/*
  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 | {15, 15, w^2 - 2*w - 5}>;

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^8 - 5*x^7 - 30*x^6 + 161*x^5 + 234*x^4 - 1406*x^3 - 793*x^2 + 3840*x + 1696;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 1, e, -455/35303*e^7 + 4572/35303*e^6 + 2828/35303*e^5 - 119576/35303*e^4 + 174499/35303*e^3 + 572784/35303*e^2 - 931225/35303*e - 355608/35303, -753/35303*e^7 + 1980/35303*e^6 + 21517/35303*e^5 - 50395/35303*e^4 - 142919/35303*e^3 + 189959/35303*e^2 + 286095/35303*e + 252398/35303, -3049/141212*e^7 - 5907/141212*e^6 + 74107/70606*e^5 + 153287/141212*e^4 - 1089199/70606*e^3 - 463695/70606*e^2 + 8616513/141212*e + 996463/35303, 279/70606*e^7 + 5033/70606*e^6 - 13902/35303*e^5 - 134073/70606*e^4 + 273809/35303*e^3 + 345476/35303*e^2 - 2112789/70606*e - 386146/35303, -4869/141212*e^7 + 12381/141212*e^6 + 79763/70606*e^5 - 325017/141212*e^4 - 740201/70606*e^3 + 752479/70606*e^2 + 4891613/141212*e + 323128/35303, 2567/141212*e^7 - 13923/141212*e^6 - 28073/70606*e^5 + 374427/141212*e^4 + 10575/70606*e^3 - 984723/70606*e^2 + 1309121/141212*e + 436544/35303, 1170/35303*e^7 - 1670/35303*e^6 - 42575/35303*e^5 + 40187/35303*e^4 + 464123/35303*e^3 - 111186/35303*e^2 - 1640050/35303*e - 719604/35303, -6451/70606*e^7 + 26105/70606*e^6 + 83050/35303*e^5 - 685375/70606*e^4 - 387417/35303*e^3 + 1629298/35303*e^2 + 893443/70606*e - 418402/35303, 1543/70606*e^7 + 9867/70606*e^6 - 52590/35303*e^5 - 254077/70606*e^4 + 946280/35303*e^3 + 653654/35303*e^2 - 8114929/70606*e - 1599316/35303, 13195/141212*e^7 - 26679/141212*e^6 - 217521/70606*e^5 + 678767/141212*e^4 + 2041503/70606*e^3 - 1280071/70606*e^2 - 13381107/141212*e - 1305172/35303, 455/35303*e^7 - 4572/35303*e^6 - 2828/35303*e^5 + 119576/35303*e^4 - 174499/35303*e^3 - 572784/35303*e^2 + 895922/35303*e + 496820/35303, -521/141212*e^7 + 3761/141212*e^6 - 3269/70606*e^5 - 86721/141212*e^4 + 255743/70606*e^3 + 152661/70606*e^2 - 3105343/141212*e - 322616/35303, 1132/35303*e^7 + 3212/35303*e^6 - 60805/35303*e^5 - 89597/35303*e^4 + 959826/35303*e^3 + 610977/35303*e^2 - 3995836/35303*e - 2177872/35303, 6527/70606*e^7 - 35869/70606*e^6 - 64820/35303*e^5 + 944943/70606*e^4 - 108286/35303*e^3 - 2316158/35303*e^2 + 3747523/70606*e + 1594246/35303, -493/35303*e^7 + 9454/35303*e^6 - 15402/35303*e^5 - 249360/35303*e^4 + 670202/35303*e^3 + 1259644/35303*e^2 - 3216405/35303*e - 1602058/35303, 10815/70606*e^7 - 51645/70606*e^6 - 123225/35303*e^5 + 1335063/70606*e^4 + 238493/35303*e^3 - 3005456/35303*e^2 + 2201885/70606*e + 429834/35303, -3979/141212*e^7 + 24387/141212*e^6 + 49841/70606*e^5 - 670711/141212*e^4 - 189675/70606*e^3 + 1891483/70606*e^2 - 15389/141212*e - 701726/35303, -417/35303*e^7 - 310/35303*e^6 + 21058/35303*e^5 + 10208/35303*e^4 - 321204/35303*e^3 - 114076/35303*e^2 + 1353955/35303*e + 961448/35303, -12639/141212*e^7 + 74163/141212*e^6 + 109341/70606*e^5 - 1916215/141212*e^4 + 643843/70606*e^3 + 4415715/70606*e^2 - 11630005/141212*e - 1298238/35303, -734/35303*e^7 - 461/35303*e^6 + 30632/35303*e^5 + 14497/35303*e^4 - 373119/35303*e^3 - 118168/35303*e^2 + 1146261/35303*e + 487290/35303, 7791/70606*e^7 - 31035/70606*e^6 - 103508/35303*e^5 + 824939/70606*e^4 + 564185/35303*e^3 - 2043283/35303*e^2 - 2113405/70606*e + 592894/35303, 1999/35303*e^7 - 13414/35303*e^6 - 27632/35303*e^5 + 350150/35303*e^4 - 384364/35303*e^3 - 1639562/35303*e^2 + 2644215/35303*e + 1097262/35303, -3727/35303*e^7 + 5018/35303*e^6 + 125815/35303*e^5 - 122191/35303*e^4 - 1210298/35303*e^3 + 333541/35303*e^2 + 3680352/35303*e + 1731774/35303, -9481/141212*e^7 + 17757/141212*e^6 + 144063/70606*e^5 - 431893/141212*e^4 - 1132777/70606*e^3 + 676161/70606*e^2 + 5886641/141212*e + 598864/35303, 1599/35303*e^7 - 14050/35303*e^6 - 16999/35303*e^5 + 373826/35303*e^4 - 437733/35303*e^3 - 1902983/35303*e^2 + 2536271/35303*e + 1403024/35303, 1468/35303*e^7 + 922/35303*e^6 - 61264/35303*e^5 - 28994/35303*e^4 + 781541/35303*e^3 + 306942/35303*e^2 - 2963279/35303*e - 1680640/35303, -5831/70606*e^7 + 5909/70606*e^6 + 110995/35303*e^5 - 136043/70606*e^4 - 1293059/35303*e^3 + 70947/35303*e^2 + 9660567/70606*e + 2551910/35303, 1246/35303*e^7 - 11434/35303*e^6 - 6115/35303*e^5 + 299755/35303*e^4 - 527283/35303*e^3 - 1520209/35303*e^2 + 3000916/35303*e + 2196932/35303, -715/35303*e^7 - 2902/35303*e^6 + 39747/35303*e^5 + 79389/35303*e^4 - 638622/35303*e^3 - 496901/35303*e^2 + 2606578/35303*e + 1357636/35303, 647/70606*e^7 + 39509/70606*e^6 - 87281/35303*e^5 - 1051139/70606*e^4 + 2078336/35303*e^3 + 2753578/35303*e^2 - 18258509/70606*e - 3886230/35303, -7365/70606*e^7 + 39479/70606*e^6 + 66338/35303*e^5 - 1027375/70606*e^4 + 320420/35303*e^3 + 2431470/35303*e^2 - 6531015/70606*e - 2158830/35303, -14311/141212*e^7 + 6547/141212*e^6 + 273129/70606*e^5 - 142475/141212*e^4 - 3136739/70606*e^3 - 172439/70606*e^2 + 21832263/141212*e + 2501100/35303, -4273/70606*e^7 + 17565/70606*e^6 + 61074/35303*e^5 - 463379/70606*e^4 - 422783/35303*e^3 + 1064698/35303*e^2 + 2527579/70606*e + 532492/35303, 13979/141212*e^7 - 8487/141212*e^6 - 271011/70606*e^5 + 220023/141212*e^4 + 3204437/70606*e^3 - 275613/70606*e^2 - 23551443/141212*e - 1932998/35303, 8032/35303*e^7 - 21120/35303*e^6 - 253050/35303*e^5 + 561082/35303*e^4 + 2171691/35303*e^3 - 2673451/35303*e^2 - 6546677/35303*e - 1185984/35303, -12117/70606*e^7 + 51693/70606*e^6 + 152798/35303*e^5 - 1344843/70606*e^4 - 645461/35303*e^3 + 3040708/35303*e^2 + 1232651/70606*e + 266020/35303, -3455/70606*e^7 - 35501/70606*e^6 + 138371/35303*e^5 + 940569/70606*e^4 - 2628223/35303*e^3 - 2493064/35303*e^2 + 22124023/70606*e + 4481452/35303, -6075/70606*e^7 - 18485/70606*e^6 + 154660/35303*e^5 + 509617/70606*e^4 - 2332594/35303*e^3 - 