/*
  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^8 + 2*x^7 - 18*x^6 - 34*x^5 + 53*x^4 + 52*x^3 - 43*x^2 - 7*x + 2;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -1, -1666/5283*e^7 - 3703/5283*e^6 + 29008/5283*e^5 + 6977/587*e^4 - 71540/5283*e^3 - 98834/5283*e^2 + 14042/1761*e + 15760/5283, 1484/5283*e^7 + 3920/5283*e^6 - 24596/5283*e^5 - 7448/587*e^4 + 43525/5283*e^3 + 118048/5283*e^2 - 7225/1761*e - 34460/5283, 147/587*e^7 + 344/587*e^6 - 2525/587*e^5 - 5765/587*e^4 + 5898/587*e^3 + 7961/587*e^2 - 5478/587*e - 113/587, -203/1761*e^7 - 503/1761*e^6 + 3431/1761*e^5 + 2691/587*e^4 - 7474/1761*e^3 - 6745/1761*e^2 + 4059/587*e - 5686/1761, -1610/5283*e^7 - 2957/5283*e^6 + 29276/5283*e^5 + 5677/587*e^4 - 89335/5283*e^3 - 83614/5283*e^2 + 18853/1761*e + 6884/5283, 238/5283*e^7 + 529/5283*e^6 - 4144/5283*e^5 - 829/587*e^4 + 10220/5283*e^3 - 9277/5283*e^2 - 5528/1761*e + 9824/5283, -899/1761*e^7 - 1976/1761*e^6 + 15446/1761*e^5 + 10911/587*e^4 - 36118/1761*e^3 - 39682/1761*e^2 + 9506/587*e - 1030/1761, -2852/5283*e^7 - 5540/5283*e^6 + 52766/5283*e^5 + 10526/587*e^4 - 175609/5283*e^3 - 154003/5283*e^2 + 56491/1761*e + 8270/5283, 2002/5283*e^7 + 2896/5283*e^6 - 37966/5283*e^5 - 5385/587*e^4 + 139109/5283*e^3 + 47513/5283*e^2 - 43289/1761*e + 15512/5283, 1769/5283*e^7 + 3377/5283*e^6 - 32666/5283*e^5 - 6475/587*e^4 + 108904/5283*e^3 + 101545/5283*e^2 - 39061/1761*e - 29444/5283, 2944/5283*e^7 + 8275/5283*e^6 - 46288/5283*e^5 - 15429/587*e^4 + 45620/5283*e^3 + 206177/5283*e^2 + 5878/1761*e - 22852/5283, -1196/5283*e^7 - 3857/5283*e^6 + 16163/5283*e^5 + 6800/587*e^4 + 25712/5283*e^3 - 44302/5283*e^2 - 27152/1761*e - 11188/5283, -1598/5283*e^7 - 5816/5283*e^6 + 22541/5283*e^5 + 10933/587*e^4 + 15908/5283*e^3 - 171673/5283*e^2 - 10682/1761*e + 47246/5283, 3874/5283*e^7 + 5947/5283*e^6 - 74290/5283*e^5 - 11358/587*e^4 + 280094/5283*e^3 + 151769/5283*e^2 - 82715/1761*e - 17050/5283, -7138/5283*e^7 - 15466/5283*e^6 + 125839/5283*e^5 + 29268/587*e^4 - 330443/5283*e^3 - 422276/5283*e^2 + 64943/1761*e + 58924/5283, 10630/5283*e^7 + 23494/5283*e^6 - 183844/5283*e^5 - 43582/587*e^4 + 436265/5283*e^3 + 525317/5283*e^2 - 85319/1761*e - 41842/5283, 7016/5283*e^7 + 17237/5283*e^6 - 118121/5283*e^5 - 32222/587*e^4 + 236947/5283*e^3 + 434401/5283*e^2 - 45550/1761*e - 66002/5283, -34/1761*e^7 - 1585/1761*e^6 - 2930/1761*e^5 + 8741/587*e^4 + 56653/1761*e^3 - 43706/1761*e^2 - 21684/587*e + 7150/1761, 1166/1761*e^7 + 1319/1761*e^6 - 23099/1761*e^5 - 7577/587*e^4 + 95959/1761*e^3 + 27595/1761*e^2 - 27857/587*e + 3616/1761, 3743/5283*e^7 + 11183/5283*e^6 - 60200/5283*e^5 - 21189/587*e^4 + 74647/5283*e^3 + 342202/5283*e^2 + 9458/1761*e - 83456/5283, -5183/5283*e^7 - 11498/5283*e^6 + 91799/5283*e^5 + 