/*
  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 | {29, 29, -w^4 + 4*w^2 + 1}>;

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^6 - 24*x^4 + 152*x^2 - 128;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e, 1/8*e^5 - 2*e^3 + 6*e, 3/16*e^5 - 3*e^3 + 17/2*e, -1, 0, -1/16*e^5 + e^3 - 7/2*e, -1/16*e^5 + e^3 - 5/2*e, -1/2*e^4 + 5*e^2 + 8, 8, 1/16*e^5 - 17/2*e, -3/4*e^4 + 11*e^2 - 23, -2*e^2 + 16, 1/16*e^5 - 2*e^3 + 29/2*e, e^3 - 11*e, -4, -1/16*e^5 + e^3 + 1/2*e, 6, 1/8*e^5 - 3*e^3 + 19*e, -1/2*e^4 + 5*e^2 + 10, -3/16*e^5 + 4*e^3 - 35/2*e, -1/8*e^5 + 2*e^3 - 10*e, 1/2*e^4 - 8*e^2 + 10, -1/8*e^5 + e^3 + 8*e, 1/2*e^4 - 6*e^2 + 4, -e^2 + 4, 2*e^2 - 12, -1/4*e^5 + 3*e^3 - 2*e, -e^3 + 10*e, 3/16*e^5 - 3*e^3 + 9/2*e, e^3 - 12*e, 4, 12, -e^4 + 14*e^2 - 24, 1/2*e^4 - 7*e^2 + 8, 1/2*e^4 - 5*e^2 - 6, -1/4*e^5 + 5*e^3 - 21*e, -e^2, -1/2*e^4 + 8*e^2 - 8, 1/2*e^4 - 6*e^2 + 12, -1/2*e^4 + 9*e^2 - 20, -3/16*e^5 + 5*e^3 - 63/2*e, 1/2*e^4 - 9*e^2 + 32, -1/8*e^5 + 2*e^3 - 6*e, 3/16*e^5 - 4*e^3 + 45/2*e, -1/4*e^5 + 6*e^3 - 28*e, -1/2*e^4 + 5*e^2 + 16, -1/16*e^5 + e^3 - 15/2*e, 3/2*e^4 - 20*e^2 + 30, -1/8*e^5 + e^3 + 4*e, 1/8*e^5 - 2*e^3 + 6*e, -2*e^3 + 24*e, 1/2*e^4 - 4*e^2 - 12, -3*e^3 + 32*e, -1/2*e^4 + 11*e^2 - 40, -1/8*e^5 + 14*e, 1/2*e^4 - 6*e^2 + 6, 3/16*e^5 - 2*e^3 - 7/2*e, -3/2*e^4 + 22*e^2 - 50, -3/16*e^5 + 4*e^3 - 35/2*e, -e^4 + 13*e^2 - 10, -e^4 + 10*e^2 + 16, -1/2*e^4 + 7*e^2 - 32, -1/16*e^5 + 3*e^3 - 53/2*e, -e^4 + 10*e^2 + 8, 1/16*e^5 - 35/2*e, -e^4 + 14*e^2 - 12, -3/16*e^5 + e^3 + 39/2*e, 1/2*e^5 - 7*e^3 + 9*e, 1/2*e^4 - 8*e^2 - 4, 5/8*e^5 - 10*e^3 + 27*e, -1/4*e^5 + 3*e^3 + 3*e, 5/8*e^5 - 11*e^3 + 41*e, 1/2*e^4 - 8*e^2 + 30, -1/8*e^5 + 16*e, e^4 - 15*e^2 + 18, 1/8*e^5 - 2*e^3, e^4 - 16*e^2 + 30, 3/2*e^4 - 23*e^2 + 52, 1/16*e^5 - 4*e^3 + 75/2*e, -1/2*e^4 + 8*e^2 - 6, -1/2*e^4 + 3*e^2 + 16, -3/8*e^5 + 8*e^3 - 38*e, -1/8*e^5 + 4*e^3 - 33*e, 1/2*e^4 - 8*e^2 + 4, -1/8*e^5 + 3*e^3 - 16*e, 1/2*e^4 - 13*e^2 + 48, 5/16*e^5 - 4*e^3 - 5/2*e, 1/2*e^4 - 11*e^2 + 46, -3/2*e^4 + 20*e^2 - 38, 3/4*e^5 - 13*e^3 + 43*e, 1/2*e^4 - 7*e^2 + 28, 3/16*e^5 - 45/2*e, -e^4 + 12*e^2 - 14, -11/16*e^5 + 12*e^3 - 93/2*e, -1/4*e^5 + 3*e^3 - 2*e, -e^3 + 6*e, -3/2*e^4 + 24*e^2 - 62, -1/16*e^5 + e^3 - 1/2*e, 1/16*e^5 - 19/2*e, 3/4*e^5 - 12*e^3 + 38*e, -7/16*e^5 + 9*e^3 - 93/2*e, -3/2*e^4 + 16*e^2 + 24, 1/8*e^5 - 4*e^3 + 27*e, 1/8*e^5 - 5*e^3 + 41*e, -1/2*e^4 + 5*e^2 + 16, 2*e^4 - 30*e^2 + 72, 1/2*e^4 - 4*e^2 - 12, e^4 - 14*e^2 + 32, 1/2*e^5 - 9*e^3 + 39*e, 1/2*e^5 - 10*e^3 + 51*e, 2*e^2 - 10, -e^4 + 20*e^2 - 68, -17/16*e^5 + 18*e^3 - 121/2*e, -3/2*e^4 + 19*e^2 - 26, -5*e, -3/16*e^5 + 4*e^3 - 33/2*e, -17/16*e^5 + 18*e^3 - 123/2*e, -5/16*e^5 + 4*e^3 - 15/2*e, -e^4 + 11*e^2 + 4, -2*e^2 + 20, 7/16*e^5 - 5*e^3 - 5/2*e, -e^4 + 15*e^2 - 38, -1/4*e^5 + 5*e^3 - 18*e, -3/16*e^5 - e^3 + 65/2*e, -1/2*e^5 + 11*e^3 - 57*e, -1/2*e^4 + 7*e^2 + 20, -1/2*e^4 + 10*e^2 - 54, -1/2*e^4 + 6*e^2 + 4, 3/4*e^5 - 14*e^3 + 50*e, -2*e^4 + 28*e^2 - 32, 2*e^4 - 30*e^2 + 78, -1/4*e^5 + 7*e^3 - 52*e, -1/16*e^5 + 3*e^3 - 33/2*e, -3*e^2 + 38, 1/4*e^5 - 6*e^3 + 43*e, -1/16*e^5 - 3*e^3 + 87/2*e, -6*e^2 + 58, e^3 - 24*e, e^4 - 12*e^2 + 14, 1/2*e^4 - 4*e^2 - 12, -1/2*e^4 + 10*e^2 - 18, -3/2*e^4 + 19*e^2 - 20, 3/16*e^5 - 5*e^3 + 71/2*e, -1/2*e^4 + 6*e^2 - 6, -1/4*e^5 + 3*e^3 - 2*e, -1/2*e^5 + 10*e^3 - 51*e, -3/2*e^4 + 18*e^2 - 20, 1/16*e^5 - 35/2*e, -3/2*e^4 + 20*e^2 - 42, 7/16*e^5 - 7*e^3 + 37/2*e, 5/8*e^5 - 13*e^3 + 67*e, 1/4*e^5 - 8*e^3 + 51*e, -1/8*e^5 - 3*e^3 + 57*e, -13/16*e^5 + 14*e^3 - 85/2*e, -2*e^2 + 12];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, -w^4 + 4*w^2 + 1}>] := 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;