/*
  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![-1, -3, 3, 7, -3, -3, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {53,53,4*w^5 - 14*w^4 - 4*w^3 + 27*w^2 - 4*w - 6}>;

primesArray := [
[8, 2, w^5 - 4*w^4 + 9*w^2 - w - 3],
[8, 2, w^4 - 3*w^3 - 2*w^2 + 5*w],
[9, 3, -w^5 + 3*w^4 + 3*w^3 - 7*w^2 - 2*w + 1],
[19, 19, -w^5 + 3*w^4 + 3*w^3 - 7*w^2 - 3*w + 1],
[19, 19, -w + 2],
[37, 37, 2*w^5 - 6*w^4 - 5*w^3 + 11*w^2 + 3*w - 2],
[37, 37, -w^3 + 3*w^2 + w - 3],
[53, 53, -w^5 + 2*w^4 + 4*w^3 - 2*w - 2],
[53, 53, 2*w^5 - 7*w^4 - 2*w^3 + 14*w^2 - 2*w - 5],
[53, 53, 2*w^5 - 6*w^4 - 5*w^3 + 12*w^2 + 2*w - 3],
[53, 53, -4*w^5 + 14*w^4 + 4*w^3 - 27*w^2 + 4*w + 6],
[71, 71, -w^5 + 2*w^4 + 5*w^3 - 3*w^2 - 3*w - 1],
[71, 71, 2*w^5 - 8*w^4 + w^3 + 16*w^2 - 7*w - 6],
[73, 73, -3*w^5 + 10*w^4 + 5*w^3 - 20*w^2 - 2*w + 6],
[73, 73, -w^5 + 3*w^4 + 2*w^3 - 4*w^2 - 2],
[73, 73, w^3 - 3*w^2 - 2*w + 2],
[73, 73, w^4 - 2*w^3 - 5*w^2 + 4*w + 4],
[89, 89, -w^5 + 3*w^4 + 2*w^3 - 5*w^2 + 3],
[89, 89, 3*w^5 - 10*w^4 - 5*w^3 + 21*w^2 - 5],
[107, 107, 2*w^5 - 6*w^4 - 5*w^3 + 12*w^2 + 2*w - 2],
[107, 107, -2*w^5 + 7*w^4 + 2*w^3 - 14*w^2 + 2*w + 6],
[109, 109, 2*w^5 - 6*w^4 - 5*w^3 + 11*w^2 + 2*w],
[109, 109, -2*w^4 + 4*w^3 + 9*w^2 - 4*w - 4],
[127, 127, w^5 - 3*w^4 - 3*w^3 + 7*w^2 + 4*w - 4],
[127, 127, w^5 - 3*w^4 - 3*w^3 + 7*w^2 + 4*w - 2],
[127, 127, -w^5 + 2*w^4 + 6*w^3 - 6*w^2 - 6*w + 2],
[127, 127, -2*w^5 + 8*w^4 - 18*w^2 + 5*w + 7],
[163, 163, -3*w^5 + 10*w^4 + 5*w^3 - 20*w^2 - 2*w + 5],
[163, 163, -2*w^5 + 6*w^4 + 5*w^3 - 11*w^2 - 4*w + 4],
[163, 163, -3*w^5 + 9*w^4 + 8*w^3 - 19*w^2 - 4*w + 5],
[163, 163, -w^5 + 3*w^4 + 2*w^3 - 4*w^2 - 2*w - 2],
[179, 179, -2*w^5 + 7*w^4 + 2*w^3 - 14*w^2 + 4*w + 6],
[179, 179, w^5 - 2*w^4 - 6*w^3 + 6*w^2 + 4*w - 4],
[179, 179, w^5 - 5*w^4 + 4*w^3 + 9*w^2 - 9*w - 1],
[179, 179, -3*w^5 + 12*w^4 - w^3 - 25*w^2 + 9*w + 9],
[181, 181, -w^5 + 4*w^4 - 10*w^2 + 5*w + 4],
[181, 181, w^5 - 3*w^4 - 3*w^3 + 6*w^2 + 5*w - 1],
[197, 197, -w^5 + w^4 + 8*w^3 + w^2 - 11*w - 3],
[197, 197, 3*w^5 - 10*w^4 - 5*w^3 + 21*w^2 + w - 5],
[199, 199, 3*w^5 - 10*w^4 - 5*w^3 + 20*w^2 - 3],
