod a = u / v [invertible]; w mod d in (Z/d)^*; c1+c2= q2, d2|c1, d1|c2
    5075             :  * The orbit has cardinal C = a phi(d) <= N */
    5076             : static GEN
    5077          28 : eisf(long N, long C, long a, long d1, GEN Z2, long c1, long c2,
    5078             :      long q1, long u)
    5079             : {
    5080          28 :   GEN m = cgetg(C+1, t_VEC);
    5081          28 :   long v, n = 1, l = lg(Z2);
    5082          56 :   for (v = 1; v <= a; v++)
    5083          28 :     if (ugcd(v,a)==1)
    5084             :     {
    5085          28 :       long w1 = Fl_div(u, v, a), vq1 = v * q1, i, j;
    5086          56 :       for (i = 0; i < d1; i++, w1 += a)
    5087             :       { /* w1 defined mod a*d1, lifts u/v (mod a) */
    5088          56 :         for (j = 1; j < l; j++)
    5089          28 :           if (Z2[j])
    5090             :           {
    5091          28 :             long wq2 = (c1 * w1 + c2 * j) % N;
    5092          28 :             if (wq2 <= 0) wq2 += N;
    5093          28 :             gel(m, n++) = mkvecsmall2(vq1, wq2);
    5094             :           }
    5095             :       }
    5096             :     }
    5097          28 :   return m;
    5098             : }
    5099             : 
    5100             : /* basis for Gamma_0(N)-invariant functions attached to cusps */
    5101             : static GEN
    5102          28 : eisspace(long N, long k, long s)
    5103             : {
    5104          28 :   GEN v, D, F = factoru(N);
    5105             :   long l, n, i, j;
    5106          28 :   D = divisorsu_fact(F); l = lg(D);
    5107          28 :   n = mfnumcuspsu_fact(F);
    5108          28 :   v = cgetg((k==2)? n: n+1, t_VEC);
    5109          56 :   for (i = (k==2)? 2: 1, j = 1; i < l; i++) /* remove d = 1 if k = 2 */
    5110             :   {
    5111          28 :     long d = D[i], Nd = D[l-i], a = ugcd(d, Nd), q1, q2, d1, d2, C, c1, c2, u;
    5112             :     GEN Z2;
    5113             : 
    5114          56 :     if (s < 0 && a <= 2) continue;
    5115          28 :     q1 = N / a;
    5116          28 :     q2 = q1 / d;
    5117          28 :     d2 = u_ppo(d/a, a);
    5118          28 :     d1 = d / d2;
    5119          28 :     C = eulerphiu(d) * a;
    5120          28 :     Z2 = coprimes_zv(d2);
    5121             :     /* d = d1d2, (d2,a) = 1; d1 and a have same prime divisors */
    5122          28 :     (void)cbezout(d1, d2, &c2, &c1);
    5123          28 :     c2 *= d1 * q2;
    5124          28 :     c1 *= d2 * q2;
    5125          28 :     if (a <= 2)
    5126             :     { /* sigma.(C cusp attached to [q1,q2,u]) = C */
    5127          28 :       gel(v, j++) = eisf(N,C,a,d1,Z2,c1,c2, N/a, 1);
    5128          28 :       continue;
    5129             :     }
    5130           0 :     for (u = 1; 2*u < a; u++)
    5131             :     {
    5132           0 :       if (ugcd(u,a) != 1) continue;
    5133           0 :       gel(v, j++) = eisf(N,C,a,d1,Z2,c1,c2, q1, u);
    5134           0 :       if (!s) gel(v, j++) = eisf(N,C,a,d1,Z2,c1,c2, q1, a-u);
    5135             :     }
    5136             :   }
    5137          28 :   if (s) setlg(v, j);
    5138          28 :   return v;
    5139             : }
    5140             : 
    5141             : /* action of g on V_k */
    5142             : static GEN
    5143         168 : act(GEN P, GEN g, long k)
    5144             : {
    5145         168 :   GEN a = gcoeff(g,1,1), b = gcoeff(g,1,2), V1, V2, Q;
    5146         168 :   GEN c = gcoeff(g,2,1), d = gcoeff(g,2,2);
    5147             :   long i;
    5148         168 :   if (k == 2) return P;
    5149         168 :   V1 = RgX_powers(deg1pol_shallow(c, a, 0), k-2); /* V1[i] = (a + c Y)^i */
    5150         168 :   V2 = RgX_powers(deg1pol_shallow(d, b, 0), k-2); /* V2[j] = (b + d Y)^j */
    5151         168 :   Q = gmul(RgX_coeff(P,0), gel(V1, k-2));
    5152        2520 :   for (i = 1; i < k-2; i++)
    5153             :   {
    5154        2352 :     GEN v1 = gel(V1, k-2-i);
    5155        2352 :     GEN v2 = gel(V2, i);
