/*
  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, -5, 2, 8, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {79, 79, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 9*w + 5}>;

primesArray := [
[9, 3, -w^2 + 2],
[19, 19, -w^3 + 4*w],
[19, 19, w^3 - 3*w - 1],
[29, 29, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 3],
[29, 29, w^4 - w^3 - 4*w^2 + 2*w + 2],
[31, 31, -w^4 + 4*w^2 + w - 3],
[41, 41, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 11*w - 4],
[49, 7, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 11*w + 6],
[59, 59, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5],
[59, 59, w^5 - w^4 - 4*w^3 + 3*w^2 + 3*w - 3],
[59, 59, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 9*w - 4],
[61, 61, -w^5 + w^4 + 5*w^3 - 3*w^2 - 7*w + 1],
[64, 2, -2],
[71, 71, -2*w^5 + 2*w^4 + 10*w^3 - 8*w^2 - 11*w + 6],
[79, 79, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 9*w + 5],
[79, 79, -w^4 - w^3 + 5*w^2 + 4*w - 3],
[79, 79, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 8*w + 4],
[81, 3, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 13*w - 8],
[89, 89, 2*w^5 - 2*w^4 - 9*w^3 + 6*w^2 + 8*w - 1],
[101, 101, -w^4 - w^3 + 5*w^2 + 3*w - 3],
[101, 101, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 4],
[109, 109, -2*w^5 + 4*w^4 + 8*w^3 - 14*w^2 - 6*w + 5],
[109, 109, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 7],
[125, 5, w^5 - w^4 - 5*w^3 + 4*w^2 + 5*w],
[131, 131, -w^2 - w + 3],
[131, 131, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 3*w - 4],
[131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 5],
[131, 131, -2*w^5 + 4*w^4 + 8*w^3 - 15*w^2 - 6*w + 7],
[139, 139, w^4 - 3*w^3 - 3*w^2 + 9*w],
[149, 149, 2*w^5 - 3*w^4 - 9*w^3 + 12*w^2 + 7*w - 5],
[149, 149, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 2*w - 2],
[151, 151, 3*w^5 - 4*w^4 - 13*w^3 + 14*w^2 + 10*w - 5],
[151, 151, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 12*w + 6],
[151, 151, 4*w^5 - 5*w^4 - 18*w^3 + 18*w^2 + 15*w - 9],
[151, 151, 2*w^5 - 2*w^4 - 9*w^3 + 7*w^2 + 8*w - 5],
[169, 13, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 13*w + 1],
[179, 179, w^3 - w^2 - 2*w + 3],
[181, 181, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 3],
[181, 181, -2*w^4 + w^3 + 8*w^2 - 3*w - 2],