[199, 199, -2*w^5 + 7*w^4 + 3*w^3 - 17*w^2 + 10],
[199, 199, -w^5 + 3*w^4 + 3*w^3 - 8*w^2 - 2*w + 2],
[199, 199, 2*w^5 - 7*w^4 - 4*w^3 + 19*w^2 + 2*w - 7],
[233, 233, w^5 - 3*w^4 - 2*w^3 + 5*w^2 - w + 1],
[233, 233, -2*w^5 + 7*w^4 + 2*w^3 - 14*w^2 + 3*w + 6],
[233, 233, 3*w^5 - 10*w^4 - 4*w^3 + 18*w^2 - w - 4],
[233, 233, w^5 - 3*w^4 - 3*w^3 + 7*w^2 + 4*w - 3],
[233, 233, 3*w^5 - 9*w^4 - 7*w^3 + 16*w^2 + 3*w - 3],
[251, 251, 3*w^5 - 9*w^4 - 8*w^3 + 19*w^2 + 5*w - 4],
[251, 251, 2*w^5 - 5*w^4 - 7*w^3 + 6*w^2 + 5*w + 3],
[269, 269, -w^5 + 4*w^4 + w^3 - 13*w^2 + w + 7],
[269, 269, -2*w^4 + 6*w^3 + 3*w^2 - 6*w - 1],
[271, 271, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + w - 7],
[271, 271, -4*w^5 + 12*w^4 + 10*w^3 - 23*w^2 - 5*w + 4],
[289, 17, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 6*w + 5],
[289, 17, -2*w^5 + 7*w^4 + 3*w^3 - 16*w^2 + 4],
[289, 17, -w^5 + 2*w^4 + 6*w^3 - 5*w^2 - 6*w + 2],
[307, 307, -2*w^5 + 9*w^4 - 3*w^3 - 20*w^2 + 7*w + 7],
[307, 307, -4*w^5 + 12*w^4 + 10*w^3 - 22*w^2 - 7*w + 3],
[343, 7, -w^4 + 4*w^3 - 6*w - 1],
[343, 7, 3*w^5 - 10*w^4 - 5*w^3 + 21*w^2 + 2*w - 7],
[359, 359, 2*w^5 - 8*w^4 + w^3 + 16*w^2 - 6*w - 8],
[359, 359, 4*w^5 - 13*w^4 - 8*w^3 + 27*w^2 + 4*w - 8],
[361, 19, 2*w^4 - 6*w^3 - 4*w^2 + 8*w + 1],
[361, 19, -2*w^5 + 7*w^4 + 3*w^3 - 16*w^2 + 3],
[379, 379, 4*w^5 - 13*w^4 - 8*w^3 + 28*w^2 + w - 7],
[379, 379, w^5 - 5*w^4 + 2*w^3 + 15*w^2 - 7*w - 9],
[379, 379, w^3 - 3*w^2 - 3*w + 5],
[379, 379, w^5 - 3*w^4 - 4*w^3 + 9*w^2 + 5*w - 2],
[397, 397, 2*w^5 - 8*w^4 + 19*w^2 - 7*w - 7],
[397, 397, -3*w^5 + 11*w^4 + 2*w^3 - 23*w^2 + 6*w + 7],
[397, 397, -2*w^5 + 7*w^4 + 3*w^3 - 15*w^2 - 3*w + 4],
[397, 397, 3*w^5 - 9*w^4 - 7*w^3 + 15*w^2 + 5*w - 3],
[433, 433, -w^5 + 2*w^4 + 6*w^3 - 5*w^2 - 8*w],
[433, 433, 4*w^5 - 12*w^4 - 10*w^3 + 23*w^2 + 5*w - 7],
[433, 433, 2*w^5 - 6*w^4 - 5*w^3 + 10*w^2 + 5*w + 2],
[433, 433, 3*w^5 - 10*w^4 - 5*w^3 + 21*w^2 - w - 9],
[433, 433, w^5 - 4*w^4 + 9*w^2 - 6],
[433, 433, 2*w^5 - 8*w^4 + w^3 + 15*w^2 - 5*w - 6],
[449, 449, w^4 - 3*w^3 - w^2 + 3*w - 3],
[449, 449, 2*w^5 - 8*w^4 + 19*w^2 - 5*w - 6],
