/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![5, -1, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {21, 21, w^2 - w - 6}>;

primesArray := [
[3, 3, -w^3 + 2*w^2 + 5*w - 3],
[5, 5, w],
[7, 7, -w^3 + 2*w^2 + 4*w - 3],
[13, 13, -w^2 + w + 4],
[13, 13, -w^3 + 2*w^2 + 5*w - 2],
[16, 2, 2],
[17, 17, -w + 2],
[19, 19, -w^3 + 2*w^2 + 3*w - 2],
[27, 3, w^3 - 3*w^2 - 4*w + 7],
[31, 31, w^3 - 3*w^2 - 3*w + 7],
[31, 31, -w^3 + w^2 + 6*w + 1],
[41, 41, w^2 - w - 1],
[43, 43, 2*w^3 - 5*w^2 - 6*w + 4],
[47, 47, -w^3 + 2*w^2 + 5*w - 1],
[53, 53, -w - 3],
[53, 53, -w^3 + 3*w^2 + 3*w - 6],
[59, 59, 2*w^2 - 3*w - 6],
[59, 59, w^3 - w^2 - 7*w - 3],
[79, 79, 2*w^3 - 4*w^2 - 8*w + 3],
[79, 79, w^2 - 2*w - 1],
[101, 101, 2*w^3 - 4*w^2 - 9*w + 4],
[103, 103, -w^3 + 3*w^2 + 4*w - 2],
[103, 103, w^2 - 3*w - 2],
[107, 107, -w^3 + w^2 + 5*w - 1],
[109, 109, -w^3 + 2*w^2 + 4*w - 6],
[121, 11, -w^3 + 2*w^2 + 6*w - 3],
[121, 11, w^2 - 3*w - 3],
[125, 5, -w^3 + w^2 + 7*w + 1],
[127, 127, 2*w^3 - 5*w^2 - 5*w + 7],
[127, 127, 2*w^3 - 5*w^2 - 7*w + 6],
[167, 167, -2*w^3 + 5*w^2 + 7*w - 11],
[169, 13, -w^2 + w + 8],
[181, 181, -2*w^3 + 5*w^2 + 8*w - 7],
[193, 193, -w^3 + 3*w^2 + 3*w - 1],
[197, 197, 2*w^3 - 4*w^2 - 7*w + 8],
[197, 197, 3*w^3 - 7*w^2 - 10*w + 9],
[199, 199, w^3 - 2*w^2 - 3*w - 3],
[211, 211, -w^3 + w^2 + 7*w - 2],
[211, 211, w^3 - w^2 - 6*w + 1],
[227, 227, 2*w^3 - 3*w^2 - 9*w + 3],
[229, 229, 3*w^3 - 6*w^2 - 13*w + 8],
[239, 239, 2*w^3 - 4*w^2 - 9*w + 3],
[239, 239, 2*w^3 - 4*w^2 - 6*w + 1],
[241, 241, 3*w^3 - 7*w^2 - 9*w + 6],
[241, 241, -2*w^3 + 4*w^2 + 6*w - 3],
[251, 251, -4*w^3 + 10*w^2 + 15*w - 16],
[251, 251, -w^3 + 3*w^2 - 3],
[271, 271, -w^3 + 4*w^2 + 4*w - 11],
[271, 271, -3*w^3 + 9*w^2 + 6*w - 13],
[277, 277, -w^3 + 3*w^2 + 2*w - 9],
[281, 281, -w^3 + 4*w^2 + w - 8],
[293, 293, 3*w^3 - 7*w^2 - 11*w + 13],
[293, 293, w^2 - 6],
[293, 293, -4*w^3 + 11*w^2 + 9*w - 14],
[293, 293, 3*w^3 - 5*w^2 - 15*w + 6],
[311, 311, -2*w^3 + 4*w^2 + 7*w - 4],
[313, 313, -w^3 + 4*w^2 + 3*w - 13],
[313, 313, -w^3 + w^2 + 6*w - 2],
[317, 317, w^3 - 4*w^2 - w + 9],
[337, 337, 3*w^3 - 7*w^2 - 8*w + 7],
[343, 7, -2*w^3 + 3*w^2 + 9*w + 1],
[359, 359, -w^3 + 2*w^2 + 4*w - 7],
[359, 359, 2*w^3 - 3*w^2 - 10*w + 4],
[367, 367, 2*w^3 - 4*w^2 - 5*w + 2],
[367, 367, 2*w^3 - 3*w^2 - 11*w - 2],
[367, 367, 2*w - 7],
[367, 367, w^3 - 7*w - 4],
[373, 373, -w^3 + 4*w^2 + w - 14],
[373, 373, -2*w^3 + 4*w^2 + 10*w - 11],
[379, 379, w^3 - 2*w^2 - 2*w - 1],
[383, 383, w^3 - 8*w - 4],
[397, 397, 4*w^3 - 10*w^2 - 12*w + 7],
[397, 397, -2*w^3 + 3*w^2 + 9*w - 2],
[401, 401, -w^3 + 6*w + 8],
[409, 409, 2*w^3 - 3*w^2 - 11*w + 4],
[419, 419, 3*w^3 - 6*w^2 - 15*w + 7],
[419, 419, -w^2 + 4*w + 8],
[431, 431, 5*w^3 - 15*w^2 - 10*w + 18],
[431, 431, 3*w^3 - 4*w^2 - 16*w - 2],
[439, 439, w^3 - w^2 - 4*w - 4],
[443, 443, -4*w^3 + 9*w^2 + 15*w - 16],
[461, 461, 2*w^3 - 2*w^2 - 13*w - 1],
[467, 467, 2*w^3 - 6*w^2 - 2*w + 7],
[467, 467, 3*w^3 - 3*w^2 - 19*w - 6],
[479, 479, w^2 - 8],
[479, 479, -3*w^3 + 7*w^2 + 14*w - 11],
[491, 491, 3*w^3 - 5*w^2 - 16*w + 2],
[499, 499, -w^2 + w - 2],
[509, 509, -w^3 + 5*w^2 - w - 11],
[509, 509, w^3 - 5*w - 1],
[521, 521, -w^3 + 4*w^2 + 3*w - 7],
[521, 521, -w^3 + 2*w^2 + 7*w - 4],
[541, 541, -2*w^3 + 4*w^2 + 11*w - 6],
[557, 557, -w^3 + w^2 + 8*w + 3],
[557, 557, w^3 - 2*w^2 - 2*w - 2],
[563, 563, 2*w^3 - 2*w^2 - 10*w - 3],
[563, 563, 3*w^3 - 5*w^2 - 15*w + 1],
[569, 569, 2*w^2 - 3*w - 4],
[569, 569, 3*w^3 - 8*w^2 - 9*w + 7],
[571, 571, -w^3 + 4*w^2 - 11],
[571, 571, 3*w^3 - 8*w^2 - 11*w + 11],
[577, 577, 3*w^3 - 8*w^2 - 5*w + 11],
[577, 577, -4*w^3 + 11*w^2 + 9*w - 9],
[593, 593, 2*w^3 - 7*w^2 - 2*w + 9],
[601, 601, -w^3 + 3*w^2 + 5*w - 9],
[613, 613, 3*w^3 - 5*w^2 - 14*w + 8],
[613, 613, -w^3 + 2*w^2 + 7*w - 3],
[617, 617, -2*w^3 + 6*w^2 + 5*w - 4],
[617, 617, -w^3 + 3*w^2 - w - 6],