1610917/35303*e^2 + 18324447/70606*e + 4358422/35303, 529/35303*e^7 - 12221/35303*e^6 + 14092/35303*e^5 + 327720/35303*e^4 - 632567/35303*e^3 - 1752419/35303*e^2 + 2649975/35303*e + 1957930/35303, -831/3284*e^7 + 4007/3284*e^6 + 9441/1642*e^5 - 105235/3284*e^4 - 16379/1642*e^3 + 254101/1642*e^2 - 204993/3284*e - 63574/821, -9777/70606*e^7 + 48353/70606*e^6 + 110223/35303*e^5 - 1264469/70606*e^4 - 181338/35303*e^3 + 3035431/35303*e^2 - 2329873/70606*e - 1512674/35303, 22635/141212*e^7 + 16573/141212*e^6 - 491263/70606*e^5 - 473077/141212*e^4 + 6575769/70606*e^3 + 1877721/70606*e^2 - 50824867/141212*e - 4782530/35303, 70/821*e^7 - 135/821*e^6 - 2456/821*e^5 + 3492/821*e^4 + 25698/821*e^3 - 13157/821*e^2 - 89646/821*e - 47474/821, 3283/35303*e^7 - 29730/35303*e^6 - 15517/35303*e^5 + 779689/35303*e^4 - 1407350/35303*e^3 - 3846631/35303*e^2 + 7789099/35303*e + 4046402/35303, 1880/35303*e^7 - 11132/35303*e^6 - 25263/35303*e^5 + 291177/35303*e^4 - 388150/35303*e^3 - 1406116/35303*e^2 + 2745547/35303*e + 1450704/35303, 1956/35303*e^7 - 20896/35303*e^6 + 11197/35303*e^5 + 550745/35303*e^4 - 1379556/35303*e^3 - 2815139/35303*e^2 + 7386513/35303*e + 4226028/35303, 7375/35303*e^7 - 14751/35303*e^6 - 246794/35303*e^5 + 366617/35303*e^4 + 2376253/35303*e^3 - 1243975/35303*e^2 - 7982880/35303*e - 3656106/35303, 2335/141212*e^7 - 51007/141212*e^6 + 38909/70606*e^5 + 1328631/141212*e^4 - 1605187/70606*e^3 - 3072327/70606*e^2 + 13808733/141212*e + 734002/35303, 3352/35303*e^7 - 14440/35303*e^6 - 82750/35303*e^5 + 365031/35303*e^4 + 315199/35303*e^3 - 1522647/35303*e^2 - 127689/35303*e + 139100/35303, 10164/35303*e^7 - 16318/35303*e^6 - 358089/35303*e^5 + 412295/35303*e^4 + 3741530/35303*e^3 - 1421573/35303*e^2 - 13167499/35303*e - 5363866/35303, -4067/141212*e^7 - 23765/141212*e^6 + 114203/70606*e^5 + 596933/141212*e^4 - 1800773/70606*e^3 - 1252277/70606*e^2 + 12901531/141212*e + 587058/35303, 6357/70606*e^7 - 32609/70606*e^6 - 83301/35303*e^5 + 880869/70606*e^4 + 423598/35303*e^3 - 2310796/35303*e^2 - 1544379/70606*e + 989346/35303, -14437/70606*e^7 + 33883/70606*e^6 + 222467/35303*e^5 - 840371/70606*e^4 - 1834609/35303*e^3 + 1475409/35303*e^2 + 10929173/70606*e + 2614274/35303, -10568/35303*e^7 + 55215/35303*e^6 + 223733/35303*e^5 - 1444648/35303*e^4 - 9892/35303*e^3 + 6877075/35303*e^2 - 4293655/35303*e - 3242734/35303, -19519/70606*e^7 + 77345/70606*e^6 + 250209/35303*e^5 - 2017351/70606*e^4 - 1163705/35303*e^3 + 4663868/35303*e^2 + 2950435/70606*e - 475286/35303, 16511/141212*e^7 - 73655/141212*e^6 - 187635/70606*e^5 + 1918619/141212*e^4 + 344529/70606*e^3 - 4478035/70606*e^2 + 3473529/141212*e + 864724/35303, 26799/141212*e^7 - 79891/141212*e^6 - 414045/70606*e^5 + 2094811/141212*e^4 + 3439225/70606*e^3 - 4762659/70606*e^2 - 21376867/141212*e - 1128862/35303, -6499/70606*e^7 + 6259/70606*e^6 + 130994/35303*e^5 - 