[449, 449, -w^5 + 3*w^4 + 4*w^3 - 9*w^2 - 6*w + 3],
[449, 449, 3*w^5 - 10*w^4 - 5*w^3 + 21*w^2 - 2*w - 5],
[467, 467, 3*w^5 - 12*w^4 + 28*w^2 - 7*w - 10],
[467, 467, -3*w^5 + 9*w^4 + 8*w^3 - 19*w^2 - 3*w + 4],
[487, 487, -2*w^5 + 7*w^4 + 3*w^3 - 15*w^2 - 3*w + 5],
[487, 487, -4*w^5 + 13*w^4 + 8*w^3 - 27*w^2 - 4*w + 7],
[487, 487, 2*w^5 - 8*w^4 + 19*w^2 - 7*w - 8],
[487, 487, -w^5 + 4*w^4 + w^3 - 12*w^2 + 2*w + 5],
[503, 503, w^5 - 3*w^4 - 3*w^3 + 8*w^2 - 6],
[503, 503, w^4 - 3*w^3 - 3*w^2 + 6*w],
[521, 521, -w^5 + 2*w^4 + 5*w^3 - 2*w^2 - 6*w + 2],
[521, 521, -2*w^5 + 7*w^4 + 2*w^3 - 13*w^2],
[523, 523, -3*w^5 + 12*w^4 - w^3 - 25*w^2 + 8*w + 7],
[523, 523, -2*w^5 + 7*w^4 + w^3 - 11*w^2 + 4*w + 4],
[523, 523, 3*w^5 - 9*w^4 - 7*w^3 + 16*w^2 + 2*w],
[523, 523, -w^5 + w^4 + 8*w^3 - w^2 - 8*w + 1],
[541, 541, w^4 - 4*w^3 + 6*w],
[541, 541, -2*w^4 + 7*w^3 + w^2 - 10*w],
[541, 541, -4*w^5 + 12*w^4 + 11*w^3 - 26*w^2 - 6*w + 8],
[541, 541, -w^5 + w^4 + 8*w^3 - 8*w - 4],
[557, 557, 2*w^5 - 8*w^4 + w^3 + 16*w^2 - 8*w - 6],
[557, 557, w^4 - 2*w^3 - 4*w^2 + 4],
[577, 577, w^5 - 5*w^4 + 3*w^3 + 11*w^2 - 4*w - 5],
[577, 577, 4*w^5 - 12*w^4 - 10*w^3 + 23*w^2 + 6*w - 5],
[577, 577, w^5 - w^4 - 9*w^3 + 3*w^2 + 12*w - 1],
[577, 577, -2*w^5 + 7*w^4 + 3*w^3 - 15*w^2 - 3*w + 3],
[593, 593, -3*w^5 + 8*w^4 + 10*w^3 - 13*w^2 - 9*w + 2],
[593, 593, -2*w^5 + 7*w^4 + w^3 - 10*w^2 + w - 1],
[593, 593, -2*w^5 + 5*w^4 + 9*w^3 - 12*w^2 - 9*w + 5],
[593, 593, -w^5 + 2*w^4 + 6*w^3 - 5*w^2 - 5*w + 2],
[613, 613, w^5 - 5*w^4 + 3*w^3 + 11*w^2 - 8*w - 2],
[613, 613, 4*w^5 - 12*w^4 - 10*w^3 + 23*w^2 + 6*w - 6],
[613, 613, 3*w^5 - 11*w^4 - 3*w^3 + 25*w^2 - 8],
[613, 613, 3*w^5 - 10*w^4 - 4*w^3 + 18*w^2 - 4],
[631, 631, w^4 - 4*w^3 + 2*w^2 + 4*w - 4],
[631, 631, -2*w^5 + 7*w^4 + 2*w^3 - 14*w^2 + 3*w + 7],
[631, 631, -3*w^5 + 10*w^4 + 4*w^3 - 17*w^2 + 3],
[631, 631, -2*w^5 + 9*w^4 - 3*w^3 - 21*w^2 + 9*w + 8],
[647, 647, w^4 - 3*w^3 - 2*w^2 + 5*w - 3],
[647, 647, -w^4 + 4*w^3 - 9*w + 1],
[683, 683, 3*w^5 - 10*w^4 - 4*w^3 + 19*w^2 - 3*w - 2],
[683, 683, -5*w^5 + 15*w^4 + 13*w^3 - 30*w^2 - 8*w + 6],
[701, 701, 2*w^5 - 8*w^4 + w^3 + 16*w^2 - 5*w - 6],