[191, 191, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 4],
[191, 191, -w^3 + 2*w^2 + 3*w - 4],
[191, 191, -2*w^5 + w^4 + 11*w^3 - 3*w^2 - 12*w],
[191, 191, w^4 - 2*w^3 - 4*w^2 + 7*w],
[199, 199, -3*w^5 + 4*w^4 + 15*w^3 - 14*w^2 - 16*w + 6],
[199, 199, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 12*w + 7],
[211, 211, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 8*w + 5],
[211, 211, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 8],
[229, 229, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 10*w + 6],
[229, 229, w^4 - 4*w^2 - 1],
[229, 229, w^5 - w^4 - 3*w^3 + 3*w^2 - 3*w - 2],
[239, 239, -2*w^5 + 3*w^4 + 9*w^3 - 11*w^2 - 8*w + 8],
[241, 241, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 3*w - 2],
[241, 241, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w + 1],
[251, 251, -3*w^5 + 3*w^4 + 15*w^3 - 10*w^2 - 17*w + 3],
[271, 271, -2*w^5 + 4*w^4 + 9*w^3 - 14*w^2 - 8*w + 2],
[271, 271, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 4*w],
[271, 271, w^5 - 2*w^4 - 3*w^3 + 7*w^2 - 2*w - 4],
[289, 17, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 12*w - 9],
[311, 311, -2*w^5 + 3*w^4 + 7*w^3 - 9*w^2 - 3*w + 2],
[331, 331, 3*w^5 - 5*w^4 - 14*w^3 + 19*w^2 + 14*w - 12],
[349, 349, -4*w^5 + 6*w^4 + 18*w^3 - 21*w^2 - 17*w + 7],
[349, 349, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[359, 359, -2*w^5 + 4*w^4 + 9*w^3 - 15*w^2 - 10*w + 9],
[361, 19, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 6*w - 4],
[361, 19, -3*w^5 + 5*w^4 + 13*w^3 - 18*w^2 - 10*w + 9],
[379, 379, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 14*w + 7],
[379, 379, 2*w^5 - 4*w^4 - 10*w^3 + 15*w^2 + 11*w - 5],
[389, 389, -5*w^5 + 7*w^4 + 22*w^3 - 25*w^2 - 17*w + 11],
[389, 389, w^5 - 3*w^4 - 3*w^3 + 12*w^2 - 2*w - 5],
[401, 401, -4*w^5 + 6*w^4 + 18*w^3 - 23*w^2 - 15*w + 10],
[401, 401, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - w + 4],
[401, 401, 4*w^5 - 5*w^4 - 20*w^3 + 19*w^2 + 21*w - 11],
[401, 401, -3*w^5 + 3*w^4 + 16*w^3 - 12*w^2 - 18*w + 6],
[409, 409, -2*w^4 + 2*w^3 + 7*w^2 - 6*w - 2],
[409, 409, -w^5 + 3*w^4 + 5*w^3 - 13*w^2 - 5*w + 9],
[419, 419, 2*w^5 - 2*w^4 - 10*w^3 + 7*w^2 + 10*w - 6],
[419, 419, -w^5 + 7*w^3 - w^2 - 10*w + 4],
[419, 419, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 9],
