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 v2.12.1 lcov report (development 25579-8c4672f557) Lines: 2819 3025 93.2 %
Date: 2020-07-09 06:03:45 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. It is distributed in the hope that it will be useful, but WITHOUT
       8             : ANY WARRANTY WHATSOEVER.
       9             : 
      10             : Check the License for details. You should have received a copy of it, along
      11             : with the package; see the file 'COPYING'. If not, write to the Free Software
      12             : Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */
      13             : 
      14             : #include "pari.h"
      15             : #include "paripriv.h"
      16             : 
      17             : /* Adapted from shp_package/moments by Robert Pollack
      18             :  * http://www.math.mcgill.ca/darmon/programs/shp/shp.html */
      19             : static GEN mskinit(ulong N, long k, long sign);
      20             : static GEN mshecke_i(GEN W, ulong p);
      21             : static GEN ZSl2_star(GEN v);
      22             : static GEN getMorphism(GEN W1, GEN W2, GEN v);
      23             : static GEN voo_act_Gl2Q(GEN g, long k);
      24             : 
      25             : /* Input: P^1(Z/NZ) (formed by create_p1mod)
      26             :    Output: # P^1(Z/NZ) */
      27             : static long
      28        8316 : p1_size(GEN p1N) { return lg(gel(p1N,1)) - 1; }
      29             : static ulong
      30    53559009 : p1N_get_N(GEN p1N) { return gel(p1N,3)[2]; }
      31             : static GEN
      32    24978562 : p1N_get_hash(GEN p1N) { return gel(p1N,2); }
      33             : static GEN
      34        2219 : p1N_get_fa(GEN p1N) { return gel(p1N,4); }
      35             : static GEN
      36        2107 : p1N_get_div(GEN p1N) { return gel(p1N,5); }
      37             : static GEN
      38    22239623 : p1N_get_invsafe(GEN p1N) { return gel(p1N,6); }
      39             : static GEN
      40     7062188 : p1N_get_inverse(GEN p1N) { return gel(p1N,7); }
      41             : 
      42             : /* ms-specific accessors */
      43             : /* W = msinit, return the output of msinit_N */
      44             : static GEN
      45     5144244 : get_msN(GEN W) { return lg(W) == 4? gel(W,1): W; }
      46             : static GEN
      47     3044860 : msN_get_p1N(GEN W) { return gel(W,1); }
      48             : static GEN
      49      209062 : msN_get_genindex(GEN W) { return gel(W,5); }
      50             : static GEN
      51    39969097 : msN_get_E2fromE1(GEN W) { return gel(W,7); }
      52             : static GEN
      53        1393 : msN_get_annT2(GEN W) { return gel(W,8); }
      54             : static GEN
      55        1393 : msN_get_annT31(GEN W) { return gel(W,9); }
      56             : static GEN
      57        1358 : msN_get_singlerel(GEN W) { return gel(W,10); }
      58             : static GEN
      59      755426 : msN_get_section(GEN W) { return gel(W,12); }
      60             : 
      61             : static GEN
      62       87948 : ms_get_p1N(GEN W) { return msN_get_p1N(get_msN(W)); }
      63             : static long
      64       75362 : ms_get_N(GEN W) { return p1N_get_N(ms_get_p1N(W)); }
      65             : static GEN
      66        1680 : ms_get_hashcusps(GEN W) { W = get_msN(W); return gel(W,16); }
      67             : static GEN
      68       17059 : ms_get_section(GEN W) { return msN_get_section(get_msN(W)); }
      69             : static GEN
      70      204050 : ms_get_genindex(GEN W) { return msN_get_genindex(get_msN(W)); }
      71             : static long
      72      199444 : ms_get_nbgen(GEN W) { return lg(ms_get_genindex(W))-1; }
      73             : static long
      74     2428335 : ms_get_nbE1(GEN W)
      75             : {
      76             :   GEN W11;
      77     2428335 :   W = get_msN(W); W11 = gel(W,11);
      78     2428335 :   return W11[4] - W11[3];
      79             : }
      80             : 
      81             : /* msk-specific accessors */
      82             : static long
      83         126 : msk_get_dim(GEN W) { return gmael(W,3,2)[2]; }
      84             : static GEN
      85       82516 : msk_get_basis(GEN W) { return gmael(W,3,1); }
      86             : static long
      87       46984 : msk_get_weight(GEN W) { return gmael(W,3,2)[1]; }
      88             : static long
      89       21434 : msk_get_sign(GEN W)
      90             : {
      91       21434 :   GEN t = gel(W,2);
      92       21434 :   return typ(t)==t_INT? 0: itos(gel(t,1));
      93             : }
      94             : static GEN
      95         959 : msk_get_star(GEN W) { return gmael(W,2,2); }
      96             : static GEN
      97        3710 : msk_get_starproj(GEN W) { return gmael(W,2,3); }
      98             : 
      99             : static int
     100         434 : is_Qevproj(GEN x)
     101         434 : { return typ(x) == t_VEC && lg(x) == 5 && typ(gel(x,1)) == t_MAT; }
     102             : long
     103         224 : msdim(GEN W)
     104             : {
     105         224 :   if (is_Qevproj(W)) return lg(gel(W,1)) - 1;
     106         210 :   checkms(W);
     107         203 :   if (!msk_get_sign(W)) return msk_get_dim(W);
     108          91 :   return lg(gel(msk_get_starproj(W), 1)) - 1;
     109             : }
     110             : long
     111          14 : msgetlevel(GEN W) { checkms(W); return ms_get_N(W); }
     112             : long
     113          14 : msgetweight(GEN W) { checkms(W); return msk_get_weight(W); }
     114             : long
     115          28 : msgetsign(GEN W) { checkms(W); return msk_get_sign(W); }
     116             : 
     117             : void
     118       36547 : checkms(GEN W)
     119             : {
     120       36547 :   if (typ(W) != t_VEC || lg(W) != 4
     121       36547 :       || typ(gel(W,1)) != t_VEC || lg(gel(W,1)) != 17)
     122           7 :     pari_err_TYPE("checkms [please apply msinit]", W);
     123       36540 : }
     124             : 
     125             : /** MODULAR TO SYM **/
     126             : 
     127             : /* q a t_FRAC or t_INT */
     128             : static GEN
     129     2221884 : Q_log_init(ulong N, GEN q)
     130             : {
     131             :   long l, n;
     132             :   GEN Q;
     133             : 
     134     2221884 :   q = gboundcf(q, 0);
     135     2221884 :   l = lg(q);
     136     2221884 :   Q = cgetg(l, t_VECSMALL);
     137     2221884 :   Q[1] = 1;
     138    21320124 :   for (n=2; n <l; n++) Q[n] = umodiu(gel(q,n), N);
     139    19099094 :   for (n=3; n < l; n++)
     140    16877210 :     Q[n] = Fl_add(Fl_mul(Q[n], Q[n-1], N), Q[n-2], N);
     141     2221884 :   return Q;
     142             : }
     143             : 
     144             : /** INIT MODSYM STRUCTURE, WEIGHT 2 **/
     145             : 
     146             : /* num = [Gamma : Gamma_0(N)] = N * Prod_{p|N} (1+p^-1) */
     147             : static ulong
     148        2107 : count_Manin_symbols(ulong N, GEN P)
     149             : {
     150        2107 :   long i, l = lg(P);
     151        2107 :   ulong num = N;
     152        5467 :   for (i = 1; i < l; i++) { ulong p = P[i]; num *= p+1; num /= p; }
     153        2107 :   return num;
     154             : }
     155             : /* returns the list of "Manin symbols" (c,d) in (Z/NZ)^2, (c,d,N) = 1
     156             :  * generating H^1(X_0(N), Z) */
     157             : static GEN
     158        2107 : generatemsymbols(ulong N, ulong num, GEN divN)
     159             : {
     160        2107 :   GEN ret = cgetg(num+1, t_VEC);
     161        2107 :   ulong c, d, curn = 0;
     162             :   long i, l;
     163             :   /* generate Manin-symbols in two lists: */
     164             :   /* list 1: (c:1) for 0 <= c < N */
     165      231973 :   for (c = 0; c < N; c++) gel(ret, ++curn) = mkvecsmall2(c, 1);
     166        2107 :   if (N == 1) return ret;
     167             :   /* list 2: (c:d) with 1 <= c < N, c | N, 0 <= d < N, gcd(d,N) > 1, gcd(c,d)=1.
     168             :    * Furthermore, d != d0 (mod N/c) with c,d0 already in the list */
     169        2079 :   l = lg(divN) - 1;
     170             :   /* c = 1 first */
     171        2079 :   gel(ret, ++curn) = mkvecsmall2(1,0);
     172      227759 :   for (d = 2; d < N; d++)
     173      225680 :     if (ugcd(d,N) != 1UL)
     174       80871 :       gel(ret, ++curn) = mkvecsmall2(1,d);
     175             :   /* omit c = 1 (first) and c = N (last) */
     176        6370 :   for (i=2; i < l; i++)
     177             :   {
     178             :     ulong Novc, d0;
     179        4291 :     c = divN[i];
     180        4291 :     Novc = N / c;
     181      120575 :     for (d0 = 2; d0 <= Novc; d0++)
     182             :     {
     183      116284 :       ulong k, d = d0;
     184      116284 :       if (ugcd(d, Novc) == 1UL) continue;
     185      146902 :       for (k = 0; k < c; k++, d += Novc)
     186      134554 :         if (ugcd(c,d) == 1UL)
     187             :         {
     188       27356 :           gel(ret, ++curn) = mkvecsmall2(c,d);
     189       27356 :           break;
     190             :         }
     191             :     }
     192             :   }
     193        2079 :   if (curn != num) pari_err_BUG("generatemsymbols [wrong number of symbols]");
     194        2079 :   return ret;
     195             : }
     196             : 
     197             : static GEN
     198        2107 : inithashmsymbols(ulong N, GEN symbols)
     199             : {
     200        2107 :   GEN H = zerovec(N);
     201        2107 :   long k, l = lg(symbols);
     202             :   /* skip the (c:1), 0 <= c < N and (1:0) */
     203      110334 :   for (k=N+2; k < l; k++)
     204             :   {
     205      108227 :     GEN s = gel(symbols, k);
     206      108227 :     ulong c = s[1], d = s[2], Novc = N/c;
     207      108227 :     if (gel(H,c) == gen_0) gel(H,c) = const_vecsmall(Novc+1,0);
     208      108227 :     if (c != 1) { d %= Novc; if (!d) d = Novc; }
     209      108227 :     mael(H, c, d) = k;
     210             :   }
     211        2107 :   return H;
     212             : }
     213             : 
     214             : /** Helper functions for Sl2(Z) / Gamma_0(N) **/
     215             : /* M a 2x2 ZM in SL2(Z) */
     216             : GEN
     217     1237117 : SL2_inv_shallow(GEN M)
     218             : {
     219     1237117 :   GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
     220     1237117 :   GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
     221     1237117 :   return mkmat22(d,negi(b), negi(c),a);
     222             : }
     223             : /* SL2_inv(M)[2] */
     224             : static GEN
     225        3514 : SL2_inv2(GEN M)
     226             : {
     227        3514 :   GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
     228        3514 :   return mkcol2(negi(b),a);
     229             : }
     230             : /* M a 2x2 mat2 in SL2(Z) */
     231             : static GEN
     232      735553 : sl2_inv(GEN M)
     233             : {
     234      735553 :   long a=coeff(M,1,1), b=coeff(M,1,2), c=coeff(M,2,1), d=coeff(M,2,2);
     235      735553 :   return mkvec2(mkvecsmall2(d, -c), mkvecsmall2(-b, a));
     236             : }
     237             : /* Return the mat2 [a,b; c,d], not a zm to avoid GP problems */
     238             : static GEN
     239     1924573 : mat2(long a, long b, long c, long d)
     240     1924573 : { return mkvec2(mkvecsmall2(a,c), mkvecsmall2(b,d)); }
     241             : static GEN
     242      406882 : mat2_to_ZM(GEN M)
     243             : {
     244      406882 :   GEN A = gel(M,1), B = gel(M,2);
     245      406882 :   retmkmat2(mkcol2s(A[1],A[2]), mkcol2s(B[1],B[2]));
     246             : }
     247             : 
     248             : /* Input: a = 2-vector = path = {r/s,x/y}
     249             :  * Output: either [r,x;s,y] or [-r,x;-s,y], whichever has determinant > 0 */
     250             : static GEN
     251      118209 : path_to_ZM(GEN a)
     252             : {
     253      118209 :   GEN v = gel(a,1), w = gel(a,2);
     254      118209 :   long r = v[1], s = v[2], x = w[1], y = w[2];
     255      118209 :   if (cmpii(mulss(r,y), mulss(x,s)) < 0) { r = -r; s = -s; }
     256      118209 :   return mkmat22(stoi(r),stoi(x),stoi(s),stoi(y));
     257             : }
     258             : static GEN
     259     1196195 : path_to_zm(GEN a)
     260             : {
     261     1196195 :   GEN v = gel(a,1), w = gel(a,2);
     262     1196195 :   long r = v[1], s = v[2], x = w[1], y = w[2];
     263     1196195 :   if (cmpii(mulss(r,y), mulss(x,s)) < 0) { r = -r; s = -s; }
     264     1196195 :   return mat2(r,x,s,y);
     265             : }
     266             : /* path from c1 to c2 */
     267             : static GEN
     268      687715 : mkpath(GEN c1, GEN c2) { return mat2(c1[1], c2[1], c1[2], c2[2]); }
     269             : static long
     270      972104 : cc(GEN M) { GEN v = gel(M,1); return v[2]; }
     271             : static long
     272      972104 : dd(GEN M) { GEN v = gel(M,2); return v[2]; }
     273             : 
     274             : /*Input: a,b = 2 paths, N = integer
     275             :  *Output: 1 if the a,b are \Gamma_0(N)-equivalent; 0 otherwise */
     276             : static int
     277      112196 : gamma_equiv(GEN a, GEN b, ulong N)
     278             : {
     279      112196 :   pari_sp av = avma;
     280      112196 :   GEN m = path_to_zm(a);
     281      112196 :   GEN n = path_to_zm(b);
     282      112196 :   GEN d = subii(mulss(cc(m),dd(n)), mulss(dd(m),cc(n)));
     283      112196 :   return gc_bool(av, umodiu(d, N) == 0);
     284             : }
     285             : /* Input: a,b = 2 paths that are \Gamma_0(N)-equivalent, N = integer
     286             :  * Output: M in \Gamma_0(N) such that Mb=a */
     287             : static GEN
     288       58660 : gamma_equiv_matrix(GEN a, GEN b)
     289             : {
     290       58660 :   GEN m = path_to_ZM(a);
     291       58660 :   GEN n = path_to_ZM(b);
     292       58660 :   return ZM_mul(m, SL2_inv_shallow(n));
     293             : }
     294             : 
     295             : /*************/
     296             : /* P^1(Z/NZ) */
     297             : /*************/
     298             : /* a != 0 in Z/NZ. Return v in (Z/NZ)^* such that av = gcd(a, N) (mod N)*/
     299             : static ulong
     300      429520 : Fl_inverse(ulong a, ulong N) { ulong g; return Fl_invgen(a,N,&g); }
     301             : 
     302             : /* Input: N = integer
     303             :  * Output: creates P^1(Z/NZ) = [symbols, H, N]
     304             :  *   symbols: list of vectors [x,y] that give a set of representatives
     305             :  *            of P^1(Z/NZ)
     306             :  *   H: an M by M grid whose value at the r,c-th place is the index of the
     307             :  *      "standard representative" equivalent to [r,c] occurring in the first
     308             :  *      list. If gcd(r,c,N) > 1 the grid has value 0. */
     309             : static GEN
     310        2107 : create_p1mod(ulong N)
     311             : {
     312        2107 :   GEN fa = factoru(N), div = divisorsu_fact(fa);
     313        2107 :   ulong i, nsym = count_Manin_symbols(N, gel(fa,1));
     314        2107 :   GEN symbols = generatemsymbols(N, nsym, div);
     315        2107 :   GEN H = inithashmsymbols(N,symbols);
     316        2107 :   GEN invsafe = cgetg(N, t_VECSMALL), inverse = cgetg(N, t_VECSMALL);
     317      229866 :   for (i = 1; i < N; i++)
     318             :   {
     319      227759 :     invsafe[i] = Fl_invsafe(i,N);
     320      227759 :     inverse[i] = Fl_inverse(i,N);
     321             :   }
     322        2107 :   return mkvecn(7, symbols, H, utoipos(N), fa, div, invsafe, inverse);
     323             : }
     324             : 
     325             : /* Let (c : d) in P1(Z/NZ).
     326             :  * If c = 0 return (0:1). If d = 0 return (1:0).
     327             :  * Else replace by (cu : du), where u in (Z/NZ)^* such that C := cu = gcd(c,N).
     328             :  * In create_p1mod(), (c : d) is represented by (C:D) where D = du (mod N/c)
     329             :  * is smallest such that gcd(C,D) = 1. Return (C : du mod N/c), which need
     330             :  * not belong to P1(Z/NZ) ! A second component du mod N/c = 0 is replaced by
     331             :  * N/c in this case to avoid problems with array indices */
     332             : static void
     333    24978562 : p1_std_form(long *pc, long *pd, GEN p1N)
     334             : {
     335    24978562 :   ulong N = p1N_get_N(p1N);
     336             :   ulong u;
     337    24978562 :   *pc = umodsu(*pc, N); if (!*pc) { *pd = 1; return; }
     338    22503047 :   *pd = umodsu(*pd, N); if (!*pd) { *pc = 1; return; }
     339    22239623 :   u = p1N_get_invsafe(p1N)[*pd];
     340    22239623 :   if (u) { *pc = Fl_mul(*pc,u,N); *pd = 1; return; } /* (d,N) = 1 */
     341             : 
     342     7062188 :   u = p1N_get_inverse(p1N)[*pc];
     343     7062188 :   if (u > 1) { *pc = Fl_mul(*pc,u,N); *pd = Fl_mul(*pd,u,N); }
     344             :   /* c | N */
     345     7062188 :   if (*pc != 1) *pd %= (N / *pc);
     346     7062188 :   if (!*pd) *pd = N / *pc;
     347             : }
     348             : 
     349             : /* Input: v = [x,y] = elt of P^1(Z/NZ) = class in Gamma_0(N) \ PSL2(Z)
     350             :  * Output: returns the index of the standard rep equivalent to v */
     351             : static long
     352    24978562 : p1_index(long x, long y, GEN p1N)
     353             : {
     354    24978562 :   ulong N = p1N_get_N(p1N);
     355    24978562 :   GEN H = p1N_get_hash(p1N);
     356             : 
     357    24978562 :   p1_std_form(&x, &y, p1N);
     358    24978562 :   if (y == 1) return x+1;
     359     7325612 :   if (y == 0) return N+1;
     360     7062188 :   if (mael(H,x,y) == 0) pari_err_BUG("p1_index");
     361     7062188 :   return mael(H,x,y);
     362             : }
     363             : 
     364             : /* Cusps for \Gamma_0(N) */
     365             : 
     366             : /* \sum_{d | N} \phi(gcd(d, N/d)), using multiplicativity. fa = factor(N) */
     367             : ulong
     368        2191 : mfnumcuspsu_fact(GEN fa)
     369             : {
     370        2191 :   GEN P = gel(fa,1), E = gel(fa,2);
     371        2191 :   long i, l = lg(P);
     372        2191 :   ulong T = 1;
     373        5649 :   for (i = 1; i < l; i++)
     374             :   {
     375        3458 :     long e = E[i], e2 = e >> 1; /* floor(E[i] / 2) */
     376        3458 :     ulong p = P[i];
     377        3458 :     if (odd(e))
     378        3059 :       T *= 2 * upowuu(p, e2);
     379             :     else
     380         399 :       T *= (p+1) * upowuu(p, e2-1);
     381             :   }
     382        2191 :   return T;
     383             : }
     384             : ulong
     385           7 : mfnumcuspsu(ulong n)
     386           7 : { pari_sp av = avma; return gc_ulong(av, mfnumcuspsu_fact( factoru(n) )); }
     387             : /* \sum_{d | N} \phi(gcd(d, N/d)), using multiplicativity. fa = factor(N) */
     388             : GEN
     389          14 : mfnumcusps_fact(GEN fa)
     390             : {
     391          14 :   GEN P = gel(fa,1), E = gel(fa,2), T = gen_1;
     392          14 :   long i, l = lg(P);
     393          35 :   for (i = 1; i < l; i++)
     394             :   {
     395          21 :     GEN p = gel(P,i), c;
     396          21 :     long e = itos(gel(E,i)), e2 = e >> 1; /* floor(E[i] / 2) */
     397          21 :     if (odd(e))
     398           0 :       c = shifti(powiu(p, e2), 1);
     399             :     else
     400          21 :       c = mulii(addiu(p,1), powiu(p, e2-1));
     401          21 :     T = T? mulii(T, c): c;
     402             :   }
     403          14 :   return T? T: gen_1;
     404             : }
     405             : GEN
     406          21 : mfnumcusps(GEN n)
     407             : {
     408          21 :   pari_sp av = avma;
     409          21 :   GEN F = check_arith_pos(n,"mfnumcusps");
     410          21 :   if (!F)
     411             :   {
     412          14 :     if (lgefint(n) == 3) return utoi( mfnumcuspsu(n[2]) );
     413           7 :     F = absZ_factor(n);
     414             :   }
     415          14 :   return gerepileuptoint(av, mfnumcusps_fact(F));
     416             : }
     417             : 
     418             : /* to each cusp in \Gamma_0(N) P1(Q), represented by p/q, we associate a
     419             :  * unique index. Canonical representative: (1:0) or (p:q) with q | N, q < N,
     420             :  * p defined modulo d := gcd(N/q,q), (p,d) = 1.
     421             :  * Return [[N, nbcusps], H, cusps]*/
     422             : static GEN
     423        2107 : inithashcusps(GEN p1N)
     424             : {
     425        2107 :   ulong N = p1N_get_N(p1N);
     426        2107 :   GEN div = p1N_get_div(p1N), H = zerovec(N+1);
     427        2107 :   long k, ind, l = lg(div), ncusp = mfnumcuspsu_fact(p1N_get_fa(p1N));
     428        2107 :   GEN cusps = cgetg(ncusp+1, t_VEC);
     429             : 
     430        2107 :   gel(H,1) = mkvecsmall2(0/*empty*/, 1/* first cusp: (1:0) */);
     431        2107 :   gel(cusps, 1) = mkvecsmall2(1,0);
     432        2107 :   ind = 2;
     433        8477 :   for (k=1; k < l-1; k++) /* l-1: remove q = N */
     434             :   {
     435        6370 :     ulong p, q = div[k], d = ugcd(q, N/q);
     436        6370 :     GEN h = const_vecsmall(d+1,0);
     437        6370 :     gel(H,q+1) = h ;
     438       16044 :     for (p = 0; p < d; p++)
     439        9674 :       if (ugcd(p,d) == 1)
     440             :       {
     441        7784 :         h[p+1] = ind;
     442        7784 :         gel(cusps, ind) = mkvecsmall2(p,q);
     443        7784 :         ind++;
     444             :       }
     445             :   }
     446        2107 :   return mkvec3(mkvecsmall2(N,ind-1), H, cusps);
     447             : }
     448             : /* c = [p,q], (p,q) = 1, return a canonical representative for
     449             :  * \Gamma_0(N)(p/q) */
     450             : static GEN
     451      203469 : cusp_std_form(GEN c, GEN S)
     452             : {
     453      203469 :   long p, N = gel(S,1)[1], q = umodsu(c[2], N);
     454             :   ulong u, d;
     455      203469 :   if (q == 0) return mkvecsmall2(1, 0);
     456      201761 :   p = umodsu(c[1], N);
     457      201761 :   u = Fl_inverse(q, N);
     458      201761 :   q = Fl_mul(q,u, N);
     459      201761 :   d = ugcd(q, N/q);
     460      201761 :   return mkvecsmall2(Fl_div(p % d,u % d, d), q);
     461             : }
     462             : /* c = [p,q], (p,q) = 1, return the index of the corresponding cusp.
     463             :  * S from inithashcusps */
     464             : static ulong
     465      203469 : cusp_index(GEN c, GEN S)
     466             : {
     467             :   long p, q;
     468      203469 :   GEN H = gel(S,2);
     469      203469 :   c = cusp_std_form(c, S);
     470      203469 :   p = c[1]; q = c[2];
     471      203469 :   if (!mael(H,q+1,p+1)) pari_err_BUG("cusp_index");
     472      203469 :   return mael(H,q+1,p+1);
     473             : }
     474             : 
     475             : /* M a square invertible ZM, return a ZM iM such that iM M = M iM = d.Id */
     476             : static GEN
     477        3066 : ZM_inv_denom(GEN M)
     478             : {
     479        3066 :   GEN diM, iM = ZM_inv(M, &diM);
     480        3066 :   return mkvec2(iM, diM);
     481             : }
     482             : /* return M^(-1) v, dinv = ZM_inv_denom(M) OR Qevproj_init(M) */
     483             : static GEN
     484      744023 : ZC_apply_dinv(GEN dinv, GEN v)
     485             : {
     486             :   GEN x, c, iM;
     487      744023 :   if (lg(dinv) == 3)
     488             :   {
     489      665917 :     iM = gel(dinv,1);
     490      665917 :     c = gel(dinv,2);
     491             :   }
     492             :   else
     493             :   { /* Qevproj_init */
     494       78106 :     iM = gel(dinv,2);
     495       78106 :     c = gel(dinv,3);
     496       78106 :     v = typ(v) == t_MAT? rowpermute(v, gel(dinv,4))
     497       78106 :                        : vecpermute(v, gel(dinv,4));
     498             :   }
     499      744023 :   x = RgM_RgC_mul(iM, v);
     500      744023 :   if (!isint1(c)) x = RgC_Rg_div(x, c);
     501      744023 :   return x;
     502             : }
     503             : 
     504             : /* M an n x d ZM of rank d (basis of a Q-subspace), n >= d.
     505             :  * Initialize a projector on M */
     506             : GEN
     507        5488 : Qevproj_init(GEN M)
     508             : {
     509             :   GEN v, perm, MM, iM, diM;
     510        5488 :   v = ZM_indexrank(M); perm = gel(v,1);
     511        5488 :   MM = rowpermute(M, perm); /* square invertible */
     512        5488 :   iM = ZM_inv(MM, &diM);
     513        5488 :   return mkvec4(M, iM, diM, perm);
     514             : }
     515             : 
     516             : /* same with typechecks */
     517             : static GEN
     518         721 : Qevproj_init0(GEN M)
     519             : {
     520         721 :   switch(typ(M))
     521             :   {
     522         665 :     case t_VEC:
     523         665 :       if (lg(M) == 5) return M;
     524           0 :       break;
     525          49 :     case t_COL:
     526          49 :       M = mkmat(M);/*fall through*/
     527          56 :     case t_MAT:
     528          56 :       M = Q_primpart(M);
     529          56 :       RgM_check_ZM(M,"Qevproj_init");
     530          56 :       return Qevproj_init(M);
     531             :   }
     532           0 :   pari_err_TYPE("Qevproj_init",M);
     533             :   return NULL;/*LCOV_EXCL_LINE*/
     534             : }
     535             : 
     536             : /* T an n x n QM, pro = Qevproj_init(M), pro2 = Qevproj_init(M2); TM \subset M2.
     537             :  * Express these column vectors on M2's basis */
     538             : static GEN
     539        3661 : Qevproj_apply2(GEN T, GEN pro, GEN pro2)
     540             : {
     541        3661 :   GEN M = gel(pro,1), iM = gel(pro2,2), ciM = gel(pro2,3), perm = gel(pro2,4);
     542        3661 :   return RgM_Rg_div(RgM_mul(iM, RgM_mul(rowpermute(T,perm), M)), ciM);
     543             : }
     544             : /* T an n x n QM, stabilizing d-dimensional Q-vector space spanned by the
     545             :  * d columns of M, pro = Qevproj_init(M). Return dxd matrix of T acting on M */
     546             : GEN
     547        3031 : Qevproj_apply(GEN T, GEN pro) { return Qevproj_apply2(T, pro, pro); }
     548             : /* Qevproj_apply(T,pro)[,k] */
     549             : GEN
     550         819 : Qevproj_apply_vecei(GEN T, GEN pro, long k)
     551             : {
     552         819 :   GEN M = gel(pro,1), iM = gel(pro,2), ciM = gel(pro,3), perm = gel(pro,4);
     553         819 :   GEN v = RgM_RgC_mul(iM, RgM_RgC_mul(rowpermute(T,perm), gel(M,k)));
     554         819 :   return RgC_Rg_div(v, ciM);
     555             : }
     556             : 
     557             : static int
     558         434 : cmp_dim(void *E, GEN a, GEN b)
     559             : {
     560             :   long k;
     561             :   (void)E;
     562         434 :   a = gel(a,1);
     563         434 :   b = gel(b,1); k = lg(a)-lg(b);
     564         434 :   return k? ((k > 0)? 1: -1): 0;
     565             : }
     566             : 
     567             : /* FIXME: could use ZX_roots for deglim = 1 */
     568             : static GEN
     569         343 : ZX_factor_limit(GEN T, long deglim, long *pl)
     570             : {
     571         343 :   GEN fa = ZX_factor(T), P, E;
     572             :   long i, l;
     573         343 :   P = gel(fa,1); *pl = l = lg(P);
     574         343 :   if (deglim <= 0) return fa;
     575         224 :   E = gel(fa,2);
     576         567 :   for (i = 1; i < l; i++)
     577         406 :     if (degpol(gel(P,i)) > deglim) break;
     578         224 :   setlg(P,i);
     579         224 :   setlg(E,i); return fa;
     580             : }
     581             : 
     582             : /* Decompose the subspace H (Qevproj format) in simple subspaces.
