/*
  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, 0, 8, 4, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 8*w}>;

primesArray := [
[5, 5, w - 1],
[11, 11, w^3 - 3*w],
[19, 19, -w^5 + w^4 + 5*w^3 - 4*w^2 - 5*w],
[25, 5, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 8*w],
[29, 29, -w^4 + 4*w^2 + 1],
[31, 31, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 3*w - 4],
[31, 31, -w^2 + 2],
[59, 59, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w - 2],
[59, 59, w^4 - w^3 - 5*w^2 + 3*w + 4],
[59, 59, -w^4 - w^3 + 4*w^2 + 5*w],
[61, 61, w^5 - w^4 - 4*w^3 + 5*w^2 - 4],
[64, 2, 2],
[71, 71, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w],
[71, 71, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 + w + 4],
[71, 71, 2*w^5 - 3*w^4 - 9*w^3 + 13*w^2 + 4*w - 4],
[71, 71, w^3 - 5*w],
[79, 79, w^5 - w^4 - 4*w^3 + 4*w^2 + w - 2],
[89, 89, w^3 - w^2 - 4*w + 1],
[101, 101, -2*w^5 + 3*w^4 + 11*w^3 - 13*w^2 - 12*w + 2],
[101, 101, 2*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 20*w + 4],
[101, 101, w^5 - w^4 - 5*w^3 + 5*w^2 + 2*w - 3],
[101, 101, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 1],
[109, 109, -w^5 + w^4 + 7*w^3 - 4*w^2 - 11*w - 1],
[121, 11, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 4],
[131, 131, w^4 - 4*w^2 + w - 1],
[131, 131, -2*w^5 + 2*w^4 + 10*w^3 - 9*w^2 - 7*w + 2],
[139, 139, w^4 + w^3 - 3*w^2 - 4*w - 4],
[149, 149, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 8*w + 2],
[149, 149, 2*w^5 - 11*w^3 - 2*w^2 + 11*w + 6],
[151, 151, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 16*w - 6],
[169, 13, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 6*w - 3],
[179, 179, w^5 - 2*w^4 - 7*w^3 + 9*w^2 + 12*w - 1],
[179, 179, 2*w^5 - 4*w^4 - 9*w^3 + 18*w^2 + 4*w - 7],
[179, 179, -3*w^5 + 3*w^4 + 15*w^3 - 13*w^2 - 12*w + 3],
[179, 179, -w^5 + 3*w^4 + 5*w^3 - 14*w^2 - 5*w + 6],
[181, 181, w^3 + w^2 - 4*w - 6],
[191, 191, -w^5 + w^4 + 4*w^3 - 5*w^2 + w + 3],
[191, 191, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 3],
[191, 191, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 3],
[191, 191, -w^5 + w^4 + 7*w^3 - 4*w^2 - 12*w - 2],
[199, 199, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w - 1],
[199, 199, w^4 + w^3 - 5*w^2 - 3*w + 3],
[199, 199, -w^5 + w^4 + 3*w^3 - 4*w^2 + 3*w + 2],
[199, 199, w^2 + w - 3],
[211, 211, 3*w^5 - 4*w^4 - 16*w^3 + 17*w^2 + 16*w - 3],
[239, 239, -3*w^5 + 3*w^4 + 16*w^3 - 12*w^2 - 16*w + 3],
[239, 239, 4*w^5 - 6*w^4 - 21*w^3 + 25*w^2 + 19*w - 5],