[421, 421, 3*w^5 - 3*w^4 - 16*w^3 + 12*w^2 + 18*w - 7],
[431, 431, 4*w^5 - 7*w^4 - 17*w^3 + 25*w^2 + 13*w - 10],
[431, 431, -2*w^5 + 3*w^4 + 7*w^3 - 10*w^2 - w + 5],
[431, 431, 4*w^5 - 6*w^4 - 19*w^3 + 23*w^2 + 17*w - 10],
[439, 439, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 11*w + 4],
[461, 461, w^5 - 2*w^4 - 3*w^3 + 6*w^2 + 2],
[461, 461, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 7],
[479, 479, -2*w^5 + w^4 + 9*w^3 - 2*w^2 - 8*w],
[479, 479, 2*w^5 - 2*w^4 - 10*w^3 + 8*w^2 + 12*w - 7],
[491, 491, -3*w^5 + 3*w^4 + 14*w^3 - 12*w^2 - 12*w + 7],
[491, 491, -3*w^5 + 3*w^4 + 15*w^3 - 12*w^2 - 14*w + 8],
[491, 491, -2*w^2 + w + 6],
[499, 499, -3*w^5 + 3*w^4 + 14*w^3 - 10*w^2 - 13*w + 6],
[509, 509, -2*w^5 + 2*w^4 + 8*w^3 - 6*w^2 - 5*w],
[521, 521, 3*w^5 - 4*w^4 - 14*w^3 + 15*w^2 + 15*w - 9],
[521, 521, -3*w^5 + 5*w^4 + 14*w^3 - 19*w^2 - 15*w + 9],
[521, 521, -4*w^5 + 5*w^4 + 18*w^3 - 17*w^2 - 16*w + 6],
[529, 23, w^5 - 3*w^4 - 5*w^3 + 11*w^2 + 7*w - 5],
[541, 541, -3*w^5 + 4*w^4 + 14*w^3 - 15*w^2 - 14*w + 10],
[569, 569, -w^5 - w^4 + 6*w^3 + 4*w^2 - 6*w - 1],
[571, 571, -2*w^4 + w^3 + 9*w^2 - 3*w - 4],
[599, 599, -3*w^5 + 3*w^4 + 14*w^3 - 11*w^2 - 11*w + 4],
[601, 601, w^4 - 2*w^2 - w - 2],
[619, 619, 2*w^5 - 3*w^4 - 9*w^3 + 11*w^2 + 10*w - 5],
[619, 619, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 3*w - 3],
[619, 619, -3*w^5 + 5*w^4 + 13*w^3 - 20*w^2 - 10*w + 10],
[619, 619, w^5 - 2*w^4 - 5*w^3 + 10*w^2 + 5*w - 8],
[619, 619, -4*w^5 + 6*w^4 + 19*w^3 - 23*w^2 - 17*w + 11],
[619, 619, -4*w^5 + 6*w^4 + 17*w^3 - 22*w^2 - 13*w + 10],
[631, 631, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 5*w - 6],
[641, 641, -w^3 + w^2 + 3*w - 5],
[641, 641, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + 3*w + 4],
[659, 659, -w^5 + 3*w^4 + 5*w^3 - 12*w^2 - 5*w + 5],
[661, 661, -2*w^5 + 3*w^4 + 8*w^3 - 12*w^2 - 3*w + 8],
[661, 661, 2*w^5 - 3*w^4 - 8*w^3 + 10*w^2 + 6*w - 1],
[691, 691, -w^3 - w^2 + 5*w + 1],
[691, 691, -3*w^5 + 4*w^4 + 13*w^3 - 13*w^2 - 10*w + 6],
[691, 691, 3*w^5 - 4*w^4 - 15*w^3 + 15*w^2 + 14*w - 6],
[701, 701, 3*w^5 - 6*w^4 - 13*w^3 + 23*w^2 + 10*w - 10],
[701, 701, -3*w^5 + 6*w^4 + 13*w^3 - 23*w^2 - 11*w + 9],
[709, 709, 3*w^5 - 6*w^4 - 13*w^3 + 22*w^2 + 11*w - 10],