[701, 701, 4*w^5 - 14*w^4 - 4*w^3 + 27*w^2 - 4*w - 8],
[719, 719, 2*w^5 - 5*w^4 - 9*w^3 + 12*w^2 + 10*w - 3],
[719, 719, -4*w^5 + 12*w^4 + 10*w^3 - 22*w^2 - 6*w + 3],
[719, 719, -w^4 + 3*w^3 + 2*w^2 - 2*w - 3],
[719, 719, 2*w^4 - 6*w^3 - 2*w^2 + 5*w],
[739, 739, w^4 - 2*w^3 - 6*w^2 + 4*w + 3],
[739, 739, w^5 - 4*w^4 - w^3 + 13*w^2 - w - 9],
[739, 739, w^3 - 4*w^2 + 7],
[739, 739, -w^3 + 4*w^2 + w - 4],
[757, 757, 4*w^5 - 13*w^4 - 7*w^3 + 24*w^2 + 2*w - 4],
[757, 757, 2*w^5 - 8*w^4 + w^3 + 17*w^2 - 9*w - 6],
[773, 773, w^5 - 4*w^4 + w^3 + 8*w^2 - 6*w - 5],
[773, 773, 5*w^5 - 17*w^4 - 8*w^3 + 37*w^2 - 11],
[773, 773, 3*w^5 - 7*w^4 - 13*w^3 + 12*w^2 + 10*w - 2],
[773, 773, -w^5 + 4*w^4 + w^3 - 11*w^2 + 3],
[773, 773, -3*w^5 + 10*w^4 + 6*w^3 - 23*w^2 - 3*w + 6],
[773, 773, -2*w^5 + 10*w^4 - 5*w^3 - 26*w^2 + 13*w + 14],
[809, 809, 2*w^5 - 5*w^4 - 7*w^3 + 7*w^2 + 3*w + 2],
[809, 809, w^5 - 5*w^4 + 4*w^3 + 9*w^2 - 8*w - 5],
[811, 811, -2*w^5 + 5*w^4 + 7*w^3 - 7*w^2 - 4*w - 2],
[811, 811, 3*w^5 - 10*w^4 - 6*w^3 + 23*w^2 + 4*w - 6],
[827, 827, -w^5 + 3*w^4 + 3*w^3 - 6*w^2 - 5*w + 3],
[827, 827, w^5 - 4*w^4 + 10*w^2 - 5*w - 6],
[881, 881, -2*w^4 + 7*w^3 + 2*w^2 - 12*w - 1],
[881, 881, -w^5 + 2*w^4 + 7*w^3 - 8*w^2 - 9*w + 5],
[881, 881, 3*w^5 - 9*w^4 - 9*w^3 + 21*w^2 + 7*w - 5],
[881, 881, -w^5 + 2*w^4 + 5*w^3 - 2*w^2 - 3*w - 2],
[881, 881, 2*w^5 - 8*w^4 + 20*w^2 - 8*w - 9],
[881, 881, 2*w^5 - 7*w^4 - 2*w^3 + 13*w^2 - 2*w - 5],
[883, 883, -4*w^5 + 12*w^4 + 9*w^3 - 20*w^2 - 4*w + 2],
[883, 883, 2*w^3 - 5*w^2 - 7*w + 7],
[883, 883, -4*w^5 + 15*w^4 - 26*w^2 + 12*w + 1],
[883, 883, -w^5 + 4*w^4 - 2*w^3 - 4*w^2 + 6*w + 1],
[883, 883, 4*w^5 - 12*w^4 - 10*w^3 + 23*w^2 + 4*w - 3],
[883, 883, -4*w^5 + 13*w^4 + 6*w^3 - 22*w^2 + 4],
[919, 919, w^4 - 4*w^3 + w^2 + 4*w - 3],
[919, 919, 4*w^5 - 13*w^4 - 7*w^3 + 25*w^2 + 3*w - 7],
[971, 971, -w^5 + 3*w^4 + 2*w^3 - 4*w^2 - w - 4],
[971, 971, w^5 - 2*w^4 - 6*w^3 + 4*w^2 + 8*w - 2],
[971, 971, 2*w^5 - 5*w^4 - 7*w^3 + 6*w^2 + 7*w - 1],
[971, 971, w^5 - 3*w^4 - 2*w^3 + 4*w^2 + w - 3],
[991, 991, 2*w^4 - 6*w^3 - 5*w^2 + 11*w + 1],