[239, 239, -w^5 + w^4 + 7*w^3 - 5*w^2 - 10*w + 2],
[241, 241, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 4*w + 2],
[251, 251, 3*w^5 - 4*w^4 - 16*w^3 + 18*w^2 + 15*w - 5],
[251, 251, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 2],
[251, 251, -3*w^5 + 4*w^4 + 16*w^3 - 17*w^2 - 15*w + 2],
[251, 251, -w^5 + 2*w^4 + 4*w^3 - 9*w^2 - w + 4],
[269, 269, w^5 - w^4 - 6*w^3 + 6*w^2 + 7*w - 5],
[271, 271, 3*w^5 - 3*w^4 - 15*w^3 + 12*w^2 + 11*w - 1],
[271, 271, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 2*w],
[281, 281, w^2 - w - 4],
[281, 281, -3*w^5 + 4*w^4 + 16*w^3 - 18*w^2 - 14*w + 4],
[281, 281, w^5 - w^4 - 6*w^3 + 4*w^2 + 9*w],
[281, 281, -3*w^5 + 3*w^4 + 14*w^3 - 13*w^2 - 8*w + 4],
[289, 17, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 10*w - 1],
[311, 311, -w^5 - w^4 + 6*w^3 + 6*w^2 - 8*w - 5],
[311, 311, -2*w^5 + 3*w^4 + 10*w^3 - 13*w^2 - 8*w + 7],
[331, 331, 2*w^5 - 3*w^4 - 11*w^3 + 12*w^2 + 12*w - 4],
[331, 331, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 12*w + 3],
[331, 331, -w^5 + w^4 + 4*w^3 - 3*w^2 + w - 1],
[331, 331, -2*w^5 + 5*w^4 + 10*w^3 - 21*w^2 - 8*w + 5],
[359, 359, 2*w^4 - 2*w^3 - 9*w^2 + 7*w + 3],
[361, 19, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],
[379, 379, 3*w^5 - 2*w^4 - 15*w^3 + 8*w^2 + 11*w - 1],
[379, 379, 2*w^5 - 2*w^4 - 11*w^3 + 8*w^2 + 13*w - 3],
[389, 389, w^5 + w^4 - 5*w^3 - 5*w^2 + 3*w + 4],
[389, 389, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 16*w - 5],
[389, 389, -w^5 + 4*w^4 + 5*w^3 - 17*w^2 - 4*w + 4],
[389, 389, 3*w^5 - 4*w^4 - 15*w^3 + 18*w^2 + 10*w - 6],
[401, 401, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 8*w + 3],
[401, 401, -2*w^5 + 4*w^4 + 11*w^3 - 17*w^2 - 13*w + 3],
[409, 409, w^3 + w^2 - 3*w - 5],
[419, 419, -w^5 + w^4 + 4*w^3 - 6*w^2 + 4],
[419, 419, -w^4 + 2*w^3 + 5*w^2 - 8*w - 6],
[421, 421, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 6],
[431, 431, 2*w^3 + w^2 - 9*w - 3],
[439, 439, -w^5 + 6*w^3 + w^2 - 6*w - 4],
[439, 439, w^5 - 2*w^4 - 4*w^3 + 9*w^2 - 7],
[439, 439, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 14*w - 4],
[439, 439, -2*w^5 + 3*w^4 + 11*w^3 - 11*w^2 - 13*w - 1],
[439, 439, w^4 - w^3 - 3*w^2 + 3*w - 3],
[439, 439, -3*w^5 + 4*w^4 + 16*w^3 - 16*w^2 - 17*w + 2],
[449, 449, -w^3 - 2*w^2 + 5*w + 6],
[461, 461, w^5 - w^4 - 6*w^3 + 3*w^2 + 10*w + 1],
[479, 479, 3*w^5 - 5*w^4 - 16*w^3 + 21*w^2 + 15*w - 2],
[491, 491, w^5 - w^4 - 6*w^3 + 3*w^2 + 10*w + 2],
[499, 499, w^5 - 5*w^3 - 2*w^2 + 4*w + 4],
