/*
  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![7, 2, -7, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17,17,-w + 2}>;

primesArray := [
[4, 2, w^3 - 3*w^2 - 3*w + 4],
[7, 7, w],
[7, 7, w^3 - 3*w^2 - 4*w + 5],
[9, 3, w^3 - 4*w^2 - w + 9],
[17, 17, 2*w^3 - 6*w^2 - 7*w + 8],
[17, 17, -w^3 + 3*w^2 + 4*w - 3],
[17, 17, -w + 2],
[23, 23, 2*w^3 - 7*w^2 - 4*w + 12],
[23, 23, -w^2 + 2*w + 3],
[31, 31, -2*w^3 + 7*w^2 + 5*w - 12],
[31, 31, -w^3 + 4*w^2 + 2*w - 8],
[41, 41, 3*w^3 - 10*w^2 - 7*w + 16],
[41, 41, 2*w^3 - 7*w^2 - 5*w + 10],
[49, 7, 2*w^3 - 6*w^2 - 6*w + 9],
[71, 71, w^2 - 2*w - 2],
[71, 71, 2*w^3 - 7*w^2 - 4*w + 13],
[73, 73, 3*w^3 - 11*w^2 - 5*w + 19],
[73, 73, -4*w^3 + 13*w^2 + 10*w - 17],
[79, 79, 3*w^3 - 9*w^2 - 10*w + 13],
[79, 79, -2*w^3 + 6*w^2 + 5*w - 8],
[97, 97, 2*w^3 - 6*w^2 - 8*w + 9],
[97, 97, 2*w - 1],
[113, 113, 2*w^3 - 6*w^2 - 7*w + 6],
[113, 113, -w^3 + 3*w^2 + 2*w - 1],
[137, 137, w^3 - 2*w^2 - 5*w + 2],
[137, 137, 2*w^3 - 7*w^2 - 3*w + 12],
[137, 137, -3*w^3 + 10*w^2 + 7*w - 17],
[137, 137, -w^3 + 2*w^2 + 6*w - 2],
[167, 167, -2*w^3 + 7*w^2 + 6*w - 10],
[167, 167, 3*w^3 - 9*w^2 - 10*w + 10],
[169, 13, -2*w^3 + 7*w^2 + 2*w - 9],
[169, 13, -2*w^3 + 5*w^2 + 10*w - 4],
[193, 193, -2*w^3 + 8*w^2 + 3*w - 13],
[193, 193, -3*w^3 + 11*w^2 + 6*w - 22],
[199, 199, 5*w^3 - 16*w^2 - 14*w + 26],
[199, 199, w^3 - 2*w^2 - 7*w + 3],
[223, 223, -3*w^3 + 8*w^2 + 12*w - 8],
[223, 223, 3*w^3 - 11*w^2 - 6*w + 17],
[223, 223, 2*w^3 - 8*w^2 - 3*w + 18],
[223, 223, -4*w^3 + 13*w^2 + 9*w - 18],
[233, 233, -3*w^3 + 10*w^2 + 6*w - 16],
[233, 233, -2*w^3 + 5*w^2 + 9*w - 6],
[241, 241, -3*w^3 + 9*w^2 + 8*w - 10],
[241, 241, 4*w^3 - 12*w^2 - 13*w + 15],
[257, 257, 2*w^3 - 8*w^2 - w + 16],
[257, 257, 4*w^3 - 13*w^2 - 11*w + 18],
[257, 257, -w^3 + 5*w^2 - 2*w - 9],
[257, 257, -w^3 + 2*w^2 + 4*w + 2],
[263, 263, 4*w^3 - 14*w^2 - 9*w + 27],
[263, 263, 5*w^3 - 15*w^2 - 17*w + 25],
[263, 263, -2*w^3 + 7*w^2 + 6*w - 16],
[263, 263, -2*w^3 + 7*w^2 + 6*w - 9],
[271, 271, -w - 4],
[271, 271, -w^3 + 3*w^2 + 4*w - 9],
[271, 271, 5*w^3 - 16*w^2 - 14*w + 24],
[271, 271, -4*w^3 + 14*w^2 + 9*w - 24],
[281, 281, -2*w^3 + 6*w^2 + 5*w - 4],
[281, 281, -3*w^3 + 9*w^2 + 10*w - 9],
[281, 281, 3*w^3 - 10*w^2 - 8*w + 12],
[281, 281, w^2 - w - 8],
[311, 311, w^2 - 5],
[311, 311, -4*w^3 + 13*w^2 + 12*w - 20],
[313, 313, 6*w^3 - 19*w^2 - 18*w + 31],
[313, 313, -2*w^3 + 5*w^2 + 6*w - 6],
[337, 337, 3*w^3 - 11*w^2 - 3*w + 15],
[337, 337, w^3 - w^2 - 9*w - 5],
[353, 353, -w^3 + 2*w^2 + 5*w - 4],
[353, 353, w^3 - w^2 - 8*w - 3],
[353, 353, 3*w^3 - 10*w^2 - 7*w + 19],
[353, 353, -4*w^3 + 14*w^2 + 7*w - 22],
[361, 19, -3*w^3 + 11*w^2 + 6*w - 18],
[361, 19, 2*w^3 - 8*w^2 - 3*w + 17],
[367, 367, -4*w^3 + 12*w^2 + 13*w - 18],
[367, 367, 3*w^3 - 9*w^2 - 8*w + 13],