[641, 641, 4*w^3 - 9*w^2 - 17*w + 14],
[641, 641, -5*w^3 + 12*w^2 + 18*w - 17],
[643, 643, -3*w^3 + 8*w^2 + 10*w - 13],
[647, 647, 3*w - 4],
[647, 647, -w^3 + 4*w^2 + 2*w - 9],
[653, 653, -w^3 + 2*w^2 + 5*w - 8],
[661, 661, 2*w^3 - 7*w^2 - 5*w + 21],
[661, 661, -w^3 + 2*w^2 + 3*w - 8],
[673, 673, 2*w^3 - 7*w^2 - 5*w + 9],
[673, 673, w^3 - 2*w^2 - 3*w - 4],
[677, 677, -w^3 + 4*w^2 + 3*w - 8],
[683, 683, -3*w^3 + 4*w^2 + 17*w - 1],
[683, 683, -3*w^3 + 7*w^2 + 8*w - 2],
[691, 691, w^2 - 4*w - 2],
[701, 701, 3*w^3 - 6*w^2 - 11*w + 3],
[709, 709, -w^3 + 4*w^2 + 3*w - 17],
[709, 709, w^3 + w^2 - 10*w - 8],
[709, 709, -3*w^3 + 10*w^2 + 2*w - 16],
[709, 709, w^3 - w^2 - 8*w + 1],
[727, 727, -w^3 + 5*w^2 - w - 13],
[727, 727, 2*w^3 - 4*w^2 - 11*w + 2],
[733, 733, -2*w^3 + 3*w^2 + 10*w - 6],
[733, 733, 2*w^3 - 5*w^2 - 5*w + 9],
[739, 739, -3*w^3 + 6*w^2 + 12*w - 8],
[743, 743, 3*w^3 - 4*w^2 - 17*w - 4],
[743, 743, 2*w^3 - 3*w^2 - 8*w + 1],
[743, 743, -3*w^3 + 7*w^2 + 11*w - 7],
[743, 743, -3*w^3 + 6*w^2 + 11*w - 9],
[757, 757, 2*w^3 - 5*w^2 - 6*w + 11],
[757, 757, -w^3 + w^2 + 8*w + 2],
[761, 761, -5*w^3 + 14*w^2 + 13*w - 12],
[769, 769, -3*w^2 + 4*w + 13],
[773, 773, 3*w^3 - 4*w^2 - 18*w + 3],
[797, 797, 4*w^3 - 7*w^2 - 21*w + 8],
[797, 797, -3*w^3 + 7*w^2 + 11*w - 16],
[809, 809, -2*w^2 + 5*w + 11],
[827, 827, w - 6],
[829, 829, -w^3 - w^2 + 11*w + 8],
[841, 29, -3*w^3 + 6*w^2 + 14*w - 6],
[841, 29, -2*w^3 + 5*w^2 + 8*w - 4],
[853, 853, 3*w^3 - 6*w^2 - 9*w + 8],
[857, 857, 2*w^2 - 3*w - 3],
[859, 859, -w^3 + w^2 + 4*w + 7],
[863, 863, 2*w^3 - 3*w^2 - 7*w - 2],
[863, 863, -3*w^3 + 6*w^2 + 10*w - 11],
[881, 881, -3*w^2 + 4*w + 12],
[881, 881, w^2 - 2*w + 3],
[883, 883, 4*w^3 - 13*w^2 - 6*w + 17],
[883, 883, -2*w^3 + 4*w^2 + 12*w - 3],
[887, 887, -3*w^3 + 5*w^2 + 16*w - 8],
[887, 887, 3*w^3 - 5*w^2 - 14*w + 3],
[911, 911, -w^3 + 10*w + 1],
[919, 919, 5*w^3 - 13*w^2 - 14*w + 11],
[929, 929, 2*w^3 - 3*w^2 - 12*w - 4],
[937, 937, -3*w - 7],
[941, 941, -w^3 + 3*w^2 + 3*w - 12],
[947, 947, 2*w^3 - 6*w^2 - 9*w + 8],
[947, 947, -4*w^3 + 7*w^2 + 19*w - 6],
[953, 953, 3*w^2 - w - 9],
[961, 31, 4*w^3 - 12*w^2 - 5*w + 17],
[971, 971, 3*w^3 - 6*w^2 - 16*w + 12],
[983, 983, w^3 + w^2 - 7*w - 12],
[991, 991, -2*w^3 + 7*w^2 + 3*w - 12],
[991, 991, -2*w^3 + 4*w^2 + 12*w - 9],
[997, 997, 3*w^3 - 5*w^2 - 14*w + 9],
[997, 997, -2*w^3 + 3*w^2 + 11*w - 7],
[1013, 1013, -2*w^3 + 6*w^2 + 3*w - 11],
[1019, 1019, -w^3 + 5*w^2 + 2*w - 8],
[1021, 1021, -w^3 + 5*w^2 - w - 17],
[1021, 1021, 3*w^3 - 6*w^2 - 8*w + 3],
[1033, 1033, -2*w^3 + 5*w^2 + 6*w - 13],
[1033, 1033, 3*w^3 - 6*w^2 - 11*w + 4],
[1051, 1051, 3*w^3 - 7*w^2 - 10*w + 13],
[1061, 1061, 2*w^2 - 5*w - 13],
[1063, 1063, -w^3 + w^2 + 5*w + 8],
[1069, 1069, -5*w^3 + 12*w^2 + 19*w - 18],
[1069, 1069, w^3 - 5*w^2 - w + 9],
[1097, 1097, -3*w^3 + 9*w^2 + 8*w - 12],
[1097, 1097, -2*w^3 + 6*w^2 + 7*w - 12],
[1103, 1103, -3*w^3 + 4*w^2 + 15*w - 3],
[1103, 1103, -w^3 + 3*w^2 + 5*w - 11],
[1109, 1109, 5*w^3 - 11*w^2 - 16*w + 6],
[1129, 1129, -w^2 + 3*w - 4],
[1129, 1129, 2*w^3 - 2*w^2 - 10*w - 9],
[1151, 1151, 4*w^3 - 10*w^2 - 14*w + 13],
[1153, 1153, -2*w^3 + 5*w^2 + 11*w - 12],
[1153, 1153, -2*w^3 + 8*w^2 - 13],
[1153, 1153, -w^3 + 3*w^2 + 5*w - 12],
[1153, 1153, 2*w^3 - w^2 - 11*w - 4],
[1163, 1163, -w^3 + w^2 + 9*w - 4],
[1163, 1163, 2*w^3 - 4*w^2 - 10*w - 1],
[1171, 1171, 3*w^3 - 4*w^2 - 18*w - 1],
[1193, 1193, w^3 - 4*w^2 + 2*w + 6],
[1193, 1193, w^3 - 2*w^2 - 5*w - 4],
[1201, 1201, 2*w^3 - 4*w^2 - 9*w - 3],
[1201, 1201, 4*w^3 - 10*w^2 - 13*w + 11],
[1213, 1213, -4*w^3 + 7*w^2 + 19*w - 11],
[1217, 1217, -2*w^3 + 8*w^2 + 2*w - 13],
[1229, 1229, -4*w^3 + 13*w^2 + 6*w - 19],
[1229, 1229, -4*w^3 + 8*w^2 + 16*w - 13],
[1229, 1229, -3*w^3 + 9*w^2 + 6*w - 16],
[1229, 1229, 2*w^3 - 3*w^2 - 6*w - 1],
[1231, 1231, -3*w^3 + 6*w^2 + 16*w - 8],