154401/70606*e^4 - 1612809/35303*e^3 + 145594/35303*e^2 + 12027765/70606*e + 3015862/35303, -10/821*e^7 + 723/821*e^6 - 2464/821*e^5 - 19030/821*e^4 + 69867/821*e^3 + 98523/821*e^2 - 326501/821*e - 159060/821, 6421/35303*e^7 - 29683/35303*e^6 - 141473/35303*e^5 + 761287/35303*e^4 + 125669/35303*e^3 - 3348076/35303*e^2 + 2367519/35303*e + 1034486/35303, 16901/70606*e^7 - 97747/70606*e^6 - 153544/35303*e^5 + 2555701/70606*e^4 - 678394/35303*e^3 - 6120504/35303*e^2 + 13753099/70606*e + 4363148/35303, -448/35303*e^7 + 14821/35303*e^6 - 34691/35303*e^5 - 398531/35303*e^4 + 1132056/35303*e^3 + 2099924/35303*e^2 - 5248305/35303*e - 2145702/35303, -3371/35303*e^7 - 18422/35303*e^6 + 214847/35303*e^5 + 487955/35303*e^4 - 3756511/35303*e^3 - 2723313/35303*e^2 + 15965842/35303*e + 7322062/35303, 11319/35303*e^7 - 55080/35303*e^6 - 264790/35303*e^5 + 1443619/35303*e^4 + 686149/35303*e^3 - 6872949/35303*e^2 + 1226556/35303*e + 2240970/35303, 13119/141212*e^7 - 16915/141212*e^6 - 235751/70606*e^5 + 419199/141212*e^4 + 2572509/70606*e^3 - 522605/70606*e^2 - 19363587/141212*e - 1893094/35303, -29021/141212*e^7 + 134961/141212*e^6 + 362293/70606*e^5 - 3518741/141212*e^4 - 1491077/70606*e^3 + 8132951/70606*e^2 + 3254241/141212*e - 79010/35303, -13667/141212*e^7 - 39029/141212*e^6 + 383011/70606*e^5 + 1023945/141212*e^4 - 6253925/70606*e^3 - 2859997/70606*e^2 + 52674475/141212*e + 5104658/35303, 15089/141212*e^7 - 158525/141212*e^6 - 2899/70606*e^5 + 4118849/141212*e^4 - 4043797/70606*e^3 - 9775981/70606*e^2 + 42757651/141212*e + 4340482/35303, -7045/35303*e^7 + 18806/35303*e^6 + 223018/35303*e^5 - 495589/35303*e^4 - 1907705/35303*e^3 + 2369467/35303*e^2 + 5579542/35303*e + 493120/35303, -4087/141212*e^7 - 73221/141212*e^6 + 228321/70606*e^5 + 1918449/141212*e^4 - 4853169/70606*e^3 - 4977151/70606*e^2 + 42988411/141212*e + 4679850/35303, 7235/70606*e^7 - 7913/70606*e^6 - 136540/35303*e^5 + 226631/70606*e^4 + 1550351/35303*e^3 - 624840/35303*e^2 - 10569537/70606*e - 837250/35303, -3443/70606*e^7 + 22415/70606*e^6 + 20476/35303*e^5 - 586643/70606*e^4 + 431759/35303*e^3 + 1443465/35303*e^2 - 6024763/70606*e - 1743170/35303, 4757/35303*e^7 + 150/35303*e^6 - 194676/35303*e^5 - 12911/35303*e^4 + 2434173/35303*e^3 + 432143/35303*e^2 - 8966149/35303*e - 4070678/35303, 2145/141212*e^7 - 26597/141212*e^6 + 28637/70606*e^5 + 609105/141212*e^4 - 1372065/70606*e^3 - 719723/70606*e^2 + 13891611/141212*e + 217378/35303, -1037/70606*e^7 - 15417/70606*e^6 + 53190/35303*e^5 + 414057/70606*e^4 - 1126019/35303*e^3 - 1181715/35303*e^2 + 10308937/70606*e + 2876468/35303, -5327/70606*e^7 - 32829/70606*e^6 + 172431/35303*e^5 + 890391/70606*e^4 - 3041885/35303*e^3 - 2566509/35303*e^2 + 25877799/70606*e + 5749074/35303, -6179/35303*e^7 + 20889/35303*e^6 + 175815/35303*e^5 - 540493/35303*e^4 - 1184773/35303*e^3 + 2422408/35303*e^2 + 2991825/35303*e - 