[991, 991, -w^5 + w^4 + 9*w^3 - 3*w^2 - 11*w + 2],
[991, 991, w^4 - 3*w^3 - w^2 + w - 2],
[991, 991, -2*w^4 + 6*w^3 + 4*w^2 - 7*w - 1],
[1061, 1061, -w - 3],
[1061, 1061, 3*w^5 - 12*w^4 + w^3 + 24*w^2 - 6*w - 8],
[1061, 1061, -4*w^5 + 12*w^4 + 11*w^3 - 26*w^2 - 7*w + 6],
[1061, 1061, -w^5 + 4*w^4 + w^3 - 12*w^2 + 3],
[1061, 1061, w^5 - 3*w^4 - 3*w^3 + 7*w^2 + 3*w - 6],
[1061, 1061, 4*w^5 - 16*w^4 + w^3 + 35*w^2 - 14*w - 10],
[1063, 1063, 4*w^5 - 12*w^4 - 10*w^3 + 22*w^2 + 7*w - 4],
[1063, 1063, 3*w^5 - 10*w^4 - 5*w^3 + 21*w^2 - 2*w - 7],
[1097, 1097, -4*w^5 + 15*w^4 + 2*w^3 - 32*w^2 + 5*w + 11],
[1097, 1097, -3*w^5 + 10*w^4 + 4*w^3 - 17*w^2 + 4],
[1097, 1097, -w^5 + 2*w^4 + 4*w^3 + w^2 - 4*w - 6],
[1097, 1097, w^5 - w^4 - 8*w^3 + w^2 + 9*w],
[1117, 1117, -2*w^5 + 9*w^4 - 3*w^3 - 21*w^2 + 11*w + 7],
[1117, 1117, 2*w^5 - 8*w^4 + 17*w^2 - w - 4],
[1117, 1117, -3*w^5 + 10*w^4 + 4*w^3 - 18*w^2 + 2*w + 6],
[1117, 1117, -2*w^5 + 6*w^4 + 4*w^3 - 9*w^2 - 2],
[1151, 1151, 2*w^5 - 4*w^4 - 9*w^3 + 3*w^2 + 3*w + 2],
[1151, 1151, -5*w^5 + 18*w^4 + 5*w^3 - 40*w^2 + 6*w + 17],
[1151, 1151, 4*w^5 - 12*w^4 - 10*w^3 + 24*w^2 + 4*w - 7],
[1151, 1151, w^5 - 4*w^4 + w^3 + 4*w^2 + 2*w + 2],
[1151, 1151, -w^5 + 6*w^4 - 5*w^3 - 16*w^2 + 9*w + 8],
[1151, 1151, w^5 - 11*w^3 - 2*w^2 + 12*w + 5],
[1153, 1153, -2*w^5 + 7*w^4 + 3*w^3 - 18*w^2 + 3*w + 9],
[1153, 1153, 3*w^5 - 10*w^4 - 5*w^3 + 21*w^2 - 3],
[1153, 1153, w^5 - 4*w^4 + 11*w^2 - 5*w - 6],
[1153, 1153, -w^5 + 3*w^4 + 2*w^3 - 5*w^2 + 5],
[1187, 1187, -2*w^5 + 7*w^4 + 3*w^3 - 18*w^2 + 4*w + 8],
[1187, 1187, 3*w^5 - 9*w^4 - 9*w^3 + 23*w^2 + 3*w - 12],
[1187, 1187, -w^4 + 3*w^3 + 4*w^2 - 8*w - 4],
[1187, 1187, -2*w^4 + 6*w^3 + 6*w^2 - 11*w - 3],
[1223, 1223, -4*w^5 + 13*w^4 + 7*w^3 - 24*w^2 - 4*w + 4],
[1223, 1223, 2*w^5 - 8*w^4 + 18*w^2 - 2*w - 5],
[1277, 1277, 2*w^5 - 9*w^4 + 3*w^3 + 21*w^2 - 11*w - 8],
[1277, 1277, -w^5 + 5*w^4 - 3*w^3 - 13*w^2 + 10*w + 7],
[1277, 1277, 2*w^5 - 8*w^4 + 17*w^2 - 2*w - 5],
[1277, 1277, 2*w^5 - 8*w^4 + 17*w^2 - w - 5],
[1277, 1277, -3*w^5 + 12*w^4 - 28*w^2 + 8*w + 11],
[1277, 1277, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 + 2*w + 5],
