/////////////////////////////////////////////////////////// // lchizero.m - MAGMA program // // // // Date: August, 1999 // // // // Author: William A. Stein (was@math.berkeley.edu) // /////////////////////////////////////////////////////////// /************************************************************* * lchinz[p,i] = [divisors d of p-1 such that * L(A_{f_i},chi_d,1)=0, where * chi_d:(Z/pZ)^*-->C^* is one of the * conjugate characters of degree d. * * The computation was done using modular symbols. * We consider L(A_f,chi_d,1)=/=0 if the value * of a vector dual to f on the modular symbol * s = sum_{a in (Z/pZ)^*} chi_d(a) {0,-1/a} * is 0. * * This computation was done using the C++ program HECKE. * * The eigenforms in all tables are ordered in the same * way, which extends the ordering used by Cremona in his * tables of elliptic curves. * We sort the factors of $S_2(p)$ as follows. * First by dimension, with smallest dimension first. * Within each dimension, sort in binary order, * by the sign of the Atkin-Lehner involution * with - corresponding to 0 and + to 1. * Let $l$ be the smallest prime not equal to $p$. * Within each of the Atkin-Lehner classes, sort by * the magnitudes of the $K(f)/\Q$-trace of * $a_l$ breaking ties by letting the positive trace be first. * If there are still any ties, repeat the final step with the * next smallest prime not equal to $p$, etc. *************************************************************/ LChiZero := [[] : i in [1..997]]; LChiZero[11] := [[]]; LChiZero[13] := []; LChiZero[17] := [[2]]; LChiZero[19] := [[]]; LChiZero[23] := [[]]; LChiZero[29] := [[2]]; LChiZero[31] := [[]]; LChiZero[37] := [[1, 2, 6],[2]]; LChiZero[41] := [[2]]; LChiZero[43] := [[1],[]]; LChiZero[47] := [[]]; LChiZero[53] := [[1, 2],[2]]; LChiZero[59] := [[]]; LChiZero[61] := [[1, 2],[2]]; LChiZero[67] := [[],[1],[]]; LChiZero[71] := [[],[]]; LChiZero[73] := [[2, 3, 6],[1, 2],[2]]; LChiZero[79] := [[1],[]]; LChiZero[83] := [[1],[]]; LChiZero[89] := [[1, 2],[2],[2]]; LChiZero[97] := [[1, 2],[2]]; LChiZero[101] := [[1, 2, 4],[2]]; LChiZero[103] := [[1],[]]; LChiZero[107] := [[1],[]]; LChiZero[109] := [[2],[1, 2],[2]]; LChiZero[113] := [[2],[2],[1, 2],[2]]; LChiZero[127] := [[1],[]]; LChiZero[131] := [[1, 2],[]]; LChiZero[137] := [[1, 2],[2]]; LChiZero[139] := [[3],[1],[]]; LChiZero[149] := [[1, 2],[2]]; LChiZero[151] := [[1],[],[]]; LChiZero[157] := [[1, 2],[2]]; LChiZero[163] := [[1],[1],[]]; LChiZero[167] := [[1],[]]; LChiZero[173] := [[1, 2],[2]]; LChiZero[179] := [[],[1],[]]; LChiZero[181] := [[1, 2],[2]]; LChiZero[191] := [[1],[]]; LChiZero[193] := [[1, 2],[1, 2],[2]]; LChiZero[197] := [[1, 2],[1, 2],[2]]; LChiZero[199] := [[],[1],[]]; LChiZero[211] := [[],[1],[1],[]]; LChiZero[223] := [[1],[1],[]]; LChiZero[227] := [[1],[],[],[1],[]]; LChiZero[229] := [[1, 2, 3],[1, 2],[2]]; LChiZero[233] := [[2],[1, 2],[2]]; LChiZero[239] := [[1],[]]; LChiZero[241] := [[1, 2],[2]]; LChiZero[251] := [[1],[]]; LChiZero[257] := [[1, 2],[2]]; LChiZero[263] := [[1],[]]; LChiZero[269] := [[1, 2],[1, 2],[2]]; LChiZero[271] := [[1],[]]; LChiZero[277] := [[1, 2],[2],[1, 2],[2]]; LChiZero[281] := [[1, 2],[2]]; LChiZero[283] := [[1],[]]; LChiZero[293] := [[1, 2],[2]]; LChiZero[307] := [[],[3],[],[],[],[],[1]]; LChiZero[311] := [[1],[]]; LChiZero[313] := [[2],[1, 2],[2]]; LChiZero[317] := [[1, 2],[2]]; LChiZero[331] := [[1],[1],[1],[]]; LChiZero[337] := [[1, 