/*
  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, -6, 3, 10, -3, -3, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {51, 51, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - 3*w}>;

primesArray := [
[3, 3, w - 1],
[17, 17, -w^2 + 2*w + 1],
[17, 17, -w^3 + w^2 + 4*w],
[19, 19, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + w - 4],
[19, 19, w^2 - w - 1],
[37, 37, w^4 - 2*w^3 - 3*w^2 + 3*w + 2],
[37, 37, w^4 - 4*w^3 + w^2 + 7*w - 3],
[53, 53, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 9*w - 6],
[53, 53, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 3*w + 2],
[64, 2, -2],
[71, 71, -w^5 + 4*w^4 - w^3 - 8*w^2 + 3*w - 1],
[71, 71, 2*w^5 - 5*w^4 - 8*w^3 + 15*w^2 + 12*w - 6],
[71, 71, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 9*w - 7],
[73, 73, -2*w^5 + 6*w^4 + 5*w^3 - 16*w^2 - 7*w + 3],
[73, 73, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 6*w + 6],
[89, 89, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 8*w - 4],
[89, 89, w^5 - w^4 - 9*w^3 + 7*w^2 + 16*w - 6],
[89, 89, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + 3*w - 3],
[89, 89, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 10*w - 6],
[107, 107, w^5 - 2*w^4 - 7*w^3 + 10*w^2 + 12*w - 6],
[107, 107, 2*w^5 - 5*w^4 - 7*w^3 + 14*w^2 + 9*w - 5],
[107, 107, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 6*w - 3],
[107, 107, w^4 - 2*w^3 - 3*w^2 + 4*w + 1],
[109, 109, 2*w^5 - 5*w^4 - 8*w^3 + 15*w^2 + 13*w - 6],
[109, 109, -w^4 + 4*w^3 - w^2 - 9*w + 3],
[127, 127, w^4 - 3*w^3 + 5*w - 4],
[127, 127, -w^5 + 3*w^4 + w^3 - 6*w^2 + w + 3],
[127, 127, 2*w^5 - 5*w^4 - 8*w^3 + 16*w^2 + 11*w - 6],
[127, 127, w^5 - 4*w^4 + 2*w^3 + 7*w^2 - 7*w],
[163, 163, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 5],
[163, 163, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 2*w + 5],
[163, 163, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 8*w + 4],
[163, 163, w^5 - 4*w^4 + 10*w^2 - w - 4],
[179, 179, -w^5 + 3*w^4 + 3*w^3 - 7*w^2 - 7*w - 1],
[179, 179, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 5*w - 7],
[179, 179, -2*w^5 + 5*w^4 + 10*w^3 - 19*w^2 - 18*w + 10],
[179, 179, -2*w^5 + 5*w^4 + 9*w^3 - 17*w^2 - 14*w + 9],
[197, 197, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 3*w - 7],
[197, 197, -w^5 + 4*w^4 - w^3 - 7*w^2 + 3*w - 2],
[197, 197, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 3],
[197, 197, -2*w^5 + 8*w^4 - 2*w^3 - 17*w^2 + 10*w + 1],
[197, 197, -w^4 + 3*w^3 + 2*w^2 - 6*w - 2],
[197, 197, w^5 - 4*w^4 + w^3 + 9*w^2 - 4*w - 2],
[199, 199, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 10*w + 4],
[199, 199, -w^4 + 3*w^3 + 2*w^2 - 9*w + 1],
[233, 233, 2*w^5 - 4*w^4 - 12*w^3 + 17*w^2 + 19*w - 9],
[233, 233, -w^5 + w^4 + 8*w^3 - 6*w^2 - 13*w + 6],
[233, 233, 2*w^5 - 6*w^4 - 6*w^3 + 19*w^2 + 9*w - 7],
[233, 233, w^5 - 4*w^4 + 11*w^2 - 4*w - 2],
[251, 251, -2*w^5 + 5*w^4 + 10*w^3 - 18*w^2 - 18*w + 7],
[251, 251, w^3 - w^2 - 3*w - 2],
[251, 251, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 3*w - 3],
[251, 251, -w^5 + 4*w^4 - 11*w^2 + 4*w + 3],
[269, 269, w^5 - 3*w^4 - w^3 + 6*w^2 - 2*w - 3],
[269, 269, 2*w^5 - 7*w^4 - 3*w^3 + 20*w^2 + 3*w - 8],
[271, 271, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 2*w - 7],
[271, 271, -w^4 + 4*w^3 - w^2 - 10*w + 6],
[289, 17, -w^5 + 3*w^4 + w^3 - 5*w^2 - 2],
[289, 17, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 6*w - 4],
[307, 307, 2*w^5 - 5*w^4 - 10*w^3 + 19*w^2 + 16*w - 8],
[307, 307, w^5 - 4*w^4 + 2*w^3 + 7*w^2 - 6*w - 1],
[307, 307, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 3*w + 2],
[307, 307, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 7*w - 9],
[361, 19, 2*w^5 - 6*w^4 - 6*w^3 + 18*w^2 + 8*w - 5],
[361, 19, w^5 - 3*w^4 - w^3 + 5*w^2 + 3],
[379, 379, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 9*w - 5],
[379, 379, -2*w + 3],
[397, 397, -w^5 + 5*w^4 - 6*w^3 - 6*w^2 + 14*w - 5],
[397, 397, -w^5 + 3*w^4 + w^3 - 4*w^2 - w - 5],
[397, 397, 3*w^5 - 9*w^4 - 7*w^3 + 23*w^2 + 9*w - 6],
[397, 397, -w^4 + 5*w^3 - 2*w^2 - 10*w + 1],
[431, 431, -3*w^5 + 8*w^4 + 11*w^3 - 24*w^2 - 18*w + 9],
[431, 431, 2*w^5 - 6*w^4 - 3*w^3 + 13*w^2 + w - 3],
[431, 431, -3*w^5 + 8*w^4 + 10*w^3 - 23*w^2 - 14*w + 8],
[431, 431, -2*w^5 + 6*w^4 + 4*w^3 - 14*w^2 - 5*w],
[433, 433, -3*w^5 + 9*w^4 + 8*w^3 - 25*w^2 - 11*w + 9],
[433, 433, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 - 3*w + 7],
[433, 433, 2*w^5 - 5*w^4 - 7*w^3 + 13*w^2 + 11*w - 4],
[433, 433, -w^4 + 4*w^3 + w^2 - 10*w - 2],