22081/587*e^4 - 251776/5283*e^3 - 356971/5283*e^2 + 69094/1761*e + 68678/5283, 1985/1761*e^7 + 4745/1761*e^6 - 34148/1761*e^5 - 26753/587*e^4 + 78505/1761*e^3 + 124276/1761*e^2 - 16563/587*e - 7328/1761, -6685/5283*e^7 - 13771/5283*e^6 + 118573/5283*e^5 + 25754/587*e^4 - 323732/5283*e^3 - 324440/5283*e^2 + 75716/1761*e + 5614/5283, 3857/1761*e^7 + 7796/1761*e^6 - 68711/1761*e^5 - 44085/587*e^4 + 191314/1761*e^3 + 196834/1761*e^2 - 42488/587*e - 15236/1761, -1427/587*e^7 - 2972/587*e^6 + 25330/587*e^5 + 50469/587*e^4 - 69426/587*e^3 - 76854/587*e^2 + 47655/587*e + 6348/587, 5380/5283*e^7 + 14311/5283*e^6 - 87460/5283*e^5 - 26780/587*e^4 + 130025/5283*e^3 + 376928/5283*e^2 - 7610/1761*e - 81412/5283, -1069/1761*e^7 - 2857/1761*e^6 + 16645/1761*e^5 + 15287/587*e^4 - 13481/1761*e^3 - 38087/1761*e^2 - 3820/587*e - 14588/1761, -2788/5283*e^7 - 8461/5283*e^6 + 43261/5283*e^5 + 15665/587*e^4 - 29909/5283*e^3 - 204533/5283*e^2 - 20023/1761*e + 19258/5283, 11050/5283*e^7 + 23806/5283*e^6 - 192400/5283*e^5 - 44527/587*e^4 + 479783/5283*e^3 + 565505/5283*e^2 - 106469/1761*e - 45016/5283, -2662/5283*e^7 - 4141/5283*e^6 + 54430/5283*e^5 + 8631/587*e^4 - 242966/5283*e^3 - 212552/5283*e^2 + 75770/1761*e + 67966/5283, -176/5283*e^7 - 3854/5283*e^6 - 6880/5283*e^5 + 6853/587*e^4 + 154040/5283*e^3 - 69721/5283*e^2 - 68705/1761*e - 14368/5283, 10348/5283*e^7 + 19360/5283*e^6 - 186703/5283*e^5 - 36197/587*e^4 + 559649/5283*e^3 + 426032/5283*e^2 - 156401/1761*e - 12994/5283, -252/587*e^7 - 422/587*e^6 + 5251/587*e^5 + 8038/587*e^4 - 24702/587*e^3 - 25052/587*e^2 + 25156/587*e + 11766/587, -998/1761*e^7 - 842/1761*e^6 + 20381/1761*e^5 + 5269/587*e^4 - 92992/1761*e^3 - 31243/1761*e^2 + 28789/587*e + 26108/1761, -823/587*e^7 - 1886/587*e^6 + 14468/587*e^5 + 32368/587*e^4 - 37585/587*e^3 - 56175/587*e^2 + 27938/587*e + 8056/587, 3448/5283*e^7 + 9706/5283*e^6 - 54442/5283*e^5 - 17737/587*e^4 + 54521/5283*e^3 + 205799/5283*e^2 + 5152/1761*e - 81604/5283, 1585/1761*e^7 + 4951/1761*e^6 - 24490/1761*e^5 - 28162/587*e^4 + 18443/1761*e^3 + 150404/1761*e^2 - 1787/587*e - 46066/1761, -613/587*e^7 - 1730/587*e^6 + 9603/587*e^5 + 28996/587*e^4 - 8782/587*e^3 - 43125/587*e^2 - 3200/587*e + 10578/587, 10736/5283*e^7 + 23774/5283*e^6 - 187865/5283*e^5 - 44701/587*e^4 + 472204/5283*e^3 + 618277/5283*e^2 - 91747/1761*e - 127322/5283, 8914/5283*e^7 + 14974/5283*e^6 - 166396/5283*e^5 - 27981/587*e^4 + 564575/5283*e^3 + 327611/5283*e^2 - 137522/1761*e + 29390/5283, -1166/1761*e^7 - 1319/1761*e^6 + 23099/1761*e^5 + 8164/587*e^4 - 95959/1761*e^3 - 55771/1761*e^2 + 21987/587*e + 31604/1761, 997/1761*e^7 + 640/1761*e^6 - 20260/1761*e^5 - 3648/587*e^4 + 91706/1761*e^3 + 2921/1761*e^2 - 25007/587*e + 4868/1761, -10274/5283*e^7 - 25544/5283*e^6 + 174227/5283*e^5 + 48064/587*e^4 - 369391/5283*e^3 - 693466/5283*e^2 + 80809/1761*e + 122774/5283, 329/5283*e^7 - 2221/5283*e^6 - 11633/5283*e^5 + 4396/587*e^4 + 111397/5283*e^3 - 108695/5283*e^2 - 45037/1761*e + 19174/5283, -2933/5283*e^7 - 770/5283*e^6 + 62567/5283*e^5 + 876/587*e^4 - 316756/5283*e^3 + 108887/5283*e^2 + 95161/1761*e - 103042/5283, -5248/5283*e^7 - 8779/5283*e^6 + 97903/5283*e^5 + 16504/587*e^4 - 340649/5283*e^3 - 193883/5283*e^2 + 121214/1761*e - 13472/5283, 11900/5283*e^7 + 26450/5283*e^6 - 201917/5283*e^5 - 49081/587*e^4 + 431755/5283*e^3 + 576901/5283*e^2 - 72124/1761*e - 5402/5283, -5059/5283*e^7 - 7582/5283*e^6 + 96166/5283*e^5 + 14171/587*e^4 - 361745/5283*e^3 - 145157/5283*e^2 + 135470/1761*e - 35504/5283, 10814/5283*e^7 + 23681/5283*e^6 - 186737/5283*e^5 - 43996/587*e^4 + 440437/5283*e^3 + 524005/5283*e^2 - 82090/1761*e - 18176/5283, 9247/5283*e^7 + 18844/5283*e^6 - 164425/5283*e^5 - 35334/587*e^4 + 453947/5283*e^3 + 460757/5283*e^2 - 107876/1761*e - 121126/5283, 274/587*e^7 + 170/587*e^6 - 5565/587*e^5 - 2320/587*e^4 + 25405/587*e^3 - 8717/587*e^2 - 32337/587*e + 8814/587, -4801/1761*e^7 - 10057/1761*e^6 + 84319/1761*e^5 + 56327/587*e^4 - 220145/1761*e^3 - 234281/1761*e^2 + 51954/587*e + 6370/1761, 12683/5283*e^7 + 26126/5283*e^6 - 222698/5283*e^5 - 48935/587*e^4 + 577564/5283*e^3 + 622540/5283*e^2 - 117214/1761*e - 84602/5283, -7160/5283*e^7 - 14627/5283*e^6 + 124979/5283*e^5 + 27263/587*e^4 - 311188/5283*e^3 - 333916/5283*e^2 + 53053/1761*e + 99392/5283, -3926/1761*e^7 - 7646/1761*e^6 + 70016/1761*e^5 + 42643/587*e^4 - 195520/1761*e^3 - 164644/1761*e^2 + 41644/587*e + 638/1761, -1931/1761*e^7 - 4403/1761*e^6 + 32897/1761*e^5 + 24334/587*e^4 - 72457/1761*e^3 - 93499/1761*e^2 + 19546/587*e + 7574/1761, 4123/5283*e^7 + 13981/5283*e^6 - 62155/5283*e^5 - 26153/587*e^4 + 19178/5283*e^3 + 394160/5283*e^2 + 12796/1761*e - 90856/5283, 560/587*e^7 + 1003/587*e^6 - 10821/587*e^5 - 17797/587*e^4 + 42175/587*e^3 + 37735/587*e^2 - 47619/587*e - 14798/587, 4832/5283*e^7 + 9275/5283*e^6 - 91592/5283*e^5 - 18503/587*e^4 + 333973/5283*e^3 + 374404/5283*e^2 - 102865/1761*e - 121346/5283, -7360/5283*e^7 - 12763/5283*e^6 + 136852/5283*e^5 + 23604/587*e^4 - 462728/5283*e^3 - 248651/5283*e^2 + 147317/1761*e - 48530/5283, -1120/1761*e^7 - 2593/1761*e^6 + 17533/1761*e^5 + 13430/587*e^4 - 15671/1761*e^3 - 10313/1761*e^2 - 8170/587*e - 33800/1761, -8330/5283*e^7 - 18515/5283*e^6 + 145040/5283*e^5 + 34885/587*e^4 - 362983/5283*e^3 - 473038/5283*e^2 + 93103/1761*e + 4838/5283, -1699/587*e^7 - 3912/587*e^6 + 29479/587*e^5 + 65956/587*e^4 - 71714/587*e^3 - 99543/587*e^2 + 50843/587*e + 10718/587, -358/1761*e^7 - 3637/1761*e^6 - 707/1761*e^5 + 20320/587*e^4 + 97849/1761*e^3 - 