    5156        2352 :     Q = gadd(Q, gmul(RgX_coeff(P,i), RgX_mul(v1,v2)));
    5157             :   }
    5158         168 :   return gadd(Q, gmul(RgX_coeff(P,k-2), gel(V2,k-2)));
    5159             : }
    5160             : 
    5161             : static long
    5162         420 : co_get_N(GEN co) { return gel(co,1)[1]; }
    5163             : static long
    5164         504 : co_get_k(GEN co) { return gel(co,1)[2]; }
    5165             : static GEN
    5166         140 : co_get_B(GEN co) { return gel(co,2); }
    5167             : static GEN
    5168         112 : co_get_BD(GEN co) { return gel(co,3); }
    5169             : static GEN
    5170          84 : co_get_C(GEN co) { return gel(co,4); }
    5171             : 
    5172             : /* N g^(-1) . eval on g([0,a]_oo)=g([pi_oo(0),pi_oo(a)]), fg = f|g */
    5173             : static GEN
    5174         252 : evalcap(GEN co, GEN fg, GEN a)
    5175             : {
    5176         252 :   long n, t, l = lg(fg), N = co_get_N(co), k = co_get_k(co);
    5177             :   GEN P, B, z, T;
    5178             :   pari_sp av;
    5179         252 :   if (isintzero(a)) return gen_0;
    5180             :   /* (a+y)^(k-1) - y^(k-1) */
    5181          56 :   P = gsub(gpowgs(deg1pol_shallow(gen_1, a, 0), k-1), pol_xn(k-1, 0));
    5182          56 :   B = co_get_B(co); z = gen_0;
    5183          56 :   av = avma; T = zero_zv(N);
    5184         112 :   for (n = 1; n < l; n++)
    5185             :   {
    5186          56 :     GEN v = gel(fg, n);
    5187          56 :     t = v[1]; T[t]++;
    5188             :   }
    5189         112 :   for (t = 1; t <= N; t++)
    5190             :   {
    5191          56 :     long c = T[t];
    5192          56 :     if (c)
    5193             :     {
    5194          56 :       GEN u = gmael(B, k, t);
    5195          56 :       if (c != 1) u = gmulsg(c, u);
    5196          56 :       z = gadd(z, u);
    5197             :     }
    5198             :   }
    5199          56 :   if (co_get_BD(co)) z = gmul(co_get_BD(co),z);
    5200          56 :   z = gerepileupto(av, gdivgs(z, -k * (k-1)));
    5201          56 :   return RgX_Rg_mul(P, z);
    5202             : };
    5203             : 
    5204             : /* eval N g^(-1) * Psi(f) on g{oo,0}, fg = f|g */
    5205             : static GEN
    5206          84 : evalcup(GEN co, GEN fg)
    5207             : {
    5208          84 :   long j, n, k = co_get_k(co), l = lg(fg);
    5209          84 :   GEN B = co_get_B(co), C = co_get_C(co), P = cgetg(k+1, t_POL);
    5210          84 :   P[1] = evalvarn(0);
    5211        1428 :   for (j = 2; j <= k; j++) gel(P,j) = gen_0;
    5212         168 :   for (n = 1; n < l; n++)
    5213             :   {
    5214          84 :     GEN v = gel(fg,n);
    5215          84 :     long t = v[1], s = v[2];
    5216        1428 :     for (j = 1; j < k; j++)
    5217             :     {
    5218        1344 :       long j1 = k-j;
    5219        1344 :       GEN u = gmael(B, j1, t);
    5220        1344 :       GEN v = gmael(B, j, s);
    5221        1344 :       gel(P, j1+1) = gadd(gel(P, j1+1), gmul(u,v));
    5222             :     }
    5223             :   }
    5224        1428 :   for (j = 1; j < k; j++) gel(P, j+1) = gmul(gel(C,j), gel(P, j+1));
    5225          84 :   return normalizepol(P);
    5226             : }
    5227             : 
    5228             : /* Manin-Stevens algorithm, prepare for [pi_0(oo),pi_r(oo)] */
    5229             : static GEN
    5230          84 : evalmanin(GEN r)
    5231             : {
    5232          84 :   GEN fr = gboundcf(r, 0), pq, V;
    5233          84 :   long j, n = lg(fr)-1; /* > 0 */
    5234          84 :   V = cgetg(n+2, t_VEC);
    5235          84 :   gel(V,1) = gel(fr,1); /* a_0; tau_{-1} = id */
    5236          84 :   if (n == 1)
    5237             :   { /* r integer, can happen iff N = 1 */
    5238          84 :     gel(V,2) = mkvec2(gen_0, mkmat22(negi(r), gen_1, gen_m1, gen_0));
    5239          84 :     return V;
    5240             :   }