     583             :  * Eg for H = msnew */
     584             : static GEN
     585         266 : mssplit_i(GEN W, GEN H, long deglim)
     586             : {
     587         266 :   ulong p, N = ms_get_N(W);
     588             :   long first, dim;
     589             :   forprime_t S;
     590         266 :   GEN T1 = NULL, T2 = NULL, V;
     591         266 :   dim = lg(gel(H,1))-1;
     592         266 :   V = vectrunc_init(dim+1);
     593         266 :   if (!dim) return V;
     594         259 :   (void)u_forprime_init(&S, 2, ULONG_MAX);
     595         259 :   vectrunc_append(V, H);
     596         259 :   first = 1; /* V[1..first-1] contains simple subspaces */
     597         399 :   while ((p = u_forprime_next(&S)))
     598             :   {
     599             :     GEN T;
     600             :     long j, lV;
     601         399 :     if (N % p == 0) continue;
     602         336 :     if (T1 && T2) {
     603          21 :       T = RgM_add(T1,T2);
     604          21 :       T2 = NULL;
     605             :     } else {
     606         315 :       T2 = T1;
     607         315 :       T1 = T = mshecke(W, p, NULL);
     608             :     }
     609         336 :     lV = lg(V);
     610         679 :     for (j = first; j < lV; j++)
     611             :     {
     612         343 :       pari_sp av = avma;
     613             :       long lP;
     614         343 :       GEN Vj = gel(V,j), P = gel(Vj,1);
     615         343 :       GEN TVj = Qevproj_apply(T, Vj); /* c T | V_j */
     616         343 :       GEN ch = QM_charpoly_ZX(TVj), fa = ZX_factor_limit(ch,deglim, &lP);
     617         343 :       GEN F = gel(fa, 1), E = gel(fa, 2);
     618         343 :       long k, lF = lg(F);
     619         343 :       if (lF == 2 && lP == 2)
     620             :       {
     621         336 :         if (isint1(gel(E,1)))
     622             :         { /* simple subspace */
     623         168 :           swap(gel(V,first), gel(V,j));
     624         168 :           first++;
     625             :         }
     626             :         else
     627           0 :           set_avma(av);
     628             :       }
     629         175 :       else if (lF == 1) /* discard V[j] */
     630           7 :       { swap(gel(V,j), gel(V,lg(V)-1)); setlg(V, lg(V)-1); }
     631             :       else
     632             :       { /* can split Vj */
     633             :         GEN pows;
     634         168 :         long D = 1;
     635         658 :         for (k = 1; k < lF; k++)
     636             :         {
     637         490 :           long d = degpol(gel(F,k));
     638         490 :           if (d > D) D = d;
     639             :         }
     640             :         /* remove V[j] */
     641         168 :         swap(gel(V,j), gel(V,lg(V)-1)); setlg(V, lg(V)-1);
     642         168 :         pows = RgM_powers(TVj, minss((long)2*sqrt((double)D), D));
     643         658 :         for (k = 1; k < lF; k++)
     644             :         {
     645         490 :           GEN f = gel(F,k);
     646         490 :           GEN K = QM_ker( RgX_RgMV_eval(f, pows)) ; /* Ker f(TVj) */
     647         490 :           GEN p = vec_Q_primpart( RgM_mul(P, K) );
     648         490 :           vectrunc_append(V, Qevproj_init(p));
     649         490 :           if (lg(K) == 2 || isint1(gel(E,k)))
     650             :           { /* simple subspace */
     651         406 :             swap(gel(V,first), gel(V, lg(V)-1));
     652         406 :             first++;
     653             :           }
     654             :         }
     655         168 :         if (j < first) j = first;
     656             :       }
     657             :     }
     658         336 :     if (first >= lg(V)) {
     659         259 :       gen_sort_inplace(V, NULL, cmp_dim, NULL);
     660         259 :       return V;
     661             :     }
     662             :   }
     663           0 :   pari_err_BUG("subspaces not found");
     664             :   return NULL;/*LCOV_EXCL_LINE*/
     665             : }
     666             : GEN
     667         266 : mssplit(GEN W, GEN H, long deglim)
     668             : {
     669         266 :   pari_sp av = avma;
     670         266 :   checkms(W);
     671         266 :   if (!msk_get_sign(W))
     672           0 :     pari_err_DOMAIN("mssplit","abs(sign)","!=",gen_1,gen_0);
     673         266 :   if (!H) H = msnew(W);
     674         266 :   H = Qevproj_init0(H);
     675         266 :   return gerepilecopy(av, mssplit_i(W,H,deglim));
     676             : }
     677             : 
     678             : /* proV = Qevproj_init of a Hecke simple subspace, return [ a_n, n <= B ] */
     679             : static GEN
     680         245 : msqexpansion_i(GEN W, GEN proV, ulong B)
     681             : {
     682         245 :   ulong p, N = ms_get_N(W), sqrtB;
     683         245 :   long i, d, k = msk_get_weight(W);
     684             :   forprime_t S;
     685         245 :   GEN T1=NULL, T2=NULL, TV=NULL, ch=NULL, v, dTiv, Tiv, diM, iM, L;
     686         245 :   switch(B)
     687             :   {
     688           0 :     case 0: return cgetg(1,t_VEC);
     689           0 :     case 1: return mkvec(gen_1);
     690             :   }
     691         245 :   (void)u_forprime_init(&S, 2, ULONG_MAX);
     692         357 :   while ((p = u_forprime_next(&S)))
     693             :   {
     694             :     GEN T;
     695         357 :     if (N % p == 0) continue;
     696         266 :     if (T1 && T2)
     697             :     {
     698           0 :       T = RgM_add(T1,T2);
     699           0 :       T2 = NULL;
     700             :     }
     701             :     else
     702             :     {
     703         266 :       T2 = T1;
     704         266 :       T1 = T = mshecke(W, p, NULL);
     705             :     }
     706         266 :     TV = Qevproj_apply(T, proV); /* T | V */
     707         266 :     ch = QM_charpoly_ZX(TV);
     708         266 :     if (ZX_is_irred(ch)) break;
     709          21 :     ch = NULL;
     710             :   }
     711         245 :   if (!ch) pari_err_BUG("q-Expansion not found");
     712             :   /* T generates the Hecke algebra (acting on V) */
     713         245 :   d = degpol(ch);
     714         245 :   v = vec_ei(d, 1); /* take v = e_1 */
     715         245 :   Tiv = cgetg(d+1, t_MAT); /* Tiv[i] = T^(i-1)v */
     716         245 :   gel(Tiv, 1) = v;
     717         343 :   for (i = 2; i <= d; i++) gel(Tiv, i) = RgM_RgC_mul(TV, gel(Tiv,i-1));
     718         245 :   Tiv = Q_remove_denom(Tiv, &dTiv);
     719         245 :   iM = ZM_inv(Tiv, &diM);
     720         245 :   if (dTiv) diM = gdiv(diM, dTiv);
     721         245 :   L = const_vec(B,NULL);
     722         245 :   sqrtB = usqrt(B);
     723         245 :   gel(L,1) = d > 1? mkpolmod(gen_1,ch): gen_1;
     724        2471 :   for (p = 2; p <= B; p++)
     725             :   {
     726        2226 :     pari_sp av = avma;
     727             :     GEN T, u, Tv, ap, P;
     728             :     ulong m;
     729        2226 :     if (gel(L,p)) continue;  /* p not prime */
     730         819 :     T = mshecke(W, p, NULL);
     731         819 :     Tv = Qevproj_apply_vecei(T, proV, 1); /* Tp.v */
     732             :     /* Write Tp.v = \sum u_i T^i v */
     733         819 :     u = RgC_Rg_div(RgM_RgC_mul(iM, Tv), diM);
     734         819 :     ap = gerepilecopy(av, RgV_to_RgX(u, 0));
     735         819 :     if (d > 1)
     736         399 :       ap = mkpolmod(ap,ch);
     737             :     else
     738         420 :       ap = simplify_shallow(ap);
     739         819 :     gel(L,p) = ap;
     740         819 :     if (!(N % p))
     741             :     { /* p divides the level */
     742         147 :       ulong C = B/p;
     743         546 :       for (m=1; m<=C; m++)
     744         399 :         if (gel(L,m)) gel(L,m*p) = gmul(gel(L,m), ap);
     745         147 :       continue;
     746             :     }
     747         672 :     P = powuu(p,k-1);
     748         672 :     if (p <= sqrtB) {
     749         119 :       ulong pj, oldpj = 1;
     750         546 :       for (pj = p; pj <= B; oldpj=pj, pj *= p)
     751             :       {
     752         427 :         GEN apj = (pj==p)? ap
     753         427 :                          : gsub(gmul(ap,gel(L,oldpj)), gmul(P,gel(L,oldpj/p)));
     754         427 :         gel(L,pj) = apj;
     755        3136 :         for (m = B/pj; m > 1; m--)
     756        2709 :           if (gel(L,m) && m%p) gel(L,m*pj) = gmul(gel(L,m), apj);
     757             :       }
     758             :     } else {
     759         553 :       gel(L,p) = ap;
     760        1092 :       for (m = B/p; m > 1; m--)
     761         539 :         if (gel(L,m)) gel(L,m*p) = gmul(gel(L,m), ap);
     762             :     }
     763             :   }
     764         245 :   return L;
     765             : }
     766             : GEN
     767         245 : msqexpansion(GEN W, GEN proV, ulong B)
     768             : {
     769         245 :   pari_sp av = avma;
     770         245 :   checkms(W);
     771         245 :   proV = Qevproj_init0(proV);
     772         245 :   return gerepilecopy(av, msqexpansion_i(W,proV,B));
     773             : }
     774             : 
     775             : static GEN
     776         217 : Qevproj_apply0(GEN T, GEN pro)
     777             : {
     778         217 :   GEN iM = gel(pro,2), perm = gel(pro,4);
     779         217 :   return vec_Q_primpart(ZM_mul(iM, rowpermute(T,perm)));
     780             : }
     781             : /* T a ZC or ZM */
     782             : GEN
     783        4186 : Qevproj_down(GEN T, GEN pro)
     784             : {
     785        4186 :   GEN iM = gel(pro,2), ciM = gel(pro,3), perm = gel(pro,4);
     786        4186 :   if (typ(T) == t_COL)
     787        4186 :     return RgC_Rg_div(ZM_ZC_mul(iM, vecpermute(T,perm)), ciM);
     788             :   else
     789           0 :     return RgM_Rg_div(ZM_mul(iM, rowpermute(T,perm)), ciM);
     790             : }
     791             : 
     792             : static GEN
     793         287 : Qevproj_star(GEN W, GEN H)
     794             : {
     795         287 :   long s = msk_get_sign(W);
     796         287 :   if (s)
     797             :   { /* project on +/- component */
     798         217 :     GEN A = RgM_mul(msk_get_star(W), H);
     799         217 :     A = (s > 0)? gadd(A, H): gsub(A, H);
     800             :     /* Im(star + sign) = Ker(star - sign) */
     801         217 :     H = QM_image_shallow(A);
     802         217 :     H = Qevproj_apply0(H, msk_get_starproj(W));
     803             :   }
     804         287 :   return H;
     805             : }
     806             : 
     807             : static GEN
     808        3059 : Tp_matrices(ulong p)
     809             : {
     810        3059 :   GEN v = cgetg(p+2, t_VEC);
     811             :   ulong i;
     812       28707 :   for (i = 1; i <= p; i++) gel(v,i) = mat2(1, i-1, 0, p);
     813        3059 :   gel(v,i) = mat2(p, 0, 0, 1);
     814        3059 :   return v;
     815             : }
     816             : static GEN
     817         987 : Up_matrices(ulong p)
     818             : {
     819         987 :   GEN v = cgetg(p+1, t_VEC);
     820             :   ulong i;
     821        6300 :   for (i = 1; i <= p; i++) gel(v,i) = mat2(1, i-1, 0, p);
     822         987 :   return v;
     823             : }
     824             : 
     825             : /* M = N/p. Classes of Gamma_0(M) / Gamma_O(N) when p | M */
     826             : static GEN
     827         182 : NP_matrices(ulong M, ulong p)
     828             : {
     829         182 :   GEN v = cgetg(p+1, t_VEC);
     830             :   ulong i;
     831        1365 :   for (i = 1; i <= p; i++) gel(v,i) = mat2(1, 0, (i-1)*M, 1);
     832         182 :   return v;
     833             : }
     834             : /* M = N/p. Extra class of Gamma_0(M) / Gamma_O(N) when p \nmid M */
     835             : static GEN
     836          98 : NP_matrix_extra(ulong M, ulong p)
     837             : {
     838          98 :   long w,z, d = cbezout(p, -M, &w, &z);
     839          98 :   if (d != 1) return NULL;
     840          98 :   return mat2(w,z,M,p);
     841             : }
     842             : static GEN
     843         112 : WQ_matrix(long N, long Q)
     844             : {
     845         112 :   long w,z, d = cbezout(Q, N/Q, &w, &z);
     846         112 :   if (d != 1) return NULL;
     847         112 :   return mat2(Q,1,-N*z,Q*w);
     848             : }
     849             : 
     850             : GEN
     851         287 : msnew(GEN W)
     852             : {
     853         287 :   pari_sp av = avma;
     854         287 :   GEN S = mscuspidal(W, 0);
     855         287 :   ulong N = ms_get_N(W);
     856         287 :   long s = msk_get_sign(W), k = msk_get_weight(W);
     857         287 :   if (N > 1 && (!uisprime(N) || (k == 12 || k > 14)))
     858             :   {
     859         112 :     GEN p1N = ms_get_p1N(W), P = gel(p1N_get_fa(p1N), 1);
     860         112 :     long i, nP = lg(P)-1;
     861         112 :     GEN v = cgetg(2*nP + 1, t_COL);
     862         112 :     S = gel(S,1); /* Q basis */
     863         294 :     for (i = 1; i <= nP; i++)
     864             :     {
     865         182 :       pari_sp av = avma, av2;
     866         182 :       long M = N/P[i];
     867         182 :       GEN T1,Td, Wi = mskinit(M, k, s);
     868         182 :       GEN v1 = NP_matrices(M, P[i]);
     869         182 :       GEN vd = Up_matrices(P[i]);
     870             :       /* p^2 \nmid N */
     871         182 :       if (M % P[i])
     872             :       {
     873          98 :         v1 = vec_append(v1, NP_matrix_extra(M,P[i]));
     874          98 :         vd = vec_append(vd, WQ_matrix(N,P[i]));
     875             :       }
     876         182 :       T1 = getMorphism(W, Wi, v1);
     877         182 :       Td = getMorphism(W, Wi, vd);
     878         182 :       if (s)
     879             :       {
     880         168 :         T1 = Qevproj_apply2(T1, msk_get_starproj(W), msk_get_starproj(Wi));
     881         168 :         Td = Qevproj_apply2(Td, msk_get_starproj(W), msk_get_starproj(Wi));
     882             :       }
     883         182 :       av2 = avma;
     884         182 :       T1 = RgM_mul(T1,S);
     885         182 :       Td = RgM_mul(Td,S);  /* multiply by S = restrict to mscusp */
     886         182 :       gerepileallsp(av, av2, 2, &T1, &Td);
     887         182 :       gel(v,2*i-1) = T1;
     888         182 :       gel(v,2*i)   = Td;
     889             :     }
     890         112 :     S = ZM_mul(S, QM_ker(matconcat(v))); /* Snew */
     891         112 :     S = Qevproj_init(vec_Q_primpart(S));
     892             :   }
     893         287 :   return gerepilecopy(av, S);
     894             : }
     895             : 
     896             : /* Solve the Manin relations for a congruence subgroup \Gamma by constructing
     897             :  * a well-formed fundamental domain for the action of \Gamma on upper half
     898             :  * space. See
     899             :  * Pollack and Stevens, Overconvergent modular symbols and p-adic L-functions
     900             :  * Annales scientifiques de l'ENS 44, fascicule 1 (2011), 1-42
     901             :  * http://math.bu.edu/people/rpollack/Papers/Overconvergent_modular_symbols_and_padic_Lfunctions.pdf
     902             :  *
     903             :  * FIXME: Implemented for \Gamma = \Gamma_0(N) only. */
     904             : 
     905             : /* linked lists */
     906             : typedef struct list_t { GEN data; struct list_t *next; } list_t;
     907             : static list_t *
     908      115164 : list_new(GEN x)
     909             : {
     910      115164 :   list_t *L = (list_t*)stack_malloc(sizeof(list_t));
     911      115164 :   L->data = x;
     912      115164 :   L->next = NULL; return L;
     913             : }
     914             : static void
     915      113085 : list_insert(list_t *L, GEN x)
     916             : {
     917      113085 :   list_t *l = list_new(x);
     918      113085 :   l->next = L->next;
     919      113085 :   L->next = l;
     920      113085 : }
     921             : 
     922             : /*Input: N > 1, p1N = P^1(Z/NZ)
     923             :  *Output: a connected fundamental domain for the action of \Gamma_0(N) on
     924             :  *  upper half space.  When \Gamma_0(N) is torsion free, the domain has the
     925             :  *  property that all of its vertices are cusps.  When \Gamma_0(N) has
     926             :  *  three-torsion, 2 extra triangles need to be added.
     927             :  *
     928             :  * The domain is constructed by beginning with the triangle with vertices 0,1
     929             :  * and oo.  Each adjacent triangle is successively tested to see if it contains
     930             :  * points not \Gamma_0(N) equivalent to some point in our region.  If a
     931             :  * triangle contains new points, it is added to the region.  This process is
     932             :  * continued until the region can no longer be extended (and still be a
     933             :  * fundamental domain) by added an adjacent triangle.  The list of cusps
     934             :  * between 0 and 1 are then returned
     935             :  *
     936             :  * Precisely, the function returns a list such that the elements of the list
     937             :  * with odd index are the cusps in increasing order.  The even elements of the
     938             :  * list are either an "x" or a "t".  A "t" represents that there is an element
     939             :  * of order three such that its fixed point is in the triangle directly
     940             :  * adjacent to the our region with vertices given by the cusp before and after
     941             :  * the "t".  The "x" represents that this is not the case. */
     942             : enum { type_X, type_DO /* ? */, type_T };
     943             : static GEN
     944        2079 : form_list_of_cusps(ulong N, GEN p1N)
     945             : {
     946        2079 :   pari_sp av = avma;
     947        2079 :   long i, position, nbC = 2;
     948             :   GEN v, L;
     949             :   list_t *C, *c;
     950             :   /* Let t be the index of a class in PSL2(Z) / \Gamma in our fixed enumeration
     951             :    * v[t] != 0 iff it is the class of z tau^r for z a previous alpha_i
     952             :    * or beta_i.
     953             :    * For \Gamma = \Gamma_0(N), the enumeration is given by p1_index.
     954             :    * We write cl(gamma) = the class of gamma mod \Gamma */
     955        2079 :   v = const_vecsmall(p1_size(p1N), 0);
     956        2079 :   i = p1_index( 0, 1, p1N); v[i] = 1;
     957        2079 :   i = p1_index( 1,-1, p1N); v[i] = 2;
     958        2079 :   i = p1_index(-1, 0, p1N); v[i] = 3;
     959             :   /* the value is unused [debugging]: what matters is whether it is != 0 */
     960        2079 :   position = 4;
     961             :   /* at this point, Fund = R, v contains the classes of Id, tau, tau^2 */
     962             : 
     963        2079 :   C  = list_new(mkvecsmall3(0,1, type_X));
     964        2079 :   list_insert(C, mkvecsmall3(1,1,type_DO));
     965             :   /* C is a list of triples[a,b,t], where c = a/b is a cusp, and t is the type
     966             :    * of the path between c and the PREVIOUS cusp in the list, coded as
     967             :    *   type_DO = "?", type_X = "x", type_T = "t"
     968             :    * Initially, C = [0/1,"?",1/1]; */
     969             : 
     970             :   /* loop through the current set of cusps C and check to see if more cusps
     971             :    * should be added */
     972             :   for (;;)
     973       10367 :   {
     974       12446 :     int done = 1;
     975      542836 :     for (c = C; c; c = c->next)
     976             :     {
     977             :       GEN cusp1, cusp2, gam;
     978             :       long pos, b1, b2, b;
     979             : 
     980      542836 :       if (!c->next) break;
     981      530390 :       cusp1 = c->data; /* = a1/b1 */
     982      530390 :       cusp2 = (c->next)->data; /* = a2/b2 */
     983      530390 :       if (cusp2[3] != type_DO) continue;
     984             : 
     985             :       /* gam (oo -> 0) = (cusp2 -> cusp1), gam in PSL2(Z) */
     986      224091 :       gam = path_to_zm(mkpath(cusp2, cusp1)); /* = [a2,a1;b2,b1] */
     987             :       /* we have normalized the cusp representation so that a1 b2 - a2 b1 = 1 */
     988      224091 :       b1 = coeff(gam,2,1); b2 = coeff(gam,2,2);
     989             :       /* gam.1  = (a1 + a2) / (b1 + b2) */
     990      224091 :       b = b1 + b2;
     991             :       /* Determine whether the adjacent triangle *below* (cusp1->cusp2)
     992             :        * should be added */
     993      224091 :       pos = p1_index(b1,b2, p1N); /* did we see cl(gam) before ? */
     994      224091 :       if (v[pos])
     995      112196 :         cusp2[3] = type_X; /* NO */
     996             :       else
     997             :       { /* YES */
     998             :         ulong B1, B2;
     999      111895 :         v[pos] = position;
    1000      111895 :         i = p1_index(-(b1+b2), b1, p1N); v[i] = position+1;
    1001      111895 :         i = p1_index(b2, -(b1+b2), p1N); v[i] = position+2;
    1002             :         /* add cl(gam), cl(gam*TAU), cl(gam*TAU^2) to v */
    1003      111895 :         position += 3;
    1004             :         /* gam tau gam^(-1) in \Gamma ? */
    1005      111895 :         B1 = umodsu(b1, N);
    1006      111895 :         B2 = umodsu(b2, N);
    1007      111895 :         if ((Fl_sqr(B2,N) + Fl_sqr(B1,N) + Fl_mul(B1,B2,N)) % N == 0)
    1008         889 :           cusp2[3] = type_T;
    1009             :         else
    1010             :         {
    1011      111006 :           long a1 = coeff(gam, 1,1), a2 = coeff(gam, 1,2);
    1012      111006 :           long a = a1 + a2; /* gcd(a,b) = 1 */
    1013      111006 :           list_insert(c, mkvecsmall3(a,b,type_DO));
    1014      111006 :           c = c->next;
    1015      111006 :           nbC++;
    1016      111006 :           done = 0;
    1017             :         }
    1018             :       }
    1019             :     }
    1020       12446 :     if (done) break;
    1021             :   }
    1022        2079 :   L = cgetg(nbC+1, t_VEC); i = 1;
    1023      117243 :   for (c = C; c; c = c->next) gel(L,i++) = c->data;
    1024        2079 :   return gerepilecopy(av, L);
    1025             : }
    1026             : 
    1027             : /* W an msN. M in PSL2(Z). Return index of M in P1^(Z/NZ) = Gamma0(N) \ PSL2(Z),
    1028             :  * and M0 in Gamma_0(N) such that M = M0 * M', where M' = chosen
    1029             :  * section( PSL2(Z) -> P1^(Z/NZ) ). */
    1030             : static GEN
    1031      498463 : Gamma0N_decompose(GEN W, GEN M, long *index)
    1032             : {
    1033      498463 :   GEN p1N = msN_get_p1N(W), W3 = gel(W,3), section = msN_get_section(W);
    1034             :   GEN A;
    1035      498463 :   ulong N = p1N_get_N(p1N);
    1036      498463 :   ulong c = umodiu(gcoeff(M,2,1), N);
    1037      498463 :   ulong d = umodiu(gcoeff(M,2,2), N);
    1038      498463 :   long s, ind = p1_index(c, d, p1N); /* as an elt of P1(Z/NZ) */
    1039      498463 :   *index = W3[ind]; /* as an elt of F, E2, ... */
    1040      498463 :   M = ZM_zm_mul(M, sl2_inv(gel(section,ind)));
    1041             :   /* normalize mod +/-Id */
    1042      498463 :   A = gcoeff(M,1,1);
    1043      498463 :   s = signe(A);
    1044      498463 :   if (s < 0)
    1045      237125 :     M = ZM_neg(M);
    1046      261338 :   else if (!s)
    1047             :   {
    1048         378 :     GEN C = gcoeff(M,2,1);
    1049         378 :     if (signe(C) < 0) M = ZM_neg(M);
    1050             :   }
    1051      498463 :   return M;
    1052             : }
    1053             : /* W an msN; as above for a path. Return [[ind], M] */
    1054             : static GEN
    1055      234486 : path_Gamma0N_decompose(GEN W, GEN path)
    1056             : {
    1057      234486 :   GEN p1N = msN_get_p1N(W);
    1058      234486 :   GEN p1index_to_ind = gel(W,3);
    1059      234486 :   GEN section = msN_get_section(W);
    1060      234486 :   GEN M = path_to_zm(path);
    1061      234486 :   long p1index = p1_index(cc(M), dd(M), p1N);
    1062      234486 :   long ind = p1index_to_ind[p1index];
    1063      234486 :   GEN M0 = ZM_zm_mul(mat2_to_ZM(M), sl2_inv(gel(section,p1index)));
    1064      234486 :   return mkvec2(mkvecsmall(ind), M0);
    1065             : }
    1066             : 
    1067             : /*Form generators of H_1(X_0(N),{cusps},Z)
    1068             : *
    1069             : *Input: N = integer > 1, p1N = P^1(Z/NZ)
    1070             : *Output: [cusp_list,E,F,T2,T3,E1] where
    1071             : *  cusps_list = list of cusps describing fundamental domain of
    1072             : *    \Gamma_0(N).
    1073             : *  E = list of paths in the boundary of the fundamental domains and oriented
    1074             : *    clockwise such that they do not contain a point
    1075             : *    fixed by an element of order 2 and they are not an edge of a
    1076             : *    triangle containing a fixed point of an element of order 3
    1077             : *  F = list of paths in the interior of the domain with each
    1078             : *    orientation appearing separately
    1079             : * T2 = list of paths in the boundary of domain containing a point fixed
    1080             : *    by an element of order 2 (oriented clockwise)
    1081             : * T3 = list of paths in the boundard of domain which are the edges of
    1082             : *    some triangle containing a fixed point of a matrix of order 3 (both
    1083             : *    orientations appear)
    1084             : * E1 = a sublist of E such that every path in E is \Gamma_0(N)-equivalent to
    1085             : *    either an element of E1 or the flip (reversed orientation) of an element
    1086             : *    of E1.
    1087             : * (Elements of T2 are \Gamma_0(N)-equivalent to their own flip.)
    1088             : *
    1089             : * sec = a list from 1..#p1N of matrices describing a section of the map
    1090             : *   SL_2(Z) to P^1(Z/NZ) given by [a,b;c,d]-->[c,d].
    1091             : *   Given our fixed enumeration of P^1(Z/NZ), the j-th element of the list
    1092             : *   represents the image of the j-th element of P^1(Z/NZ) under the section. */
    1093             : 
    1094             : /* insert path in set T */
    1095             : static void
    1096      340144 : set_insert(hashtable *T, GEN path)
    1097      340144 : { hash_insert(T, path,  (void*)(T->nb + 1)); }
    1098             : 
    1099             : static GEN
    1100       18711 : hash_to_vec(hashtable *h)
    1101             : {
    1102       18711 :   GEN v = cgetg(h->nb + 1, t_VEC);
    1103             :   ulong i;
    1104     2696764 :   for (i = 0; i < h->len; i++)
    1105             :   {
    1106     2678053 :     hashentry *e = h->table[i];
    1107     3242064 :     while (e)
    1108             :     {
    1109      564011 :       GEN key = (GEN)e->key;
    1110      564011 :       long index = (long)e->val;
    1111      564011 :       gel(v, index) = key;
    1112      564011 :       e = e->next;
    1113             :     }
    1114             :   }
    1115       18711 :   return v;
    1116             : }
    1117             : 
    1118             : static long
    1119      173082 : path_to_p1_index(GEN path, GEN p1N)
    1120             : {
    1121      173082 :   GEN M = path_to_zm(path);
    1122      173082 :   return p1_index(cc(M), dd(M), p1N);
    1123             : }
    1124             : 
    1125             : /* Pollack-Stevens sets */
    1126             : typedef struct PS_sets_t {
    1127             :   hashtable *F, *T2, *T31, *T32, *E1, *E2;
    1128             :   GEN E2fromE1, stdE1;
    1129             : } PS_sets_t;
    1130             : 
    1131             : static hashtable *
    1132       16142 : set_init(long max)
    1133       16142 : { return hash_create(max, (ulong(*)(void*))&hash_GEN,
    1134             :                           (int(*)(void*,void*))&gidentical, 1); }
    1135             : /* T = E2fromE1[i] = [c, gamma] */
    1136             : static ulong
    1137    40032230 : E2fromE1_c(GEN T) { return itou(gel(T,1)); }
    1138             : static GEN
    1139      579957 : E2fromE1_Zgamma(GEN T) { return gel(T,2); }
    1140             : static GEN
    1141       57694 : E2fromE1_gamma(GEN T) { return gcoeff(gel(T,2),1,1); }
    1142             : 
    1143             : static void
    1144      115388 : insert_E(GEN path, PS_sets_t *S, GEN p1N)
    1145             : {
    1146      115388 :   GEN rev = vecreverse(path);
    1147      115388 :   long std = path_to_p1_index(rev, p1N);
    1148      115388 :   GEN v = gel(S->stdE1, std);
    1149      115388 :   if (v)
    1150             :   { /* [s, p1], where E1[s] is the path p1 = vecreverse(path) mod \Gamma */
    1151       57694 :     GEN gamma, p1 = gel(v,2);
    1152       57694 :     long r, s = itou(gel(v,1));
    1153             : 
    1154       57694 :     set_insert(S->E2, path);
    1155       57694 :     r = S->E2->nb;
    1156       57694 :     if (gel(S->E2fromE1, r) != gen_0) pari_err_BUG("insert_E");
    1157             : 
    1158       57694 :     gamma = gamma_equiv_matrix(rev, p1);
    1159             :     /* reverse(E2[r]) = gamma * E1[s] */
    1160       57694 :     gel(S->E2fromE1, r) = mkvec2(utoipos(s), to_famat_shallow(gamma,gen_m1));
    1161             :   }
    1162             :   else
    1163             :   {
    1164       57694 :     set_insert(S->E1, path);
    1165       57694 :     std = path_to_p1_index(path, p1N);
    1166       57694 :     gel(S->stdE1, std) = mkvec2(utoipos(S->E1->nb), path);
    1167             :   }
    1168      115388 : }
    1169             : 
    1170             : static GEN
    1171        8316 : cusp_infinity(void) { return mkvecsmall2(1,0); }
    1172             : 
    1173             : static void
    1174        2079 : form_E_F_T(ulong N, GEN p1N, GEN *pC, PS_sets_t *S)
    1175             : {
    1176        2079 :   GEN C, cusp_list = form_list_of_cusps(N, p1N);
    1177        2079 :   long nbgen = lg(cusp_list)-1, nbmanin = p1_size(p1N), r, s, i;
    1178             :   hashtable *F, *T2, *T31, *T32, *E1, *E2;
    1179             : 
    1180        2079 :   *pC = C = cgetg(nbgen+1, t_VEC);
    1181      117243 :   for (i = 1; i <= nbgen; i++)
    1182             :   {
    1183      115164 :     GEN c = gel(cusp_list,i);
    1184      115164 :     gel(C,i) = mkvecsmall2(c[1], c[2]);
    1185             :   }
    1186        2079 :   S->F  = F  = set_init(nbmanin);
    1187        2079 :   S->E1 = E1 = set_init(nbgen);
    1188        2079 :   S->E2 = E2 = set_init(nbgen);
    1189        2079 :   S->T2 = T2 = set_init(nbgen);
    1190        2079 :   S->T31 = T31 = set_init(nbgen);
    1191        2079 :   S->T32 = T32 = set_init(nbgen);
    1192             : 
    1193             :   /* T31 represents the three torsion paths going from left to right */
    1194             :   /* T32 represents the three torsion paths going from right to left */
    1195      115164 :   for (r = 1; r < nbgen; r++)
    1196             :   {
    1197      113085 :     GEN c2 = gel(cusp_list,r+1);
    1198      113085 :     if (c2[3] == type_T)
    1199             :     {
    1200         889 :       GEN c1 = gel(cusp_list,r), path = mkpath(c1,c2), path2 = vecreverse(path);
    1201         889 :       set_insert(T31, path);
    1202         889 :       set_insert(T32, path2);
    1203             :     }
    1204             :   }
    1205             : 
    1206             :   /* to record relations between E2 and E1 */
    1207        2079 :   S->E2fromE1 = zerovec(nbgen);
    1208        2079 :   S->stdE1 = const_vec(nbmanin, NULL);
    1209             : 
    1210             :   /* Assumption later: path [oo,0] is E1[1], path [1,oo] is E2[1] */
    1211             :   {
    1212        2079 :     GEN oo = cusp_infinity();
    1213        2079 :     GEN p1 = mkpath(oo, mkvecsmall2(0,1)); /* [oo, 0] */
    1214        2079 :     GEN p2 = mkpath(mkvecsmall2(1,1), oo); /* [1, oo] */
    1215        2079 :     insert_E(p1, S, p1N);
    1216        2079 :     insert_E(p2, S, p1N);
    1217             :   }
    1218             : 
    1219      115164 :   for (r = 1; r < nbgen; r++)
    1220             :   {
    1221      113085 :     GEN c1 = gel(cusp_list,r);
    1222    22298080 :     for (s = r+1; s <= nbgen; s++)
    1223             :     {
    1224    22184995 :       pari_sp av = avma;
    1225    22184995 :       GEN c2 = gel(cusp_list,s), path;
    1226    22184995 :       GEN d = subii(mulss(c1[1],c2[2]), mulss(c1[2],c2[1]));
    1227    22184995 :       set_avma(av);
    1228    22184995 :       if (!is_pm1(d)) continue;
    1229             : 
    1230      224091 :       path = mkpath(c1,c2);
    1231      224091 :       if (r+1 == s)
    1232             :       {
    1233      113085 :         GEN w = path;
    1234      113085 :         ulong hash = T31->hash(w); /* T31, T32 use the same hash function */
    1235      113085 :         if (!hash_search2(T31, w, hash) && !hash_search2(T32, w, hash))
    1236             :         {
    1237      112196 :           if (gamma_equiv(path, vecreverse(path), N))
    1238         966 :             set_insert(T2, path);
    1239             :           else
    1240      111230 :             insert_E(path, S, p1N);
    1241             :         }
    1242             :       } else {
    1243      111006 :         set_insert(F, mkvec2(path, mkvecsmall2(r,s)));
    1244      111006 :         set_insert(F, mkvec2(vecreverse(path), mkvecsmall2(s,r)));
    1245             :       }
    1246             :     }
    1247             :   }
    1248        2079 :   setlg(S->E2fromE1, E2->nb+1);
    1249        2079 : }
    1250             : 
    1251             : /* v = \sum n_i g_i, g_i in Sl(2,Z), return \sum n_i g_i^(-1) */
    1252             : static GEN
    1253      845705 : ZSl2_star(GEN v)
    1254             : {
    1255             :   long i, l;
    1256             :   GEN w, G;
    1257      845705 :   if (typ(v) == t_INT) return v;
    1258      845705 :   G = gel(v,1);
    1259      845705 :   w = cgetg_copy(G, &l);
    1260     2015363 :   for (i = 1; i < l; i++)
    1261             :   {
    1262     1169658 :     GEN g = gel(G,i);
    1263     1169658 :     if (typ(g) == t_MAT) g = SL2_inv_shallow(g);
    1264     1169658 :     gel(w,i) = g;
    1265             :   }
    1266      845705 :   return ZG_normalize(mkmat2(w, gel(v,2)));
    1267             : }
    1268             : 
    1269             : /* Input: h = set of unimodular paths, p1N = P^1(Z/NZ) = Gamma_0(N)\PSL2(Z)
    1270             :  * Output: Each path is converted to a matrix and then an element of P^1(Z/NZ)
    1271             :  * Append the matrix to W[12], append the index that represents
    1272             :  * these elements of P^1 (the classes mod Gamma_0(N) via our fixed
    1273             :  * enumeration to W[2]. */
    1274             : static void
    1275       12474 : paths_decompose(GEN W, hashtable *h, int flag)
    1276             : {
    1277       12474 :   GEN p1N = ms_get_p1N(W), section = ms_get_section(W);
    1278       12474 :   GEN v = hash_to_vec(h);
    1279       12474 :   long i, l = lg(v);
    1280      352618 :   for (i = 1; i < l; i++)
    1281             :   {
    1282      340144 :     GEN e = gel(v,i);
    1283      340144 :     GEN M = path_to_zm(flag? gel(e,1): e);
    1284      340144 :     long index = p1_index(cc(M), dd(M), p1N);
    1285      340144 :     vecsmalltrunc_append(gel(W,2), index);
    1286      340144 :     gel(section, index) = M;
    1287             :   }
    1288       12474 : }
    1289             : static void
    1290        2079 : fill_W2_W12(GEN W, PS_sets_t *S)
    1291             : {
    1292        2079 :   GEN p1N = msN_get_p1N(W);
    1293        2079 :   long n = p1_size(p1N);
    1294        2079 :   gel(W, 2) = vecsmalltrunc_init(n+1);
    1295        2079 :   gel(W,12) = cgetg(n+1, t_VEC);
    1296             :   /* F contains [path, [index cusp1, index cusp2]]. Others contain paths only */
    1297        2079 :   paths_decompose(W, S->F, 1);
    1298        2079 :   paths_decompose(W, S->E2, 0);
    1299        2079 :   paths_decompose(W, S->T32, 0);
    1300        2079 :   paths_decompose(W, S->E1, 0);
    1301        2079 :   paths_decompose(W, S->T2, 0);
    1302        2079 :   paths_decompose(W, S->T31, 0);
    1303        2079 : }
    1304             : 
    1305             : /* x t_VECSMALL, corresponds to a map x(i) = j, where 1 <= j <= max for all i
    1306             :  * Return y s.t. y[j] = i or 0 (not in image) */
    1307             : static GEN
    1308        4158 : reverse_list(GEN x, long max)
    1309             : {
    1310        4158 :   GEN y = const_vecsmall(max, 0);
    1311        4158 :   long r, lx = lg(x);
    1312      403851 :   for (r = 1; r < lx; r++) y[ x[r] ] = r;
    1313        4158 :   return y;
    1314             : }
    1315             : 
    1316             : /* go from C[a] to C[b]; return the indices of paths
    1317             :  * E.g. if a < b
    1318             :  *   (C[a]->C[a+1], C[a+1]->C[a+2], ... C[b-1]->C[b])
    1319             :  * (else reverse direction)
    1320             :  * = b - a paths */
    1321             : static GEN
    1322      216818 : F_indices(GEN W, long a, long b)
    1323             : {
    1324      216818 :   GEN v = cgetg(labs(b-a) + 1, t_VEC);
    1325      216818 :   long s, k = 1;
    1326      216818 :   if (a < b) {
    1327      108409 :     GEN index_forward = gel(W,13);
    1328      841127 :     for (s = a; s < b; s++) gel(v,k++) = gel(index_forward,s);
    1329             :   } else {
    1330      108409 :     GEN index_backward = gel(W,14);
    1331      841127 :     for (s = a; s > b; s--) gel(v,k++) = gel(index_backward,s);
    1332             :   }
    1333      216818 :   return v;
    1334             : }
    1335             : /* go from C[a] to C[b] via oo; return the indices of paths
    1336             :  * E.g. if a < b
    1337             :  *   (C[a]->C[a-1], ... C[2]->C[1],
    1338             :  *    C[1]->oo, oo-> C[end],
    1339             :  *    C[end]->C[end-1], ... C[b+1]->C[b])
    1340             :  *  a-1 + 2 + end-(b+1)+1 = end - b + a + 1 paths  */
    1341             : static GEN
    1342        5194 : F_indices_oo(GEN W, long end, long a, long b)
    1343             : {
    1344        5194 :   GEN index_oo = gel(W,15);
    1345        5194 :   GEN v = cgetg(end-labs(b-a)+1 + 1, t_VEC);
    1346        5194 :   long s, k = 1;
    1347             : 
    1348        5194 :   if (a < b) {
    1349        2597 :     GEN index_backward = gel(W,14);
    1350        2597 :     for (s = a; s > 1; s--) gel(v,k++) = gel(index_backward,s);
    1351        2597 :     gel(v,k++) = gel(index_backward,1); /* C[1] -> oo */
    1352        2597 :     gel(v,k++) = gel(index_oo,2); /* oo -> C[end] */
    1353       45542 :     for (s = end; s > b; s--) gel(v,k++) = gel(index_backward,s);
    1354             :   } else {
    1355        2597 :     GEN index_forward = gel(W,13);
    1356       45542 :     for (s = a; s < end; s++) gel(v,k++) = gel(index_forward,s);
    1357        2597 :     gel(v,k++) = gel(index_forward,end); /* C[end] -> oo */
    1358        2597 :     gel(v,k++) = gel(index_oo,1); /* oo -> C[1] */
    1359        2597 :     for (s = 1; s < b; s++) gel(v,k++) = gel(index_forward,s);
    1360             :   }
    1361        5194 :   return v;
    1362             : }
    1363             : /* index of oo -> C[1], oo -> C[end] */
    1364             : static GEN
    1365        2079 : indices_oo(GEN W, GEN C)
    1366             : {
    1367        2079 :   long end = lg(C)-1;
    1368        2079 :   GEN w, v = cgetg(2+1, t_VEC), oo = cusp_infinity();
    1369        2079 :   w = mkpath(oo, gel(C,1)); /* oo -> C[1]=0 */
    1370        2079 :   gel(v,1) = path_Gamma0N_decompose(W, w);
    1371        2079 :   w = mkpath(oo, gel(C,end)); /* oo -> C[end]=1 */
    1372        2079 :   gel(v,2) = path_Gamma0N_decompose(W, w);
    1373        2079 :   return v;
    1374             : }
    1375             : 
    1376             : /* index of C[1]->C[2], C[2]->C[3], ... C[end-1]->C[end], C[end]->oo
    1377             :  * Recall that C[1] = 0, C[end] = 1 */
    1378             : static GEN
    1379        2079 : indices_forward(GEN W, GEN C)
    1380             : {
    1381        2079 :   long s, k = 1, end = lg(C)-1;
    1382        2079 :   GEN v = cgetg(end+1, t_VEC);
    1383      117243 :   for (s = 1; s <= end; s++)
    1384             :   {
    1385      115164 :     GEN w = mkpath(gel(C,s), s == end? cusp_infinity(): gel(C,s+1));
    1386      115164 :     gel(v,k++) = path_Gamma0N_decompose(W, w);
    1387             :   }
    1388        2079 :   return v;
    1389             : }
    1390             : /* index of C[1]->oo, C[2]->C[1], ... C[end]->C[end-1] */
    1391             : static GEN
    1392        2079 : indices_backward(GEN W, GEN C)
    1393             : {
    1394        2079 :   long s, k = 1, end = lg(C)-1;
    1395        2079 :   GEN v = cgetg(end+1, t_VEC);
    1396      117243 :   for (s = 1; s <= end; s++)
    1397             :   {
    1398      115164 :     GEN w = mkpath(gel(C,s), s == 1? cusp_infinity(): gel(C,s-1));
    1399      115164 :     gel(v,k++) = path_Gamma0N_decompose(W, w);
    1400             :   }
    1401        2079 :   return v;
    1402             : }
    1403             : 
    1404             : /*[0,-1;1,-1]*/
    1405             : static GEN
    1406        2163 : mkTAU()
    1407        2163 : { return mkmat22(gen_0,gen_m1, gen_1,gen_m1); }
    1408             : /* S */
    1409             : static GEN
    1410          84 : mkS()
    1411          84 : { return mkmat22(gen_0,gen_1, gen_m1,gen_0); }
    1412             : /* N = integer > 1. Returns data describing Delta_0 = Z[P^1(Q)]_0 seen as
    1413             :  * a Gamma_0(N) - module. */
    1414             : static GEN
    1415        2107 : msinit_N(ulong N)
    1416             : {
    1417             :   GEN p1N, C, vecF, vecT2, vecT31, TAU, W, W2, singlerel, annT2, annT31;
    1418             :   GEN F_index;
    1419             :   ulong r, s, width;
    1420             :   long nball, nbgen, nbp1N;
    1421             :   hashtable *F, *T2, *T31, *T32, *E1, *E2;
    1422             :   PS_sets_t S;
    1423             : 
    1424        2107 :   W = zerovec(16);
    1425        2107 :   gel(W,1) = p1N = create_p1mod(N);
    1426        2107 :   gel(W,16)= inithashcusps(p1N);
    1427        2107 :   TAU = mkTAU();
    1428        2107 :   if (N == 1)
    1429             :   {
    1430          28 :     gel(W,5) = mkvecsmall(1);
    1431             :     /* cheat because sets are not disjoint if N=1 */
    1432          28 :     gel(W,11) = mkvecsmall5(0, 0, 1, 1, 2);
    1433          28 :     gel(W,12) = mkvec(mat2(1,0,0,1));
    1434          28 :     gel(W,8) = mkvec( mkmat22(gen_1,gen_1, mkS(),gen_1) );
    1435          28 :     gel(W,9) = mkvec( mkmat2(mkcol3(gen_1,TAU,ZM_sqr(TAU)),
    1436             :                              mkcol3(gen_1,gen_1,gen_1)) );
    1437          28 :     return W;
    1438             :   }
    1439        2079 :   nbp1N = p1_size(p1N);
    1440        2079 :   form_E_F_T(N,p1N, &C, &S);
    1441        2079 :   E1  = S.E1;
    1442        2079 :   E2  = S.E2;
    1443        2079 :   T31 = S.T31;
    1444        2079 :   T32 = S.T32;
    1445        2079 :   F   = S.F;
    1446        2079 :   T2  = S.T2;
    1447        2079 :   nbgen = lg(C)-1;
    1448             : 
    1449             :  /* Put our paths in the order: F,E2,T32,E1,T2,T31
    1450             :   * W2[j] associates to the j-th element of this list its index in P1. */
    1451        2079 :   fill_W2_W12(W, &S);
    1452        2079 :   W2 = gel(W, 2);
    1453        2079 :   nball = lg(W2)-1;
    1454        2079 :   gel(W,3) = reverse_list(W2, nbp1N);
    1455        2079 :   gel(W,5) = vecslice(gel(W,2), F->nb + E2->nb + T32->nb + 1, nball);
    1456        2079 :   gel(W,4) = reverse_list(gel(W,5), nbp1N);
    1457        2079 :   gel(W,13) = indices_forward(W, C);
    1458        2079 :   gel(W,14) = indices_backward(W, C);
    1459        2079 :   gel(W,15) = indices_oo(W, C);
    1460       10395 :   gel(W,11) = mkvecsmall5(F->nb,
    1461        2079 :                           F->nb + E2->nb,
    1462        2079 :                           F->nb + E2->nb + T32->nb,
    1463        2079 :                           F->nb + E2->nb + T32->nb + E1->nb,
    1464        2079 :                           F->nb + E2->nb + T32->nb + E1->nb + T2->nb);
    1465             :   /* relations between T32 and T31 [not stored!]
