/*
  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![18, 0, -12, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {1, 1, 1}>;

primesArray := [
[2, 2, -1/3*w^3 - 1/3*w^2 + 3*w + 4],
[7, 7, 1/3*w^3 + 1/3*w^2 - 3*w - 3],
[7, 7, -1/3*w^2 + w + 1],
[7, 7, 1/3*w^2 + w - 1],
[7, 7, 1/3*w^3 - 1/3*w^2 - 3*w + 3],
[9, 3, w - 3],
[41, 41, 1/3*w^3 - 1/3*w^2 - 3*w + 1],
[41, 41, -1/3*w^2 + w + 3],
[41, 41, 1/3*w^2 + w - 3],
[41, 41, -1/3*w^3 - 1/3*w^2 + 3*w + 1],
[47, 47, 1/3*w^3 + 1/3*w^2 - 4*w - 1],
[47, 47, -1/3*w^3 + 1/3*w^2 + 2*w - 3],
[47, 47, 1/3*w^3 + 1/3*w^2 - 2*w - 3],
[47, 47, 1/3*w^3 - 1/3*w^2 - 4*w + 1],
[89, 89, -1/3*w^3 + 2/3*w^2 + 3*w - 3],
[89, 89, 2/3*w^2 + w - 5],
[89, 89, 2/3*w^2 - w - 5],
[89, 89, 1/3*w^3 + 2/3*w^2 - 3*w - 3],
[97, 97, 2/3*w^3 - 1/3*w^2 - 6*w + 5],
[97, 97, -w^3 - 5/3*w^2 + 10*w + 15],
[97, 97, -5/3*w^3 - 5/3*w^2 + 16*w + 21],
[97, 97, -2/3*w^3 - 1/3*w^2 + 6*w + 5],
[103, 103, -4/3*w^3 - 4/3*w^2 + 13*w + 17],
[103, 103, 1/3*w^3 + w^2 - 5*w - 7],
[103, 103, -w^3 - w^2 + 9*w + 11],
[103, 103, -2/3*w^3 - 4/3*w^2 + 7*w + 11],
[137, 137, -1/3*w^3 - 1/3*w^2 + 3*w - 1],
[137, 137, 1/3*w^2 + w - 5],
[137, 137, 1/3*w^2 - w - 5],
[137, 137, 1/3*w^3 - 1/3*w^2 - 3*w - 1],
[151, 151, w^2 - w - 5],
[151, 151, 1/3*w^3 + w^2 - 3*w - 7],
[151, 151, -1/3*w^3 + w^2 + 3*w - 7],
[151, 151, w^2 + w - 5],
[191, 191, 1/3*w^3 + 2/3*w^2 - 4*w - 3],
[191, 191, 1/3*w^3 + 2/3*w^2 - 2*w - 5],
[191, 191, -1/3*w^3 + 2/3*w^2 + 2*w - 5],
[191, 191, -1/3*w^3 + 2/3*w^2 + 4*w - 3],
[193, 193, 2/3*w^3 + w^2 - 8*w - 7],
[193, 193, -1/3*w^3 - 1/3*w^2 + 4*w + 7],
[193, 193, -7/3*w^3 - 13/3*w^2 + 26*w + 37],
[193, 193, -2/3*w^3 + w^2 + 8*w - 7],
[199, 199, 5/3*w^3 + 2*w^2 - 17*w - 23],
[199, 199, -2/3*w^3 - 5/3*w^2 + 9*w + 13],
[199, 199, -2/3*w^3 - 1/3*w^2 + 5*w + 5],
[199, 199, -w^3 - 2*w^2 + 11*w + 17],
[233, 233, 1/3*w^3 - 4/3*w^2 - 3*w + 5],
[233, 233, 1/3*w^3 - 4/3*w^2 - 5*w + 7],
[233, 233, 4/3*w^2 + w - 11],
[233, 233, 1/3*w^3 + 4/3*w^2 - 3*w - 5],
[239, 239, 1/3*w^3 + w^2 - 4*w - 5],
[239, 239, 4/3*w^3 + 4/3*w^2 - 14*w - 17],
[239, 239, 4/3*w^3 + 2/3*w^2 - 12*w - 13],
[239, 239, -1/3*w^3 + w^2 + 4*w - 5],
[241, 241, -2/3*w^3 + 6*w + 1],
[241, 241, 2/3*w^3 - 1/3*w^2 - 6*w + 3],
[241, 241, -2/3*w^3 - 1/3*w^2 + 6*w + 3],
[241, 241, 2/3*w^3 - 6*w + 1],
[281, 281, -w^3 - w^2 + 9*w + 13],
[281, 281, -2/3*w^3 - 2/3*w^2 + 5*w + 7],
[281, 281, -2/3*w^3 + 2/3*w^2 + 5*w - 7],
[281, 281, -7/3*w^3 - 3*w^2 + 23*w + 31],
[289, 17, w^2 - 5],
[289, 17, w^2 - 7],
[337, 337, -w^3 + 4/3*w^2 + 8*w - 7],
[337, 337, -2*w + 7],
[337, 337, 2/3*w^3 - 2*w^2 - 4*w + 11],
[337, 337, 1/3*w^3 + 2/3*w^2 - 6*w + 1],
[383, 383, 2/3*w^3 + 5/3*w^2 - 8*w - 11],
[383, 383, -w^3 - 2/3*w^2 + 10*w + 11],
[383, 383, -5/3*w^3 - 4/3*w^2 + 16*w + 19],
[383, 383, -4/3*w^3 - 7/3*w^2 + 14*w + 19],