[1279, 1279, 2*w^5 - 9*w^4 + 3*w^3 + 21*w^2 - 9*w - 9],
[1279, 1279, -2*w^5 + 5*w^4 + 9*w^3 - 13*w^2 - 8*w + 6],
[1279, 1279, w^5 - 5*w^4 + 3*w^3 + 12*w^2 - 6*w - 6],
[1279, 1279, -2*w^4 + 6*w^3 + 3*w^2 - 7*w],
[1297, 1297, -2*w^5 + 6*w^4 + 6*w^3 - 15*w^2 - 4*w + 4],
[1297, 1297, -2*w^5 + 7*w^4 + 2*w^3 - 14*w^2 + 5*w + 4],
[1367, 1367, -3*w^5 + 8*w^4 + 10*w^3 - 14*w^2 - 8*w + 4],
[1367, 1367, -w^5 + 2*w^4 + 5*w^3 - 3*w^2 - 2*w - 3],
[1367, 1367, -2*w^5 + 9*w^4 - 4*w^3 - 18*w^2 + 10*w + 7],
[1367, 1367, -4*w^5 + 14*w^4 + 4*w^3 - 27*w^2 + 2*w + 6],
[1367, 1367, -2*w^5 + 4*w^4 + 11*w^3 - 8*w^2 - 10*w + 1],
[1367, 1367, w^5 - 5*w^4 + 4*w^3 + 9*w^2 - 10*w - 5],
[1369, 37, 2*w^5 - 5*w^4 - 9*w^3 + 12*w^2 + 8*w - 4],
[1369, 37, -3*w^5 + 11*w^4 + 3*w^3 - 25*w^2 + 2*w + 7],
[1423, 1423, 2*w^5 - 9*w^4 + 3*w^3 + 20*w^2 - 9*w - 7],
[1423, 1423, -3*w^5 + 12*w^4 - 27*w^2 + 5*w + 10],
[1439, 1439, 5*w^5 - 17*w^4 - 6*w^3 + 31*w^2 - 7],
[1439, 1439, 2*w^5 - 8*w^4 + 19*w^2 - 7*w - 9],
[1439, 1439, 2*w^5 - 7*w^4 - 3*w^3 + 15*w^2 + 3*w - 6],
[1439, 1439, -5*w^5 + 14*w^4 + 15*w^3 - 25*w^2 - 12*w + 5],
[1459, 1459, -4*w^5 + 14*w^4 + 4*w^3 - 26*w^2 + 3*w + 1],
[1459, 1459, 4*w^5 - 13*w^4 - 7*w^3 + 25*w^2 - w - 5],
[1459, 1459, 3*w^5 - 8*w^4 - 11*w^3 + 17*w^2 + 7*w - 3],
[1459, 1459, w^5 - 5*w^4 + w^3 + 17*w^2 - 3*w - 6],
[1493, 1493, -4*w^5 + 12*w^4 + 10*w^3 - 23*w^2 - 4*w + 5],
[1493, 1493, w^5 - 4*w^4 + 2*w^3 + 4*w^2 - 6*w + 1],
[1493, 1493, -2*w^5 + 7*w^4 + 3*w^3 - 17*w^2 + 3*w + 7],
[1493, 1493, -2*w^5 + 6*w^4 + 6*w^3 - 13*w^2 - 7*w + 4],
[1511, 1511, -2*w^5 + 6*w^4 + 6*w^3 - 13*w^2 - 8*w + 4],
[1511, 1511, -4*w^5 + 13*w^4 + 8*w^3 - 28*w^2 - 2*w + 6],
[1531, 1531, -3*w^5 + 10*w^4 + 6*w^3 - 25*w^2 + w + 14],
[1531, 1531, -3*w^5 + 13*w^4 - 3*w^3 - 30*w^2 + 12*w + 12],
[1549, 1549, -2*w^5 + 7*w^4 + 3*w^3 - 15*w^2 - 5*w + 8],
[1549, 1549, w^5 - 3*w^4 - w^3 + w^2 + 5],
[1549, 1549, 3*w^5 - 9*w^4 - 7*w^3 + 15*w^2 + 4*w - 3],
[1549, 1549, -w^5 + 7*w^4 - 9*w^3 - 15*w^2 + 16*w + 3],
[1567, 1567, -3*w^5 + 11*w^4 + 2*w^3 - 22*w^2 + w + 6],
[1567, 1567, w^5 - w^4 - 8*w^3 + 11*w + 1],