    1466             :    * T32[i] = - T31[i] */
    1467             : 
    1468             :   /* relations of F */
    1469        2079 :   width = E1->nb + T2->nb + T31->nb;
    1470             :   /* F_index[r] = [index_1, ..., index_k], where index_i is the p1_index()
    1471             :    * of the elementary unimodular path between 2 consecutive cusps
    1472             :    * [in E1,E2,T2,T31 or T32] */
    1473        2079 :   F_index = cgetg(F->nb+1, t_VEC);
    1474        2079 :   vecF = hash_to_vec(F);
    1475      224091 :   for (r = 1; r <= F->nb; r++)
    1476             :   {
    1477      222012 :     GEN w = gel(gel(vecF,r), 2);
    1478      222012 :     long a = w[1], b = w[2], d = labs(b - a);
    1479             :     /* c1 = cusp_list[a],  c2 = cusp_list[b], ci != oo */
    1480      222012 :     gel(F_index,r) = (nbgen-d >= d-1)? F_indices(W, a,b)
    1481      222012 :                                      : F_indices_oo(W, lg(C)-1,a,b);
    1482             :   }
    1483             : 
    1484        2079 :   singlerel = cgetg(width+1, t_VEC);
    1485             :   /* form the single boundary relation */
    1486       59773 :   for (s = 1; s <= E2->nb; s++)
    1487             :   { /* reverse(E2[s]) = gamma * E1[c] */
    1488       57694 :     GEN T = gel(S.E2fromE1,s), gamma = E2fromE1_gamma(T);
    1489       57694 :     gel(singlerel, E2fromE1_c(T)) = mkmat22(gen_1,gen_1, gamma,gen_m1);
    1490             :   }
    1491        3934 :   for (r = E1->nb + 1; r <= width; r++) gel(singlerel, r) = gen_1;
    1492             : 
    1493             :   /* form the 2-torsion relations */
    1494        2079 :   annT2 = cgetg(T2->nb+1, t_VEC);
    1495        2079 :   vecT2 = hash_to_vec(T2);
    1496        3045 :   for (r = 1; r <= T2->nb; r++)
    1497             :   {
    1498         966 :     GEN w = gel(vecT2,r);
    1499         966 :     GEN gamma = gamma_equiv_matrix(vecreverse(w), w);
    1500         966 :     gel(annT2, r) = mkmat22(gen_1,gen_1, gamma,gen_1);
    1501             :   }
    1502             : 
    1503             :   /* form the 3-torsion relations */
    1504        2079 :   annT31 = cgetg(T31->nb+1, t_VEC);
    1505        2079 :   vecT31 = hash_to_vec(T31);
    1506        2968 :   for (r = 1; r <= T31->nb; r++)
    1507             :   {
    1508         889 :     GEN M = path_to_ZM( vecreverse(gel(vecT31,r)) );
    1509         889 :     GEN gamma = ZM_mul(ZM_mul(M, TAU), SL2_inv_shallow(M));
    1510         889 :     gel(annT31, r) = mkmat2(mkcol3(gen_1,gamma,ZM_sqr(gamma)),
    1511             :                             mkcol3(gen_1,gen_1,gen_1));
    1512             :   }
    1513        2079 :   gel(W,6) = F_index;
    1514        2079 :   gel(W,7) = S.E2fromE1;
    1515        2079 :   gel(W,8) = annT2;
    1516        2079 :   gel(W,9) = annT31;
    1517        2079 :   gel(W,10)= singlerel;
    1518        2079 :   return W;
    1519             : }
    1520             : static GEN
    1521         112 : cusp_to_P1Q(GEN c) { return c[2]? sstoQ(c[1], c[2]): mkoo(); }
    1522             : static GEN
    1523          21 : mspathgens_i(GEN W)
    1524             : {
    1525             :   GEN R, r, g, section, gen, annT2, annT31;
    1526             :   long i, l;
    1527          21 :   checkms(W); W = get_msN(W);
    1528          21 :   section = msN_get_section(W);
    1529          21 :   gen = ms_get_genindex(W);
    1530          21 :   l = lg(gen);
    1531          21 :   g = cgetg(l,t_VEC);
    1532          77 :   for (i = 1; i < l; i++)
    1533             :   {
    1534          56 :     GEN p = gel(section,gen[i]);
    1535          56 :     gel(g,i) = mkvec2(cusp_to_P1Q(gel(p,1)), cusp_to_P1Q(gel(p,2)));
    1536             :   }
    1537          21 :   annT2 = msN_get_annT2(W);
    1538          21 :   annT31= msN_get_annT31(W);
    1539          21 :   if (ms_get_N(W) == 1)
    1540             :   {
    1541           7 :     R = cgetg(3, t_VEC);
    1542           7 :     gel(R,1) = mkvec( mkvec2(gel(annT2,1), gen_1) );
    1543           7 :     gel(R,2) = mkvec( mkvec2(gel(annT31,1), gen_1) );
    1544             :   }
    1545             :   else
    1546             :   {
    1547          14 :     GEN singlerel = msN_get_singlerel(W);
    1548          14 :     long j, nbT2 = lg(annT2)-1, nbT31 = lg(annT31)-1, nbE1 = ms_get_nbE1(W);
    1549          14 :     R = cgetg(nbT2+nbT31+2, t_VEC);
    1550          14 :     l = lg(singlerel);
    1551          14 :     r = cgetg(l, t_VEC);
    1552          42 :     for (i = 1; i <= nbE1; i++)
    1553          28 :       gel(r,i) = mkvec2(gel(singlerel, i), utoi(i));
    1554          35 :     for (; i < l; i++)
    1555          21 :       gel(r,i) = mkvec2(gen_1, utoi(i));
    1556          14 :     gel(R,1) = r; j = 2;
    1557          35 :     for (i = 1; i <= nbT2; i++,j++)
    1558          21 :       gel(R,j) = mkvec( mkvec2(gel(annT2,i), utoi(i + nbE1)) );
    1559          14 :     for (i = 1; i <= nbT31; i++,j++)
    1560           0 :       gel(R,j) = mkvec( mkvec2(gel(annT31,i), utoi(i + nbE1 + nbT2)) );
    1561             :   }
    1562          21 :   return mkvec2(g,R);
    1563             : }
    1564             : GEN
    1565          21 : mspathgens(GEN W)
    1566             : {
    1567          21 :   pari_sp av = avma;
    1568          21 :   return gerepilecopy(av, mspathgens_i(W));
    1569             : }
    1570             : /* Modular symbols in weight k: Hom_Gamma(Delta, Q[x,y]_{k-2}) */
    1571             : /* A symbol phi is represented by the {phi(g_i)}, {phi(g'_i)}, {phi(g''_i)}
    1572             :  * where the {g_i, g'_i, g''_i} are the Z[\Gamma]-generators of Delta,
    1573             :  * g_i corresponds to E1, g'_i to T2, g''_i to T31.
    1574             :  */
    1575             : 
    1576             : /* FIXME: export. T^1, ..., T^n */
    1577             : static GEN
    1578      701834 : RgX_powers(GEN T, long n)
    1579             : {
    1580      701834 :   GEN v = cgetg(n+1, t_VEC);
    1581             :   long i;
    1582      701834 :   gel(v, 1) = T;
    1583     1643600 :   for (i = 1; i < n; i++) gel(v,i+1) = RgX_mul(gel(v,i), T);
    1584      701834 :   return v;
    1585             : }
    1586             : 
    1587             : /* g = [a,b;c,d] a mat2. Return (X^{k-2} | g)(X,Y)[X = 1]. */
    1588             : static GEN
    1589        2604 : voo_act_Gl2Q(GEN g, long k)
    1590             : {
    1591        2604 :   GEN mc = stoi(-coeff(g,2,1)), d = stoi(coeff(g,2,2));
    1592        2604 :   return RgX_to_RgC(gpowgs(deg1pol_shallow(mc, d, 0), k-2), k-1);
    1593             : }
    1594             : 
    1595             : struct m_act {
    1596             :   long dim, k, p;
    1597             :   GEN q;
    1598             :   GEN(*act)(struct m_act *,GEN);
    1599             : };
    1600             : 
    1601             : /* g = [a,b;c,d]. Return (P | g)(X,Y)[X = 1] = P(dX - cY, -b X + aY)[X = 1],
    1602             :  * for P = X^{k-2}, X^{k-3}Y, ..., Y^{k-2} */
    1603             : GEN
    1604      350749 : RgX_act_Gl2Q(GEN g, long k)
    1605             : {
    1606             :   GEN a,b,c,d, V1,V2,V;
    1607             :   long i;
    1608      350749 :   if (k == 2) return matid(1);
    1609      350749 :   a = gcoeff(g,1,1); b = gcoeff(g,1,2);
    1610      350749 :   c = gcoeff(g,2,1); d = gcoeff(g,2,2);
    1611      350749 :   V1 = RgX_powers(deg1pol_shallow(gneg(c), d, 0), k-2); /* d - c Y */
    1612      350749 :   V2 = RgX_powers(deg1pol_shallow(a, gneg(b), 0), k-2); /*-b + a Y */
    1613      350749 :   V = cgetg(k, t_MAT);
    1614      350749 :   gel(V,1)   = RgX_to_RgC(gel(V1, k-2), k-1);
    1615      819280 :   for (i = 1; i < k-2; i++)
    1616             :   {
    1617      468531 :     GEN v1 = gel(V1, k-2-i); /* (d-cY)^(k-2-i) */
    1618      468531 :     GEN v2 = gel(V2, i); /* (-b+aY)^i */
    1619      468531 :     gel(V,i+1) = RgX_to_RgC(RgX_mul(v1,v2), k-1);
    1620             :   }
    1621      350749 :   gel(V,k-1) = RgX_to_RgC(gel(V2, k-2), k-1);
    1622      350749 :   return V; /* V[i+1] = X^i | g */
    1623             : }
    1624             : /* z in Z[Gl2(Q)], return the matrix of z acting on V */
    1625             : static GEN
    1626      600649 : act_ZGl2Q(GEN z, struct m_act *T, hashtable *H)
    1627             : {
    1628      600649 :   GEN S = NULL, G, E;
    1629             :   pari_sp av;
    1630             :   long l, j;
    1631             :   /* paranoia: should not occur */
    1632      600649 :   if (typ(z) == t_INT) return scalarmat_shallow(z, T->dim);
    1633      600649 :   G = gel(z,1); l = lg(G);
    1634      600649 :   E = gel(z,2); av = avma;
    1635     1770307 :   for (j = 1; j < l; j++)
    1636             :   {
    1637     1169658 :     GEN M, g = gel(G,j), n = gel(E,j);
    1638     1169658 :     if (typ(g) == t_INT) /* = 1 */
    1639        3948 :       M = n; /* n*Id_dim */
    1640             :     else
    1641             :     { /*search in H succeeds because of preload*/
    1642     1165710 :       M = H? (GEN)hash_search(H,g)->val: T->act(T,g);
    1643     1165710 :       if (is_pm1(n))
    1644     1158185 :       { if (signe(n) < 0) M = RgM_neg(M); }
    1645             :       else
    1646        7525 :         M = RgM_Rg_mul(M, n);
    1647             :     }
    1648     1169658 :     if (!S) { S = M; continue; }
    1649      569009 :     S = gadd(S, M);
    1650      569009 :     if (gc_needed(av,1))
    1651             :     {
    1652           0 :       if(DEBUGMEM>1) pari_warn(warnmem,"act_ZGl2Q, j = %ld",j);
    1653           0 :       S = gerepileupto(av, S);
    1654             :     }
    1655             :   }
    1656      600649 :   return gerepilecopy(av, S);
    1657             : }
    1658             : static GEN
    1659      350602 : _RgX_act_Gl2Q(struct m_act *S, GEN z) { return RgX_act_Gl2Q(z, S->k); }
    1660             : /* acting on (X^{k-2},...,Y^{k-2}) */
    1661             : GEN
    1662       60907 : RgX_act_ZGl2Q(GEN z, long k)
    1663             : {
    1664             :   struct m_act T;
    1665       60907 :   T.k = k;
    1666       60907 :   T.dim = k-1;
    1667       60907 :   T.act=&_RgX_act_Gl2Q;
    1668       60907 :   return act_ZGl2Q(z, &T, NULL);
    1669             : }
    1670             : 
    1671             : /* First pass, identify matrices in Sl_2 to convert to operators;
    1672             :  * insert operators in hashtable. This allows GC in act_ZGl2Q */
    1673             : static void
    1674     1069894 : hash_preload(GEN M, struct m_act *S, hashtable *H)
    1675             : {
    1676     1069894 :   if (typ(M) != t_INT)
    1677             :   {
    1678     1069894 :     ulong h = H->hash(M);
    1679     1069894 :     hashentry *e = hash_search2(H, M, h);
    1680     1069894 :     if (!e) hash_insert2(H, M, S->act(S,M), h);
    1681             :   }
    1682     1069894 : }
    1683             : /* z a sparse operator */
    1684             : static void
    1685      539728 : hash_vecpreload(GEN z, struct m_act *S, hashtable *H)
    1686             : {
    1687      539728 :   GEN G = gel(z,1);
    1688      539728 :   long i, l = lg(G);
    1689     1609622 :   for (i = 1; i < l; i++) hash_preload(gel(G,i), S, H);
    1690      539728 : }
    1691             : static void
    1692       40677 : ZGl2QC_preload(struct m_act *S, GEN v, hashtable *H)
    1693             : {
    1694       40677 :   GEN val = gel(v,2);
    1695       40677 :   long i, l = lg(val);
    1696      580405 :   for (i = 1; i < l; i++) hash_vecpreload(gel(val,i), S, H);
    1697       40677 : }
    1698             : /* Given a sparse vector of elements in Z[G], convert it to a (sparse) vector
    1699             :  * of operators on V (given by t_MAT) */
    1700             : static void
    1701       40691 : ZGl2QC_to_act(struct m_act *S, GEN v, hashtable *H)
    1702             : {
    1703       40691 :   GEN val = gel(v,2);
    1704       40691 :   long i, l = lg(val);
    1705      580433 :   for (i = 1; i < l; i++) gel(val,i) = act_ZGl2Q(gel(val,i), S, H);
    1706       40691 : }
    1707             : 
    1708             : /* For all V[i] in Z[\Gamma], find the P such that  P . V[i]^* = 0;
    1709             :  * write P in basis X^{k-2}, ..., Y^{k-2} */
    1710             : static GEN
    1711        1260 : ZGV_tors(GEN V, long k)
    1712             : {
    1713        1260 :   long i, l = lg(V);
    1714        1260 :   GEN v = cgetg(l, t_VEC);
    1715        1764 :   for (i = 1; i < l; i++)
    1716             :   {
    1717         504 :     GEN a = ZSl2_star(gel(V,i));
    1718         504 :     gel(v,i) = ZM_ker(RgX_act_ZGl2Q(a,k));
    1719             :   }
    1720        1260 :   return v;
    1721             : }
    1722             : 
    1723             : static long
    1724   110641321 : set_from_index(GEN W11, long i)
    1725             : {
    1726   110641321 :   if (i <= W11[1]) return 1;
    1727    96328113 :   if (i <= W11[2]) return 2;
    1728    56359394 :   if (i <= W11[3]) return 3;
    1729    56171556 :   if (i <= W11[4]) return 4;
    1730     2309825 :   if (i <= W11[5]) return 5;
    1731      255626 :   return 6;
    1732             : }
    1733             : 
    1734             : /* det M = 1 */
    1735             : static void
    1736     1535667 : treat_index(GEN W, GEN M, long index, GEN v)
    1737             : {
    1738     1535667 :   GEN W11 = gel(W,11);
    1739     1535667 :   long shift = W11[3]; /* #F + #E2 + T32 */
    1740     1535667 :   switch(set_from_index(W11, index))
    1741             :   {
    1742      251167 :     case 1: /*F*/
    1743             :     {
    1744      251167 :       GEN F_index = gel(W,6), ind = gel(F_index, index);
    1745      251167 :       long j, l = lg(ind);
    1746     1288371 :       for (j = 1; j < l; j++)
    1747             :       {
    1748     1037204 :         GEN IND = gel(ind,j), M0 = gel(IND,2);
    1749     1037204 :         long index = mael(IND,1,1);
    1750     1037204 :         treat_index(W, ZM_mul(M,M0), index, v);
    1751             :       }
    1752      251167 :       break;
    1753             :     }
    1754             : 
    1755      579957 :     case 2: /*E2, E2[r] + gamma * E1[s] = 0 */
    1756             :     {
    1757      579957 :       long r = index - W11[1];
    1758      579957 :       GEN z = gel(msN_get_E2fromE1(W), r);
    1759             : 
    1760      579957 :       index = E2fromE1_c(z);
    1761      579957 :       M = G_ZG_mul(M, E2fromE1_Zgamma(z)); /* M * (-gamma) */
    1762      579957 :       gel(v, index) = ZG_add(gel(v, index), M);
    1763      579957 :       break;
    1764             :     }
    1765             : 
    1766        5922 :     case 3: /*T32, T32[i] = -T31[i] */
    1767             :     {
    1768        5922 :       long T3shift = W11[5] - W11[2]; /* #T32 + #E1 + #T2 */
    1769        5922 :       index += T3shift;
    1770        5922 :       index -= shift;
    1771        5922 :       gel(v, index) = ZG_add(gel(v, index), to_famat_shallow(M,gen_m1));
    1772        5922 :       break;
    1773             :     }
    1774      698621 :     default: /*E1,T2,T31*/
    1775      698621 :       index -= shift;
    1776      698621 :       gel(v, index) = ZG_add(gel(v, index), to_famat_shallow(M,gen_1));
    1777      698621 :       break;
    1778             :   }
    1779     1535667 : }
    1780             : static void
    1781   109105654 : treat_index_trivial(GEN v, GEN W, long index)
    1782             : {
    1783   109105654 :   GEN W11 = gel(W,11);
    1784   109105654 :   long shift = W11[3]; /* #F + #E2 + T32 */
    1785   109105654 :   switch(set_from_index(W11, index))
    1786             :   {
    1787    14062041 :     case 1: /*F*/
    1788             :     {
    1789    14062041 :       GEN F_index = gel(W,6), ind = gel(F_index, index);
    1790    14062041 :       long j, l = lg(ind);
    1791    99889426 :       for (j = 1; j < l; j++)
    1792             :       {
    1793    85827385 :         GEN IND = gel(ind,j);
    1794    85827385 :         treat_index_trivial(v, W, mael(IND,1,1));
    1795             :       }
    1796    14062041 :       break;
    1797             :     }
    1798             : 
    1799    39388762 :     case 2: /*E2, E2[r] + gamma * E1[s] = 0 */
    1800             :     {
    1801    39388762 :       long r = index - W11[1];
    1802    39388762 :       long s = E2fromE1_c(gel(msN_get_E2fromE1(W), r));
    1803    39388762 :       v[s]--;
    1804    39388762 :       break;
    1805             :     }
    1806             : 
    1807     2473205 :     case 3: case 5: case 6: /*T32,T2,T31*/
    1808     2473205 :       break;
    1809             : 
    1810    53181646 :     case 4: /*E1*/
    1811    53181646 :       v[index-shift]++;
    1812    53181646 :       break;
    1813             :   }
    1814   109105654 : }
    1815             : 
    1816             : static GEN
    1817      178213 : M2_log(GEN W, GEN M)
    1818             : {
    1819      178213 :   GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
    1820      178213 :   GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
    1821             :   GEN  u, v, D, V;
    1822             :   long index, s;
    1823             : 
    1824      178213 :   W = get_msN(W);
    1825      178213 :   V = zerovec(ms_get_nbgen(W));
    1826             : 
    1827      178213 :   D = subii(mulii(a,d), mulii(b,c));
    1828      178213 :   s = signe(D);
    1829      178213 :   if (!s) return V;
    1830      176869 :   if (is_pm1(D))
    1831             :   { /* shortcut, no need to apply Manin's trick */
    1832       63399 :     if (s < 0) { b = negi(b); d = negi(d); }
    1833       63399 :     M = Gamma0N_decompose(W, mkmat22(a,b, c,d), &index);
    1834       63399 :     treat_index(W, M, index, V);
    1835             :   }
    1836             :   else
    1837             :   {
    1838             :     GEN U, B, P, Q, PQ, C1,C2;
    1839             :     long i, l;
    1840      113470 :     (void)bezout(a,c,&u,&v);
    1841      113470 :     B = addii(mulii(b,u), mulii(d,v));
    1842             :     /* [u,v;-c,a] [a,b; c,d] = [1,B; 0,D], i.e. M = U [1,B;0,D] */
    1843      113470 :     U = mkmat22(a,negi(v), c,u);
    1844             : 
    1845             :     /* {1/0 -> B/D} as \sum g_i, g_i unimodular paths */
    1846      113470 :     PQ = ZV_allpnqn( gboundcf(gdiv(B,D), 0) );
    1847      113470 :     P = gel(PQ,1); l = lg(P);
    1848      113470 :     Q = gel(PQ,2);
    1849      113470 :     C1 = gel(U,1);
    1850      548534 :     for (i = 1; i < l; i++, C1 = C2)
    1851             :     {
    1852             :       GEN M;
    1853      435064 :       C2 = ZM_ZC_mul(U, mkcol2(gel(P,i), gel(Q,i)));
    1854      435064 :       if (!odd(i)) C1 = ZC_neg(C1);
    1855      435064 :       M = Gamma0N_decompose(W, mkmat2(C1,C2), &index);
    1856      435064 :       treat_index(W, M, index, V);
    1857             :     }
    1858             :   }
    1859      176869 :   return V;
    1860             : }
    1861             : 
    1862             : /* express +oo->q=a/b in terms of the Z[G]-generators, trivial action */
    1863             : static void
    1864     2221884 : Q_log_trivial(GEN v, GEN W, GEN q)
    1865             : {
    1866     2221884 :   GEN Q, W3 = gel(W,3), p1N = msN_get_p1N(W);
    1867     2221884 :   ulong c,d, N = p1N_get_N(p1N);
    1868             :   long i, lx;
    1869             : 
    1870     2221884 :   Q = Q_log_init(N, q);
    1871     2221884 :   lx = lg(Q);
    1872     2221884 :   c = 0;
    1873    23542008 :   for (i = 1; i < lx; i++, c = d)
    1874             :   {
    1875             :     long index;
    1876    21320124 :     d = Q[i];
    1877    21320124 :     if (c && !odd(i)) c = N - c;
    1878    21320124 :     index = W3[ p1_index(c,d,p1N) ];
    1879    21320124 :     treat_index_trivial(v, W, index);
    1880             :   }
    1881     2221884 : }
    1882             : static void
    1883      804069 : M2_log_trivial(GEN V, GEN W, GEN M)
    1884             : {
    1885      804069 :   GEN p1N = gel(W,1), W3 = gel(W,3);
    1886      804069 :   ulong N = p1N_get_N(p1N);
    1887      804069 :   GEN a = gcoeff(M,1,1), b = gcoeff(M,1,2);
    1888      804069 :   GEN c = gcoeff(M,2,1), d = gcoeff(M,2,2);
    1889             :   GEN  u, v, D;
    1890             :   long index, s;
    1891             : 
    1892      804069 :   D = subii(mulii(a,d), mulii(b,c));
    1893      804069 :   s = signe(D);
    1894      811293 :   if (!s) return;
    1895      804062 :   if (is_pm1(D))
    1896             :   { /* shortcut, not need to apply Manin's trick */
    1897      301000 :     if (s < 0) d = negi(d);
    1898      301000 :     index = W3[ p1_index(umodiu(c,N),umodiu(d,N),p1N) ];
    1899      301000 :     treat_index_trivial(V, W, index);
    1900             :   }
    1901             :   else
    1902             :   {
    1903             :     GEN U, B, P, Q, PQ;
    1904             :     long i, l;
    1905      503062 :     if (!signe(c)) { Q_log_trivial(V,W,gdiv(b,d)); return; }
    1906      495838 :     (void)bezout(a,c,&u,&v);
    1907      495838 :     B = addii(mulii(b,u), mulii(d,v));
    1908             :     /* [u,v;-c,a] [a,b; c,d] = [1,B; 0,D], i.e. M = U [1,B;0,D] */
    1909      495838 :     U = mkvec2(c, u);
    1910             : 
    1911             :     /* {1/0 -> B/D} as \sum g_i, g_i unimodular paths */
    1912      495838 :     PQ = ZV_allpnqn( gboundcf(gdiv(B,D), 0) );
    1913      495838 :     P = gel(PQ,1); l = lg(P);
    1914      495838 :     Q = gel(PQ,2);
    1915     2152983 :     for (i = 1; i < l; i++, c = d)
    1916             :     {
    1917     1657145 :       d = addii(mulii(gel(U,1),gel(P,i)), mulii(gel(U,2),gel(Q,i)));
    1918     1657145 :       if (!odd(i)) c = negi(c);
    1919     1657145 :       index = W3[ p1_index(umodiu(c,N),umodiu(d,N),p1N) ];
    1920     1657145 :       treat_index_trivial(V, W, index);
    1921             :     }
    1922             :   }
    1923             : }
    1924             : 
    1925             : static GEN
    1926       16772 : cusp_to_ZC(GEN c)
    1927             : {
    1928       16772 :   switch(typ(c))
    1929             :   {
    1930          35 :     case t_INFINITY:
    1931          35 :       return mkcol2(gen_1,gen_0);
    1932          91 :     case t_INT:
    1933          91 :       return mkcol2(c,gen_1);
    1934         140 :     case t_FRAC:
    1935         140 :       return mkcol2(gel(c,1),gel(c,2));
    1936       16506 :     case t_VECSMALL:
    1937       16506 :       return mkcol2(stoi(c[1]), stoi(c[2]));
    1938           0 :     default:
    1939           0 :       pari_err_TYPE("mspathlog",c);
    1940             :       return NULL;/*LCOV_EXCL_LINE*/
    1941             :   }
    1942             : }
    1943             : static GEN
    1944        8386 : path2_to_M2(GEN p)
    1945        8386 : { return mkmat2(cusp_to_ZC(gel(p,1)), cusp_to_ZC(gel(p,2))); }
    1946             : static GEN
    1947       16149 : path_to_M2(GEN p)
    1948             : {
    1949       16149 :   if (lg(p) != 3) pari_err_TYPE("mspathlog",p);
    1950       16142 :   switch(typ(p))
    1951             :   {
    1952        9870 :     case t_MAT:
    1953        9870 :       RgM_check_ZM(p,"mspathlog");
    1954        9870 :       break;
    1955        6272 :     case t_VEC:
    1956        6272 :       p = path2_to_M2(p);
    1957        6272 :       break;
    1958           0 :     default: pari_err_TYPE("mspathlog",p);
    1959             :   }
    1960       16142 :   return p;
    1961             : }
    1962             : /* Expresses path p as \sum x_i g_i, where the g_i are our distinguished
    1963             :  * generators and x_i \in Z[\Gamma]. Returns [x_1,...,x_n] */
    1964             : GEN
    1965       12691 : mspathlog(GEN W, GEN p)
    1966             : {
    1967       12691 :   pari_sp av = avma;
    1968       12691 :   checkms(W);
    1969       12691 :   return gerepilecopy(av, M2_log(W, path_to_M2(p)));
    1970             : }
    1971             : 
    1972             : /** HECKE OPERATORS **/
    1973             : /* [a,b;c,d] * cusp */
    1974             : static GEN
    1975     1922172 : cusp_mul(long a, long b, long c, long d, GEN cusp)
    1976             : {
    1977     1922172 :   long x = cusp[1], y = cusp[2];
    1978     1922172 :   long A = a*x+b*y, B = c*x+d*y, u = cgcd(A,B);
    1979     1922172 :   if (u != 1) { A /= u; B /= u; }
    1980     1922172 :   return mkcol2s(A, B);
    1981             : }
    1982             : /* f in Gl2(Q), act on path (zm), return path_to_M2(f.path) */
    1983             : static GEN
    1984      961086 : Gl2Q_act_path(GEN f, GEN path)
    1985             : {
    1986      961086 :   long a = coeff(f,1,1), b = coeff(f,1,2);
    1987      961086 :   long c = coeff(f,2,1), d = coeff(f,2,2);
    1988      961086 :   GEN c1 = cusp_mul(a,b,c,d, gel(path,1));
    1989      961086 :   GEN c2 = cusp_mul(a,b,c,d, gel(path,2));
    1990      961086 :   return mkmat2(c1,c2);
    1991             : }
    1992             : 
    1993             : static GEN
    1994     2417037 : init_act_trivial(GEN W) { return const_vecsmall(ms_get_nbE1(W), 0); }
    1995             : static GEN
    1996        3444 : mspathlog_trivial(GEN W, GEN p)
    1997             : {
    1998             :   GEN v;
    1999        3444 :   W = get_msN(W);
    2000        3444 :   v = init_act_trivial(W);
    2001        3444 :   M2_log_trivial(v, W, path_to_M2(p));
    2002        3437 :   return v;
    2003             : }
    2004             : 
    2005             : /* map from W1=Hom(Delta_0(N1),Q) -> W2=Hom(Delta_0(N2),Q), weight 2,
    2006             :  * trivial action. v a t_VEC of Gl2_Q (\sum v[i] in Z[Gl2(Q)]).
    2007             :  * Return the matrix attached to the action of v. */
    2008             : static GEN
    2009        3668 : getMorphism_trivial(GEN WW1, GEN WW2, GEN v)
    2010             : {
    2011        3668 :   GEN T, section, gen, W1 = get_msN(WW1), W2 = get_msN(WW2);
    2012             :   long j, lv, d2;
    2013        3668 :   if (ms_get_N(W1) == 1) return cgetg(1,t_MAT);
    2014        3668 :   if (ms_get_N(W2) == 1) return zeromat(0, ms_get_nbE1(W1));
    2015        3668 :   section = msN_get_section(W2);
    2016        3668 :   gen = msN_get_genindex(W2);
    2017        3668 :   d2 = ms_get_nbE1(W2);
    2018        3668 :   T = cgetg(d2+1, t_MAT);
    2019        3668 :   lv = lg(v);
    2020      202601 :   for (j = 1; j <= d2; j++)
    2021             :   {
    2022      198933 :     GEN w = gel(section, gen[j]);
    2023      198933 :     GEN t = init_act_trivial(W1);
    2024      198933 :     pari_sp av = avma;
    2025             :     long l;
    2026      999565 :     for (l = 1; l < lv; l++) M2_log_trivial(t, W1, Gl2Q_act_path(gel(v,l), w));
    2027      198933 :     gel(T,j) = t; set_avma(av);
    2028             :   }
    2029        3668 :   return shallowtrans(zm_to_ZM(T));
    2030             : }
    2031             : 
    2032             : static GEN
    2033      165522 : RgV_sparse(GEN v, GEN *pind)
    2034             : {
    2035             :   long i, l, k;
    2036      165522 :   GEN w = cgetg_copy(v,&l), ind = cgetg(l, t_VECSMALL);
    2037    17143112 :   for (i = k = 1; i < l; i++)
    2038             :   {
    2039    16977590 :     GEN c = gel(v,i);
    2040    16977590 :     if (typ(c) == t_INT) continue;
    2041      784798 :     gel(w,k) = c; ind[k] = i; k++;
    2042             :   }
    2043      165522 :   setlg(w,k); setlg(ind,k);
    2044      165522 :   *pind = ind; return w;
    2045             : }
    2046             : 
    2047             : static int
    2048      162568 : mat2_isidentity(GEN M)
    2049             : {
    2050      162568 :   GEN A = gel(M,1), B = gel(M,2);
    2051      162568 :   return A[1] == 1 && A[2] == 0 && B[1] == 0 && B[2] == 1;
    2052             : }
    2053             : /* path a mat22/mat22s, return log(f.path)^* . f in sparse form */
    2054             : static GEN
    2055      165522 : M2_logf(GEN Wp, GEN path, GEN f)
    2056             : {
    2057      165522 :   pari_sp av = avma;
    2058             :   GEN ind, L;
    2059             :   long i, l;
    2060      165522 :   if (f)
    2061      160454 :     path = Gl2Q_act_path(f, path);
    2062        5068 :   else if (typ(gel(path,1)) == t_VECSMALL)
    2063        2114 :     path = path2_to_M2(path);
    2064      165522 :   L = M2_log(Wp, path);
    2065      165522 :   L = RgV_sparse(L,&ind); l = lg(L);
    2066      950320 :   for (i = 1; i < l; i++) gel(L,i) = ZSl2_star(gel(L,i));
    2067      165522 :   if (f) ZGC_G_mul_inplace(L, mat2_to_ZM(f));
    2068      165522 :   return gerepilecopy(av, mkvec2(ind,L));
    2069             : }
    2070             : 
    2071             : static hashtable *
    2072        3668 : Gl2act_cache(long dim) { return set_init(dim*10); }
    2073             : 
    2074             : /* f zm/ZM in Gl_2(Q), acts from the left on Delta, which is generated by
    2075             :  * (g_i) as Z[Gamma1]-module, and by (G_i) as Z[Gamma2]-module.