[449, 449, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 + w + 3],
[449, 449, -w^4 + 4*w^3 - 2*w^2 - 5*w + 5],
[449, 449, -w^5 + 4*w^4 - w^3 - 8*w^2 + 3*w - 2],
[449, 449, -w^3 + 5*w + 3],
[449, 449, w^3 - 5*w],
[449, 449, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - w - 6],
[467, 467, -2*w^4 + 7*w^3 - 16*w + 7],
[467, 467, -2*w^4 + 6*w^3 + w^2 - 10*w + 3],
[487, 487, -w^5 + 2*w^4 + 6*w^3 - 10*w^2 - 8*w + 9],
[487, 487, 2*w^5 - 6*w^4 - 5*w^3 + 15*w^2 + 7*w - 3],
[487, 487, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 8*w - 6],
[487, 487, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 8*w + 1],
[503, 503, 2*w^5 - 7*w^4 - 3*w^3 + 19*w^2 + 3*w - 6],
[503, 503, 2*w^5 - 6*w^4 - 5*w^3 + 18*w^2 + 4*w - 6],
[503, 503, 2*w^5 - 5*w^4 - 9*w^3 + 17*w^2 + 13*w - 7],
[503, 503, -w^5 + 4*w^4 - 10*w^2 + 3*w - 1],
[521, 521, w^5 - 4*w^4 + 11*w^2 - w - 6],
[521, 521, w^5 - w^4 - 9*w^3 + 8*w^2 + 15*w - 6],
[521, 521, -w^5 + 4*w^4 - 2*w^3 - 8*w^2 + 9*w],
[521, 521, -2*w^5 + 4*w^4 + 11*w^3 - 15*w^2 - 17*w + 9],
[523, 523, 3*w^5 - 8*w^4 - 10*w^3 + 24*w^2 + 12*w - 10],
[523, 523, -w^4 + 3*w^3 - w^2 - 4*w + 5],
[523, 523, -2*w^5 + 4*w^4 + 11*w^3 - 14*w^2 - 17*w + 7],
[523, 523, -w^5 + 5*w^4 - 4*w^3 - 9*w^2 + 9*w + 1],
[577, 577, -3*w^5 + 8*w^4 + 10*w^3 - 22*w^2 - 14*w + 7],
[577, 577, -2*w^5 + 7*w^4 + w^3 - 15*w^2 + 2*w + 2],
[593, 593, 2*w^5 - 5*w^4 - 9*w^3 + 17*w^2 + 15*w - 10],
[593, 593, w^3 - w^2 - 2*w - 2],
[593, 593, -3*w^5 + 9*w^4 + 9*w^3 - 27*w^2 - 14*w + 10],
[593, 593, 2*w^5 - 6*w^4 - 6*w^3 + 19*w^2 + 8*w - 7],
[613, 613, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 12*w - 5],
[613, 613, w^5 - w^4 - 9*w^3 + 8*w^2 + 13*w - 7],
[613, 613, w^5 - 5*w^4 + 3*w^3 + 11*w^2 - 6*w - 2],
[613, 613, -w^4 + 3*w^3 + w^2 - 5*w + 4],
[613, 613, w^5 - w^4 - 9*w^3 + 7*w^2 + 14*w - 4],
[613, 613, w^5 - 2*w^4 - 7*w^3 + 11*w^2 + 12*w - 8],
[631, 631, 2*w^5 - 7*w^4 + 15*w^2 - 6*w - 2],
[631, 631, -3*w^5 + 8*w^4 + 10*w^3 - 23*w^2 - 12*w + 10],
[631, 631, w^5 - 3*w^4 + w^3 + 3*w^2 - 6*w + 3],
[631, 631, -3*w^5 + 8*w^4 + 13*w^3 - 29*w^2 - 21*w + 15],
[683, 683, -w^5 + 4*w^4 - w^3 - 8*w^2 + 2*w - 1],
[683, 683, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 11*w - 4],
[701, 701, -w^5 + 3*w^4 + w^3 - 7*w^2 + 3*w + 2],
[701, 701, 2*w^2 - 3*w - 4],
[719, 719, 2*w^5 - 5*w^4 - 8*w^3 + 16*w^2 + 13*w - 8],
[719, 719, w^5 - 5*w^4 + 4*w^3 + 10*w^2 - 10*w - 1],
[719, 719, w^5 - w^4 - 9*w^3 + 6*w^2 + 16*w - 2],
[719, 719, 2*w^5 - 5*w^4 - 9*w^3 + 18*w^2 + 15*w - 10],
[739, 739, -2*w^5 + 9*w^4 - 6*w^3 - 15*w^2 + 16*w - 7],
[739, 739, 4*w^5 - 9*w^4 - 19*w^3 + 30*w^2 + 28*w - 14],
[757, 757, 2*w^5 - 5*w^4 - 9*w^3 + 16*w^2 + 17*w - 7],
[757, 757, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 6*w - 6],
[757, 757, 2*w^5 - 6*w^4 - 7*w^3 + 20*w^2 + 11*w - 6],
[757, 757, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 1],
[773, 773, 2*w^5 - 8*w^4 + 3*w^3 + 16*w^2 - 11*w],
[773, 773, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 12*w - 6],
[809, 809, 2*w^5 - 6*w^4 - 3*w^3 + 13*w^2 + 2*w - 3],
[809, 809, -2*w^3 + 5*w^2 + 4*w - 6],
[811, 811, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],
[811, 811, 2*w^5 - 7*w^4 - w^3 + 17*w^2 - 3*w - 6],
[827, 827, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 4],
[827, 827, w^5 - 4*w^4 + w^3 + 10*w^2 - 5*w - 5],
[829, 829, 2*w^5 - 4*w^4 - 11*w^3 + 15*w^2 + 18*w - 9],
[829, 829, 2*w^5 - 6*w^4 - 7*w^3 + 19*w^2 + 13*w - 7],
[829, 829, -2*w^5 + 9*w^4 - 5*w^3 - 18*w^2 + 15*w],
[829, 829, 3*w^5 - 9*w^4 - 9*w^3 + 28*w^2 + 11*w - 11],
[829, 829, w^3 - 2*w^2 - 3*w - 1],
[829, 829, -2*w^5 + 4*w^4 + 12*w^3 - 17*w^2 - 20*w + 10],
[863, 863, -w^4 + 4*w^3 + w^2 - 11*w],
[863, 863, w^4 - 4*w^3 - w^2 + 11*w + 1],
[919, 919, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 4*w + 1],
[919, 919, -w^5 + 4*w^4 - w^3 - 8*w^2 + 6*w - 4],
[919, 919, -w^4 + 5*w^3 - w^2 - 12*w + 1],
[919, 919, -2*w^5 + 5*w^4 + 7*w^3 - 13*w^2 - 11*w + 6],
[937, 937, 2*w^5 - 6*w^4 - 3*w^3 + 12*w^2 + 3*w],
[937, 937, 2*w^5 - 4*w^4 - 10*w^3 + 12*w^2 + 16*w - 5],
[937, 937, w^5 - 4*w^4 + 8*w^2 + w + 1],
[937, 937, 2*w^4 - 8*w^3 + 2*w^2 + 16*w - 5],
[953, 953, -w^5 + 4*w^4 - 11*w^2 + 3*w + 7],
[953, 953, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 13*w - 7],
[953, 953, -w^5 + 3*w^4 + w^3 - 7*w^2 + 4*w + 1],
[953, 953, 2*w^5 - 8*w^4 + 3*w^3 + 16*w^2 - 11*w - 1],
[971, 971, w^4 - w^3 - 5*w^2 + 2*w + 4],
[971, 971, -w^4 + w^3 + 4*w^2 - w + 1],