[1583, 1583, -w^5 + 2*w^4 + 6*w^3 - 5*w^2 - 9*w + 2],
[1583, 1583, 2*w^5 - 7*w^4 - 3*w^3 + 16*w^2 + 3*w - 7],
[1583, 1583, -w^5 + 4*w^4 - 11*w^2 + 6*w + 7],
[1583, 1583, w^5 - 4*w^4 + 11*w^2 - 6*w - 5],
[1601, 1601, w^2 - w - 5],
[1601, 1601, -3*w^5 + 13*w^4 - 3*w^3 - 30*w^2 + 11*w + 11],
[1601, 1601, w^5 - 6*w^4 + 6*w^3 + 12*w^2 - 9*w - 2],
[1601, 1601, -w^5 + 3*w^4 + 2*w^3 - 4*w^2 + w - 3],
[1601, 1601, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 5*w - 4],
[1601, 1601, -2*w^5 + 7*w^4 + 3*w^3 - 17*w^2 + w + 4],
[1619, 1619, 3*w^4 - 8*w^3 - 9*w^2 + 10*w + 5],
[1619, 1619, -3*w^5 + 12*w^4 - 27*w^2 + 8*w + 7],
[1619, 1619, w^5 - 6*w^4 + 7*w^3 + 10*w^2 - 12*w - 1],
[1619, 1619, 5*w^5 - 18*w^4 - 6*w^3 + 41*w^2 - 13],
[1621, 1621, -4*w^5 + 12*w^4 + 10*w^3 - 23*w^2 - 4*w + 4],
[1621, 1621, -3*w^5 + 10*w^4 + 5*w^3 - 20*w^2 - 3*w + 3],
[1621, 1621, w^5 - 2*w^4 - 5*w^3 + 2*w^2 + 7*w - 2],
[1621, 1621, -w^5 + 4*w^4 - 2*w^3 - 4*w^2 + 6*w],
[1657, 1657, -3*w^5 + 9*w^4 + 8*w^3 - 20*w^2 - 2*w + 6],
[1657, 1657, 2*w^5 - 8*w^4 + w^3 + 17*w^2 - 8*w - 8],
[1693, 1693, -4*w^5 + 16*w^4 - w^3 - 34*w^2 + 11*w + 14],
[1693, 1693, w^5 - 6*w^4 + 5*w^3 + 15*w^2 - 7*w - 9],
[1709, 1709, 2*w^5 - 10*w^4 + 6*w^3 + 23*w^2 - 15*w - 10],
[1709, 1709, -2*w^5 + 7*w^4 + 4*w^3 - 18*w^2 - 6*w + 9],
[1709, 1709, -2*w^5 + 8*w^4 - w^3 - 16*w^2 + 10*w + 5],
[1709, 1709, w^5 - 4*w^4 + 2*w^3 + 5*w^2 - 8*w],
[1747, 1747, 2*w^5 - 5*w^4 - 7*w^3 + 7*w^2 + 6*w - 2],
[1747, 1747, -5*w^5 + 17*w^4 + 7*w^3 - 34*w^2 + 8],
[1783, 1783, 2*w^5 - 6*w^4 - 6*w^3 + 15*w^2 + 3*w - 3],
[1783, 1783, -w^4 + 3*w^3 + 3*w^2 - 5*w - 6],
[1801, 1801, -5*w^5 + 15*w^4 + 14*w^3 - 33*w^2 - 8*w + 11],
[1801, 1801, 4*w^5 - 15*w^4 - w^3 + 29*w^2 - 9*w - 8],
[1871, 1871, 3*w^5 - 10*w^4 - 6*w^3 + 23*w^2 + 6*w - 7],
[1871, 1871, -7*w^5 + 24*w^4 + 10*w^3 - 50*w^2 + w + 14],
[1873, 1873, 3*w^5 - 7*w^4 - 13*w^3 + 12*w^2 + 10*w - 4],
[1873, 1873, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 4*w + 1],
[1889, 1889, w^5 - 7*w^4 + 8*w^3 + 18*w^2 - 13*w - 8],
[1889, 1889, -3*w^5 + 13*w^4 - 4*w^3 - 26*w^2 + 11*w + 7],
[1907, 1907, -w^5 + 3*w^4 + w^2 + 3*w - 3],