    2076             :  * We have f.G_j = \sum_i \lambda_{i,j} g_i,   \lambda_{i,j} in Z[Gamma1]
    2077             :  * For phi in Hom_Gamma1(D,V), g in D, phi | f is in Hom_Gamma2(D,V) and
    2078             :  *  (phi | f)(G_j) = phi(f.G_j) | f
    2079             :  *                 = phi( \sum_i \lambda_{i,j} g_i ) | f
    2080             :  *                 = \sum_i phi(g_i) | (\lambda_{i,j}^* f)
    2081             :  *                 = \sum_i phi(g_i) | \mu_{i,j}(f)
    2082             :  * More generally
    2083             :  *  (\sum_k (phi |v_k))(G_j) = \sum_i phi(g_i) | \Mu_{i,j}
    2084             :  * with \Mu_{i,j} = \sum_k \mu{i,j}(v_k)
    2085             :  * Return the \Mu_{i,j} matrix as vector of sparse columns of operators on V */
    2086             : static GEN
    2087        3192 : init_dual_act(GEN v, GEN W1, GEN W2, struct m_act *S)
    2088             : {
    2089        3192 :   GEN section = ms_get_section(W2), gen = ms_get_genindex(W2);
    2090             :   /* HACK: the actions we consider in dimension 1 are trivial and in
    2091             :    * characteristic != 2, 3 => torsion generators are 0
    2092             :    * [satisfy e.g. (1+gamma).g = 0 => \phi(g) | 1+gamma  = 0 => \phi(g) = 0 */
    2093        3192 :   long j, lv = lg(v), dim = S->dim == 1? ms_get_nbE1(W2): lg(gen)-1;
    2094        3192 :   GEN T = cgetg(dim+1, t_VEC);
    2095        3192 :   hashtable *H = Gl2act_cache(dim);
    2096       40929 :   for (j = 1; j <= dim; j++)
    2097             :   {
    2098       37737 :     pari_sp av = avma;
    2099       37737 :     GEN w = gel(section, gen[j]); /* path_to_zm( E1/T2/T3 element ) */
    2100       37737 :     GEN t = NULL;
    2101             :     long k;
    2102      200305 :     for (k = 1; k < lv; k++)
    2103             :     {
    2104      162568 :       GEN tk, f = gel(v,k);
    2105      162568 :       if (typ(gel(f,1)) != t_VECSMALL) f = ZM_to_zm(f);
    2106      162568 :       if (mat2_isidentity(f)) f = NULL;
    2107      162568 :       tk = M2_logf(W1, w, f); /* mu_{.,j}(v[k]) as sparse vector */
    2108      162568 :       t = t? ZGCs_add(t, tk): tk;
    2109             :     }
    2110       37737 :     gel(T,j) = gerepilecopy(av, t);
    2111             :   }
    2112       40929 :   for (j = 1; j <= dim; j++)
    2113             :   {
    2114       37737 :     ZGl2QC_preload(S, gel(T,j), H);
    2115       37737 :     ZGl2QC_to_act(S, gel(T,j), H);
    2116             :   }
    2117        3192 :   return T;
    2118             : }
    2119             : 
    2120             : /* modular symbol given by phi[j] = \phi(G_j)
    2121             :  * \sum L[i]*phi[i], L a sparse column of operators */
    2122             : static GEN
    2123      354354 : dense_act_col(GEN col, GEN phi)
    2124             : {
    2125      354354 :   GEN s = NULL, colind = gel(col,1), colval = gel(col,2);
    2126      354354 :   long i, l = lg(colind), lphi = lg(phi);
    2127     5630121 :   for (i = 1; i < l; i++)
    2128             :   {
    2129     5278490 :     long a = colind[i];
    2130             :     GEN t;
    2131     5278490 :     if (a >= lphi) break; /* happens if k=2: torsion generator t omitted */
    2132     5275767 :     t = gel(phi, a); /* phi(G_a) */
    2133     5275767 :     t = RgM_RgC_mul(gel(colval,i), t);
    2134     5275767 :     s = s? RgC_add(s, t): t;
    2135             :   }
    2136      354354 :   return s;
    2137             : }
    2138             : /* modular symbol given by \phi( G[ind[j]] ) = val[j]
    2139             :  * \sum L[i]*phi[i], L a sparse column of operators */
    2140             : static GEN
    2141      779093 : sparse_act_col(GEN col, GEN phi)
    2142             : {
    2143      779093 :   GEN s = NULL, colind = gel(col,1), colval = gel(col,2);
    2144      779093 :   GEN ind = gel(phi,2), val = gel(phi,3);
    2145      779093 :   long a, l = lg(ind);
    2146      779093 :   if (lg(gel(phi,1)) == 1) return RgM_RgC_mul(gel(colval,1), gel(val,1));
    2147     3033205 :   for (a = 1; a < l; a++)
    2148             :   {
    2149     2254413 :     GEN t = gel(val, a); /* phi(G_i) */
    2150     2254413 :     long i = zv_search(colind, ind[a]);
    2151     2254413 :     if (!i) continue;
    2152      540603 :     t = RgM_RgC_mul(gel(colval,i), t);
    2153      540603 :     s = s? RgC_add(s, t): t;
    2154             :   }
    2155      778792 :   return s;
    2156             : }
    2157             : static int
    2158       69139 : phi_sparse(GEN phi) { return typ(gel(phi,1)) == t_VECSMALL; }
    2159             : /* phi in Hom_Gamma1(Delta, V), return the matrix whose colums are the
    2160             :  *   \sum_i phi(g_i) | \mu_{i,j} = (phi|f)(G_j),
    2161             :  * see init_dual_act. */
    2162             : static GEN
    2163       69139 : dual_act(long dimV, GEN act, GEN phi)
    2164             : {
    2165       69139 :   long l = lg(act), j;
    2166       69139 :   GEN v = cgetg(l, t_MAT);
    2167       69139 :   GEN (*ACT)(GEN,GEN) = phi_sparse(phi)? sparse_act_col: dense_act_col;
    2168     1199254 :   for (j = 1; j < l; j++)
    2169             :   {
    2170     1130115 :     pari_sp av = avma;
    2171     1130115 :     GEN s = ACT(gel(act,j), phi);
    2172     1130115 :     gel(v,j) = s? gerepileupto(av,s): zerocol(dimV);
    2173             :   }
    2174       69139 :   return v;
    2175             : }
    2176             : 
    2177             : /* in level N > 1 */
    2178             : static void
    2179       59087 : msk_get_st(GEN W, long *s, long *t)
    2180       59087 : { GEN st = gmael(W,3,3); *s = st[1]; *t = st[2]; }
    2181             : static GEN
    2182       59087 : msk_get_link(GEN W) { return gmael(W,3,4); }
    2183             : static GEN
    2184       59402 : msk_get_inv(GEN W) { return gmael(W,3,5); }
    2185             : /* \phi in Hom(Delta, V), \phi(G_k) = phi[k]. Write \phi as
    2186             :  *   \sum_{i,j} mu_{i,j} phi_{i,j}, mu_{i,j} in Q */
    2187             : static GEN
    2188       59402 : getMorphism_basis(GEN W, GEN phi)
    2189             : {
    2190       59402 :   GEN R, Q, Ls, T0, T1, Ts, link, basis, inv = msk_get_inv(W);
    2191             :   long i, j, r, s, t, dim, lvecT;
    2192             : 
    2193       59402 :   if (ms_get_N(W) == 1) return ZC_apply_dinv(inv, gel(phi,1));
    2194       59087 :   lvecT = lg(phi);
    2195       59087 :   basis = msk_get_basis(W);
    2196       59087 :   dim = lg(basis)-1;
    2197       59087 :   R = zerocol(dim);
    2198       59087 :   msk_get_st(W, &s, &t);
    2199       59087 :   link = msk_get_link(W);
    2200      789922 :   for (r = 2; r < lvecT; r++)
    2201             :   {
    2202             :     GEN Tr, L;
    2203      730835 :     if (r == s) continue;
    2204      671748 :     Tr = gel(phi,r); /* Phi(G_r), r != 1,s */
    2205      671748 :     L = gel(link, r);
    2206      671748 :     Q = ZC_apply_dinv(gel(inv,r), Tr);
    2207             :     /* write Phi(G_r) as sum_{a,b} mu_{a,b} Phi_{a,b}(G_r) */
    2208     3510668 :     for (j = 1; j < lg(L); j++) gel(R, L[j]) = gel(Q,j);
    2209             :   }
    2210       59087 :   Ls = gel(link, s);
    2211       59087 :   T1 = gel(phi,1); /* Phi(G_1) */
    2212       59087 :   gel(R, Ls[t]) = gel(T1, 1);
    2213             : 
    2214       59087 :   T0 = NULL;
    2215      789922 :   for (i = 2; i < lg(link); i++)
    2216             :   {
    2217             :     GEN L;
    2218      730835 :     if (i == s) continue;
    2219      671748 :     L = gel(link,i);
    2220     3510668 :     for (j =1 ; j < lg(L); j++)
    2221             :     {
    2222     2838920 :       long n = L[j]; /* phi_{i,j} = basis[n] */
    2223     2838920 :       GEN mu_ij = gel(R, n);
    2224     2838920 :       GEN phi_ij = gel(basis, n), pols = gel(phi_ij,3);
    2225     2838920 :       GEN z = RgC_Rg_mul(gel(pols, 3), mu_ij);
    2226     2838920 :       T0 = T0? RgC_add(T0, z): z; /* += mu_{i,j} Phi_{i,j} (G_s) */
    2227             :     }
    2228             :   }
    2229       59087 :   Ts = gel(phi,s); /* Phi(G_s) */
    2230       59087 :   if (T0) Ts = RgC_sub(Ts, T0);
    2231             :   /* solve \sum_{j!=t} mu_{s,j} Phi_{s,j}(G_s) = Ts */
    2232       59087 :   Q = ZC_apply_dinv(gel(inv,s), Ts);
    2233      234080 :   for (j = 1; j < t; j++) gel(R, Ls[j]) = gel(Q,j);
    2234             :   /* avoid mu_{s,t} */
    2235       59906 :   for (j = t; j < lg(Q); j++) gel(R, Ls[j+1]) = gel(Q,j);
    2236       59087 :   return R;
    2237             : }
    2238             : 
    2239             : /* a = s(g_i) for some modular symbol s; b in Z[G]
    2240             :  * return s(b.g_i) = b^* . s(g_i) */
    2241             : static GEN
    2242      115626 : ZGl2Q_act_s(GEN b, GEN a, long k)
    2243             : {
    2244      115626 :   if (typ(b) == t_INT)
    2245             :   {
    2246       58604 :     if (!signe(b)) return gen_0;
    2247          14 :     switch(typ(a))
    2248             :     {
    2249          14 :       case t_POL:
    2250          14 :         a = RgX_to_RgC(a, k-1); /*fall through*/
    2251          14 :       case t_COL:
    2252          14 :         a = RgC_Rg_mul(a,b);
    2253          14 :         break;
    2254           0 :       default: a = scalarcol_shallow(b,k-1);
    2255             :     }
    2256             :   }
    2257             :   else
    2258             :   {
    2259       57022 :     b = RgX_act_ZGl2Q(ZSl2_star(b), k);
    2260       57022 :     switch(typ(a))
    2261             :     {
    2262          63 :       case t_POL:
    2263          63 :         a = RgX_to_RgC(a, k-1); /*fall through*/
    2264       45262 :       case t_COL:
    2265       45262 :         a = RgM_RgC_mul(b,a);
    2266       45262 :         break;
    2267       11760 :       default: a = RgC_Rg_mul(gel(b,1),a);
    2268             :     }
    2269             :   }
    2270       57036 :   return a;
    2271             : }
    2272             : 
    2273             : static int
    2274          21 : checksymbol(GEN W, GEN s)
    2275             : {
    2276             :   GEN t, annT2, annT31, singlerel;
    2277             :   long i, k, l, nbE1, nbT2, nbT31;
    2278          21 :   k = msk_get_weight(W);
    2279          21 :   W = get_msN(W);
    2280          21 :   nbE1 = ms_get_nbE1(W);
    2281          21 :   singlerel = gel(W,10);
    2282          21 :   l = lg(singlerel);
    2283          21 :   if (k == 2)
    2284             :   {
    2285           0 :     for (i = nbE1+1; i < l; i++)
    2286           0 :       if (!gequal0(gel(s,i))) return 0;
    2287           0 :     return 1;
    2288             :   }
    2289          21 :   annT2 = msN_get_annT2(W); nbT2 = lg(annT2)-1;
    2290          21 :   annT31 = msN_get_annT31(W);nbT31 = lg(annT31)-1;
    2291          21 :   t = NULL;
    2292          84 :   for (i = 1; i < l; i++)
    2293             :   {
    2294          63 :     GEN a = gel(s,i);
    2295          63 :     a = ZGl2Q_act_s(gel(singlerel,i), a, k);
    2296          63 :     t = t? gadd(t, a): a;
    2297             :   }
    2298          21 :   if (!gequal0(t)) return 0;
    2299          14 :   for (i = 1; i <= nbT2; i++)
    2300             :   {
    2301           0 :     GEN a = gel(s,i + nbE1);
    2302           0 :     a = ZGl2Q_act_s(gel(annT2,i), a, k);
    2303           0 :     if (!gequal0(a)) return 0;
    2304             :   }
    2305          28 :   for (i = 1; i <= nbT31; i++)
    2306             :   {
    2307          14 :     GEN a = gel(s,i + nbE1 + nbT2);
    2308          14 :     a = ZGl2Q_act_s(gel(annT31,i), a, k);
    2309          14 :     if (!gequal0(a)) return 0;
    2310             :   }
    2311          14 :   return 1;
    2312             : }
    2313             : GEN
    2314          56 : msissymbol(GEN W, GEN s)
    2315             : {
    2316             :   long k, nbgen;
    2317          56 :   checkms(W);
    2318          56 :   k = msk_get_weight(W);
    2319          56 :   nbgen = ms_get_nbgen(W);
    2320          56 :   switch(typ(s))
    2321             :   {
    2322          21 :     case t_VEC: /* values s(g_i) */
    2323          21 :       if (lg(s)-1 != nbgen) return gen_0;
    2324          21 :       break;
    2325          28 :     case t_COL:
    2326          28 :       if (msk_get_sign(W))
    2327             :       {
    2328           0 :         GEN star = gel(msk_get_starproj(W), 1);
    2329           0 :         if (lg(star) == lg(s)) return gen_1;
    2330             :       }
    2331          28 :       if (k == 2) /* on the dual basis of (g_i) */
    2332             :       {
    2333           0 :         if (lg(s)-1 != nbgen) return gen_0;
    2334             :       }
    2335             :       else
    2336             :       {
    2337          28 :         GEN basis = msk_get_basis(W);
    2338          28 :         return (lg(s) == lg(basis))? gen_1: gen_0;
    2339             :       }
    2340           0 :       break;
    2341           7 :     case t_MAT:
    2342             :     {
    2343           7 :       long i, l = lg(s);
    2344           7 :       GEN v = cgetg(l, t_VEC);
    2345          21 :       for (i = 1; i < l; i++) gel(v,i) = msissymbol(W,gel(s,i))? gen_1: gen_0;
    2346           7 :       return v;
    2347             :     }
    2348           0 :     default: return gen_0;
    2349             :   }
    2350          21 :   return checksymbol(W,s)? gen_1: gen_0;
    2351             : }
    2352             : 
    2353             : /* map op: W1 = Hom(Delta_0(N1),V) -> W2 = Hom(Delta_0(N2),V), given by
    2354             :  * \sum v[i], v[i] in Gl2(Q) */
    2355             : static GEN
    2356        6377 : getMorphism(GEN W1, GEN W2, GEN v)
    2357             : {
    2358             :   struct m_act S;
    2359             :   GEN B1, M, act;
    2360        6377 :   long a, l, k = msk_get_weight(W1);
    2361        6377 :   if (k == 2) return getMorphism_trivial(W1,W2,v);
    2362        2709 :   S.k = k;
    2363        2709 :   S.dim = k-1;
    2364        2709 :   S.act = &_RgX_act_Gl2Q;
    2365        2709 :   act = init_dual_act(v,W1,W2,&S);
    2366        2709 :   B1 = msk_get_basis(W1);
    2367        2709 :   l = lg(B1); M = cgetg(l, t_MAT);
    2368       61180 :   for (a = 1; a < l; a++)
    2369             :   {
    2370       58471 :     pari_sp av = avma;
    2371       58471 :     GEN phi = dual_act(S.dim, act, gel(B1,a));
    2372       58471 :     GEN D = getMorphism_basis(W2, phi);
    2373       58471 :     gel(M,a) = gerepilecopy(av, D);
    2374             :   }
    2375        2709 :   return M;
    2376             : }
    2377             : static GEN
    2378        5215 : msendo(GEN W, GEN v) { return getMorphism(W, W, v); }
    2379             : 
    2380             : static GEN
    2381        2527 : endo_project(GEN W, GEN e, GEN H)
    2382             : {
    2383        2527 :   if (msk_get_sign(W)) e = Qevproj_apply(e, msk_get_starproj(W));
    2384        2527 :   if (H) e = Qevproj_apply(e, Qevproj_init0(H));
    2385        2527 :   return e;
    2386             : }
    2387             : static GEN
    2388        3381 : mshecke_i(GEN W, ulong p)
    2389             : {
    2390        3381 :   GEN v = ms_get_N(W) % p? Tp_matrices(p): Up_matrices(p);
    2391        3381 :   return msendo(W,v);
    2392             : }
    2393             : GEN
    2394        2478 : mshecke(GEN W, long p, GEN H)
    2395             : {
    2396        2478 :   pari_sp av = avma;
    2397             :   GEN T;
    2398        2478 :   checkms(W);
    2399        2478 :   if (p <= 1) pari_err_PRIME("mshecke",stoi(p));
    2400        2478 :   T = mshecke_i(W,p);
    2401        2478 :   T = endo_project(W,T,H);
    2402        2478 :   return gerepilecopy(av, T);
    2403             : }
    2404             : 
    2405             : static GEN
    2406          42 : msatkinlehner_i(GEN W, long Q)
    2407             : {
    2408          42 :   long N = ms_get_N(W);
    2409             :   GEN v;
    2410          42 :   if (Q == 1) return matid(msk_get_dim(W));
    2411          28 :   if (Q == N) return msendo(W, mkvec(mat2(0,1,-N,0)));
    2412          21 :   if (N % Q) pari_err_DOMAIN("msatkinlehner","N % Q","!=",gen_0,stoi(Q));
    2413          14 :   v = WQ_matrix(N, Q);
    2414          14 :   if (!v) pari_err_DOMAIN("msatkinlehner","gcd(Q,N/Q)","!=",gen_1,stoi(Q));
    2415          14 :   return msendo(W,mkvec(v));
    2416             : }
    2417             : GEN
    2418          42 : msatkinlehner(GEN W, long Q, GEN H)
    2419             : {
    2420          42 :   pari_sp av = avma;
    2421             :   GEN w;
    2422             :   long k;
    2423          42 :   checkms(W);
    2424          42 :   k = msk_get_weight(W);
    2425          42 :   if (Q <= 0) pari_err_DOMAIN("msatkinlehner","Q","<=",gen_0,stoi(Q));
    2426          42 :   w = msatkinlehner_i(W,Q);
    2427          35 :   w = endo_project(W,w,H);
    2428          35 :   if (k > 2 && Q != 1) w = RgM_Rg_div(w, powuu(Q,(k-2)>>1));
    2429          35 :   return gerepilecopy(av, w);
    2430             : }
    2431             : 
    2432             : static GEN
    2433        1813 : msstar_i(GEN W) { return msendo(W, mkvec(mat2(-1,0,0,1))); }
    2434             : GEN
    2435          14 : msstar(GEN W, GEN H)
    2436             : {
    2437          14 :   pari_sp av = avma;
    2438             :   GEN s;
    2439          14 :   checkms(W);
    2440          14 :   s = msstar_i(W);
    2441          14 :   s = endo_project(W,s,H);
    2442          14 :   return gerepilecopy(av, s);
    2443             : }
    2444             : 
    2445             : #if 0
    2446             : /* is \Gamma_0(N) cusp1 = \Gamma_0(N) cusp2 ? */
    2447             : static int
    2448             : iscuspeq(ulong N, GEN cusp1, GEN cusp2)
    2449             : {
    2450             :   long p1, q1, p2, q2, s1, s2, d;
    2451             :   p1 = cusp1[1]; p2 = cusp2[1];
    2452             :   q1 = cusp1[2]; q2 = cusp2[2];
    2453             :   d = Fl_mul(umodsu(q1,N),umodsu(q2,N), N);
    2454             :   d = ugcd(d, N);
    2455             : 
    2456             :   s1 = q1 > 2? Fl_inv(umodsu(p1,q1), q1): 1;
    2457             :   s2 = q2 > 2? Fl_inv(umodsu(p2,q2), q2): 1;
    2458             :   return Fl_mul(s1,q2,d) == Fl_mul(s2,q1,d);
    2459             : }
    2460             : #endif
    2461             : 
    2462             : /* return E_c(r) */
    2463             : static GEN
    2464        2604 : get_Ec_r(GEN c, long k)
    2465             : {
    2466        2604 :   long p = c[1], q = c[2], u, v;
    2467             :   GEN gr;
    2468        2604 :   (void)cbezout(p, q, &u, &v);
    2469        2604 :   gr = mat2(p, -v, q, u); /* g . (1:0) = (p:q) */
    2470        2604 :   return voo_act_Gl2Q(sl2_inv(gr), k);
    2471             : }
    2472             : /* N > 1; returns the modular symbol attached to the cusp c := p/q via the rule
    2473             :  * E_c(path from a to b in Delta_0) := E_c(b) - E_c(a), where
    2474             :  * E_c(r) := 0 if r != c mod Gamma
    2475             :  *           v_oo | gamma_r^(-1)
    2476             :  * where v_oo is stable by T = [1,1;0,1] (i.e x^(k-2)) and
    2477             :  * gamma_r . (1:0) = r, for some gamma_r in SL_2(Z) * */
    2478             : static GEN
    2479         462 : msfromcusp_trivial(GEN W, GEN c)
    2480             : {
    2481         462 :   GEN section = ms_get_section(W), gen = ms_get_genindex(W);
    2482         462 :   GEN S = ms_get_hashcusps(W);
    2483         462 :   long j, ic = cusp_index(c, S), l = ms_get_nbE1(W)+1;
    2484         462 :   GEN phi = cgetg(l, t_COL);
    2485       90356 :   for (j = 1; j < l; j++)
    2486             :   {
    2487       89894 :     GEN vj, g = gel(section, gen[j]); /* path_to_zm(generator) */
    2488       89894 :     GEN c1 = gel(g,1), c2 = gel(g,2);
    2489       89894 :     long i1 = cusp_index(c1, S);
    2490       89894 :     long i2 = cusp_index(c2, S);
    2491       89894 :     if (i1 == ic)
    2492        3290 :       vj = (i2 == ic)?  gen_0: gen_1;
    2493             :     else
    2494       86604 :       vj = (i2 == ic)? gen_m1: gen_0;
    2495       89894 :     gel(phi, j) = vj;
    2496             :   }
    2497         462 :   return phi;
    2498             : }
    2499             : static GEN
    2500        1393 : msfromcusp_i(GEN W, GEN c)
    2501             : {
    2502             :   GEN section, gen, S, phi;
    2503        1393 :   long j, ic, l, k = msk_get_weight(W);
    2504        1393 :   if (k == 2)
    2505             :   {
    2506         462 :     long N = ms_get_N(W);
    2507         462 :     return N == 1? cgetg(1,t_COL): msfromcusp_trivial(W, c);
    2508             :   }
    2509         931 :   k = msk_get_weight(W);
    2510         931 :   section = ms_get_section(W);
    2511         931 :   gen = ms_get_genindex(W);
    2512         931 :   S = ms_get_hashcusps(W);
    2513         931 :   ic = cusp_index(c, S);
    2514         931 :   l = lg(gen);
    2515         931 :   phi = cgetg(l, t_COL);
    2516       12075 :   for (j = 1; j < l; j++)
    2517             :   {
    2518       11144 :     GEN vj = NULL, g = gel(section, gen[j]); /* path_to_zm(generator) */
    2519       11144 :     GEN c1 = gel(g,1), c2 = gel(g,2);
    2520       11144 :     long i1 = cusp_index(c1, S);
    2521       11144 :     long i2 = cusp_index(c2, S);
    2522       11144 :     if (i1 == ic) vj = get_Ec_r(c1, k);
    2523       11144 :     if (i2 == ic)
    2524             :     {
    2525        1302 :       GEN s = get_Ec_r(c2, k);
    2526        1302 :       vj = vj? gsub(vj, s): gneg(s);
    2527             :     }
    2528       11144 :     if (!vj) vj = zerocol(k-1);
    2529       11144 :     gel(phi, j) = vj;
    2530             :   }
    2531         931 :   return getMorphism_basis(W, phi);
    2532             : }
    2533             : GEN
    2534          28 : msfromcusp(GEN W, GEN c)
    2535             : {
    2536          28 :   pari_sp av = avma;
    2537             :   long N;
    2538          28 :   checkms(W);
    2539          28 :   N = ms_get_N(W);
    2540          28 :   switch(typ(c))
    2541             :   {
    2542           7 :     case t_INFINITY:
    2543           7 :       c = mkvecsmall2(1,0);
    2544           7 :       break;
    2545          14 :     case t_INT:
    2546          14 :       c = mkvecsmall2(smodis(c,N), 1);
    2547          14 :       break;
    2548           7 :     case t_FRAC:
    2549           7 :       c = mkvecsmall2(smodis(gel(c,1),N), smodis(gel(c,2),N));
    2550           7 :       break;
    2551           0 :     default:
    2552           0 :       pari_err_TYPE("msfromcusp",c);
    2553             :   }
    2554          28 :   return gerepilecopy(av, msfromcusp_i(W,c));
    2555             : }
    2556             : 
    2557             : static GEN
    2558         287 : mseisenstein_i(GEN W)
    2559             : {
    2560         287 :   GEN M, S = ms_get_hashcusps(W), cusps = gel(S,3);
    2561         287 :   long i, l = lg(cusps);
    2562         287 :   if (msk_get_weight(W)==2) l--;
    2563         287 :   M = cgetg(l, t_MAT);
    2564        1652 :   for (i = 1; i < l; i++) gel(M,i) = msfromcusp_i(W, gel(cusps,i));
    2565         287 :   return Qevproj_init(Qevproj_star(W, QM_image_shallow(M)));
    2566             : }
    2567             : GEN
    2568          21 : mseisenstein(GEN W)
    2569             : {
    2570          21 :   pari_sp av = avma;
    2571          21 :   checkms(W); return gerepilecopy(av, mseisenstein_i(W));
    2572             : }
    2573             : 
    2574             : /* upper bound for log_2 |charpoly(T_p|S)|, where S is a cuspidal subspace of
    2575             :  * dimension d, k is the weight */
    2576             : #if 0
    2577             : static long
    2578             : TpS_char_bound(ulong p, long k, long d)
    2579             : { /* |eigenvalue| <= 2 p^(k-1)/2 */
    2580             :   return d * (2 + (log2((double)p)*(k-1))/2);
    2581             : }
    2582             : #endif
    2583             : static long
    2584         266 : TpE_char_bound(ulong p, long k, long d)
    2585             : { /* |eigenvalue| <= 2 p^(k-1) */
    2586         266 :   return d * (2 + log2((double)p)*(k-1));
    2587             : }
    2588             : 
    2589             : static GEN eisker(GEN M);
    2590             : static int
    2591         294 : use_Petersson(long N, long k, long s)
    2592             : {
    2593         294 :   if (!s)
    2594             :   {
    2595          70 :     if (N == 1)  return 1;
    2596          49 :     if (N <= 3)  return k >= 42;
    2597          42 :     if (N == 4)  return k >= 30;
    2598          42 :     if (N == 5)  return k >= 20;
    2599          42 :     if (N <= 10) return k >= 14;
    2600          35 :     if (N <= 16) return k >= 10;
    2601           7 :     if (N <= 28) return k >= 8;
    2602           7 :     if (N <= 136 || N == 180 || N == 200 || N == 225) return k >= 6;
    2603           0 :     return k >= 4;
    2604             :   }
    2605         224 :   if (s < 0)
    2606             :   {
    2607           0 :     if (N <= 64 || N == 100 || N == 128 || N == 144 || N == 225
    2608           0 :         || N == 351 || N == 375) return k >= 8;
    2609           0 :     return k >= 6;
    2610             :   }
    2611         224 :   if (N == 1) return 1;
    2612         217 :   if (N == 2) return k >= 56;
    2613         217 :   if (N == 3) return k >= 68;
    2614         182 :   if (N == 4) return k >= 78;
    2615         175 :   if (N == 5) return k >= 38;
    2616         147 :   if (N == 6) return k >= 24;
    2617         147 :   if (N == 7) return k >= 44;
    2618         140 :   if (N <= 9) return k >= 28;
    2619         133 :   if (N <= 13) return k >= 20;
    2620          98 :   if (N <= 21 || N == 50) return k >= 14;
    2621          70 :   if (N == 24 || N == 25) return k >= 16;
    2622          70 :   if (N <= 58 || N == 63 || N == 72 || N == 84 || N == 208 || N == 224) return k >= 10;
    2623          42 :   if (N <= 128 || N == 144 || N == 145 || N == 160 || N == 168 || N == 175 ||
    2624          21 :       N == 180 || N == 252 || N == 253 || N == 273 || N == 320 || N == 335 ||
    2625          42 :       N == 336 || N == 345 || N == 360) return k >= 8;
    2626          21 :   return k >= 6;
    2627             : }
    2628             : /* eisspace^-(N) = 0 */
    2629             : static int
    2630          49 : isminustriv(GEN F)
    2631             : {
    2632          49 :   GEN P = gel(F,1), E = gel(F,2);
    2633          49 :   long i = 1, l = lg(P);
    2634          49 :   if (l == 1) return 1;
    2635          49 :   if (P[1] == 2)
    2636             :   {
    2637           7 :     if (E[1] >= 4) return 0;
    2638           7 :     i++;
    2639             :   }
    2640          98 :   for (; i < l; i++)
    2641          49 :     if (E[i] > 1) return 0;
    2642          49 :   return 1;
    2643             : }
    2644             : 
    2645             : GEN
    2646         343 : mscuspidal(GEN W, long flag)
    2647             : {
    2648         343 :   pari_sp av = avma;
    2649             :   GEN M, E, S;
    2650             :   ulong p, N;
    2651             :   long k, s;
    2652             : 
    2653         343 :   checkms(W);
    2654         343 :   N = ms_get_N(W);
    2655         343 :   k = msk_get_weight(W);
    2656         343 :   s = msk_get_sign(W);
    2657         343 :   E = flag? mseisenstein_i(W): NULL;
    2658         343 :   if (s < 0 && isminustriv(factoru(N))) M = matid(msdim(W));
    2659         294 :   else if (use_Petersson(N, k, s)) M = eisker(W);
    2660             :   else
    2661             :   {
    2662             :     GEN T, TE, chE;
    2663             :     forprime_t F;
    2664             :     long bit;
    2665             :     pari_timer ti;
    2666             : 
    2667         266 :     if (!E) E = mseisenstein_i(W);
    2668         266 :     (void)u_forprime_init(&F, 2, ULONG_MAX);
    2669         392 :     while ((p = u_forprime_next(&F)))
    2670         392 :       if (N % p) break;
    2671         266 :     if (DEBUGLEVEL) timer_start(&ti);
    2672         266 :     T = mshecke(W, p, NULL);
    2673         266 :     if (DEBUGLEVEL) timer_printf(&ti,"Tp, p = %ld", p);
    2674         266 :     TE = Qevproj_apply(T, E); /* T_p | E */
    2675         266 :     if (DEBUGLEVEL) timer_printf(&ti,"Qevproj_init(E)");
    2676         266 :     bit = TpE_char_bound(p, k, lg(TE)-1);
    2677         266 :     chE = QM_charpoly_ZX_bound(TE, bit);
    2678         266 :     chE = ZX_radical(chE);
    2679         266 :     M = RgX_RgM_eval(chE, T);
    2680         266 :     M = QM_image_shallow(M);
    2681             :   }
    2682         343 :   S = Qevproj_init(M);
    2683         343 :   return gerepilecopy(av, flag? mkvec2(S,E): S);
    2684             : }
    2685             : 
    2686             : /** INIT ELLSYM STRUCTURE **/
    2687             : /* V a vector of ZM. If all of them have 0 last row, return NULL.
    2688             :  * Otherwise return [m,i,j], where m = V[i][last,j] contains the value
    2689             :  * of smallest absolute value */
    2690             : static GEN
    2691         945 : RgMV_find_non_zero_last_row(long offset, GEN V)
    2692             : {
    2693         945 :   long i, lasti = 0, lastj = 0, lV = lg(V);
    2694         945 :   GEN m = NULL;
    2695        4109 :   for (i = 1; i < lV; i++)
    2696             :   {
    2697        3164 :     GEN M = gel(V,i);
    2698        3164 :     long j, n, l = lg(M);
    2699        3164 :     if (l == 1) continue;
    2700        2849 :     n = nbrows(M);
    2701       13860 :     for (j = 1; j < l; j++)
    2702             :     {
    2703       11011 :       GEN a = gcoeff(M, n, j);
    2704       11011 :       if (!gequal0(a) && (!m || abscmpii(a, m) < 0))
    2705             :       {
    2706        1596 :         m = a; lasti = i; lastj = j;
    2707        1596 :         if (is_pm1(m)) goto END;
    2708             :       }
    2709             :     }
    2710             :   }
    2711         945 : END:
    2712         945 :   if (!m) return NULL;
    2713         630 :   return mkvec2(m, mkvecsmall2(lasti+offset, lastj));
    2714             : }
    2715             : /* invert the d_oo := (\gamma_oo - 1) operator, acting on
    2716             :  * [x^(k-2), ..., y^(k-2)] */
    2717             : static GEN
    2718         630 : Delta_inv(GEN doo, long k)
    2719             : {
    2720         630 :   GEN M = RgX_act_ZGl2Q(doo, k);
    2721         630 :   M = RgM_minor(M, k-1, 1); /* 1st column and last row are 0 */
    2722         630 :   return ZM_inv_denom(M);
    2723             : }
    2724             : /* The ZX P = \sum a_i x^i y^{k-2-i} is given by the ZV [a_0, ..., a_k-2]~,
    2725             :  * return Q and d such that P = doo Q + d y^k-2, where d in Z and Q */
    2726             : static GEN
    2727       12873 : doo_decompose(GEN dinv, GEN P, GEN *pd)
    2728             : {
    2729       12873 :   long l = lg(P); *pd = gel(P, l-1);
    2730       12873 :   P = vecslice(P, 1, l-2);
    2731       12873 :   return shallowconcat(gen_0, ZC_apply_dinv(dinv, P));
    2732             : }
    2733             : 
    2734             : static GEN
    2735       12873 : get_phi_ij(long i,long j,long n, long s,long t,GEN P_st,GEN Q_st,GEN d_st,
    2736             :            GEN P_ij, GEN lP_ij, GEN dinv)
    2737             : {
    2738             :   GEN ind, pols;
    2739       12873 :   if (i == s && j == t)
    2740             :   {
    2741         630 :     ind = mkvecsmall(1);
    2742         630 :     pols = mkvec(scalarcol_shallow(gen_1, lg(P_st)-1)); /* x^{k-2} */
    2743             :   }
    2744             :   else
    2745             :   {
    2746       12243 :     GEN d_ij, Q_ij = doo_decompose(dinv, lP_ij, &d_ij);
    2747       12243 :     GEN a = ZC_Z_mul(P_ij, d_st);
    2748       12243 :     GEN b = ZC_Z_mul(P_st, negi(d_ij));
    2749       12243 :     GEN c = RgC_sub(RgC_Rg_mul(Q_ij, d_st), RgC_Rg_mul(Q_st, d_ij));
    2750       12243 :     if (i == s) { /* j != t */
    2751        1659 :       ind = mkvecsmall2(1, s);
    2752        1659 :       pols = mkvec2(c, ZC_add(a, b));
    2753             :     } else {
    2754       10584 :       ind = mkvecsmall3(1, i, s);
    2755       10584 :       pols = mkvec3(c, a, b); /* image of g_1, g_i, g_s */
    2756             :     }
    2757       12243 :     pols = Q_primpart(pols);
    2758             :   }
    2759       12873 :   return mkvec3(mkvecsmall3(i,j,n), ind, pols);
    2760             : }
    2761             : 
    2762             : static GEN
    2763        1155 : mskinit_trivial(GEN WN)
    2764             : {
    2765        1155 :   long dim = ms_get_nbE1(WN);
    2766        1155 :   return mkvec3(WN, gen_0, mkvec2(gen_0,mkvecsmall2(2, dim)));
    2767             : }
    2768             : /* sum of #cols of the matrices contained in V */
    2769             : static long
    2770        1260 : RgMV_dim(GEN V)
    2771             : {
    2772        1260 :   long l = lg(V), d = 0, i;
    2773        1764 :   for (i = 1; i < l; i++) d += lg(gel(V,i)) - 1;
    2774        1260 :   return d;
    2775             : }
    2776             : static GEN
    2777         630 : mskinit_nontrivial(GEN WN, long k)
    2778             : {
    2779         630 :   GEN annT2 = gel(WN,8), annT31 = gel(WN,9), singlerel = gel(WN,10);
    2780             :   GEN link, basis, monomials, Inv;
    2781         630 :   long nbE1 = ms_get_nbE1(WN);
    2782         630 :   GEN dinv = Delta_inv(ZG_neg( ZSl2_star(gel(singlerel,1)) ), k);
    2783         630 :   GEN p1 = cgetg(nbE1+1, t_VEC), remove;
    2784         630 :   GEN p2 = ZGV_tors(annT2, k);
    2785         630 :   GEN p3 = ZGV_tors(annT31, k);
    2786         630 :   GEN gentor = shallowconcat(p2, p3);
    2787             :   GEN P_st, lP_st, Q_st, d_st;
    2788             :   long n, i, dim, s, t, u;
    2789         630 :   gel(p1, 1) = cgetg(1,t_MAT); /* dummy */
    2790        3381 :   for (i = 2; i <= nbE1; i++) /* skip 1st element = (\gamma_oo-1)g_oo */
    2791             :   {
    2792        2751 :     GEN z = gel(singlerel, i);
    2793        2751 :     gel(p1, i) = RgX_act_ZGl2Q(ZSl2_star(z), k);
    2794             :   }
    2795         630 :   remove = RgMV_find_non_zero_last_row(nbE1, gentor);
    2796         630 :   if (!remove) remove = RgMV_find_non_zero_last_row(0, p1);
    2797         630 :   if (!remove) pari_err_BUG("msinit [no y^k-2]");
    2798         630 :   remove = gel(remove,2); /* [s,t] */
    2799         630 :   s = remove[1];
    2800         630 :   t = remove[2];
    2801             :   /* +1 because of = x^(k-2), but -1 because of Manin relation */
    2802         630 :   dim = (k-1)*(nbE1-1) + RgMV_dim(p2) + RgMV_dim(p3);
    2803             :   /* Let (g_1,...,g_d) be the Gamma-generators of Delta, g_1 = g_oo.
