/*
  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, -4, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {131,131,w^4 - w^3 - 5*w^2 + w + 5}>;

primesArray := [
[11, 11, w^4 + w^3 - 4*w^2 - 3*w + 2],
[23, 23, -w^4 + 3*w^2 + 1],
[23, 23, -w^4 + 3*w^2 + w - 2],
[23, 23, w^4 - w^3 - 3*w^2 + 3*w + 2],
[23, 23, -w^4 + w^3 + 4*w^2 - 3*w - 1],
[23, 23, -w^2 + w + 3],
[32, 2, 2],
[43, 43, -2*w^4 + w^3 + 6*w^2 - 2*w - 1],
[43, 43, -w^4 + 2*w^2 + w + 1],
[43, 43, w^3 + w^2 - 4*w - 2],
[43, 43, 2*w^4 - w^3 - 7*w^2 + 3*w + 3],
[43, 43, w^4 - w^3 - 4*w^2 + 4*w + 2],
[67, 67, 2*w^4 - 7*w^2 + 2],
[67, 67, w^4 - 2*w^3 - 3*w^2 + 6*w + 2],
[67, 67, 2*w^4 - 7*w^2 - w + 4],
[67, 67, w^4 - 2*w^3 - 4*w^2 + 6*w + 2],
[67, 67, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[89, 89, w^3 + w^2 - 4*w - 1],
[89, 89, -2*w^4 + w^3 + 7*w^2 - 3*w - 2],
[89, 89, -w^4 + w^3 + 4*w^2 - 4*w - 3],
[89, 89, -w^4 + 2*w^2 + w + 2],
[89, 89, 2*w^4 - w^3 - 6*w^2 + 2*w + 2],
[109, 109, -w^3 + 2*w^2 + 3*w - 3],
[109, 109, w^4 - 4*w^2 - 2*w + 3],
[109, 109, -2*w^4 + 2*w^3 + 7*w^2 - 4*w - 3],
[109, 109, 2*w^3 - 5*w - 1],
[109, 109, -w^4 - w^3 + 5*w^2 + 2*w - 4],
[131, 131, w^4 - 3*w^3 - 2*w^2 + 7*w],
[131, 131, -w^4 + 2*w^3 + 5*w^2 - 7*w - 5],
[131, 131, 2*w^4 - 2*w^3 - 6*w^2 + 3*w + 2],
[131, 131, w^4 - 2*w^3 - 3*w^2 + 7*w],
[131, 131, 2*w^4 + w^3 - 8*w^2 - 3*w + 4],
[197, 197, 3*w^4 - w^3 - 10*w^2 + w + 5],
[197, 197, 2*w^4 - 7*w^2 + w + 1],
[197, 197, -3*w^4 + 2*w^3 + 10*w^2 - 5*w - 5],
[197, 197, -2*w^4 - w^3 + 9*w^2 + 2*w - 5],
[197, 197, -3*w^4 + 3*w^3 + 10*w^2 - 6*w - 5],
[199, 199, w^4 + 2*w^3 - 5*w^2 - 5*w + 3],
[199, 199, -2*w^4 + w^3 + 7*w^2 - 4*w - 2],
[199, 199, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],
[199, 199, 2*w^4 - 8*w^2 + w + 4],
[199, 199, 3*w^4 - w^3 - 10*w^2 + 2*w + 4],
[241, 241, w^4 - 2*w^3 - w^2 + 3*w - 3],
[241, 241, -w^4 + 3*w^3 + 2*w^2 - 9*w],
[241, 241, 3*w^4 - 11*w^2 - 2*w + 6],
[241, 241, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],
[241, 241, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6],
[243, 3, -3],