[709, 709, -w^5 + 5*w^3 - 4*w - 2],
[709, 709, -w^5 + w^4 + 7*w^3 - 6*w^2 - 11*w + 5],
[719, 719, -2*w^5 + 3*w^4 + 9*w^3 - 12*w^2 - 7*w + 4],
[751, 751, w^4 + w^3 - 5*w^2 - 6*w + 4],
[769, 769, 4*w^5 - 5*w^4 - 19*w^3 + 18*w^2 + 18*w - 10],
[769, 769, 4*w^5 - 6*w^4 - 17*w^3 + 22*w^2 + 13*w - 11],
[809, 809, w^5 - w^4 - 4*w^3 + 5*w^2 + w - 7],
[809, 809, 2*w^5 - 5*w^4 - 9*w^3 + 19*w^2 + 10*w - 7],
[809, 809, -w^5 + 5*w^3 + w^2 - 7*w],
[809, 809, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 15*w + 11],
[811, 811, -2*w^5 + 4*w^4 + 8*w^3 - 13*w^2 - 6*w + 5],
[811, 811, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - w],
[811, 811, 2*w^5 - 4*w^4 - 9*w^3 + 16*w^2 + 10*w - 9],
[811, 811, 2*w^5 - 4*w^4 - 10*w^3 + 16*w^2 + 10*w - 7],
[821, 821, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 5*w - 2],
[829, 829, 3*w^5 - 5*w^4 - 13*w^3 + 19*w^2 + 12*w - 9],
[839, 839, 2*w^5 - 4*w^4 - 7*w^3 + 14*w^2 + w - 6],
[839, 839, 4*w^5 - 5*w^4 - 18*w^3 + 19*w^2 + 14*w - 10],
[841, 29, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 10*w + 5],
[841, 29, -w^5 + 3*w^4 + 6*w^3 - 12*w^2 - 10*w + 6],
[859, 859, 3*w^5 - 3*w^4 - 15*w^3 + 10*w^2 + 14*w - 2],
[881, 881, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 4],
[911, 911, -4*w^5 + 5*w^4 + 19*w^3 - 20*w^2 - 17*w + 11],
[919, 919, 2*w^5 - 3*w^4 - 10*w^3 + 13*w^2 + 9*w - 10],
[929, 929, -w^5 + w^4 + 5*w^3 - 2*w^2 - 8*w - 1],
[941, 941, -5*w^5 + 7*w^4 + 23*w^3 - 26*w^2 - 20*w + 11],
[961, 31, -3*w^5 + 4*w^4 + 16*w^3 - 16*w^2 - 20*w + 8],
[971, 971, w^5 - 5*w^3 + 5*w - 4],
[991, 991, 3*w^5 - 5*w^4 - 12*w^3 + 18*w^2 + 8*w - 6],
[991, 991, 3*w^5 - 5*w^4 - 14*w^3 + 18*w^2 + 12*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + 3*x^4 - 11*x^3 - 33*x^2 + 12*x + 40;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^2 + e - 6, e^2 - e - 6, -1/2*e^4 + 13/2*e^2 - 12, -1/2*e^3 + 11/2*e + 3, -1/2*e^4 - e^3 + 11/2*e^2 + 10*e - 7, -1/2*e^3 + 7/2*e + 3, 5, 1/2*e^4 - 9/2*e^2 + e + 4, 1/2*e^4 - e^3 - 11/2*e^2 + 10*e + 9, 1/2*e^4 - 13/2*e^2 - 2*e + 7, e^4 + e^3 - 10*e^2 - 9*e + 10, -1/2*e^4 + 9/2*e^2 - 3*e + 1, e^4 + e^3 - 13*e^2 - 8*e + 25, -1, -1/2*e^4 + 15/2*e^2 - 22, -1/2*e^4 - 1/2*e^3 + 9/2*e^2 + 7/2*e - 1, -1/2*e^4 + e^3 + 9/2*e^2 - 9*e + 2, -1/2*e^4 + 13/2*e^2 - 16, -1/2*e^4 - e^3 + 