    2804             :    * We describe modular symbols by the collection phi(g_1), ..., phi(g_d)
    2805             :    * \in V := Q[x,y]_{k-2}, with right Gamma action.
    2806             :    * For each i = 1, .., d, let V_i \subset V be the Q-vector space of
    2807             :    * allowed values for phi(g_i): with basis (P^{i,j}) given by the monomials
    2808             :    * x^(j-1) y^{k-2-(j-1)}, j = 1 .. k-1
    2809             :    * (g_i in E_1) or the solution of the torsion equations (1 + gamma)P = 0
    2810             :    * (g_i in T2) or (1 + gamma + gamma^2)P = 0 (g_i in T31). All such P
    2811             :    * are chosen in Z[x,y] with Q_content 1.
    2812             :    *
    2813             :    * The Manin relation (singlerel) is of the form \sum_i \lambda_i g_i = 0,
    2814             :    * where \lambda_i = 1 if g_i in T2 or T31, and \lambda_i = (1 - \gamma_i)
    2815             :    * for g_i in E1.
    2816             :    *
    2817             :    * If phi \in Hom_Gamma(Delta, V), it is defined by phi(g_i) := P_i in V
    2818             :    * with \sum_i P_i . \lambda_i^* = 0, where (\sum n_i g_i)^* :=
    2819             :    * \sum n_i \gamma_i^(-1).
    2820             :    *
    2821             :    * We single out gamma_1 / g_1 (g_oo in Pollack-Stevens paper) and
    2822             :    * write P_{i,j} \lambda_i^* =  Q_{i,j} (\gamma_1 - 1)^* + d_{i,j} y^{k-2}
    2823             :    * where d_{i,j} is a scalar and Q_{i,j} in V; we normalize Q_{i,j} to
    2824             :    * that the coefficient of x^{k-2} is 0.
    2825             :    *
    2826             :    * There exist (s,t) such that d_{s,t} != 0.
    2827             :    * A Q-basis of the (dual) space of modular symbols is given by the
    2828             :    * functions phi_{i,j}, 2 <= i <= d, 1 <= j <= k-1, mapping
    2829             :    *  g_1 -> d_{s,t} Q_{i,j} - d_{i,j} Q_{s,t} + [(i,j)=(s,t)] x^{k-2}
    2830             :    * If i != s
    2831             :    *   g_i -> d_{s,t} P_{i,j}
    2832             :    *   g_s -> - d_{i,j} P_{s,t}
    2833             :    * If i = s, j != t
    2834             :    *   g_i -> d_{s,t} P_{i,j} - d_{i,j} P_{s,t}
    2835             :    * And everything else to 0. Again we normalize the phi_{i,j} such that
    2836             :    * their image has content 1. */
    2837         630 :   monomials = matid(k-1); /* represent the monomials x^{k-2}, ... , y^{k-2} */
    2838         630 :   if (s <= nbE1) /* in E1 */
    2839             :   {
    2840         315 :     P_st = gel(monomials, t);
    2841         315 :     lP_st = gmael(p1, s, t); /* P_{s,t} lambda_s^* */
    2842             :   }
    2843             :   else /* in T2, T31 */
    2844             :   {
    2845         315 :     P_st = gmael(gentor, s - nbE1, t);
    2846         315 :     lP_st = P_st;
    2847             :   }
    2848         630 :   Q_st = doo_decompose(dinv, lP_st, &d_st);
    2849         630 :   basis = cgetg(dim+1, t_VEC);
    2850         630 :   link = cgetg(nbE1 + lg(gentor), t_VEC);
    2851         630 :   gel(link,1) = cgetg(1,t_VECSMALL); /* dummy */
    2852         630 :   n = 1;
    2853        3381 :   for (i = 2; i <= nbE1; i++)
    2854             :   {
    2855        2751 :     GEN L = cgetg(k, t_VECSMALL);
    2856             :     long j;
    2857             :     /* link[i][j] = n gives correspondance between phi_{i,j} and basis[n] */
    2858        2751 :     gel(link,i) = L;
    2859       14056 :     for (j = 1; j < k; j++)
    2860             :     {
    2861       11305 :       GEN lP_ij = gmael(p1, i, j); /* P_{i,j} lambda_i^* */
    2862       11305 :       GEN P_ij = gel(monomials,j);
    2863       11305 :       L[j] = n;
    2864       11305 :       gel(basis, n) = get_phi_ij(i,j,n, s,t, P_st, Q_st, d_st, P_ij, lP_ij, dinv);
    2865       11305 :       n++;
    2866             :     }
    2867             :   }
    2868        1134 :   for (u = 1; u < lg(gentor); u++,i++)
    2869             :   {
    2870         504 :     GEN V = gel(gentor,u);
    2871         504 :     long j, lV = lg(V);
    2872         504 :     GEN L = cgetg(lV, t_VECSMALL);
    2873         504 :     gel(link,i) = L;
    2874        2072 :     for (j = 1; j < lV; j++)
    2875             :     {
    2876        1568 :       GEN lP_ij = gel(V, j); /* P_{i,j} lambda_i^* = P_{i,j} */
    2877        1568 :       GEN P_ij = lP_ij;
    2878        1568 :       L[j] = n;
    2879        1568 :       gel(basis, n) = get_phi_ij(i,j,n, s,t, P_st, Q_st, d_st, P_ij, lP_ij, dinv);
    2880        1568 :       n++;
    2881             :     }
    2882             :   }
    2883         630 :   Inv = cgetg(lg(link), t_VEC);
    2884         630 :   gel(Inv,1) = cgetg(1, t_MAT); /* dummy */
    2885        3885 :   for (i = 2; i < lg(link); i++)
    2886             :   {
    2887        3255 :     GEN M, inv, B = gel(link,i);
    2888        3255 :     long j, lB = lg(B);
    2889        3255 :     if (i == s) { B = vecsplice(B, t); lB--; } /* remove phi_st */
    2890        3255 :     M = cgetg(lB, t_MAT);
    2891       15498 :     for (j = 1; j < lB; j++)
    2892             :     {
    2893       12243 :       GEN phi_ij = gel(basis, B[j]), pols = gel(phi_ij,3);
    2894       12243 :       gel(M, j) = gel(pols, 2); /* phi_ij(g_i) */
    2895             :     }
    2896        3255 :     if (i <= nbE1 && i != s) /* maximal rank k-1 */
    2897        2436 :       inv = ZM_inv_denom(M);
    2898             :     else /* i = s (rank k-2) or from torsion: rank k/3 or k/2 */
    2899         819 :       inv = Qevproj_init(M);
    2900        3255 :     gel(Inv,i) = inv;
    2901             :   }
    2902         630 :   return mkvec3(WN, gen_0, mkvec5(basis, mkvecsmall2(k, dim), mkvecsmall2(s,t),
    2903             :                                   link, Inv));
    2904             : }
    2905             : static GEN
    2906        1799 : add_star(GEN W, long sign)
    2907             : {
    2908        1799 :   GEN s = msstar_i(W);
    2909        1799 :   GEN K = sign? QM_ker(gsubgs(s, sign)): cgetg(1,t_MAT);
    2910        1799 :   gel(W,2) = mkvec3(stoi(sign), s, Qevproj_init(K));
    2911        1799 :   return W;
    2912             : }
    2913             : /* WN = msinit_N(N) */
    2914             : static GEN
    2915        1799 : mskinit(ulong N, long k, long sign)
    2916             : {
    2917        1799 :   GEN W, WN = msinit_N(N);
    2918        1799 :   if (N == 1)
    2919             :   {
    2920          14 :     GEN basis, M = RgXV_to_RgM(mfperiodpolbasis(k, 0), k-1);
    2921          14 :     GEN T = cgetg(1, t_VECSMALL), ind = mkvecsmall(1);
    2922          14 :     long i, l = lg(M);
    2923          14 :     basis = cgetg(l, t_VEC);
    2924          70 :     for (i = 1; i < l; i++) gel(basis,i) = mkvec3(T, ind, mkvec(gel(M,i)));
    2925          14 :     W = mkvec3(WN, gen_0, mkvec5(basis, mkvecsmall2(k, l-1), mkvecsmall2(0,0),
    2926             :                                  gen_0, Qevproj_init(M)));
    2927             :   }
    2928             :   else
    2929        1785 :     W = k == 2? mskinit_trivial(WN)
    2930        1785 :               : mskinit_nontrivial(WN, k);
    2931        1799 :   return add_star(W, sign);
    2932             : }
    2933             : GEN
    2934         476 : msinit(GEN N, GEN K, long sign)
    2935             : {
    2936         476 :   pari_sp av = avma;
    2937             :   GEN W;
    2938             :   long k;
    2939         476 :   if (typ(N) != t_INT) pari_err_TYPE("msinit", N);
    2940         476 :   if (typ(K) != t_INT) pari_err_TYPE("msinit", K);
    2941         476 :   k = itos(K);
    2942         476 :   if (k < 2) pari_err_DOMAIN("msinit","k", "<", gen_2,K);
    2943         476 :   if (odd(k)) pari_err_IMPL("msinit [odd weight]");
    2944         476 :   if (signe(N) <= 0) pari_err_DOMAIN("msinit","N", "<=", gen_0,N);
    2945         476 :   W = mskinit(itou(N), k, sign);
    2946         476 :   return gerepilecopy(av, W);
    2947             : }
    2948             : 
    2949             : /* W = msinit, xpm integral modular symbol of weight 2, c t_FRAC
    2950             :  * Return image of <oo->c> */
    2951             : GEN
    2952     2214660 : mseval2_ooQ(GEN W, GEN xpm, GEN c)
    2953             : {
    2954     2214660 :   pari_sp av = avma;
    2955             :   GEN v;
    2956     2214660 :   W = get_msN(W);
    2957     2214660 :   v = init_act_trivial(W);
    2958     2214660 :   Q_log_trivial(v, W, c); /* oo -> (a:b), c = a/b */
    2959     2214660 :   return gerepileuptoint(av, ZV_zc_mul(xpm, v));
    2960             : }
    2961             : 
    2962             : static GEN
    2963       20314 : eval_single(GEN s, long k, GEN B, long v)
    2964             : {
    2965             :   long i, l;
    2966       20314 :   GEN A = cgetg_copy(s,&l);
    2967      135863 :   for (i=1; i<l; i++) gel(A,i) = ZGl2Q_act_s(gel(B,i), gel(s,i), k);
    2968       20314 :   A = RgV_sum(A);
    2969       20314 :   if (is_vec_t(typ(A))) A = RgV_to_RgX(A, v);
    2970       20314 :   return A;
    2971             : }
    2972             : /* Evaluate symbol s on mspathlog B (= sum p_i g_i, p_i in Z[G]). Allow
    2973             :  * s = t_MAT [ collection of symbols, return a vector ]*/
    2974             : static GEN
    2975       16072 : mseval_by_values(GEN W, GEN s, GEN p, long v)
    2976             : {
    2977       16072 :   long i, l, k = msk_get_weight(W);
    2978             :   GEN A;
    2979       16072 :   if (k == 2)
    2980             :   { /* trivial represention: don't bother with Z[G] */
    2981        3444 :     GEN B = mspathlog_trivial(W,p);
    2982        3437 :     if (typ(s) != t_MAT) return RgV_zc_mul(s,B);
    2983        3367 :     l = lg(s); A = cgetg(l, t_VEC);
    2984       10101 :     for (i = 1; i < l; i++) gel(A,i) = RgV_zc_mul(gel(s,i), B);
    2985             :   }
    2986             :   else
    2987             :   {
    2988       12628 :     GEN B = mspathlog(W,p);
    2989       12628 :     if (typ(s) != t_MAT) return eval_single(s, k, B, v);
    2990         812 :     l = lg(s); A = cgetg(l, t_VEC);
    2991        9310 :     for (i = 1; i < l; i++) gel(A,i) = eval_single(gel(s,i), k, B, v);
    2992             :   }
    2993        4179 :   return A;
    2994             : }
    2995             : 
    2996             : /* express symbol on the basis phi_{i,j} */
    2997             : static GEN
    2998       20692 : symtophi(GEN W, GEN s)
    2999             : {
    3000       20692 :   GEN e, basis = msk_get_basis(W);
    3001       20692 :   long i, l = lg(basis);
    3002       20692 :   if (lg(s) != l) pari_err_TYPE("mseval",s);
    3003       20692 :   e = zerovec(ms_get_nbgen(W));
    3004      313670 :   for (i=1; i<l; i++)
    3005             :   {
    3006      292978 :     GEN phi, ind, pols, c = gel(s,i);
    3007             :     long j, m;
    3008      292978 :     if (gequal0(c)) continue;
    3009      122696 :     phi = gel(basis,i);
    3010      122696 :     ind = gel(phi,2); m = lg(ind);
    3011      122696 :     pols = gel(phi,3);
    3012      470806 :     for (j=1; j<m; j++)
    3013             :     {
    3014      348110 :       long t = ind[j];
    3015      348110 :       gel(e,t) = gadd(gel(e,t), gmul(c, gel(pols,j)));
    3016             :     }
    3017             :   }
    3018       20692 :   return e;
    3019             : }
    3020             : /* evaluate symbol s on path p */
    3021             : GEN
    3022       17031 : mseval(GEN W, GEN s, GEN p)
    3023             : {
    3024       17031 :   pari_sp av = avma;
    3025       17031 :   long i, k, l, v = 0;
    3026       17031 :   checkms(W);
    3027       17031 :   k = msk_get_weight(W);
    3028       17031 :   switch(typ(s))
    3029             :   {
    3030           7 :     case t_VEC: /* values s(g_i) */
    3031           7 :       if (lg(s)-1 != ms_get_nbgen(W)) pari_err_TYPE("mseval",s);
    3032           7 :       if (!p) return gcopy(s);
    3033           0 :       v = gvar(s);
    3034           0 :       break;
    3035       12831 :     case t_COL:
    3036       12831 :       if (msk_get_sign(W))
    3037             :       {
    3038         399 :         GEN star = gel(msk_get_starproj(W), 1);
    3039         399 :         if (lg(star) == lg(s)) s = RgM_RgC_mul(star, s);
    3040             :       }
    3041       12831 :       if (k == 2) /* on the dual basis of (g_i) */
    3042             :       {
    3043         637 :         if (lg(s)-1 != ms_get_nbE1(W)) pari_err_TYPE("mseval",s);
    3044         637 :         if (!p) return gtrans(s);
    3045             :       }
    3046             :       else
    3047       12194 :         s = symtophi(W,s);
    3048       12271 :       break;
    3049        4193 :     case t_MAT:
    3050        4193 :       l = lg(s);
    3051        4193 :       if (!p)
    3052             :       {
    3053           7 :         GEN v = cgetg(l, t_VEC);
    3054          28 :         for (i = 1; i < l; i++) gel(v,i) = mseval(W, gel(s,i), NULL);
    3055           7 :         return v;
    3056             :       }
    3057        4186 :       if (l == 1) return cgetg(1, t_VEC);
    3058        4179 :       if (msk_get_sign(W))
    3059             :       {
    3060          84 :         GEN star = gel(msk_get_starproj(W), 1);
    3061          84 :         if (lg(star) == lgcols(s)) s = RgM_mul(star, s);
    3062             :       }
    3063        4179 :       if (k == 2)
    3064        3367 :       { if (nbrows(s) != ms_get_nbE1(W)) pari_err_TYPE("mseval",s); }
    3065             :       else
    3066             :       {
    3067         812 :         GEN t = cgetg(l, t_MAT);
    3068        9310 :         for (i = 1; i < l; i++) gel(t,i) = symtophi(W,gel(s,i));
    3069         812 :         s = t;
    3070             :       }
    3071        4179 :       break;
    3072           0 :     default: pari_err_TYPE("mseval",s);
    3073             :   }
    3074       16450 :   if (p)
    3075       16072 :     s = mseval_by_values(W, s, p, v);
    3076             :   else
    3077             :   {
    3078         378 :     l = lg(s);
    3079        3675 :     for (i = 1; i < l; i++)
    3080             :     {
    3081        3297 :       GEN c = gel(s,i);
    3082        3297 :       if (!isintzero(c)) gel(s,i) = RgV_to_RgX(gel(s,i), v);
    3083             :     }
    3084             :   }
    3085       16443 :   return gerepilecopy(av, s);
    3086             : }
    3087             : 
    3088             : static GEN
    3089        1792 : allxpm(GEN W, GEN xpm, long f)
    3090             : {
    3091        1792 :   GEN v, L = coprimes_zv(f);
    3092        1792 :   long a, nonzero = 0;
    3093        1792 :   v = const_vec(f, NULL);
    3094        6797 :   for (a = 1; a <= f; a++)
    3095             :   {
    3096             :     GEN c;
    3097        5005 :     if (!L[a]) continue;
    3098        3696 :     c = mseval2_ooQ(W, xpm, sstoQ(a, f));
    3099        3696 :     if (!gequal0(c)) { gel(v,a) = c; nonzero = 1; }
    3100             :   }
    3101        1792 :   return nonzero? v: NULL;
    3102             : }
    3103             : /* \sum_{a mod f} chi(a) x(a/f) */
    3104             : static GEN
    3105         854 : seval(GEN G, GEN chi, GEN vx)
    3106             : {
    3107         854 :   GEN vZ, T, s = gen_0, go = zncharorder(G,chi);
    3108         854 :   long i, l = lg(vx), o = itou(go);
    3109         854 :   T = polcyclo(o,0);
    3110         854 :   vZ = mkvec2(RgXQ_powers(RgX_rem(pol_x(0), T), o-1, T), go);
    3111        3458 :   for (i = 1; i < l; i++)
    3112             :   {
    3113        2604 :     GEN x = gel(vx,i);
    3114        2604 :     if (x) s = gadd(s, gmul(x, znchareval(G, chi, utoi(i), vZ)));
    3115             :   }
    3116         854 :   return gequal0(s)? NULL: poleval(s, rootsof1u_cx(o, DEFAULTPREC));
    3117             : }
    3118             : 
    3119             : /* Let W = msinit(conductor(E), 2), xpm an integral modular symbol with the same
    3120             :  * eigenvalues as L_E. There exist a unique C such that
    3121             :  *   C*L(E,(D/.),1)_{xpm} = L(E,(D/.),1) / w1(E_D) != 0,
    3122             :  * for all D fundamental, sign(D) = s, and such that E_D has rank 0.
    3123             :  * Return C * ellQtwist_bsdperiod(E,s) */
    3124             : static GEN
    3125         854 : ell_get_Cw(GEN LE, GEN W, GEN xpm, long s)
    3126             : {
    3127         854 :   long f, NE = ms_get_N(W);
    3128         854 :   const long bit = 64;
    3129             : 
    3130         854 :   for (f = 1;; f++)
    3131        1750 :   { /* look for chi with conductor f coprime to N(E) and parity s
    3132             :      * such that L(E,chi,1) != 0 */
    3133        2604 :     pari_sp av = avma;
    3134             :     GEN vchi, vx, G;
    3135             :     long l, i;
    3136        2604 :     if ((f & 3) == 2 || ugcd(NE,f) != 1) continue;
    3137        1792 :     vx = allxpm(W, xpm, f); if (!vx) continue;
    3138         854 :     G = znstar0(utoipos(f),1);
    3139         854 :     vchi = chargalois(G,NULL); l = lg(vchi);
    3140        1470 :     for (i = 1; i < l; i++)
    3141             :     {
    3142        1470 :       pari_sp av2 = avma;
    3143        1470 :       GEN tau, z, S, L, chi = gel(vchi,i);
    3144        1470 :       long o = zncharisodd(G,chi);
    3145        1470 :       if ((s > 0 && o) || (s < 0 && !o)
    3146        1470 :           || itos(zncharconductor(G, chi)) != f) continue;
    3147         854 :       S = seval(G, chi, vx);
    3148         854 :       if (!S) { set_avma(av2); continue; }
    3149             : 
    3150         854 :       L = lfuntwist(LE, mkvec2(G, zncharconj(G,chi)));
    3151         854 :       z = lfun(L, gen_1, bit);
    3152         854 :       tau = znchargauss(G, chi, gen_1, bit);
    3153         854 :       return gdiv(gmul(z, tau), S); /* C * w */
    3154             :     }
    3155           0 :     set_avma(av);
    3156             :   }
    3157             : }
    3158             : static GEN
    3159         742 : ell_get_scale(GEN LE, GEN W, long sign, GEN x)
    3160             : {
    3161         742 :   if (sign)
    3162         630 :     return ell_get_Cw(LE, W, gel(x,1), sign);
    3163             :   else
    3164             :   {
    3165         112 :     GEN Cwp = ell_get_Cw(LE, W, gel(x,1), 1);
    3166         112 :     GEN Cwm = ell_get_Cw(LE, W, gel(x,2),-1);
    3167         112 :     return mkvec2(Cwp, Cwm);
    3168             :   }
    3169             : }
    3170             : /* E minimal */
    3171             : static GEN
    3172         931 : msfromell_scale(GEN x, GEN Cw, GEN E, long s)
    3173             : {
    3174         931 :   GEN B = int2n(32);
    3175         931 :   if (s)
    3176             :   {
    3177         630 :     GEN C = gdiv(Cw, ellQtwist_bsdperiod(E,s));
    3178         630 :     return ZC_Q_mul(gel(x,1), bestappr(C,B));
    3179             :   }
    3180             :   else
    3181             :   {
    3182         301 :     GEN xp = gel(x,1), Cp = gdiv(gel(Cw,1), ellQtwist_bsdperiod(E, 1)), L;
    3183         301 :     GEN xm = gel(x,2), Cm = gdiv(gel(Cw,2), ellQtwist_bsdperiod(E,-1));
    3184         301 :     xp = ZC_Q_mul(xp, bestappr(Cp,B));
    3185         301 :     xm = ZC_Q_mul(xm, bestappr(Cm,B));
    3186         301 :     if (signe(ell_get_disc(E)) > 0)
    3187         133 :       L = mkmat2(xp, xm); /* E(R) has 2 connected components */
    3188             :     else
    3189         168 :       L = mkmat2(gsub(xp,xm), gmul2n(xm,1));
    3190         301 :     return mkvec3(xp, xm, L);
    3191             :   }
    3192             : }
    3193             : /* v != 0 */
    3194             : static GEN
    3195         854 : Flc_normalize(GEN v, ulong p)
    3196             : {
    3197         854 :   long i, l = lg(v);
    3198        1512 :   for (i = 1; i < l; i++)
    3199        1512 :     if (v[i])
    3200             :     {
    3201         854 :       if (v[i] != 1) v = Flv_Fl_div(v, v[i], p);
    3202         854 :       return v;
    3203             :     }
    3204           0 :   return NULL;
    3205             : }
    3206             : /* K \cap Ker M  [F_l vector spaces]. K = NULL means full space */
    3207             : static GEN
    3208         903 : msfromell_ker(GEN K, GEN M, ulong l)
    3209             : {
    3210         903 :   GEN B, Ml = ZM_to_Flm(M, l);
    3211         903 :   if (K) Ml = Flm_mul(Ml, K, l);
    3212         903 :   B = Flm_ker(Ml, l);
    3213         903 :   if (!K) K = B;
    3214         161 :   else if (lg(B) < lg(K))
    3215         147 :     K = Flm_mul(K, B, l);
    3216         903 :   return K;
    3217             : }
    3218             : /* K = \cap_p Ker(T_p - a_p), 2-dimensional. Set *xl to the 1-dimensional
    3219             :  * Fl-basis  such that star . xl = sign . xl if sign != 0 and
    3220             :  * star * xl[1] = xl[1]; star * xl[2] = -xl[2] if sign = 0 */
    3221             : static void
    3222         742 : msfromell_l(GEN *pxl, GEN K, GEN star, long sign, ulong l)
    3223             : {
    3224         742 :   GEN s = ZM_to_Flm(star, l);
    3225         742 :   GEN a = gel(K,1), Sa = Flm_Flc_mul(s,a,l);
    3226         742 :   GEN b = gel(K,2);
    3227         742 :   GEN t = Flv_add(a,Sa,l), xp, xm;
    3228         742 :   if (zv_equal0(t))
    3229             :   {
    3230         140 :     xm = a;
    3231         140 :     xp = Flv_add(b,Flm_Flc_mul(s,b,l), l);
    3232             :   }
    3233             :   else
    3234             :   {
    3235         602 :     xp = t; t = Flv_sub(a, Sa, l);
    3236         602 :     xm = zv_equal0(t)? Flv_sub(b, Flm_Flc_mul(s,b,l), l): t;
    3237             :   }
    3238             :   /* xp = 0 on Im(S - 1), xm = 0 on Im(S + 1) */
    3239         742 :   if (sign > 0)
    3240         518 :     *pxl = mkmat(Flc_normalize(xp, l));
    3241         224 :   else if (sign < 0)
    3242         112 :     *pxl = mkmat(Flc_normalize(xm, l));
    3243             :   else
    3244         112 :     *pxl = mkmat2(Flc_normalize(xp, l), Flc_normalize(xm, l));
    3245         742 : }
    3246             : /* return a primitive symbol */
    3247             : static GEN
    3248         742 : msfromell_ratlift(GEN x, GEN q)
    3249             : {
    3250         742 :   GEN B = sqrti(shifti(q,-1));
    3251         742 :   GEN r = FpM_ratlift(x, q, B, B, NULL);
    3252         742 :   if (r) r = Q_primpart(r);
    3253         742 :   return r;
    3254             : }
    3255             : static int
    3256         742 : msfromell_check(GEN x, GEN vT, GEN star, long sign)
    3257             : {
    3258             :   long i, l;
    3259             :   GEN sx;
    3260         742 :   if (!x) return 0;
    3261         742 :   l = lg(vT);
    3262        1645 :   for (i = 1; i < l; i++)
    3263             :   {
    3264         903 :     GEN T = gel(vT,i);
    3265         903 :     if (!gequal0(ZM_mul(T, x))) return 0; /* fail */
    3266             :   }
    3267         742 :   sx = ZM_mul(star,x);
    3268         742 :   if (sign)
    3269         630 :     return ZV_equal(gel(sx,1), sign > 0? gel(x,1): ZC_neg(gel(x,1)));
    3270             :   else
    3271         112 :     return ZV_equal(gel(sx,1),gel(x,1)) && ZV_equal(gel(sx,2),ZC_neg(gel(x,2)));
    3272             : }
    3273             : GEN
    3274         742 : msfromell(GEN E0, long sign)
    3275             : {
    3276         742 :   pari_sp av = avma, av2;
    3277         742 :   GEN T, Cw, E, NE, star, q, vT, xl, xr, W, x = NULL, K = NULL;
    3278             :   long lE, single;
    3279             :   ulong p, l, N;
    3280             :   forprime_t S, Sl;
    3281             : 
    3282         742 :   if (typ(E0) != t_VEC) pari_err_TYPE("msfromell",E0);
    3283         742 :   lE = lg(E0);
    3284         742 :   if (lE == 1) return cgetg(1,t_VEC);
    3285         742 :   single = (typ(gel(E0,1)) != t_VEC);
    3286         742 :   E = single ? E0: gel(E0,1);
    3287         742 :   NE = ellQ_get_N(E);
    3288             :   /* must make it integral for ellap; we have minimal model at hand */
    3289         742 :   T = obj_check(E, Q_MINIMALMODEL); if (lg(T) != 2) E = gel(T,3);
    3290         742 :   N = itou(NE); av2 = avma;
    3291         742 :   W = gerepilecopy(av2, mskinit(N,2,0));
    3292         742 :   star = msk_get_star(W);
    3293         742 :   (void)u_forprime_init(&Sl, 1UL<<29, ULONG_MAX);
    3294             :   /* loop for p <= count_Manin_symbols(N) / 6 would be enough */
    3295         742 :   (void)u_forprime_init(&S, 2, ULONG_MAX);
    3296         742 :   vT = cgetg(1, t_VEC);
    3297         742 :   l = u_forprime_next(&Sl);
    3298        1267 :   while( (p = u_forprime_next(&S)) )
    3299             :   {
    3300             :     GEN M;
    3301        1267 :     if (N % p == 0) continue;
    3302         903 :     av2 = avma;
    3303         903 :     M = RgM_Rg_sub_shallow(mshecke_i(W, p), ellap(E, utoipos(p)));
    3304         903 :     M = gerepilecopy(av2, M);
    3305         903 :     vT = vec_append(vT, M); /* for certification at the end */
    3306         903 :     K = msfromell_ker(K, M, l);
    3307         903 :     if (lg(K) == 3) break;
    3308             :   }
    3309         742 :   if (!p) pari_err_BUG("msfromell: ran out of primes");
    3310             : 
    3311             :   /* mod one l should be enough */
    3312         742 :   msfromell_l(&xl, K, star, sign, l);
    3313         742 :   x = ZM_init_CRT(xl, l);
    3314         742 :   q = utoipos(l);
    3315         742 :   xr = msfromell_ratlift(x, q);
    3316             :   /* paranoia */
    3317         742 :   while (!msfromell_check(xr, vT, star, sign) && (l = u_forprime_next(&Sl)) )
    3318             :   {
    3319           0 :     GEN K = NULL;
    3320           0 :     long i, lvT = lg(vT);
    3321           0 :     for (i = 1; i < lvT; i++)
    3322             :     {
    3323           0 :       K = msfromell_ker(K, gel(vT,i), l);
    3324           0 :       if (lg(K) == 3) break;
    3325             :     }
    3326           0 :     if (i >= lvT) { x = NULL; continue; }
    3327           0 :     msfromell_l(&xl, K, star, sign, l);
    3328           0 :     ZM_incremental_CRT(&x, xl, &q, l);
    3329           0 :     xr = msfromell_ratlift(x, q);
    3330             :   }
    3331             :   /* linear form = 0 on all Im(Tp - ap) and Im(S - sign) if sign != 0 */
    3332         742 :   Cw = ell_get_scale(lfuncreate(E), W, sign, xr);
    3333         742 :   if (single)
    3334         693 :     x = msfromell_scale(xr, Cw, E, sign);
    3335             :   else
    3336             :   { /* assume all E0[i] isogenous, given by minimal models */
    3337          49 :     GEN v = cgetg(lE, t_VEC);
    3338             :     long i;
    3339         287 :     for (i=1; i<lE; i++) gel(v,i) = msfromell_scale(xr, Cw, gel(E0,i), sign);
    3340          49 :     x = v;
    3341             :   }
    3342         742 :   return gerepilecopy(av, mkvec2(W, x));
    3343             : }
    3344             : 
    3345             : GEN
    3346          21 : msfromhecke(GEN W, GEN v, GEN H)
    3347             : {
    3348          21 :   pari_sp av = avma;
    3349          21 :   long i, l = lg(v);
    3350          21 :   GEN K = NULL;
    3351          21 :   checkms(W);
    3352          21 :   if (typ(v) != t_VEC) pari_err_TYPE("msfromhecke",v);
    3353          49 :   for (i = 1; i < l; i++)
    3354             :   {
    3355          28 :     GEN K2, T, p, P, c = gel(v,i);
    3356          28 :     if (typ(c) != t_VEC || lg(c) != 3) pari_err_TYPE("msfromhecke",v);
    3357          28 :     p = gel(c,1);
    3358          28 :     if (typ(p) != t_INT) pari_err_TYPE("msfromhecke",v);
    3359          28 :     P = gel(c,2);
    3360          28 :     switch(typ(P))
    3361             :     {
    3362          21 :       case t_INT:
    3363          21 :         P = deg1pol_shallow(gen_1, negi(P), 0);
    3364          21 :         break;
    3365           7 :       case t_POL:
    3366           7 :         if (RgX_is_ZX(P)) break;
    3367             :       default:
    3368           0 :         pari_err_TYPE("msfromhecke",v);
    3369             :     };
    3370          28 :     T = mshecke(W, itos(p), H);
    3371          28 :     T = Q_primpart(RgX_RgM_eval(P, T));
    3372          28 :     if (K) T = ZM_mul(T,K);
    3373          28 :     K2 = ZM_ker(T);
    3374          28 :     if (!K) K = K2;
    3375           7 :     else if (lg(K2) < lg(K)) K = ZM_mul(K,K2);
    3376             :   }
    3377          21 :   return gerepilecopy(av, K);
    3378             : }
    3379             : 
    3380             : /* OVERCONVERGENT MODULAR SYMBOLS */
    3381             : 
    3382             : static GEN
    3383        2933 : mspadic_get_Wp(GEN W) { return gel(W,1); }
    3384             : static GEN
    3385         483 : mspadic_get_Tp(GEN W) { return gel(W,2); }
    3386             : static GEN
    3387         483 : mspadic_get_bin(GEN W) { return gel(W,3); }
    3388             : static GEN
    3389         476 : mspadic_get_actUp(GEN W) { return gel(W,4); }
    3390             : static GEN
    3391         476 : mspadic_get_q(GEN W) { return gel(W,5); }
    3392             : static long
    3393        1456 : mspadic_get_p(GEN W) { return gel(W,6)[1]; }
    3394             : static long
    3395        1211 : mspadic_get_n(GEN W) { return gel(W,6)[2]; }
    3396             : static long
    3397         161 : mspadic_get_flag(GEN W) { return gel(W,6)[3]; }
    3398             : static GEN
    3399         483 : mspadic_get_M(GEN W) { return gel(W,7); }
    3400             : static GEN
    3401         483 : mspadic_get_C(GEN W) { return gel(W,8); }
    3402             : static long
    3403         973 : mspadic_get_weight(GEN W) { return msk_get_weight(mspadic_get_Wp(W)); }
    3404             : 
    3405             : void
    3406         980 : checkmspadic(GEN W)
    3407             : {
    3408         980 :   if (typ(W) != t_VEC || lg(W) != 9) pari_err_TYPE("checkmspadic",W);
    3409         980 :   checkms(mspadic_get_Wp(W));
    3410         980 : }
    3411             : 
    3412             : /* f in M_2(Z) \cap GL_2(Q), p \nmid a [ and for the result to mean anything
    3413             :  * p | c, but not needed here]. Return the matrix M in M_D(Z), D = M+k-1
    3414             :  * such that, if v = \int x^i d mu, i < D, is a vector of D moments of mu,
    3415             :  * then M * v is the vector of moments of mu | f  mod p^D */
    3416             : static GEN
    3417      276073 : moments_act_i(struct m_act *S, GEN f)
    3418             : {
    3419      276073 :   long j, k = S->k, D = S->dim;
    3420      276073 :   GEN a = gcoeff(f,1,1), b = gcoeff(f,1,2);
    3421      276073 :   GEN c = gcoeff(f,2,1), d = gcoeff(f,2,2);
    3422      276073 :   GEN u, z, q = S->q, mat = cgetg(D+1, t_MAT);
    3423             : 
    3424      276073 :   a = modii(a,q);
    3425      276073 :   c = modii(c,q);
    3426      276073 :   z = FpX_powu(deg1pol(c,a,0), k-2, q); /* (a+cx)^(k-2) */
    3427             :   /* u := (b+dx) / (a+cx) mod (q,x^D) = (b/a +d/a*x) / (1 - (-c/a)*x) */
    3428      276073 :   if (!equali1(a))
    3429             :   {
    3430      271229 :     GEN ai = Fp_inv(a,q);
    3431      271229 :     b = Fp_mul(b,ai,q);
    3432      271229 :     c = Fp_mul(c,ai,q);
    3433      271229 :     d = Fp_mul(d,ai,q);
    3434             :   }
    3435      276073 :   u = deg1pol_shallow(d, b, 0);
    3436             :   /* multiply by 1 / (1 - (-c/a)*x) */
    3437      276073 :   if (signe(c))
    3438             :   {
    3439      269640 :     GEN C = Fp_neg(c,q), v = cgetg(D+2,t_POL);
    3440      269640 :     v[1] = evalsigne(1)|evalvarn(0);
    3441      269640 :     gel(v, 2) = gen_1; gel(v, 3) = C;
    3442     1405138 :     for (j = 4; j < D+2; j++)
    3443             :     {
    3444     1329027 :       GEN t = Fp_mul(gel(v,j-1), C, q);
    3445     1329027 :       if (!signe(t)) { setlg(v,j); break; }
    3446     1135498 :       gel(v,j) = t;
    3447             :     }
    3448      269640 :     u = FpXn_mul(u, v, D, q);
    3449             :   }
    3450     2369024 :   for (j = 1; j <= D; j++)
    3451             :   {
    3452     2092951 :     gel(mat,j) = RgX_to_RgC(z, D); /* (a+cx)^(k-2) * ((b+dx)/(a+cx))^(j-1) */
    3453     2092951 :     if (j != D) z = FpXn_mul(z, u, D, q);
    3454             :   }
    3455      276073 :   return shallowtrans(mat);
    3456             : }
    3457             : static GEN
    3458      275611 : moments_act(struct m_act *S, GEN f)
    3459      275611 : { pari_sp av = avma; return gerepilecopy(av, moments_act_i(S,f)); }
    3460             : static GEN
    3461         483 : init_moments_act(GEN W, long p, long n, GEN q, GEN v)
    3462             : {
    3463             :   struct m_act S;
    3464         483 :   long k = msk_get_weight(W);
    3465         483 :   S.p = p;
    3466         483 :   S.k = k;
    3467         483 :   S.q = q;
    3468         483 :   S.dim = n+k-1;
    3469         483 :   S.act = &moments_act; return init_dual_act(v,W,W,&S);
    3470             : }
    3471             : 
    3472             : static void
    3473        6762 : clean_tail(GEN phi, long c, GEN q)
    3474             : {
    3475        6762 :   long a, l = lg(phi);
    3476      214438 :   for (a = 1; a < l; a++)
    3477             :   {
    3478      207676 :     GEN P = FpV_red(gel(phi,a), q); /* phi(G_a) = vector of moments */
    3479      207676 :     long j, lP = lg(P);
    3480     1007825 :     for (j = c; j < lP; j++) gel(P,j) = gen_0; /* reset garbage to 0 */
    3481      207676 :     gel(phi,a) = P;
    3482             :   }
    3483        6762 : }
    3484             : /* concat z to all phi[i] */
    3485             : static GEN
    3486         630 : concat2(GEN phi, GEN z)
    3487             : {
    3488             :   long i, l;
    3489         630 :   GEN v = cgetg_copy(phi,&l);
    3490       29022 :   for (i = 1; i < l; i++) gel(v,i) = shallowconcat(gel(phi,i), z);
    3491         630 :   return v;
    3492             : }
    3493             : static GEN
    3494         630 : red_mod_FilM(GEN phi, ulong p, long k, long flag)
    3495             : {
    3496             :   long a, l;
    3497         630 :   GEN den = gen_1, v = cgetg_copy(phi, &l);
    3498         630 :   if (flag)
    3499             :   {
    3500         343 :     phi = Q_remove_denom(phi, &den);
    3501         343 :     if (!den) { den = gen_1; flag = 0; }
    3502             :   }
    3503       29386 :   for (a = 1; a < l; a++)
    3504             :   {
    3505       28756 :     GEN P = gel(phi,a), q = den;
    3506             :     long j;
    3507      207676 :     for (j = lg(P)-1; j >= k+1; j--)
    3508             :     {
    3509      178920 :       q = muliu(q,p);
    3510      178920 :       gel(P,j) = modii(gel(P,j),q);
    3511             :     }
    3512       28756 :     q = muliu(q,p);
    3513       93380 :     for (     ; j >= 1; j--)
    3514       64624 :       gel(P,j) = modii(gel(P,j),q);
    3515       28756 :     gel(v,a) = P;
    3516             :   }
    3517         630 :   if (flag) v = gdiv(v, den);
    3518         630 :   return v;
    3519             : }
    3520             : 
    3521             : /* denom(C) | p^(2(k-1) - v_p(ap)) */
    3522             : static GEN
    3523         154 : oms_dim2(GEN W, GEN phi, GEN C, GEN ap)
    3524             : {
    3525         154 :   long t, i, k = mspadic_get_weight(W);
    3526         154 :   long p = mspadic_get_p(W), n = mspadic_get_n(W);
    3527         154 :   GEN phi1 = gel(phi,1), phi2 = gel(phi,2);
    3528         154 :   GEN v, q = mspadic_get_q(W);
    3529         154 :   GEN act = mspadic_get_actUp(W);
    3530             : 
    3531         154 :   t = signe(ap)? Z_lval(ap,p) : k-1;
    3532         154 :   phi1 = concat2(phi1, zerovec(n));
    3533         154 :   phi2 = concat2(phi2, zerovec(n));
    3534        2107 :   for (i = 1; i <= n; i++)
    3535             :   {
    3536        1953 :     phi1 = dual_act(k-1, act, phi1);
    3537        1953 :     phi1 = dual_act(k-1, act, phi1);
    3538        1953 :     clean_tail(phi1, k + i*t, q);
    3539             : 
    3540        1953 :     phi2 = dual_act(k-1, act, phi2);
    3541        1953 :     phi2 = dual_act(k-1, act, phi2);
    3542        1953 :     clean_tail(phi2, k + i*t, q);
    3543             :   }
    3544         154 :   C = gpowgs(C,n);
    3545         154 :   v = RgM_RgC_mul(C, mkcol2(phi1,phi2));
    3546         154 :   phi1 = red_mod_FilM(gel(v,1), p, k, 1);
    3547         154 :   phi2 = red_mod_FilM(gel(v,2), p, k, 1);
    3548         154 :   return mkvec2(phi1,phi2);
    3549             : }
    3550             : 
    3551             : /* flag = 0 iff alpha is a p-unit */
    3552             : static GEN
    3553         322 : oms_dim1(GEN W, GEN phi, GEN alpha, long flag)
    3554             : {
    3555         322 :   long i, k = mspadic_get_weight(W);
    3556         322 :   long p = mspadic_get_p(W), n = mspadic_get_n(W);
    3557         322 :   GEN q = mspadic_get_q(W);
    3558         322 :   GEN act = mspadic_get_actUp(W);
    3559         322 :   phi = concat2(phi, zerovec(n));
    3560        3178 :   for (i = 1; i <= n; i++)
    3561             :   {
    3562        2856 :     phi = dual_act(k-1, act, phi);
    3563        2856 :     clean_tail(phi, k + i, q);
    3564             :   }
    3565         322 :   phi = gmul(lift_shallow(gpowgs(alpha,n)), phi);
    3566         322 :   phi = red_mod_FilM(phi, p, k, flag);
    3567         322 :   return mkvec(phi);
    3568             : }
    3569             : 
    3570             : /* lift polynomial P in RgX[X,Y]_{k-2} to a distribution \mu such that
    3571             :  * \int (Y - X z)^(k-2) d\mu(z) = P(X,Y)
    3572             :  * Return the t_VEC of k-1 first moments of \mu: \int z^i d\mu(z), 0<= i < k-1.
