/*
  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 | {59, 59, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*w - 2}>;

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^7 - 2*x^6 - 16*x^5 + 32*x^4 + 35*x^3 - 76*x^2 - 20*x + 48;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/2*e^6 - 1/2*e^5 + 9*e^4 + 8*e^3 - 63/2*e^2 - 25/2*e + 24, -2, 3/2*e^6 - 26*e^4 - e^3 + 161/2*e^2 + 2*e - 64, -3/2*e^6 + 26*e^4 + e^3 - 161/2*e^2 - 2*e + 62, -9/4*e^6 + 3/2*e^5 + 37*e^4 - 21*e^3 - 371/4*e^2 + 24*e + 64, 5/2*e^6 - e^5 - 41*e^4 + 13*e^3 + 199/2*e^2 - 11*e - 62, 1, 9/4*e^6 - 5/2*e^5 - 36*e^4 + 36*e^3 + 311/4*e^2 - 47*e - 44, -e^6 + e^5 + 16*e^4 - 14*e^3 - 35*e^2 + 13*e + 20, -3/2*e^6 + 26*e^4 + e^3 - 161/2*e^2 - 4*e + 56, -7/4*e^6 + 2*e^5 + 28*e^4 - 29*e^3 - 241/4*e^2 + 77/2*e + 33, -1/2*e^6 + 8*e^4 - 35/2*e^2 + 5*e + 8, 5/4*e^6 - 1/2*e^5 - 21*e^4 + 7*e^3 + 231/4*e^2 - 10*e - 44, -7/2*e^6 + 5/2*e^5 + 57*e^4 - 35*e^3 - 271/2*e^2 + 79/2*e + 84, -e^4 + 16*e^2 - 2*e - 28, -5/2*e^6 + 3/2*e^5 + 41*e^4 - 21*e^3 - 201/2*e^2 + 49/2*e + 68, 4*e^6 - e^5 - 66*e^4 + 12*e^3 + 165*e^2 - 5*e - 102, -1/2*e^6 + 9*e^4 - 65/2*e^2 + e + 34, 7/2*e^6 - 5/2*e^5 - 57*e^4 + 35*e^3 + 269/2*e^2 - 83/2*e - 88, 13/4*e^6 - 3/2*e^5 - 55*e^4 + 21*e^3 + 623/4*e^2 - 33*e - 118, -5/2*e^6 + 3/2*e^5 + 41*e^4 - 21*e^3 - 201/2*e^2 + 53/2*e + 58, 4*e^6 - 68*e^4 - 2*e^3 + 194*e^2 + 4*e - 134, 3*e^6 - 2*e^5 - 50*e^4 + 28*e^3 + 133*e^2 - 34*e - 100, 5/2*e^6 - e^5 - 40*e^4 + 13*e^3 + 171/2*e^2 - 9*e - 40, 9/4*e^6 - 1/2*e^5 - 38*e^4 + 6*e^3 + 431/4*e^2 - 5*e - 84, 11/2*e^6 - 3/2*e^5 - 93*e^4 + 19*e^3 + 521/2*e^2 - 37/2*e - 182, -23/4*e^6 + 5/2*e^5 + 97*e^4 - 35*e^3 - 1085/4*e^2 + 51*e + 198, 29/4*e^6 - 9/2*e^5 - 119*e^4 + 63*e^3 + 1175/4*e^2 - 77*e - 198, 21/2*e^6 - 7/2*e^5 - 178*e^4 + 47*e^3 + 1015/2*e^2 - 125/2*e - 380, 9/2*e^6 - 3*e^5 - 73*e^4 + 42*e^3 + 337/2*e^2 - 46*e - 106, 3/2*e^6 + e^5 - 30*e^4 - 15*e^3 + 281/2*e^2 + 11*e - 144, -15/2*e^6 + 4*e^5 + 125*e^4 - 56*e^3 - 663/2*e^2 + 71*e + 236, 2*e^6 - e^5 - 35*e^4 + 15*e^3 + 113*e^2 - 33*e - 92, 39/4*e^6 - 7/2*e^5 - 165*e^4 + 46*e^3 + 1865/4*e^2 - 51*e - 348, -15/2*e^6 + 4*e^5 + 125*e^4 - 56*e^3 - 659/2*e^2 + 76*e + 222, -5/2*e^6 + 3*e^5 + 40*e^4 - 43*e^3 - 171/2*e^2 + 46*e + 50, -19/4*e^6 + 3/2*e^5 + 78*e^4 - 18*e^3 - 765/4*e^2 + e + 120, 3/2*e^6 - 2*e^5 - 23*e^4 + 29*e^3 + 71/2*e^2 - 37*e - 4, 1/2*e^6 + 2*e^5 - 11*e^4 - 30*e^3 + 121/2*e^2 + 36*e - 64, 10*e^6 - 5*e^5 - 166*e^4 + 69*e^3 + 432*e^2 - 88*e - 296, 11/4*e^6 - 7/2*e^5 - 43*e^4 + 50*e^3 + 321/4*e^2 - 58*e - 40, -6*e^6 + 4*e^5 + 97*e^4 - 56*e^3 - 219*e^2 + 68*e + 120, -5/2*e^6 + 2*e^5 + 42*e^4 - 30*e^3 - 231/2*e^2 + 51*e + 82, -17/2*e^6 + 4*e^5 + 140*e^4 - 54*e^3 - 695/2*e^2 + 60*e + 214, -47/4*e^6 + 11/2*e^5 + 197*e^4 - 76*e^3 - 2133/4*e^2 + 100*e + 372, 8*e^6 - 3*e^5 - 134*e^4 + 40*e^3 + 362*e^2 - 47*e - 244, -9/2*e^6 + 3*e^5 + 74*e^4 - 43*e^3 - 365/2*e^2 + 59*e + 110, -45/4*e^6 + 15/2*e^5 + 185*e^4 - 106*e^3 - 1847/4*e^2 + 132*e + 314, 19/2*e^6 - 13/2*e^5 - 156*e^4 + 92*e^3 + 767/2*e^2 - 229/2*e - 252, -7*e^6 + 2*e^5 + 122*e^4 - 27*e^3 - 387*e^2 + 41*e + 318, -9/4*e^6 + 5/2*e^5 + 35*e^4 - 37*e^3 - 251/4*e^2 + 56*e + 20, 19/4*e^6 - 3/2*e^5 - 81*e^4 + 20*e^3 + 945/4*e^2 - 25*e - 180, 61/4*e^6 - 13/2*e^5 - 255*e^4 + 88*e^3 + 2743/4*e^2 - 103*e - 486, 9/2*e^6 - 5/2*e^5 - 73*e^4 + 34*e^3 + 335/2*e^2 - 65/2*e - 102, 13/2*e^6 - 4*e^5 - 106*e^4 + 55*e^3 + 507/2*e^2 - 58*e - 166, -9/2*e^6 + 2*e^5 + 74*e^4 - 27*e^3 - 363/2*e^2 + 30*e + 106, -e^6 + 3*e^5 + 14*e^4 - 45*e^3 - 6*e^2 + 64*e - 20, 1/2*e^6 - 1/2*e^5 - 7*e^4 + 6*e^3 + 3/2*e^2 + 7/2*e + 4, -8*e^6 + 3*e^5 + 134*e^4 - 39*e^3 - 364*e^2 + 32*e + 270, -29/2*e^6 + 11/2*e^5 + 244*e^4 - 74*e^3 - 1341/2*e^2 + 177/2*e + 476, -15/2*e^6 + 3/2*e^5 + 129*e^4 - 19*e^3 - 777/2*e^2 + 57/2*e + 306, 7/2*e^6 - 5/2*e^5 - 59*e^4 + 37*e^3 + 325/2*e^2 - 131/2*e - 114, -2*e^6 - 2*e^5 + 38*e^4 + 30*e^3 - 156*e^2 - 28*e + 148, 17/2*e^6 - 3*e^5 - 145*e^4 + 40*e^3 + 845/2*e^2 - 44*e - 328, -35/2*e^6 + 11*e^5 + 286*e^4 - 154*e^3 - 1377/2*e^2 + 182*e + 432, -8*e^6 + 4*e^5 + 133*e^4 - 54*e^3 - 349*e^2 + 53*e + 256, 3/2*e^6 - e^5 - 28*e^4 + 15*e^3 + 225/2*e^2 - 25*e - 114, 37/2*e^6 - 15/2*e^5 - 312*e^4 + 102*e^3 + 1737/2*e^2 - 255/2*e - 642, -5/2*e^6 + 4*e^5 + 38*e^4 - 58*e^3 - 115/2*e^2 + 71*e + 32, -33/2*e^6 + 7*e^5 + 