[1907, 1907, -w^5 + 6*w^4 - 6*w^3 - 14*w^2 + 11*w + 6],
[1907, 1907, -w^5 + w^4 + 9*w^3 - 4*w^2 - 9*w + 3],
[1907, 1907, -6*w^5 + 19*w^4 + 12*w^3 - 36*w^2 - 3*w + 6],
[1979, 1979, -w^5 + 6*w^4 - 7*w^3 - 10*w^2 + 13*w + 4],
[1979, 1979, -4*w^5 + 14*w^4 + 5*w^3 - 29*w^2 + 8],
[1979, 1979, w^5 - w^4 - 10*w^3 + 7*w^2 + 11*w - 4],
[1979, 1979, w^5 - 5*w^4 + 2*w^3 + 14*w^2 - 6*w - 6],
[1997, 1997, w^5 - 6*w^4 + 7*w^3 + 10*w^2 - 12*w - 2],
[1997, 1997, -w^5 + 3*w^4 + w^3 - 3*w^2 + 4*w - 3],
[1999, 1999, -5*w^5 + 17*w^4 + 5*w^3 - 29*w^2 + 6*w + 1],
[1999, 1999, -5*w^5 + 15*w^4 + 11*w^3 - 25*w^2 - 2*w + 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x;
K := Rationals(); e := 1;

heckeEigenvaluesArray := [3, 3, -2, -8, 4, -10, 6, -2, 6, -6, 1, -8, -16, -6, -2, 6, 2, 10, 2, 0, -8, -6, -14, -16, -8, 8, -16, -8, -4, -8, 12, 12, -24, 0, 20, -10, 10, -14, -6, 0, -4, -8, 16, 6, -2, -6, 18, -18, -12, 28, -10, 2, 24, -24, 10, -14, -30, -16, -16, -20, 0, 8, 28, -30, -10, 20, 8, 36, 8, -2, -14, -26, -2, -34, -2, 6, -2, 22, -22, 38, 14, -6, 34, 12, 28, -16, 12, 16, 24, 24, 0, 10, -42, -12, -20, -44, 32, -14, -26, 22, -10, -10, 2, -42, -2, -26, -42, -14, -46, -14, 6, 22, 26, -38, -38, -16, 32, -16, -16, 8, 12, -28, 44, -6, -26, 28, -12, -40, 48, -12, 36, 12, 24, -6, 26, -22, 6, 26, -10, 6, 42, 10, 42, -52, 20, 28, -4, 30, 30, -34, -30, 46, 18, -4, -20, -28, -36, -28, 20, 24, -24, -48, 28, -48, 12, 0, -32, 48, 32, -38, 42, 6, 26, 62, 10, -20, -36, -22, -22, -2, -2, 34, 46, 26, 2, 20, -48, 28, 0, -48, 40, -14, -42, 50, -46, 20, -20, 56, 28, 24, 52, -38, 58, -30, 2, -42, -18, 24, -20, 64, -40, 38, 22, 0, 8, -40, 4, 24, 8, 30, 10, 72, -20, 0, -24, 40, 64, -16, -16, 56, -52, 18, 66, -54, -54, -28, -8, -28, -36, 22, -62, -38, -2, -68, -8, 16, 32, -8, 72, -42, 54, -42, 46, 30, -14, -4, 24, -76, 52, -18, -54, 34, -26, -38, -18, -34, -66, 46, 34, 34, -10, -12, -44, 40, -64, 22, -50, -40, 60, -34, -30, -78, 50, 28, 60, -36, -48, 24, 8, 20, -8, 78, 22, -72, -8];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {53,53,4*w^5 - 14*w^4 - 4*w^3 + 27*w^2 - 4*w - 6}>] := -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;