[1231, 1231, -3*w^2 + w + 12],
[1237, 1237, -2*w^3 + 5*w^2 + 4*w - 9],
[1237, 1237, -w^2 - 3],
[1237, 1237, -w^3 + 6*w - 3],
[1237, 1237, -4*w - 1],
[1249, 1249, -2*w^3 + 4*w^2 + 5*w - 8],
[1283, 1283, 3*w - 8],
[1289, 1289, 2*w^3 - w^2 - 14*w - 6],
[1289, 1289, 4*w^3 - 9*w^2 - 13*w + 8],
[1291, 1291, w^3 - w^2 - 3*w - 4],
[1297, 1297, 4*w^3 - 9*w^2 - 15*w + 9],
[1303, 1303, -w^3 + 4*w^2 - 13],
[1321, 1321, 3*w^3 - 6*w^2 - 13*w + 3],
[1321, 1321, -3*w^3 + 6*w^2 + 11*w - 7],
[1367, 1367, -2*w^3 + 6*w^2 + 4*w - 13],
[1369, 37, 2*w^2 - w - 11],
[1369, 37, w^3 - 2*w^2 - w - 2],
[1373, 1373, -w^3 + 3*w^2 + 8*w - 6],
[1381, 1381, -w^3 + 3*w^2 - 9],
[1381, 1381, 3*w^3 - 8*w^2 - 8*w + 14],
[1427, 1427, 3*w^3 - 7*w^2 - 9*w + 12],
[1429, 1429, 4*w^3 - 9*w^2 - 11*w + 8],
[1433, 1433, -w^3 + 8*w + 12],
[1447, 1447, -4*w^3 + 10*w^2 + 15*w - 14],
[1447, 1447, 6*w^3 - 14*w^2 - 21*w + 19],
[1451, 1451, w^3 - w^2 - 9*w - 4],
[1459, 1459, 3*w^3 - 7*w^2 - 13*w + 9],
[1481, 1481, -3*w^2 + 3*w + 19],
[1481, 1481, -w^3 + 5*w^2 - 3*w - 14],
[1487, 1487, -2*w^3 + 6*w^2 + w - 6],
[1489, 1489, -3*w^3 + 3*w^2 + 19*w + 3],
[1499, 1499, 3*w^2 - 4*w - 7],
[1511, 1511, 4*w^3 - 6*w^2 - 20*w + 3],
[1523, 1523, 8*w^3 - 23*w^2 - 17*w + 27],
[1531, 1531, -w^3 + 2*w^2 + 3*w - 9],
[1531, 1531, 3*w^3 - 9*w^2 - 8*w + 19],
[1549, 1549, -5*w^3 + 11*w^2 + 17*w - 18],
[1553, 1553, 2*w^2 - 9],
[1567, 1567, 2*w^3 - 7*w^2 - 4*w + 22],
[1579, 1579, -2*w^3 + 4*w^2 + 8*w + 3],
[1579, 1579, -3*w^3 + 5*w^2 + 13*w - 1],
[1607, 1607, w^3 - w^2 - 6*w - 8],
[1609, 1609, -4*w^3 + 7*w^2 + 19*w - 2],
[1619, 1619, -5*w^3 + 11*w^2 + 13*w - 9],
[1627, 1627, 4*w^3 - 9*w^2 - 16*w + 11],
[1657, 1657, 2*w^3 - 8*w^2 - w + 12],
[1657, 1657, -4*w^3 + 11*w^2 + 15*w - 17],
[1667, 1667, -w^3 + 5*w^2 - 6],
[1667, 1667, -3*w^3 + 6*w^2 + 10*w - 6],
[1697, 1697, w^3 + w^2 - 9*w - 7],
[1697, 1697, 3*w^3 - 6*w^2 - 13*w + 2],
[1697, 1697, w^3 - 6*w + 1],
[1697, 1697, -5*w^3 + 11*w^2 + 22*w - 17],
[1709, 1709, w^3 - w^2 - 8*w - 9],
[1709, 1709, -w^3 + 3*w^2 - 13],
[1723, 1723, 2*w^3 - 3*w^2 - 7*w - 3],
[1733, 1733, -w^3 + 4*w^2 + 3*w - 19],
[1741, 1741, -w^3 + 5*w^2 - 14],
[1747, 1747, -4*w^3 + 6*w^2 + 22*w + 1],
[1747, 1747, 2*w^3 - 5*w^2 - 9*w + 2],
[1753, 1753, -2*w^3 + 5*w^2 + 3*w - 1],
[1759, 1759, -w^3 + 2*w^2 + 4*w - 9],
[1783, 1783, -3*w^3 + 8*w^2 + 5*w - 12],
[1783, 1783, -2*w^3 + 6*w^2 + 3*w - 12],
[1783, 1783, 2*w^3 - 7*w^2 - 4*w + 17],
[1783, 1783, 2*w^3 - 6*w^2 - w + 7],
[1787, 1787, -5*w^3 + 8*w^2 + 25*w - 3],
[1823, 1823, w^3 - 8*w - 1],
[1823, 1823, -3*w^3 + 5*w^2 + 14*w + 6],
[1831, 1831, -3*w^3 + 8*w^2 + 14*w - 18],
[1847, 1847, w^3 - 3*w^2 - w + 14],
[1847, 1847, -2*w^3 + 2*w^2 + 11*w + 11],
[1861, 1861, 2*w^2 - w - 12],
[1871, 1871, 4*w^3 - 8*w^2 - 19*w + 9],
[1871, 1871, -w^3 + 5*w^2 - w - 7],
[1873, 1873, -3*w^3 + 5*w^2 + 13*w - 4],
[1877, 1877, -4*w^3 + 8*w^2 + 16*w - 7],
[1877, 1877, 3*w^3 - 5*w^2 - 15*w - 2],
[1879, 1879, -w^3 + 4*w^2 - w - 12],
[1907, 1907, w^3 - w^2 - 10*w - 6],
[1931, 1931, -w^3 + 5*w^2 - w - 19],
[1931, 1931, -6*w^3 + 17*w^2 + 12*w - 22],
[1951, 1951, 4*w^3 - 10*w^2 - 14*w + 21],
[1951, 1951, -2*w^3 + 4*w^2 + 13*w - 13],
[1993, 1993, 2*w^3 - 2*w^2 - 11*w + 1],
[1993, 1993, -w^3 + 6*w + 14]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 + x^8 - 39*x^7 - 29*x^6 + 510*x^5 + 212*x^4 - 2635*x^3 - 414*x^2 + 4376*x + 208;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, -1, 207969/19459292*e^8 + 1115551/19459292*e^7 - 5533797/19459292*e^6 - 33368695/19459292*e^5 + 8066193/4864823*e^4 + 66693608/4864823*e^3 - 16629179/19459292*e^2 - 146209830/4864823*e - 28354999/4864823, 229337/9729646*e^8 + 891125/9729646*e^7 - 6764415/9729646*e^6 - 25711093/9729646*e^5 + 26720335/4864823*e^4 + 95721983/4864823*e^3 - 121583571/9729646*e^2 - 189463940/4864823*e - 7273224/4864823, 141379/19459292*e^8 + 855999/19459292*e^7 - 