276*e^4 - 95*e^3 - 1477/2*e^2 + 113*e + 512, 5/2*e^6 - 2*e^5 - 44*e^4 + 29*e^3 + 295/2*e^2 - 41*e - 148, 7/2*e^6 + 2*e^5 - 61*e^4 - 32*e^3 + 383/2*e^2 + 47*e - 146, 2*e^6 - 6*e^5 - 29*e^4 + 89*e^3 + 29*e^2 - 116*e - 6, -29/2*e^6 + 8*e^5 + 238*e^4 - 110*e^3 - 1171/2*e^2 + 123*e + 372, -39/2*e^6 + 9*e^5 + 325*e^4 - 123*e^3 - 1715/2*e^2 + 144*e + 598, -21/2*e^6 + 7*e^5 + 170*e^4 - 98*e^3 - 775/2*e^2 + 113*e + 216, -1/2*e^6 + 3/2*e^5 + 7*e^4 - 22*e^3 - 3/2*e^2 + 59/2*e - 16, 15/2*e^6 - 6*e^5 - 120*e^4 + 84*e^3 + 519/2*e^2 - 89*e - 144, -7/2*e^6 + 5/2*e^5 + 56*e^4 - 34*e^3 - 239/2*e^2 + 53/2*e + 60, 17/2*e^6 - 4*e^5 - 141*e^4 + 54*e^3 + 729/2*e^2 - 55*e - 242, 2*e^4 - 29*e^2 - 2*e + 36, 3*e^6 - 4*e^5 - 48*e^4 + 59*e^3 + 105*e^2 - 77*e - 82, 11/2*e^6 - 2*e^5 - 94*e^4 + 26*e^3 + 557/2*e^2 - 25*e - 232, -41/4*e^6 + 13/2*e^5 + 171*e^4 - 92*e^3 - 1819/4*e^2 + 119*e + 328, 17/2*e^6 - 146*e^4 - 5*e^3 + 871/2*e^2 + 17*e - 326, -27/2*e^6 + 13/2*e^5 + 227*e^4 - 91*e^3 - 1253/2*e^2 + 247/2*e + 462, 21/2*e^6 - 4*e^5 - 174*e^4 + 53*e^3 + 887/2*e^2 - 56*e - 282, -33/2*e^6 + 11/2*e^5 + 279*e^4 - 73*e^3 - 1575/2*e^2 + 177/2*e + 596, 26*e^6 - 14*e^5 - 431*e^4 + 195*e^3 + 1113*e^2 - 244*e - 750, -14*e^6 + 6*e^5 + 236*e^4 - 83*e^3 - 654*e^2 + 113*e + 466, -16*e^6 + 12*e^5 + 263*e^4 - 170*e^3 - 652*e^2 + 210*e + 444, 20*e^6 - 13*e^5 - 330*e^4 + 183*e^3 + 834*e^2 - 226*e - 576, 12*e^6 - 7*e^5 - 199*e^4 + 98*e^3 + 517*e^2 - 125*e - 366, 11*e^6 - 190*e^4 - 6*e^3 + 581*e^2 + 10*e - 456, 3/4*e^6 - 1/2*e^5 - 13*e^4 + 7*e^3 + 145/4*e^2 - 9*e - 10, 33/4*e^6 - 5/2*e^5 - 139*e^4 + 32*e^3 + 1539/4*e^2 - 23*e - 278, 19*e^6 - 9*e^5 - 317*e^4 + 123*e^3 + 842*e^2 - 143*e - 586, 17/4*e^6 - 7/2*e^5 - 73*e^4 + 51*e^3 + 875/4*e^2 - 79*e - 170, 73/4*e^6 - 17/2*e^5 - 301*e^4 + 115*e^3 + 3019/4*e^2 - 127*e - 478, -57/4*e^6 + 9/2*e^5 + 243*e^4 - 59*e^3 - 2835/4*e^2 + 76*e + 548, -17/2*e^6 + 9*e^5 + 135*e^4 - 129*e^3 - 559/2*e^2 + 157*e + 142, -15/4*e^6 + 7/2*e^5 + 62*e^4 - 52*e^3 - 637/4*e^2 + 85*e + 104, 18*e^6 - 9*e^5 - 297*e^4 + 123*e^3 + 750*e^2 - 134*e - 504, 7/4*e^6 - 1/2*e^5 - 31*e^4 + 5*e^3 + 421/4*e^2 + 7*e - 106, -6*e^6 + 4*e^5 + 100*e^4 - 56*e^3 - 