/*
  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 | {29, 29, w^4 - w^3 - 4*w^2 + 2*w + 2}>;

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^4 - 6*x^3 - 9*x^2 + 102*x - 139;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^3 - 2*e^2 - 18*e + 37, e - 1, -e^3 + 3*e^2 + 17*e - 44, -1, -e^3 + 5/2*e^2 + 35/2*e - 87/2, 1/2*e^3 - 2*e^2 - 9*e + 59/2, e^3 - 7/2*e^2 - 33/2*e + 101/2, -1/2*e^2 - 3/2*e + 13/2, 3/2*e^3 - 3*e^2 - 26*e + 107/2, -1/2*e^3 + 7*e + 1/2, e^3 - 3*e^2 - 17*e + 48, -1/2*e^3 + 5/2*e^2 + 17/2*e - 36, e^2 + 3*e - 19, -e^2 + 20, -3/2*e^3 + 4*e^2 + 27*e - 145/2, -e^3 + e^2 + 17*e - 20, -2*e^3 + 5*e^2 + 33*e - 77, e^3 - 4*e^2 - 16*e + 61, -3*e^3 + 7*e^2 + 51*e - 118, -e^3 + 7/2*e^2 + 29/2*e - 109/2, e^2 + e - 11, 3*e^3 - 8*e^2 - 51*e + 133, -3*e^3 + 8*e^2 + 50*e - 129, e^3 - e^2 - 15*e + 14, -2*e^3 + 3*e^2 + 34*e - 52, -3*e^3 + 7*e^2 + 52*e - 120, 2*e + 2, -2*e + 6, -2*e + 10, 3/2*e^2 + 1/2*e - 59/2, 3/2*e^3 - e^2 - 26*e + 47/2, 2*e^3 - 4*e^2 - 34*e + 78, e^3 - 17*e + 4, 2*e^3 - 4*e^2 - 36*e + 76, -e^3 + 17*e - 6, -2*e^3 + 6*e^2 + 35*e - 99, 2*e^3 - 6*e^2 - 38*e + 100, 4*e^3 - 11*e^2 - 68*e + 180, -2*e^3 + 6*e^2 + 32*e - 90, -e^3 + 9/2*e^2 + 35/2*e - 123/2, -1/2*e^3 + 5*e - 3/2, -1/2*e^3 + 7*e + 1/2, -3*e^3 + 8*e^2 + 54*e - 139, 2*e^3 - 6*e^2 - 37*e + 108, -e^3 + e^2 + 15*e - 22, -3*e^3 + 6*e^2 + 51*e - 101, 4*e^3 - 9*e^2 - 67*e + 145, 4*e^3 - 10*e^2 - 68*e + 166, -e^3 + 2*e^2 + 16*e - 15, -e^2 + e + 5, e^3 - 3*e^2 - 21*e + 48, e^3 - 4*e^2 - 20*e + 61, e^3 - 3*e^2 - 15*e + 58, 2*e + 10, -1/2*e^2 - 11/2*e + 33/2, -e^3 + 11/2*e^2 + 37/2*e - 149/2, 2*e^3 - 3*e^2 - 37*e + 69, 3*e^3 - 7*e^2 - 51*e + 130, -5/2*e^3 + 6*e^2 + 45*e - 183/2, 2*e^3 - 6*e^2 - 39*e + 108, -3*e^3 + 8*e^2 + 47*e - 128, 2*e^3 - 5*e^2 - 36*e + 97, 2*e^2 + 6*e - 40, -3*e^2 + e + 35, -e^2 + 3*e + 15, 2*e^3 - 8*e^2 - 34*e + 112, -7/2*e^3 + 10*e^2 + 67*e - 349/2, 7/2*e^3 - 10*e^2 - 61*e + 337/2, 4*e^3 - 12*e^2 - 76*e + 210, e^3 - 3*e^2 - 22*e + 48, e^3 - 3*e^2 - 17*e + 68, 3*e^3 - 9*e^2 - 50*e + 130, 2*e^3 - 7*e^2 - 32*e + 99, -4*e^3 + 7*e^2 + 69*e - 123, -2*e^2 + 26, 4*e^3 - 9*e^2 - 70*e + 156, -4*e^3 + 11*e^2 + 69*e - 