[383, 383, 5*w^3 - 17*w^2 - 12*w + 30],
[383, 383, 2*w^2 - 3*w - 5],
[383, 383, 3*w^3 - 9*w^2 - 11*w + 11],
[383, 383, -w^3 + 3*w^2 + w - 1],
[401, 401, 3*w^3 - 11*w^2 - 8*w + 23],
[401, 401, 4*w^3 - 14*w^2 - 11*w + 22],
[431, 431, 2*w^3 - 5*w^2 - 10*w + 8],
[431, 431, 6*w^3 - 20*w^2 - 15*w + 29],
[433, 433, -4*w^3 + 12*w^2 + 13*w - 17],
[433, 433, -3*w^3 + 9*w^2 + 8*w - 12],
[433, 433, 3*w^3 - 11*w^2 - 6*w + 20],
[433, 433, -2*w^3 + 8*w^2 + 3*w - 15],
[439, 439, -3*w^3 + 11*w^2 + 6*w - 19],
[439, 439, -2*w^3 + 8*w^2 + 3*w - 16],
[449, 449, w^3 - 4*w^2 + w + 11],
[449, 449, -w^3 + 2*w^2 + 7*w - 6],
[457, 457, w^2 - 4*w - 3],
[457, 457, 4*w^3 - 15*w^2 - 6*w + 27],
[457, 457, w^2 - 4*w - 2],
[457, 457, -2*w^3 + 6*w^2 + 8*w - 15],
[479, 479, -4*w^3 + 14*w^2 + 9*w - 29],
[479, 479, -w^3 + 5*w^2 - 6],
[487, 487, -w^3 + 6*w^2 - 3*w - 16],
[487, 487, -5*w^3 + 18*w^2 + 9*w - 29],
[503, 503, -2*w^3 + 7*w^2 + 8*w - 15],
[503, 503, 4*w^3 - 13*w^2 - 14*w + 20],
[521, 521, 3*w^3 - 10*w^2 - 9*w + 10],
[521, 521, -2*w^3 + 7*w^2 + 3*w - 16],
[529, 23, w^3 - 3*w^2 - 3*w - 1],
[599, 599, -4*w^3 + 12*w^2 + 14*w - 15],
[599, 599, 2*w^3 - 6*w^2 - 4*w + 5],
[599, 599, 4*w^3 - 13*w^2 - 12*w + 19],
[599, 599, w^2 - 6],
[601, 601, -4*w^3 + 12*w^2 + 11*w - 19],
[601, 601, 5*w^3 - 15*w^2 - 16*w + 24],
[607, 607, -w^3 + 4*w^2 + w - 13],
[607, 607, -w^3 + 4*w^2 + w - 2],
[617, 617, -4*w^3 + 13*w^2 + 11*w - 16],
[617, 617, 4*w^3 - 13*w^2 - 13*w + 20],
[625, 5, -5],
[631, 631, -3*w^3 + 8*w^2 + 11*w - 9],
[631, 631, 5*w^3 - 16*w^2 - 13*w + 24],
[631, 631, w^3 - 3*w^2 - 4*w - 1],
[631, 631, w - 6],
[641, 641, -3*w^3 + 12*w^2 + w - 17],
[641, 641, w^3 - 6*w^2 + 5*w + 18],
[673, 673, 4*w^3 - 15*w^2 - 6*w + 26],
[673, 673, -2*w^3 + 9*w^2 - 19],
[719, 719, 2*w^2 - 4*w - 5],
[719, 719, 4*w^3 - 14*w^2 - 8*w + 25],
[727, 727, 3*w^3 - 8*w^2 - 11*w + 8],
[727, 727, 6*w^3 - 18*w^2 - 19*w + 20],
[727, 727, -5*w^3 + 14*w^2 + 19*w - 17],
[727, 727, -5*w^3 + 16*w^2 + 13*w - 23],
[743, 743, 4*w^3 - 14*w^2 - 7*w + 23],
[743, 743, w^3 - w^2 - 8*w - 2],
[751, 751, 3*w^3 - 12*w^2 - 3*w + 23],
[751, 751, 3*w^3 - 12*w^2 - 3*w + 22],
[761, 761, 3*w^3 - 10*w^2 - 6*w + 22],
[761, 761, 3*w^3 - 10*w^2 - 6*w + 18],
[761, 761, -5*w^3 + 16*w^2 + 17*w - 25],
[761, 761, -2*w^3 + 5*w^2 + 9*w - 8],
[809, 809, -3*w^3 + 8*w^2 + 13*w - 3],
[809, 809, -2*w^3 + 6*w^2 + 3*w - 12],
[823, 823, -w^3 + 4*w^2 + 2*w - 2],
[823, 823, 2*w^3 - 7*w^2 - 5*w + 18],
[839, 839, 4*w^3 - 13*w^2 - 14*w + 27],
[839, 839, -7*w^3 + 23*w^2 + 19*w - 41],
[857, 857, -3*w^3 + 10*w^2 + 9*w - 12],
[857, 857, -w^3 + 4*w^2 + 3*w - 13],
[863, 863, 3*w^3 - 11*w^2 - 2*w + 20],
[863, 863, -2*w^3 + 4*w^2 + 13*w - 5],
[863, 863, 6*w^3 - 19*w^2 - 18*w + 27],
[863, 863, -2*w^3 + 5*w^2 + 6*w - 2],