[263, 263, -w^4 + 4*w^2 + w + 1],
[263, 263, 2*w^4 - w^3 - 8*w^2 + w + 2],
[263, 263, 3*w^4 - 3*w^3 - 9*w^2 + 7*w + 2],
[263, 263, -2*w^3 + w^2 + 4*w - 4],
[263, 263, -3*w^2 + w + 5],
[307, 307, 2*w^4 - w^3 - 6*w^2 + 3*w - 2],
[307, 307, w^4 - 2*w^2 - 2*w - 4],
[307, 307, -3*w^4 + 2*w^3 + 11*w^2 - 3*w - 6],
[307, 307, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 6],
[307, 307, -w^4 + 3*w^3 + 4*w^2 - 7*w - 3],
[331, 331, -w^4 - 2*w^3 + 3*w^2 + 7*w - 1],
[331, 331, -w^4 + 3*w^3 + 4*w^2 - 8*w - 4],
[331, 331, w^4 - w^2 - 4],
[331, 331, 3*w^4 - 2*w^3 - 9*w^2 + 5*w + 2],
[331, 331, -3*w^4 + 2*w^3 + 11*w^2 - 6*w - 5],
[353, 353, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],
[353, 353, -2*w^4 - w^3 + 7*w^2 + 4*w - 4],
[353, 353, -w^4 + 3*w^3 + 3*w^2 - 9*w - 3],
[353, 353, -3*w^4 + 11*w^2 + w - 7],
[353, 353, w^4 - w^3 - 6*w^2 + 3*w + 4],
[373, 373, w^3 + 2*w^2 - 5*w - 3],
[373, 373, 2*w^4 - 3*w^3 - 6*w^2 + 6*w],
[373, 373, -w^4 + w^3 + 3*w^2 - 3*w + 3],
[373, 373, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 5],
[373, 373, -w^4 + w^3 + 4*w^2 - 3*w - 6],
[397, 397, -w^4 - w^3 + 6*w^2 + 3*w - 5],
[397, 397, 2*w^4 - 2*w^3 - 7*w^2 + 3*w + 3],
[397, 397, -w^4 + 5*w^2 + 2*w - 5],
[397, 397, w^4 + 2*w^3 - 5*w^2 - 5*w + 4],
[397, 397, -w^4 + 3*w^3 + 3*w^2 - 7*w - 2],
[419, 419, 3*w^3 - w^2 - 8*w],
[419, 419, 2*w^4 - 3*w^3 - 6*w^2 + 6*w + 1],
[419, 419, w^4 - 2*w^3 - 5*w^2 + 7*w + 4],
[419, 419, -2*w^4 - w^3 + 9*w^2 + 3*w - 7],
[419, 419, -w^4 + 3*w^3 + w^2 - 8*w + 1],
[439, 439, -w^4 + w^3 + 5*w^2 - 5*w - 5],
[439, 439, 2*w^4 - 9*w^2 - 2*w + 8],
[439, 439, -3*w^4 + 2*w^3 + 11*w^2 - 5*w - 3],
[439, 439, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 4],
[439, 439, -w^4 + 3*w^3 + 3*w^2 - 6*w - 3],
[461, 461, -w^4 + 5*w^2 + 2*w - 6],
[461, 461, -w^4 + 3*w^2 - w + 3],
[461, 461, -w^4 - 2*w^3 + 5*w^2 + 5*w - 5],
[461, 461, -2*w^4 + 7*w^2 + w - 6],
[461, 461, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],
[463, 463, 2*w^4 - 2*w^3 - 8*w^2 + 5*w + 3],
[463, 463, w^4 + w^3 - 3*w^2 - 2*w - 2],
[463, 463, 2*w^4 - 7*w^2 - 2*w + 5],
[463, 463, w^4 - 2*w^3 - 2*w^2 + 6*w + 1],