    5241           0 :   pq = contfracpnqn(fr,n-1);
    5242           0 :   fr = vec_append(fr, gdiv(negi(gcoeff(pq,2,n-1)), gcoeff(pq,2,n)));
    5243           0 :   for (j = 0; j < n; j++)
    5244             :   {
    5245           0 :     GEN v1 = gel(pq, j+1), v2 = (j == 0)? col_ei(2,1): gel(pq, j);
    5246           0 :     GEN z = gel(fr,j+2);
    5247           0 :     if (!odd(j)) { v1 = ZC_neg(v1); z = gneg(z); }
    5248           0 :     gel(V,j+2) = mkvec2(z, mkmat2(v1,v2)); /* [a_{j+1}, tau_j] */
    5249             :   }
    5250           0 :   return V;
    5251             : }
    5252             : 
    5253             : /* evaluate N * Psi(f) on
    5254             :   g[pi_oo(0),pi_r(oo)]=g[pi_oo(0),pi_0(oo)] + g[pi_0(oo),pi_r(oo)] */
    5255             : static GEN
    5256          84 : evalhull(GEN co, GEN f, GEN r)
    5257             : {
    5258          84 :   GEN V = evalmanin(r), res = evalcap(co,f,gel(V,1));
    5259          84 :   long j, l = lg(V), N = co_get_N(co);
    5260         168 :   for (j = 2; j < l; j++)
    5261             :   {
    5262          84 :     GEN v = gel(V,j), t = gel(v,2); /* in SL_2(Z) */
    5263          84 :     GEN ft = actf(N, f, t), a = gel(v,1); /* in Q */
    5264             :     /* t([pi_0(oo),pi_oo(a)]) */
    5265          84 :     res = gsub(res, act(gsub(evalcup(co,ft), evalcap(co,ft,a)), t, co_get_k(co)));
    5266             :   }
    5267          84 :   return res;
    5268             : };
    5269             : 
    5270             : /* evaluate N * cocycle at g in Gamma_0(N), f Gamma_0(N)-invariant */
    5271             : static GEN
    5272          84 : eiscocycle(GEN co, GEN f, GEN g)
    5273             : {
    5274          84 :   pari_sp av = avma;
    5275          84 :   GEN a = gcoeff(g,1,1), b = gcoeff(g,1,2);
    5276          84 :   GEN c = gcoeff(g,2,1), d = gcoeff(g,2,2), P;
    5277          84 :   long N = co_get_N(co);
    5278          84 :   if (!signe(c))
    5279           0 :     P = evalcap(co,f, gdiv(negi(b),a));
    5280             :   else
    5281             :   {
    5282          84 :     GEN gi = SL2_inv_shallow(g);
    5283          84 :     P = gsub(evalhull(co, f, gdiv(negi(d),c)),
    5284             :              act(evalcap(co, actf(N,f,gi), gdiv(a,c)), gi, co_get_k(co)));
    5285             :   }
    5286          84 :   return gerepileupto(av, P);
    5287             : }
    5288             : 
    5289             : static GEN
    5290          28 : eisCocycle(GEN co, GEN D, GEN f)
    5291             : {
    5292          28 :   GEN V = gel(D,1), Ast = gel(D,2), G = gel(D,3);
    5293          28 :   long i, j, n = lg(G)-1;
    5294          28 :   GEN VG = cgetg(n+1, t_VEC);
    5295          84 :   for (i = j = 1; i <= n; i++)
    5296             :   {
    5297          56 :     GEN c, g, d, s = gel(V,i);
    5298          56 :     if (i > Ast[i]) continue;
    5299          56 :     g = SL2_inv_shallow(gel(G,i));
    5300          56 :     c = eiscocycle(co,f,g);
    5301          56 :     if (i < Ast[i]) /* non elliptic */
    5302           0 :       d = gen_1;
    5303             :     else
    5304             :     { /* i = Ast[i] */
    5305          56 :       GEN g2 = ZM_sqr(g);
    5306          56 :       if (ZM_isdiagonal(g2)) d = gen_2; /* \pm Id */
    5307             :       else
    5308             :       {
    5309          28 :         c = gadd(c, eiscocycle(co,f,g2));
    5310          28 :         d = utoipos(3);
    5311             :       }
    5312             :     }
    5313          56 :     gel(VG, j++) = mkvec3(d, s, c);
    5314             :   }
    5315          28 :   setlg(VG, j); return VG;
    5316             : };
    5317             : 
    5318             : /* F=modular symbol, Eis = cocycle attached to f invariant function
    5319             :  * by Gamma_0(N); CD = binomial_init(k-2) */
    5320             : static GEN
    5321          84 : eispetersson(GEN M, GEN F, GEN Eis, GEN CD)
    5322             : {
    5323          84 :   pari_sp av = avma;
    5324          84 :   long i, l = lg(Eis);