2],[2]]; LChiZero[347] := [[1],[1],[1],[]]; LChiZero[349] := [[1, 2],[2]]; LChiZero[353] := [[2, 4],[2],[1, 2],[2]]; LChiZero[359] := [[1],[1, 2],[1],[]]; LChiZero[367] := [[1],[]]; LChiZero[373] := [[1, 2, 6],[1, 2],[2]]; LChiZero[379] := [[1],[]]; LChiZero[383] := [[1],[1],[]]; LChiZero[389] := [[1, 2, 4],[1, 2],[1, 2],[1, 2],[2]]; LChiZero[397] := [[1, 2],[2],[2],[2],[1, 2]]; LChiZero[401] := [[1, 2],[2]]; LChiZero[409] := [[1, 2],[2]]; LChiZero[419] := [[1],[]]; LChiZero[421] := [[1, 2],[2]]; LChiZero[431] := [[1],[],[1],[],[1],[]]; LChiZero[433] := [[1, 2],[2],[1, 2],[2]]; LChiZero[439] := [[1],[1],[]]; LChiZero[443] := [[1],[1],[2],[1],[]]; LChiZero[449] := [[1, 2],[2]]; LChiZero[457] := [[1, 2],[1, 2],[2]]; LChiZero[461] := [[1, 2],[1, 2],[1, 2],[2]]; LChiZero[463] := [[1],[]]; LChiZero[467] := [[1, 2],[1],[]]; LChiZero[479] := [[1],[]]; LChiZero[487] := [[],[],[],[],[1]]; LChiZero[491] := [[1],[1],[]]; LChiZero[499] := [[1],[1],[]]; LChiZero[503] := [[1],[],[],[],[1],[]]; LChiZero[509] := [[1, 2],[2]]; LChiZero[521] := [[1, 2],[2]]; LChiZero[523] := [[1],[1],[]]; LChiZero[541] := [[1, 2],[2]]; LChiZero[547] := [[1],[1],[]]; LChiZero[557] := [[1, 2],[2],[1, 2],[2]]; LChiZero[563] := [[1],[1],[1],[1],[]]; LChiZero[569] := [[1, 2],[2]]; LChiZero[571] := [[],[1],[],[],[1],[],[1],[1],[]]; LChiZero[577] := [[2],[2],[2],[2],[1, 2]]; LChiZero[587] := [[1],[1],[]]; LChiZero[593] := [[1, 2],[2, 8],[2],[1, 2],[2]]; LChiZero[599] := [[1],[1],[]]; LChiZero[601] := [[1, 2],[2]]; LChiZero[607] := [[1],[1],[1],[]]; LChiZero[613] := [[1, 2],[1, 2],[2]]; LChiZero[617] := [[1, 2],[2]]; LChiZero[619] := [[1],[]]; LChiZero[631] := [[1],[]]; LChiZero[641] := [[1, 2],[2]]; LChiZero[643] := [[1, 3],[1],[]]; LChiZero[647] := [[1],[1],[]]; LChiZero[653] := [[1, 2],[1, 2],[2]]; LChiZero[659] := [[1],[],[1],[]]; LChiZero[661] := [[2],[1, 2],[2]]; LChiZero[673] := [[2],[2],[2],[1, 2]]; LChiZero[677] := [[1, 2],[1, 2],[1, 2],[2]]; LChiZero[683] := [[],[1],[]]; LChiZero[691] := [[1],[]]; LChiZero[701] := [[2],[1, 2],[2]]; LChiZero[709] := [[1, 2],[1, 2],[2]]; LChiZero[719] := [[1],[1],[]]; LChiZero[727] := [[1],[]]; LChiZero[733] := [[2],[1, 2],[1, 2],[2]]; LChiZero[739] := [[],[1],[1],[]]; LChiZero[743] := [[1],[]]; LChiZero[751] := [[1],[]]; LChiZero[757] := [[1, 2],[2]]; LChiZero[761] := [[1, 2],[1, 2],[2]]; LChiZero[769] := [[1, 2],[2]]; LChiZero[773] := [[1, 2],[1, 2],[2]]; LChiZero[787] := [[1],[]]; LChiZero[797] := [[1, 2],[2],[1, 2],[2]]; LChiZero[809] := [[1, 2],[1, 2],[2]]; LChiZero[811] := [[1],[1],[]]; LChiZero[821] := [[1, 2],[2]]; LChiZero[823] := [[1],[]]; LChiZero[827] := [[1],[1],[1],[]]; LChiZero[829] := [[1, 2],[1, 2],[2]]; LChiZero[839] := [[1],[]]; LChiZero[853] := [[1, 2],[2]]; LChiZero[857] := [[1, 2],[2]]; LChiZero[859] := [[1],[]]; LChiZero[863] := [[1],[1],[]]; LChiZero[877] := [[1, 2],[1, 2],[2]]; LChiZero[881] := [[1, 2],[2]]; LChiZero[883] := [[1],[]]; LChiZero[887] := [[1],[1],[]]; LChiZero[907] := [[1],[]]; LChiZero[911] := [[1],[1],[]]; LChiZero[919] := [[1],[1],[]]; LChiZero[929] := [[1, 2],[1, 2],[2]]; LChiZero[937] := [[1, 2],[2]]; LChiZero[941] := [[1, 2],[2]]; LChiZero[947] := [[1],[]]; LChiZero[953] := [[1, 2],[2]]; LChiZero[967] := [[1],[]]; LChiZero[971] := [[1],[]]; LChiZero[977] := [[1, 2],[2]]; LChiZero[983] := [[1],[]]; LChiZero[991] := [[1],[]]; LChiZero[997] := [[1, 2],[1, 2],[1, 2],[1, 2],[1, 2],[1, 2],[1, 2],[2]];