/*
  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 | {19,19,w - 2}>;

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^10 - 5*x^9 - 47*x^8 + 255*x^7 + 607*x^6 - 4151*x^5 - 249*x^4 + 21345*x^3 - 23248*x^2 + 620*x + 40;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-15045/48802816*e^9 - 75109/24401408*e^8 + 1485861/48802816*e^7 + 1064641/6100352*e^6 - 47913955/48802816*e^5 - 77509073/24401408*e^4 + 613146543/48802816*e^3 + 222899961/12200704*e^2 - 659963821/12200704*e + 9073247/6100352, e, 154809/97605632*e^9 - 320103/48802816*e^8 - 8089113/97605632*e^7 + 4019579/12200704*e^6 + 131841935/97605632*e^5 - 253987147/48802816*e^4 - 584492683/97605632*e^3 + 612840579/24401408*e^2 - 304448063/24401408*e + 41938821/12200704, 203845/48802816*e^9 - 477531/24401408*e^8 - 9725477/48802816*e^7 + 6089087/6100352*e^6 + 131494499/48802816*e^5 - 395910511/24401408*e^4 - 177918383/48802816*e^3 + 1014050951/12200704*e^2 - 1039628883/12200704*e + 23235617/6100352, 1, 144813/97605632*e^9 - 638835/48802816*e^8 - 5819789/97605632*e^7 + 8335447/12200704*e^6 + 34661435/97605632*e^5 - 560065831/48802816*e^4 + 892122649/97605632*e^3 + 1511114687/24401408*e^2 - 2125375659/24401408*e - 29496983/12200704, -33635/97605632*e^9 + 350077/48802816*e^8 + 262531/97605632*e^7 - 4549913/12200704*e^6 + 40766667/97605632*e^5 + 303621641/48802816*e^4 - 873512119/97605632*e^3 - 820586609/24401408*e^2 + 1190376645/24401408*e + 110023833/12200704, -1209529/97605632*e^9 + 2683623/48802816*e^8 + 58792729/97605632*e^7 - 34171195/12200704*e^6 - 828381071/97605632*e^5 + 2222901963/48802816*e^4 + 1739502987/97605632*e^3 - 5716193283/24401408*e^2 + 5292181823/24401408*e - 25647493/12200704, 114785/97605632*e^9 + 36801/48802816*e^8 - 6975425/97605632*e^7 - 423181/12200704*e^6 + 148093223/97605632*e^5 + 25400125/48802816*e^4 - 1261728131/97605632*e^3 - 53474501/24401408*e^2 + 853334953/24401408*e - 91403251/12200704, -114785/97605632*e^9 - 36801/48802816*e^8 + 6975425/97605632*e^7 + 423181/12200704*e^6 - 148093223/97605632*e^5 - 25400125/48802816*e^4 + 1261728131/97605632*e^3 + 77875909/24401408*e^2 - 853334953/24401408*e - 67205901/12200704, -949931/97605632*e^9 + 2081589/48802816*e^8 + 45997515/97605632*e^7 - 26258929/12200704*e^6 - 645626413/97605632*e^5 + 1688236257/48802816*e^4 + 1369897505/97605632*e^3 - 4282954505/24401408*e^2 + 4020141117/24401408*e - 36741391/12200704, 442747/48802816*e^9 - 1058405/24401408*e^8 - 21493659/48802816*e^7 + 13334913/6100352*e^6 + 302777309/48802816*e^5 - 854005585/24401408*e^4 - 654766993/48802816*e^3 + 2150560185/12200704*e^2 - 1882278765/12200704*e + 363039/6100352, 4633/1525088*e^9 - 10411/762544*e^8 - 217089/1525088*e^7 + 131135/190636*e^6 + 2930639/1525088*e^5 - 8365263/762544*e^4 - 5850739/1525088*e^3 + 20518319/381272*e^2 - 17497771/381272*e + 1374241/190636, 1377879/97605632*e^9 - 3185289/48802816*e^8 - 66441975/97605632*e^7 + 40554101/12200704*e^6 + 920808929/97605632*e^5 - 2629450213/48802816*e^4 - 1700805797/97605632*e^3 + 6697146285/24401408*e^2 - 6286006577/24401408*e + 108148299/12200704, -979959/97605632*e^9 + 2757225/48802816*e^8 + 44841879/97605632*e^7 - 35017557/12200704*e^6 - 532194625/97605632*e^5 + 2273702213/48802816*e^4 - 783953275/97605632*e^3 - 5847543693/24401408*e^2 + 6998851729/24401408*e - 49844843/12200704, -1179501/97605632*e^9 + 2007987/48802816*e^8 + 59948365/97605632*e^7 - 25412567/12200704*e^6 - 941812859/97605632*e^5 + 1637436007/48802816*e^4 + 3893353767/97605632*e^3 - 4151604095/24401408*e^2 + 2313471211/24401408*e - 12544041/12200704, 1269709/97605632*e^9 - 2383187/48802816*e^8 - 64736173/97605632*e^7 + 29912631/12200704*e^6 + 1020036891/97605632*e^5 - 1912865671/48802816*e^4 - 4192089159/97605632*e^3 + 4824593439/24401408*e^2 - 2701129355/24401408*e + 38157321/12200704, 552503/97605632*e^9 - 1593897/48802816*e^8 - 25270743/97605632*e^7 + 20513621/12200704*e^6 + 297650433/97605632*e^5 - 1351886853/48802816*e^4 + 536285883/97605632*e^3 + 3539216589/24401408*e^2 - 4204633425/24401408*e - 80631381/12200704, -201883/97605632*e^9 + 45765/48802816*e^8 + 11748027/97605632*e^7 - 114497/12200704*e^6 - 247374781/97605632*e^5 - 28646607/48802816*e^4 + 2242540785/97605632*e^3 + 166866599/24401408*e^2 - 1824446003/24401408*e + 173290945/12200704, -328513/48802816*e^9 + 544863/24401408*e^8 + 16760993/48802816*e^7 - 6828787/6100352*e^6 - 265365319/48802816*e^5 + 435507555/24401408*e^4 + 1137894435/48802816*e^3 - 1088488347/12200704*e^2 + 543774199/12200704*e - 28738029/6100352, 239391/48802816*e^9 - 856385/24401408*e^8 - 9883583/48802816*e^7 + 11204429/6100352*e^6 + 72704345/48802816*e^5 - 754681021/24401408*e^4 + 1158439683/48802816*e^3 + 2031352005/12200704*e^2 - 2973850025/12200704*e - 5848077/6100352, -318329/24401408*e^9 + 514407/12200704*e^8 + 16538585/24401408*e^7 - 6425979/3050176*e^6 - 273175503/24401408*e^5 + 407755339/12200704*e^4 + 1368106955/24401408*e^3 - 1020147715/6100352*e^2 + 215998143/6100352*e + 33443195/3050176, -26419/3050176*e^9 + 80805/1525088*e^8 + 1179571/3050176*e^7 - 1034843/381272*e^6 - 12892069/3050176*e^5 + 67784625/1525088*e^4 - 45068039/3050176*e^3 - 175649781/762544*e^2 + 221513229/762544*e - 263759/381272, -109445/12200704*e^9 + 201211/6100352*e^8 + 5605669/12200704*e^7 - 2512223/1525088*e^6 - 89704227/12200704*e^5 + 159200719/6100352*e^4 + 395532463/12200704*e^3 - 395575495/3050176*e^2 + 183732179/3050176*e + 5119519/1525088, 254111/48802816*e^9 - 485889/24401408*e^8 - 13545151/48802816*e^7 + 6111597/6100352*e^6 + 232742873/48802816*e^5 - 391119101/24401408*e^4 - 1242184701/48802816*e^3 + 987533061/12200704*e^2 - 179411753/12200704*e - 27234189/6100352, 15045/48802816*e^9 + 75109/24401408*e^8 - 1485861/48802816*e^7 - 1064641/6100352*e^6 + 47913955/48802816*e^5 + 77509073/24401408*e^4 - 588745135/48802816*e^3 - 235100665/12200704*e^2 + 544057133/12200704*e + 106833441/6100352, 347459/48802816*e^9 - 852509/24401408*e^8 - 15425635/48802816*e^7 + 11027577/6100352*e^6 + 169189973/48802816*e^5 - 732449065/24401408*e^4 + 494972951/48802816*e^3 + 1935498641/12200704*e^2 - 2598882661/12200704*e + 15980615/6100352, 33991/24401408*e^9 - 232537/12200704*e^8 - 150503/24401408*e^7 + 3134149/3050176*e^6 - 48261391/24401408*e^5 - 219432437/12200704*e^4 + 1019720843/24401408*e^3 + 618009981/6100352*e^2 - 1382943937/6100352*e + 17821627/3050176, -340389/24401408*e^9 + 667835/12200704*e^8 + 16862213/24401408*e^7 - 8522303/3050176*e^6 - 252093187/24401408*e^5 + 554819791/12200704*e^4 + 852014415/24401408*e^3 - 1428725351/6100352*e^2 + 