    5325          84 :   GEN res = gen_0;
    5326         252 :   for (i = 1; i < l; i++)
    5327             :   {
    5328         168 :     GEN e = gel(Eis,i), Q = mseval(M, F, gel(e,2)), z = bil(gel(e,3), Q, CD);
    5329         168 :     long d = itou(gel(e,1));
    5330         168 :     res = gadd(res, d == 1? z: gdivgs(z,d));
    5331             :   }
    5332          84 :   return gerepileupto(av, gdiv(simplify_shallow(res), gel(CD,2)));
    5333             : };
    5334             : 
    5335             : /*vB[j][i] = {i/N} */
    5336             : static GEN
    5337          28 : get_bern(long N, long k)
    5338             : {
    5339          28 :   GEN vB = cgetg(k+1, t_VEC), gN = utoipos(N);
    5340             :   long i, j; /* no need for j = 0 */
    5341         504 :   for (j = 1; j <= k; j++)
    5342             :   {
    5343         476 :     GEN c, B = RgX_rescale(bernpol(j, 0), gN);
    5344         476 :     gel(vB, j) = c = cgetg(N+1, t_VEC);
    5345         476 :     for (i = 1; i < N; i++) gel(c,i) = poleval(B, utoipos(i));
    5346         476 :     gel(c,N) = gel(B,2); /* B(0) */
    5347             :   }
    5348          28 :   return vB;
    5349             : }
    5350             : GEN
    5351          28 : eisker_worker(GEN Ei, GEN M, GEN D, GEN co, GEN CD)
    5352             : {
    5353          28 :   pari_sp av = avma;
    5354          28 :   long j, n = msdim(M), s = msk_get_sign(M);
    5355          28 :   GEN V, Eis = eisCocycle(co, D, Ei), v = cgetg(n+1, t_VEC);
    5356             : 
    5357          28 :   V = s? gel(msk_get_starproj(M), 1): matid(n);
    5358             :   /* T is multiplied by N * BD^2: same Ker */
    5359         112 :   for (j = 1; j <= n; j++) gel(v,j) = eispetersson(M, gel(V,j), Eis, CD);
    5360          28 :   return gerepileupto(av, v);
    5361             : }
    5362             : /* vC = vecbinomial(k-2); vC[j] = binom(k-2,j-1) = vC[k-j], j = 1..k-1, k even.
    5363             :  * C[k-j+1] = (-1)^(j-1) binom(k-2, j-1) / (j(k-j)) = C[j+1] */
    5364             : static GEN
    5365          28 : get_C(GEN vC, long k)
    5366             : {
    5367          28 :   GEN C = cgetg(k, t_VEC);
    5368          28 :   long j, k2 = k/2;
    5369         266 :   for (j = 1; j <= k2; j++)
    5370             :   {
    5371         238 :     GEN c = gel(vC, j);
    5372         238 :     if (!odd(j)) c = negi(c);
    5373         238 :     gel(C,k-j) = gel(C, j) = gdivgs(c, j*(k-j));
    5374             :   }
    5375          28 :   return C;
    5376             : }
    5377             : static GEN
    5378          28 : eisker(GEN M)
    5379             : {
    5380          28 :   long N = ms_get_N(M), k = msk_get_weight(M), s = msk_get_sign(M);
    5381          28 :   GEN worker, vC, co, CD, D, B, BD, T, E = eisspace(N, k, s);
    5382          28 :   long i, j, m = lg(E)-1, n = msdim(M), pending = 0;
    5383             :   struct pari_mt pt;
    5384             : 
    5385          28 :   if (m == 0) return matid(n);
    5386          28 :   vC = vecbinomial(k-2);
    5387          28 :   T = zeromatcopy(m, n);
    5388          28 :   D = mspolygon(M, 0);
    5389          28 :   B = Q_remove_denom(get_bern(N,k), &BD);
    5390          28 :   co = mkvec4(mkvecsmall2(N,k), B, BD, get_C(vC, k));
    5391          28 :   CD = binomial_init(k-2, vC);
    5392          28 :   worker = snm_closure(is_entry("_eisker_worker"), mkvec4(M, D, co, CD));
    5393          28 :   mt_queue_start_lim(&pt, worker, m);
    5394          56 :   for (i = 1; i <= m || pending; i++)
    5395             :   {
    5396             :     long workid;
    5397             :     GEN done;
    5398          28 :     mt_queue_submit(&pt, i, i<=m? mkvec(gel(E,i)): NULL);
    5399          28 :     done = mt_queue_get(&pt, &workid, &pending);
    5400         112 :     if (done) for (j = 1; j <= n; j++) gcoeff(T,workid,j) = gel(done,j);
    5401             :   }
    5402          28 :   mt_queue_end(&pt); return QM_ker(T);
    5403             : }

Generated by: LCOV version 1.13