[881, 881, -3*w^3 + 10*w^2 + 5*w - 17],
[881, 881, -3*w^3 + 8*w^2 + 13*w - 12],
[887, 887, -3*w^3 + 9*w^2 + 11*w - 9],
[887, 887, w^3 - 3*w^2 - w - 1],
[911, 911, 4*w^3 - 14*w^2 - 10*w + 17],
[911, 911, 5*w^3 - 17*w^2 - 12*w + 22],
[919, 919, -4*w^3 + 15*w^2 + 7*w - 30],
[919, 919, -2*w^2 + 7*w + 2],
[919, 919, w^3 - 5*w^2 + 4*w + 13],
[919, 919, 3*w^3 - 12*w^2 - 4*w + 20],
[929, 929, -4*w^3 + 13*w^2 + 9*w - 22],
[929, 929, 3*w^3 - 8*w^2 - 12*w + 12],
[961, 31, 4*w^3 - 12*w^2 - 12*w + 17],
[967, 967, -2*w^3 + 6*w^2 + 7*w - 2],
[967, 967, -3*w^3 + 8*w^2 + 15*w - 6],
[967, 967, w^3 - 2*w^2 - 5*w - 6],
[967, 967, 7*w^3 - 22*w^2 - 21*w + 36],
[983, 983, 5*w^3 - 17*w^2 - 13*w + 25],
[983, 983, w^3 - 5*w^2 - w + 15],
[991, 991, -6*w^3 + 21*w^2 + 13*w - 34],
[991, 991, 7*w^3 - 22*w^2 - 21*w + 33],
[1009, 1009, w^2 - 2*w + 3],
[1009, 1009, -2*w^3 + 7*w^2 + 4*w - 18],
[1031, 1031, -4*w^3 + 15*w^2 + 4*w - 26],
[1031, 1031, 3*w^2 - 8*w - 9],
[1033, 1033, 2*w^3 - 4*w^2 - 13*w + 3],
[1033, 1033, -5*w^3 + 15*w^2 + 19*w - 23],
[1033, 1033, w^3 - w^2 - 8*w - 10],
[1033, 1033, 4*w^3 - 14*w^2 - 7*w + 15],
[1049, 1049, -7*w^3 + 24*w^2 + 16*w - 34],
[1049, 1049, -5*w^3 + 16*w^2 + 18*w - 24],
[1063, 1063, w^3 - 4*w^2 + 2*w + 8],
[1063, 1063, w^3 - 2*w^2 - 4*w - 8],
[1063, 1063, 7*w^3 - 24*w^2 - 17*w + 39],
[1063, 1063, w^3 - 2*w^2 - 7*w - 6],
[1103, 1103, w^3 - 5*w^2 + 3*w + 17],
[1103, 1103, -w^3 + 5*w^2 - 3*w - 3],
[1103, 1103, -w^3 + 5*w^2 - 2*w - 6],
[1103, 1103, -2*w^3 + 8*w^2 + w - 19],
[1129, 1129, -6*w^3 + 18*w^2 + 20*w - 27],
[1129, 1129, -4*w^3 + 12*w^2 + 10*w - 17],
[1151, 1151, w^3 - w^2 - 7*w - 1],
[1151, 1151, 5*w^3 - 17*w^2 - 11*w + 29],
[1153, 1153, -2*w^3 + 8*w^2 + 3*w - 23],
[1153, 1153, -6*w^3 + 18*w^2 + 20*w - 25],
[1193, 1193, -2*w^3 + 5*w^2 + 9*w - 10],
[1193, 1193, 3*w^3 - 10*w^2 - 6*w + 20],
[1201, 1201, 6*w^3 - 19*w^2 - 16*w + 24],
[1201, 1201, 4*w^3 - 11*w^2 - 14*w + 9],
[1217, 1217, 7*w^3 - 22*w^2 - 21*w + 30],
[1217, 1217, 7*w^3 - 23*w^2 - 18*w + 39],
[1223, 1223, -2*w^3 + 9*w^2 - 15],
[1223, 1223, -3*w^3 + 9*w^2 + 13*w - 9],
[1223, 1223, -5*w^3 + 17*w^2 + 12*w - 34],
[1223, 1223, 4*w^3 - 15*w^2 - 6*w + 30],
[1231, 1231, -3*w^3 + 7*w^2 + 12*w - 3],
[1231, 1231, -8*w^3 + 26*w^2 + 21*w - 38],
[1289, 1289, -4*w^3 + 14*w^2 + 7*w - 24],
[1289, 1289, -w^3 + w^2 + 8*w + 1],
[1297, 1297, -4*w^3 + 12*w^2 + 11*w - 15],
[1297, 1297, -5*w^3 + 15*w^2 + 16*w - 20],
[1303, 1303, -3*w^3 + 10*w^2 + 7*w - 23],
[1303, 1303, -w^3 + 2*w^2 + 5*w - 8],
[1319, 1319, -8*w^3 + 28*w^2 + 15*w - 43],
[1319, 1319, w^3 + w^2 - 12*w - 12],
[1321, 1321, w^2 - 2*w - 10],
[1321, 1321, 2*w^3 - 7*w^2 - 4*w + 5],
[1367, 1367, -5*w^3 + 17*w^2 + 14*w - 26],
[1367, 1367, 2*w^3 - 8*w^2 - 5*w + 19],
[1399, 1399, -2*w^3 + 7*w^2 + 3*w - 4],
[1399, 1399, w^3 - 2*w^2 - 6*w - 6],