[971, 971, -2*w^5 + 5*w^4 + 10*w^3 - 20*w^2 - 15*w + 9],
[971, 971, w^4 - 2*w^3 - 4*w^2 + 6*w + 1],
[991, 991, -w - 3],
[991, 991, -2*w^5 + 7*w^4 + 3*w^3 - 19*w^2 - 5*w + 6],
[991, 991, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 5*w + 7],
[991, 991, 2*w^5 - 7*w^4 - 2*w^3 + 16*w^2 + 2*w],
[1009, 1009, 2*w^4 - 6*w^3 - 2*w^2 + 11*w],
[1009, 1009, -w^5 + w^4 + 9*w^3 - 7*w^2 - 15*w + 6],
[1061, 1061, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 4*w + 2],
[1061, 1061, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + 2*w - 7],
[1063, 1063, 2*w^5 - 5*w^4 - 10*w^3 + 18*w^2 + 18*w - 9],
[1063, 1063, -2*w^5 + 5*w^4 + 8*w^3 - 15*w^2 - 14*w + 8],
[1063, 1063, -w^5 + 4*w^4 - 11*w^2 + 4*w + 5],
[1063, 1063, w^5 - 2*w^4 - 7*w^3 + 10*w^2 + 14*w - 5],
[1117, 1117, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 7*w + 5],
[1117, 1117, -w^5 + 2*w^4 + 3*w^3 - 2*w^2 - 3*w - 3],
[1117, 1117, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 5*w - 8],
[1117, 1117, w^5 - 4*w^4 + 3*w^3 + 4*w^2 - 7*w + 4],
[1151, 1151, -w^4 + 4*w^3 - 10*w - 1],
[1151, 1151, w^5 - 3*w^4 - w^3 + 8*w^2 - 3*w - 3],
[1151, 1151, 3*w^5 - 8*w^4 - 8*w^3 + 19*w^2 + 8*w - 3],
[1151, 1151, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 12*w - 5],
[1171, 1171, w^5 - 2*w^4 - 5*w^3 + 5*w^2 + 7*w + 1],
[1171, 1171, 3*w^5 - 10*w^4 - 4*w^3 + 25*w^2 + 3*w - 7],
[1171, 1171, -w^5 + 5*w^4 - 3*w^3 - 11*w^2 + 8*w + 3],
[1171, 1171, -2*w^5 + 5*w^4 + 7*w^3 - 16*w^2 - 7*w + 6],
[1187, 1187, -3*w^5 + 8*w^4 + 12*w^3 - 27*w^2 - 18*w + 11],
[1187, 1187, w^4 - 2*w^3 - 2*w^2 + 4*w - 3],
[1259, 1259, 2*w^5 - 6*w^4 - 5*w^3 + 16*w^2 + 9*w - 5],
[1259, 1259, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 9*w - 7],
[1277, 1277, -3*w^5 + 11*w^4 + w^3 - 25*w^2 + 2*w + 3],
[1277, 1277, -w^4 + w^3 + 5*w^2 - w - 6],
[1277, 1277, 2*w^5 - 6*w^4 - 4*w^3 + 13*w^2 + 5*w + 1],
[1277, 1277, w^4 - 5*w^3 + 3*w^2 + 9*w - 7],
[1277, 1277, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - 4*w - 1],
[1277, 1277, 3*w^5 - 9*w^4 - 6*w^3 + 22*w^2 + 5*w - 4],
[1279, 1279, -2*w^5 + 4*w^4 + 12*w^3 - 17*w^2 - 18*w + 11],
[1279, 1279, -3*w^5 + 11*w^4 - 23*w^2 + 5*w + 2],
[1279, 1279, w^3 + w^2 - 5*w - 2],
[1279, 1279, -2*w^5 + 6*w^4 + 7*w^3 - 21*w^2 - 9*w + 9],
[1279, 1279, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 9*w + 4],
[1279, 1279, -2*w^3 + 5*w^2 + 3*w - 8],
[1297, 1297, -2*w^5 + 4*w^4 + 11*w^3 - 14*w^2 - 19*w + 7],
[1297, 1297, -w^5 + w^4 + 10*w^3 - 9*w^2 - 19*w + 7],
[1331, 11, -3*w^5 + 7*w^4 + 16*w^3 - 26*w^2 - 28*w + 9],
[1331, 11, w^3 - w^2 - w - 3],
[1367, 1367, 2*w^5 - 6*w^4 - 4*w^3 + 13*w^2 + 5*w],
[1367, 1367, w^3 - 6*w],
[1367, 1367, 3*w^5 - 9*w^4 - 6*w^3 + 22*w^2 + 5*w - 5],
[1367, 1367, w^3 - 6*w - 2],
[1369, 37, 2*w^5 - 6*w^4 - 7*w^3 + 20*w^2 + 10*w - 8],
[1369, 37, 3*w^5 - 9*w^4 - 8*w^3 + 25*w^2 + 10*w - 8],
[1423, 1423, 3*w^5 - 10*w^4 - 5*w^3 + 26*w^2 + 6*w - 3],
[1423, 1423, -2*w^4 + 7*w^3 - w^2 - 13*w + 4],
[1459, 1459, -w^5 + 4*w^4 - 3*w^3 - 4*w^2 + 8*w - 5],
[1459, 1459, 2*w^5 - 5*w^4 - 6*w^3 + 11*w^2 + 8*w],
[1459, 1459, -2*w^5 + 7*w^4 + 4*w^3 - 22*w^2 - 2*w + 7],
[1459, 1459, -3*w^5 + 8*w^4 + 14*w^3 - 29*w^2 - 24*w + 11],
[1493, 1493, -3*w^5 + 9*w^4 + 9*w^3 - 25*w^2 - 16*w + 6],
[1493, 1493, -2*w^5 + 3*w^4 + 15*w^3 - 14*w^2 - 27*w + 9],
[1493, 1493, w^5 - 4*w^4 + 2*w^3 + 5*w^2 - 5*w + 3],
[1493, 1493, -3*w^5 + 8*w^4 + 9*w^3 - 21*w^2 - 11*w + 7],
[1531, 1531, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 + w + 4],
[1531, 1531, 2*w^5 - 6*w^4 - 6*w^3 + 17*w^2 + 11*w - 7],
[1549, 1549, w^5 - 4*w^4 - 3*w^3 + 14*w^2 + 7*w - 4],
[1549, 1549, -4*w^5 + 10*w^4 + 18*w^3 - 34*w^2 - 28*w + 15],
[1567, 1567, -3*w^5 + 6*w^4 + 17*w^3 - 23*w^2 - 25*w + 9],
[1567, 1567, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 12*w + 4],
[1583, 1583, w^5 - 2*w^4 - 8*w^3 + 11*w^2 + 15*w - 7],
[1583, 1583, w^5 - 4*w^4 - w^3 + 13*w^2 - w - 7],
[1601, 1601, -w^4 + 4*w^3 - 3*w^2 - 6*w + 7],
[1601, 1601, 3*w^5 - 8*w^4 - 9*w^3 + 22*w^2 + 10*w - 8],
[1619, 1619, 2*w^5 - 5*w^4 - 11*w^3 + 19*w^2 + 21*w - 6],
[1619, 1619, w^5 - 2*w^4 - 8*w^3 + 13*w^2 + 12*w - 12],
[1619, 1619, -4*w^5 + 9*w^4 + 20*w^3 - 32*w^2 - 31*w + 13],
[1619, 1619, w^5 - 2*w^4 - 6*w^3 + 10*w^2 + 10*w - 12],
[1637, 1637, -w^5 + 2*w^4 + 4*w^3 - 4*w^2 - 5*w - 3],
[1637, 1637, w^5 - 4*w^4 + 2*w^3 + 6*w^2 - 5*w + 4],
[1637, 1637, 4*w^5 - 10*w^4 - 17*w^3 + 31*w^2 + 29*w - 12],
[1637, 1637, -w^5 + 5*w^4 - 5*w^3 - 8*w^2 + 15*w - 1],
[1657, 1657, w^4 - 2*w^3 - w^2 + w - 4],