[509, 509, -w^5 + 2*w^4 + 5*w^3 - 10*w^2 - 4*w + 4],
[509, 509, -w^5 + w^4 + 6*w^3 - 6*w^2 - 7*w + 4],
[521, 521, -3*w^5 + 5*w^4 + 16*w^3 - 22*w^2 - 14*w + 5],
[541, 541, -w^4 + w^3 + 5*w^2 - 2*w - 5],
[541, 541, 2*w^5 - 3*w^4 - 9*w^3 + 14*w^2 + 4*w - 5],
[541, 541, -2*w^5 + 3*w^4 + 9*w^3 - 13*w^2 - 2*w + 4],
[569, 569, 2*w^5 - w^4 - 11*w^3 + 5*w^2 + 11*w - 3],
[571, 571, w^5 - w^4 - 6*w^3 + 2*w^2 + 8*w + 4],
[571, 571, 3*w^5 - 5*w^4 - 15*w^3 + 22*w^2 + 10*w - 7],
[599, 599, -w^4 + w^3 + 3*w^2 - 4*w],
[599, 599, -w^4 + 6*w^2 + w - 4],
[601, 601, -2*w^5 + 3*w^4 + 10*w^3 - 12*w^2 - 7*w + 4],
[619, 619, -3*w^5 + 4*w^4 + 15*w^3 - 16*w^2 - 11*w + 2],
[631, 631, w^4 + 2*w^3 - 5*w^2 - 8*w + 2],
[631, 631, -w^5 + 4*w^4 + 5*w^3 - 18*w^2 - 4*w + 6],
[631, 631, w^4 + 2*w^3 - 4*w^2 - 8*w + 1],
[641, 641, -w^5 - w^4 + 5*w^3 + 5*w^2 - 5*w - 2],
[641, 641, -w^4 + 6*w^2 + w - 5],
[641, 641, w^5 - 2*w^4 - 4*w^3 + 9*w^2 + 2*w - 5],
[659, 659, -w^4 - 2*w^3 + 5*w^2 + 8*w - 1],
[661, 661, 2*w^5 - 2*w^4 - 9*w^3 + 8*w^2 + 3*w + 1],
[661, 661, 2*w^5 - 3*w^4 - 11*w^3 + 12*w^2 + 10*w - 1],
[661, 661, w^5 - 3*w^4 - 6*w^3 + 12*w^2 + 9*w],
[661, 661, -2*w^5 + 3*w^4 + 12*w^3 - 13*w^2 - 14*w + 1],
[691, 691, w^5 - 6*w^3 - w^2 + 9*w + 1],
[691, 691, 3*w^5 - 5*w^4 - 15*w^3 + 21*w^2 + 13*w - 5],
[691, 691, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 6*w + 2],
[691, 691, 2*w^5 - 3*w^4 - 12*w^3 + 12*w^2 + 16*w - 2],
[701, 701, -w^5 - w^4 + 6*w^3 + 4*w^2 - 8*w - 2],
[709, 709, -3*w^5 + 3*w^4 + 14*w^3 - 12*w^2 - 8*w + 3],
[709, 709, w^5 + w^4 - 6*w^3 - 4*w^2 + 8*w + 1],
[709, 709, 2*w^5 - 4*w^4 - 10*w^3 + 16*w^2 + 9*w - 2],
[709, 709, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 14*w - 3],
[719, 719, w^4 - w^3 - 6*w^2 + 3*w + 4],
[729, 3, -3],
[739, 739, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w - 5],
[751, 751, -2*w^5 + 3*w^4 + 11*w^3 - 13*w^2 - 12*w + 1],
[751, 751, -w^5 + w^4 + 4*w^3 - 4*w^2 - 2*w - 1],
[761, 761, -2*w^5 + 2*w^4 + 12*w^3 - 9*w^2 - 15*w],
[769, 769, -3*w^5 + 5*w^4 + 17*w^3 - 21*w^2 - 19*w + 3],
[769, 769, -2*w^5 + 2*w^4 + 9*w^3 - 7*w^2 - 5*w - 1],
[769, 769, -2*w^5 + 4*w^4 + 10*w^3 - 18*w^2 - 9*w + 7],
[769, 769, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 14*w + 2],
[811, 811, -3*w^5 + 2*w^4 + 16*w^3 - 9*w^2 - 15*w + 2],
[821, 821, 2*w^3 - w^2 - 8*w + 1],
[829, 829, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 1],
[829, 829, 2*w^5 - 2*w^4 - 10*w^3 + 9*w^2 + 8*w - 6],
[839, 839, -w^4 - 2*w^3 + 5*w^2 + 6*w - 1],
[841, 29, -2*w^5 + 2*w^4 + 12*w^3 - 9*w^2 - 16*w + 2],