165, -e^3 + 9/2*e^2 + 43/2*e - 159/2, 3*e^3 - 9*e^2 - 49*e + 134, 11/2*e^3 - 15*e^2 - 92*e + 499/2, -e^3 + 6*e^2 + 21*e - 97, e^3 - 5*e^2 - 24*e + 95, -e^3 + 5*e^2 + 19*e - 74, 3*e^3 - 9*e^2 - 53*e + 154, e^3 - 2*e^2 - 21*e + 27, 2*e^3 - 3*e^2 - 39*e + 65, -e^3 - e^2 + 12*e + 15, 3*e^3 - 7*e^2 - 54*e + 136, -6*e^3 + 13*e^2 + 105*e - 235, 2*e^3 - 5*e^2 - 37*e + 89, -7/2*e^3 + 10*e^2 + 65*e - 365/2, 3*e^3 - 7*e^2 - 52*e + 108, -e^3 + 3*e^2 + 17*e - 26, 4*e^3 - 6*e^2 - 68*e + 98, -5*e^3 + 10*e^2 + 92*e - 185, 2*e^3 - 9/2*e^2 - 75/2*e + 177/2, e^3 - 2*e^2 - 22*e + 17, -3*e^3 + 3*e^2 + 52*e - 57, 3*e^3 - 5*e^2 - 50*e + 99, -6*e^3 + 14*e^2 + 104*e - 254, -3*e^3 + 7*e^2 + 55*e - 118, 2*e^3 - 6*e^2 - 35*e + 68, 5/2*e^3 - 3*e^2 - 40*e + 101/2, 2*e^3 - 15/2*e^2 - 73/2*e + 251/2, 3*e^3 - 8*e^2 - 60*e + 139, e^3 - 4*e^2 - 12*e + 73, -3*e^2 + 2*e + 31, -e^3 + e^2 + 25*e - 16, e^3 + e^2 - 13*e - 4, -6*e^3 + 13*e^2 + 99*e - 217, -3*e^3 + 9*e^2 + 58*e - 162, 4*e^3 - 10*e^2 - 66*e + 160, -9/2*e^3 + 9*e^2 + 80*e - 333/2, -7*e^3 + 19*e^2 + 115*e - 306, 5*e^3 - 14*e^2 - 86*e + 223, 4*e^3 - 9*e^2 - 63*e + 127, 8*e^3 - 21*e^2 - 134*e + 342, -5*e^3 + 13*e^2 + 86*e - 239, -2*e^3 + 5*e^2 + 34*e - 65, 2*e^3 - 9*e^2 - 36*e + 132, -4*e^3 + 12*e^2 + 71*e - 194, -e^3 + 3*e^2 + 15*e - 66, -e^3 + e^2 + 15*e - 46, -e^3 + 19*e - 41, 3*e^3 - 13/2*e^2 - 99/2*e + 199/2, -e^3 - e^2 + 13*e + 10, 4*e^3 - 10*e^2 - 72*e + 176, -e^3 + 4*e^2 + 7*e - 67, 11/2*e^3 - 12*e^2 - 89*e + 393/2, 3*e^3 - 10*e^2 - 58*e + 179, -5*e^3 + 17*e^2 + 89*e - 264, 4*e^2 - 60, 2*e^3 - 11/2*e^2 - 61/2*e + 191/2, -2*e^3 + 11/2*e^2 + 77/2*e - 215/2, 3*e^3 - 4*e^2 - 57*e + 94, 11/2*e^3 - 12*e^2 - 87*e + 365/2, 2*e^3 - 3*e^2 - 41*e + 67, 13/2*e^3 - 16*e^2 - 109*e + 519/2, 3*e^3 - 4*e^2 - 49*e + 46, -4*e^3 + 15*e^2 + 75*e - 229, 6*e^3 - 12*e^2 - 108*e + 222, -2*e^2 + 2*e + 32, 3*e^2 + 3*e - 69, -5*e^3 + 12*e^2 + 84*e - 185, 2*e^3 - 4*e^2 - 29*e + 41, -3*e^3 + 7*e^2 + 51*e - 100, 4*e^3 - 25/2*e^2 - 151/2*e + 453/2, 4*e^3 - 23/2*e^2 - 133/2*e + 371/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, w^4 - w^3 - 4*w^2 + 2*w + 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;