[1657, 1657, 2*w^5 - 5*w^4 - 8*w^3 + 17*w^2 + 9*w - 7],
[1693, 1693, w^5 - 2*w^4 - 9*w^3 + 14*w^2 + 15*w - 8],
[1693, 1693, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + w],
[1709, 1709, 2*w^5 - 7*w^4 - 3*w^3 + 18*w^2 + 4*w - 3],
[1709, 1709, 3*w^5 - 8*w^4 - 11*w^3 + 25*w^2 + 14*w - 9],
[1709, 1709, -w^5 + w^4 + 11*w^3 - 11*w^2 - 20*w + 9],
[1709, 1709, -w^5 + w^4 + 7*w^3 - 3*w^2 - 12*w + 1],
[1747, 1747, -w^5 + 5*w^4 - 3*w^3 - 9*w^2 + 7*w - 4],
[1747, 1747, -3*w^5 + 11*w^4 - 24*w^2 + 5*w + 3],
[1783, 1783, -w^4 + 3*w^3 + 2*w^2 - 5*w - 4],
[1783, 1783, -2*w^5 + 7*w^4 + 2*w^3 - 18*w^2 - w + 4],
[1871, 1871, -w^5 + 3*w^4 + 4*w^3 - 11*w^2 - 9*w + 6],
[1871, 1871, 2*w^5 - 8*w^4 + w^3 + 17*w^2 - 3*w - 2],
[1871, 1871, 2*w^5 - 6*w^4 - 5*w^3 + 16*w^2 + 9*w - 6],
[1871, 1871, -3*w^5 + 11*w^4 - w^3 - 23*w^2 + 12*w - 3],
[1873, 1873, -w^5 + w^4 + 8*w^3 - 5*w^2 - 11*w + 4],
[1873, 1873, 3*w^5 - 6*w^4 - 15*w^3 + 19*w^2 + 22*w - 9],
[1889, 1889, 2*w^5 - 5*w^4 - 8*w^3 + 14*w^2 + 13*w - 5],
[1889, 1889, -2*w^5 + 5*w^4 + 8*w^3 - 15*w^2 - 12*w + 8],
[1889, 1889, 2*w^5 - 7*w^4 - w^3 + 16*w^2 - 3*w - 5],
[1889, 1889, w^5 - 4*w^4 + w^3 + 8*w^2 - 4*w - 3],
[1907, 1907, -3*w^5 + 6*w^4 + 18*w^3 - 25*w^2 - 28*w + 12],
[1907, 1907, 2*w^5 - 5*w^4 - 7*w^3 + 14*w^2 + 8*w - 1],
[1979, 1979, 2*w^4 - 9*w^3 + 5*w^2 + 18*w - 11],
[1979, 1979, 4*w^5 - 10*w^4 - 14*w^3 + 27*w^2 + 20*w - 8],
[1997, 1997, 3*w^5 - 8*w^4 - 11*w^3 + 25*w^2 + 14*w - 10],
[1997, 1997, -4*w^5 + 13*w^4 + 5*w^3 - 32*w^2 + w + 4],
[1997, 1997, 2*w^5 - 7*w^4 - 3*w^3 + 18*w^2 + 4*w - 4],
[1997, 1997, -4*w^5 + 11*w^4 + 15*w^3 - 35*w^2 - 24*w + 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^7 - 12*x^6 + 30*x^5 + 115*x^4 - 607*x^3 + 841*x^2 - 279*x - 113;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, -1, e, -7651/25513*e^6 + 78129/25513*e^5 - 90885/25513*e^4 - 1032306/25513*e^3 + 2792322/25513*e^2 - 1596439/25513*e - 380647/25513, 2452/25513*e^6 - 24532/25513*e^5 + 24075/25513*e^4 + 337587/25513*e^3 - 831282/25513*e^2 + 337419/25513*e + 169168/25513, -3960/25513*e^6 + 39661/25513*e^5 - 39464/25513*e^4 - 536091/25513*e^3 + 1329958/25513*e^2 - 662802/25513*e - 213608/25513, 10851/25513*e^6 - 110436/25513*e^5 + 126383/25513*e^4 + 1460099/25513*e^3 - 3916773/25513*e^2 + 2173012/25513*e + 592431/25513, 12374/25513*e^6 - 131417/25513*e^5 + 189855/25513*e^4 + 1681218/25513*e^3 - 5160557/25513*e^2 + 3308624/25513*e + 789985/25513, -392/25513*e^6 + 3256/25513*e^5 + 5474/25513*e^4 - 72241/25513*e^3 + 2335/25513*e^2 + 301158/25513*e - 87685/25513, 8092/25513*e^6 - 81792/25513*e^5 + 91105/25513*e^4 + 1084875/25513*e^3 - 2900190/25513*e^2 + 1608440/25513*e + 565399/25513, -14876/25513*e^6 + 158447/25513*e^5 - 229633/25513*e^4 - 2034658/25513*e^3 + 6247397/25513*e^2 - 3940080/25513*e - 1112351/25513, 4451/25513*e^6 - 45822/25513*e^5 + 55387/25513*e^4 + 604513/25513*e^3 - 1667871/25513*e^2 + 917814/25513*e + 449506/25513, -2870/25513*e^6 + 31128/25513*e^5 - 49218/25513*e^4 - 389497/25513*e^3 + 1264499/25513*e^2 - 942304/25513*e + 69651/25513, 19058/25513*e^6 - 203076/25513*e^5 + 296977/25513*e^4 + 2599295/25513*e^3 - 8047674/25513*e^2 + 5103216/25513*e + 1438749/25513, -15168/25513*e^6 + 156707/25513*e^5 - 192753/25513*e^4 - 2075193/25513*e^3 + 5764129/25513*e^2 - 3025335/25513*e - 1327361/25513, -23517/25513*e^6 + 240113/25513*e^5 - 279358/25513*e^4 - 3181852/25513*e^3 + 8619575/25513*e^2 - 4694456/25513*e - 1520887/25513, 73/25513*e^6 + 435/25513*e^5 - 9220/25513*e^4 + 16512/25513*e^3 + 95304/25513*e^2 - 375386/25513*e + 347152/25513, 3799/25513*e^6 - 34679/25513*e^5 + 9821/25513*e^4 + 505965/25513*e^3 - 851997/25513*e^2 + 118598/25513*e + 208119/25513, -6239/25513*e^6 + 59632/25513*e^5 - 41353/25513*e^4 - 825460/25513*e^3 + 1770941/25513*e^2 - 710994/25513*e - 35384/25513, -1607/25513*e^6 + 18034/25513*e^5 - 36786/25513*e^4 - 194685/25513*e^3 + 834254/25513*e^2 - 881553/25513*e + 139016/25513, 3595/25513*e^6 - 41836/25513*e^5 + 85564/25513*e^4 + 504557/25513*e^3 - 1959556/25513*e^2 + 1366134/25513*e + 391323/25513, 16591/25513*e^6 - 172692/25513*e^5 + 224819/25513*e^4 + 2264606/25513*e^3 - 6547823/25513*e^2 + 3598386/25513*e + 1601214/25513, -22340/25513*e^6 + 228254/25513*e^5 - 260258/25513*e^4 - 3057169/25513*e^3 + 8130876/25513*e^2 - 4086272/25513*e - 1762987/25513, -25911/25513*e^6 + 270932/25513*e^5 - 360736/25513*e^4 - 3525542/25513*e^3 + 10291269/25513*e^2 - 6064412/25513*e - 1929738/25513, -11292/25513*e^6 + 114099/25513*e^5 - 126603/25513*e^4 - 