11/2*e^2 + 6*e - 12, -e^4 - 1/2*e^3 + 12*e^2 + 13/2*e - 17, -e^2 + 2*e + 6, -e, -3/2*e^4 - e^3 + 35/2*e^2 + 12*e - 18, e^3 + e^2 - 10*e - 4, -1/2*e^4 + 9/2*e^2 + 3*e + 10, 1/2*e^4 - 13/2*e^2 - 2*e + 7, e^4 + e^3 - 12*e^2 - 8*e + 22, -e^3 - e^2 + 8*e + 14, 3/2*e^4 + e^3 - 37/2*e^2 - 8*e + 36, -1/2*e^4 + 9/2*e^2 + 2*e - 4, -e^4 + 12*e^2 - 3*e - 22, 1/2*e^4 + 2*e^3 - 11/2*e^2 - 18*e + 12, e^4 + e^3 - 14*e^2 - 8*e + 32, 3/2*e^4 + e^3 - 33/2*e^2 - 9*e + 22, 3/2*e^4 + 2*e^3 - 35/2*e^2 - 16*e + 26, e^3 - e^2 - 14*e + 4, -1/2*e^3 + 7/2*e + 7, e^4 + 1/2*e^3 - 9*e^2 - 7/2*e - 5, -1/2*e^4 - e^3 + 15/2*e^2 + 11*e - 20, -e^4 - 2*e^3 + 12*e^2 + 17*e - 12, 3/2*e^4 - 37/2*e^2 - 4*e + 28, -5/2*e^4 - 5/2*e^3 + 55/2*e^2 + 45/2*e - 37, -1/2*e^4 + 5/2*e^2 - 2*e + 21, -1/2*e^4 - 1/2*e^3 + 15/2*e^2 + 9/2*e - 9, -3*e^2 - e + 22, -e^4 + 11*e^2 + 6*e - 6, 1/2*e^4 + 3/2*e^3 - 9/2*e^2 - 17/2*e - 5, e^4 - 1/2*e^3 - 13*e^2 - 1/2*e + 29, e^4 - 14*e^2 - e + 35, -1/2*e^4 + e^3 + 15/2*e^2 - 13*e - 18, 1/2*e^4 + 2*e^3 - 13/2*e^2 - 20*e + 6, 1/2*e^4 + e^3 - 13/2*e^2 - 10*e + 8, -e^3 + 5*e - 2, 1/2*e^4 + 3/2*e^3 - 13/2*e^2 - 21/2*e + 15, -e^3 + e^2 + 4*e - 18, -1/2*e^4 - 2*e^3 + 7/2*e^2 + 18*e - 2, -2*e^4 - 5/2*e^3 + 20*e^2 + 43/2*e - 19, -e^4 + 9*e^2 + 4, e^4 - e^3 - 13*e^2 + 7*e + 42, 1/2*e^4 + e^3 - 11/2*e^2 - 2*e + 14, e^4 - 2*e^3 - 13*e^2 + 18*e + 30, 2*e^4 - 27*e^2 - 3*e + 68, 3/2*e^4 + 4*e^3 - 31/2*e^2 - 31*e + 18, -2*e^4 - 2*e^3 + 26*e^2 + 17*e - 44, e^3 - e^2 - 13*e - 7, -1/2*e^4 + e^3 + 13/2*e^2 - 15*e - 13, 2*e^4 + e^3 - 22*e^2 - 7*e + 17, -2*e^3 + 18*e + 5, 1/2*e^4 + e^3 - 7/2*e^2 - 6*e + 8, -1/2*e^4 - 4*e^3 + 9/2*e^2 + 30*e, 1/2*e^4 + 4*e^3 - 11/2*e^2 - 33*e + 10, -7/2*e^4 - 3*e^3 + 75/2*e^2 + 26*e - 44, -3/2*e^4 + 35/2*e^2 + e - 40, 1/2*e^4 - e^3 - 19/2*e^2 + 3*e + 45, 3/2*e^4 + e^3 - 37/2*e^2 + e + 42, 2*e^4 + e^3 - 17*e^2 - 4*e - 6, 5/2*e^4 + 5/2*e^3 - 57/2*e^2 - 47/2*e + 33, 1/2*e^4 + 1/2*e^3 - 13/2*e^2 - 23/2*e + 15, -1/2*e^4 - e^3 + 11/2*e^2 + 11*e - 28, -3/2*e^4 - 2*e^3 + 41/2*e^2 + 15*e - 45, -5/2*e^4 - 3*e^3 + 53/2*e^2 + 22*e - 34, -1/2*e^4 + 4*e^3 + 15/2*e^2 - 34*e - 8, -3*e^4 - 3*e^3 + 35*e^2 + 32*e - 52, 1/2*e^4 + e^3 - 7/2*e^2 - 4*e + 4, -5/2*e^4 + e^3 + 65/2*e^2 - 4*e - 66, -3/2*e^4 + 1/2*e^3 + 39/2*e^2 - 27/2*e - 49, -e^4 + e^3 + 12*e^2 - 13*e - 15, 1/2*e^4 + 5/2*e^3 - 15/2*e^2 - 45/2*e + 9, -e^4 + 3*e^3 + 14*e^2 - 24*e - 28, -2*e^4 - 2*e^3 + 18*e^2 + 16*e + 8, -5/2*e^4 - 3*e^3 + 49/2*e^2 + 20*e - 6, e^4 - 3*e^3 - 9*e^2 + 27*e - 3, -e^4 + 14*e^2 - 2*e - 48, 2*e^3 + 2*e^2 - 18*e - 12, e^3 + e^2 - 6*e - 23, 1/2*e^4 + e^3 - 15/2*e^2 - 14*e + 10, 1/2*e^4 - 3*e^3 - 7/2*e^2 + 26*e - 4, e^4 - e^3 - 11*e^2 + 13*e + 36, e^3 - e^2 - 14*e + 32, -3/2*e^4 - 5/2*e^3 + 35/2*e^2 + 27/2*e - 25, 2*e^4 + 2*e^3 - 26*e^2 - 26*e + 50, -1/2*e^4 + 13/2*e^2 - 6*e - 7, 2*e^3 + e^2 - 21*e + 4, 1/2*e^4 - 1/2*e^3 - 5/2*e^2 + 15/2*e - 1, 5/2*e^4 - 61/2*e^2 - 2*e + 47, 3/2*e^4 - 3/2*e^3 - 35/2*e^2 + 21/2*e + 1, -7/2*e^4 - 3/2*e^3 + 77/2*e^2 + 23/2*e - 61, 1/2*e^4 - 7/2*e^2 - 3*e - 30, -e^4 - 2*e^3 + 10*e^2 + 18*e + 5, -7/2*e^4 + 89/2*e^2 + 3*e - 91, -7/2*e^4 - 2*e^3 + 77/2*e^2 + 27*e - 32, 5/2*e^3 + 2*e^2 - 35/2*e - 27, 1/2*e^4 + 3/2*e^3 - 7/2*e^2 - 15/2*e + 9, 1/2*e^4 + e^3 + 3/2*e^2 - 7*e - 23, 2*e^4 + e^3 - 23*e^2 - 12*e + 14, e^4 + 2*e^3 - 9*e^2 - 22*e + 6, e^4 - 2*e^3 - 18*e^2 + 16*e + 57, 3*e^4 + 5/2*e^3 - 33*e^2 - 59/2*e + 39, e^4 + 3*e^3 - 13*e^2 - 19*e + 34, -3*e^4 - 3*e^3 + 34*e^2 + 27*e - 66, 1/2*e^4 - 23/2*e^2 + 2*e + 36, -5/2*e^4 - e^3 + 63/2*e^2 + 11*e - 32, -5/2*e^4 - 11/2*e^3 + 49/2*e^2 + 93/2*e - 19, -e^3 - 4*e^2 + e + 26, 2*e^4 + 2*e^3 - 20*e^2 - 24*e + 26, 1/2*e^4 + e^3 - 5/2*e^2 - 15*e - 20, 1/2*e^4 + 2*e^3 - 13/2*e^2 - 27*e - 2, -1/2*e^4 - 1/2*e^2 - 3*e + 16, -e^4 + 2*e^3 + 10*e^2 - 13*e - 4, -7/2*e^4 - e^3 + 75/2*e^2 + 2*e - 64, 3*e^4 + e^3 - 33*e^2 - 14*e + 21, 1/2*e^4 + 1/2*e^3 - 3/2*e^2 - 5/2*e - 3, e^3 - 3*e^2 - 5*e + 22, 7/2*e^4 - 79/2*e^2 + 4*e + 56, -2*e^4 + e^3 + 26*e^2 - 11*e - 62, -2*e^4 - 3*e^3 + 15*e^2 + 26*e + 10, 1/2*e^4 - 9/2*e^2 - 12*e - 4, -7/2*e^3 - 5*e^2 + 47/2*e + 29, 3*e^4 + 4*e^3 - 32*e^2 - 41*e + 38, e^3 - 7*e + 10, 1/2*e^4 + 5/2*e^3 - 7/2*e^2 - 41/2*e + 15, 7/2*e^4 + 5/2*e^3 - 81/2*e^2 - 49/2*e + 41, -5*e^4 - 5*e^3 + 54*e^2 + 44*e - 65, e^4 - e^3 - 12*e^2 + 11*e + 19, -e^4 - e^3 + 11*e^2 + 21*e - 9, 1/2*e^4 + 2*e^3 - 17/2*e^2 - 22*e - 5, -3/2*e^4 + e^3 + 27/2*e^2 - 10*e - 8, -3*e^4 - e^3 + 40*e^2 + 14*e - 72];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {79, 79, -3*w^5 + 4*w^4 + 13*w^3 - 14*w^2 - 9*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;