[431, 431, -2/3*w^3 + 1/3*w^2 + 4*w - 5],
[431, 431, -2/3*w^3 - 1/3*w^2 + 8*w - 1],
[431, 431, 2/3*w^3 - 1/3*w^2 - 8*w - 1],
[431, 431, 2/3*w^3 + 1/3*w^2 - 4*w - 5],
[433, 433, 2/3*w^3 + 7/3*w^2 - 10*w - 15],
[433, 433, -3*w^3 - 10/3*w^2 + 30*w + 39],
[433, 433, -1/3*w^3 - 4/3*w^2 + 4*w + 9],
[433, 433, -4/3*w^3 - 5/3*w^2 + 12*w + 15],
[439, 439, -4/3*w^3 - 1/3*w^2 + 11*w + 13],
[439, 439, 1/3*w^3 + 2/3*w^2 - 5*w - 5],
[439, 439, -3*w^3 - 10/3*w^2 + 29*w + 37],
[439, 439, -4/3*w^3 - 7/3*w^2 + 13*w + 19],
[479, 479, 1/3*w^2 - 2*w - 5],
[479, 479, 2/3*w^3 + 1/3*w^2 - 6*w + 1],
[479, 479, 2/3*w^3 - 1/3*w^2 - 6*w - 1],
[479, 479, -1/3*w^2 - 2*w + 5],
[487, 487, 1/3*w^3 + 4/3*w^2 - 3*w - 7],
[487, 487, 4/3*w^2 - w - 9],
[487, 487, -4/3*w^2 - w + 9],
[487, 487, 1/3*w^3 - 4/3*w^2 - 3*w + 7],
[521, 521, 2/3*w^3 - 2/3*w^2 - 5*w - 3],
[521, 521, 5/3*w^3 + 7/3*w^2 - 17*w - 21],
[521, 521, -7/3*w^3 - 11/3*w^2 + 25*w + 33],
[521, 521, -4/3*w^3 - 2/3*w^2 + 13*w + 15],
[529, 23, -1/3*w^2 - 3],
[529, 23, 1/3*w^2 - 7],
[569, 569, 2/3*w^3 + 2/3*w^2 - 5*w + 1],
[569, 569, 1/3*w^3 - 2/3*w^2 - 5*w + 9],
[569, 569, -1/3*w^3 - 2/3*w^2 + 5*w + 9],
[569, 569, -2/3*w^3 + 2/3*w^2 + 5*w + 1],
[577, 577, -w^3 + 1/3*w^2 + 8*w + 3],
[577, 577, -1/3*w^3 - 1/3*w^2 + 6*w + 7],
[577, 577, 1/3*w^3 - 1/3*w^2 - 6*w + 7],
[577, 577, w^3 + 1/3*w^2 - 8*w + 3],
[617, 617, -2/3*w^3 - 5/3*w^2 + 5*w + 7],
[617, 617, -5/3*w^2 + 3*w + 5],
[617, 617, -8/3*w^3 - 13/3*w^2 + 27*w + 37],
[617, 617, -w^3 - 5/3*w^2 + 9*w + 15],
[625, 5, -5],
[631, 631, -1/3*w^3 - 5/3*w^2 + 3*w + 7],
[631, 631, -5/3*w^2 - w + 13],
[631, 631, 5/3*w^2 - w - 13],
[631, 631, 1/3*w^3 - 5/3*w^2 - 3*w + 7],
[673, 673, 2/3*w^3 + 1/3*w^2 - 8*w - 1],
[673, 673, -2/3*w^3 - 1/3*w^2 + 4*w + 3],
[673, 673, -2/3*w^3 + 1/3*w^2 + 4*w - 3],
[673, 673, 2/3*w^3 - 1/3*w^2 - 8*w + 1],
[719, 719, 1/3*w^3 - 1/3*w^2 - 6*w + 9],
[719, 719, w^3 - 1/3*w^2 - 8*w - 5],
[719, 719, w^3 + 1/3*w^2 - 8*w + 5],
[719, 719, -1/3*w^3 - 1/3*w^2 + 6*w + 9],
[727, 727, 2/3*w^3 + 1/3*w^2 - 5*w - 1],
[727, 727, -1/3*w^3 - 1/3*w^2 + 5*w + 3],
[727, 727, 1/3*w^3 - 1/3*w^2 - 5*w + 3],
[727, 727, -2/3*w^3 + 1/3*w^2 + 5*w - 1],
[761, 761, -2/3*w^3 + 1/3*w^2 + 3*w + 3],
[761, 761, 7/3*w^3 + 10/3*w^2 - 23*w - 33],
[761, 761, 3*w^3 + 10/3*w^2 - 29*w - 39],
[761, 761, -2*w^3 - 5/3*w^2 + 17*w + 21],
[769, 769, -4/3*w^3 - w^2 + 12*w + 13],
[769, 769, -w^3 - 2/3*w^2 + 10*w + 13],
[769, 769, -w^3 - 8/3*w^2 + 12*w + 19],
[769, 769, -2*w^3 - 3*w^2 + 22*w + 29],
[809, 809, -2/3*w^3 - 2/3*w^2 + 7*w + 3],
[809, 809, w^2 + w - 11],
[809, 809, w^2 - w - 11],
[809, 809, 2/3*w^3 - 2/3*w^2 - 7*w + 3],
[823, 823, 1/3*w^3 - 5*w - 1],
[823, 823, 2/3*w^3 - 5*w - 1],
[823, 823, 2/3*w^3 - 5*w + 1],
[823, 823, -1/3*w^3 + 5*w - 1],
[857, 857, -5/3*w^2 + w + 15],
[857, 857, -1/3*w^3 - 5/3*w^2 + 3*w + 5],