1512668/25513*e^3 + 4024641/25513*e^2 - 2210526/25513*e - 700644/25513, 8402/25513*e^6 - 92177/25513*e^5 + 157848/25513*e^4 + 1127035/25513*e^3 - 3943774/25513*e^2 + 2926312/25513*e + 591656/25513, 9665/25513*e^6 - 105271/25513*e^5 + 170280/25513*e^4 + 1321847/25513*e^3 - 4374019/25513*e^2 + 2961550/25513*e + 712047/25513, -1485/25513*e^6 + 18062/25513*e^5 - 40312/25513*e^4 - 223358/25513*e^3 + 926077/25513*e^2 - 510059/25513*e - 590363/25513, 15083/25513*e^6 - 157563/25513*e^5 + 209430/25513*e^4 + 2040589/25513*e^3 - 5972608/25513*e^2 + 3732237/25513*e + 995488/25513, 392/823*e^6 - 4079/823*e^5 + 5225/823*e^4 + 54135/823*e^3 - 153767/823*e^2 + 73307/823*e + 45712/823, -5468/25513*e^6 + 54790/25513*e^5 - 54853/25513*e^4 - 734595/25513*e^3 + 1828634/25513*e^2 - 1013698/25513*e - 104970/25513, -9811/25513*e^6 + 104401/25513*e^5 - 151840/25513*e^4 - 1354871/25513*e^3 + 4183411/25513*e^2 - 2287317/25513*e - 1227760/25513, -19067/25513*e^6 + 196382/25513*e^5 - 233980/25513*e^4 - 2612864/25513*e^3 + 7076216/25513*e^2 - 3706494/25513*e - 1204051/25513, -7456/25513*e^6 + 78592/25513*e^5 - 103631/25513*e^4 - 1044467/25513*e^3 + 2953936/25513*e^2 - 1396227/25513*e - 584283/25513, 5213/25513*e^6 - 57358/25513*e^5 + 104884/25513*e^4 + 656296/25513*e^3 - 2505097/25513*e^2 + 2445553/25513*e + 53332/25513, 25788/25513*e^6 - 268869/25513*e^5 + 354253/25513*e^4 + 3493177/25513*e^3 - 10156528/25513*e^2 + 6069222/25513*e + 1761057/25513, -15667/25513*e^6 + 154083/25513*e^5 - 135670/25513*e^4 - 2096146/25513*e^3 + 4904020/25513*e^2 - 2413007/25513*e - 175371/25513, 6170/25513*e^6 - 68431/25513*e^5 + 116122/25513*e^4 + 848996/25513*e^3 - 2849040/25513*e^2 + 2030412/25513*e + 295451/25513, 26104/25513*e^6 - 266287/25513*e^5 + 302459/25513*e^4 + 3544383/25513*e^3 - 9395884/25513*e^2 + 5027218/25513*e + 1562814/25513, -675/25513*e^6 + 8210/25513*e^5 - 20643/25513*e^4 - 73694/25513*e^3 + 413986/25513*e^2 - 716592/25513*e + 279023/25513, -12456/25513*e^6 + 124288/25513*e^5 - 117638/25513*e^4 - 1711299/25513*e^3 + 4119308/25513*e^2 - 1587542/25513*e - 978978/25513, -2329/25513*e^6 + 22469/25513*e^5 - 17592/25513*e^4 - 305222/25513*e^3 + 696541/25513*e^2 - 265690/25513*e - 77026/25513, -371/25513*e^6 + 10371/25513*e^5 - 64980/25513*e^4 - 40580/25513*e^3 + 1035942/25513*e^2 - 1376054/25513*e + 70546/25513, 10054/25513*e^6 - 102254/25513*e^5 + 109266/25513*e^4 + 1389565/25513*e^3 - 3518071/25513*e^2 + 1595186/25513*e + 580313/25513, -18091/25513*e^6 + 196606/25513*e^5 - 313214/25513*e^4 - 2485066/25513*e^3 + 8116956/25513*e^2 - 5301369/25513*e - 1732379/25513, 30432/25513*e^6 - 310046/25513*e^5 + 351363/25513*e^4 + 4142044/25513*e^3 - 11031066/25513*e^2 + 5643686/25513*e + 2302873/25513, -9317/25513*e^6 + 91967/25513*e^5 - 80377/25513*e^4 - 1256413/25513*e^3 + 2910676/25513*e^2 - 1426333/25513*e + 69486/25513, -20785/25513*e^6 + 216900/25513*e^5 - 284706/25513*e^4 - 2847335/25513*e^3 + 8234925/25513*e^2 - 4481032/25513*e - 1708229/25513, 16393/25513*e^6 - 166882/25513*e^5 + 182025/25513*e^4 + 2246731/25513*e^3 - 5774615/25513*e^2 + 2817715/25513*e + 1126197/25513, -4159/25513*e^6 + 47562/25513*e^5 - 92267/25513*e^4 - 563978/25513*e^3 + 2151139/25513*e^2 - 1858072/25513*e - 362061/25513, -52789/25513*e^6 + 548334/25513*e^5 - 705235/25513*e^4 - 7165313/25513*e^3 + 20513256/25513*e^2 - 11854132/25513*e - 3872002/25513, -9934/25513*e^6 + 106464/25513*e^5 - 158323/25513*e^4 - 1361723/25513*e^3 + 4190587/25513*e^2 - 2614176/25513*e - 299382/25513, -18996/25513*e^6 + 200999/25513*e^5 - 288731/25513*e^4 - 2565350/25513*e^3 + 7879778/25513*e^2 - 5064156/25513*e - 994674/25513, 15379/25513*e^6 - 164187/25513*e^5 + 238099/25513*e^4 + 2121172/25513*e^3 - 6497648/25513*e^2 + 3965628/25513*e + 1562587/25513, -49183/25513*e^6 + 509010/25513*e^5 - 637137/25513*e^4 - 6678189/25513*e^3 + 18765874/25513*e^2 - 10800392/25513*e - 3016316/25513, -3194/25513*e^6 + 19761/25513*e^5 + 75582/25513*e^4 - 444260/25513*e^3 - 490063/25513*e^2 + 1987560/25513*e - 283206/25513, -11121/25513*e^6 + 113720/25513*e^5 - 124435/25513*e^4 - 1535500/25513*e^3 + 3908879/25513*e^2 - 1883055/25513*e - 567566/25513, -20469/25513*e^6 + 219482/25513*e^5 - 336500/25513*e^4 - 2770616/25513*e^3 + 8919030/25513*e^2 - 5931244/25513*e - 1523777/25513, 5825/25513*e^6 - 61400/25513*e^5 + 81759/25513*e^4 + 813598/25513*e^3 - 2295006/25513*e^2 + 1075654/25513*e + 371162/25513, -33548/25513*e^6 + 345300/25513*e^5 - 413547/25513*e^4 - 4596271/25513*e^3 + 12539534/25513*e^2 - 