[859, 859, -w^5 + 2*w^4 + 4*w^3 - 10*w^2 - w + 4],
[859, 859, 2*w^5 - 3*w^4 - 10*w^3 + 11*w^2 + 7*w + 1],
[881, 881, -2*w^5 + 3*w^4 + 9*w^3 - 14*w^2 - 4*w + 4],
[881, 881, 2*w^5 - 2*w^4 - 10*w^3 + 10*w^2 + 6*w - 5],
[911, 911, w^5 - w^4 - 4*w^3 + 5*w^2 - 2*w - 5],
[911, 911, w^3 - 4*w - 4],
[929, 929, -w^5 + 3*w^4 + 6*w^3 - 12*w^2 - 9*w + 1],
[941, 941, 2*w^4 - 10*w^2 + w + 6],
[941, 941, 3*w^5 - 3*w^4 - 15*w^3 + 14*w^2 + 10*w - 5],
[941, 941, -w^5 + 5*w^3 + w^2 - 2*w - 4],
[961, 31, -w^4 - w^3 + 6*w^2 + 4*w - 4],
[961, 31, 2*w^5 - 3*w^4 - 12*w^3 + 12*w^2 + 15*w - 3],
[971, 971, -2*w^5 + 3*w^4 + 12*w^3 - 13*w^2 - 14*w],
[991, 991, w^5 - 3*w^4 - 4*w^3 + 14*w^2 + w - 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 4*x^3 - 9*x - 5;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 2*e^3 + 5*e^2 - 7*e - 10, -e^3 - 4*e^2 + e + 5, 1, -e^2 - 4*e + 3, -3*e^3 - 9*e^2 + 7*e + 11, -e^3 - e^2 + 5*e, -2*e - 7, -e^2 + e + 11, -4*e^3 - 11*e^2 + 15*e + 20, -e^3 - e^2 + 5*e - 4, -2*e^3 - 5*e^2 + 11*e + 9, 4*e^3 + 11*e^2 - 11*e - 20, e^3 + 3*e^2 + 2*e - 4, 3*e^3 + 10*e^2 - 9*e - 20, -2*e^3 - 5*e^2 + 7*e + 5, e^3 + 6*e^2 + 5*e - 11, 2*e^3 + 6*e^2 - 8*e - 15, -2*e^3 - 5*e^2 + 4*e - 5, -4*e^3 - 13*e^2 + 9*e + 20, 4*e^3 + 11*e^2 - 15*e - 23, -3*e^3 - 11*e^2 + 3*e + 15, -5*e^3 - 14*e^2 + 16*e + 20, 4*e^3 + 11*e^2 - 17*e - 29, 6*e^3 + 16*e^2 - 14*e - 25, e^3 - 11*e - 4, -4*e^3 - 9*e^2 + 12*e + 10, -7*e^3 - 21*e^2 + 18*e + 34, -e^3 + 15*e - 3, -2*e^3 - 7*e^2 + 11*e + 20, 2*e - 2, 4*e^3 + 14*e^2 - 4*e - 17, 3*e^2 + e - 13, -6*e^3 - 12*e^2 + 27*e + 24, 2*e^3 + 5*e^2 - 10*e - 18, 2*e^3 + 5*e^2 - 2*e - 7, 2*e^3 + 9*e^2 + 3*e - 14, e^3 + 8*e^2 + 8*e - 14, -2*e^3 - 3*e^2 + 7*e + 1, 2*e^3 + 4*e^2 - 10*e + 4, 7*e^3 + 20*e^2 - 30*e - 46, 2*e^3 + 11*e^2 + 7*e - 18, e^3 + e^2 - 4*e - 8, 2*e^3 + 3*e^2 - 15*e - 18, 6*e^3 + 16*e^2 - 24*e - 39, -2*e^3 - e^2 + 17*e - 1, e^3 - e^2 - 11*e + 10, 2*e^3 + 5*e^2 - 11*e - 15, -4*e^3 - 8*e^2 + 11*e - 8, -6*e^3 - 18*e^2 + 16*e + 41, e^3 + 5*e^2 - e - 1, -2*e^3 - 8*e^2 - 2*e + 21, 9*e^3 + 28*e^2 - 22*e - 49, 5*e^3 + 16*e^2 - 14*e - 45, 12*e^3 + 34*e^2 - 34*e - 59, e^3 - 3*e^2 - 18*e + 9, e^3 + 9*e^2 + 8*e - 12, 5*e^3 + 16*e^2 - 5*e - 16, -7*e^3 - 18*e^2 + 33*e + 45, -4*e^3 - 10*e^2 + 5*e + 11, 4*e^3 + 9*e^2 - 19*e - 25, 10*e^3 + 32*e^2 - 19*e - 55, -8*e^3 - 27*e^2 + 8*e + 46, -e^3 + 3*e - 10, -3*e^3 - 8*e^2 + 12*e + 5, 3*e^3 + 6*e^2 - 11*e - 5, 5*e^3 + 21*e^2 - 2*e - 45, -7*e^3 - 15*e^2 + 26*e + 20, -e^3 - 5*e - 23, 2*e^3 - 15*e - 6, 2*e^3 + 7*e^2 - 18, -5*e^3 - 11*e^2 + 21*e + 27, 7*e^3 + 20*e^2 - 18*e - 31, 2*e^3 + 11*e^2 + 2*e - 30, 3*e^3 + 2*e^2 - 17*e + 6, e^3 - 2*e^2 - 13*e + 14, -9*e^3 - 27*e^2 + 19*e + 53, -11*e^3 - 33*e^2 + 22*e + 43, 5*e^3 + 16*e^2 - 12*e - 19, -11*e^3 - 35*e^2 + 33*e + 66, -2*e^3 - 11*e^2 - 6*e + 25, -10*e^3 - 24*e^2 + 42*e + 51, -6*e^3 - 19*e^2 + 13*e + 27, e^3 - 3*e + 18, -15*e^3 - 50*e^2 + 32*e + 76, -9*e^3 - 23*e^2 + 26*e + 43, 7*e^3 + 15*e^2 - 36*e - 39, -9*e^3 - 28*e^2 + 17*e + 39, 8*e^3 + 23*e^2 - 16*e - 33, 11*e^3 + 26*e^2 - 41*e - 47, 7*e^3 + 22*e^2 - 5*e - 24, 10*e^3 + 34*e^2 - 17*e - 65, 8*e^3 + 28*e^2 - 12*e - 47, -8*e^3 - 29*e^2 + 6*e + 44, 4*e^2 + 3*e - 38, 5*e^3 + 11*e^2 - 18*e - 37, 4*e^3 + 12*e^2 - 19*e - 42, -7*e^3 - 24*e^2 + 12*e + 44, 11*e^3 + 25*e^2 - 42*e - 46, -13*e^3 - 37*e^2 + 46*e + 76, 4*e^3 + 11*e^2 - 24*e - 27, 9*e^3 + 29*e^2 - 20*e - 39, -9*e^3 - 29*e^2 + 8*e + 47, 13*e^3 + 39*e^2 - 39*e - 67, 2*e^3 + 9*e^2 - 7*e - 29, -e^3 - 4*e^2 - 2*e + 29, -9*e^3 - 25*e^2 + 35*e + 45, 16*e^3 + 39*e^2 - 65*e - 85, 6*e^3 + 11*e^2 - 27*e - 34, -4*e^3 - 9*e^2 + 27*e + 20, 3*e^2 + 3*e - 30, -4*e^3 - 14*e^2 + 19*e + 32, -5*e^3 - 19*e^2 + 4*e + 15, 5*e^3 + 8*e^2 - 20*e + 10, 6*e^3 + 20*e^2 - 7*e - 39, 7*e^2 + 21*e - 21, 2*e^3 + 13*e^2 + 3*e - 43, -e^3 - 12*e^2 - 3*e + 52, 5*e^3 + 10*e^2 - 22*e - 13, 4*e^2 - 2*e - 23, 5*e^3 + 19*e^2 - 6*e - 53, -3*e^3 - 9*e^2 + 13*e + 14, 3*e^3 + 7*e^2 - 23*e - 33, 9*e^3 + 28*e^2 - 34*e - 56, -15*e^3 - 44*e^2 + 55*e + 96, -11*e^3 - 35*e^2 + 33*e + 62, 3*e^3 + 14*e^2 + 6*e - 23, -12*e^3 - 25*e^2 + 56*e + 63, 3*e^3 + 13*e^2 + 12*e - 35, e^2 - 4*e - 32, 8*e^3 + 16*e^2 - 44*e - 37, 5*e^3 + 7*e^2 - 35*e + 2, -10*e^3 - 33*e^2 + 30*e + 63, -5*e^3 - 21*e^2 + 2*e + 49, 2*e^2 - 3*e - 22, -3*e^2 - 8*e + 18, 4*e^3 + 13*e^2 - 7*e - 3, 10*e^3 + 29*e^2 - 13*e - 38, 7*e^3 + 17*e^2 - 33*e - 63, 14*e^3 + 43*e^2 - 37*e - 74, -5*e^3 - 17*e^2 + 31*e + 64, 9*e^3 + 15*e^2 - 35*e - 5, e^3 + 2*e^2 + 7*e - 7, -9*e^3 - 27*e^2 + 16*e + 19, 14*e^3 + 44*e^2 - 37*e - 71, 2*e^3 + 9*e^2 + 2*e - 8, e^3 + 10*e^2 + 8*e - 20, -6*e^3 - 20*e^2 + 14*e + 30, -5*e^3 - 20*e^2 - 4*e + 38, -6*e^3 - 23*e^2 + e + 38, -2*e^3 - 18*e^2 - 18*e + 37, 5*e^3 + 9*e^2 - 43*e - 15, e^3 + e^2 - 17*e - 26, -17*e^3 - 53*e^2 + 43*e + 80, -17*e^3 - 48*e^2 + 62*e + 108, 12*e^3 + 32*e^2 - 30*e - 36];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {25, 5, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 8*w}>] := -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;