[857, 857, 1/3*w^3 - 5/3*w^2 - 3*w + 5],
[857, 857, 5/3*w^2 + w - 15],
[863, 863, -2/3*w^3 - 5/3*w^2 + 10*w + 15],
[863, 863, -5/3*w^3 - 8/3*w^2 + 18*w + 27],
[863, 863, -3*w^3 - 14/3*w^2 + 32*w + 45],
[863, 863, -5/3*w^2 + 4*w + 9],
[911, 911, 1/3*w^3 - 2/3*w^2 - 4*w - 1],
[911, 911, -1/3*w^3 + 2/3*w^2 + 2*w - 9],
[911, 911, 1/3*w^3 + 2/3*w^2 - 2*w - 9],
[911, 911, -1/3*w^3 - 2/3*w^2 + 4*w - 1],
[919, 919, w^3 + 1/3*w^2 - 9*w - 1],
[919, 919, -w^3 - 2/3*w^2 + 9*w + 7],
[919, 919, w^3 - 2/3*w^2 - 9*w + 7],
[919, 919, -w^3 + 1/3*w^2 + 9*w - 1],
[953, 953, -w^3 - w^2 + 9*w + 5],
[953, 953, -w^2 - 3*w + 7],
[953, 953, w^2 - 3*w - 7],
[953, 953, w^3 - w^2 - 9*w + 5],
[961, 31, 4/3*w^2 - 7],
[961, 31, 4/3*w^2 - 9],
[967, 967, -w^3 - 1/3*w^2 + 9*w + 5],
[967, 967, w^3 + 2/3*w^2 - 9*w - 5],
[967, 967, -w^3 + 2/3*w^2 + 9*w - 5],
[967, 967, w^3 - 1/3*w^2 - 9*w + 5],
[1009, 1009, 1/3*w^3 + 5/3*w^2 - 4*w - 9],
[1009, 1009, -8/3*w^3 - 10/3*w^2 + 26*w + 33],
[1009, 1009, 4/3*w^3 + 8/3*w^2 - 16*w - 21],
[1009, 1009, 7/3*w^3 + 5/3*w^2 - 22*w - 27],
[1049, 1049, -2/3*w^3 + 2/3*w^2 + 5*w - 9],
[1049, 1049, -1/3*w^3 - 2/3*w^2 + 5*w - 1],
[1049, 1049, 1/3*w^3 - 2/3*w^2 - 5*w - 1],
[1049, 1049, 2/3*w^3 + 2/3*w^2 - 5*w - 9],
[1063, 1063, w^3 + 2/3*w^2 - 9*w - 13],
[1063, 1063, 2*w^3 + w^2 - 17*w - 19],
[1063, 1063, -2/3*w^3 - w^2 + 9*w + 11],
[1063, 1063, -3*w^3 - 14/3*w^2 + 31*w + 43],
[1097, 1097, -2/3*w^3 + 1/3*w^2 + 7*w + 1],
[1097, 1097, 1/3*w^3 + 1/3*w^2 - w - 5],
[1097, 1097, -1/3*w^3 + 1/3*w^2 + w - 5],
[1097, 1097, 2/3*w^3 + 1/3*w^2 - 7*w + 1],
[1103, 1103, 2/3*w^2 + 2*w - 7],
[1103, 1103, 2/3*w^3 + 2/3*w^2 - 6*w - 1],
[1103, 1103, -2/3*w^3 + 2/3*w^2 + 6*w - 1],
[1103, 1103, 2/3*w^2 - 2*w - 7],
[1151, 1151, -2/3*w^3 + 4*w - 7],
[1151, 1151, 2/3*w^3 - 8*w - 7],
[1151, 1151, -2/3*w^3 + 8*w - 7],
[1151, 1151, 2/3*w^3 - 4*w - 7],
[1153, 1153, -1/3*w^3 + 6*w - 5],
[1153, 1153, -w^3 + 8*w + 5],
[1153, 1153, -w^3 + 8*w - 5],
[1153, 1153, 1/3*w^3 - 6*w - 5],
[1193, 1193, -1/3*w^3 - 2/3*w^2 + 5*w - 3],
[1193, 1193, 2/3*w^3 + 2/3*w^2 - 5*w - 11],
[1193, 1193, -2/3*w^3 + 2/3*w^2 + 5*w - 11],
[1193, 1193, 1/3*w^3 - 2/3*w^2 - 5*w - 3],
[1201, 1201, -7/3*w^3 - 4/3*w^2 + 20*w + 23],
[1201, 1201, -w^3 + 2/3*w^2 + 8*w - 1],
[1201, 1201, w^3 + 2/3*w^2 - 8*w - 1],
[1201, 1201, 1/3*w^3 + 2/3*w^2 - 6*w - 7],
[1249, 1249, -2/3*w^3 + 2*w^2 + 6*w - 11],
[1249, 1249, 2*w^2 + 2*w - 13],
[1249, 1249, 2*w^2 - 2*w - 13],
[1249, 1249, 2/3*w^3 + 2*w^2 - 6*w - 11],
[1289, 1289, w^3 - 2/3*w^2 - 11*w + 3],
[1289, 1289, 1/3*w^3 + w^2 - 5*w - 11],
[1289, 1289, -1/3*w^3 + w^2 + 5*w - 11],
[1289, 1289, -w^3 - 2/3*w^2 + 11*w + 3],
[1297, 1297, -1/3*w^3 + 4*w - 7],
[1297, 1297, -1/3*w^3 + 2*w - 7],
[1297, 1297, 1/3*w^3 - 2*w - 7],
[1297, 1297, 1/3*w^3 - 4*w - 7],
[1303, 1303, 1/3*w^3 - 2/3*w^2 - 5*w + 13],
