/*
  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 | {4, 2, 1/3*w^2 - 2}>;

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 + 8*x^3 - 4*x^2 - 128*x - 164;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, -3/2*e^3 - 4*e^2 + 28*e + 42, 1/2*e^3 + e^2 - 10*e - 12, e^3 + 3*e^2 - 19*e - 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6*e^3 + 17*e^2 - 110*e - 186, -3*e^3 - 9*e^2 + 52*e + 102, -5*e^3 - 15*e^2 + 92*e + 178, 2*e^3 + 7*e^2 - 34*e - 78, -13/2*e^3 - 19*e^2 + 118*e + 212, 6*e^3 + 18*e^2 - 109*e - 208, 5*e^3 + 15*e^2 - 89*e - 170, -9/2*e^3 - 14*e^2 + 80*e + 158, 0, 0, 0, 0, -4*e^3 - 10*e^2 + 75*e + 104, -5/2*e^3 - 7*e^2 + 50*e + 84, 7/2*e^3 + 10*e^2 - 68*e - 122, 3*e^3 + 7*e^2 - 57*e - 74, 0, 0, 0, 0, 8*e^3 + 23*e^2 - 148*e - 264, -4*e^3 - 13*e^2 + 72*e + 152, -6*e^3 - 17*e^2 + 108*e + 184, 2*e^3 + 7*e^2 - 32*e - 80, -9*e^3 - 24*e^2 + 171*e + 272, 3/2*e^3 + 6*e^2 - 30*e - 82, 3/2*e^3 + 3*e^2 - 24*e - 16, 6*e^3 + 15*e^2 - 117*e - 166, 0, 0, 0, 0, 0, 0, 0, 0, 10*e^3 + 29*e^2 - 188*e - 352, -3*e^2 - 4*e + 32, -10*e^3 - 27*e^2 + 184*e + 280, e^2 + 8*e - 16, 0, 0, 0, 0, -4*e^3 - 12*e^2 + 72*e + 122, 4*e^3 + 12*e^2 - 72*e - 150, -5*e^3 - 13*e^2 + 94*e + 122, 3*e^3 + 9*e^2 - 58*e - 130, -e^3 - 3*e^2 + 22*e + 22, 3*e^3 + 7*e^2 - 58*e - 94, 0, 0, 0, 0, 0, 0, 0, 0, 9*e^3 + 23*e^2 - 168*e - 242, 4*e^3 + 11*e^2 - 82*e - 134, -8*e^3 - 23*e^2 + 154*e + 278, -5*e^3 - 11*e^2 + 96*e + 114, e^3 + 4*e^2 - 11*e - 64, -19/2*e^3 - 26*e^2 + 174*e + 258, -3/2*e^3 - 7*e^2 + 24*e + 64, 10*e^3 + 29*e^2 - 187*e - 362, 0, 0, 0, 0, 2*e^3 + 6*e^2 - 29*e - 68, -25/2*e^3 - 34*e^2 + 232*e + 362, 3/2*e^3 + e^2 - 34*e - 16, 9*e^3 + 27*e^2 - 169*e - 334, 0, 0, 0, 0, -14*e^3 - 42*e^2 + 252*e + 474, 14*e^3 + 42*e^2 - 252*e - 478, 0, 0, 0, 0, e^3 + 5*e^2 - 26*e - 82, -13*e^3 - 33*e^2 + 250*e + 370, 15*e^3 + 39*e^2 - 286*e - 430, -3*e^3 - 11*e^2 + 62*e + 158, 0, 0, 0, 0, -10, 15/2*e^3 + 21*e^2 - 140*e - 240, -4*e^3 - 11*e^2 + 71*e + 114, -3*e^3 - 10*e^2 + 55*e + 120, -1/2*e^3 + 14*e - 2, 2*e^3 + 5*e^2 - 34*e - 66, 3*e^3 + 7*e^2 - 60*e - 98, -7*e^3 - 19*e^2 + 132*e + 194, 2*e^3 + 7*e^2 - 38*e - 110, 0, 0, 0, 0, -25/2*e^3 - 37*e^2 + 224*e + 424, 15*e^3 + 44*e^2 - 273*e - 488, 10*e^3 + 31*e^2 - 177*e - 342, -25/2*e^3 - 38*e^2 + 226*e + 446, 0, 0, 0, 0, 2*e^3 + e^2 - 44*e - 8, e^2 + 12*e - 8, 14*e^3 + 41*e^2 - 264*e - 496, -16*e^3 - 43*e^2 + 296*e + 456, 0, 0, 0, 0, -15/2*e^3 - 24*e^2 + 132*e + 290, 9*e^3 + 27*e^2 - 159*e - 286, 12*e^3 + 36*e^2 - 219*e - 400, -27/2*e^3 - 39*e^2 + 246*e + 452, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3*e^3 + 12*e^2 - 57*e - 148, 15/2*e^3 + 18*e^2 - 150*e - 190, -33/2*e^3 - 45*e^2 + 312*e + 524, 6*e^3 + 15*e^2 - 105*e - 130, 0, 0, 0, 0, -9*e^3 - 27*e^2 + 162*e + 326, 9*e^3 + 27*e^2 - 162*e - 286, -4*e^3 - 10*e^2 + 71*e + 60, -9/2*e^3 - 11*e^2 + 90*e + 88, 19/2*e^3 + 26*e^2 - 180*e - 334, -e^3 - 5*e^2 + 19*e + 34, 3*e^2 + 6*e - 22, -3*e^3 - 9*e^2 + 48*e + 110, -9*e^3 - 27*e^2 + 168*e + 338, 12*e^3 + 33*e^2 - 222*e - 346, 0, 0, 0, 0, -6*e^3 - 19*e^2 + 103*e + 246, 15/2*e^3 + 24*e^2 - 134*e - 246, 23/2*e^3 + 33*e^2 - 208*e - 332, -13*e^3 - 38*e^2 + 239*e + 468, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*e^3 + 6*e^2 - 44*e - 110, -6*e^3 - 14*e^2 + 116*e + 122, 10*e^3 + 26*e^2 - 188*e - 310, -6*e^3 - 18*e^2 + 116*e + 194, 0, 0, 0, 0, 5*e^3 + 19*e^2 - 86*e - 206, -19*e^3 - 55*e^2 + 354*e + 662, -3*e^3 - 11*e^2 + 42*e + 142, 17*e^3 + 47*e^2 - 310*e - 486, 5*e^2 + 4*e - 54, 4*e^3 + 9*e^2 - 88*e - 86, -18*e^3 - 51*e^2 + 340*e + 618, 14*e^3 + 37*e^2 - 256*e - 366, 0, 0, 0, 0, 2*e^3 + 7*e^2 - 28*e - 120, -12*e^3 - 33*e^2 + 220*e + 312, -2*e^3 - 9*e^2 + 32*e + 64, 12*e^3 + 35*e^2 - 224*e - 456, -9*e^3 - 30*e^2 + 165*e + 356, -3/2*e^3 + 42*e - 10, 45/2*e^3 + 63*e^2 - 420*e - 724, -12*e^3 - 33*e^2 + 213*e + 338, -14*e^3 - 45*e^2 + 249*e + 510, 25/2*e^3 + 39*e^2 - 216*e - 448, 49/2*e^3 + 72*e^2 - 450*e - 838, -23*e^3 - 66*e^2 + 417*e + 720, 0, 0, 0, 0, 0, 0, 0, 0, 13*e^3 + 38*e^2 - 251*e - 472, -35/2*e^3 - 45*e^2 + 328*e + 472, 21/2*e^3 + 24*e^2 - 202*e - 262, -6*e^3 - 17*e^2 + 125*e + 206, 0, 0, 0, 0, 0, 0, 0, 0, -e^3 - 3*e^2 + 4*e + 34, 22*e^3 + 59*e^2 - 410*e - 622, -6*e^3 - 11*e^2 + 122*e + 134, -15*e^3 - 45*e^2 + 284*e + 566, 51/2*e^3 + 72*e^2 - 464*e - 774, -10*e^3 - 32*e^2 + 167*e + 372, -27*e^3 - 79*e^2 + 499*e + 930, 23/2*e^3 + 39*e^2 - 202*e - 440, 0, 0, 0, 0, 0, 0, 0, 0, -18*e^3 - 46*e^2 + 348*e + 526, -2*e^3 - 10*e^2 + 44*e + 166, -2*e^3 - 2*e^2 + 28*e - 10, 22*e^3 + 58*e^2 - 420*e - 642, -4*e^3 - 19*e^2 + 65*e + 210, 57/2*e^3 + 82*e^2 - 534*e - 986, 1/2*e^3 + 5*e^2 + 12*e - 76, -25*e^3 - 68*e^2 + 457*e + 700, 0, 0, 0, 0, 17/2*e^3 + 28*e^2 - 168*e - 378, -26*e^3 - 68*e^2 + 493*e + 740, 19*e^3 + 47*e^2 - 367*e - 530, -3/2*e^3 - 7*e^2 + 42*e + 112, 0, 0, 0, 0, 6*e^3 + 21*e^2 - 120*e - 280, -24*e^3 - 63*e^2 + 456*e + 704, 18*e^3 + 45*e^2 - 348*e - 496, -3*e^2 + 12*e + 80, 33/2*e^3 + 45*e^2 - 316*e - 508, 3*e^3 + 4*e^2 - 55*e - 4, -8*e^3 - 19*e^2 + 145*e + 194, -23/2*e^3 - 30*e^2 + 226*e + 358, 0, 0, 0, 0];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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