/*
  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 | {9,3,1/2*w^3 + 1/2*w^2 - 7/2*w - 9/2}>;

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 - 5*x^2 - 8*x + 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e^3 + e^2 - 4*e - 1, -e^3 - e^2 + 5*e + 3, -e^3 - e^2 + 6*e + 4, -1, e^2 + e, -2*e^2 - 2*e + 6, -2*e^3 - 4*e^2 + 10*e + 8, -2*e^3 - e^2 + 10*e + 5, e^3 + e^2 - 8*e - 2, 3*e^3 + 2*e^2 - 16*e - 10, -2*e^3 + 14*e + 3, e^2 - e + 2, e^2 + 5*e - 4, -e^3 + 3*e - 5, e^3 + 5*e^2 - 3*e - 15, e^3 - e^2 - 7*e - 1, 2*e^2 - 3*e - 16, -e^3 - 2*e^2 + 4, -2*e^2 - e + 13, 4*e^2 + 2*e - 18, -2*e^2 + 2*e + 15, -e^3 - 3*e^2 + 6*e + 20, e^3 - 3*e + 13, 3*e^3 + 5*e^2 - 18*e - 15, -2*e^2 - 3*e + 9, -2*e^2 + 6, 2*e^3 + 2*e^2 - 11*e - 4, -e^3 + 3*e + 1, e^3 + 5*e^2 + 3*e - 15, -e^3 + 3*e^2 + 5*e - 13, -4*e^2 - 5*e + 12, 2*e^3 + e^2 - 8*e - 1, -2*e^3 - 4*e^2 + 8*e + 10, 2*e^3 - 5*e + 10, 3*e^3 - 2*e^2 - 20*e + 8, -2*e^3 + 10*e - 11, -2*e^3 - 6*e^2 + 10*e + 22, 2*e^3 - 12*e + 14, 3*e^3 + e^2 - 15*e - 3, 3*e^3 + 8*e^2 - 14*e - 20, 2*e^2 - 5*e - 20, 2*e^3 - e^2 - 17*e + 6, -3*e^3 + 13*e - 11, 6*e^3 + 7*e^2 - 24*e - 22, -2*e^3 - 6*e^2 + 14*e + 12, 3*e^3 + 5*e^2 - 13*e - 13, 3*e^3 + 4*e^2 - 12*e - 28, -e^3 - 2*e^2 + 3*e + 15, e^3 + 3*e^2 + 3*e - 1, -3*e^3 - 2*e^2 + 13*e - 10, -2*e^3 + 3*e^2 + 3*e - 26, -e^3 + e^2 - 13, -5*e^3 - 4*e^2 + 16*e, 2*e^3 - 8*e + 18, -3*e^3 - e^2 + 21*e + 13, -2*e^3 - 3*e^2 + 3*e + 14, 8*e^3 + 9*e^2 - 35*e - 24, 7*e^3 + 10*e^2 - 34*e - 28, -2*e^3 - 2*e^2 + 14*e - 1, 6*e^2 + 6*e - 8, e^3 + 3*e^2 - 8*e - 10, -4*e^3 - 8*e^2 + 13*e + 21, -2*e^3 + 4*e^2 + 12*e - 28, 2*e^3 + 6*e^2 - 6*e - 27, -e^3 + 2*e^2 + 8*e - 2, 6*e^3 + 6*e^2 - 34*e - 16, -5*e^2 - 6*e + 22, -e^3 + 3*e^2 - 2*e - 18, -e^3 + 13*e - 3, -2*e^2 - 2, e^3 + 6*e^2 - 7*e - 20, 6*e^3 + 2*e^2 - 22*e + 12, 2*e^3 - 4*e^2 - 20*e + 22, -e^3 - 10*e^2 + e + 37, -2*e^3 + 2*e^2 + 9*e - 12, -6*e^3 - 2*e^2 + 24*e - 1, -10*e^2 - 6*e + 30, 2*e^3 + 4*e^2 - 12*e - 16, -4*e^3 + 18*e + 4, 2*e^3 - 15*e + 4, 2*e^3 - 2*e^2 - 14*e + 2, 5*e^3 + 13*e^2 - 26*e - 31, 2*e^3 + 7*e^2 - 2*e - 17, 6*e^3 + 8*e^2 - 44*e - 25, -e^3 - 7*e^2 + 3*e + 35, -3*e^3 - 3*e^2 + 12*e + 33, -4*e^3 - 8*e^2 + 16*e + 2, 2*e^3 - 2*e^2 - 2*e + 8, 4*e^3 + 15*e^2 - 12*e - 34, 3*e^3 - 