[1409, 1409, w^3 - 2*w^2 - 3*w - 3],
[1409, 1409, 3*w^3 - 10*w^2 - 5*w + 18],
[1409, 1409, -5*w^3 + 16*w^2 + 15*w - 22],
[1409, 1409, -3*w^3 + 8*w^2 + 13*w - 13],
[1423, 1423, 7*w^3 - 24*w^2 - 18*w + 44],
[1423, 1423, -8*w^3 + 26*w^2 + 21*w - 40],
[1433, 1433, -7*w^3 + 24*w^2 + 17*w - 45],
[1433, 1433, -3*w^3 + 12*w^2 + 3*w - 29],
[1439, 1439, -2*w^3 + 6*w^2 + 4*w - 3],
[1439, 1439, -4*w^3 + 12*w^2 + 14*w - 13],
[1471, 1471, -w - 6],
[1471, 1471, -w^3 + 3*w^2 + 4*w - 11],
[1471, 1471, w^3 - 2*w^2 - 8*w + 4],
[1471, 1471, w^2 - 5*w - 4],
[1489, 1489, -2*w^3 + 9*w^2 + 2*w - 17],
[1489, 1489, -6*w^3 + 21*w^2 + 14*w - 38],
[1543, 1543, -7*w^3 + 24*w^2 + 19*w - 45],
[1543, 1543, 2*w^3 - 3*w^2 - 15*w - 4],
[1543, 1543, -5*w^3 + 18*w^2 + 6*w - 26],
[1543, 1543, -3*w^3 + 12*w^2 + 7*w - 20],
[1553, 1553, -4*w^3 + 14*w^2 + 11*w - 20],
[1553, 1553, -3*w^3 + 11*w^2 + 8*w - 25],
[1559, 1559, 2*w^2 - 4*w - 3],
[1559, 1559, 4*w^3 - 14*w^2 - 8*w + 27],
[1567, 1567, -4*w^3 + 15*w^2 + 3*w - 20],
[1567, 1567, w^3 - 12*w - 10],
[1601, 1601, -3*w^3 + 10*w^2 + 10*w - 12],
[1601, 1601, 2*w^3 - 7*w^2 - 7*w + 18],
[1607, 1607, -7*w^3 + 23*w^2 + 20*w - 36],
[1607, 1607, -6*w^3 + 19*w^2 + 18*w - 26],
[1607, 1607, 2*w^2 - w - 9],
[1607, 1607, -2*w^3 + 5*w^2 + 6*w - 1],
[1609, 1609, -4*w^3 + 15*w^2 + 8*w - 24],
[1609, 1609, 4*w^3 - 15*w^2 - 8*w + 31],
[1663, 1663, -6*w^3 + 19*w^2 + 13*w - 26],
[1663, 1663, -7*w^3 + 20*w^2 + 26*w - 26],
[1681, 41, -5*w^3 + 15*w^2 + 15*w - 17],
[1697, 1697, -w^3 + 9*w + 6],
[1697, 1697, -7*w^3 + 24*w^2 + 15*w - 39],
[1721, 1721, -5*w^3 + 18*w^2 + 13*w - 38],
[1721, 1721, -w^3 + 6*w^2 - 5*w - 20],
[1721, 1721, -w^3 + w^2 + 6*w + 11],
[1721, 1721, 6*w^3 - 20*w^2 - 15*w + 24],
[1753, 1753, -3*w^3 + 9*w^2 + 13*w - 17],
[1753, 1753, w^3 - 3*w^2 - 7*w + 3],
[1753, 1753, -6*w^3 + 22*w^2 + 12*w - 45],
[1753, 1753, -6*w^3 + 21*w^2 + 16*w - 40],
[1759, 1759, -2*w^3 + 6*w^2 + w - 8],
[1759, 1759, 7*w^3 - 21*w^2 - 26*w + 33],
[1759, 1759, -3*w^3 + 10*w^2 + 10*w - 26],
[1759, 1759, -5*w^3 + 17*w^2 + 12*w - 37],
[1783, 1783, -5*w^3 + 18*w^2 + 11*w - 29],
[1783, 1783, 4*w^3 - 15*w^2 - 7*w + 26],
[1783, 1783, 3*w^3 - 12*w^2 - 5*w + 26],
[1783, 1783, -3*w^3 + 12*w^2 + 4*w - 24],
[1801, 1801, 8*w^3 - 24*w^2 - 27*w + 33],
[1801, 1801, 4*w^3 - 12*w^2 - 16*w + 19],
[1801, 1801, 4*w - 1],
[1801, 1801, 7*w^3 - 21*w^2 - 23*w + 33],
[1831, 1831, w^3 - 2*w^2 - 8*w + 2],
[1831, 1831, w^2 - 5*w - 2],
[1847, 1847, 6*w^3 - 20*w^2 - 15*w + 25],
[1847, 1847, w^3 - w^2 - 6*w - 10],
[1849, 43, -6*w^3 + 21*w^2 + 14*w - 37],
[1849, 43, -2*w^3 + 9*w^2 + 2*w - 18],
[1871, 1871, -4*w^3 + 13*w^2 + 8*w - 22],
[1871, 1871, 4*w^3 - 13*w^2 - 12*w + 15],
[1871, 1871, 4*w^3 - 11*w^2 - 16*w + 17],
[1871, 1871, w^2 - 10],
[1873, 1873, -w^3 + 5*w^2 - 20],