[1303, 1303, -11/3*w^3 - 13/3*w^2 + 37*w + 49],
[1303, 1303, -w^3 - 7/3*w^2 + 11*w + 19],
[1303, 1303, -1/3*w^3 - 2/3*w^2 + 5*w + 13],
[1399, 1399, 4/3*w^3 + 2/3*w^2 - 11*w + 1],
[1399, 1399, -1/3*w^3 - 2/3*w^2 + 7*w + 9],
[1399, 1399, 1/3*w^3 - 2/3*w^2 - 7*w + 9],
[1399, 1399, -4/3*w^3 + 2/3*w^2 + 11*w + 1],
[1433, 1433, w^3 - 9*w - 7],
[1433, 1433, 4/3*w^3 + 11/3*w^2 - 17*w - 25],
[1433, 1433, -4/3*w^3 - 7/3*w^2 + 13*w + 17],
[1433, 1433, -w^3 + 9*w - 7],
[1439, 1439, -1/3*w^3 - 4/3*w^2 + 6*w + 11],
[1439, 1439, -2*w^3 - 3*w^2 + 22*w + 31],
[1439, 1439, 4/3*w^3 + 3*w^2 - 16*w - 25],
[1439, 1439, -5/3*w^3 - 10/3*w^2 + 20*w + 29],
[1447, 1447, 1/3*w^2 + w - 9],
[1447, 1447, 1/3*w^3 - 1/3*w^2 - 3*w - 5],
[1447, 1447, -1/3*w^3 - 1/3*w^2 + 3*w - 5],
[1447, 1447, 1/3*w^2 - w - 9],
[1481, 1481, -2/3*w^3 + 5/3*w^2 + 7*w - 9],
[1481, 1481, 1/3*w^3 + 5/3*w^2 - w - 11],
[1481, 1481, -1/3*w^3 + 5/3*w^2 + w - 11],
[1481, 1481, -2/3*w^3 - 5/3*w^2 + 7*w + 9],
[1487, 1487, 2/3*w^3 + 8/3*w^2 - 10*w - 15],
[1487, 1487, -w^3 - 1/3*w^2 + 10*w + 9],
[1487, 1487, -w^3 + 1/3*w^2 + 10*w - 9],
[1487, 1487, -2*w^3 - 10/3*w^2 + 20*w + 27],
[1489, 1489, -1/3*w^3 + 5/3*w^2 + 2*w - 9],
[1489, 1489, -1/3*w^3 + 5/3*w^2 + 4*w - 11],
[1489, 1489, 1/3*w^3 + 5/3*w^2 - 4*w - 11],
[1489, 1489, 1/3*w^3 + 5/3*w^2 - 2*w - 9],
[1543, 1543, -4*w^3 - 13/3*w^2 + 39*w + 49],
[1543, 1543, 4*w^3 + 5*w^2 - 41*w - 55],
[1543, 1543, 2/3*w^3 - 2/3*w^2 - 9*w + 1],
[1543, 1543, 4/3*w^3 + 5/3*w^2 - 15*w - 17],
[1583, 1583, -1/3*w^3 + w^2 + 4*w + 1],
[1583, 1583, 1/3*w^3 + w^2 - 2*w - 13],
[1583, 1583, -1/3*w^3 + w^2 + 2*w - 13],
[1583, 1583, -1/3*w^3 - w^2 + 4*w - 1],
[1721, 1721, w^3 - 1/3*w^2 - 7*w + 9],
[1721, 1721, 2/3*w^3 + 1/3*w^2 - 9*w + 5],
[1721, 1721, -2/3*w^3 + 1/3*w^2 + 9*w + 5],
[1721, 1721, w^3 + 1/3*w^2 - 7*w - 9],
[1777, 1777, -8/3*w^3 - 3*w^2 + 28*w + 37],
[1777, 1777, -w^3 - 2/3*w^2 + 8*w + 7],
[1777, 1777, -w^3 + 2/3*w^2 + 8*w - 7],
[1777, 1777, -2/3*w^3 - 3*w^2 + 10*w + 19],
[1783, 1783, -1/3*w^2 - w - 5],
[1783, 1783, -1/3*w^3 + 1/3*w^2 + 3*w - 9],
[1783, 1783, 1/3*w^3 + 1/3*w^2 - 3*w - 9],
[1783, 1783, -1/3*w^2 + w - 5],
[1823, 1823, 2/3*w^3 - 4/3*w^2 - 4*w + 9],
[1823, 1823, 2/3*w^3 - 4/3*w^2 - 8*w + 7],
[1823, 1823, 2/3*w^3 + 4/3*w^2 - 8*w - 7],
[1823, 1823, 2/3*w^3 + 4/3*w^2 - 4*w - 9],
[1831, 1831, -w - 7],
[1831, 1831, -1/3*w^3 + 3*w - 7],
[1831, 1831, 1/3*w^3 - 3*w - 7],
[1831, 1831, w - 7],
[1871, 1871, -4*w^3 - 13/3*w^2 + 38*w + 49],
[1871, 1871, -7/3*w^3 - 10/3*w^2 + 22*w + 31],
[1871, 1871, -7/3*w^3 - 4/3*w^2 + 20*w + 25],
[1871, 1871, 2/3*w^3 + 1/3*w^2 - 4*w - 7],
[1873, 1873, -1/3*w^2 + 4*w - 3],
[1873, 1873, 4/3*w^3 + 1/3*w^2 - 12*w - 7],
[1873, 1873, 4/3*w^3 - 1/3*w^2 - 12*w + 7],
[1873, 1873, 1/3*w^2 + 4*w + 3],
[1879, 1879, -1/3*w^3 + 7/3*w^2 + 5*w - 11],