4049221/19459292*e^6 - 24359787/19459292*e^5 + 13557339/9729646*e^4 + 44732927/4864823*e^3 - 11438733/19459292*e^2 - 162275241/9729646*e - 40114725/4864823, -94865/4864823*e^8 - 298166/4864823*e^7 + 2715470/4864823*e^6 + 8374850/4864823*e^5 - 20186045/4864823*e^4 - 57776644/4864823*e^3 + 42408100/4864823*e^2 + 99071967/4864823*e - 14119444/4864823, 74912/4864823*e^8 + 306016/4864823*e^7 - 2287143/4864823*e^6 - 9340516/4864823*e^5 + 19221041/4864823*e^4 + 77607673/4864823*e^3 - 46674406/4864823*e^2 - 177932447/4864823*e - 12134482/4864823, 299947/9729646*e^8 + 913571/9729646*e^7 - 8671751/9729646*e^6 - 26469339/9729646*e^5 + 32872796/4864823*e^4 + 97438978/4864823*e^3 - 131667995/9729646*e^2 - 186612441/4864823*e - 25270600/4864823, 295575/9729646*e^8 + 1190801/9729646*e^7 - 8647629/9729646*e^6 - 34755079/9729646*e^5 + 32825588/4864823*e^4 + 134688664/4864823*e^3 - 123092593/9729646*e^2 - 299896237/4864823*e - 29628328/4864823, -229337/9729646*e^8 - 891125/9729646*e^7 + 6764415/9729646*e^6 + 25711093/9729646*e^5 - 26720335/4864823*e^4 - 95721983/4864823*e^3 + 131313217/9729646*e^2 + 189463940/4864823*e - 21915714/4864823, -308607/4864823*e^8 - 1032071/4864823*e^7 + 9146817/4864823*e^6 + 30139471/4864823*e^5 - 73147503/4864823*e^4 - 229151701/4864823*e^3 + 167841826/4864823*e^2 + 471684183/4864823*e + 22720228/4864823, -427755/19459292*e^8 - 1381637/19459292*e^7 + 12528147/19459292*e^6 + 39383721/19459292*e^5 - 23891774/4864823*e^4 - 71666650/4864823*e^3 + 177854717/19459292*e^2 + 150483902/4864823*e + 40757323/4864823, 49541/19459292*e^8 - 63621/19459292*e^7 - 199897/19459292*e^6 + 2476877/19459292*e^5 - 5002387/4864823*e^4 - 9197070/4864823*e^3 + 149899597/19459292*e^2 + 34574116/4864823*e - 34217893/4864823, -138221/9729646*e^8 - 221013/4864823*e^7 + 2403126/4864823*e^6 + 6615551/4864823*e^5 - 50765273/9729646*e^4 - 52671051/4864823*e^3 + 181406299/9729646*e^2 + 215530777/9729646*e - 52067899/4864823, -120470/4864823*e^8 - 1061131/9729646*e^7 + 6532449/9729646*e^6 + 31161023/9729646*e^5 - 41117479/9729646*e^4 - 120658605/4864823*e^3 + 21627865/4864823*e^2 + 519261997/9729646*e + 42286667/4864823, 71756/4864823*e^8 + 447015/9729646*e^7 - 3983099/9729646*e^6 - 14224093/9729646*e^5 + 27948549/9729646*e^4 + 61623015/4864823*e^3 - 24723509/4864823*e^2 - 305225367/9729646*e - 22765495/4864823, -180179/19459292*e^8 + 28707/19459292*e^7 + 5660875/19459292*e^6 - 2946367/19459292*e^5 - 12558436/4864823*e^4 + 16432039/4864823*e^3 + 114728521/19459292*e^2 - 77408087/4864823*e + 6570813/4864823, -115397/4864823*e^8 - 338069/9729646*e^7 + 7353295/9729646*e^6 + 8898351/9729646*e^5 - 67612613/9729646*e^4 - 23370348/4864823*e^3 + 99700526/4864823*e^2 + 7701027/9729646*e - 79952125/4864823, 182795/9729646*e^8 + 278806/4864823*e^7 - 2786469/4864823*e^6 - 8236680/4864823*e^5 + 47294789/9729646*e^4 + 64879062/4864823*e^3 - 119975363/9729646*e^2 - 319476509/9729646*e + 2288849/4864823, -236851/4864823*e^8 - 1617127/9729646*e^7 + 14310535/9729646*e^6 + 46054849/9729646*e^5 - 118346457/9729646*e^4 - 167528686/4864823*e^3 + 143118317/4864823*e^2 + 638142999/9729646*e - 45267/4864823, 590583/9729646*e^8 + 2059259/9729646*e^7 - 17743895/9729646*e^6 - 60196349/9729646*e^5 + 73576540/4864823*e^4 + 228130359/4864823*e^3 - 380671647/9729646*e^2 - 466238328/4864823*e + 26890438/4864823, 136577/9729646*e^8 + 408295/9729646*e^7 - 4125097/9729646*e^6 - 13947753/9729646*e^5 + 17435624/4864823*e^4 + 68725433/4864823*e^3 - 94335827/9729646*e^2 - 193819679/4864823*e + 33511432/4864823, 408369/4864823*e^8 + 1711830/4864823*e^7 - 11651183/4864823*e^6 - 49663196/4864823*e^5 + 83616187/4864823*e^4 + 374397098/4864823*e^3 - 144931022/4864823*e^2 - 763041106/4864823*e - 40499088/4864823, 99371/4864823*e^8 + 337354/4864823*e^7 - 2753684/4864823*e^6 - 10192377/4864823*e^5 + 17924383/4864823*e^4 + 82300609/4864823*e^3 - 6390263/4864823*e^2 - 189905443/4864823*e - 91499496/4864823, -187706/4864823*e^8 - 827045/4864823*e^7 + 5290697/4864823*e^6 + 24248633/4864823*e^5 - 37186052/4864823*e^4 - 187492266/4864823*e^3 + 63648012/4864823*e^2 + 414153424/4864823*e + 12836206/4864823, -504677/9729646*e^8 - 784550/4864823*e^7 + 7513335/4864823*e^6 + 22299453/4864823*e^5 - 118365551/9729646*e^4 - 161034953/4864823*e^3 + 231840935/9729646*e^2 + 627412845/9729646*e + 69421717/4864823, 388659/9729646*e^8 + 1857827/9729646*e^7 - 11030583/9729646*e^6 - 53230545/9729646*e^5 + 38971469/4864823*e^4 + 196374506/4864823*e^3 - 126309009/9729646*e^2 - 373942077/4864823*e - 68211766/4864823, -683477/19459292*e^8 - 2858507/19459292*e^7 + 20397305/19459292*e^6 + 86927779/19459292*e^5 - 40841606/4864823*e^4 - 180292021/4864823*e^3 + 368190975/19459292*e^2 + 424614902/4864823*e + 30665885/4864823, -394189/9729646*e^8 - 669017/4864823*e^7 + 6060203/4864823*e^6 + 19512394/4864823*e^5 - 103347869/9729646*e^4 - 149414376/4864823*e^3 + 267731521/9729646*e^2 + 644268951/9729646*e - 63291531/4864823, 731747/9729646*e^8 + 2552791/9729646*e^7 - 21843115/9729646*e^6 - 75207249/9729646*e^5 + 86664214/4864823*e^4 + 291033571/4864823*e^3 - 345414991/9729646*e^2 - 616832828/4864823*e - 115910282/4864823, 171518/4864823*e^8 + 1806533/9729646*e^7 - 8991831/9729646*e^6 - 53271543/9729646*e^5 + 48885917/9729646*e^4 + 206868412/4864823*e^3 + 7916941/4864823*e^2 - 868479921/9729646*e - 137840815/4864823, 39607/19459292*e^8 + 294793/19459292*e^7 - 1333475/19459292*e^6 - 8961393/19459292*e^5 + 3267145/4864823*e^4 + 16540258/4864823*e^3 - 65956309/19459292*e^2 - 8709814/4864823*e + 62276011/4864823, -19344/4864823*e^8 - 6287/4864823*e^7 + 1090375/4864823*e^6 - 158669/4864823*e^5 - 17989860/4864823*e^4 + 3391488/4864823*e^3 + 88651027/4864823*e^2 + 368404/4864823*e - 57798454/4864823, -333619/4864823*e^8 - 1497912/4864823*e^7 + 9235858/4864823*e^6 + 43678743/4864823*e^5 - 61256316/4864823*e^4 - 331478717/4864823*e^3 + 81376129/4864823*e^2 + 673239884/4864823*e + 94270910/4864823, 202329/4864823*e^8 + 431677/4864823*e^7 - 6392857/4864823*e^6 - 12923550/4864823*e^5 + 56498244/4864823*e^4 + 101586706/4864823*e^3 - 122162195/4864823*e^2 - 222489702/4864823*e - 139965334/4864823, -460860/4864823*e^8 - 1643635/4864823*e^7 + 13540891/4864823*e^6 + 47279316/4864823*e^5 - 106928548/4864823*e^4 - 345638246/4864823*e^3 + 257184489/4864823*e^2 + 634842318/4864823*e - 53102000/4864823, -828393/9729646*e^8 - 2482097/9729646*e^7 + 24891101/9729646*e^6 + 73481483/9729646*e^5 - 100704378/4864823*e^4 - 287124373/4864823*e^3 + 417285129/9729646*e^2 + 627264486/4864823*e + 111855630/4864823, 391387/9729646*e^8 + 773433/4864823*e^7 - 5186775/4864823*e^6 - 22830728/4864823*e^5 + 62143329/9729646*e^4 + 175179202/4864823*e^3 - 47813939/9729646*e^2 - 712884435/9729646*e - 82410505/4864823, -306347/19459292*e^8 - 1068557/19459292*e^7 + 6730867/19459292*e^6 + 28485097/19459292*e^5 - 1591625/4864823*e^4 - 43344787/4864823*e^3 - 251614279/19459292*e^2 + 67079122/4864823*e + 94017111/4864823, -146103/9729646*e^8 - 538645/9729646*e^7 + 3674705/9729646*e^6 + 16038969/9729646*e^5 - 10074341/4864823*e^4 - 62035672/4864823*e^3 + 23695559/9729646*e^2 + 117351797/4864823*e - 8921418/4864823, 1111179/9729646*e^8 + 3969775/9729646*e^7 - 32847269/9729646*e^6 - 115453853/9729646*e^5 + 127427680/4864823*e^4 + 436321470/4864823*e^3 - 487334639/9729646*e^2 - 897990334/4864823*e - 160533670/4864823, -330623/9729646*e^8 - 1176037/9729646*e^7 + 9864707/9729646*e^6 + 35906121/9729646*e^5 - 38451629/4864823*e^4 - 148721613/4864823*e^3 + 122572279/9729646*e^2 + 355364640/4864823*e + 103465594/4864823, 17496/4864823*e^8 + 216938/4864823*e^7 - 750813/4864823*e^6 - 7322725/4864823*e^5 + 11273616/4864823*e^4 + 73502171/4864823*e^3 - 69323551/4864823*e^2 - 211765901/4864823*e + 