/*
  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![31, 0, -12, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, w^2 - 6}>;

primesArray := [
[4, 2, -1/2*w^3 - 1/2*w^2 + 7/2*w + 11/2],
[5, 5, -1/2*w^3 + w^2 + 5/2*w - 4],
[5, 5, -1/2*w^2 - w + 3/2],
[9, 3, -1/2*w^3 + 1/2*w^2 + 7/2*w - 9/2],
[9, 3, -1/2*w^3 - 1/2*w^2 + 7/2*w + 9/2],
[19, 19, -1/2*w^2 + w + 5/2],
[19, 19, 1/2*w^2 + w - 5/2],
[29, 29, -3/2*w^2 - w + 17/2],
[29, 29, -3/2*w^2 + w + 17/2],
[31, 31, 1/2*w^3 - 7/2*w],
[59, 59, 1/2*w^2 + w - 11/2],
[59, 59, 1/2*w^2 - w - 11/2],
[61, 61, -1/2*w^3 + w^2 + 7/2*w - 5],
[61, 61, 1/2*w^3 + w^2 - 7/2*w - 5],
[71, 71, -1/2*w^3 - 2*w^2 + 11/2*w + 13],
[71, 71, 3/2*w^3 + 2*w^2 - 23/2*w - 18],
[79, 79, 1/2*w^3 + 1/2*w^2 - 7/2*w - 1/2],
[79, 79, -2*w^2 + w + 10],
[79, 79, 2*w^2 + w - 10],
[79, 79, 1/2*w^3 - 1/2*w^2 - 7/2*w + 1/2],
[89, 89, 1/2*w^3 - 2*w^2 - 5/2*w + 11],
[89, 89, -1/2*w^2 + w + 15/2],
[109, 109, -w^3 + 7*w + 3],
[109, 109, 1/2*w^3 - 2*w^2 - 9/2*w + 12],
[109, 109, w^3 + w^2 - 8*w - 11],
[109, 109, w^3 - 7*w + 3],
[121, 11, 3/2*w^2 - 19/2],
[121, 11, -1/2*w^2 + 13/2],
[131, 131, -1/2*w^3 + 3/2*w^2 + 7/2*w - 15/2],
[131, 131, -1/2*w^3 + 9/2*w + 3],
[179, 179, -w^3 - 5/2*w^2 + 9*w + 35/2],
[179, 179, 1/2*w^3 + 2*w^2 - 7/2*w - 13],
[179, 179, -1/2*w^3 + 2*w^2 + 7/2*w - 13],
[179, 179, 3/2*w^3 + 4*w^2 - 27/2*w - 29],
[181, 181, -1/2*w^3 + w^2 + 5/2*w - 7],
[181, 181, 1/2*w^3 + w^2 - 5/2*w - 7],
[191, 191, 2*w^2 + w - 12],
[191, 191, 2*w^2 - w - 12],
[211, 211, w^3 - w^2 - 7*w + 4],
[211, 211, -w^3 - w^2 + 7*w + 4],
[229, 229, 3/2*w^2 + w - 23/2],
[229, 229, 3/2*w^2 - w - 23/2],
[241, 241, 1/2*w^3 + 3/2*w^2 - 9/2*w - 17/2],
[241, 241, -1/2*w^3 + 3/2*w^2 + 9/2*w - 17/2],
[251, 251, w^3 - 5*w + 3],
[251, 251, 3/2*w^3 - 23/2*w + 2],
[251, 251, 5/2*w^2 + 3*w - 23/2],
[251, 251, w^3 + 3*w^2 - 9*w - 24],
[269, 269, 1/2*w^2 + 2*w - 9/2],
[269, 269, 1/2*w^2 - 2*w - 9/2],
[271, 271, 1/2*w^3 - 1/2*w^2 - 3/2*w - 3/2],
[271, 271, w^3 + 3/2*w^2 - 6*w - 15/2],
[271, 271, 1/2*w^3 + 3*w^2 - 9/2*w - 16],
[271, 271, -1/2*w^3 - 2*w^2 + 5/2*w + 13],
[281, 281, w^3 - 5/2*w^2 - 8*w + 33/2],
[281, 281, w^2 + 2*w - 6],
[281, 281, w^2 - 2*w - 6],
[281, 281, w^3 + 5/2*w^2 - 8*w - 33/2],