[1879, 1879, 2/3*w^3 + 7/3*w^2 - 5*w - 17],
[1879, 1879, -2/3*w^3 + 7/3*w^2 + 5*w - 17],
[1879, 1879, 1/3*w^3 + 7/3*w^2 - 5*w - 11],
[1913, 1913, 1/3*w^3 + 1/3*w^2 - w - 7],
[1913, 1913, -2/3*w^3 - 1/3*w^2 + 7*w - 3],
[1913, 1913, 2/3*w^3 - 1/3*w^2 - 7*w - 3],
[1913, 1913, -1/3*w^3 + 1/3*w^2 + w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 4*x^3 - 28*x^2 - 64*x - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/10*e^3 + 3/10*e^2 - 13/5*e - 14/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 14/5, -1/10*e^3 - 3/10*e^2 + 18/5*e + 24/5, -1/10*e^3 - 3/10*e^2 + 18/5*e + 24/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 14/5, -1/5*e^3 - 3/5*e^2 + 26/5*e + 28/5, -1/10*e^3 + 1/5*e^2 + 18/5*e - 26/5, -1/10*e^3 - 4/5*e^2 + 8/5*e + 54/5, -1/10*e^3 - 4/5*e^2 + 8/5*e + 54/5, -1/10*e^3 + 1/5*e^2 + 18/5*e - 26/5, 1/5*e^3 + 3/5*e^2 - 26/5*e - 28/5, 1/5*e^3 + 3/5*e^2 - 26/5*e - 28/5, 1/5*e^3 + 3/5*e^2 - 26/5*e - 28/5, 1/5*e^3 + 3/5*e^2 - 26/5*e - 28/5, -2/5*e^3 - 6/5*e^2 + 52/5*e + 56/5, -2/5*e^3 - 6/5*e^2 + 52/5*e + 56/5, -2/5*e^3 - 6/5*e^2 + 52/5*e + 56/5, -2/5*e^3 - 6/5*e^2 + 52/5*e + 56/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 48/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 28/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 28/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 48/5, 1/10*e^3 + 3/10*e^2 - 18/5*e + 16/5, -1/10*e^3 - 3/10*e^2 + 18/5*e + 54/5, -1/10*e^3 - 3/10*e^2 + 18/5*e + 54/5, 1/10*e^3 + 3/10*e^2 - 18/5*e + 16/5, 3/10*e^3 + 2/5*e^2 - 44/5*e - 2/5, 3/10*e^3 + 7/5*e^2 - 34/5*e - 82/5, 3/10*e^3 + 7/5*e^2 - 34/5*e - 82/5, 3/10*e^3 + 2/5*e^2 - 44/5*e - 2/5, -1/10*e^3 - 3/10*e^2 + 18/5*e - 66/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 104/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 104/5, -1/10*e^3 - 3/10*e^2 + 18/5*e - 66/5, e^2 + 2*e - 16, -e^2 - 2*e + 16, -e^2 - 2*e + 16, e^2 + 2*e - 16, -1/5*e^3 - 3/5*e^2 + 36/5*e - 2/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 78/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 78/5, -1/5*e^3 - 3/5*e^2 + 36/5*e - 2/5, -3/10*e^3 - 9/10*e^2 + 54/5*e + 22/5, 3/10*e^3 + 9/10*e^2 - 54/5*e - 92/5, 3/10*e^3 + 9/10*e^2 - 54/5*e - 92/5, -3/10*e^3 - 9/10*e^2 + 54/5*e + 22/5, 2/5*e^3 + 1/5*e^2 - 62/5*e + 24/5, 2/5*e^3 + 11/5*e^2 - 42/5*e - 136/5, 2/5*e^3 + 11/5*e^2 - 42/5*e - 136/5, 2/5*e^3 + 1/5*e^2 - 62/5*e + 24/5, 3/5*e^3 + 9/5*e^2 - 78/5*e - 84/5, 3/5*e^3 + 9/5*e^2 - 78/5*e - 84/5, 3/5*e^3 + 9/5*e^2 - 78/5*e - 84/5, 3/5*e^3 + 9/5*e^2 - 78/5*e - 84/5, -1/5*e^3 - 3/5*e^2 + 36/5*e - 2/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 78/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 78/5, -1/5*e^3 - 3/5*e^2 + 36/5*e - 2/5, 2/5*e^3 + 6/5*e^2 - 52/5*e - 56/5, 2/5*e^3 + 6/5*e^2 - 52/5*e - 56/5, 2/5*e^3 + 6/5*e^2 - 52/5*e - 56/5, 2/5*e^3 + 