4*e^2 - 24*e + 16, -6*e^2 - 14*e + 29, -2*e^3 - 2*e^2 + 4*e, -e^3 - 7*e^2 + 7*e + 37, e^3 - 2*e^2 - 14*e + 6, 4*e^3 - 2*e^2 - 32*e + 2, 7*e^3 + e^2 - 35*e - 5, -3*e^3 - 3*e^2 + 25*e + 21, 4*e^3 + 7*e^2 - 26*e - 13, -2*e^3 + 2*e^2 + 23*e - 14, -8*e^3 - 13*e^2 + 32*e + 25, -e^3 - 3*e^2 + 4*e + 11, 3*e^3 + 7*e^2 - 8*e - 9, -2*e^3 + 2*e^2 + 8*e + 10, -2*e^2 - 10*e - 6, -4*e^3 - 8*e^2 + 24*e + 14, -7*e^3 - 9*e^2 + 35*e + 21, -e^3 - 9*e^2 + 11*e + 39, -e^2 + 36, -3*e^3 + 7*e^2 + 19*e - 29, -2*e^3 + 2*e^2 + 16*e + 14, -5*e^3 - 11*e^2 + 29*e + 15, 5*e^3 + 11*e^2 - 34*e - 26, -2*e^3 - 4*e^2 + 2*e - 6, -2*e^3 + 4*e^2 + 22*e - 19, 10*e^3 + 10*e^2 - 62*e - 34, 4*e^3 + 5*e^2 - 25*e + 6, 3*e^3 - 7*e^2 - 24*e + 39, -10*e^3 - 8*e^2 + 54*e + 40, 2*e^3 + 4*e^2 - 18*e - 20, -6*e^3 - 4*e^2 + 42*e + 6, 3*e^3 + 5*e^2 - 12*e - 26, 6*e^3 + 11*e^2 - 15*e - 32, -e^3 + 6*e^2 + e - 15, 9*e^3 + 7*e^2 - 50*e - 20, -10*e^3 - 14*e^2 + 50*e + 26, 7*e^3 + 11*e^2 - 29*e - 27, 3*e^3 - 5*e^2 - 32*e + 14, -e^3 + e^2 + 8*e - 10, 2*e^3 + 4*e^2 - 17*e, 8*e^3 + 4*e^2 - 35*e + 12, 2*e^3 + 8*e^2 - 12*e - 29, 4*e^3 - 2*e^2 - 30*e - 12, 6*e^3 + 9*e^2 - 43*e - 18, -6*e^3 - 6*e^2 + 38*e + 36, 2*e^3 - 14*e + 24, -6*e^3 - 14*e^2 + 28*e + 40, 10*e^2 - 2*e - 39, -5*e^3 - 10*e^2 + 18*e + 28, -3*e^3 - 2*e^2 + 9*e - 36, 4*e^3 - e^2 - 38*e + 13, 3*e^3 - 10*e^2 - 24*e + 34, -6*e^3 + 2*e^2 + 30*e - 23, -3*e^2 - 14*e - 11, 2*e^3 + 2*e^2 - 14*e - 27, 4*e^3 + 9*e^2 - 17*e - 20, -6*e^3 - 4*e^2 + 31*e + 27, -8*e^3 - 2*e^2 + 29*e - 27, -2*e^3 + 16*e^2 + 23*e - 45, 4*e^3 + 6*e^2 - 16*e + 12, 4*e^3 + 3*e^2 - 19*e + 18, -6*e^3 + 2*e^2 + 33*e - 4, -2*e^2 + 10*e - 3, -4*e^3 - 2*e^2 + 28*e + 36, -10*e^3 - 14*e^2 + 46*e + 36, -4*e^3 + 3*e^2 + 16*e - 19, 6*e^3 + 8*e^2 - 36*e - 8, 2*e^2 - 6*e - 26, -5*e^3 - 5*e^2 + 12*e + 15, -2*e^3 - 2*e^2 + 18*e - 24, -2*e^3 - 6*e^2 - 5*e + 25, -e^2 - 5*e + 36, 3*e^3 - 4*e^2 - 13*e + 8, -6*e^3 - 7*e^2 + 26*e + 35, -8*e^3 - 5*e^2 + 58*e + 11, -6*e^3 - 14*e^2 + 34*e + 34, -9*e^3 - 14*e^2 + 25*e + 46, 5*e^3 + 15*e^2 - 26*e - 44, -2*e^2 - 10*e + 8, 8*e^3 + 6*e^2 - 22*e - 5, 5*e^3 + 15*e^2 - 20*e - 50];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9,3,1/2*w^3 + 1/2*w^2 - 7/2*w - 9/2}>] := 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;