[311, 311, -5/2*w^3 - 3*w^2 + 37/2*w + 30],
[311, 311, 2*w^2 - 2*w - 13],
[331, 331, -1/2*w^3 + 7/2*w - 5],
[331, 331, -w^3 + 5/2*w^2 + 7*w - 29/2],
[331, 331, w^3 + 5/2*w^2 - 7*w - 29/2],
[331, 331, 1/2*w^3 - 7/2*w - 5],
[349, 349, 1/2*w^3 - 2*w^2 - 11/2*w + 16],
[349, 349, w^3 + 3*w^2 - 6*w - 19],
[349, 349, 1/2*w^3 - w^2 - 1/2*w - 1],
[349, 349, 1/2*w^3 + 2*w^2 - 11/2*w - 16],
[359, 359, 1/2*w^3 + 3/2*w^2 - 11/2*w - 15/2],
[359, 359, w^3 - 4*w^2 - 2*w + 16],
[361, 19, -1/2*w^2 + 15/2],
[379, 379, 1/2*w^3 + 5/2*w^2 - 11/2*w - 37/2],
[379, 379, -w^3 + 1/2*w^2 + 6*w - 15/2],
[389, 389, w^3 - 1/2*w^2 - 7*w + 5/2],
[389, 389, w^3 + 1/2*w^2 - 7*w - 5/2],
[401, 401, 1/2*w^3 + 3/2*w^2 - 5/2*w - 21/2],
[401, 401, -1/2*w^3 + 3/2*w^2 + 5/2*w - 21/2],
[409, 409, -2*w^3 - 5*w^2 + 18*w + 38],
[409, 409, 1/2*w^3 - w^2 - 13/2*w + 1],
[419, 419, -1/2*w^3 + 1/2*w^2 + 5/2*w - 19/2],
[419, 419, 1/2*w^3 + 1/2*w^2 - 5/2*w - 19/2],
[431, 431, 1/2*w^3 - 3/2*w^2 - 7/2*w + 11/2],
[431, 431, 1/2*w^3 + 3/2*w^2 - 7/2*w - 11/2],
[439, 439, -1/2*w^3 + 5/2*w^2 + 5/2*w - 27/2],
[439, 439, -1/2*w^3 - 5/2*w^2 + 5/2*w + 27/2],
[449, 449, w^3 - 1/2*w^2 - 7*w + 7/2],
[449, 449, w^3 + 1/2*w^2 - 7*w - 7/2],
[461, 461, w^3 - 7/2*w^2 - 5*w + 37/2],
[461, 461, -1/2*w^2 + 2*w + 21/2],
[479, 479, -w^3 + 4*w^2 + 4*w - 20],
[479, 479, 1/2*w^3 - 3/2*w + 8],
[491, 491, 1/2*w^3 - 5/2*w^2 - 3/2*w + 29/2],
[491, 491, 1/2*w^3 + 1/2*w^2 - 1/2*w - 7/2],
[491, 491, 1/2*w^3 + w^2 - 11/2*w - 8],
[491, 491, -1/2*w^3 + w^2 + 5/2*w - 12],
[499, 499, 3*w^2 - w - 21],
[499, 499, 1/2*w^3 + 5/2*w^2 - 11/2*w - 29/2],
[499, 499, w^3 + 5*w^2 - 11*w - 32],
[499, 499, 3*w^2 + w - 21],
[509, 509, -w^3 + 3/2*w^2 + 4*w - 3/2],
[509, 509, 1/2*w^3 + 3*w^2 - 3/2*w - 18],
[529, 23, 1/2*w^3 - 2*w^2 - 11/2*w + 11],
[529, 23, 1/2*w^3 + 2*w^2 - 11/2*w - 11],
[569, 569, 1/2*w^3 + w^2 - 11/2*w - 3],
[569, 569, -1/2*w^3 + w^2 + 11/2*w - 3],
[571, 571, 5/2*w^2 + 2*w - 33/2],
[571, 571, 2*w^3 - 5/2*w^2 - 16*w + 49/2],
[571, 571, 2*w^3 + 5/2*w^2 - 16*w - 49/2],
[571, 571, 5/2*w^2 - 2*w - 33/2],
[599, 599, 1/2*w^3 - 1/2*w^2 - 11/2*w - 3/2],
[599, 599, -1/2*w^3 - 1/2*w^2 + 11/2*w - 3/2],
[601, 601, 2*w^3 + 3/2*w^2 - 15*w - 35/2],