6/5*e^2 - 52/5*e - 56/5, 2/5*e^3 + 6/5*e^2 - 72/5*e - 26/5, -2/5*e^3 - 6/5*e^2 + 72/5*e + 126/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 88/5, 1/5*e^3 + 3/5*e^2 - 36/5*e + 12/5, 1/5*e^3 + 3/5*e^2 - 36/5*e + 12/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 88/5, -e^2 - 2*e + 16, e^2 + 2*e - 16, e^2 + 2*e - 16, -e^2 - 2*e + 16, -4/5*e^3 - 7/5*e^2 + 114/5*e + 32/5, -4/5*e^3 - 17/5*e^2 + 94/5*e + 192/5, -4/5*e^3 - 17/5*e^2 + 94/5*e + 192/5, -4/5*e^3 - 7/5*e^2 + 114/5*e + 32/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 28/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 48/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 48/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 28/5, -1/10*e^3 - 3/10*e^2 + 18/5*e - 6/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 44/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 44/5, -1/10*e^3 - 3/10*e^2 + 18/5*e - 6/5, -3/5*e^3 - 4/5*e^2 + 88/5*e + 4/5, -3/5*e^3 - 14/5*e^2 + 68/5*e + 164/5, -3/5*e^3 - 14/5*e^2 + 68/5*e + 164/5, -3/5*e^3 - 4/5*e^2 + 88/5*e + 4/5, -1/10*e^3 - 3/10*e^2 + 18/5*e + 54/5, 1/10*e^3 + 3/10*e^2 - 18/5*e + 16/5, 1/10*e^3 + 3/10*e^2 - 18/5*e + 16/5, -1/10*e^3 - 3/10*e^2 + 18/5*e + 54/5, 1/2*e^3 + 3*e^2 - 10*e - 38, 1/2*e^3 - 16*e + 10, 1/2*e^3 - 16*e + 10, 1/2*e^3 + 3*e^2 - 10*e - 38, -2/5*e^3 - 6/5*e^2 + 72/5*e + 186/5, 2/5*e^3 + 6/5*e^2 - 72/5*e + 34/5, -3/10*e^3 - 2/5*e^2 + 44/5*e + 2/5, -3/10*e^3 - 7/5*e^2 + 34/5*e + 82/5, -3/10*e^3 - 7/5*e^2 + 34/5*e + 82/5, -3/10*e^3 - 2/5*e^2 + 44/5*e + 2/5, e^3 + 3*e^2 - 36*e - 40, -e^3 - 3*e^2 + 36*e + 36, -e^3 - 3*e^2 + 36*e + 36, e^3 + 3*e^2 - 36*e - 40, 3/5*e^3 + 19/5*e^2 - 58/5*e - 244/5, 3/5*e^3 - 1/5*e^2 - 98/5*e + 76/5, 3/5*e^3 - 1/5*e^2 - 98/5*e + 76/5, 3/5*e^3 + 19/5*e^2 - 58/5*e - 244/5, 38, 7/10*e^3 + 21/10*e^2 - 126/5*e - 68/5, -7/10*e^3 - 21/10*e^2 + 126/5*e + 198/5, -7/10*e^3 - 21/10*e^2 + 126/5*e + 198/5, 7/10*e^3 + 21/10*e^2 - 126/5*e - 68/5, -1/5*e^3 - 3/5*e^2 + 36/5*e - 32/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 108/5, 1/5*e^3 + 3/5*e^2 - 36/5*e - 108/5, -1/5*e^3 - 3/5*e^2 + 36/5*e - 32/5, e^3 + 2*e^2 - 28*e - 12, e^3 + 4*e^2 - 24*e - 44, e^3 + 4*e^2 - 24*e - 44, e^3 + 2*e^2 - 28*e - 12, -1/10*e^3 - 3/10*e^2 + 18/5*e - 36/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 74/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 74/5, -1/10*e^3 - 3/10*e^2 + 18/5*e - 36/5, -13/10*e^3 - 22/5*e^2 + 164/5*e + 222/5, -13/10*e^3 - 17/5*e^2 + 174/5*e + 142/5, -13/10*e^3 - 17/5*e^2 + 174/5*e + 142/5, -13/10*e^3 - 22/5*e^2 + 164/5*e + 222/5, e^3 + 3*e^2 - 36*e - 34, -e^3 - 3*e^2 + 36*e + 42, -e^3 - 3*e^2 + 36*e + 42, e^3 + 3*e^2 - 36*e - 34, 3/10*e^3 + 7/5*e^2 - 34/5*e - 82/5, 3/10*e^3 + 2/5*e^2 - 44/5*e - 2/5, 3/10*e^3 + 2/5*e^2 - 44/5*e - 2/5, 3/10*e^3 + 7/5*e^2 - 34/5*e - 82/5, 