[463, 463, -w^3 + 2*w^2 + w - 5],
[571, 571, w^4 - w^3 - 3*w^2 + 2*w - 3],
[571, 571, 3*w^4 - 2*w^3 - 10*w^2 + 6*w + 2],
[571, 571, -2*w^4 + 2*w^3 + 8*w^2 - 7*w - 6],
[571, 571, 2*w^4 - 5*w^2 - 2*w - 2],
[571, 571, 3*w^4 - w^3 - 9*w^2 + 2*w + 3],
[593, 593, 3*w^4 - w^3 - 9*w^2 + w + 3],
[593, 593, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3],
[593, 593, -2*w^4 + 2*w^3 + 7*w^2 - 7*w - 4],
[593, 593, w^4 - w^2 - 2*w - 4],
[593, 593, 3*w^4 - w^3 - 10*w^2 + 3*w + 2],
[617, 617, -w^4 - w^3 + 6*w^2 + 4*w - 5],
[617, 617, 3*w^3 - w^2 - 7*w],
[617, 617, 2*w^4 - w^3 - 8*w^2 + 6],
[617, 617, 2*w^4 - 3*w^3 - 7*w^2 + 6*w + 3],
[617, 617, -w^4 - 2*w^3 + 6*w^2 + 5*w - 6],
[659, 659, -w^4 + w^3 + 4*w^2 - 5*w - 4],
[659, 659, 2*w^3 + w^2 - 7*w],
[659, 659, -3*w^4 + 2*w^3 + 9*w^2 - 4*w - 4],
[659, 659, 3*w^4 - w^3 - 11*w^2 + 3*w + 3],
[659, 659, w^4 - w^2 - w - 5],
[661, 661, -5*w^4 + 2*w^3 + 18*w^2 - 4*w - 9],
[661, 661, -w^4 + 3*w^3 + 6*w^2 - 9*w - 8],
[661, 661, w^4 + 2*w^3 - 2*w^2 - 7*w - 2],
[661, 661, -w^4 + w^3 + 5*w^2 - 2*w - 9],
[661, 661, -3*w^4 + 2*w^3 + 12*w^2 - 2*w - 8],
[683, 683, -w^4 + w^3 + 4*w^2 - 5*w - 3],
[683, 683, 2*w^3 + w^2 - 7*w - 1],
[683, 683, w^4 - w^2 - w - 4],
[683, 683, 3*w^4 - 2*w^3 - 9*w^2 + 4*w + 3],
[683, 683, 3*w^4 - w^3 - 11*w^2 + 3*w + 4],
[727, 727, -5*w^4 + 3*w^3 + 19*w^2 - 7*w - 9],
[727, 727, -2*w^4 - 2*w^3 + 7*w^2 + 9*w - 4],
[727, 727, 3*w^4 + 2*w^3 - 13*w^2 - 6*w + 6],
[727, 727, w^4 - 2*w^2 + 3*w - 4],
[727, 727, -2*w^4 + 5*w^3 + 6*w^2 - 14*w - 4],
[769, 769, -2*w^4 - w^3 + 9*w^2 + 2*w - 7],
[769, 769, -w^4 + 4*w^2 + 3*w - 4],
[769, 769, -w^3 + 3*w^2 + 3*w - 4],
[769, 769, 3*w^4 - 3*w^3 - 10*w^2 + 6*w + 3],
[769, 769, 3*w^3 - 8*w - 2],
[857, 857, w^4 - 5*w^3 - 2*w^2 + 14*w - 1],
[857, 857, 4*w^4 - w^3 - 13*w^2 + 4],
[857, 857, -2*w^4 + 4*w^3 + 6*w^2 - 10*w - 5],
[857, 857, 4*w^4 + w^3 - 14*w^2 - 5*w + 6],
[857, 857, 4*w^4 + w^3 - 16*w^2 - 4*w + 8],
[859, 859, 3*w^4 - w^3 - 11*w^2 + 5*w + 5],
[859, 859, -2*w^4 + 5*w^3 + 6*w^2 - 13*w - 1],
[859, 859, -2*w^4 - 2*w^3 + 5*w^2 + 7*w + 1],