[601, 601, 5/2*w^3 + 3*w^2 - 39/2*w - 29],
[619, 619, 5/2*w^3 + 5/2*w^2 - 37/2*w - 51/2],
[619, 619, w^3 - 1/2*w^2 - 5*w + 11/2],
[619, 619, 3/2*w^2 - 2*w - 23/2],
[619, 619, w^3 + 3*w^2 - 10*w - 21],
[631, 631, -5/2*w^3 + 33/2*w + 13],
[631, 631, 3/2*w^3 - 5*w^2 - 13/2*w + 22],
[631, 631, -2*w^3 - 2*w^2 + 14*w + 23],
[631, 631, -5/2*w^2 - 5*w + 13/2],
[641, 641, -1/2*w^3 + 1/2*w^2 + 11/2*w - 5/2],
[641, 641, 1/2*w^3 + 1/2*w^2 - 11/2*w - 5/2],
[659, 659, -3/2*w^3 - w^2 + 23/2*w + 14],
[659, 659, 3/2*w^3 + 5/2*w^2 - 19/2*w - 27/2],
[661, 661, -1/2*w^3 + 1/2*w^2 + 11/2*w - 9/2],
[661, 661, w^3 + w^2 - 8*w - 5],
[661, 661, -1/2*w^3 + 2*w^2 + 7/2*w - 8],
[661, 661, -1/2*w^3 - 1/2*w^2 + 11/2*w + 9/2],
[691, 691, 1/2*w^3 - 3/2*w^2 - 3/2*w + 19/2],
[691, 691, 1/2*w^3 + 3/2*w^2 - 3/2*w - 19/2],
[709, 709, 1/2*w^3 - 2*w^2 - 3/2*w + 12],
[709, 709, 1/2*w^3 + 2*w^2 - 3/2*w - 12],
[751, 751, 3/2*w^3 - w^2 - 21/2*w + 11],
[751, 751, 3/2*w^3 + w^2 - 21/2*w - 11],
[761, 761, 2*w^3 + 11/2*w^2 - 17*w - 85/2],
[761, 761, -3/2*w^3 - 4*w^2 + 17/2*w + 23],
[769, 769, w^3 - 7/2*w^2 - 8*w + 45/2],
[769, 769, -1/2*w^3 - 5/2*w^2 + 5/2*w + 35/2],
[769, 769, -1/2*w^3 + 5/2*w^2 + 5/2*w - 35/2],
[769, 769, w^3 + 7/2*w^2 - 8*w - 45/2],
[809, 809, w^3 - 7/2*w^2 - 7*w + 41/2],
[809, 809, w^3 + 7/2*w^2 - 7*w - 41/2],
[811, 811, -1/2*w^3 - 9/2*w^2 + 13/2*w + 59/2],
[811, 811, 3*w^3 + 5*w^2 - 24*w - 45],
[821, 821, -2*w^3 - 7/2*w^2 + 12*w + 33/2],
[821, 821, w^3 - 4*w^2 - w + 13],
[829, 829, -w^3 + 1/2*w^2 + 8*w - 1/2],
[829, 829, w^3 + 1/2*w^2 - 8*w - 1/2],
[839, 839, w^3 - 3/2*w^2 - 7*w + 13/2],
[839, 839, w^3 + 3/2*w^2 - 7*w - 13/2],
[841, 29, 5/2*w^2 - 27/2],
[859, 859, 1/2*w^3 - 7/2*w - 6],
[859, 859, -3/2*w^3 + 2*w^2 + 21/2*w - 15],
[859, 859, 3/2*w^3 + 2*w^2 - 21/2*w - 15],
[859, 859, 1/2*w^3 - 7/2*w + 6],
[881, 881, -3/2*w^3 + 21/2*w + 8],
[881, 881, -2*w^3 + 1/2*w^2 + 13*w + 15/2],
[919, 919, -3/2*w^3 + 5/2*w^2 + 15/2*w - 19/2],
[919, 919, 3/2*w^3 + w^2 - 21/2*w - 4],
[929, 929, 1/2*w^3 + w^2 - 9/2*w - 1],
[929, 929, 1/2*w^3 - w^2 - 9/2*w + 1],
[941, 941, 2*w^3 + 7*w^2 - 19*w - 51],
[941, 941, 5/2*w^2 - 3*w - 33/2],
[961, 31, -w^2 + 12],
[971, 971, 3/2*w^2 - 3*w - 25/2],
[971, 971, w^3 + 7/2*w^2 - 6*w - 39/2],