    3573             :  *   \sum_j (-1)^(k-2-j) binomial(k-2,j) Y^j \int z^(k-2-j) d\mu(z) = P(1,Y)
    3574             :  * Input is P(1,Y), bin = vecbinomial(k-2): bin[j] = binomial(k-2,j-1) */
    3575             : static GEN
    3576       38626 : RgX_to_moments(GEN P, GEN bin)
    3577             : {
    3578       38626 :   long j, k = lg(bin);
    3579             :   GEN Pd, Bd;
    3580       38626 :   if (typ(P) != t_POL) P = scalarpol(P,0);
    3581       38626 :   P = RgX_to_RgC(P, k-1); /* deg <= k-2 */
    3582       38626 :   settyp(P, t_VEC);
    3583       38626 :   Pd = P+1;  /* Pd[i] = coeff(P,i) */
    3584       38626 :   Bd = bin+1;/* Bd[i] = binomial(k-2,i) */
    3585       46249 :   for (j = 1; j < k-2; j++)
    3586             :   {
    3587        7623 :     GEN c = gel(Pd,j);
    3588        7623 :     if (odd(j)) c = gneg(c);
    3589        7623 :     gel(Pd,j) = gdiv(c, gel(Bd,j));
    3590             :   }
    3591       38626 :   return vecreverse(P);
    3592             : }
    3593             : static GEN
    3594         882 : RgXC_to_moments(GEN v, GEN bin)
    3595             : {
    3596             :   long i, l;
    3597         882 :   GEN w = cgetg_copy(v,&l);
    3598       39508 :   for (i=1; i<l; i++) gel(w,i) = RgX_to_moments(gel(v,i),bin);
    3599         882 :   return w;
    3600             : }
    3601             : 
    3602             : /* W an mspadic, assume O[2] is integral, den is the cancelled denominator
    3603             :  * or NULL, L = log(path)^* in sparse form */
    3604             : static GEN
    3605        2954 : omseval_int(struct m_act *S, GEN PHI, GEN L, hashtable *H)
    3606             : {
    3607             :   long i, l;
    3608        2954 :   GEN v = cgetg_copy(PHI, &l);
    3609        2954 :   ZGl2QC_to_act(S, L, H); /* as operators on V */
    3610        6286 :   for (i = 1; i < l; i++)
    3611             :   {
    3612        3332 :     GEN T = dense_act_col(L, gel(PHI,i));
    3613        3332 :     gel(v,i) = T? FpC_red(T,S->q): zerocol(S->dim);
    3614             :   }
    3615        2954 :   return v;
    3616             : }
    3617             : 
    3618             : GEN
    3619          14 : msomseval(GEN W, GEN phi, GEN path)
    3620             : {
    3621             :   struct m_act S;
    3622          14 :   pari_sp av = avma;
    3623             :   GEN v, Wp;
    3624             :   long n, vden;
    3625          14 :   checkmspadic(W);
    3626          14 :   if (typ(phi) != t_COL || lg(phi) != 4)  pari_err_TYPE("msomseval",phi);
    3627          14 :   vden = itos(gel(phi,2));
    3628          14 :   phi = gel(phi,1);
    3629          14 :   n = mspadic_get_n(W);
    3630          14 :   Wp= mspadic_get_Wp(W);
    3631          14 :   S.k = mspadic_get_weight(W);
    3632          14 :   S.p = mspadic_get_p(W);
    3633          14 :   S.q = powuu(S.p, n+vden);
    3634          14 :   S.dim = n + S.k - 1;
    3635          14 :   S.act = &moments_act;
    3636          14 :   path = path_to_M2(path);
    3637          14 :   v = omseval_int(&S, phi, M2_logf(Wp,path,NULL), NULL);
    3638          14 :   return gerepilecopy(av, v);
    3639             : }
    3640             : /* W = msinit(N,k,...); if flag < 0 or flag >= k-1, allow all symbols;
    3641             :  * else commit to v_p(a_p) <= flag (ordinary if flag = 0)*/
    3642             : GEN
    3643         490 : mspadicinit(GEN W, long p, long n, long flag)
    3644             : {
    3645         490 :   pari_sp av = avma;
    3646             :   long a, N, k;
    3647             :   GEN P, C, M, bin, Wp, Tp, q, pn, actUp, teich, pas;
    3648             : 
    3649         490 :   checkms(W);
    3650         490 :   N = ms_get_N(W);
    3651         490 :   k = msk_get_weight(W);
    3652         490 :   if (flag < 0) flag = 1; /* worst case */
    3653         357 :   else if (flag >= k) flag = k-1;
    3654             : 
    3655         490 :   bin = vecbinomial(k-2);
    3656         490 :   Tp = mshecke(W, p, NULL);
    3657         490 :   if (N % p == 0)
    3658             :   {
    3659          91 :     if ((N/p) % p == 0) pari_err_IMPL("mspadicinit when p^2 | N");
    3660             :     /* a_p != 0 */
    3661          84 :     Wp = W;
    3662          84 :     M = gen_0;
    3663          84 :     flag = (k-2) / 2; /* exact valuation */
    3664             :     /* will multiply by matrix with denominator p^(k-2)/2 in mspadicint.
    3665             :      * Except if p = 2 (multiply by alpha^2) */
    3666          84 :     if (p == 2) n += k-2; else n += (k-2)/2;
    3667          84 :     pn = powuu(p,n);
    3668             :     /* For accuracy mod p^n, oms_dim1 require p^(k/2*n) */
    3669          84 :     q = powiu(pn, k/2);
    3670             :   }
    3671             :   else
    3672             :   { /* p-stabilize */
    3673         399 :     long s = msk_get_sign(W);
    3674             :     GEN M1, M2;
    3675             : 
    3676         399 :     Wp = mskinit(N*p, k, s);
    3677         399 :     M1 = getMorphism(W, Wp, mkvec(mat2(1,0,0,1)));
    3678         399 :     M2 = getMorphism(W, Wp, mkvec(mat2(p,0,0,1)));
    3679         399 :     if (s)
    3680             :     {
    3681         147 :       GEN SW = msk_get_starproj(W), SWp = msk_get_starproj(Wp);
    3682         147 :       M1 = Qevproj_apply2(M1, SW, SWp);
    3683         147 :       M2 = Qevproj_apply2(M2, SW, SWp);
    3684             :     }
    3685         399 :     M = mkvec2(M1,M2);
    3686         399 :     n += Z_lval(Q_denom(M), p); /*den. introduced by p-stabilization*/
    3687             :     /* in supersingular case: will multiply by matrix with denominator p^k
    3688             :      * in mspadicint. Except if p = 2 (multiply by alpha^2) */
    3689         399 :     if (flag) { if (p == 2) n += 2*k-2; else n += k; }
    3690         399 :     pn = powuu(p,n);
    3691             :     /* For accuracy mod p^n, supersingular require p^((2k-1-v_p(a_p))*n) */
    3692         399 :     if (flag) /* k-1 also takes care of a_p = 0. Worst case v_p(a_p) = flag */
    3693         231 :       q = powiu(pn, 2*k-1 - flag);
    3694             :     else
    3695         168 :       q = pn;
    3696             :   }
    3697         483 :   actUp = init_moments_act(Wp, p, n, q, Up_matrices(p));
    3698             : 
    3699         483 :   if (p == 2) C = gen_0;
    3700             :   else
    3701             :   {
    3702         427 :     pas = matpascal(n);
    3703         427 :     teich = teichmullerinit(p, n+1);
    3704         427 :     P = gpowers(utoipos(p), n);
    3705         427 :     C = cgetg(p, t_VEC);
    3706        2317 :     for (a = 1; a < p; a++)
    3707             :     { /* powb[j+1] = ((a - w(a)) / p)^j mod p^n */
    3708        1890 :       GEN powb = Fp_powers(diviuexact(subui(a, gel(teich,a)), p), n, pn);
    3709        1890 :       GEN Ca = cgetg(n+2, t_VEC);
    3710        1890 :       long j, r, ai = Fl_inv(a, p); /* a^(-1) */
    3711        1890 :       gel(C,a) = Ca;
    3712       22134 :       for (j = 0; j <= n; j++)
    3713             :       {
    3714       20244 :         GEN Caj = cgetg(j+2, t_VEC);
    3715       20244 :         GEN atij = gel(teich, Fl_powu(ai,j,p));/* w(a)^(-j) = w(a^(-j) mod p) */
    3716       20244 :         gel(Ca,j+1) = Caj;
    3717      158200 :         for (r = 0; r <= j; r++)
    3718             :         {
    3719      137956 :           GEN c = Fp_mul(gcoeff(pas,j+1,r+1), gel(powb, j-r+1), pn);
    3720      137956 :           c = Fp_mul(c,atij,pn); /* binomial(j,r)*b^(j-r)*w(a)^(-j) mod p^n */
    3721      137956 :           gel(Caj,r+1) = mulii(c, gel(P,j+1)); /* p^j * c mod p^(n+j) */
    3722             :         }
    3723             :       }
    3724             :     }
    3725             :   }
    3726         483 :   return gerepilecopy(av, mkvecn(8, Wp,Tp, bin, actUp, q,
    3727             :                                  mkvecsmall3(p,n,flag), M, C));
    3728             : }
    3729             : 
    3730             : #if 0
    3731             : /* assume phi an ordinary OMS */
    3732             : static GEN
    3733             : omsactgl2(GEN W, GEN phi, GEN M)
    3734             : {
    3735             :   GEN q, Wp, act;
    3736             :   long p, k, n;
    3737             :   checkmspadic(W);
    3738             :   Wp = mspadic_get_Wp(W);
    3739             :   p = mspadic_get_p(W);
    3740             :   k = mspadic_get_weight(W);
    3741             :   n = mspadic_get_n(W);
    3742             :   q = mspadic_get_q(W);
    3743             :   act = init_moments_act(Wp, p, n, q, M);
    3744             :   phi = gel(phi,1);
    3745             :   return dual_act(k-1, act, gel(phi,1));
    3746             : }
    3747             : #endif
    3748             : 
    3749             : static GEN
    3750         483 : eigenvalue(GEN T, GEN x)
    3751             : {
    3752         483 :   long i, l = lg(x);
    3753         637 :   for (i = 1; i < l; i++)
    3754         637 :     if (!isintzero(gel(x,i))) break;
    3755         483 :   if (i == l) pari_err_DOMAIN("mstooms", "phi", "=", gen_0, x);
    3756         483 :   return gdiv(RgMrow_RgC_mul(T,x,i), gel(x,i));
    3757             : }
    3758             : 
    3759             : /* p coprime to ap, return unit root of x^2 - ap*x + p^(k-1), accuracy p^n */
    3760             : GEN
    3761         532 : mspadic_unit_eigenvalue(GEN ap, long k, GEN p, long n)
    3762             : {
    3763         532 :   GEN sqrtD, D = subii(sqri(ap), shifti(powiu(p,k-1),2));
    3764         532 :   if (absequaliu(p,2))
    3765             :   {
    3766          35 :     n++; sqrtD = Zp_sqrt(D, p, n);
    3767          35 :     if (mod4(sqrtD) != mod4(ap)) sqrtD = negi(sqrtD);
    3768             :   }
    3769             :   else
    3770         497 :     sqrtD = Zp_sqrtlift(D, ap, p, n);
    3771             :   /* sqrtD = ap (mod p) */
    3772         532 :   return gmul2n(gadd(ap, cvtop(sqrtD,p,n)), -1);
    3773             : }
    3774             : 
    3775             : /* W = msinit(N,k,...); phi = T_p/U_p - eigensymbol */
    3776             : GEN
    3777         483 : mstooms(GEN W, GEN phi)
    3778             : {
    3779         483 :   pari_sp av = avma;
    3780             :   GEN Wp, bin, Tp, c, alpha, ap, phi0, M;
    3781             :   long k, p, vden;
    3782             : 
    3783         483 :   checkmspadic(W);
    3784         483 :   if (typ(phi) != t_COL)
    3785             :   {
    3786         161 :     if (!is_Qevproj(phi)) pari_err_TYPE("mstooms",phi);
    3787         161 :     phi = gel(phi,1);
    3788         161 :     if (lg(phi) != 2) pari_err_TYPE("mstooms [dim_Q (eigenspace) > 1]",phi);
    3789         161 :     phi = gel(phi,1);
    3790             :   }
    3791             : 
    3792         483 :   Wp = mspadic_get_Wp(W);
    3793         483 :   Tp = mspadic_get_Tp(W);
    3794         483 :   bin = mspadic_get_bin(W);
    3795         483 :   k = msk_get_weight(Wp);
    3796         483 :   p = mspadic_get_p(W);
    3797         483 :   M = mspadic_get_M(W);
    3798             : 
    3799         483 :   phi = Q_remove_denom(phi, &c);
    3800         483 :   ap = eigenvalue(Tp, phi);
    3801         483 :   vden = c? Z_lvalrem(c, p, &c): 0;
    3802             : 
    3803         483 :   if (typ(M) == t_INT)
    3804             :   { /* p | N */
    3805             :     GEN c1;
    3806          84 :     alpha = ap;
    3807          84 :     alpha = ginv(alpha);
    3808          84 :     phi0 = mseval(Wp, phi, NULL);
    3809          84 :     phi0 = RgXC_to_moments(phi0, bin);
    3810          84 :     phi0 = Q_remove_denom(phi0, &c1);
    3811          84 :     if (c1) { vden += Z_lvalrem(c1, p, &c1); c = mul_denom(c,c1); }
    3812          84 :     if (umodiu(ap,p)) /* p \nmid a_p */
    3813          49 :       phi = oms_dim1(W, phi0, alpha, 0);
    3814             :     else
    3815             :     {
    3816          35 :       phi = oms_dim1(W, phi0, alpha, 1);
    3817          35 :       phi = Q_remove_denom(phi, &c1);
    3818          35 :       if (c1) { vden += Z_lvalrem(c1, p, &c1); c = mul_denom(c,c1); }
    3819             :     }
    3820             :   }
    3821             :   else
    3822             :   { /* p-stabilize */
    3823             :     GEN M1, M2, phi1, phi2, c1;
    3824         399 :     if (typ(M) != t_VEC || lg(M) != 3) pari_err_TYPE("mstooms",W);
    3825         399 :     M1 = gel(M,1);
    3826         399 :     M2 = gel(M,2);
    3827             : 
    3828         399 :     phi1 = RgM_RgC_mul(M1, phi);
    3829         399 :     phi2 = RgM_RgC_mul(M2, phi);
    3830         399 :     phi1 = mseval(Wp, phi1, NULL);
    3831         399 :     phi2 = mseval(Wp, phi2, NULL);
    3832             : 
    3833         399 :     phi1 = RgXC_to_moments(phi1, bin);
    3834         399 :     phi2 = RgXC_to_moments(phi2, bin);
    3835         399 :     phi = Q_remove_denom(mkvec2(phi1,phi2), &c1);
    3836         399 :     phi1 = gel(phi,1);
    3837         399 :     phi2 = gel(phi,2);
    3838         399 :     if (c1) { vden += Z_lvalrem(c1, p, &c1); c = mul_denom(c,c1); }
    3839             :     /* all polynomials multiplied by c p^vden */
    3840         399 :     if (umodiu(ap, p))
    3841             :     {
    3842         238 :       alpha = mspadic_unit_eigenvalue(ap, k, utoipos(p), mspadic_get_n(W));
    3843         238 :       alpha = ginv(alpha);
    3844         238 :       phi0 = gsub(phi1, gmul(lift_shallow(alpha),phi2));
    3845         238 :       phi = oms_dim1(W, phi0, alpha, 0);
    3846             :     }
    3847             :     else
    3848             :     { /* p | ap, alpha = [a_p, -1; p^(k-1), 0] */
    3849         161 :       long flag = mspadic_get_flag(W);
    3850         161 :       if (!flag || (signe(ap) && Z_lval(ap,p) < flag))
    3851           7 :         pari_err_TYPE("mstooms [v_p(ap) > mspadicinit flag]", phi);
    3852         154 :       alpha = mkmat22(ap,gen_m1, powuu(p, k-1),gen_0);
    3853         154 :       alpha = ginv(alpha);
    3854         154 :       phi = oms_dim2(W, mkvec2(phi1,phi2), gsqr(alpha), ap);
    3855         154 :       phi = Q_remove_denom(phi, &c1);
    3856         154 :       if (c1) { vden += Z_lvalrem(c1, p, &c1); c = mul_denom(c,c1); }
    3857             :     }
    3858             :   }
    3859         476 :   if (vden) c = mul_denom(c, powuu(p,vden));
    3860         476 :   if (p == 2) alpha = gsqr(alpha);
    3861         476 :   if (c) alpha = gdiv(alpha,c);
    3862         476 :   if (typ(alpha) == t_MAT)
    3863             :   { /* express in basis (omega,-p phi(omega)) */
    3864         154 :     gcoeff(alpha,2,1) = gdivgs(gcoeff(alpha,2,1), -p);
    3865         154 :     gcoeff(alpha,2,2) = gdivgs(gcoeff(alpha,2,2), -p);
    3866             :     /* at the end of mspadicint we shall multiply result by [1,0;0,-1/p]*alpha
    3867             :      * vden + k is the denominator of this matrix */
    3868             :   }
    3869             :   /* phi is integral-valued */
    3870         476 :   return gerepilecopy(av, mkcol3(phi, stoi(vden), alpha));
    3871             : }
    3872             : 
    3873             : /* HACK: the v[j] have different lengths */
    3874             : static GEN
    3875        2156 : FpVV_dotproduct(GEN v, GEN w, GEN p)
    3876             : {
    3877        2156 :   long j, l = lg(v);
    3878        2156 :   GEN T = cgetg(l, t_VEC);
    3879       26026 :   for (j = 1; j < l; j++) gel(T,j) = FpV_dotproduct(gel(v,j),w,p);
    3880        2156 :   return T;
    3881             : }
    3882             : 
    3883             : /* \int (-4z)^j given \int z^j */
    3884             : static GEN
    3885          98 : twistmoment_m4(GEN v)
    3886             : {
    3887             :   long i, l;
    3888          98 :   GEN w = cgetg_copy(v, &l);
    3889        2009 :   for (i = 1; i < l; i++)
    3890             :   {
    3891        1911 :     GEN c = gel(v,i);
    3892        1911 :     if (i > 1) c = gmul2n(c, (i-1)<<1);
    3893        1911 :     gel(w,i) = odd(i)? c: gneg(c);
    3894             :   }
    3895          98 :   return w;
    3896             : }
    3897             : /* \int (4z)^j given \int z^j */
    3898             : static GEN
    3899          98 : twistmoment_4(GEN v)
    3900             : {
    3901             :   long i, l;
    3902          98 :   GEN w = cgetg_copy(v, &l);
    3903        2009 :   for (i = 1; i < l; i++)
    3904             :   {
    3905        1911 :     GEN c = gel(v,i);
    3906        1911 :     if (i > 1) c = gmul2n(c, (i-1)<<1);
    3907        1911 :     gel(w,i) = c;
    3908             :   }
    3909          98 :   return w;
    3910             : }
    3911             : /* W an mspadic, phi eigensymbol, p \nmid D. Return C(x) mod FilM */
    3912             : GEN
    3913         483 : mspadicmoments(GEN W, GEN PHI, long D)
    3914             : {
    3915         483 :   pari_sp av = avma;
    3916         483 :   long na, ia, b, lphi, aD = labs(D), pp, p, k, n, vden;
    3917             :   GEN Wp, Dact, vL, v, C, pn, phi;
    3918             :   struct m_act S;
    3919             :   hashtable *H;
    3920             : 
    3921         483 :   checkmspadic(W);
    3922         483 :   Wp = mspadic_get_Wp(W);
    3923         483 :   p = mspadic_get_p(W);
    3924         483 :   k = mspadic_get_weight(W);
    3925         483 :   n = mspadic_get_n(W);
    3926         483 :   C = mspadic_get_C(W);
    3927         483 :   if (typ(PHI) != t_COL || lg(PHI) != 4 || typ(gel(PHI,1)) != t_VEC)
    3928         476 :     PHI = mstooms(W, PHI);
    3929         476 :   vden = itos( gel(PHI,2) );
    3930         476 :   phi = gel(PHI,1); lphi = lg(phi);
    3931         476 :   if (p == 2) { na = 2; pp = 4; }
    3932         420 :   else        { na = p-1; pp = p; }
    3933         476 :   pn = powuu(p, n + vden);
    3934             : 
    3935         476 :   S.p = p;
    3936         476 :   S.k = k;
    3937         476 :   S.q = pn;
    3938         476 :   S.dim = n+k-1;
    3939         476 :   S.act = &moments_act;
    3940         476 :   H = Gl2act_cache(ms_get_nbgen(Wp));
    3941         476 :   if (D == 1) Dact = NULL;
    3942             :   else
    3943             :   {
    3944          63 :     GEN gaD = utoi(aD), Dk = Fp_pows(stoi(D), 2-k, pn);
    3945          63 :     if (!sisfundamental(D)) pari_err_TYPE("mspadicmoments", stoi(D));
    3946          63 :     if (D % p == 0) pari_err_DOMAIN("mspadicmoments","p","|", stoi(D), utoi(p));
    3947          63 :     Dact = cgetg(aD, t_VEC);
    3948         532 :     for (b = 1; b < aD; b++)
    3949             :     {
    3950         469 :       GEN z = NULL;
    3951         469 :       long s = kross(D, b);
    3952         469 :       if (s)
    3953             :       {
    3954         462 :         pari_sp av2 = avma;
    3955             :         GEN d;
    3956         462 :         z = moments_act_i(&S, mkmat22(gaD,utoipos(b), gen_0,gaD));
    3957         462 :         d = s > 0? Dk: Fp_neg(Dk, pn);
    3958         924 :         z = equali1(d)? gerepilecopy(av2, z)
    3959         462 :                       : gerepileupto(av2, FpM_Fp_mul(z, d, pn));
    3960             :       }
    3961         469 :       gel(Dact,b) = z;
    3962             :     }
    3963             :   }
    3964         476 :   vL = cgetg(na+1,t_VEC);
    3965             :   /* first pass to precompute log(paths), preload matrices and allow GC later */
    3966        2464 :   for (ia = 1; ia <= na; ia++)
    3967             :   {
    3968             :     GEN path, La;
    3969        1988 :     long a = (p == 2 && ia == 2)? -1: ia;
    3970        1988 :     if (Dact)
    3971             :     { /* twist by D */
    3972         224 :       La = cgetg(aD, t_VEC);
    3973        1442 :       for (b = 1; b < aD; b++)
    3974             :       {
    3975        1218 :         GEN Actb = gel(Dact,b);
    3976        1218 :         if (!Actb) continue;
    3977             :         /* oo -> a/pp + b/|D|*/
    3978        1176 :         path = mkmat22(gen_1, addii(mulss(a, aD), muluu(pp, b)),
    3979             :                        gen_0, muluu(pp, aD));
    3980        1176 :         gel(La,b) = M2_logf(Wp,path,NULL);
    3981        1176 :         ZGl2QC_preload(&S, gel(La,b), H);
    3982             :       }
    3983             :     }
    3984             :     else
    3985             :     {
    3986        1764 :       path = mkmat22(gen_1,stoi(a), gen_0, utoipos(pp));
    3987        1764 :       La = M2_logf(Wp,path,NULL);
    3988        1764 :       ZGl2QC_preload(&S, La, H);
    3989             :     }
    3990        1988 :     gel(vL,ia) = La;
    3991             :   }
    3992         476 :   v = cgetg(na+1,t_VEC);
    3993             :   /* second pass, with GC */
    3994        2464 :   for (ia = 1; ia <= na; ia++)
    3995             :   {
    3996        1988 :     pari_sp av2 = avma;
    3997        1988 :     GEN vca, Ca = gel(C,ia), La = gel(vL,ia), va = cgetg(lphi, t_VEC);
    3998             :     long i;
    3999        1988 :     if (!Dact) vca = omseval_int(&S, phi, La, H);
    4000             :     else
    4001             :     { /* twist by D */
    4002         224 :       vca = cgetg(lphi,t_VEC);
    4003        1442 :       for (b = 1; b < aD; b++)
    4004             :       {
    4005        1218 :         GEN T, Actb = gel(Dact,b);
    4006        1218 :         if (!Actb) continue;
    4007        1176 :         T = omseval_int(&S, phi, gel(La,b), H);
    4008        2352 :         for (i = 1; i < lphi; i++)
    4009             :         {
    4010        1176 :           GEN z = FpM_FpC_mul(Actb, gel(T,i), pn);
    4011        1176 :           gel(vca,i) = b==1? z: ZC_add(gel(vca,i), z);
    4012             :         }
    4013             :       }
    4014             :     }
    4015        1988 :     if (p != 2)
    4016        4032 :     { for (i=1; i<lphi; i++) gel(va,i) = FpVV_dotproduct(Ca,gel(vca,i),pn); }
    4017         112 :     else if (ia == 1) /* \tilde{a} = 1 */
    4018         154 :     { for (i=1; i<lphi; i++) gel(va,i) = twistmoment_4(gel(vca,i)); }
    4019             :     else /* \tilde{a} = -1 */
    4020         154 :     { for (i=1; i<lphi; i++) gel(va,i) = twistmoment_m4(gel(vca,i)); }
    4021        1988 :     gel(v,ia) = gerepilecopy(av2, va);
    4022             :   }
    4023         476 :   return gerepilecopy(av, mkvec3(v, gel(PHI,3), mkvecsmall4(p,n+vden,n,D)));
    4024             : }
    4025             : static void
    4026        1918 : checkoms(GEN v)
    4027             : {
    4028        1918 :   if (typ(v) != t_VEC || lg(v) != 4 || typ(gel(v,1)) != t_VEC
    4029        1918 :       || typ(gel(v,3))!=t_VECSMALL)
    4030           0 :     pari_err_TYPE("checkoms [apply mspadicmoments]", v);
    4031        1918 : }
    4032             : static long
    4033        4284 : oms_get_p(GEN oms) { return gel(oms,3)[1]; }
    4034             : static long
    4035        4186 : oms_get_n(GEN oms) { return gel(oms,3)[2]; }
    4036             : static long
    4037        2464 : oms_get_n0(GEN oms) { return gel(oms,3)[3]; }
    4038             : static long
    4039        1918 : oms_get_D(GEN oms) { return gel(oms,3)[4]; }
    4040             : static int
    4041          98 : oms_is_supersingular(GEN oms) { GEN v = gel(oms,1); return lg(gel(v,1)) == 3; }
    4042             : 
    4043             : /* sum(j = 1, n, (-1)^(j+1)/j * x^j) */
    4044             : static GEN
    4045         784 : log1x(long n)
    4046             : {
    4047         784 :   long i, l = n+3;
    4048         784 :   GEN v = cgetg(l, t_POL);
    4049         784 :   v[1] = evalvarn(0)|evalsigne(1); gel(v,2) = gen_0;
    4050        8904 :   for (i = 3; i < l; i++) gel(v,i) = ginv(stoi(odd(i)? i-2: 2-i));
    4051         784 :   return v;
    4052             : }
    4053             : 
    4054             : /* S = (1+x)^zk log(1+x)^logj (mod x^(n+1)) */
    4055             : static GEN
    4056        1820 : xlog1x(long n, long zk, long logj, long *pteich)
    4057             : {
    4058        1820 :   GEN S = logj? RgXn_powu_i(log1x(n), logj, n+1): NULL;
    4059        1820 :   if (zk)
    4060             :   {
    4061        1183 :     GEN L = deg1pol_shallow(gen_1, gen_1, 0); /* x+1 */
    4062        1183 :     *pteich += zk;
    4063        1183 :     if (zk < 0) { L = RgXn_inv(L,n+1); zk = -zk; }
    4064        1183 :     if (zk != 1) L = RgXn_powu_i(L, zk, n+1);
    4065        1183 :     S = S? RgXn_mul(S, L, n+1): L;
    4066             :   }
    4067        1820 :   return S;
    4068             : }
    4069             : 
    4070             : /* oms from mspadicmoments; integrate teichmuller^i * S(x) [S = NULL: 1]*/
    4071             : static GEN
    4072        2366 : mspadicint(GEN oms, long teichi, GEN S)
    4073             : {
    4074        2366 :   pari_sp av = avma;
    4075        2366 :   long p = oms_get_p(oms), n = oms_get_n(oms), n0 = oms_get_n0(oms);
    4076        2366 :   GEN vT = gel(oms,1), alpha = gel(oms,2), gp = utoipos(p);
    4077        2366 :   long loss = S? Z_lval(Q_denom(S), p): 0;
    4078        2366 :   long nfinal = minss(n-loss, n0);
    4079        2366 :   long i, la, l = lg(gel(vT,1));
    4080        2366 :   GEN res = cgetg(l, t_COL), teich = NULL;
    4081             : 
    4082        2366 :   if (S) S = RgX_to_RgC(S,lg(gmael(vT,1,1))-1);
    4083        2366 :   if (p == 2)
    4084             :   {
    4085         448 :     la = 3; /* corresponds to [1,-1] */
    4086         448 :     teichi &= 1;
    4087             :   }
    4088             :   else
    4089             :   {
    4090        1918 :     la = p; /* corresponds to [1,2,...,p-1] */
    4091        1918 :     teichi = umodsu(teichi, p-1);
    4092        1918 :     if (teichi) teich = teichmullerinit(p, n);
    4093             :   }
    4094        5446 :   for (i=1; i<l; i++)
    4095             :   {
    4096        3080 :     pari_sp av2 = avma;
    4097        3080 :     GEN s = gen_0;
    4098             :     long ia;
    4099       14756 :     for (ia = 1; ia < la; ia++)
    4100             :     { /* Ta[j+1] correct mod p^n */
    4101       11676 :       GEN Ta = gmael(vT,ia,i), v = S? RgV_dotproduct(Ta, S): gel(Ta,1);
    4102       11676 :       if (teichi && ia != 1)
    4103             :       {
    4104        3843 :         if (p != 2)
    4105        3626 :           v = gmul(v, gel(teich, Fl_powu(ia,teichi,p)));
    4106             :         else
    4107         217 :           if (teichi) v = gneg(v);
    4108             :       }
    4109       11676 :       s = gadd(s, v);
    4110             :     }
    4111        3080 :     s = gadd(s, zeropadic(gp,nfinal));
    4112        3080 :     gel(res,i) = gerepileupto(av2, s);
    4113             :   }
    4114        2366 :   return gerepileupto(av, gmul(alpha, res));
    4115             : }
    4116             : /* integrate P = polynomial in log(x); vlog[j+1] = mspadicint(0,log(1+x)^j) */
    4117             : static GEN
    4118         539 : mspadicint_RgXlog(GEN P, GEN vlog)
    4119             : {
    4120         539 :   long i, d = degpol(P);
    4121         539 :   GEN s = gmul(gel(P,2), gel(vlog,1));
    4122        1848 :   for (i = 1; i <= d; i++) s = gadd(s, gmul(gel(P,i+2), gel(vlog,i+1)));
    4123         539 :   return s;
    4124             : };
    4125             : 
    4126             : /* oms from mspadicmoments */
    4127             : GEN
    4128          98 : mspadicseries(GEN oms, long teichi)
    4129             : {
    4130          98 :   pari_sp av = avma;
    4131             :   GEN S, L, X, vlog, s, s2, u, logu, bin;
    4132             :   long j, p, m, n, step, stop;
    4133          98 :   checkoms(oms);
    4134          98 :   n = oms_get_n0(oms);
    4135          98 :   if (n < 1)
    4136             :   {
    4137           0 :     s = zeroser(0,0);
    4138           0 :     if (oms_is_supersingular(oms)) s = mkvec2(s,s);
    4139           0 :     return gerepilecopy(av, s);
    4140             :   }
    4141          98 :   p = oms_get_p(oms);
    4142          98 :   vlog = cgetg(n+1, t_VEC);
    4143          98 :   step = p == 2? 2: 1;
    4144          98 :   stop = 0;
    4145          98 :   S = NULL;
    4146          98 :   L = log1x(n);
    4147         644 :   for (j = 0; j < n; j++)
    4148             :   {
    4149         616 :     if (j) stop += step + u_lval(j,p); /* = step*j + v_p(j!) */
    4150         616 :     if (stop >= n) break;
    4151             :     /* S = log(1+x)^j */
    4152         546 :     gel(vlog,j+1) = mspadicint(oms,teichi,S);
    4153         546 :     S = S? RgXn_mul(S, L, n+1): L;
    4154             :   }
    4155          98 :   m = j;
    4156          98 :   u = utoipos(p == 2? 5: 1+p);
    4157          98 :   logu = glog(cvtop(u, utoipos(p), 4*m), 0);
    4158          98 :   X = gdiv(pol_x(0), logu);
    4159          98 :   s = cgetg(m+1, t_VEC);
    4160          98 :   s2 = oms_is_supersingular(oms)? cgetg(m+1, t_VEC): NULL;
    4161          98 :   bin = pol_1(0);
    4162         539 :   for (j = 0; j < m; j++)
    4163             :   { /* bin = binomial(x/log(1+p+O(p^(4*n))), j) mod x^m */
    4164         539 :     GEN a, v = mspadicint_RgXlog(bin, vlog);
    4165         539 :     int done = 1;
    4166         539 :     gel(s,j+1) = a = gel(v,1);
    4167         539 :     if (!gequal0(a) || valp(a) > 0) done = 0; else setlg(s,j+1);
    4168         539 :     if (s2)
    4169             :     {
    4170         119 :       gel(s2,j+1) = a = gel(v,2);
    4171         119 :       if (!gequal0(a) || valp(a) > 0) done = 0; else setlg(s2,j+1);
    4172             :     }
    4173         539 :     if (done || j == m-1) break;
    4174         441 :     bin = RgXn_mul(bin, gdivgs(gsubgs(X, j), j+1), m);
    4175             :   }
    4176          98 :   s = RgV_to_ser(s,0,lg(s)+1);
    4177          98 :   if (s2) { s2 = RgV_to_ser(s2,0,lg(s2)+1); s = mkvec2(s, s2); }
    4178          98 :   if (kross(oms_get_D(oms), p) >= 0) return gerepilecopy(av, s);
    4179           7 :   return gerepileupto(av, gneg(s));
    4180             : }
    4181             : void
    4182        1911 : mspadic_parse_chi(GEN s, GEN *s1, GEN *s2)
    4183             : {
    4184        1911 :   if (!s) *s1 = *s2 = gen_0;
    4185        1778 :   else switch(typ(s))
    4186             :   {
    4187        1274 :     case t_INT: *s1 = *s2 = s; break;
    4188         504 :     case t_VEC:
    4189         504 :       if (lg(s) == 3)
    4190             :       {
    4191         504 :         *s1 = gel(s,1);
    4192         504 :         *s2 = gel(s,2);
    4193         504 :         if (typ(*s1) == t_INT && typ(*s2) == t_INT) break;
    4194             :       }
    4195           0 :     default: pari_err_TYPE("mspadicL",s);
    4196           0 :              *s1 = *s2 = NULL;
    4197             :   }
    4198        1911 : }
    4199             : /* oms from mspadicmoments
    4200             :  * r-th derivative of L(f,chi^s,psi) in direction <chi>
    4201             :    - s \in Z_p \times \Z/(p-1)\Z, s-> chi^s=<\chi>^s_1 omega^s_2)
    4202             :    - Z -> Z_p \times \Z/(p-1)\Z par s-> (s, s mod p-1).