[859, 859, -w^3 + w^2 + 6*w + 1],
[859, 859, 4*w^4 - w^3 - 13*w^2 + 3*w + 4],
[881, 881, 2*w^4 + w^3 - 11*w^2 - 3*w + 11],
[881, 881, w^4 - 6*w^2 - 3*w + 5],
[881, 881, -2*w^4 + 5*w^3 + 6*w^2 - 12*w - 1],
[881, 881, -2*w^4 - 3*w^3 + 9*w^2 + 8*w - 4],
[881, 881, 3*w^4 + w^3 - 12*w^2 - 3*w + 7],
[947, 947, 2*w^4 + 3*w^3 - 7*w^2 - 8*w],
[947, 947, 2*w^4 - 3*w^3 - 6*w^2 + 4*w],
[947, 947, 2*w^4 + 2*w^3 - 7*w^2 - 8*w + 4],
[947, 947, -w^4 + 2*w^3 - 7*w + 3],
[947, 947, -4*w^4 + 5*w^3 + 14*w^2 - 12*w - 5],
[967, 967, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],
[967, 967, -2*w^4 - w^3 + 7*w^2 + 4*w - 5],
[967, 967, w^4 - 2*w^2 + w - 4],
[967, 967, w^4 - w^3 - 2*w^2 - 3],
[967, 967, -2*w^4 + 3*w^3 + 6*w^2 - 8*w - 4],
[991, 991, -w^4 + 4*w^3 + 2*w^2 - 12*w - 1],
[991, 991, w^4 + 3*w^3 - 3*w^2 - 11*w - 1],
[991, 991, 2*w^3 - 2*w^2 - 5*w + 8],
[991, 991, -2*w^4 + 3*w^3 + 4*w^2 - 5*w + 3],
[991, 991, 4*w^4 - 4*w^3 - 11*w^2 + 9*w]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 - 6*x^4 - 13*x^3 + 146*x^2 - 295*x + 164;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/2*e^3 - 1/2*e^2 - 23/2*e + 20, -e^4 + 3*e^3 + 22*e^2 - 81*e + 55, -e^4 + 3*e^3 + 22*e^2 - 81*e + 55, 3/2*e^4 - 19/4*e^3 - 133/4*e^2 + 499/4*e - 81, 3/2*e^4 - 5*e^3 - 33*e^2 + 263/2*e - 91, -1/4*e^4 + 3/4*e^3 + 25/4*e^2 - 39/2*e + 5, 1/2*e^3 - 1/2*e^2 - 19/2*e + 16, -e^4 + 3*e^3 + 22*e^2 - 79*e + 51, -e^4 + 3*e^3 + 22*e^2 - 79*e + 51, -1/4*e^4 + e^3 + 6*e^2 - 101/4*e + 14, -e^4 + 7/2*e^3 + 43/2*e^2 - 185/2*e + 71, -e^4 + 4*e^3 + 21*e^2 - 104*e + 91, 7/2*e^4 - 23/2*e^3 - 151/2*e^2 + 303*e - 225, 1/2*e^4 - 2*e^3 - 11*e^2 + 105/2*e - 40, -5/4*e^4 + 4*e^3 + 28*e^2 - 425/4*e + 69, 1/2*e^4 - 7/4*e^3 - 45/4*e^2 + 183/4*e - 30, 1/2*e^4 - 3/2*e^3 - 23/2*e^2 + 41*e - 22, 2*e^4 - 13/2*e^3 - 89/2*e^2 + 339/2*e - 101, -2*e^4 + 13/2*e^3 + 87/2*e^2 - 343/2*e + 124, -3*e^4 + 19/2*e^3 + 131/2*e^2 - 509/2*e + 179, -1/2*e^4 + 5/4*e^3 + 43/4*e^2 - 141/4*e + 23, 5/2*e^4 - 17/2*e^3 - 107/2*e^2 + 224*e - 176, 7/4*e^4 - 6*e^3 - 38*e^2 + 627/4*e - 119, e^4 - 13/4*e^3 - 83/4*e^2 + 351/4*e - 75, -2*e^4 + 7*e^3 + 43*e^2 - 183*e + 140, -5/4*e^4 + 9/2*e^3 + 55/2*e^2 - 463/4*e + 83, 11/2*e^4 - 37/2*e^3 - 239/2*e^2 + 485*e - 360, -5/2*e^4 + 15/2*e^3 + 113/2*e^2 - 199*e + 114, e^4 - 3*e^3 - 23*e^2 + 77*e - 32, 11/2*e^4 - 37/2*e^3 - 239/2*e^2 + 485*e - 360, 1, -2*e^4 + 11/2*e^3 + 91/2*e^2 - 293/2*e + 72, -5*e^4 + 16*e^3 + 109*e^2 - 424*e + 297, -1/2*e^4 + 3/2*e^3 + 23/2*e^2 - 44*e + 21, -e^4 + 5/2*e^3 + 43/2*e^2 - 137/2*e + 40, 3/4*e^3 + 1/4*e^2 - 57/4*e + 12, 1/2*e^4 - 2*e^3 - 10*e^2 + 105/2*e - 52, 1/2*e^4 - 9/4*e^3 - 47/4*e^2 + 209/4*e - 15, 1/4*e^4 - 3/2*e^3 - 7/2*e^2 + 147/4*e - 54, -13/4*e^4 + 23/2*e^3 + 141/2*e^2 - 1203/4*e + 227, -2*e^4 + 6*e^3 + 45*e^2 - 160*e + 94, 4*e^4 - 12*e^3 - 88*e^2 + 325*e - 219, 9/2*e^4 - 29/2*e^3 - 197/2*e^2 + 385*e - 275, -e^4 + 4*e^3 + 20*e^2 - 105*e + 96, -1/2*e^4 + e^3 + 12*e^2 - 53/2*e - 3, 3/4*e^4 - e^3 - 18*e^2 + 127/4*e + 4, 1/2*e^4 - 3/2*e^3 - 21/2*e^2 + 42*e - 24, -3/2*e^4 + 9/2*e^3 + 67/2*e^2 - 121*e + 70, -4*e^4 + 14*e^3 + 86*e^2 - 363*e + 288, -3/4*e^4 + 5/2*e^3 + 35/2*e^2 - 269/4*e + 33, 7/2*e^4 - 47/4*e^3 - 313/4*e^2 + 1215/4*e - 185, 2*e^2 - 24, -5/2*e^4 + 7*e^3 + 56*e^2 - 373/2*e + 101, -6*e^4 + 41/2*e^3 + 259/2*e^2 - 1077/2*e + 414, e^4 - 3*e^3 - 22*e^2 + 82*e - 60, -3/4*e^4 + 3*e^3 + 17*e^2 - 307/4*e + 49, 13/2*e^4 - 43/2*e^3 - 285/2*e^2 + 567*e - 408, -9/4*e^4 + 8*e^3 + 48*e^2 - 837/4*e + 163, -5/4*e^4 + 11/2*e^3 + 51/2*e^2 - 559/4*e + 128, 9/2*e^4 - 15*e^3 - 99*e^2 + 777/2*e - 266, -5/2*e^4 + 8*e^3 + 55*e^2 - 425/2*e + 153, 2*e^4 - 13/2*e^3 - 91/2*e^2 + 347/2*e - 88, -1/2*e^4 + 12*e^2 - 13/2*e - 28, -13/2*e^4 + 21*e^3 + 143*e^2 - 1103/2*e + 376, 3/2*e^4 - 15/4*e^3 - 129/4*e^2 + 415/4*e - 71, -3/2*e^4 + 5*e^3 + 33*e^2 - 253/2*e + 84, 1/2*e^4 - 5/2*e^3 - 21/2*e^2 + 63*e - 63, -11/2*e^4 + 19*e^3 + 120*e^2 - 989/2*e + 369, 6*e^4 - 79/4*e^3 - 529/4*e^2 + 2057/4*e - 341, -9/2*e^4 + 29/2*e^3 + 197/2*e^2 - 387*e + 265, 9/4*e^4 - 13/2*e^3 - 99/2*e^2 + 699/4*e - 119, -5/2*e^4 + 17/2*e^3 + 105/2*e^2 - 228*e + 202, 1/2*e^4 - 2*e^3 - 11*e^2 + 101/2*e - 11, 6*e^4 - 20*e^3 - 130*e^2 + 529*e - 386, -3*e^4 + 10*e^3 + 67*e^2 - 264*e + 