[991, 991, 1/2*w^3 - 3/2*w^2 - 13/2*w + 13/2],
[991, 991, w^3 + 5/2*w^2 - 5*w - 29/2],
[991, 991, w^3 - 5/2*w^2 - 5*w + 29/2],
[991, 991, -3/2*w^3 + 2*w^2 + 17/2*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 2*x^3 - 24*x^2 - 62*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-2, -1, 1, e, 1/6*e^3 + 1/6*e^2 - 25/6*e - 31/6, -1/12*e^3 + 5/12*e^2 + 7/12*e - 83/12, -1/4*e^3 + 1/4*e^2 + 23/4*e + 17/4, 1/3*e^3 - 1/6*e^2 - 22/3*e - 41/6, -1/6*e^3 + 1/3*e^2 + 13/6*e - 13/3, 1/6*e^3 - 5/6*e^2 - 19/6*e + 47/6, 1/4*e^3 + 1/4*e^2 - 19/4*e - 43/4, -1/4*e^3 - 1/4*e^2 + 19/4*e + 19/4, -1/4*e^3 + 1/4*e^2 + 11/4*e + 5/4, -7/12*e^3 - 1/12*e^2 + 157/12*e + 139/12, -1/4*e^3 - 3/4*e^2 + 19/4*e + 57/4, -1/4*e^3 - 3/4*e^2 + 19/4*e + 57/4, 1/3*e^3 + 1/3*e^2 - 16/3*e - 31/3, -7/12*e^3 - 1/12*e^2 + 145/12*e + 115/12, 5/12*e^3 - 1/12*e^2 - 83/12*e - 101/12, 1/2*e^3 + 1/2*e^2 - 21/2*e - 31/2, -5/12*e^3 + 7/12*e^2 + 119/12*e + 35/12, 1/12*e^3 - 11/12*e^2 + 5/12*e + 161/12, e^3 - e^2 - 20*e - 11, -2/3*e^3 + 1/3*e^2 + 38/3*e + 23/3, -2/3*e^3 + 1/3*e^2 + 38/3*e + 23/3, 5/6*e^3 - 7/6*e^2 - 89/6*e + 37/6, 1/12*e^3 + 1/12*e^2 - 31/12*e - 91/12, 1/6*e^3 + 1/6*e^2 - 31/6*e - 97/6, 2/3*e^3 - 5/6*e^2 - 44/3*e - 43/6, 1/3*e^3 - 7/6*e^2 - 13/3*e + 91/6, 1/12*e^3 - 17/12*e^2 + 5/12*e + 239/12, -1/6*e^3 + 5/6*e^2 + 31/6*e - 83/6, -1/6*e^3 - 7/6*e^2 + 31/6*e + 73/6, 5/12*e^3 - 13/12*e^2 - 119/12*e + 43/12, 7/12*e^3 - 17/12*e^2 - 145/12*e - 1/12, -5/12*e^3 + 19/12*e^2 + 83/12*e - 253/12, -1/6*e^3 + 5/6*e^2 + 31/6*e - 101/6, -1/6*e^3 - 7/6*e^2 + 31/6*e + 55/6, -1/2*e^3 + e^2 + 23/2*e + 8, -1/6*e^3 + 4/3*e^2 + 7/6*e - 79/3, 2/3*e^3 - 11/6*e^2 - 35/3*e + 41/6, -5/6*e^3 + 5/3*e^2 + 101/6*e - 35/3, -11/6*e^3 + 7/6*e^2 + 221/6*e + 143/6, -3/2*e^3 + 3/2*e^2 + 53/2*e + 3/2, -4/3*e^3 + 2/3*e^2 + 70/3*e + 46/3, -1/12*e^3 - 7/12*e^2 + 67/12*e + 109/12, 7/12*e^3 + 1/12*e^2 - 181/12*e - 139/12, 5/3*e^3 - 1/3*e^2 - 101/3*e - 77/3, -3/4*e^3 - 5/4*e^2 + 69/4*e + 59/4, -1/4*e^3 - 3/4*e^2 + 7/4*e + 117/4, -1/12*e^3 + 23/12*e^2 + 31/12*e - 305/12, 3/2*e^3 - 61/2*e - 21, 7/6*e^3 - 1/3*e^2 - 121/6*e - 68/3, -1/12*e^3 - 25/12*e^2 + 31/12*e + 319/12, 2/3*e^3 + 1/6*e^2 - 44/3*e - 157/6, 5/12*e^3 - 31/12*e^2 - 47/12*e + 385/12, 13/12*e^3 - 23/12*e^2 - 295/12*e - 7/12, 1/3*e^3 - 1/6*e^2 - 13/3*e + 49/6, -3/2*e^3 + 1/2*e^2 + 69/2*e + 59/2, 1/2*e^3 - 3/2*e^2 - 7/2*e + 39/2, -1/2*e^3 + 1/2*e^2 + 19/2*e - 17/2, -2/3*e^3 + 1/3*e^2 + 50/3*e + 59/3, e^2 - 4*e - 25, 1/2*e^3 - 1/2*e^2 - 19/2*e - 27/2, -5/4*e^3 + 3/4*e^2 + 119/4*e + 95/4, 17/12*e^3 - 1/12*e^2 - 383/12*e - 389/12, -7/12*e^3 + 11/12*e^2 + 73/12*e - 113/12, 1/4*e^3 - 7/4*e^2 + 5/4*e + 117/4, -7/12*e^3 - 13/12*e^2 + 109/12*e + 331/12, -11/12*e^3 - 17/12*e^2 + 233/12*e + 383/12, -1/2*e^3 - 1/2*e^2 + 31/2*e + 17/2, -1/4*e^3 + 3/4*e^2 + 19/4*e + 31/4, 1/4*e^3 - 3/4*e^2 - 19/4*e + 73/4, -1/12*e^3 + 17/12*e^2 - 5/12*e - 383/12, -5/12*e^3 + 13/12*e^2 + 119/12*e + 101/12, 1/6*e^3 + 1/6*e^2 - 31/6*e + 23/6, 1/6*e^3 + 1/6*e^2 - 31/6*e + 23/6, -13/12*e^3 + 5/12*e^2 + 343/12*e + 205/12, 1/4*e^3 + 7/4*e^2 - 51/4*e - 121/4, -11/12*e^3 + 7/12*e^2 + 197/12*e - 37/12, 13/12*e^3 - 5/12*e^2 - 259/12*e - 313/12, 1/6*e^3 - 5/6*e^2 - 31/6*e + 11/6, -1/6*e^3 - 7/6*e^2 + 31/6*e + 145/6, 4/3*e^3 - 7/6*e^2 - 79/3*e - 131/6, -7/6*e^3 + 4/3*e^2 + 127/6*e - 28/3, 1/12*e^3 + 31/12*e^2 - 31/12*e - 313/12, -1/12*e^3 + 29/12*e^2 + 31/12*e - 467/12, -1/6*e^3 - 7/6*e^2 - 5/6*e + 163/6, 5/6*e^3 + 11/6*e^2 - 119/6*e - 179/6, 2/3*e^3 + 1/6*e^2 - 53/3*e - 139/6, 1/6*e^3 + 2/3*e^2 - 49/6*e - 62/3, e^3 - 25*e - 18, -5/6*e^3 + 2/3*e^2 + 119/6*e - 17/3, -1/6*e^3 + 4/3*e^2 - 5/6*e - 7/3, e^2 - 6*e - 13, -1/3*e^3 - 5/6*e^2 + 22/3*e + 95/6, -5/6*e^3 + 13/6*e^2 + 77/6*e - 151/6, -4/3*e^3 + 5/3*e^2 + 85/3*e + 43/3, 1/6*e^3 + 2/3*e^2 - 13/6*e - 38/3, -1/6*e^3 - 7/6*e^2 + 31/6*e + 37/6, -1/6*e^3 + 5/6*e^2 + 31/6*e - 119/6, 1/6*e^3 - 5/6*e^2 - 43/6*e - 25/6, -1/2*e^3 - 3/2*e^2 + 27/2*e + 81/2, 2*e^3 - 44*e - 42, e^3 - e^2 - 13*e + 1, 17/12*e^3 - 31/12*e^2 - 311/12*e + 193/12, 7/12*e^3 + 1/12*e^2 - 157/12*e + 149/12, 1/4*e^3 - 1/4*e^2 - 11/4*e - 101/4, -19/12*e^3 + 29/12*e^2 + 373/12*e + 61/12, -17/12*e^3 + 13/12*e^2 + 383/12*e + 389/12, -7/12*e^3 + 23/12*e^2 + 73/12*e - 425/12, 3/4*e^3 - 9/4*e^2 - 49/4*e + 39/4, 13/12*e^3 - 23/12*e^2 - 271/12*e + 137/12, 1/2*e^3 + 3/2*e^2 - 23/2*e - 93/2, -3/2*e^3 + 2*e^2 + 51/2*e - 6, 2*e^3 - 3/2*e^2 - 41*e - 47/2, 1/6*e^3 + 7/6*e^2 - 7/6*e - 