    4203             :  */
    4204             : GEN
    4205        1820 : mspadicL(GEN oms, GEN s, long r)
    4206             : {
    4207        1820 :   pari_sp av = avma;
    4208             :   GEN s1, s2, z, S;
    4209             :   long p, n, teich;
    4210        1820 :   checkoms(oms);
    4211        1820 :   p = oms_get_p(oms);
    4212        1820 :   n = oms_get_n(oms);
    4213        1820 :   mspadic_parse_chi(s, &s1,&s2);
    4214        1820 :   teich = umodiu(subii(s2,s1), p==2? 2: p-1);
    4215        1820 :   S = xlog1x(n, itos(s1), r, &teich);
    4216        1820 :   z = mspadicint(oms, teich, S);
    4217        1820 :   if (lg(z) == 2) z = gel(z,1);
    4218        1820 :   if (kross(oms_get_D(oms), p) < 0) z = gneg(z);
    4219        1820 :   return gerepilecopy(av, z);
    4220             : }
    4221             : 
    4222             : /****************************************************************************/
    4223             : 
    4224             : struct siegel
    4225             : {
    4226             :   GEN V, Ast;
    4227             :   long N; /* level */
    4228             :   long oo; /* index of the [oo,0] path */
    4229             :   long k1, k2; /* two distinguished indices */
    4230             :   long n; /* #W, W = initial segment [in siegelstepC] already normalized */
    4231             : };
    4232             : 
    4233             : static void
    4234         378 : siegel_init(struct siegel *C, GEN M)
    4235             : {
    4236             :   GEN CPI, CP, MM, V, W, Ast;
    4237         378 :   GEN m = gel(M,11), M2 = gel(M,2), S = msN_get_section(M);
    4238         378 :   GEN E2fromE1 = msN_get_E2fromE1(M);
    4239         378 :   long m0 = lg(M2)-1;
    4240         378 :   GEN E2  = vecslice(M2, m[1]+1, m[2]);/* E2 */
    4241         378 :   GEN E1T = vecslice(M2, m[3]+1, m0); /* E1,T2,T31 */
    4242         378 :   GEN L = shallowconcat(E1T, E2);
    4243         378 :   long i, l = lg(L), n = lg(E1T)-1, lE = lg(E2);
    4244             : 
    4245         378 :   Ast = cgetg(l, t_VECSMALL);
    4246        6195 :   for (i = 1; i < lE; ++i)
    4247             :   {
    4248        5817 :     long j = E2fromE1_c(gel(E2fromE1,i));
    4249        5817 :     Ast[n+i] = j;
    4250        5817 :     Ast[j] = n+i;
    4251             :   }
    4252         686 :   for (; i<=n; ++i) Ast[i] = i;
    4253         378 :   MM = cgetg (l,t_VEC);
    4254             : 
    4255       12320 :   for (i = 1; i < l; i++)
    4256             :   {
    4257       11942 :     GEN c = gel(S, L[i]);
    4258       11942 :     long c12, c22, c21 = ucoeff(c,2,1);
    4259       11942 :     if (!c21) { gel(MM,i) = gen_0; continue; }
    4260       11564 :     c22 = ucoeff(c,2,2);
    4261       11564 :     if (!c22) { gel(MM,i) = gen_m1; continue; }
    4262       11186 :     c12 = ucoeff(c,1,2);
    4263       11186 :     gel(MM,i) = sstoQ(c12, c22); /* right extremity > 0 */
    4264             :   }
    4265         378 :   CP = indexsort(MM);
    4266         378 :   CPI = cgetg(l, t_VECSMALL);
    4267         378 :   V = cgetg(l, t_VEC);
    4268         378 :   W = cgetg(l, t_VECSMALL);
    4269       12320 :   for (i = 1; i < l; ++i)
    4270             :   {
    4271       11942 :     gel(V,i) = mat2_to_ZM(gel(S, L[CP[i]]));
    4272       11942 :     CPI[CP[i]] = i;
    4273             :   }
    4274       12320 :   for (i = 1; i < l; ++i) W[CPI[i]] = CPI[Ast[i]];
    4275         378 :   C->V = V;
    4276         378 :   C->Ast = W;
    4277         378 :   C->n = 0;
    4278         378 :   C->oo = 2;
    4279         378 :   C->N = ms_get_N(M);
    4280         378 : }
    4281             : 
    4282             : static double
    4283           0 : ZMV_size(GEN v)
    4284             : {
    4285           0 :   long i, l = lg(v);
    4286           0 :   GEN z = cgetg(l, t_VECSMALL);
    4287           0 :   for (i = 1; i < l; i++) z[i] = gexpo(gel(v,i));
    4288           0 :   return ((double)zv_sum(z)) / (4*(l-1));
    4289             : }
    4290             : 
    4291             : /* apply permutation perm to struct S. Don't follow k1,k2 */
    4292             : static void
    4293        5558 : siegel_perm0(struct siegel *S, GEN perm)
    4294             : {
    4295        5558 :   pari_sp av = avma;
    4296        5558 :   long i, l = lg(S->V);
    4297        5558 :   GEN V2 = cgetg(l, t_VEC), Ast2 = cgetg(l, t_VECSMALL);
    4298        5558 :   GEN V = S->V, Ast = S->Ast;
    4299             : 
    4300      267078 :   for (i = 1; i < l; i++) gel(V2,perm[i]) = gel(V,i);
    4301      267078 :   for (i = 1; i < l; i++) Ast2[perm[i]] = perm[Ast[i]];
    4302      267078 :   for (i = 1; i < l; i++) { S->Ast[i] = Ast2[i]; gel(V,i) = gel(V2,i); }
    4303        5558 :   set_avma(av); S->oo = perm[S->oo];
    4304        5558 : }
    4305             : /* apply permutation perm to full struct S */
    4306             : static void
    4307        5194 : siegel_perm(struct siegel *S, GEN perm)
    4308             : {
    4309        5194 :   siegel_perm0(S, perm);
    4310        5194 :   S->k1 = perm[S->k1];
    4311        5194 :   S->k2 = perm[S->k2];
    4312        5194 : }
    4313             : /* cyclic permutation of lg = l-1 moving a -> 1, a+1 -> 2, etc.  */
    4314             : static GEN
    4315        2884 : rotate_perm(long l, long a)
    4316             : {
    4317        2884 :   GEN p = cgetg(l, t_VECSMALL);
    4318        2884 :   long i, j = 1;
    4319       86905 :   for (i = a; i < l; i++) p[i] = j++;
    4320       49329 :   for (i = 1; i < a; i++) p[i] = j++;
    4321        2884 :   return p;
    4322             : }
    4323             : 
    4324             : /* a1 < c1 <= a2 < c2*/
    4325             : static GEN
    4326        2520 : basic_op_perm(long l, long a1, long a2, long c1, long c2)
    4327             : {
    4328        2520 :   GEN p = cgetg(l, t_VECSMALL);
    4329        2520 :   long i, j = 1;
    4330        2520 :   p[a1] = j++;
    4331       22568 :   for (i = c1; i < a2; i++)   p[i] = j++;
    4332       32284 :   for (i = a1+1; i < c1; i++) p[i] = j++;
    4333        2520 :   p[a2] = j++;
    4334       44891 :   for (i = c2; i < l; i++)    p[i] = j++;
    4335        2520 :   for (i = 1; i < a1; i++)    p[i] = j++;
    4336       29855 :   for (i = a2+1; i < c2; i++) p[i] = j++;
    4337        2520 :   return p;
    4338             : }
    4339             : static GEN
    4340         154 : basic_op_perm_elliptic(long l, long a1)
    4341             : {
    4342         154 :   GEN p = cgetg(l, t_VECSMALL);
    4343         154 :   long i, j = 1;
    4344         154 :   p[a1] = j++;
    4345        2660 :   for (i = 1; i < a1; i++)   p[i] = j++;
    4346        3990 :   for (i = a1+1; i < l; i++) p[i] = j++;
    4347         154 :   return p;
    4348             : }
    4349             : static GEN
    4350       14616 : ZM2_rev(GEN T) { return mkmat2(gel(T,2), ZC_neg(gel(T,1))); }
    4351             : 
    4352             : /* In place, V = vector of consecutive paths, between x <= y.
    4353             :  * V[x..y-1] <- g*V[x..y-1] */
    4354             : static void
    4355        5733 : path_vec_mul(GEN V, long x, long y, GEN g)
    4356             : {
    4357             :   long j;
    4358             :   GEN M;
    4359        5733 :   if (x == y) return;
    4360        3360 :   M = gel(V,x); gel(V,x) = ZM_mul(g,M);
    4361       37709 :   for (j = x+1; j < y; j++) /* V[j] <- g*V[j], optimized */
    4362             :   {
    4363       34349 :     GEN Mnext = gel(V,j); /* Mnext[,1] = M[,2] */
    4364       34349 :     GEN gM = gel(V,j-1), u = gel(gM,2);
    4365       34349 :     if (!ZV_equal(gel(M,2), gel(Mnext,1))) u = ZC_neg(u);
    4366       34349 :     gel(V,j) = mkmat2(u, ZM_ZC_mul(g,gel(Mnext,2)));
    4367       34349 :     M = Mnext;
    4368             :   }
    4369             : }
    4370             : 
    4371        4830 : static long prev(GEN V, long i) { return (i == 1)? lg(V)-1: i-1; }
    4372        4830 : static long next(GEN V, long i) { return (i == lg(V)-1)? 1: i+1; }
    4373             : static GEN
    4374       19810 : ZM_det2(GEN u, GEN v)
    4375             : {
    4376       19810 :   GEN a = gel(u,1), c = gel(u,2);
    4377       19810 :   GEN b = gel(v,1), d = gel(v,2); return subii(mulii(a,d), mulii(b,c));
    4378             : }
    4379             : static GEN
    4380       14616 : ZM2_det(GEN T) { return ZM_det2(gel(T,1),gel(T,2)); }
    4381             : static long
    4382        5194 : ZM_det2_sign(GEN u, GEN v)
    4383             : {
    4384        5194 :   pari_sp av = avma;
    4385        5194 :   long s = signe(ZM_det2(u, v));
    4386        5194 :   return gc_long(av, s);
    4387             : }
    4388             : static void
    4389        4466 : fill1(GEN V, long a)
    4390             : {
    4391        4466 :   long p = prev(V,a), n = next(V,a);
    4392        4466 :   GEN u = gmael(V,p,2), v = gmael(V,n,1);
    4393        4466 :   if (ZM_det2_sign(u,v) < 0) v = ZC_neg(v);
    4394        4466 :   gel(V,a) = mkmat2(u, v);
    4395        4466 : }
    4396             : /* a1 < a2 */
    4397             : static void
    4398        2520 : fill2(GEN V, long a1, long a2)
    4399             : {
    4400        2520 :   if (a2 != a1+1) { fill1(V,a1); fill1(V,a2); } /* non adjacent, reconnect */
    4401             :   else
    4402             :   { /* parabolic */
    4403         364 :     long p = prev(V,a1), n = next(V,a2);
    4404         364 :     GEN u, v, C = gmael(V,a1,2), mC = NULL; /* = \pm V[a2][1] */
    4405         364 :     u = gmael(V,p,2); v = C;
    4406         364 :     if (ZM_det2_sign(u,v) < 0) v = mC = ZC_neg(C);
    4407         364 :     gel(V,a1) = mkmat2(u,v);
    4408         364 :     v = gmael(V,n,1); u = C;
    4409         364 :     if (ZM_det2_sign(u,v) < 0) u = mC? mC: ZC_neg(C);
    4410         364 :     gel(V,a2) = mkmat2(u,v);
    4411             :   }
    4412        2520 : }
    4413             : 
    4414             : /* DU = det(U), return g = T*U^(-1) or NULL if not in Gamma0(N); if N = 0,
    4415             :  * only test whether g is integral */
    4416             : static GEN
    4417       14903 : ZM2_div(GEN T, GEN U, GEN DU, long N)
    4418             : {
    4419       14903 :   GEN a=gcoeff(U,1,1), b=gcoeff(U,1,2), c=gcoeff(U,2,1), d=gcoeff(U,2,2);
    4420       14903 :   GEN e=gcoeff(T,1,1), f=gcoeff(T,1,2), g=gcoeff(T,2,1), h=gcoeff(T,2,2);
    4421             :   GEN A, B, C, D, r;
    4422             : 
    4423       14903 :   C = dvmdii(subii(mulii(d,g), mulii(c,h)), DU, &r);
    4424       14903 :   if (r != gen_0 || (N && smodis(C,N))) return NULL;
    4425       14616 :   A = dvmdii(subii(mulii(d,e), mulii(c,f)), DU, &r);
    4426       14616 :   if (r != gen_0) return NULL;
    4427       14616 :   B = dvmdii(subii(mulii(a,f), mulii(b,e)), DU, &r);
    4428       14616 :   if (r != gen_0) return NULL;
    4429       14616 :   D = dvmdii(subii(mulii(a,h), mulii(g,b)), DU, &r);
    4430       14616 :   if (r != gen_0) return NULL;
    4431       14616 :   return mkmat22(A,B,C,D);
    4432             : }
    4433             : 
    4434             : static GEN
    4435       14616 : get_g(struct siegel *S, long a1)
    4436             : {
    4437       14616 :   pari_sp av = avma;
    4438       14616 :   long a2 = S->Ast[a1];
    4439       14616 :   GEN a = gel(S->V,a1), ar = ZM2_rev(gel(S->V,a2)), Dar = ZM2_det(ar);
    4440       14616 :   GEN g = ZM2_div(a, ar, Dar, S->N);
    4441       14616 :   if (!g)
    4442             :   {
    4443         287 :     GEN tau = mkmat22(gen_0,gen_m1, gen_1,gen_m1); /*[0,-1;1,-1]*/
    4444         287 :     g = ZM2_div(ZM_mul(ar, tau), ar, Dar, 0);
    4445             :   }
    4446       14616 :   return gerepilecopy(av, g);
    4447             : }
    4448             : /* input V = (X1 a X2 | X3 a^* X4) + Ast
    4449             :  * a1 = index of a
    4450             :  * a2 = index of a^*, inferred from a1. We must have a != a^*
    4451             :  * c1 = first cut [ index of first path in X3 ]
    4452             :  * c2 = second cut [ either in X4 or X1, index of first path ]
    4453             :  * Assume a < a^* (cf Paranoia below): c1 or c2 must be in
    4454             :  *    ]a,a^*], and the other in the "complement" ]a^*,a] */
    4455             : static void
    4456        2520 : basic_op(struct siegel *S, long a1, long c1, long c2)
    4457             : {
    4458             :   pari_sp av;
    4459        2520 :   long l = lg(S->V), a2 = S->Ast[a1];
    4460             :   GEN g;
    4461             : 
    4462        2520 :   if (a1 == a2)
    4463             :   { /* a = a^* */
    4464           0 :     g = get_g(S, a1);
    4465           0 :     path_vec_mul(S->V, a1+1, l, g);
    4466           0 :     av = avma;
    4467           0 :     siegel_perm(S, basic_op_perm_elliptic(l, a1));
    4468             :     /* fill the hole left at a1, reconnect the path */
    4469           0 :     set_avma(av); fill1(S->V, a1); return;
    4470             :   }
    4471             : 
    4472             :   /* Paranoia: (a,a^*) conjugate, call 'a' the first one */
    4473        2520 :   if (a2 < a1) lswap(a1,a2);
    4474             :   /* Now a1 < a2 */
    4475        2520 :   if (c1 <= a1 || c1 > a2) lswap(c1,c2); /* ensure a1 < c1 <= a2 */
    4476        2520 :   if (c2 < a1)
    4477             :   { /* if cut c2 is in X1 = X11|X12, rotate to obtain
    4478             :        (a X2 | X3 a^* X4 X11|X12): then a1 = 1 */
    4479             :     GEN p;
    4480        2520 :     av = avma; p = rotate_perm(l, a1);
    4481        2520 :     siegel_perm(S, p);
    4482        2520 :     a1 = 1; /* = p[a1] */
    4483        2520 :     a2 = S->Ast[1]; /* > a1 */
    4484        2520 :     c1 = p[c1];
    4485        2520 :     c2 = p[c2]; set_avma(av);
    4486             :   }
    4487             :   /* Now a1 < c1 <= a2 < c2; a != a^* */
    4488        2520 :   g = get_g(S, a1);
    4489        2520 :   if (S->oo >= c1 && S->oo < c2) /* W inside [c1..c2[ */
    4490         539 :   { /* c2 -> c1 excluding a1 */
    4491         539 :     GEN gi = SL2_inv_shallow(g); /* g a^* = a; gi a = a^* */
    4492         539 :     path_vec_mul(S->V, 1, a1, gi);
    4493         539 :     path_vec_mul(S->V, a1+1, c1, gi);
    4494         539 :     path_vec_mul(S->V, c2, l, gi);
    4495             :   }
    4496             :   else
    4497             :   { /* c1 -> c2 excluding a2 */
    4498        1981 :     path_vec_mul(S->V, c1, a2, g);
    4499        1981 :     path_vec_mul(S->V, a2+1, c2, g);
    4500             :   }
    4501        2520 :   av = avma;
    4502        2520 :   siegel_perm(S, basic_op_perm(l, a1,a2, c1,c2));
    4503        2520 :   set_avma(av);
    4504             :   /* fill the holes left at a1,a2, reconnect the path */
    4505        2520 :   fill2(S->V, a1, a2);
    4506             : }
    4507             : /* a = a^* (elliptic case) */
    4508             : static void
    4509         154 : basic_op_elliptic(struct siegel *S, long a1)
    4510             : {
    4511             :   pari_sp av;
    4512         154 :   long l = lg(S->V);
    4513         154 :   GEN g = get_g(S, a1);
    4514         154 :   path_vec_mul(S->V, a1+1, l, g);
    4515         154 :   av = avma; siegel_perm(S, basic_op_perm_elliptic(l, a1));
    4516             :   /* fill the hole left at a1 (now at 1), reconnect the path */
    4517         154 :   set_avma(av); fill1(S->V, 1);
    4518         154 : }
    4519             : 
    4520             : /* input V = W X a b Y a^* Z b^* T, W already normalized
    4521             :  * X = [n+1, k1-1], Y = [k2+1, Ast[k1]-1],
    4522             :  * Z = [Ast[k1]+1, Ast[k2]-1], T = [Ast[k2]+1, oo].
    4523             :  * Assume that X doesn't start by c c^* or a b a^* b^*. */
    4524             : static void
    4525        1057 : siegelstep(struct siegel *S)
    4526             : {
    4527        1057 :   if (S->Ast[S->k1] == S->k1)
    4528             :   {
    4529         154 :     basic_op_elliptic(S, S->k1);
    4530         154 :     S->n++;
    4531             :   }
    4532         903 :   else if (S->Ast[S->k1] == S->k1+1)
    4533             :   {
    4534         364 :     basic_op(S, S->k1, S->Ast[S->k1], 1); /* 1: W starts there */
    4535         364 :     S->n += 2;
    4536             :   }
    4537             :   else
    4538             :   {
    4539         539 :     basic_op(S, S->k2, S->Ast[S->k1], 1); /* 1: W starts there */
    4540         539 :     basic_op(S, S->k1, S->k2, S->Ast[S->k2]);
    4541         539 :     basic_op(S, S->Ast[S->k2], S->k2, S->Ast[S->k1]);
    4542         539 :     basic_op(S, S->k1, S->Ast[S->k1], S->Ast[S->k2]);
    4543         539 :     S->n += 4;
    4544             :   }
    4545        1057 : }
    4546             : 
    4547             : /* normalize hyperbolic polygon */
    4548             : static void
    4549         301 : mssiegel(struct siegel *S)
    4550             : {
    4551         301 :   pari_sp av = avma;
    4552             :   long k, t, nv;
    4553             : #ifdef COUNT
    4554             :   long countset[16];
    4555             :   for (k = 0; k < 16; k++) countset[k] = 0;
    4556             : #endif
    4557             : 
    4558         301 :   nv = lg(S->V)-1;
    4559         301 :   if (DEBUGLEVEL>1) err_printf("nv = %ld, expo = %.2f\n", nv,ZMV_size(S->V));
    4560         301 :   t = 0;
    4561        2205 :   while (S->n < nv)
    4562             :   {
    4563        1904 :     if (S->Ast[S->n+1] == S->n+1) { S->n++; continue; }
    4564        1778 :     if (S->Ast[S->n+1] == S->n+2) { S->n += 2; continue; }
    4565        1134 :     if (S->Ast[S->n+1] == S->n+3 && S->Ast[S->n+2] == S->n+4) { S->n += 4; continue; }
    4566        1057 :     k = nv;
    4567        1127 :     while (k > S->n)
    4568             :     {
    4569        1127 :       if (S->Ast[k] == k) { k--; continue; }
    4570        1099 :       if (S->Ast[k] == k-1) { k -= 2; continue; }
    4571        1057 :       if (S->Ast[k] == k-2 && S->Ast[k-1] == k-3) { k -= 4; continue; }
    4572        1057 :       break;
    4573             :     }
    4574        1057 :     if (k != nv)
    4575             :     {
    4576          63 :       pari_sp av2 = avma;
    4577          63 :       siegel_perm0(S, rotate_perm(nv+1, k+1));
    4578          63 :       set_avma(av2); S->n += nv-k;
    4579             :     }
    4580             : 
    4581        6223 :     for (k = S->n+1; k <= nv; k++)
    4582        6223 :       if (S->Ast[k] <= k) { t = S->Ast[k]; break; }
    4583        1057 :     S->k1 = t;
    4584        1057 :     S->k2 = t+1;
    4585             : #ifdef COUNT
    4586             :     countset[ ((S->k1-1 == S->n)
    4587             :               | ((S->k2 == S->Ast[S->k1]-1) << 1)
    4588             :               | ((S->Ast[S->k1] == S->Ast[S->k2]-1) << 2)
    4589             :               | ((S->Ast[S->k2] == nv) << 3)) ]++;
    4590             : #endif
    4591        1057 :     siegelstep(S);
    4592        1057 :     if (gc_needed(av,2))
    4593             :     {
    4594           0 :       if(DEBUGMEM>1) pari_warn(warnmem,"mspolygon, n = %ld",S->n);
    4595           0 :       gerepileall(av, 2, &S->V, &S->Ast);
    4596             :     }
    4597             :   }
    4598         301 :   if (DEBUGLEVEL>1) err_printf("expo = %.2f\n", ZMV_size(S->V));
    4599             : #ifdef COUNT
    4600             :   for (k = 0; k < 16; k++)
    4601             :     err_printf("%3ld: %6ld\n", k, countset[k]);
    4602             : #endif
    4603         301 : }
    4604             : 
    4605             : /* return a vector of char* */
    4606             : static GEN
    4607           0 : Ast2v(GEN Ast)
    4608             : {
    4609           0 :   long j = 0, k, l = lg(Ast);
    4610           0 :   GEN v = const_vec(l-1, NULL);
    4611           0 :   for (k=1; k < l; k++)
    4612             :   {
    4613             :     char *sj;
    4614           0 :     if (gel(v,k)) continue;
    4615           0 :     j++;
    4616           0 :     sj = stack_sprintf("$%ld$", j);
    4617           0 :     gel(v,k) = (GEN)sj;
    4618           0 :     if (Ast[k] != k) gel(v,Ast[k]) = (GEN)stack_sprintf("$%ld^*$", j);
    4619             :   }
    4620           0 :   return v;
    4621             : };
    4622             : 
    4623             : static void
    4624           0 : decorate(pari_str *s, GEN g, GEN arc, double high)
    4625             : {
    4626           0 :   double a = gtodouble(gcoeff(g,1,1)), c = gtodouble(gcoeff(g,2,1));
    4627           0 :   double d = gtodouble(gcoeff(g,2,2));
    4628           0 :   if (a + d)
    4629             :   {
    4630           0 :     double t, u, C = 360/(2*M_PI), x = (a-d) / (2*c), y = 0.8660254/fabs(c);
    4631           0 :     long D1 = itos(gcoeff(arc,2,1));
    4632           0 :     long D2 = itos(gcoeff(arc,2,2));
    4633           0 :     str_printf(s, "\\coordinate (ellpt) at (%.4f,%.4f);\n\\draw (ellpt) node {$\\bullet$}\n", x, y);
    4634           0 :     if (D1)
    4635             :     {
    4636           0 :       t = gtodouble(gcoeff(arc,1,1)) / D1;
    4637           0 :       u = (x*x + y*y - t*t)/(x-t)/2;
    4638           0 :       str_printf(s, "arc (%.4f:180:%.4f)\n", C*atan2(y,x-u), fabs(t-u));
    4639             :     }
    4640             :     else
    4641           0 :       str_printf(s, "-- (%.4f,%.4f)\n", x, high);
    4642           0 :     if (D2)
    4643             :     {
    4644           0 :       t = gtodouble(gcoeff(arc,1,2)) / D2;
    4645           0 :       u = (x*x + y*y - t*t)/(x-t)/2;
    4646           0 :       str_printf(s, "(ellpt) arc (%.4f:0:%.4f);\n", C*atan2(y,x-u), fabs(t-u));
    4647             :     }
    4648             :     else
    4649           0 :       str_printf(s, "(ellpt) -- (%.4f,%.4f);\n", x, high);
    4650             :   }
    4651             :   else
    4652           0 :     str_printf(s, "\\draw (%.4f,%.4f) node {$\\circ$};\n",a/c,fabs(1/c));
    4653           0 : }
    4654             : 
    4655             : static GEN
    4656           0 : polygon2tex(GEN V, GEN Ast, GEN G)
    4657             : {
    4658           0 :   pari_sp av = avma;
    4659             :   pari_str s;
    4660           0 :   long j, l = lg(V), flag = (l <= 16);
    4661           0 :   double d, high = (l < 4)? 1.2: 0.5;
    4662           0 :   GEN v = Ast2v(Ast), r1 = NULL, r2 = NULL;
    4663             : 
    4664           0 :   for (j = 1; j < l; j++)
    4665             :   {
    4666           0 :     GEN arc = gel(V,j);
    4667           0 :     if (!signe(gcoeff(arc,2,1)))
    4668           0 :       r1 = gdiv(gcoeff(arc,1,2), gcoeff(arc,2,2));
    4669           0 :     else if (!signe(gcoeff(arc,2,2)))
    4670           0 :       r2 = gdiv(gcoeff(arc,1,1), gcoeff(arc,2,1));
    4671             :   }
    4672           0 :   if (!r1 || !r2) pari_err_BUG("polgon2tex");
    4673           0 :   str_init(&s, 1); d = fabs(gtodouble(gsub(r1,r2)));
    4674           0 :   str_printf(&s, "\n\\begin{tikzpicture}[scale=%.2f]\n",
    4675             :                  d? (10 / d): 10);
    4676           0 :   for (j = 1; j < l; j++)
    4677             :   {
    4678           0 :     GEN arc = gel(V,j);
    4679           0 :     if (itos(gcoeff(arc,2,1)))
    4680             :     {
    4681           0 :       GEN a = gdiv(gcoeff(arc,1,1), gcoeff(arc,2,1));
    4682           0 :       double aa = gtodouble(a);
    4683           0 :       str_printf(&s, "\\draw (%.4f,0) ", aa);
    4684           0 :       if (flag || j == 2 || j == l-1)
    4685             :       {
    4686             :         long n, d;
    4687           0 :         Qtoss(a, &n, &d);
    4688           0 :         if (d == 1)
    4689           0 :           str_printf(&s, "node [below] {$%ld$}\n", n);
    4690             :         else
    4691           0 :           str_printf(&s, "node [below] {$\\frac{%ld}{%ld}$}\n", n, d);
    4692             :       }
    4693           0 :       if (itos(gcoeff(arc,2,2)))
    4694             :       {
    4695           0 :         GEN b = gdiv(gcoeff(arc,1,2),gcoeff(arc,2,2));
    4696           0 :         str_printf(&s, "arc (%s:%.4f) ", (gcmp(a,b)<0)?"180:0":"0:180",
    4697           0 :                    fabs((gtodouble(b)-aa)/2));
    4698           0 :         if (flag)
    4699           0 :           str_printf(&s, "node [midway, above] {%s} ", (char*)gel(v,j));
    4700             :       }
    4701             :       else
    4702             :       {
    4703           0 :         str_printf(&s, "-- (%.4f,%.4f) ", aa, high);
    4704           0 :         if (flag)
    4705           0 :           str_printf(&s, "node [very near end, right] {%s}",(char*)gel(v,j));
    4706             :       }
    4707             :     }
    4708             :     else
    4709             :     {
    4710           0 :       GEN b = gdiv(gcoeff(arc,1,2), gcoeff(arc,2,2));
    4711           0 :       double bb = gtodouble(b);
    4712           0 :       str_printf(&s, "\\draw (%.4f,%.4f)--(%.4f,0)\n", bb, high, bb);
    4713           0 :       if (flag)
    4714           0 :         str_printf(&s,"node [very near start, left] {%s}\n", (char*)gel(v,j));
    4715             :     }
    4716           0 :     str_printf(&s,";\n");
    4717           0 :     if (Ast[j] == j) decorate(&s, gel(G,j), arc, high);
    4718             :   }
    4719           0 :   str_printf(&s, "\n\\end{tikzpicture}");
    4720           0 :   return gerepileuptoleaf(av, strtoGENstr(s.string));
    4721             : }
    4722             : 
    4723             : static GEN
    4724           0 : circle2tex(GEN Ast, GEN G)
    4725             : {
    4726           0 :   pari_sp av = avma;
    4727           0 :   GEN v = Ast2v(Ast);
    4728             :   pari_str s;
    4729           0 :   long u, n = lg(Ast)-1;
    4730           0 :   const double ang = 360./n;
    4731             : 
    4732           0 :   if (n > 30)
    4733             :   {
    4734           0 :     v = const_vec(n, (GEN)"");
    4735           0 :     gel(v,1) = (GEN)"$(1,\\infty)$";
    4736             :   }
    4737           0 :   str_init(&s, 1);
    4738           0 :   str_puts(&s, "\n\\begingroup\n\
    4739             :   \\def\\geo#1#2{(#2:1) arc (90+#2:270+#1:{tan((#2-#1)/2)})}\n\
    4740             :   \\def\\sgeo#1#2{(#2:1) -- (#1:1)}\n\
    4741             :   \\def\\unarc#1#2#3{({#1 * #3}:1.2) node {#2}}\n\
    4742             :   \\def\\cell#1#2{({#1 * #2}:0.95) circle(0.05)}\n\
    4743             :   \\def\\link#1#2#3#4#5{\\unarc{#1}{#2}{#5}\\geo{#1*#5}{#3*#5}\\unarc{#3}{#4}{#5}}\n\
    4744             :   \\def\\slink#1#2#3#4#5{\\unarc{#1}{#2}{#5}\\sgeo{#1*#5}{#3*#5}\\unarc{#3}{#4}{#5}}");
    4745             : 
    4746           0 :   str_puts(&s, "\n\\begin{tikzpicture}[scale=4]\n");
    4747           0 :   str_puts(&s, "\\draw (0, 0) circle(1);\n");
    4748           0 :   for (u=1; u <= n; u++)
    4749             :   {
    4750           0 :     if (Ast[u] == u)
    4751             :     {
    4752           0 :       str_printf(&s,"\\draw\\unarc{%ld}{%s}{%.4f}; \\draw\\cell{%ld}{%.4f};\n",
    4753           0 :                  u, v[u], ang, u, ang);
    4754           0 :       if (ZM_isscalar(gpowgs(gel(G,u),3), NULL))
    4755           0 :         str_printf(&s,"\\fill \\cell{%ld}{%.4f};\n", u, ang);
    4756             :     }
    4757           0 :     else if(Ast[u] > u)
    4758           0 :       str_printf(&s, "\\draw \\%slink {%ld}{%s}{%ld}{%s}{%.4f};\n",
    4759           0 :                      (Ast[u] - u)*ang > 179? "s": "", u, v[u], Ast[u], v[Ast[u]], ang);
    4760             :   }
    4761           0 :   str_printf(&s, "\\end{tikzpicture}\\endgroup");
    4762           0 :   return gerepileuptoleaf(av, strtoGENstr(s.string));
    4763             : }
    4764             : 
    4765             : GEN
    4766         434 : mspolygon(GEN M, long flag)
    4767             : {
    4768         434 :   pari_sp av = avma;
    4769             :   struct siegel T;
    4770         434 :   GEN v, msN = NULL, G = NULL;
    4771         434 :   if (typ(M) == t_INT)
    4772             :   {
    4773         308 :     long N = itos(M);
    4774         308 :     if (N <= 0) pari_err_DOMAIN("msinit","N", "<=", gen_0,M);
    4775         308 :     msN = msinit_N(N);
    4776             :   }
    4777         126 :   else if (checkfarey_i(M))
    4778             :   {
    4779           0 :     T.V = gel(M,1);
    4780           0 :     T.Ast = gel(M,2);
    4781           0 :     G = gel(M,3);
    4782             :   }
    4783             :   else
    4784         126 :   { checkms(M); msN = get_msN(M); }
    4785         434 :   if (flag < 0 || flag > 3) pari_err_FLAG("mspolygon");
    4786         434 :   if (!G)
    4787             :   {
    4788         434 :     if (ms_get_N(msN) == 1)
    4789             :     {
    4790          56 :       GEN S = mkS();
    4791          56 :       T.V = mkvec2(matid(2), S);
    4792          56 :       T.Ast = mkvecsmall2(1,2);
    4793          56 :       G = mkvec2(S, mkTAU());
    4794             :     }
    4795             :     else
    4796             :     {
    4797             :       long i, l;
    4798         378 :       siegel_init(&T, msN);
    4799         378 :       l = lg(T.V);
    4800         378 :       if (flag & 1)