171, 3/2*e^4 - 7/2*e^3 - 69/2*e^2 + 96*e - 42, -11/2*e^4 + 19*e^3 + 120*e^2 - 983/2*e + 357, 9*e^4 - 30*e^3 - 196*e^2 + 791*e - 582, 11/2*e^4 - 37/2*e^3 - 235/2*e^2 + 487*e - 384, 3*e^4 - 17/2*e^3 - 131/2*e^2 + 465/2*e - 158, -5/4*e^4 + 9/2*e^3 + 51/2*e^2 - 487/4*e + 103, 1/4*e^4 - 6*e^2 - 3/4*e + 9, -9/2*e^4 + 29/2*e^3 + 197/2*e^2 - 382*e + 271, -4*e^4 + 15*e^3 + 86*e^2 - 385*e + 300, 5/2*e^4 - 31/4*e^3 - 217/4*e^2 + 819/4*e - 153, -2*e^4 + 25/4*e^3 + 175/4*e^2 - 647/4*e + 103, 7/2*e^4 - 23/2*e^3 - 151/2*e^2 + 308*e - 238, -7/2*e^4 + 11*e^3 + 78*e^2 - 589/2*e + 182, -7/2*e^4 + 25/2*e^3 + 151/2*e^2 - 325*e + 266, -15/2*e^4 + 47/2*e^3 + 329/2*e^2 - 628*e + 422, -9/4*e^4 + 15/2*e^3 + 95/2*e^2 - 811/4*e + 176, 9/2*e^4 - 14*e^3 - 100*e^2 + 749/2*e - 252, -5/2*e^4 + 17/2*e^3 + 107/2*e^2 - 230*e + 172, 3/2*e^4 - 7/2*e^3 - 69/2*e^2 + 100*e - 24, -11/4*e^4 + 21/2*e^3 + 119/2*e^2 - 1077/4*e + 207, e^4 - 13/4*e^3 - 91/4*e^2 + 343/4*e - 59, 3*e^4 - 21/2*e^3 - 129/2*e^2 + 547/2*e - 219, 13/2*e^4 - 22*e^3 - 142*e^2 + 1147/2*e - 405, -11/2*e^4 + 37/2*e^3 + 239/2*e^2 - 481*e + 366, -10*e^4 + 65/2*e^3 + 437/2*e^2 - 1715/2*e + 598, 5*e^4 - 35/2*e^3 - 217/2*e^2 + 915/2*e - 334, 15/2*e^4 - 25*e^3 - 166*e^2 + 1303/2*e - 437, 13/4*e^4 - 19/2*e^3 - 145/2*e^2 + 1035/4*e - 147, 7/2*e^4 - 13*e^3 - 77*e^2 + 679/2*e - 254, 9*e^4 - 61/2*e^3 - 393/2*e^2 + 1607/2*e - 597, -3/2*e^3 + 3/2*e^2 + 71/2*e - 56, 13/2*e^4 - 43/2*e^3 - 285/2*e^2 + 562*e - 387, -21/4*e^4 + 17*e^3 + 113*e^2 - 1805/4*e + 349, -2*e^3 + 2*e^2 + 52*e - 88, e^4 - 4*e^3 - 21*e^2 + 108*e - 103, -5*e^4 + 16*e^3 + 107*e^2 - 422*e + 331, 7*e^4 - 47/2*e^3 - 303/2*e^2 + 1235/2*e - 470, -13/2*e^4 + 39/2*e^3 + 291/2*e^2 - 524*e + 309, -1/2*e^4 + 3/2*e^3 + 19/2*e^2 - 37*e + 37, 4*e^4 - 13*e^3 - 88*e^2 + 341*e - 215, 1/4*e^3 - 9/4*e^2 - 47/4*e + 43, -2*e^4 + 15/2*e^3 + 87/2*e^2 - 375/2*e + 133, 3*e^4 - 21/2*e^3 - 131/2*e^2 + 549/2*e - 214, -5/2*e^4 + 19/2*e^3 + 111/2*e^2 - 246*e + 171, 2*e^4 - 31/4*e^3 - 165/4*e^2 + 805/4*e - 168, -11/2*e^4 + 61/4*e^3 + 491/4*e^2 - 1649/4*e + 229, 7/2*e^4 - 23/2*e^3 - 157/2*e^2 + 304*e - 