106859/1761*e^2 - 38408/587*e + 23284/1761, -1991/1761*e^7 - 2435/1761*e^6 + 38396/1761*e^5 + 13586/587*e^4 - 149617/1761*e^3 - 38863/1761*e^2 + 47473/587*e + 2996/1761, -949/5283*e^7 - 1510/5283*e^6 + 16213/5283*e^5 + 2282/587*e^4 - 37022/5283*e^3 + 37546/5283*e^2 + 9005/1761*e - 37130/5283, -2182/1761*e^7 - 4036/1761*e^6 + 38614/1761*e^5 + 22066/587*e^4 - 104678/1761*e^3 - 64988/1761*e^2 + 23758/587*e - 15158/1761, 1208/1761*e^7 + 4520/1761*e^6 - 17615/1761*e^5 - 26351/587*e^4 - 4997/1761*e^3 + 169324/1761*e^2 + 13466/587*e - 57632/1761, -2840/5283*e^7 - 3116/5283*e^6 + 56597/5283*e^5 + 5803/587*e^4 - 234139/5283*e^3 - 46591/5283*e^2 + 48088/1761*e - 88726/5283, 493/587*e^7 + 970/587*e^6 - 8584/587*e^5 - 16000/587*e^4 + 21170/587*e^3 + 17974/587*e^2 - 11208/587*e - 6820/587, -3611/5283*e^7 - 10934/5283*e^6 + 60077/5283*e^5 + 20892/587*e^4 - 105649/5283*e^3 - 354628/5283*e^2 + 21379/1761*e - 21994/5283, 218/5283*e^7 + 1772/5283*e^6 + 1798/5283*e^5 - 3132/587*e^4 - 84179/5283*e^3 + 33589/5283*e^2 + 63068/1761*e - 113798/5283, 6212/5283*e^7 + 13319/5283*e^6 - 105365/5283*e^5 - 24543/587*e^4 + 217339/5283*e^3 + 264187/5283*e^2 - 17893/1761*e - 54794/5283, 4367/1761*e^7 + 8678/1761*e^6 - 77591/1761*e^5 - 48995/587*e^4 + 216736/1761*e^3 + 213181/1761*e^2 - 57688/587*e - 27392/1761, 19504/5283*e^7 + 40954/5283*e^6 - 343639/5283*e^5 - 76169/587*e^4 + 901853/5283*e^3 + 928094/5283*e^2 - 193070/1761*e - 69508/5283, 658/5283*e^7 + 841/5283*e^6 - 12700/5283*e^5 - 1774/587*e^4 + 48455/5283*e^3 + 36194/5283*e^2 - 9068/1761*e - 25048/5283, -7634/5283*e^7 - 15281/5283*e^6 + 134786/5283*e^5 + 28623/587*e^4 - 360754/5283*e^3 - 345007/5283*e^2 + 79438/1761*e - 116044/5283, -8630/5283*e^7 - 15719/5283*e^6 + 160208/5283*e^5 + 30277/587*e^4 - 537463/5283*e^3 - 485140/5283*e^2 + 153493/1761*e + 94652/5283, 4106/5283*e^7 + 5264/5283*e^6 - 82991/5283*e^5 - 10119/587*e^4 + 377692/5283*e^3 + 131050/5283*e^2 - 178255/1761*e + 20140/5283, -7625/5283*e^7 - 13463/5283*e^6 + 138980/5283*e^5 + 24934/587*e^4 - 428425/5283*e^3 - 253882/5283*e^2 + 89425/1761*e - 109546/5283, -3715/5283*e^7 - 10810/5283*e^6 + 55051/5283*e^5 + 19952/587*e^4 - 1658/5283*e^3 - 234215/5283*e^2 - 43153/1761*e + 57886/5283, -11770/5283*e^7 - 21322/5283*e^6 + 216124/5283*e^5 + 40277/587*e^4 - 692498/5283*e^3 - 527984/5283*e^2 + 180965/1761*e + 11212/5283, 239/587*e^7 + 731/587*e^6 - 4265/587*e^5 - 13498/587*e^4 + 11506/587*e^3 + 39590/587*e^2 - 10320/587*e - 17630/587, 5933/1761*e^7 + 13313/1761*e^6 - 103304/1761*e^5 - 75121/587*e^4 + 256531/1761*e^3 + 341440/1761*e^2 - 59818/587*e - 9692/1761, -17098/5283*e^7 - 35695/5283*e^6 + 306097/5283*e^5 + 67527/587*e^4 - 877160/5283*e^3 - 957194/5283*e^2 + 240212/1761*e + 31552/5283, -11426/5283*e^7 - 