    4801             :       {
    4802         301 :         long oo2 = 0;
    4803             :         pari_sp av;
    4804         301 :         mssiegel(&T);
    4805        3451 :         for (i = 1; i < l; i++)
    4806             :         {
    4807        3451 :           GEN c = gel(T.V, i);
    4808        3451 :           GEN c22 = gcoeff(c,2,2); if (!signe(c22)) { oo2 = i; break; }
    4809             :         }
    4810         301 :         if (!oo2) pari_err_BUG("mspolygon");
    4811         301 :         av = avma; siegel_perm0(&T, rotate_perm(l, oo2));
    4812         301 :         set_avma(av);
    4813             :       }
    4814         378 :       G = cgetg(l, t_VEC);
    4815       12320 :       for (i = 1; i < l; i++) gel(G,i) = get_g(&T, i);
    4816             :     }
    4817             :   }
    4818         434 :   if (flag & 2)
    4819           0 :     v = mkvec5(T.V, T.Ast, G, polygon2tex(T.V,T.Ast,G), circle2tex(T.Ast,G));
    4820             :   else
    4821         434 :     v = mkvec3(T.V, T.Ast, G);
    4822         434 :   return gerepilecopy(av, v);
    4823             : }
    4824             : 
    4825             : #if 0
    4826             : static int
    4827             : iselliptic(GEN Ast, long i) { return i == Ast[i]; }
    4828             : static int
    4829             : isparabolic(GEN Ast, long i)
    4830             : { long i2 = Ast[i]; return (i2 == i+1 || i2 == i-1); }
    4831             : #endif
    4832             : 
    4833             : /* M from msinit, F QM maximal rank */
    4834             : GEN
    4835          77 : mslattice(GEN M, GEN F)
    4836             : {
    4837          77 :   pari_sp av = avma;
    4838             :   long i, ivB, j, k, l, lF;
    4839             :   GEN D, U, G, A, vB, m, d;
    4840             : 
    4841          77 :   checkms(M);
    4842          77 :   if (!F) F = gel(mscuspidal(M, 0), 1);
    4843             :   else
    4844             :   {
    4845          49 :     if (is_Qevproj(F)) F = gel(F,1);
    4846          49 :     if (typ(F) != t_MAT) pari_err_TYPE("mslattice",F);
    4847             :   }
    4848          77 :   lF = lg(F); if (lF == 1) return cgetg(1, t_MAT);
    4849          77 :   D = mspolygon(M,0);
    4850          77 :   k = msk_get_weight(M);
    4851          77 :   F = vec_Q_primpart(F);
    4852          77 :   if (typ(F)!=t_MAT || !RgM_is_ZM(F)) pari_err_TYPE("mslattice",F);
    4853          77 :   G = gel(D,3); l = lg(G);
    4854          77 :   A = gel(D,2);
    4855          77 :   vB = cgetg(l, t_COL);
    4856          77 :   d = mkcol2(gen_0,gen_1);
    4857          77 :   m = mkmat2(d, d);
    4858        7091 :   for (i = ivB = 1; i < l; i++)
    4859             :   {
    4860        7014 :     GEN B, vb, g = gel(G,i);
    4861        7014 :     if (A[i] < i) continue;
    4862        3514 :     gel(m,2) = SL2_inv2(g);
    4863        3514 :     vb = mseval(M, F, m);
    4864        3514 :     if (k == 2) B = vb;
    4865             :     else
    4866             :     {
    4867             :       long lB;
    4868         147 :       B = RgXV_to_RgM(vb, k-1);
    4869             :       /* add coboundaries */
    4870         147 :       B = shallowconcat(B, RgM_Rg_sub(RgX_act_Gl2Q(g, k), gen_1));
    4871             :       /* beware: the basis for RgX_act_Gl2Q is (X^(k-2),...,Y^(k-2)) */
    4872         147 :       lB = lg(B);
    4873        3444 :       for (j = 1; j < lB; j++) gel(B,j) = vecreverse(gel(B,j));
    4874             :     }
    4875        3514 :     gel(vB, ivB++) = B;
    4876             :   }
    4877          77 :   setlg(vB, ivB);
    4878          77 :   vB = shallowmatconcat(vB);
    4879          77 :   if (ZM_equal0(vB)) return gerepilecopy(av, F);
    4880             : 
    4881          77 :   (void)QM_ImQ_hnfall(vB, &U, 0);
    4882          77 :   if (k > 2) U = rowslice(U, 1, lgcols(U)-k); /* remove coboundary part */
    4883          77 :   U = Q_remove_denom(U, &d);
    4884          77 :   F = ZM_hnf(ZM_mul(F, U));
    4885          77 :   if (d) F = RgM_Rg_div(F, d);
    4886          77 :   return gerepileupto(av, F);
    4887             : }
    4888             : 
    4889             : /**** Petersson scalar product ****/
    4890             : /* TODO:
    4891             :  * Eisspace: represent functions by coordinates of non-0 entries in matrix */
    4892             : 
    4893             : /* oo -> g^(-1) oo */
    4894             : static GEN
    4895        6181 : cocycle(GEN g)
    4896        6181 : { return mkmat22(gen_1, gcoeff(g,2,2), gen_0, negi(gcoeff(g,2,1))); }
    4897             : 
    4898             : /* CD = binomial_init(k-2); return <P,Q> * D (integral) */
    4899             : static GEN
    4900       18151 : bil(GEN P, GEN Q, GEN CD)
    4901             : {
    4902       18151 :   GEN s, C = gel(CD,1);
    4903       18151 :   long i, n = lg(C)-2; /* k - 2 */
    4904       18151 :   if (!n) return gmul(P,Q);
    4905       18130 :   if (typ(P) != t_POL) P = scalarpol_shallow(P,0);
    4906       18130 :   if (typ(Q) != t_POL) Q = scalarpol_shallow(Q,0);
    4907       18130 :   s = gen_0;
    4908       37282 :   for (i = n - degpol(Q); i <= degpol(P); i++)
    4909             :   {
    4910       19152 :     GEN t = gmul(gmul(RgX_coeff(P,i), RgX_coeff(Q, n-i)), gel(C,i+1));
    4911       19152 :     s = odd(i)? gsub(s, t): gadd(s, t);
    4912             :   }
    4913       18130 :   return s;
    4914             : }
    4915             : 
    4916             : /* Let D = lcm {binomial(n,k), k = 0..n} = lcm([1..n+1]) / (n+1)
    4917             :  * Return [C, D] where C[i] = D / binomial(n,i+1), i = 0..n */
    4918             : static GEN
    4919        1379 : binomial_init(long n, GEN vC)
    4920             : {
    4921        1379 :   GEN C = vC? shallowcopy(vC): vecbinomial(n), c = C + 1;
    4922        1379 :   GEN D = diviuexact(ZV_lcm(identity_ZV(n+1)), n+1);
    4923        1379 :   long k, d = (n + 1) >> 1;
    4924             : 
    4925        1379 :   gel(c,0) = D;
    4926        2961 :   for (k = 1; k <= d; k++) gel(c, k) = diviiexact(D, gel(c, k));
    4927        2961 :   for (     ; k <= n;  k++) gel(c, k) = gel(c, n-k);
    4928        1379 :   return mkvec2(C, D);
    4929             : }
    4930             : 
    4931             : static void
    4932        1351 : mspetersson_i(GEN W, GEN F, GEN G, GEN *pvf, GEN *pvg, GEN *pC)
    4933             : {
    4934        1351 :   GEN WN = get_msN(W), annT2, annT31, section, c, vf, vg;
    4935             :   long i, n1, n2, n3;
    4936             : 
    4937        1351 :   annT2 = msN_get_annT2(WN);
    4938        1351 :   annT31 = msN_get_annT31(WN);
    4939        1351 :   section = msN_get_section(WN);
    4940             : 
    4941        1351 :   if (ms_get_N(WN) == 1)
    4942             :   {
    4943           7 :     vf = cgetg(3, t_VEC);
    4944           7 :     vg = cgetg(3, t_VEC);
    4945           7 :     gel(vf,1) = mseval(W, F, gel(section,1));
    4946           7 :     gel(vf,2) = gneg(gel(vf,1));
    4947           7 :     n1 = 0;
    4948             :   }
    4949             :   else
    4950             :   {
    4951        1344 :     GEN singlerel = msN_get_singlerel(WN);
    4952        1344 :     GEN gen = msN_get_genindex(WN);
    4953        1344 :     long l = lg(gen);
    4954        1344 :     vf = cgetg(l, t_VEC);
    4955        1344 :     vg = cgetg(l, t_VEC); /* generators of Delta ordered as E1,T2,T31 */
    4956        7476 :     for (i = 1; i < l; i++) gel(vf, i) = mseval(W, F, gel(section,gen[i]));
    4957        1344 :     n1 = ms_get_nbE1(WN); /* E1 */
    4958        7420 :     for (i = 1; i <= n1; i++)
    4959             :     {
    4960        6076 :       c = cocycle(gcoeff(gel(singlerel,i),2,1));
    4961        6076 :       gel(vg, i) = mseval(W, G, c);
    4962             :     }
    4963             :   }
    4964        1351 :   n2 = lg(annT2)-1; /* T2 */
    4965        1386 :   for (i = 1; i <= n2; i++)
    4966             :   {
    4967          35 :     c = cocycle(gcoeff(gel(annT2,i), 2,1));
    4968          35 :     gel(vg, i+n1) = gmul2n(mseval(W, G, c), -1);
    4969             :   }
    4970        1351 :   n3 = lg(annT31)-1; /* T31 */
    4971        1386 :   for (i = 1; i <= n3; i++)
    4972             :   {
    4973             :     GEN f;
    4974          35 :     c = cocycle(gcoeff(gel(annT31,i), 2,1));
    4975          35 :     f = mseval(W, G, c);
    4976          35 :     c = cocycle(gcoeff(gel(annT31,i), 3,1));
    4977          35 :     gel(vg, i+n1+n2) = gdivgs(gadd(f, mseval(W, G, c)), 3);
    4978             :   }
    4979        1351 :   *pC = binomial_init(msk_get_weight(W) - 2, NULL);
    4980        1351 :   *pvf = vf;
    4981        1351 :   *pvg = vg;
    4982        1351 : }
    4983             : 
    4984             : /* Petersson product on Hom_G(Delta_0, V_k) */
    4985             : GEN
    4986        1351 : mspetersson(GEN W, GEN F, GEN G)
    4987             : {
    4988        1351 :   pari_sp av = avma;
    4989             :   GEN vf, vg, CD, cf, cg, A;
    4990             :   long k, l, tG, tF;
    4991        1351 :   checkms(W);
    4992        1351 :   if (!F) F = matid(msdim(W));
    4993        1351 :   if (!G) G = F;
    4994        1351 :   tF = typ(F);
    4995        1351 :   tG = typ(G);
    4996        1351 :   if (tF == t_MAT && tG != t_MAT) pari_err_TYPE("mspetersson",G);
    4997        1351 :   if (tG == t_MAT && tF != t_MAT) pari_err_TYPE("mspetersson",F);
    4998        1351 :   mspetersson_i(W, F, G, &vf, &vg, &CD);
    4999        1351 :   vf = Q_primitive_part(vf, &cf);
    5000        1351 :   vg = Q_primitive_part(vg, &cg);
    5001        1351 :   A = mul_content(mul_content(cf, cg), ginv(gel(CD,2)));
    5002        1351 :   l = lg(vf);
    5003        1351 :   if (tF != t_MAT)
    5004             :   { /* <F,G>, two symbols */
    5005        1274 :     GEN s = gen_0;
    5006        7105 :     for (k = 1; k < l; k++) s = gadd(s, bil(gel(vf,k), gel(vg,k), CD));
    5007        1274 :     return gerepileupto(av, gmul(s, A));
    5008             :   }
    5009          77 :   else if (F != G)
    5010             :   { /* <(f_1,...,f_m), (g_1,...,g_n)> */
    5011           0 :     long iF, iG, lF = lg(F), lG = lg(G);
    5012           0 :     GEN M = cgetg(lG, t_MAT);
    5013           0 :     for (iG = 1; iG < lG; iG++)
    5014             :     {
    5015           0 :       GEN c = cgetg(lF, t_COL);
    5016           0 :       gel(M,iG) = c;
    5017           0 :       for (iF = 1; iF < lF; iF++)
    5018             :       {
    5019           0 :         GEN s = gen_0;
    5020           0 :         for (k = 1; k < l; k++)
    5021           0 :           s = gadd(s, bil(gmael(vf,k,iF), gmael(vg,k,iG), CD));
    5022           0 :         gel(c,iF) = s; /* M[iF,iG] = <F[iF], G[iG] > */
    5023             :       }
    5024             :     }
    5025           0 :     return gerepileupto(av, RgM_Rg_mul(M, A));
    5026             :   }
    5027             :   else
    5028             :   { /* <(f_1,...,f_n), (f_1,...,f_n)> */
    5029          77 :     long iF, iG, n = lg(F)-1;
    5030          77 :     GEN M = zeromatcopy(n,n);
    5031         693 :     for (iG = 1; iG <= n; iG++)
    5032        3192 :       for (iF = iG+1; iF <= n; iF++)
    5033             :       {
    5034        2576 :         GEN s = gen_0;
    5035       14728 :         for (k = 1; k < l; k++)
    5036       12152 :           s = gadd(s, bil(gmael(vf,k,iF), gmael(vg,k,iG), CD));
    5037        2576 :         gcoeff(M,iF,iG) = s; /* <F[iF], F[iG] > */
    5038        2576 :         gcoeff(M,iG,iF) = gneg(s);
    5039             :       }
    5040          77 :     return gerepileupto(av, RgM_Rg_mul(M, A));
    5041             :   }
    5042             : }
    5043             : 
    5044             : /* action of g in SL_2(Z/NZ) on functions f: (Z/NZ)^2 -> Q given by sparse
    5045             :  * matrix M. */
    5046             : static GEN
    5047         168 : actf(long N, GEN M, GEN g)
    5048             : {
    5049             :   long n, a, b, c, d, l;
    5050             :   GEN m;
    5051         168 :   c = umodiu(gcoeff(g,2,1), N); if (!c) return M;
    5052           0 :   d = umodiu(gcoeff(g,2,2), N);
    5053           0 :   a = umodiu(gcoeff(g,1,1), N);
    5054           0 :   b = umodiu(gcoeff(g,1,2), N);
    5055           0 :   m = cgetg_copy(M, &l);
    5056           0 :   for (n = 1; n < l; n++)
    5057             :   {
    5058           0 :     GEN v = gel(M,n);
    5059           0 :     long i = v[1], j = v[2];
    5060           0 :     long I = Fl_add(Fl_mul(a,i,N), Fl_mul(c,j,N), N);
    5061           0 :     long J = Fl_add(Fl_mul(b,i,N), Fl_mul(d,j,N), N);
    5062           0 :     if (!I) I = N;
    5063           0 :     if (!J) J = N;
    5064           0 :     gel(m,n) = mkvecsmall2(I,J);
    5065             :   }
    5066           0 :   return m;
    5067             : }
    5068             : 
    5069             : /* q1 = N/a, q2 = q1/d, (u,a) = 1. Gamma_0(N)-orbit attached to [q1,q2,u]
    5070             :  * in (Z/N)^2; set of [q1 v, q2 w], v in (Z/a)^*, w in Z/a*d,
    5071             :  * w mod a = u / v [invertible]; w mod d in (Z/d)^*; c1+c2= q2, d2|c1, d1|c2
    5072             :  * The orbit has cardinal C = a phi(d) <= N */
    5073             : static GEN
    5074          28 : eisf(long N, long C, long a, long d1, GEN Z2, long c1, long c2,
    5075             :      long q1, long u)
    5076             : {
    5077          28 :   GEN m = cgetg(C+1, t_VEC);
    5078          28 :   long v, n = 1, l = lg(Z2);
    5079          56 :   for (v = 1; v <= a; v++)
    5080          28 :     if (ugcd(v,a)==1)
    5081             :     {
    5082          28 :       long w1 = Fl_div(u, v, a), vq1 = v * q1, i, j;
    5083          56 :       for (i = 0; i < d1; i++, w1 += a)
    5084             :       { /* w1 defined mod a*d1, lifts u/v (mod a) */
    5085          56 :         for (j = 1; j < l; j++)
    5086          28 :           if (Z2[j])
    5087             :           {
    5088          28 :             long wq2 = (c1 * w1 + c2 * j) % N;
    5089          28 :             if (wq2 <= 0) wq2 += N;
    5090          28 :             gel(m, n++) = mkvecsmall2(vq1, wq2);
    5091             :           }
    5092             :       }
    5093             :     }
    5094          28 :   return m;
    5095             : }
    5096             : 
    5097             : /* basis for Gamma_0(N)-invariant functions attached to cusps */
    5098             : static GEN
    5099          28 : eisspace(long N, long k, long s)
    5100             : {
    5101          28 :   GEN v, D, F = factoru(N);
    5102             :   long l, n, i, j;
    5103          28 :   D = divisorsu_fact(F); l = lg(D);
    5104          28 :   n = mfnumcuspsu_fact(F);
    5105          28 :   v = cgetg((k==2)? n: n+1, t_VEC);
    5106          56 :   for (i = (k==2)? 2: 1, j = 1; i < l; i++) /* remove d = 1 if k = 2 */
    5107             :   {
    5108          28 :     long d = D[i], Nd = D[l-i], a = ugcd(d, Nd), q1, q2, d1, d2, C, c1, c2, u;
    5109             :     GEN Z2;
    5110             : 
    5111          56 :     if (s < 0 && a <= 2) continue;
    5112          28 :     q1 = N / a;
    5113          28 :     q2 = q1 / d;
    5114          28 :     d2 = u_ppo(d/a, a);
    5115          28 :     d1 = d / d2;
    5116          28 :     C = eulerphiu(d) * a;
    5117          28 :     Z2 = coprimes_zv(d2);
    5118             :     /* d = d1d2, (d2,a) = 1; d1 and a have same prime divisors */
    5119          28 :     (void)cbezout(d1, d2, &c2, &c1);
    5120          28 :     c2 *= d1 * q2;
    5121          28 :     c1 *= d2 * q2;
    5122          28 :     if (a <= 2)
    5123             :     { /* sigma.(C cusp attached to [q1,q2,u]) = C */
    5124          28 :       gel(v, j++) = eisf(N,C,a,d1,Z2,c1,c2, N/a, 1);
    5125          28 :       continue;
    5126             :     }
    5127           0 :     for (u = 1; 2*u < a; u++)
    5128             :     {
    5129           0 :       if (ugcd(u,a) != 1) continue;
    5130           0 :       gel(v, j++) = eisf(N,C,a,d1,Z2,c1,c2, q1, u);
    5131           0 :       if (!s) gel(v, j++) = eisf(N,C,a,d1,Z2,c1,c2, q1, a-u);
    5132             :     }
    5133             :   }
    5134          28 :   if (s) setlg(v, j);
    5135          28 :   return v;
    5136             : }
    5137             : 
    5138             : /* action of g on V_k */
    5139             : static GEN
    5140         168 : act(GEN P, GEN g, long k)
    5141             : {
    5142         168 :   GEN a = gcoeff(g,1,1), b = gcoeff(g,1,2), V1, V2, Q;
    5143         168 :   GEN c = gcoeff(g,2,1), d = gcoeff(g,2,2);
    5144             :   long i;
    5145         168 :   if (k == 2) return P;
    5146         168 :   V1 = RgX_powers(deg1pol_shallow(c, a, 0), k-2); /* V1[i] = (a + c Y)^i */
    5147         168 :   V2 = RgX_powers(deg1pol_shallow(d, b, 0), k-2); /* V2[j] = (b + d Y)^j */
    5148         168 :   Q = gmul(RgX_coeff(P,0), gel(V1, k-2));
    5149        2520 :   for (i = 1; i < k-2; i++)
    5150             :   {
    5151        2352 :     GEN v1 = gel(V1, k-2-i);
    5152        2352 :     GEN v2 = gel(V2, i);
    5153        2352 :     Q = gadd(Q, gmul(RgX_coeff(P,i), RgX_mul(v1,v2)));
    5154             :   }
    5155         168 :   return gadd(Q, gmul(RgX_coeff(P,k-2), gel(V2,k-2)));
    5156             : }
    5157             : 
    5158             : static long
    5159         420 : co_get_N(GEN co) { return gel(co,1)[1]; }
    5160             : static long
    5161         504 : co_get_k(GEN co) { return gel(co,1)[2]; }
    5162             : static GEN
    5163         140 : co_get_B(GEN co) { return gel(co,2); }
    5164             : static GEN
    5165         112 : co_get_BD(GEN co) { return gel(co,3); }
    5166             : static GEN
    5167          84 : co_get_C(GEN co) { return gel(co,4); }
    5168             : 
    5169             : /* N g^(-1) . eval on g([0,a]_oo)=g([pi_oo(0),pi_oo(a)]), fg = f|g */
    5170             : static GEN
    5171         252 : evalcap(GEN co, GEN fg, GEN a)
    5172             : {
    5173         252 :   long n, t, l = lg(fg), N = co_get_N(co), k = co_get_k(co);
    5174             :   GEN P, B, z, T;
    5175             :   pari_sp av;
    5176         252 :   if (isintzero(a)) return gen_0;
    5177             :   /* (a+y)^(k-1) - y^(k-1) */
    5178          56 :   P = gsub(gpowgs(deg1pol_shallow(gen_1, a, 0), k-1), pol_xn(k-1, 0));
    5179          56 :   B = co_get_B(co); z = gen_0;
    5180          56 :   av = avma; T = zero_zv(N);
    5181         112 :   for (n = 1; n < l; n++)
    5182             :   {
    5183          56 :     GEN v = gel(fg, n);
    5184          56 :     t = v[1]; T[t]++;
    5185             :   }
    5186         112 :   for (t = 1; t <= N; t++)
    5187             :   {
    5188          56 :     long c = T[t];
    5189          56 :     if (c)
    5190             :     {
    5191          56 :       GEN u = gmael(B, k, t);
    5192          56 :       if (c != 1) u = gmulsg(c, u);
    5193          56 :       z = gadd(z, u);
    5194             :     }
    5195             :   }
    5196          56 :   if (co_get_BD(co)) z = gmul(co_get_BD(co),z);
    5197          56 :   z = gerepileupto(av, gdivgs(z, -k * (k-1)));
    5198          56 :   return RgX_Rg_mul(P, z);
    5199             : };
    5200             : 
    5201             : /* eval N g^(-1) * Psi(f) on g{oo,0}, fg = f|g */
    5202             : static GEN
    5203          84 : evalcup(GEN co, GEN fg)
    5204             : {
    5205          84 :   long j, n, k = co_get_k(co), l = lg(fg);
    5206          84 :   GEN B = co_get_B(co), C = co_get_C(co), P = cgetg(k+1, t_POL);
    5207          84 :   P[1] = evalvarn(0);
    5208        1428 :   for (j = 2; j <= k; j++) gel(P,j) = gen_0;
    5209         168 :   for (n = 1; n < l; n++)
    5210             :   {
    5211          84 :     GEN v = gel(fg,n);
    5212          84 :     long t = v[1], s = v[2];
    5213        1428 :     for (j = 1; j < k; j++)
    5214             :     {
    5215        1344 :       long j1 = k-j;
    5216        1344 :       GEN u = gmael(B, j1, t);
    5217        1344 :       GEN v = gmael(B, j, s);
    5218        1344 :       gel(P, j1+1) = gadd(gel(P, j1+1), gmul(u,v));
    5219             :     }
    5220             :   }
    5221        1428 :   for (j = 1; j < k; j++) gel(P, j+1) = gmul(gel(C,j), gel(P, j+1));
    5222          84 :   return normalizepol(P);
    5223             : }
    5224             : 
    5225             : /* Manin-Stevens algorithm, prepare for [pi_0(oo),pi_r(oo)] */
    5226             : static GEN
    5227          84 : evalmanin(GEN r)
    5228             : {
    5229          84 :   GEN fr = gboundcf(r, 0), pq, V;
    5230          84 :   long j, n = lg(fr)-1; /* > 0 */
    5231          84 :   V = cgetg(n+2, t_VEC);
    5232          84 :   gel(V,1) = gel(fr,1); /* a_0; tau_{-1} = id */
    5233          84 :   if (n == 1)
    5234             :   { /* r integer, can happen iff N = 1 */
    5235          84 :     gel(V,2) = mkvec2(gen_0, mkmat22(negi(r), gen_1, gen_m1, gen_0));
    5236          84 :     return V;
    5237             :   }
    5238           0 :   pq = contfracpnqn(fr,n-1);
    5239           0 :   fr = vec_append(fr, gdiv(negi(gcoeff(pq,2,n-1)), gcoeff(pq,2,n)));
    5240           0 :   for (j = 0; j < n; j++)
    5241             :   {
    5242           0 :     GEN v1 = gel(pq, j+1), v2 = (j == 0)? col_ei(2,1): gel(pq, j);
    5243           0 :     GEN z = gel(fr,j+2);
    5244           0 :     if (!odd(j)) { v1 = ZC_neg(v1); z = gneg(z); }
    5245           0 :     gel(V,j+2) = mkvec2(z, mkmat2(v1,v2)); /* [a_{j+1}, tau_j] */
    5246             :   }
    5247           0 :   return V;
    5248             : }
    5249             : 
    5250             : /* evaluate N * Psi(f) on
    5251             :   g[pi_oo(0),pi_r(oo)]=g[pi_oo(0),pi_0(oo)] + g[pi_0(oo),pi_r(oo)] */
    5252             : static GEN
    5253          84 : evalhull(GEN co, GEN f, GEN r)
    5254             : {
    5255          84 :   GEN V = evalmanin(r), res = evalcap(co,f,gel(V,1));
    5256          84 :   long j, l = lg(V), N = co_get_N(co);
    5257         168 :   for (j = 2; j < l; j++)
    5258             :   {
    5259          84 :     GEN v = gel(V,j), t = gel(v,2); /* in SL_2(Z) */
    5260          84 :     GEN ft = actf(N, f, t), a = gel(v,1); /* in Q */
    5261             :     /* t([pi_0(oo),pi_oo(a)]) */
    5262          84 :     res = gsub(res, act(gsub(evalcup(co,ft), evalcap(co,ft,a)), t, co_get_k(co)));
    5263             :   }
    5264          84 :   return res;
    5265             : };
    5266             : 
    5267             : /* evaluate N * cocycle at g in Gamma_0(N), f Gamma_0(N)-invariant */
    5268             : static GEN
    5269          84 : eiscocycle(GEN co, GEN f, GEN g)
    5270             : {
    5271          84 :   pari_sp av = avma;
    5272          84 :   GEN a = gcoeff(g,1,1), b = gcoeff(g,1,2);
    5273          84 :   GEN c = gcoeff(g,2,1), d = gcoeff(g,2,2), P;
    5274          84 :   long N = co_get_N(co);
    5275          84 :   if (!signe(c))
    5276           0 :     P = evalcap(co,f, gdiv(negi(b),a));
    5277             :   else
    5278             :   {
    5279          84 :     GEN gi = SL2_inv_shallow(g);
    5280          84 :     P = gsub(evalhull(co, f, gdiv(negi(d),c)),
    5281             :              act(evalcap(co, actf(N,f,gi), gdiv(a,c)), gi, co_get_k(co)));
    5282             :   }
    5283          84 :   return gerepileupto(av, P);
    5284             : }
    5285             : 
    5286             : static GEN
    5287          28 : eisCocycle(GEN co, GEN D, GEN f)
    5288             : {
    5289          28 :   GEN V = gel(D,1), Ast = gel(D,2), G = gel(D,3);
    5290          28 :   long i, j, n = lg(G)-1;
    5291          28 :   GEN VG = cgetg(n+1, t_VEC);
    5292          84 :   for (i = j = 1; i <= n; i++)
    5293             :   {
    5294          56 :     GEN c, g, d, s = gel(V,i);
    5295          56 :     if (i > Ast[i]) continue;
    5296          56 :     g = SL2_inv_shallow(gel(G,i));
    5297          56 :     c = eiscocycle(co,f,g);
    5298          56 :     if (i < Ast[i]) /* non elliptic */
    5299           0 :       d = gen_1;
    5300             :     else
    5301             :     { /* i = Ast[i] */
    5302          56 :       GEN g2 = ZM_sqr(g);
    5303          56 :       if (ZM_isdiagonal(g2)) d = gen_2; /* \pm Id */
    5304             :       else
    5305             :       {
    5306          28 :         c = gadd(c, eiscocycle(co,f,g2));
    5307          28 :         d = utoipos(3);
    5308             :       }
    5309             :     }
    5310          56 :     gel(VG, j++) = mkvec3(d, s, c);
    5311             :   }
    5312          28 :   setlg(VG, j); return VG;
    5313             : };
    5314             : 
    5315             : /* F=modular symbol, Eis = cocycle attached to f invariant function
    5316             :  * by Gamma_0(N); CD = binomial_init(k-2) */
    5317             : static GEN
    5318          84 : eispetersson(GEN M, GEN F, GEN Eis, GEN CD)
    5319             : {
    5320          84 :   pari_sp av = avma;
    5321          84 :   long i, l = lg(Eis);
    5322          84 :   GEN res = gen_0;
    5323         252 :   for (i = 1; i < l; i++)
    5324             :   {
    5325         168 :     GEN e = gel(Eis,i), Q = mseval(M, F, gel(e,2)), z = bil(gel(e,3), Q, CD);
    5326         168 :     long d = itou(gel(e,1));
    5327         168 :     res = gadd(res, d == 1? z: gdivgs(z,d));
    5328             :   }
    5329          84 :   return gerepileupto(av, gdiv(simplify_shallow(res), gel(CD,2)));
    5330             : };
    5331             : 
    5332             : /*vB[j][i] = {i/N} */
    5333             : static GEN
    5334          28 : get_bern(long N, long k)
    5335             : {
    5336          28 :   GEN vB = cgetg(k+1, t_VEC), gN = utoipos(N);
    5337             :   long i, j; /* no need for j = 0 */
    5338         504 :   for (j = 1; j <= k; j++)
    5339             :   {
    5340         476 :     GEN c, B = RgX_rescale(bernpol(j, 0), gN);
    5341         476 :     gel(vB, j) = c = cgetg(N+1, t_VEC);
    5342         476 :     for (i = 1; i < N; i++) gel(c,i) = poleval(B, utoipos(i));
    5343         476 :     gel(c,N) = gel(B,2); /* B(0) */
    5344             :   }
    5345          28 :   return vB;
    5346             : }
    5347             : GEN
    5348          28 : eisker_worker(GEN Ei, GEN M, GEN D, GEN co, GEN CD)
    5349             : {
    5350          28 :   pari_sp av = avma;
    5351          28 :   long j, n = msdim(M), s = msk_get_sign(M);
    5352          28 :   GEN V, Eis = eisCocycle(co, D, Ei), v = cgetg(n+1, t_VEC);
    5353             : 
    5354          28 :   V = s? gel(msk_get_starproj(M), 1): matid(n);
    5355             :   /* T is multiplied by N * BD^2: same Ker */
    5356         112 :   for (j = 1; j <= n; j++) gel(v,j) = eispetersson(M, gel(V,j), Eis, CD);
    5357          28 :   return gerepileupto(av, v);
    5358             : }
    5359             : /* vC = vecbinomial(k-2); vC[j] = binom(k-2,j-1) = vC[k-j], j = 1..k-1, k even.
    5360             :  * C[k-j+1] = (-1)^(j-1) binom(k-2, j-1) / (j(k-j)) = C[j+1] */
    5361             : static GEN
    5362          28 : get_C(GEN vC, long k)
    5363             : {
    5364          28 :   GEN C = cgetg(k, t_VEC);
    5365          28 :   long j, k2 = k/2;
    5366         266 :   for (j = 1; j <= k2; j++)
    5367             :   {
    5368         238 :     GEN c = gel(vC, j);
    5369         238 :     if (!odd(j)) c = negi(c);
    5370         238 :     gel(C,k-j) = gel(C, j) = gdivgs(c, j*(k-j));
    5371             :   }
    5372          28 :   return C;
    5373             : }
    5374             : static GEN
    5375          28 : eisker(GEN M)
    5376             : {
    5377          28 :   long N = ms_get_N(M), k = msk_get_weight(M), s = msk_get_sign(M);
    5378          28 :   GEN worker, vC, co, CD, D, B, BD, T, E = eisspace(N, k, s);
    5379          28 :   long i, j, m = lg(E)-1, n = msdim(M), pending = 0;
    5380             :   struct pari_mt pt;
    5381             : 
    5382          28 :   if (m == 0) return matid(n);
    5383          28 :   vC = vecbinomial(k-2);
    5384          28 :   T = zeromatcopy(m, n);
    5385          28 :   D = mspolygon(M, 0);
    5386          28 :   B = Q_remove_denom(get_bern(N,k), &BD);
    5387          28 :   co = mkvec4(mkvecsmall2(N,k), B, BD, get_C(vC, k));
    5388          28 :   CD = binomial_init(k-2, vC);
    5389          28 :   worker = snm_closure(is_entry("_eisker_worker"), mkvec4(M, D, co, CD));
    5390          28 :   mt_queue_start_lim(&pt, worker, m);
    5391          56 :   for (i = 1; i <= m || pending; i++)
    5392             :   {
    5393             :     long workid;
    5394             :     GEN done;
    5395          28 :     mt_queue_submit(&pt, i, i<=m? mkvec(gel(E,i)): NULL);
    5396          28 :     done = mt_queue_get(&pt, &workid, &pending);
    5397         112 :     if (done) for (j = 1; j <= n; j++) gcoeff(T,workid,j) = gel(done,j);
    5398             :   }
    5399          28 :   mt_queue_end(&pt); return QM_ker(T);
    5400             : }

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