186, -9/2*e^4 + 27/2*e^3 + 195/2*e^2 - 368*e + 269, -7/2*e^4 + 11*e^3 + 75*e^2 - 597/2*e + 234, 1/2*e^4 - 14*e^2 + 17/2*e + 30, -1/2*e^4 + 14*e^2 - 1/2*e - 58, 19/2*e^4 - 115/4*e^3 - 841/4*e^2 + 3087/4*e - 488, -4*e^4 + 49/4*e^3 + 359/4*e^2 - 1303/4*e + 189, e^4 - 4*e^3 - 22*e^2 + 101*e - 90, -7*e^4 + 45/2*e^3 + 307/2*e^2 - 1193/2*e + 404, 1/2*e^4 + 1/2*e^3 - 23/2*e^2 - 8*e + 23, -3/2*e^3 + 1/2*e^2 + 59/2*e - 31, e^4 - 4*e^3 - 22*e^2 + 105*e - 96, 11/2*e^4 - 37/2*e^3 - 239/2*e^2 + 485*e - 372, -4*e^4 + 25/2*e^3 + 175/2*e^2 - 673/2*e + 252, 9/2*e^4 - 14*e^3 - 98*e^2 + 751/2*e - 244, 1/2*e^4 + 1/2*e^3 - 23/2*e^2 - 4*e + 17, -4*e^4 + 15*e^3 + 85*e^2 - 395*e + 332, 1/2*e^4 - e^3 - 11*e^2 + 55/2*e - 28, -6*e^4 + 18*e^3 + 132*e^2 - 479*e + 310, -6*e^4 + 39/2*e^3 + 261/2*e^2 - 1037/2*e + 358, 3*e^4 - 9*e^3 - 66*e^2 + 240*e - 174, 5/2*e^4 - 7*e^3 - 55*e^2 + 387/2*e - 136, -e^4 + 9/2*e^3 + 39/2*e^2 - 245/2*e + 132, -e^4 + 5/2*e^3 + 43/2*e^2 - 137/2*e + 52, 3*e^4 - 19/2*e^3 - 131/2*e^2 + 503/2*e - 168, -11/2*e^4 + 33/2*e^3 + 245/2*e^2 - 440*e + 261, 6*e^4 - 39/2*e^3 - 267/2*e^2 + 1011/2*e - 305, 27/4*e^4 - 23*e^3 - 148*e^2 + 2415/4*e - 450, 8*e^4 - 27*e^3 - 173*e^2 + 707*e - 522, -5*e^4 + 31/2*e^3 + 219/2*e^2 - 813/2*e + 263, e^4 - 5*e^3 - 22*e^2 + 123*e - 97, 11/2*e^4 - 17*e^3 - 121*e^2 + 915/2*e - 290, 1/2*e^3 - 5/2*e^2 - 43/2*e + 52, -3/2*e^4 + 7/2*e^3 + 65/2*e^2 - 97*e + 63, 37/4*e^4 - 61/2*e^3 - 397/2*e^2 + 3219/4*e - 604, 9*e^4 - 55/2*e^3 - 397/2*e^2 + 1481/2*e - 483, -17/2*e^4 + 55/2*e^3 + 381/2*e^2 - 723*e + 448, 7/4*e^4 - 4*e^3 - 41*e^2 + 415/4*e - 24, -2*e^4 + 19/4*e^3 + 181/4*e^2 - 525/4*e + 41, -2*e^4 + 25/4*e^3 + 175/4*e^2 - 687/4*e + 101, 4*e^4 - 27/2*e^3 - 179/2*e^2 + 699/2*e - 229, 4*e^4 - 23/2*e^3 - 185/2*e^2 + 619/2*e - 149, e^4 - 7/2*e^3 - 41/2*e^2 + 173/2*e - 82, 3/2*e^4 - 11/2*e^3 - 65/2*e^2 + 146*e - 130, -5/2*e^4 + 23/2*e^3 + 105/2*e^2 - 289*e + 268, 4*e^4 - 27/2*e^3 - 173/2*e^2 + 691/2*e - 264];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {131,131,w^4 - w^3 - 5*w^2 + w + 5}>] := -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;