3/10*e^3 + 9/10*e^2 - 54/5*e + 88/5, -3/10*e^3 - 9/10*e^2 + 54/5*e + 202/5, -3/10*e^3 - 9/10*e^2 + 54/5*e + 202/5, 3/10*e^3 + 9/10*e^2 - 54/5*e + 88/5, -9/10*e^3 - 16/5*e^2 + 112/5*e + 166/5, -9/10*e^3 - 11/5*e^2 + 122/5*e + 86/5, -9/10*e^3 - 11/5*e^2 + 122/5*e + 86/5, -9/10*e^3 - 16/5*e^2 + 112/5*e + 166/5, -1/5*e^3 - 13/5*e^2 + 6/5*e + 188/5, -1/5*e^3 + 7/5*e^2 + 46/5*e - 132/5, -1/5*e^3 + 7/5*e^2 + 46/5*e - 132/5, -1/5*e^3 - 13/5*e^2 + 6/5*e + 188/5, -6/5*e^3 - 23/5*e^2 + 146/5*e + 248/5, -6/5*e^3 - 13/5*e^2 + 166/5*e + 88/5, -6/5*e^3 - 13/5*e^2 + 166/5*e + 88/5, -6/5*e^3 - 23/5*e^2 + 146/5*e + 248/5, -3/10*e^3 - 9/10*e^2 + 54/5*e + 22/5, 3/10*e^3 + 9/10*e^2 - 54/5*e - 92/5, 3/10*e^3 + 9/10*e^2 - 54/5*e - 92/5, -3/10*e^3 - 9/10*e^2 + 54/5*e + 22/5, -7/10*e^3 - 18/5*e^2 + 76/5*e + 218/5, -7/10*e^3 - 3/5*e^2 + 106/5*e - 22/5, -7/10*e^3 - 3/5*e^2 + 106/5*e - 22/5, -7/10*e^3 - 18/5*e^2 + 76/5*e + 218/5, 9/5*e^3 + 27/5*e^2 - 324/5*e - 332/5, -9/5*e^3 - 27/5*e^2 + 324/5*e + 352/5, 7/10*e^3 + 21/10*e^2 - 126/5*e + 22/5, -7/10*e^3 - 21/10*e^2 + 126/5*e + 288/5, -7/10*e^3 - 21/10*e^2 + 126/5*e + 288/5, 7/10*e^3 + 21/10*e^2 - 126/5*e + 22/5, 3/5*e^3 + 9/5*e^2 - 108/5*e - 224/5, -3/5*e^3 - 9/5*e^2 + 108/5*e + 4/5, -3/5*e^3 - 9/5*e^2 + 108/5*e + 4/5, 3/5*e^3 + 9/5*e^2 - 108/5*e - 224/5, 3*e^2 + 6*e - 48, -3*e^2 - 6*e + 48, -3*e^2 - 6*e + 48, 3*e^2 + 6*e - 48, -7/10*e^3 - 21/10*e^2 + 126/5*e + 78/5, 7/10*e^3 + 21/10*e^2 - 126/5*e - 188/5, 7/10*e^3 + 21/10*e^2 - 126/5*e - 188/5, -7/10*e^3 - 21/10*e^2 + 126/5*e + 78/5, -2*e^2 - 4*e + 32, 2*e^2 + 4*e - 32, 2*e^2 + 4*e - 32, -2*e^2 - 4*e + 32, -e^2 - 2*e + 16, e^2 + 2*e - 16, e^2 + 2*e - 16, -e^2 - 2*e + 16, 1/5*e^3 - 2/5*e^2 - 36/5*e + 52/5, 1/5*e^3 + 8/5*e^2 - 16/5*e - 108/5, 1/5*e^3 + 8/5*e^2 - 16/5*e - 108/5, 1/5*e^3 - 2/5*e^2 - 36/5*e + 52/5, -2/5*e^3 - 6/5*e^2 + 72/5*e + 126/5, 2/5*e^3 + 6/5*e^2 - 72/5*e - 26/5, 2/5*e^3 + 6/5*e^2 - 72/5*e - 26/5, -2/5*e^3 - 6/5*e^2 + 72/5*e + 126/5, 3/5*e^3 + 4/5*e^2 - 88/5*e - 4/5, 3/5*e^3 + 14/5*e^2 - 68/5*e - 164/5, 3/5*e^3 + 14/5*e^2 - 68/5*e - 164/5, 3/5*e^3 + 4/5*e^2 - 88/5*e - 4/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 148/5, 1/5*e^3 + 3/5*e^2 - 36/5*e + 72/5, 1/5*e^3 + 3/5*e^2 - 36/5*e + 72/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 148/5, 1/5*e^3 + 3/5*e^2 - 36/5*e + 192/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 268/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 268/5, 1/5*e^3 + 3/5*e^2 - 36/5*e + 192/5, -3/5*e^3 - 14/5*e^2 + 68/5*e + 164/5, -3/5*e^3 - 4/5*e^2 + 88/5*e + 4/5, -3/5*e^3 - 4/5*e^2 + 88/5*e + 4/5, -3/5*e^3 - 14/5*e^2 + 68/5*e + 164/5, -e^3 - 3*e^2 + 36*e + 66, e^3 + 3*e^2 - 36*e - 10, e^3 + 3*e^2 - 36*e - 10, -e^3 - 3*e^2 + 36*e + 66, 3/10*e^3 + 9/10*e^2 - 54/5*e - 62/5, -3/10*e^3 - 9/10*e^2 + 54/5*e + 52/5, -3/10*e^3 - 9/10*e^2 + 54/5*e + 52/5, 3/10*e^3 + 9/10*e^2 - 54/5*e - 62/5, 11/10*e^3 + 33/10*e^2 - 198/5*e - 174/5, -11/10*e^3 - 33/10*e^2 + 198/5*e + 244/5, -11/10*e^3 - 33/10*e^2 + 198/5*e + 244/5, 11/10*e^3 + 33/10*e^2 - 198/5*e - 174/5, e^3 + 4*e^2 - 24*e - 44, e^3 + 2*e^2 - 28*e - 12, e^3 + 2*e^2 - 28*e - 12, e^3 + 4*e^2 - 24*e - 44, -1/5*e^3 - 18/5*e^2 - 4/5*e + 268/5, -1/5*e^3 + 12/5*e^2 + 56/5*e - 212/5, -1/5*e^3 + 12/5*e^2 + 56/5*e - 212/5, -1/5*e^3 - 18/5*e^2 - 4/5*e + 268/5, -1/2*e^3 - 3/2*e^2 + 18*e + 2, 1/2*e^3 + 3/2*e^2 - 18*e - 36, 1/2*e^3 + 3/2*e^2 - 18*e - 36, -1/2*e^3 - 3/2*e^2 + 18*e + 2, -4/5*e^3 - 17/5*e^2 + 94/5*e + 192/5, -4/5*e^3 - 7/5*e^2 + 114/5*e + 32/5, -4/5*e^3 - 7/5*e^2 + 114/5*e + 32/5, -4/5*e^3 - 17/5*e^2 + 94/5*e + 192/5, -e^3 - 2*e^2 + 28*e + 12, -e^3 - 4*e^2 + 24*e + 44, -e^3 - 4*e^2 + 24*e + 44, -e^3 - 2*e^2 + 28*e + 12, 1/5*e^3 + 3/5*e^2 - 36/5*e + 252/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 328/5, -1/5*e^3 - 3/5*e^2 + 36/5*e + 328/5, 1/5*e^3 + 3/5*e^2 - 36/5*e + 252/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 104/5, -1/10*e^3 - 3/10*e^2 + 18/5*e - 66/5, -1/10*e^3 - 3/10*e^2 + 18/5*e - 66/5, 1/10*e^3 + 3/10*e^2 - 18/5*e - 104/5, 6/5*e^3 + 18/5*e^2 - 156/5*e - 168/5, 6/5*e^3 + 18/5*e^2 - 156/5*e - 168/5, 6/5*e^3 + 18/5*e^2 - 156/5*e - 168/5, 6/5*e^3 + 18/5*e^2 - 156/5*e - 168/5, 1/5*e^3 + 3/5*e^2 - 26/5*e - 28/5, 1/5*e^3 + 3/5*e^2 - 26/5*e - 28/5, 1/5*e^3 + 3/5*e^2 - 26/5*e - 28/5, 1/5*e^3 + 3/5*e^2 - 26/5*e - 28/5, 2/5*e^3 + 6/5*e^2 - 72/5*e - 386/5, -2/5*e^3 - 6/5*e^2 + 72/5*e - 234/5, -2/5*e^3 - 6/5*e^2 + 72/5*e - 234/5, 2/5*e^3 + 6/5*e^2 - 72/5*e - 386/5, 19/10*e^3 + 57/10*e^2 - 342/5*e - 446/5, -19/10*e^3 - 57/10*e^2 + 342/5*e + 276/5, -19/10*e^3 - 57/10*e^2 + 342/5*e + 276/5, 19/10*e^3 + 57/10*e^2 - 342/5*e - 446/5, 3/5*e^3 + 14/5*e^2 - 68/5*e - 164/5, 3/5*e^3 + 4/5*e^2 - 88/5*e - 4/5, 3/5*e^3 + 4/5*e^2 - 88/5*e - 4/5, 3/5*e^3 + 14/5*e^2 - 68/5*e - 164/5, 7/10*e^3 + 21/10*e^2 - 126/5*e - 8/5, -7/10*e^3 - 21/10*e^2 + 126/5*e + 258/5, -7/10*e^3 - 21/10*e^2 + 126/5*e + 258/5, 7/10*e^3 + 21/10*e^2 - 126/5*e - 8/5, 4/5*e^3 - 8/5*e^2 - 144/5*e + 208/5, 4/5*e^3 + 32/5*e^2 - 64/5*e - 432/5, 4/5*e^3 + 32/5*e^2 - 64/5*e - 432/5, 4/5*e^3 - 8/5*e^2 - 144/5*e + 208/5, -9/5*e^3 - 27/5*e^2 + 324/5*e + 322/5, 9/5*e^3 + 27/5*e^2 - 324/5*e - 362/5, 9/5*e^3 + 27/5*e^2 - 324/5*e - 362/5, -9/5*e^3 - 27/5*e^2 + 324/5*e + 322/5, 1/2*e^3 + 3/2*e^2 - 18*e - 84, -1/2*e^3 - 3/2*e^2 + 18*e - 46, -1/2*e^3 - 3/2*e^2 + 18*e - 46, 1/2*e^3 + 3/2*e^2 - 18*e - 84, -13/5*e^3 - 39/5*e^2 + 338/5*e + 364/5, -13/5*e^3 - 39/5*e^2 + 338/5*e + 364/5, -13/5*e^3 - 39/5*e^2 + 338/5*e + 364/5, -13/5*e^3 - 39/5*e^2 + 338/5*e + 364/5];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();

// 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;