82057208/4864823, -486882/4864823*e^8 - 1662655/4864823*e^7 + 14752678/4864823*e^6 + 48168906/4864823*e^5 - 123095330/4864823*e^4 - 359255648/4864823*e^3 + 305113827/4864823*e^2 + 720098089/4864823*e + 14574804/4864823, -1986705/19459292*e^8 - 6220479/19459292*e^7 + 60444185/19459292*e^6 + 182939243/19459292*e^5 - 125892810/4864823*e^4 - 354098087/4864823*e^3 + 1173561547/19459292*e^2 + 770640215/4864823*e + 14870063/4864823, 401205/19459292*e^8 + 1422803/19459292*e^7 - 11914317/19459292*e^6 - 43411571/19459292*e^5 + 22497733/4864823*e^4 + 92519021/4864823*e^3 - 124403243/19459292*e^2 - 229925414/4864823*e - 29055101/4864823, -823345/19459292*e^8 - 2615255/19459292*e^7 + 25521981/19459292*e^6 + 75090175/19459292*e^5 - 53663103/4864823*e^4 - 137500220/4864823*e^3 + 429059631/19459292*e^2 + 275001472/4864823*e + 106468681/4864823, 1010569/9729646*e^8 + 3828935/9729646*e^7 - 29091975/9729646*e^6 - 111935873/9729646*e^5 + 109027350/4864823*e^4 + 422965813/4864823*e^3 - 456266381/9729646*e^2 - 835068158/4864823*e + 3193766/4864823, -465095/9729646*e^8 - 884498/4864823*e^7 + 6956826/4864823*e^6 + 26621182/4864823*e^5 - 110157883/9729646*e^4 - 215555274/4864823*e^3 + 196882927/9729646*e^2 + 971125901/9729646*e + 132393009/4864823, -109407/9729646*e^8 + 143251/9729646*e^7 + 3881721/9729646*e^6 - 2195853/9729646*e^5 - 20533826/4864823*e^4 - 6452157/4864823*e^3 + 106930115/9729646*e^2 + 73967389/4864823*e + 40028266/4864823, -2065975/19459292*e^8 - 6361425/19459292*e^7 + 62826587/19459292*e^6 + 187367621/19459292*e^5 - 132035724/4864823*e^4 - 362308815/4864823*e^3 + 1341440377/19459292*e^2 + 787748262/4864823*e - 104707067/4864823, -545189/4864823*e^8 - 2258272/4864823*e^7 + 15584007/4864823*e^6 + 66212632/4864823*e^5 - 113779190/4864823*e^4 - 510368849/4864823*e^3 + 206648884/4864823*e^2 + 1085851970/4864823*e + 120983910/4864823, -388687/9729646*e^8 - 1633507/9729646*e^7 + 10888309/9729646*e^6 + 46483341/9729646*e^5 - 38580093/4864823*e^4 - 166639895/4864823*e^3 + 163751407/9729646*e^2 + 295793328/4864823*e - 53298740/4864823, -477562/4864823*e^8 - 3238775/9729646*e^7 + 27617705/9729646*e^6 + 96426325/9729646*e^5 - 209215153/9729646*e^4 - 380590400/4864823*e^3 + 208524273/4864823*e^2 + 1669485775/9729646*e + 78210317/4864823, 368329/4864823*e^8 + 1411112/4864823*e^7 - 10780437/4864823*e^6 - 41491261/4864823*e^5 + 84042257/4864823*e^4 + 319900642/4864823*e^3 - 198056873/4864823*e^2 - 642279582/4864823*e + 60246040/4864823, 859287/19459292*e^8 + 2388021/19459292*e^7 - 25671327/19459292*e^6 - 67915097/19459292*e^5 + 51965522/4864823*e^4 + 118952278/4864823*e^3 - 490517837/19459292*e^2 - 194731504/4864823*e + 44062379/4864823, -787135/19459292*e^8 - 2209653/19459292*e^7 + 25344451/19459292*e^6 + 71788365/19459292*e^5 - 61403546/4864823*e^4 - 162481958/4864823*e^3 + 816784805/19459292*e^2 + 396028649/4864823*e - 92056837/4864823, 666104/4864823*e^8 + 2351138/4864823*e^7 - 19368990/4864823*e^6 - 68729868/4864823*e^5 + 149349265/4864823*e^4 + 522293673/4864823*e^3 - 329563897/4864823*e^2 - 1070098803/4864823*e + 19831512/4864823, -1167455/19459292*e^8 - 4852185/19459292*e^7 + 36006335/19459292*e^6 + 142683313/19459292*e^5 - 77664098/4864823*e^4 - 283113532/4864823*e^3 + 842390797/19459292*e^2 + 659541965/4864823*e - 75644485/4864823, -364013/4864823*e^8 - 1239702/4864823*e^7 + 10471767/4864823*e^6 + 36282593/4864823*e^5 - 77517514/4864823*e^4 - 280326393/4864823*e^3 + 142745692/4864823*e^2 + 634089346/4864823*e + 65665444/4864823, 181933/9729646*e^8 + 1208691/9729646*e^7 - 4393147/9729646*e^6 - 35852707/9729646*e^5 + 8244683/4864823*e^4 + 139013409/4864823*e^3 + 54886181/9729646*e^2 - 259677211/4864823*e - 49573088/4864823, -1488667/9729646*e^8 - 4281583/9729646*e^7 + 45389607/9729646*e^6 + 125157347/9729646*e^5 - 191888949/4864823*e^4 - 475634900/4864823*e^3 + 948575883/9729646*e^2 + 990898100/4864823*e + 4312154/4864823, -183779/4864823*e^8 - 1333151/9729646*e^7 + 11570667/9729646*e^6 + 38942195/9729646*e^5 - 104205943/9729646*e^4 - 148393034/4864823*e^3 + 156359758/4864823*e^2 + 623647595/9729646*e - 151820781/4864823, -157338/4864823*e^8 - 1058941/9729646*e^7 + 9930277/9729646*e^6 + 31994727/9729646*e^5 - 88349281/9729646*e^4 - 131300066/4864823*e^3 + 124613204/4864823*e^2 + 601746673/9729646*e - 68159919/4864823, -131330/4864823*e^8 - 796581/9729646*e^7 + 7364429/9729646*e^6 + 23468343/9729646*e^5 - 55232965/9729646*e^4 - 87948053/4864823*e^3 + 75945773/4864823*e^2 + 343045155/9729646*e - 43295017/4864823, -324350/4864823*e^8 - 1393549/4864823*e^7 + 9422840/4864823*e^6 + 40815608/4864823*e^5 - 70879261/4864823*e^4 - 313644513/4864823*e^3 + 147849277/4864823*e^2 + 644279821/4864823*e - 6749106/4864823, -210181/4864823*e^8 - 842303/9729646*e^7 + 12912417/9729646*e^6 + 27156811/9729646*e^5 - 111123533/9729646*e^4 - 122180222/4864823*e^3 + 138386812/4864823*e^2 + 667438735/9729646*e - 19998615/4864823, 322795/19459292*e^8 + 2731725/19459292*e^7 - 9331919/19459292*e^6 - 76378741/19459292*e^5 + 17294149/4864823*e^4 + 132568754/4864823*e^3 - 182224553/19459292*e^2 - 182665796/4864823*e + 25158539/4864823, -432821/9729646*e^8 - 899624/4864823*e^7 + 7292852/4864823*e^6 + 26100929/4864823*e^5 - 150622739/9729646*e^4 - 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337182247/9729646*e^5 + 282883992/4864823*e^4 + 1334815567/4864823*e^3 - 1257325515/9729646*e^2 - 2925713784/4864823*e + 63294168/4864823, -773619/9729646*e^8 - 4855965/9729646*e^7 + 21744773/9729646*e^6 + 145775933/9729646*e^5 - 74048526/4864823*e^4 - 591842942/4864823*e^3 + 168600385/9729646*e^2 + 1289949255/4864823*e + 308309598/4864823, -3140631/9729646*e^8 - 10348251/9729646*e^7 + 91179461/9729646*e^6 + 304056345/9729646*e^5 - 344637466/4864823*e^4 - 1174087732/4864823*e^3 + 1304168135/9729646*e^2 + 2571168372/4864823*e + 309491806/4864823, -1590467/4864823*e^8 - 4618469/4864823*e^7 + 47963079/4864823*e^6 + 133808537/4864823*e^5 - 397041244/4864823*e^4 - 999706431/4864823*e^3 + 946095174/4864823*e^2 + 2042237583/4864823*e + 48680842/4864823, -1863935/4864823*e^8 - 6394344/4864823*e^7 + 53922797/4864823*e^6 + 189590202/4864823*e^5 - 407967556/4864823*e^4 - 1479201608/4864823*e^3 + 815203481/4864823*e^2 + 3190679151/4864823*e + 338953456/4864823, 2453511/9729646*e^8 + 8181415/9729646*e^7 - 70181215/9729646*e^6 - 242949001/9729646*e^5 + 253412267/4864823*e^4 + 953128398/4864823*e^3 - 780899633/9729646*e^2 - 2122300640/4864823*e - 429995886/4864823, 1696633/9729646*e^8 + 6116143/9729646*e^7 - 51286135/9729646*e^6 - 182878097/9729646*e^5 + 210843167/4864823*e^4 + 736087210/4864823*e^3 - 968490611/9729646*e^2 - 1711281824/4864823*e + 44452054/4864823, 578211/4864823*e^8 + 4285513/9729646*e^7 - 35426933/9729646*e^6 - 131924783/9729646*e^5 + 311214837/9729646*e^4 + 552768777/4864823*e^3 - 469310482/4864823*e^2 - 2551170543/9729646*e + 306029785/4864823, -217738/4864823*e^8 - 1384048/4864823*e^7 + 4891132/4864823*e^6 + 38061594/4864823*e^5 - 7969138/4864823*e^4 - 256780618/4864823*e^3 - 145006930/4864823*e^2 + 435182584/4864823*e + 261459480/4864823, -1590775/4864823*e^8 - 7015772/4864823*e^7 + 46398065/4864823*e^6 + 205533983/4864823*e^5 - 354377211/4864823*e^4 - 1586074854/4864823*e^3 + 759588323/4864823*e^2 + 3324560896/4864823*e + 31643068/4864823, -586380/4864823*e^8 - 2372507/4864823*e^7 + 15123611/4864823*e^6 + 70151351/4864823*e^5 - 80481447/4864823*e^4 - 538420019/4864823*e^3 - 22177071/4864823*e^2 + 1054859174/4864823*e + 363865940/4864823, 32582/4864823*e^8 - 130721/9729646*e^7 - 390691/9729646*e^6 + 5277609/9729646*e^5 - 15448265/9729646*e^4 - 35661885/4864823*e^3 + 14370270/4864823*e^2 + 166657723/9729646*e + 69086155/4864823, -185847/4864823*e^8 - 430847/4864823*e^7 + 