[1873, 1873, -4*w^3 + 14*w^2 + 9*w - 15],
[1913, 1913, -2*w^3 + 6*w^2 + 3*w - 4],
[1913, 1913, 5*w^3 - 15*w^2 - 18*w + 19],
[1913, 1913, 6*w^3 - 20*w^2 - 11*w + 30],
[1913, 1913, 9*w^3 - 30*w^2 - 23*w + 45],
[1993, 1993, -8*w^3 + 28*w^2 + 17*w - 43],
[1993, 1993, 4*w^3 - 13*w^2 - 16*w + 19],
[1993, 1993, -6*w^3 + 21*w^2 + 14*w - 36],
[1993, 1993, -2*w^3 + 9*w^2 + 2*w - 19],
[1999, 1999, -7*w^3 + 22*w^2 + 20*w - 30],
[1999, 1999, 4*w^3 - 11*w^2 - 13*w + 10]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + 4*x^4 - 9*x^2 - 2*x + 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^4 - 3*e^3 + 3*e^2 + 7*e - 2, -e^3 - 3*e^2 + 2*e + 3, -e^2 - 2*e + 3, e^2 + 4*e, -2*e^4 - 7*e^3 + 3*e^2 + 16*e + 2, -1, -e^3 - 2*e^2 + 2*e + 3, -e^4 - 3*e^3 + 2*e^2 + 5*e + 1, -2*e^4 - 7*e^3 + e^2 + 10*e + 3, 2*e^3 + 5*e^2 - 3*e - 4, -3*e^4 - 10*e^3 + 3*e^2 + 19*e + 6, 3*e^4 + 14*e^3 + 5*e^2 - 29*e - 5, e^4 + 3*e^3 - 4*e^2 - 9*e + 5, -e^3 - e^2 + 7*e + 1, e^4 + 3*e^3 - 4*e^2 - 4*e + 11, -e^2 - 5*e + 1, 3*e^4 + 11*e^3 - e^2 - 21*e - 4, 3*e^4 + 11*e^3 - e^2 - 18*e - 4, 2*e^4 + 10*e^3 + 5*e^2 - 22*e - 8, -4*e^4 - 16*e^3 + 3*e^2 + 42*e + 2, e^4 + 4*e^3 - 4*e^2 - 15*e + 12, 3*e^4 + 11*e^3 - 3*e^2 - 31*e - 5, -e^4 - 6*e^3 - 6*e^2 + 13*e + 11, -e^4 - 3*e^3 + 4*e^2 + 10*e + 7, 2*e^4 + 3*e^3 - 10*e^2 + 2*e + 13, -2*e^4 - 7*e^3 + e^2 + 10*e - 8, 3*e^4 + 11*e^3 - e^2 - 25*e - 4, 2*e^4 + 6*e^3 - 2*e^2 - 8*e - 11, -e^3 + 11*e + 4, -e^4 - 2*e^3 + 6*e^2 + 3*e + 1, -4*e^2 - 11*e - 2, 3*e^4 + 10*e^3 - 5*e^2 - 27*e - 13, -3*e^4 - 13*e^3 - 2*e^2 + 31*e + 13, 3*e^4 + 9*e^3 - e^2 - 8*e - 5, -e^4 - 9*e^3 - 13*e^2 + 14*e + 15, 2*e^4 + 7*e^3 - e^2 - 19*e - 5, 3*e^4 + 13*e^3 + 5*e^2 - 22*e - 9, 2*e^4 + 6*e^3 - 2*e^2 - 7*e - 8, -e^4 + 9*e^2 - 3*e - 9, 2*e^4 + 9*e^3 - e^2 - 33*e - 1, e^4 + 4*e^3 + 7*e^2 + e - 12, -7*e^4 - 27*e^3 + 4*e^2 + 62*e + 13, 5*e^4 + 20*e^3 + 3*e^2 - 26*e - 9, 8*e^4 + 31*e^3 - 3*e^2 - 72*e - 7, 5*e^4 + 20*e^3 + 7*e^2 - 33*e - 16, -e^4 - 8*e^3 - 13*e^2 + 20*e + 17, 8*e^4 + 27*e^3 - 16*e^2 - 58*e + 1, -7*e^4 - 25*e^3 + 8*e^2 + 57*e + 4, 6*e^4 + 24*e^3 + 3*e^2 - 42*e - 3, -3*e^4 - 8*e^3 + 9*e^2 + 20*e + 3, 4*e^3 + 12*e^2 - 11*e - 13, -5*e^4 - 24*e^3 - 7*e^2 + 53*e - 2, e^4 + 3*e^3 - 2*e^2 - 10*e - 7, 7*e^4 + 23*e^3 - 8*e^2 - 36*e + 2, -3*e^4 - 12*e^3 + 2*e^2 + 34*e + 7, 4*e^4 + 8*e^3 - 17*e^2 - 12*e + 12, -4*e^4 - 12*e^3 + 12*e^2 + 28*e + 2, -4*e^4 - 12*e^3 + 11*e^2 + 20*e - 5, -2*e^4 - 7*e^3 - 5*e^2 - 3*e + 12, -e^4 - 8*e^3 - 4*e^2 + 36*e + 9, -3*e^4 - 15*e^3 - 9*e^2 + 26*e + 26, 10*e^4 + 40*e^3 + e^2 - 85*e - 12, -e^4 - 4*e^3 + 3*e^2 + 13*e, -e^3 - 2*e^2 + 8*e - 4, 9*e^4 + 35*e^3 + e^2 - 68*e - 14, -2*e^4 - 9*e^3 + e^2 + 25*e + 11, 9*e^4 + 34*e^3 - 6*e^2 - 68*e + 8, -6*e^4 - 24*e^3 - 7*e^2 + 32*e + 15, -3*e^4 - 16*e^3 - 15*e^2 + 19*e + 27, -2*e^4 - 6*e^3 + 12*e^2 + 21*e - 5, -7*e^4 - 22*e^3 + 13*e^2 + 49*e - 1, -3*e^4 - 8*e^3 + 7*e^2 + 13*e + 9, -10*e^3 - 26*e^2 + 20*e + 29, -5*e^4 - 15*e^3 + 13*e^2 + 24*e - 17, 8*e^4 + 29*e^3 - 11*e^2 - 66*e + 5, -3*e^4 - 14*e^3 - 15*e^2 + 14*e + 30, -4*e^4 - 16*e^3 - 3*e^2 + 25*e + 16, 6*e^4 + 23*e^3 - 40*e - 26, -3*e^2 - 9*e + 21, e^4 + 4*e^3 - 2*e^2 - 26*e + 3, -e^4 - 7*e^3 - 10*e^2 + 14*e + 15, -10*e^4 - 37*e^3 + 4*e^2 + 79*e + 17, -4*e^4 - 19*e^3 - 10*e^2 + 38*e + 8, 6*e^4 + 24*e^3 - 6*e^2 - 63*e - 7, -10*e^4 - 36*e^3 + 3*e^2 + 61*e + 13, 5*e^4 + 9*e^3 - 29*e^2 - 25*e + 22, 11*e^4 + 40*e^3 - 8*e^2 - 69*e, e^3 - 13*e, e^4 + 5*e^3 + 9*e^2 + 9*e - 20, 9*e^4 + 35*e^3 + e^2 - 64*e - 16, 8*e^4 + 34*e^3 + 3*e^2 - 71*e - 13, 3*e^3 + 3*e^2 - 15*e - 16, -2*e^3 - 6*e^2 - 8*e - 2, -3*e^4 - 5*e^3 + 16*e^2 + 10*e - 2, 3*e^4 + 5*e^3 - 21*e^2 - 12*e + 20, -6*e^4 - 24*e^3 + 54*e + 8, e^4 + 2*e^3 - 3*e^2 - 4*e - 34, -3*e^4 - 7*e^3 + 5*e^2 - 7*e + 2, 11*e^4 + 41*e^3 - 7*e^2 - 81*e - 8, -2*e^4 - 4*e^3 + e^2 - 8*e + 25, 5*e^4 + 19*e^3 + 3*e^2 - 33*e - 7, -8*e^4 - 32*e^3 - 3*e^2 + 61*e + 12, 7*e^4 + 17*e^3 - 32*e^2 - 45*e + 30, 8*e^4 + 34*e^3 - e^2 - 79*e - 7, 8*e^4 + 31*e^3 - 4*e^2 - 75*e + 3, -2*e^4 - e^3 + 22*e^2 - 39, -2*e^4 - 9*e^3 - 3*e^2 + 27*e + 19, 4*e^4 + 20*e^3 + 13*e^2 - 44*e - 21, -11*e^4 - 42*e^3 + 84*e + 25, -3*e^4 - 12*e^3 + e^2 + 23*e + 3, 12*e^4 + 47*e^3 + 2*e^2 - 88*e - 1, 13*e^4 + 42*e^3 - 23*e^2 - 84*e + 16, 7*e^4 + 23*e^3 - 5*e^2 - 27*e - 1, 14*e^4 + 46*e^3 - 25*e^2 - 91*e + 16, 2*e^3 + 4*e^2 - 24*e - 15, -e^3 - 7*e^2 + 9*e + 23, 5*e^4 + 16*e^3 - 13*e^2 - 34*e + 7, -7*e^4 - 25*e^3 + 15*e^2 + 52*e - 29, -e^4 - e^3 - e^2 - 17*e + 14, 8*e^4 + 33*e^3 + 8*e^2 - 66*e - 32, -5*e^4 - 24*e^3 - 16*e^2 + 51*e + 29, 8*e^4 + 41*e^3 + 23*e^2 - 83*e - 19, 2*e^4 + 7*e^3 + 16*e^2 + 23*e - 33, -12*e^4 - 36*e^3 + 28*e^2 + 69*e - 14, 5*e^4 + 15*e^3 - 16*e^2 - 38*e - 13, 2*e^4 + 8*e^3 - 4*e^2 - 28*e - 7, -5*e^4 - 27*e^3 - 9*e^2 + 75*e + 1, -3*e^4 - 13*e^3 - 10*e^2 + 7*e + 22, 8*e^4 + 31*e^3 - e^2 - 63*e - 12, 3*e^4 + 21*e^3 + 15*e^2 - 63*e - 31, 9*e^4 + 36*e^3 + 5*e^2 - 71*e - 8, -2*e^4 - 17*e^3 - 27*e^2 + 38*e + 44, 2*e^4 + 9*e^3 - e^2 - 21*e + 8, -4*e^4 - 25*e^3 - 22*e^2 + 55*e + 32, -3*e^4 - 12*e^3 + 6*e^2 + 32*e - 4, -9*e^4 - 27*e^3 + 21*e^2 + 57*e - 24, 6*e^4 + 31*e^3 + 18*e^2 - 65*e - 22, -14*e^4 - 45*e^3 + 30*e^2 + 108*e - 2, e^4 + 8*e^3 + 15*e^2 - 4*e - 16, -9*e^4 - 34*e^3 - 3*e^2 + 44*e + 16, -3*e^4 - e^3 + 36*e^2 + 16*e - 57, e^4 + 14*e^3 + 30*e^2 - 18*e - 17, 7*e^4 + 22*e^3 - 18*e^2 - 42*e + 12, -2*e^4 - 6*e^3 + 10*e + 12, -16*e^4 - 59*e^3 + 11*e^2 + 132*e + 12, -5*e^4 - 12*e^3 + 28*e^2 + 26*e - 21, 2*e^4 + 15*e^3 + 17*e^2 - 31*e - 35, 6*e^4 + 23*e^3 - 3*e^2 - 55*e - 8, -5*e^4 - 14*e^3 + 10*e^2 + 18*e + 17, -7*e^3 - 18*e^2 + 8*e + 10, -e^4 + 5*e^3 + 