25796/5283*e^6 + 197393/5283*e^5 + 48895/587*e^4 - 456151/5283*e^3 - 713734/5283*e^2 + 52783/1761*e + 62348/5283, 8042/5283*e^7 + 23735/5283*e^6 - 126041/5283*e^5 - 44803/587*e^4 + 114124/5283*e^3 + 689857/5283*e^2 + 32846/1761*e - 181076/5283, -9143/5283*e^7 - 24251/5283*e^6 + 148319/5283*e^5 + 44492/587*e^4 - 219826/5283*e^3 - 501925/5283*e^2 + 35050/1761*e - 53848/5283, -12206/5283*e^7 - 30149/5283*e^6 + 207245/5283*e^5 + 57107/587*e^4 - 429046/5283*e^3 - 859312/5283*e^2 + 46024/1761*e + 164846/5283, 841/587*e^7 + 1413/587*e^6 - 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33916/587*e + 3206/587, 3619/5283*e^7 + 17833/5283*e^6 - 43435/5283*e^5 - 34998/587*e^4 - 179911/5283*e^3 + 759065/5283*e^2 + 152641/1761*e - 317386/5283, 1555/587*e^7 + 3000/587*e^6 - 28491/587*e^5 - 51887/587*e^4 + 91980/587*e^3 + 90259/587*e^2 - 94078/587*e - 5504/587, -29012/5283*e^7 - 49124/5283*e^6 + 534362/5283*e^5 + 91105/587*e^4 - 1738993/5283*e^3 - 962053/5283*e^2 + 469183/1761*e + 23324/5283, -5423/1761*e^7 - 14192/1761*e^6 + 89141/1761*e^5 + 79016/587*e^4 - 148342/1761*e^3 - 332137/1761*e^2 + 24073/587*e + 39800/1761, -4934/1761*e^7 - 10508/1761*e^6 + 89846/1761*e^5 + 61774/587*e^4 - 278479/1761*e^3 - 391057/1761*e^2 + 80077/587*e + 107576/1761, 575/1761*e^7 + 3446/1761*e^6 - 4418/1761*e^5 - 19290/587*e^4 - 63566/1761*e^3 + 96277/1761*e^2 + 18969/587*e - 21578/1761, -4534/5283*e^7 - 17758/5283*e^6 + 69622/5283*e^5 + 34562/587*e^4 - 24707/5283*e^3 - 697184/5283*e^2 - 52099/1761*e + 68830/5283, -14047/5283*e^7 - 37504/5283*e^6 + 232774/5283*e^5 + 70788/587*e^4 - 392807/5283*e^3 - 1082312/5283*e^2 + 6950/1761*e + 311362/5283, -21925/5283*e^7 - 43960/5283*e^6 + 398845/5283*e^5 + 83524/587*e^4 - 1231118/5283*e^3 - 1234013/5283*e^2 + 341777/1761*e + 307954/5283, 12743/5283*e^7 + 22397/5283*e^6 - 229958/5283*e^5 - 42613/587*e^4 + 681139/5283*e^3 + 589036/5283*e^2 - 108160/1761*e - 115244/5283, 11813/5283*e^7 + 19442/5283*e^6 - 212522/5283*e^5 - 34357/587*e^4 + 621004/5283*e^3 + 104578/5283*e^2 - 105856/1761*e + 291028/5283, -927/587*e^7 - 2349/587*e^6 + 15312/587*e^5 + 38583/587*e^4 - 28688/587*e^3 - 46705/587*e^2 + 45075/587*e - 7158/587, 974/1761*e^7 - 2245/1761*e^6 - 28043/1761*e^5 + 12503/587*e^4 + 227662/1761*e^3 - 86726/1761*e^2 - 90054/587*e + 41092/1761, -10364/5283*e^7 - 12026/5283*e^6 + 216815/5283*e^5 + 24493/587*e^4 - 1045129/5283*e^3 - 495286/5283*e^2 + 352510/1761*e + 131756/5283, -3466/5283*e^7 - 18625/5283*e^6 + 40771/5283*e^5 + 35094/587*e^4 + 186481/5283*e^3 - 599369/5283*e^2 - 102610/1761*e + 68608/5283, 6469/5283*e^7 + 12403/5283*e^6 - 118852/5283*e^5 - 24681/587*e^4 + 373502/5283*e^3 + 493658/5283*e^2 - 70625/1761*e - 182698/5283];
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;