101426/35303, -857/70606*e^7 + 6051/70606*e^6 + 14612/35303*e^5 - 182627/70606*e^4 - 131705/35303*e^3 + 675360/35303*e^2 + 698611/70606*e - 470212/35303, 11461/70606*e^7 + 6573/70606*e^6 - 250694/35303*e^5 - 200727/70606*e^4 + 3350390/35303*e^3 + 935165/35303*e^2 - 25390301/70606*e - 5744266/35303, 3922/35303*e^7 - 17064/35303*e^6 - 91724/35303*e^5 + 440732/35303*e^4 + 257981/35303*e^3 - 1976010/35303*e^2 + 188525/35303*e + 548896/35303, 5994/35303*e^7 - 19418/35303*e^6 - 182812/35303*e^5 + 514375/35303*e^4 + 1447368/35303*e^3 - 2421121/35303*e^2 - 4182036/35303*e - 1044836/35303, 10867/70606*e^7 - 36029/70606*e^6 - 151629/35303*e^5 + 906937/70606*e^4 + 965850/35303*e^3 - 1751140/35303*e^2 - 4358913/70606*e - 1690216/35303, -10407/70606*e^7 + 43821/70606*e^6 + 139337/35303*e^5 - 1153043/70606*e^4 - 766591/35303*e^3 + 2696042/35303*e^2 + 3099171/70606*e - 37164/35303, 1321/35303*e^7 - 2489/35303*e^6 - 50031/35303*e^5 + 74672/35303*e^4 + 583736/35303*e^3 - 519843/35303*e^2 - 2221340/35303*e + 255304/35303, -15281/70606*e^7 + 79141/70606*e^6 + 159372/35303*e^5 - 2065559/70606*e^4 + 60107/35303*e^3 + 4794346/35303*e^2 - 6975507/70606*e - 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1103375/70606*e^4 + 2340931/35303*e^3 + 2754865/35303*e^2 - 20356507/70606*e - 4397038/35303, -4031/35303*e^7 + 8771/35303*e^6 + 121187/35303*e^5 - 207282/35303*e^4 - 916186/35303*e^3 + 603577/35303*e^2 + 2414958/35303*e + 1433196/35303, -2982/35303*e^7 + 11498/35303*e^6 + 61441/35303*e^5 - 312795/35303*e^4 + 77489/35303*e^3 + 1696588/35303*e^2 - 2328400/35303*e - 2506572/35303, 3202/35303*e^7 + 38276/35303*e^6 - 277342/35303*e^5 - 1020559/35303*e^4 + 5506791/35303*e^3 + 5584795/35303*e^2 - 24209511/35303*e - 11369752/35303, 20527/70606*e^7 - 84215/70606*e^6 - 268549/35303*e^5 + 2163857/70606*e^4 + 1372868/35303*e^3 - 4643330/35303*e^2 - 4901093/70606*e - 2803408/35303, 370/821*e^7 - 479/821*e^6 - 13099/821*e^5 + 12007/821*e^4 + 137357/821*e^3 - 37056/821*e^2 - 466806/821*e - 206600/821, 40097/141212*e^7 - 197841/141212*e^6 - 410835/70606*e^5 + 5033581/141212*e^4 - 362505/70606*e^3 - 10607957/70606*e^2 + 18970627/141212*e + 762430/35303, -8648/35303*e^7 + 1783/35303*e^6 + 342149/35303*e^5 - 19082/35303*e^4 - 4110111/35303*e^3 - 634830/35303*e^2 + 14462006/35303*e + 6487720/35303, -2730/35303*e^7 - 43174/35303*e^6 + 264089/35303*e^5 + 1153603/35303*e^4 - 5378152/35303*e^3 - 6130409/35303*e^2 + 23184595/35303*e + 9022100/35303, -19039/141212*e^7 + 63987/141212*e^6 + 335617/70606*e^5 - 1819823/141212*e^4 - 3454621/70606*e^3 + 5415011/70606*e^2 + 24205283/141212*e + 207234/35303, -20132/35303*e^7 + 107790/35303*e^6 + 407009/35303*e^5 - 2810335/35303*e^4 + 476734/35303*e^3 + 13265518/35303*e^2 - 10698617/35303*e - 7198564/35303, -17848/35303*e^7 + 93064/35303*e^6 + 374890/35303*e^5 - 2439986/35303*e^4 + 134367/35303*e^3 + 11805259/35303*e^2 - 8743567/35303*e - 7449736/35303, 13004/35303*e^7 - 54166/35303*e^6 - 324144/35303*e^5 + 1416255/35303*e^4 + 1285619/35303*e^3 - 6460081/35303*e^2 - 496925/35303*e - 142328/35303, 51563/141212*e^7 - 214207/141212*e^6 - 672407/70606*e^5 + 5632171/141212*e^4 + 3292045/70606*e^3 - 13419221/70606*e^2 - 6986703/141212*e + 785732/35303, 20275/141212*e^7 - 29543/141212*e^6 - 369873/70606*e^5 + 697459/141212*e^4 + 4083037/70606*e^3 - 535665/70606*e^2 - 30731815/141212*e - 5695992/35303, 1891/35303*e^7 - 40416/35303*e^6 + 82207/35303*e^5 + 1068251/35303*e^4 - 3144994/35303*e^3 - 5527293/35303*e^2 + 14969708/35303*e + 8149336/35303, -22541/70606*e^7 + 60537/70606*e^6 + 350302/35303*e^5 - 1558173/70606*e^4 - 2940438/35303*e^3 + 3181349/35303*e^2 + 17796741/70606*e + 4122302/35303, 15175/141212*e^7 - 72955/141212*e^6 - 218243/70606*e^5 + 1881903/141212*e^4 + 1540785/70606*e^3 - 4046317/70606*e^2 - 11279331/141212*e - 1213140/35303, 26735/141212*e^7 - 12211/141212*e^6 - 585473/70606*e^5 + 308031/141212*e^4 + 7948091/70606*e^3 - 33361/70606*e^2 - 64192671/141212*e - 6629548/35303, -20077/35303*e^7 - 3327/35303*e^6 + 838450/35303*e^5 + 121854/35303*e^4 - 10800973/35303*e^3 - 2115523/35303*e^2 + 41384620/35303*e + 18951572/35303, 9608/35303*e^7 - 28499/35303*e^6 - 282941/35303*e^5 + 731865/35303*e^4 + 2112956/35303*e^3 - 3174077/35303*e^2 - 6019705/35303*e - 739918/35303, 33143/70606*e^7 - 222313/70606*e^6 - 243124/35303*e^5 + 5783391/70606*e^4 - 2788291/35303*e^3 - 13670968/35303*e^2 + 39810795/70606*e + 9587762/35303, 14716/35303*e^7 + 6453/35303*e^6 - 613950/35303*e^5 - 211580/35303*e^4 + 7853045/35303*e^3 + 2364827/35303*e^2 - 29140130/35303*e - 14755984/35303, 18931/35303*e^7 - 126292/35303*e^6 - 314274/35303*e^5 + 3314590/35303*e^4 - 2205928/35303*e^3 - 16094570/35303*e^2 + 17315791/35303*e + 10388892/35303, 32431/70606*e^7 - 34221/70606*e^6 - 579277/35303*e^5 + 820981/70606*e^4 + 6147765/35303*e^3 - 764577/35303*e^2 - 42162863/70606*e - 9347060/35303, 42593/141212*e^7 - 13121/141212*e^6 - 856349/70606*e^5 + 299277/141212*e^4 + 10544591/70606*e^3 + 316223/70606*e^2 - 80316953/141212*e - 6907898/35303, -88641/141212*e^7 + 131953/141212*e^6 + 1568645/70606*e^5 - 3329497/141212*e^4 - 16444987/70606*e^3 + 5382513/70606*e^2 + 114552533/141212*e + 13020464/35303, -4585/35303*e^7 + 29778/35303*e^6 + 74663/35303*e^5 - 754166/35303*e^4 + 593414/35303*e^3 + 3105166/35303*e^2 - 4672290/35303*e - 607120/35303, 19595/70606*e^7 - 16503/70606*e^6 - 373191/35303*e^5 + 370557/70606*e^4 + 4304211/35303*e^3 - 55278/35303*e^2 - 30929441/70606*e - 8163104/35303, 5653/35303*e^7 - 29492/35303*e^6 - 125294/35303*e^5 + 748848/35303*e^4 + 134758/35303*e^3 - 3132251/35303*e^2 + 1601067/35303*e + 503150/35303];
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 | {5, 5, w}>] := -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;