960222387/6100352*e + 35074431/3050176, 830343/48802816*e^9 - 1844953/24401408*e^8 - 40242215/48802816*e^7 + 23326501/6100352*e^6 + 567118897/48802816*e^5 - 1506966773/24401408*e^4 - 1236524597/48802816*e^3 + 3866495997/12200704*e^2 - 3527640449/12200704*e - 22192581/6100352, 30767/3050176*e^9 - 93209/1525088*e^8 - 1327839/3050176*e^7 + 1210189/381272*e^6 + 12740713/3050176*e^5 - 80782181/1525088*e^4 + 86693475/3050176*e^3 + 215413333/762544*e^2 - 295266145/762544*e - 844773/381272, 269481/48802816*e^9 - 706167/24401408*e^8 - 12855305/48802816*e^7 + 9075147/6100352*e^6 + 168532255/48802816*e^5 - 599662875/24401408*e^4 - 67853403/48802816*e^3 + 1597752787/12200704*e^2 - 1629520975/12200704*e - 78897739/6100352, 61541/24401408*e^9 - 166971/12200704*e^8 - 2482117/24401408*e^7 + 2101231/3050176*e^6 + 19884867/24401408*e^5 - 135250511/12200704*e^4 + 206729329/24401408*e^3 + 343220615/6100352*e^2 - 557193971/6100352*e + 14190209/3050176, -1283739/48802816*e^9 + 2840261/24401408*e^8 + 62411707/48802816*e^7 - 36234561/6100352*e^6 - 882044349/48802816*e^5 + 2362012721/24401408*e^4 + 1933610993/48802816*e^3 - 6087548633/12200704*e^2 + 5460702669/12200704*e - 4238527/6100352, 12443/762544*e^9 - 12597/190636*e^8 - 626433/762544*e^7 + 631011/190636*e^6 + 9615317/762544*e^5 - 5029431/95318*e^4 - 35253835/762544*e^3 + 12665569/47659*e^2 - 33129611/190636*e - 577563/95318, -354187/97605632*e^9 - 409259/48802816*e^8 + 22904491/97605632*e^7 + 5515823/12200704*e^6 - 527914061/97605632*e^5 - 380611455/48802816*e^4 + 5036405953/97605632*e^3 + 1041116119/24401408*e^2 - 4042138787/24401408*e + 24475025/12200704, -1128503/97605632*e^9 + 1418793/48802816*e^8 + 60334551/97605632*e^7 - 17368469/12200704*e^6 - 1058040577/97605632*e^5 + 1070955525/48802816*e^4 + 6364550469/97605632*e^3 - 2569476173/24401408*e^2 - 1212580271/24401408*e + 55472981/12200704, -2193095/97605632*e^9 + 5115289/48802816*e^8 + 105052775/97605632*e^7 - 64980037/12200704*e^6 - 1433240817/97605632*e^5 + 4214759861/48802816*e^4 + 2235888373/97605632*e^3 - 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2043403731/97605632*e^5 + 4175699487/48802816*e^4 + 6754504159/97605632*e^3 - 10595057271/24401408*e^2 + 8481685187/24401408*e - 358810865/12200704, 10053669/97605632*e^9 - 22482427/48802816*e^8 - 491516165/97605632*e^7 + 284324031/12200704*e^6 + 7054815619/97605632*e^5 - 18318041487/48802816*e^4 - 17474554447/97605632*e^3 + 46504119655/24401408*e^2 - 40446909043/24401408*e + 284457345/12200704, -4486449/97605632*e^9 + 10243183/48802816*e^8 + 210787985/97605632*e^7 - 131013667/12200704*e^6 - 2747363031/97605632*e^5 + 8577639475/48802816*e^4 + 2009247219/97605632*e^3 - 22259435691/24401408*e^2 + 24452019975/24401408*e - 371761053/12200704, 3343593/97605632*e^9 - 7848503/48802816*e^8 - 164926025/97605632*e^7 + 99500107/12200704*e^6 + 2390138079/97605632*e^5 - 6419949531/48802816*e^4 - 5851632411/97605632*e^3 + 16295187987/24401408*e^2 - 14656099983/24401408*e + 261869557/12200704, 1219959/24401408*e^9 - 2684265/12200704*e^8 - 59451799/24401408*e^7 + 34469621/3050176*e^6 + 840890049/24401408*e^5 - 2260353477/12200704*e^4 - 1774176901/24401408*e^3 + 5843566605/6100352*e^2 - 5508290257/6100352*e + 67444587/3050176, 6437515/97605632*e^9 - 16384085/48802816*e^8 - 304696747/97605632*e^7 + 209417297/12200704*e^6 + 3981659469/97605632*e^5 - 13683293057/48802816*e^4 - 2251332033/97605632*e^3 + 35405509161/24401408*e^2 - 37024444253/24401408*e + 160299887/12200704, -1910481/97605632*e^9 + 1351119/48802816*e^8 + 113894449/97605632*e^7 - 14460099/12200704*e^6 - 2351373879/97605632*e^5 + 738374163/48802816*e^4 + 18978086611/97605632*e^3 - 1409510155/24401408*e^2 - 11181951129/24401408*e + 451086659/12200704, -314799/12200704*e^9 + 626049/6100352*e^8 + 16261423/12200704*e^7 - 7886621/1525088*e^6 - 264005449/12200704*e^5 + 503853277/6100352*e^4 + 1212538189/12200704*e^3 - 1264110485/3050176*e^2 + 480831353/3050176*e + 41418749/1525088, 686847/48802816*e^9 - 488545/24401408*e^8 - 37280927/48802816*e^7 + 6472013/6100352*e^6 + 669509241/48802816*e^5 - 439045981/24401408*e^4 - 4266377117/48802816*e^3 + 1182415525/12200704*e^2 + 1315884215/12200704*e - 71658285/6100352, -2760685/48802816*e^9 + 4992307/24401408*e^8 + 139918541/48802816*e^7 - 62687991/6100352*e^6 - 2190551547/48802816*e^5 + 3996802535/24401408*e^4 + 8960339879/48802816*e^3 - 9959354047/12200704*e^2 + 5595088107/12200704*e - 194327593/6100352, -94797/24401408*e^9 + 153811/12200704*e^8 + 4080557/24401408*e^7 - 2143703/3050176*e^6 - 37682075/24401408*e^5 + 152103687/12200704*e^4 - 270872249/24401408*e^3 - 417603807/6100352*e^2 + 783805195/6100352*e - 71973577/3050176, 268175/3050176*e^9 - 576169/1525088*e^8 - 12992111/3050176*e^7 + 7302151/381272*e^6 + 183037993/3050176*e^5 - 471546581/1525088*e^4 - 407849357/3050176*e^3 + 1199986641/762544*e^2 - 1103272009/762544*e + 7262467/381272, -3640811/97605632*e^9 + 8950005/48802816*e^8 + 173079819/97605632*e^7 - 113395953/12200704*e^6 - 2284275949/97605632*e^5 + 7341322657/48802816*e^4 + 1554737121/97605632*e^3 - 18842166857/24401408*e^2 + 21077574653/24401408*e - 214768207/12200704, -714515/97605632*e^9 + 2491725/48802816*e^8 + 30150707/97605632*e^7 - 32412841/12200704*e^6 - 243304709/97605632*e^5 + 2152077721/48802816*e^4 - 3251486887/97605632*e^3 - 5573239297/24401408*e^2 + 9272893077/24401408*e - 478024791/12200704, 2118447/48802816*e^9 - 5086129/24401408*e^8 - 101400271/48802816*e^7 + 64367229/6100352*e^6 + 1380299209/48802816*e^5 - 4160672109/24401408*e^4 - 2002087981/48802816*e^3 + 10653646069/12200704*e^2 - 10683129753/12200704*e + 41834179/6100352, -525493/12200704*e^9 + 812875/6100352*e^8 + 28286933/12200704*e^7 - 9944343/1525088*e^6 - 496695667/12200704*e^5 + 615102175/6100352*e^4 + 2889639231/12200704*e^3 - 1489146759/3050176*e^2 - 227109757/3050176*e + 2342383/1525088, 2571307/97605632*e^9 - 4956085/48802816*e^8 - 135116619/97605632*e^7 + 61842673/12200704*e^6 + 2231824557/97605632*e^5 - 3915070305/48802816*e^4 - 10510751393/97605632*e^3 + 9761688201/24401408*e^2 - 3730794685/24401408*e - 169870961/12200704, -443095/97605632*e^9 - 1215607/48802816*e^8 + 36395127/97605632*e^7 + 17128523/12200704*e^6 - 1054636385/97605632*e^5 - 1249393179/48802816*e^4 + 12680263973/97605632*e^3 + 3578598483/24401408*e^2 - 13217237583/24401408*e + 548342069/12200704, 1417343/97605632*e^9 - 5097953/48802816*e^8 - 57039135/97605632*e^7 + 64428269/12200704*e^6 + 425028857/97605632*e^5 - 4159896285/48802816*e^4 + 5801187043/97605632*e^3 + 10614128613/24401408*e^2 - 14966735305/24401408*e + 264384595/12200704, 274455/97605632*e^9 - 177737/48802816*e^8 - 10227639/97605632*e^7 + 2128949/12200704*e^6 + 79444513/97605632*e^5 - 117225381/48802816*e^4 + 452101403/97605632*e^3 + 112566509/24401408*e^2 - 1174979697/24401408*e + 840086539/12200704, 1609029/48802816*e^9 - 3515355/24401408*e^8 - 76948261/48802816*e^7 + 44579711/6100352*e^6 + 1057064291/48802816*e^5 - 2877719279/24401408*e^4 - 1950813871/48802816*e^3 + 7275533447/12200704*e^2 - 7025427155/12200704*e + 164457761/6100352, -60067/48802816*e^9 + 122429/24401408*e^8 + 3734211/48802816*e^7 - 1220185/6100352*e^6 - 80252917/48802816*e^5 + 36638153/24401408*e^4 + 723021577/48802816*e^3 + 158230735/12200704*e^2 - 644457979/12200704*e - 385699751/6100352, 8173/48802816*e^9 - 2022451/24401408*e^8 + 10265907/48802816*e^7 + 26401463/6100352*e^6 - 526842885/48802816*e^5 - 1776869351/24401408*e^4 + 8435474521/48802816*e^3 + 4785095359/12200704*e^2 - 10613802219/12200704*e + 145681449/6100352, -1282493/48802816*e^9 + 4776739/24401408*e^8 + 53653277/48802816*e^7 - 61015879/6100352*e^6 - 444649131/48802816*e^5 + 3978926231/24401408*e^4 - 5048940905/48802816*e^3 - 10258516591/12200704*e^2 + 14336818203/12200704*e + 54323143/6100352, 646073/24401408*e^9 - 1561639/12200704*e^8 - 29925913/24401408*e^7 + 19774411/3050176*e^6 + 372953487/24401408*e^5 - 1275912331/12200704*e^4 + 51934069/24401408*e^3 + 3237174883/6100352*e^2 - 3811427071/6100352*e + 47670789/3050176, 802331/24401408*e^9 - 1668933/12200704*e^8 - 40215099/24401408*e^7 + 21021713/3050176*e^6 + 606030397/24401408*e^5 - 1346773553/12200704*e^4 - 1962241905/24401408*e^3 + 3371506681/6100352*e^2 - 2623567693/6100352*e + 100067327/3050176, -570731/48802816*e^9 - 243275/24401408*e^8 + 36944651/48802816*e^7 + 3773615/6100352*e^6 - 840883053/48802816*e^5 - 297322527/24401408*e^4 + 7781248993/48802816*e^3 + 914483959/12200704*e^2 - 5954376195/12200704*e + 166390321/6100352, 46753/3050176*e^9 - 127239/1525088*e^8 - 2140593/3050176*e^7 + 1634027/381272*e^6 + 24890727/3050176*e^5 - 108067723/1525088*e^4 + 53015117/3050176*e^3 + 284595443/762544*e^2 - 359567615/762544*e + 23421365/381272, -7599883/97605632*e^9 + 16860373/48802816*e^8 + 365255723/97605632*e^7 - 215838417/12200704*e^6 - 5000671437/97605632*e^5 + 14125079041/48802816*e^4 + 7784550721/97605632*e^3 - 36567646377/24401408*e^2 + 37302294237/24401408*e - 253161967/12200704, 1802655/97605632*e^9 - 4822209/48802816*e^8 - 82743871/97605632*e^7 + 61774797/12200704*e^6 + 1007789273/97605632*e^5 - 4081274941/48802816*e^4 + 812760963/97605632*e^3 + 10920979525/24401408*e^2 - 12040772777/24401408*e - 567734541/12200704, -2086545/48802816*e^9 + 6348687/24401408*e^8 + 91938801/48802816*e^7 - 80916771/6100352*e^6 - 967564407/48802816*e^5 + 5264344531/24401408*e^4 - 4056769005/48802816*e^3 - 13532008459/12200704*e^2 + 17506488999/12200704*e - 109396989/6100352, 943585/48802816*e^9 - 3371455/24401408*e^8 - 39940673/48802816*e^7 + 42379859/6100352*e^6 + 355354535/48802816*e^5 - 2707171907/24401408*e^4 + 3196211965/48802816*e^3 + 6769943803/12200704*e^2 - 9846875863/12200704*e + 266421005/6100352];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19,19,w - 2}>] := -1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;