4659541/4864823*e^6 + 15616397/4864823*e^5 - 20352697/4864823*e^4 - 161610923/4864823*e^3 - 84712318/4864823*e^2 + 480198532/4864823*e + 252928848/4864823, -2087497/19459292*e^8 - 4903239/19459292*e^7 + 68139521/19459292*e^6 + 132870751/19459292*e^5 - 158145118/4864823*e^4 - 209409171/4864823*e^3 + 1647463135/19459292*e^2 + 274471575/4864823*e + 20470543/4864823, -492617/9729646*e^8 - 1140209/4864823*e^7 + 6879195/4864823*e^6 + 34675446/4864823*e^5 - 96901067/9729646*e^4 - 277918408/4864823*e^3 + 158781125/9729646*e^2 + 1094580389/9729646*e + 90815923/4864823, -1217605/4864823*e^8 - 4774427/4864823*e^7 + 35544184/4864823*e^6 + 139399439/4864823*e^5 - 273621988/4864823*e^4 - 1064631838/4864823*e^3 + 584979398/4864823*e^2 + 2221184769/4864823*e + 89620018/4864823, -717718/4864823*e^8 - 1664244/4864823*e^7 + 21892562/4864823*e^6 + 51811274/4864823*e^5 - 189533815/4864823*e^4 - 433548263/4864823*e^3 + 516895069/4864823*e^2 + 999294461/4864823*e - 179053000/4864823, -1672391/19459292*e^8 - 4346325/19459292*e^7 + 47284559/19459292*e^6 + 129735405/19459292*e^5 - 84769874/4864823*e^4 - 243743416/4864823*e^3 + 586256769/19459292*e^2 + 406917085/4864823*e + 20993833/4864823, -668452/4864823*e^8 - 2304621/4864823*e^7 + 19252869/4864823*e^6 + 67943061/4864823*e^5 - 146257643/4864823*e^4 - 529462925/4864823*e^3 + 336430403/4864823*e^2 + 1151972338/4864823*e - 230713024/4864823, -1039405/19459292*e^8 - 763087/19459292*e^7 + 35464517/19459292*e^6 + 28239867/19459292*e^5 - 99699014/4864823*e^4 - 64630408/4864823*e^3 + 1752043227/19459292*e^2 + 119139639/4864823*e - 357386703/4864823, -151250/4864823*e^8 - 1002305/4864823*e^7 + 5668171/4864823*e^6 + 32967549/4864823*e^5 - 68770087/4864823*e^4 - 309517017/4864823*e^3 + 321872718/4864823*e^2 + 787676278/4864823*e - 332161346/4864823, 4010083/19459292*e^8 + 15392041/19459292*e^7 - 126705507/19459292*e^6 - 450813429/19459292*e^5 + 287372642/4864823*e^4 + 867127909/4864823*e^3 - 3388527745/19459292*e^2 - 1773243584/4864823*e + 236891739/4864823, -815590/4864823*e^8 - 2891134/4864823*e^7 + 24270777/4864823*e^6 + 84072376/4864823*e^5 - 194460383/4864823*e^4 - 628135400/4864823*e^3 + 442817307/4864823*e^2 + 1230553536/4864823*e + 3864996/4864823, 3128155/9729646*e^8 + 10222475/9729646*e^7 - 94804865/9729646*e^6 - 297007309/9729646*e^5 + 396012905/4864823*e^4 + 1118384903/4864823*e^3 - 1998053125/9729646*e^2 - 2263131222/4864823*e + 86344178/4864823, -481644/4864823*e^8 - 1928035/4864823*e^7 + 15654014/4864823*e^6 + 58033562/4864823*e^5 - 145125391/4864823*e^4 - 476543464/4864823*e^3 + 393622878/4864823*e^2 + 1139682870/4864823*e - 34251448/4864823, 843559/4864823*e^8 + 3296321/4864823*e^7 - 23581649/4864823*e^6 - 96017844/4864823*e^5 + 162452718/4864823*e^4 + 726056950/4864823*e^3 - 265435960/4864823*e^2 - 1532050348/4864823*e + 16354492/4864823, 865471/9729646*e^8 + 399301/4864823*e^7 - 12761480/4864823*e^6 - 9214741/4864823*e^5 + 199694725/9729646*e^4 + 20787291/4864823*e^3 - 334065445/9729646*e^2 + 285105057/9729646*e - 162051549/4864823, -2165103/9729646*e^8 - 3280316/4864823*e^7 + 32886272/4864823*e^6 + 95046746/4864823*e^5 - 555640347/9729646*e^4 - 708129616/4864823*e^3 + 1477745065/9729646*e^2 + 2860866809/9729646*e - 256877521/4864823, 720457/4864823*e^8 + 1960132/4864823*e^7 - 21527123/4864823*e^6 - 60355815/4864823*e^5 + 174121671/4864823*e^4 + 503662303/4864823*e^3 - 358245464/4864823*e^2 - 1197213945/4864823*e - 197960780/4864823, 384774/4864823*e^8 + 2721755/9729646*e^7 - 22623617/9729646*e^6 - 76664407/9729646*e^5 + 173641813/9729646*e^4 + 264445947/4864823*e^3 - 148698954/4864823*e^2 - 841675853/9729646*e - 288310247/4864823];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, -w^3 + 2*w^2 + 5*w - 3}>] := 1;
ALEigenvalues[ideal<ZF | {7, 7, -w^3 + 2*w^2 + 4*w - 3}>] := 1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;