6244701/25513*e - 2825714/25513, -360/25513*e^6 + 12883/25513*e^5 - 82446/25513*e^4 - 58013/25513*e^3 + 1248116/25513*e^2 - 1662935/25513*e + 381831/25513, 14298/25513*e^6 - 148960/25513*e^5 + 184856/25513*e^4 + 1988147/25513*e^3 - 5486244/25513*e^2 + 2822895/25513*e + 1350786/25513, -3592/25513*e^6 + 35563/25513*e^5 - 30024/25513*e^4 - 500034/25513*e^3 + 1152299/25513*e^2 - 454006/25513*e - 401521/25513, -32557/25513*e^6 + 339672/25513*e^5 - 444698/25513*e^4 - 4445858/25513*e^3 + 12860679/25513*e^2 - 7258964/25513*e - 3124389/25513, 22846/25513*e^6 - 240267/25513*e^5 + 324264/25513*e^4 + 3122693/25513*e^3 - 9137358/25513*e^2 + 5534208/25513*e + 1769304/25513, 21995/25513*e^6 - 221223/25513*e^5 + 225895/25513*e^4 + 2996258/25513*e^3 - 7474790/25513*e^2 + 3412157/25513*e + 1022282/25513, -22392/25513*e^6 + 234934/25513*e^5 - 321492/25513*e^4 - 3042020/25513*e^3 + 9043666/25513*e^2 - 5362585/25513*e - 1365109/25513, -35710/25513*e^6 + 375754/25513*e^5 - 520033/25513*e^4 - 4836808/25513*e^3 + 14460290/25513*e^2 - 9065940/25513*e - 1897127/25513, 34512/25513*e^6 - 371010/25513*e^5 + 571387/25513*e^4 + 4680464/25513*e^3 - 15056224/25513*e^2 + 10160481/25513*e + 2236126/25513, 39142/25513*e^6 - 408426/25513*e^5 + 530423/25513*e^4 + 5342241/25513*e^3 - 15259140/25513*e^2 + 8778029/25513*e + 2655446/25513, -37677/25513*e^6 + 381158/25513*e^5 - 409648/25513*e^4 - 5115019/25513*e^3 + 13149431/25513*e^2 - 6635393/25513*e - 1861027/25513, 8372/25513*e^6 - 80473/25513*e^5 + 61682/25513*e^4 + 1081805/25513*e^3 - 2453558/25513*e^2 + 1535471/25513*e - 250345/25513, 36692/25513*e^6 - 362563/25513*e^5 + 322262/25513*e^4 + 4999165/25513*e^3 - 11717374/25513*e^2 + 4626442/25513*e + 2190332/25513, 31940/25513*e^6 - 325175/25513*e^5 + 366752/25513*e^4 + 4366061/25513*e^3 - 11606281/25513*e^2 + 5586374/25513*e + 2704495/25513, 45411/25513*e^6 - 479762/25513*e^5 + 667942/25513*e^4 + 6212931/25513*e^3 - 18571323/25513*e^2 + 11083968/25513*e + 3609370/25513, 2602/25513*e^6 - 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680257/25513*e^5 + 791153/25513*e^4 + 9027557/25513*e^3 - 24390079/25513*e^2 + 13089050/25513*e + 3798001/25513, 82185/25513*e^6 - 860709/25513*e^5 + 1147604/25513*e^4 + 11242177/25513*e^3 - 32768242/25513*e^2 + 18479616/25513*e + 7213319/25513, 60201/25513*e^6 - 636974/25513*e^5 + 904243/25513*e^4 + 8202973/25513*e^3 - 24902687/25513*e^2 + 15061221/25513*e + 4792951/25513, -25656/25513*e^6 + 273500/25513*e^5 - 385254/25513*e^4 - 3600321/25513*e^3 + 10661576/25513*e^2 - 5251123/25513*e - 3153750/25513, 13259/25513*e^6 - 145016/25513*e^5 + 245835/25513*e^4 + 1765366/25513*e^3 - 6234576/25513*e^2 + 5137709/25513*e + 1276856/25513, 38183/25513*e^6 - 393171/25513*e^5 + 473654/25513*e^4 + 5180543/25513*e^3 - 14155913/25513*e^2 + 8210894/25513*e + 1739779/25513, -52361/25513*e^6 + 546341/25513*e^5 - 707567/25513*e^4 - 7191874/25513*e^3 + 20544290/25513*e^2 - 11185337/25513*e - 4595544/25513, -3291/25513*e^6 + 43997/25513*e^5 - 129901/25513*e^4 - 471443/25513*e^3 + 2607025/25513*e^2 - 2204187/25513*e - 650826/25513, -20596/25513*e^6 + 229909/25513*e^5 - 408532/25513*e^4 - 2868542/25513*e^3 + 10164131/25513*e^2 - 6742901/25513*e - 3065067/25513, 59432/25513*e^6 - 610801/25513*e^5 + 733657/25513*e^4 + 8081106/25513*e^3 - 22224383/25513*e^2 + 12213095/25513*e + 4719669/25513, 8100/25513*e^6 - 73007/25513*e^5 - 7414/25513*e^4 + 1190484/25513*e^3 - 1396012/25513*e^2 - 1376479/25513*e + 529700/25513, -98287/25513*e^6 + 1005909/25513*e^5 - 1172414/25513*e^4 - 13381846/25513*e^3 + 36165616/25513*e^2 - 19054855/25513*e - 7160117/25513, 78778/25513*e^6 - 829286/25513*e^5 + 1132309/25513*e^4 + 10782940/25513*e^3 - 31867554/25513*e^2 + 18901789/25513*e + 6174417/25513, 60772/25513*e^6 - 631824/25513*e^5 + 827931/25513*e^4 + 8204913/25513*e^3 - 23848275/25513*e^2 + 14220194/25513*e + 3447235/25513, -24575/25513*e^6 + 258273/25513*e^5 - 357524/25513*e^4 - 3339731/25513*e^3 + 10058380/25513*e^2 - 5817761/25513*e - 2661306/25513, -39185/25513*e^6 + 396287/25513*e^5 - 425037/25513*e^4 - 5364549/25513*e^3 + 13852211/25513*e^2 - 6144360/25513*e - 3538299/25513, 11975/25513*e^6 - 139037/25513*e^5 + 278344/25513*e^4 + 1666458/25513*e^3 - 6429730/25513*e^2 + 5044799/25513*e + 1074773/25513, 13309/25513*e^6 - 122001/25513*e^5 + 31921/25513*e^4 + 1832245/25513*e^3 - 2994853/25513*e^2 - 359705/25513*e + 766716/25513, -47844/25513*e^6 + 515591/25513*e^5 - 808002/25513*e^4 - 6462342/25513*e^3 + 21093444/25513*e^2 - 14591135/25513*e - 3349874/25513, 26647/25513*e^6 - 279128/25513*e^5 + 379616/25513*e^4 + 3572143/25513*e^3 - 10595561/25513*e^2 + 6966751/25513*e + 992626/25513, 24479/25513*e^6 - 261641/25513*e^5 + 391667/25513*e^4 + 3322560/25513*e^3 - 10479033/25513*e^2 + 6709492/25513*e + 2171226/25513, -22743/25513*e^6 + 254511/25513*e^5 - 466935/25513*e^4 - 3086464/25513*e^3 + 11339779/25513*e^2 - 8269164/25513*e - 2901834/25513, -29240/25513*e^6 + 317848/25513*e^5 - 513797/25513*e^4 - 3969233/25513*e^3 + 13190488/25513*e^2 - 9225821/25513*e - 2442885/25513, -71022/25513*e^6 + 761219/25513*e^5 - 1164077/25513*e^4 - 9617842/25513*e^3 + 30875551/25513*e^2 - 20639836/25513*e - 4645433/25513, -67385/25513*e^6 + 708100/25513*e^5 - 964291/25513*e^4 - 9203041/25513*e^3 + 27181772/25513*e^2 - 16045772/25513*e - 4600986/25513, 45124/25513*e^6 - 466444/25513*e^5 + 601789/25513*e^4 + 6018352/25513*e^3 - 17457520/25513*e^2 + 11188739/25513*e + 1838079/25513, -3268/25513*e^6 + 21417/25513*e^5 + 74793/25513*e^4 - 445271/25513*e^3 - 562907/25513*e^2 + 1182957/25513*e + 129927/25513, 31445/25513*e^6 - 310650/25513*e^5 + 285280/25513*e^4 + 4232078/25513*e^3 - 10209034/25513*e^2 + 4846564/25513*e + 1274579/25513, -70500/25513*e^6 + 715750/25513*e^5 - 786849/25513*e^4 - 9621743/25513*e^3 + 25095960/25513*e^2 - 12277673/25513*e - 4480897/25513, 34429/25513*e^6 - 350535/25513*e^5 + 378465/25513*e^4 + 4767936/25513*e^3 - 12171524/25513*e^2 + 5471409/25513*e + 2552288/25513, 68349/25513*e^6 - 708297/25513*e^5 + 892514/25513*e^4 + 9338260/25513*e^3 - 26381772/25513*e^2 + 14399718/25513*e + 5873847/25513, -102803/25513*e^6 + 1060081/25513*e^5 - 1291587/25513*e^4 - 13999314/25513*e^3 + 38732101/25513*e^2 - 21459630/25513*e - 7584266/25513, -103851/25513*e^6 + 1082844/25513*e^5 - 1422741/25513*e^4 - 14133612/25513*e^3 + 41068878/25513*e^2 - 23743649/25513*e - 8316453/25513, 121638/25513*e^6 - 1256098/25513*e^5 + 1550675/25513*e^4 + 16509025/25513*e^3 - 45955327/25513*e^2 + 26158173/25513*e + 7425366/25513, 1119/25513*e^6 - 18146/25513*e^5 + 50890/25513*e^4 + 258351/25513*e^3 - 920903/25513*e^2 - 196215/25513*e + 558869/25513, -58610/25513*e^6 + 626883/25513*e^5 - 951062/25513*e^4 - 7913350/25513*e^3 + 25269722/25513*e^2 - 16833224/25513*e - 4146205/25513, -1621/25513*e^6 - 3718/25513*e^5 + 154757/25513*e^4 - 181775/25513*e^3 - 2040431/25513*e^2 + 2779384/25513*e + 628832/25513, -121259/25513*e^6 + 1254512/25513*e^5 - 1558701/25513*e^4 - 16541427/25513*e^3 + 46219459/25513*e^2 - 25343303/25513*e - 9113417/25513, 8174/25513*e^6 - 74663/25513*e^5 + 18888/25513*e^4 + 1114956/25513*e^3 - 1833428/25513*e^2 - 393285/25513*e + 1060548/25513, -19739/25513*e^6 + 198319/25513*e^5 - 224596/25513*e^4 - 2554470/25513*e^3 + 6989101/25513*e^2 - 4684556/25513*e - 501505/25513, 57416/25513*e^6 - 604990/25513*e^5 + 838348/25513*e^4 + 7873593/25513*e^3 - 23531761/25513*e^2 + 13054833/25513*e + 5934352/25513, -14847/25513*e^6 + 174347/25513*e^5 - 373093/25513*e^4 - 2050466/25513*e^3 + 8376974/25513*e^2 - 6306041/25513*e - 1551105/25513, -50528/25513*e^6 + 514975/25513*e^5 - 602865/25513*e^4 - 6750004/25513*e^3 + 18461026/25513*e^2 - 11461744/25513*e - 1947998/25513, -88974/25513*e^6 + 922306/25513*e^5 - 1157586/25513*e^4 - 12139968/25513*e^3 + 34110170/25513*e^2 - 19006275/25513*e - 6510146/25513, 8794/25513*e^6 - 95433/25513*e^5 + 152374/25513*e^4 + 1173763/25513*e^3 - 3895083/25513*e^2 + 3441570/25513*e - 188101/25513, -93762/25513*e^6 + 983944/25513*e^5 - 1345855/25513*e^4 - 12674270/25513*e^3 + 37708688/25513*e^2 - 23583123/25513*e - 5516425/25513, -59783/25513*e^6 + 604865/25513*e^5 - 649483/25513*e^4 - 8100037/25513*e^3 + 20846624/25513*e^2 - 10910029/25513*e - 3041756/25513, -70477/25513*e^6 + 744196/25513*e^5 - 1041389/25513*e^4 - 9621084/25513*e^3 + 28942103/25513*e^2 - 16978150/25513*e - 5868749/25513, 6624/25513*e^6 - 73764/25513*e^5 + 144407/25513*e^4 + 827617/25513*e^3 - 3376453/25513*e^2 + 3273581/25513*e + 572081/25513, 49912/25513*e^6 - 528082/25513*e^5 + 764545/25513*e^4 + 6680219/25513*e^3 - 20775395/25513*e^2 + 14464424/25513*e + 2936424/25513, 12340/25513*e^6 - 136862/25513*e^5 + 232244/25513*e^4 + 1723505/25513*e^3 - 5749106/25513*e^2 + 3448512/25513*e + 820519/25513, -54116/25513*e^6 + 567687/25513*e^5 - 771444/25513*e^4 - 7388581/25513*e^3 + 21896194/25513*e^2 - 12528011/25513*e - 4778347/25513, 20595/25513*e^6 - 227818/25513*e^5 + 398523/25513*e^4 + 2756478/25513*e^3 - 9733463/25513*e^2 + 7910108/25513*e + 1146487/25513, -76972/25513*e^6 + 803351/25513*e^5 - 1068233/25513*e^4 - 10432803/25513*e^3 + 30620327/25513*e^2 - 18483311/25513*e - 6113954/25513, 101650/25513*e^6 - 1047380/25513*e^5 + 1283086/25513*e^4 + 13808763/25513*e^3 - 38450773/25513*e^2 + 21413038/25513*e + 7948886/25513, -27633/25513*e^6 + 274301/25513*e^5 - 247394/25513*e^4 - 3800061/25513*e^3 + 8911979/25513*e^2 - 3216155/25513*e - 1944076/25513, -38743/25513*e^6 + 390533/25513*e^5 - 414808/25513*e^4 - 5250942/25513*e^3 + 13492266/25513*e^2 - 6610715/25513*e - 1996253/25513, 26916/25513*e^6 - 305834/25513*e^5 + 597276/25513*e^4 + 3586506/25513*e^3 - 13907410/25513*e^2 + 11747490/25513*e + 1396377/25513, -133055/25513*e^6 + 1376442/25513*e^5 - 1729292/25513*e^4 - 18023056/25513*e^3 + 50899506/25513*e^2 - 29677834/25513*e - 8075284/25513, -7866/25513*e^6 + 93973/25513*e^5 - 201780/25513*e^4 - 1143846/25513*e^3 + 4559662/25513*e^2 - 3072556/25513*e - 1197579/25513, 7133/25513*e^6 - 92050/25513*e^5 + 263953/25513*e^4 + 897664/25513*e^3 - 5215705/25513*e^2 + 5786723/25513*e + 415122/25513, 10379/25513*e^6 - 118491/25513*e^5 + 224092/25513*e^4 + 1454340/25513*e^3 - 5383302/25513*e^2 + 3850852/25513*e + 1397509/25513, -148878/25513*e^6 + 1550565/25513*e^5 - 2023151/25513*e^4 - 20175807/25513*e^3 + 58261253/25513*e^2 - 34899260/25513*e - 9582808/25513, 59419/25513*e^6 - 609131/25513*e^5 + 731105/25513*e^4 + 8027489/25513*e^3 - 22162020/25513*e^2 + 12589246/25513*e + 3658297/25513, 65122/25513*e^6 - 670559/25513*e^5 + 816390/25513*e^4 + 8844247/25513*e^3 - 24574362/25513*e^2 + 13880205/25513*e + 4886403/25513, 70130/25513*e^6 - 732983/25513*e^5 + 961495/25513*e^4 + 9540149/25513*e^3 - 27450194/25513*e^2 + 16776000/25513*e + 3459489/25513, 54756/25513*e^6 - 579251/25513*e^5 + 798954/25513*e^4 + 7596602/25513*e^3 - 22442548/25513*e^2 + 12025911/25513*e + 5723864/25513, 92877/25513*e^6 - 995858/25513*e^5 + 1493979/25513*e^4 + 12743200/25513*e^3 - 39976872/25513*e^2 + 24917650/25513*e + 7835984/25513, -32932/25513*e^6 + 332894/25513*e^5 - 371123/25513*e^4 - 4373408/25513*e^3 + 11588239/25513*e^2 - 6951211/25513*e - 829119/25513, 35517/25513*e^6 - 354886/25513*e^5 + 348693/25513*e^4 + 4843480/25513*e^3 - 11885907/25513*e^2 + 4745404/25513*e + 2442642/25513, 6007/25513*e^6 - 59267/25513*e^5 + 40948/25513*e^4 + 875385/25513*e^3 - 1713847/25513*e^2 - 159406/25513*e + 815525/25513, -30779/25513*e^6 + 321259/25513*e^5 - 431257/25513*e^4 - 4146440/25513*e^3 + 12293358/25513*e^2 - 7784824/25513*e - 1982242/25513, 16385/25513*e^6 - 150154/25513*e^5 + 25414/25513*e^4 + 2268687/25513*e^3 - 3349791/25513*e^2 - 397025/25513*e + 294454/25513, -73976/25513*e^6 + 789400/25513*e^5 - 1161096/25513*e^4 - 10133983/25513*e^3 + 31398851/25513*e^2 - 18929859/25513*e - 6867051/25513, 4181/25513*e^6 - 42538/25513*e^5 + 57335/25513*e^4 + 478086/25513*e^3 - 1599226/25513*e^2 + 2355856/25513*e - 418584/25513, 20109/25513*e^6 - 232112/25513*e^5 + 483671/25513*e^4 + 2687090/25513*e^3 - 11115169/25513*e^2 + 9421935/25513*e + 1752530/25513, 30786/25513*e^6 - 310383/25513*e^5 + 322729/25513*e^4 + 4191011/25513*e^3 - 10562616/25513*e^2 + 5278261/25513*e + 1456691/25513, 63797/25513*e^6 - 680901/25513*e^5 + 999816/25513*e^4 + 8717542/25513*e^3 - 26961242/25513*e^2 + 16944006/25513*e + 5061814/25513, 99020/25513*e^6 - 1033345/25513*e^5 + 1365371/25513*e^4 + 13398411/25513*e^3 - 39055880/25513*e^2 + 24224205/25513*e + 5850508/25513, 2749/25513*e^6 - 33247/25513*e^5 + 62753/25513*e^4 + 453695/25513*e^3 - 1480834/25513*e^2 + 245532/25513*e + 1659840/25513, 66219/25513*e^6 - 693729/25513*e^5 + 927725/25513*e^4 + 9060925/25513*e^3 - 26531925/25513*e^2 + 15176220/25513*e + 6557586/25513, 17414/25513*e^6 - 184214/25513*e^5 + 272553/25513*e^4 + 2314809/25513*e^3 - 7402920/25513*e^2 + 5337385/25513*e + 1286828/25513, -38673/25513*e^6 + 422754/25513*e^5 - 683672/25513*e^4 - 5315492/25513*e^3 + 17481900/25513*e^2 - 11878257/25513*e - 3144170/25513, -99998/25513*e^6 + 1062816/25513*e^5 - 1535772/25513*e^4 - 13699311/25513*e^3 + 42116627/25513*e^2 - 25244438/25513*e - 9491009/25513, -19053/25513*e^6 + 192621/25513*e^5 - 195906/25513*e^4 - 2702313/25513*e^3 + 6761776/25513*e^2 - 1575980/25513*e - 2459257/25513, 17718/25513*e^6 - 182053/25513*e^5 + 253729/25513*e^4 + 2194845/25513*e^3 - 7036094/25513*e^2 + 6565885/25513*e + 83344/25513];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w - 1}>] := 1;
ALEigenvalues[ideal<ZF | {17, 17, -w^2 + 2*w + 1}>] := 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;