33*e^2 + 6*e - 36, 5*e^4 + 24*e^3 + 25*e^2 - 23*e - 53, 6*e^4 + 19*e^3 - 9*e^2 - 40*e - 3, -11*e^4 - 41*e^3 - e^2 + 80*e + 41, -9*e^4 - 34*e^3 + 5*e^2 + 69*e - 7, -5*e^4 - 14*e^3 - 5*e + 6, -6*e^4 - 22*e^3 + 10*e^2 + 43*e - 32, 4*e^4 + 22*e^3 + 3*e^2 - 75*e - 3, 5*e^4 + 7*e^3 - 34*e^2 - 14*e + 34, -5*e^4 - 26*e^3 - 20*e^2 + 46*e + 11, -3*e^4 - 15*e^3 - 16*e^2 + 22*e + 31, 11*e^4 + 42*e^3 + 13*e^2 - 61*e - 44, 5*e^4 + 17*e^3 - 14*e^2 - 59*e + 8, e^4 - e^3 - 15*e^2 - 7*e - 13, -8*e^4 - 30*e^3 + e^2 + 41*e + 18, -10*e^4 - 46*e^3 - 18*e^2 + 88*e + 19, 7*e^4 + 34*e^3 + 24*e^2 - 62*e - 21, 8*e^3 + 24*e^2 - 18*e - 23, 16*e^4 + 59*e^3 - 8*e^2 - 119*e - 6, -3*e^4 - 12*e^3 + 8*e^2 + 40*e - 11, -10*e^4 - 42*e^3 - e^2 + 102*e - 4, 15*e^4 + 59*e^3 - 6*e^2 - 128*e - 9, -2*e^4 + e^3 + 22*e^2 + 8*e + 3, e^4 + 4*e^3 - 2*e^2 - 7*e + 36, -2*e^4 - 15*e^3 - 25*e^2 + 28*e + 45, -3*e^4 - e^3 + 22*e^2 - 11*e - 35, 9*e^4 + 28*e^3 - 18*e^2 - 52*e + 24, 6*e^4 + 13*e^3 - 36*e^2 - 36*e + 37, 18*e^4 + 65*e^3 - 14*e^2 - 133*e - 7, -2*e^4 - 16*e^3 - 24*e^2 + 29*e + 20, 9*e^4 + 34*e^3 - 13*e^2 - 88*e + 3, -5*e^4 - 22*e^3 - 6*e^2 + 62*e + 47, 3*e^4 + 13*e^3 + 9*e^2 - 31*e - 21, 23*e^4 + 86*e^3 - 2*e^2 - 162*e - 19, -17*e^4 - 70*e^3 + 5*e^2 + 172*e + 12, 4*e^3 + 9*e^2 - 29*e - 25, -6*e^4 - 31*e^3 - 12*e^2 + 80*e + 24, -14*e^4 - 52*e^3 + 10*e^2 + 100*e + 15, -13*e^4 - 57*e^3 - 9*e^2 + 125*e + 24, -20*e^4 - 76*e^3 + 10*e^2 + 160*e + 10, -e^4 + 3*e^3 + 16*e^2 - 25*e - 23, 14*e^4 + 44*e^3 - 33*e^2 - 115*e - 2, 3*e^4 + 9*e^3 - 9*e^2 - 25*e + 20, 4*e^4 + 4*e^3 - 41*e^2 - 11*e + 42, 4*e^4 + 23*e^3 + 21*e^2 - 43*e - 7, 17*e^4 + 68*e^3 - 2*e^2 - 155*e - 7, 4*e^4 + 18*e^3 + 2*e^2 - 48*e - 30, -5*e^4 - 17*e^3 + 5*e^2 + 12*e - 29, 3*e^3 + 3*e^2 - 28*e - 22, 6*e^4 + 7*e^3 - 48*e^2 - 27*e + 31, -10*e^4 - 42*e^3 - 7*e^2 + 82*e + 18, 16*e^4 + 67*e^3 + 13*e^2 - 137*e - 30, -13*e^4 - 51*e^3 + 11*e^2 + 121*e + 2, -2*e^4 - 8*e^3 - 5*e^2 + 13*e - 4, 3*e^4 - 2*e^3 - 42*e^2 - 17*e + 41, 2*e^4 + 2*e^3 + 6*e^2 + 36*e - 30, 10*e^4 + 30*e^3 - 24*e^2 - 51*e + 37, 4*e^4 + 18*e^3 + 12*e^2 - 36*e - 35, -14*e^4 - 59*e^3 - 10*e^2 + 144*e + 36, 3*e^4 + 22*e^3 + 25*e^2 - 59*e - 45, 7*e^4 + 19*e^3 - 18*e^2 - 18*e + 15, 3*e^4 + 10*e^3 - 15*e^2 - 50*e + 2, 18*e^4 + 66*e^3 - 7*e^2 - 131*e - 35, 12*e^4 + 41*e^3 - 11*e^2 - 84*e - 9, -8*e^3 - 33*e^2 + 6*e + 55, 2*e^4 + 10*e^3 + 6*e^2 + 3*e + 2, 11*e^4 + 45*e^3 - 10*e^2 - 120*e + 10, 15*e^4 + 54*e^3 - 7*e^2 - 92*e - 14, -4*e^4 - 14*e^3 + 15*e - 10, -2*e^4 - 7*e^3 - 6*e^2 - 8*e - 10, -3*e^4 - 14*e^3 - 12*e^2 + 24*e + 43, 14*e^4 + 65*e^3 + 18*e^2 - 144*e - 15, -7*e^4 - 32*e^3 - 21*e^2 + 49*e + 26, 13*e^4 + 48*e^3 - 5*e^2 - 101*e - 11, 9*e^4 + 22*e^3 - 45*e^2 - 58*e + 33, -15*e^4 - 55*e^3 - 3*e^2 + 91*e + 27, -6*e^4 - 18*e^3 + 14*e^2 + 45*e - 20, -3*e^4 - 21*e^3 - 13*e^2 + 74*e + 45, -12*e^4 - 42*e^3 + 4*e^2 + 61*e - 10, -6*e^4 - 28*e^3 - 24*e^2 + 49*e + 53, 9*e^4 + 31*e^3 - 16*e^2 - 73*e - 9, 10*e^4 + 51*e^3 + 26*e^2 - 112*e - 11, -5*e^4 - 25*e^3 - e^2 + 87*e + 4, 9*e^4 + 40*e^3 - 2*e^2 - 120*e + 2, -17*e^4 - 63*e^3 - 6*e^2 + 102*e + 34, -8*e^4 - 33*e^3 - 13*e^2 + 28*e + 30, -6*e^4 - 17*e^3 + 21*e^2 + 16*e - 26, -5*e^4 - 20*e^3 - 4*e^2 + 51*e + 51, e^4 - 8*e^3 - 21*e^2 + 47*e + 35, 18*e^4 + 81*e^3 + 23*e^2 - 166*e - 27, 6*e^4 + 30*e^3 + 21*e^2 - 60*e - 39, -11*e^4 - 38*e^3 + 5*e^2 + 52*e - 7, -3*e^3 + 10*e^2 + 31*e - 43, 13*e^4 + 40*e^3 - 15*e^2 - 50*e, -21*e^4 - 75*e^3 + 24*e^2 + 171*e + 21, -4*e^4 - 8*e^3 + 29*e^2 + 12*e - 33, 2*e^4 - 22*e^2 - 2*e + 57, -6*e^4 - 18*e^3 + 17*e^2 + 23*e - 24, 9*e^4 + 37*e^3 + 6*e^2 - 72*e - 23, e^4 + 16*e^3 + 22*e^2 - 53*e - 12, -10*e^4 - 33*e^3 + 8*e^2 + 58*e + 2, 10*e^4 + 43*e^3 + 16*e^2 - 90*e - 29, 9*e^4 + 29*e^3 - 28*e^2 - 90*e + 41, -11*e^4 - 45*e^3 + 7*e^2 + 113*e + 5, e^4 + 19*e^3 + 49*e^2 - 36*e - 65, 8*e^4 + 17*e^3 - 46*e^2 - 60*e + 28, 21*e^4 + 83*e^3 + 3*e^2 - 149*e - 21, 2*e^4 + 5*e^3 + 7*e^2 - 9*e - 76, -14*e^4 - 48*e^3 + 23*e^2 + 96*e - 9, -13*e^4 - 49*e^3 + 20*e^2 + 117*e - 8, -4*e^4 - 17*e^3 - 10*e^2 + 14*e + 32, -11*e^4 - 51*e^3 - 31*e^2 + 74*e + 46, -9*e^4 - 40*e^3 - 14*e^2 + 78*e + 30, -15*e^4 - 51*e^3 + 33*e^2 + 134*e - 29, -8*e^4 - 36*e^3 - 10*e^2 + 100*e + 57, -5*e^4 - 21*e^3 + 3*e^2 + 39*e - 5, 10*e^4 + 33*e^3 - 8*e^2 - 70*e - 37, -13*e^4 - 48*e^3 - e^2 + 86*e + 48, 13*e^4 + 48*e^3 - 4*e^2 - 85*e - 26, -14*e^4 - 62*e^3 - 15*e^2 + 127*e + 30, -15*e^4 - 56*e^3 + 12*e^2 + 109*e + 1, 8*e^4 + 36*e^3 + 2*e^2 - 121*e - 21, 2*e^4 + 5*e^3 - 19*e^2 - 48*e + 21, 13*e^4 + 52*e^3 + e^2 - 93*e - 5, -6*e^4 - 28*e^3 - 2*e^2 + 104*e + 23, 17*e^4 + 59*e^3 - 14*e^2 - 116*e - 13, 9*e^4 + 37*e^3 + 18*e^2 - 44*e - 44, 9*e^4 + 35*e^3 + 3*e^2 - 62*e - 58, 12*e^4 + 57*e^3 + 38*e^2 - 92*e - 53, -3*e^4 - 13*e^3 - 10*e^2 + 35*e + 47, -13*e^4 - 36*e^3 + 35*e^2 + 70*e - 6, 12*e^4 + 41*e^3 - 10*e^2 - 72*e - 1, -11*e^4 - 37*e^3 + 8*e^2 + 63*e + 7, 18*e^4 + 67*e^3 - 10*e^2 - 107*e - 3, 12*e^4 + 34*e^3 - 27*e^2 - 38*e + 44, -5*e^4 - 17*e^3 - 5*e^2 + 18*e + 35, 15*e^4 + 62*e^3 + 12*e^2 - 140*e - 48, 10*e^4 + 34*e^3 - 34*e^2 - 109*e + 23, 3*e^4 + 16*e^3 + 17*e^2 - 25*e - 36, -18*e^4 - 78*e^3 - 9*e^2 + 174*e + 15, 4*e^4 + 13*e^3 - 11*e^2 - 14*e + 48, 32*e^4 + 113*e^3 - 28*e^2 - 216*e - 8, -7*e^4 - 26*e^3 + 4*e^2 + 68*e + 60, -22*e^4 - 73*e^3 + 42*e^2 + 170*e + 1, 8*e^4 + 46*e^3 + 36*e^2 - 95*e - 46, -18*e^4 - 68*e^3 + 4*e^2 + 107*e + 22, 7*e^3 + 24*e^2 - 66, -5*e^3 - 11*e^2 + 26*e + 10, 15*e^4 + 39*e^3 - 60*e^2 - 81*e + 62];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17,17,-w + 2}>] := 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;