1/6, -1/3*e^3 + 7/6*e^2 + 7/3*e - 55/6, -1/6*e^3 + 5/6*e^2 - 5/6*e + 13/6, 5/6*e^3 - 1/6*e^2 - 119/6*e - 17/6, -e^3 + 1/2*e^2 + 23*e + 23/2, 5/6*e^3 + 1/3*e^2 - 119/6*e - 22/3, 1/6*e^3 - 1/3*e^2 + 5/6*e - 32/3, 3*e - 12, 1/2*e^3 + 1/2*e^2 - 25/2*e - 7/2, -2/3*e^3 + 17/6*e^2 + 32/3*e - 77/6, 1/12*e^3 + 1/12*e^2 - 43/12*e - 175/12, -1/4*e^3 - 1/4*e^2 + 27/4*e + 79/4, -e^3 + 5/2*e^2 + 21*e - 53/2, -5/12*e^3 + 7/12*e^2 + 155/12*e + 47/12, -5/12*e^3 - 17/12*e^2 + 155/12*e + 359/12, 1/6*e^3 - 29/6*e^2 + 5/6*e + 299/6, -5/6*e^3 + 25/6*e^2 + 119/6*e - 295/6, 5/6*e^3 - 7/6*e^2 - 59/6*e - 35/6, 11/6*e^3 - 1/6*e^2 - 245/6*e - 149/6, -1/12*e^3 - 1/12*e^2 - 41/12*e + 175/12, 11/12*e^3 + 11/12*e^2 - 269/12*e - 197/12, 17/12*e^3 - 1/12*e^2 - 311/12*e - 341/12, 1/3*e^3 - 13/6*e^2 - 4/3*e + 103/6, -7/6*e^3 + 4/3*e^2 + 163/6*e - 4/3, -19/12*e^3 - 1/12*e^2 + 373/12*e + 307/12, -4/3*e^3 - 5/6*e^2 + 97/3*e + 155/6, 1/6*e^3 - 1/3*e^2 + 23/6*e - 23/3, 3/2*e^3 + 3/2*e^2 - 81/2*e - 113/2, 1/2*e^3 + 1/2*e^2 - 43/2*e - 51/2, -1/3*e^3 + 13/6*e^2 + 31/3*e - 169/6, 1/3*e^3 + 17/6*e^2 - 31/3*e - 221/6, 3/4*e^3 - 3/4*e^2 - 73/4*e - 27/4, 1/12*e^3 - 17/12*e^2 + 29/12*e + 239/12, 1/2*e^3 - 3/2*e^2 - 1/2*e + 21/2, 2*e^3 - 47*e - 36, 11/12*e^3 + 11/12*e^2 - 341/12*e - 689/12, 5/3*e^3 - 5/6*e^2 - 110/3*e - 157/6, -19/12*e^3 - 1/12*e^2 + 361/12*e + 319/12, -19/12*e^3 - 1/12*e^2 + 361/12*e + 391/12, -5/6*e^3 + 5/3*e^2 + 65/6*e - 41/3, -5/12*e^3 + 13/12*e^2 + 119/12*e + 209/12, -1/12*e^3 + 17/12*e^2 - 5/12*e - 491/12, -7/3*e^3 - 1/3*e^2 + 148/3*e + 157/3, -3/2*e^3 + 1/2*e^2 + 47/2*e + 29/2, 4/3*e^3 - 5/3*e^2 - 79/3*e - 61/3, 7/6*e^3 - 11/6*e^2 - 127/6*e + 125/6, 5/4*e^3 - 11/4*e^2 - 59/4*e + 89/4, -11/4*e^3 + 5/4*e^2 + 245/4*e + 169/4, -1/12*e^3 - 1/12*e^2 + 31/12*e - 269/12, -25/12*e^3 - 1/12*e^2 + 487/12*e + 607/12, 23/12*e^3 - 1/12*e^2 - 425/12*e - 257/12, -19/12*e^3 + 53/12*e^2 + 361/12*e - 203/12, 5/3*e^3 - 7/3*e^2 - 119/3*e - 11/3, -1/3*e^3 + 11/3*e^2 - 5/3*e - 137/3, -19/12*e^3 + 53/12*e^2 + 361/12*e - 491/12];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5,5,-1/2*w^3+w^2+5/2*w-4}>] := 1;
ALEigenvalues[ideal<ZF | {5,5,-1/2*w^2-w+3/2}>] := -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;