268*e^2 + 70*e + 204, -23*e^6 + 13*e^5 + 381*e^4 - 181*e^3 - 982*e^2 + 215*e + 688, -27/2*e^6 + 12*e^5 + 216*e^4 - 172*e^3 - 929/2*e^2 + 209*e + 268, -7/2*e^6 + 9/2*e^5 + 55*e^4 - 65*e^3 - 213/2*e^2 + 155/2*e + 42, -5/2*e^6 - e^5 + 44*e^4 + 16*e^3 - 287/2*e^2 - 19*e + 108, -21/2*e^6 + 9/2*e^5 + 173*e^4 - 60*e^3 - 855/2*e^2 + 113/2*e + 258, -e^6 + 16*e^4 + e^3 - 33*e^2 + e + 22, 13/2*e^6 + 3/2*e^5 - 115*e^4 - 26*e^3 + 771/2*e^2 + 71/2*e - 314, -19/2*e^6 + 2*e^5 + 164*e^4 - 25*e^3 - 1003/2*e^2 + 38*e + 378, -31/2*e^6 + 11/2*e^5 + 261*e^4 - 72*e^3 - 1449/2*e^2 + 147/2*e + 528, -19*e^6 + 8*e^5 + 320*e^4 - 110*e^3 - 885*e^2 + 142*e + 632, 11/2*e^6 - 11/2*e^5 - 83*e^4 + 78*e^3 + 225/2*e^2 - 163/2*e - 10, 57/2*e^6 - 14*e^5 - 476*e^4 + 192*e^3 + 2551/2*e^2 - 226*e - 926, -23/2*e^6 + 8*e^5 + 193*e^4 - 114*e^3 - 1055/2*e^2 + 157*e + 382, -37/4*e^6 + 1/2*e^5 + 156*e^4 - 1731/4*e^2 - 23*e + 316, 9*e^6 - 11*e^5 - 141*e^4 + 160*e^3 + 269*e^2 - 205*e - 132, 3*e^6 - 4*e^5 - 49*e^4 + 59*e^3 + 119*e^2 - 93*e - 66, 35/2*e^6 - 19/2*e^5 - 292*e^4 + 133*e^3 + 1547/2*e^2 - 351/2*e - 528, -e^6 - 2*e^5 + 20*e^4 + 29*e^3 - 93*e^2 - 22*e + 70, -3*e^6 + 49*e^4 + 3*e^3 - 115*e^2 - 19*e + 58, 11/2*e^6 - 3/2*e^5 - 93*e^4 + 20*e^3 + 517/2*e^2 - 59/2*e - 178, 15/4*e^6 + 13/2*e^5 - 73*e^4 - 100*e^3 + 1253/4*e^2 + 139*e - 272, -93/4*e^6 + 15/2*e^5 + 393*e^4 - 98*e^3 - 4411/4*e^2 + 110*e + 818, 15/2*e^6 - 6*e^5 - 120*e^4 + 87*e^3 + 517/2*e^2 - 120*e - 140, -5*e^6 + 6*e^5 + 79*e^4 - 87*e^3 - 159*e^2 + 117*e + 96, 9*e^6 - 3*e^5 - 154*e^4 + 40*e^3 + 457*e^2 - 51*e - 372, 21/2*e^6 - 11/2*e^5 - 178*e^4 + 77*e^3 + 1009/2*e^2 - 199/2*e - 376, 13/2*e^6 - 4*e^5 - 108*e^4 + 57*e^3 + 573/2*e^2 - 80*e - 222, -11*e^6 + 5*e^5 + 186*e^4 - 69*e^3 - 521*e^2 + 88*e + 382, 35/2*e^6 - 9*e^5 - 291*e^4 + 124*e^3 + 1523/2*e^2 - 154*e - 530, -1/2*e^6 - 15/2*e^5 + 18*e^4 + 114*e^3 - 321/2*e^2 - 323/2*e + 154, 25/2*e^6 - 9*e^5 - 205*e^4 + 126*e^3 + 1005/2*e^2 - 140*e - 344, -2*e^6 - 4*e^5 + 42*e^4 + 62*e^3 - 214*e^2 - 90*e + 198, -19/4*e^6 + 5/2*e^5 + 77*e^4 - 33*e^3 - 713/4*e^2 + 29*e + 106, 23/2*e^6 - 15/2*e^5 - 192*e^4 + 105*e^3 + 1023/2*e^2 - 251/2*e - 376, -33/4*e^6 + 1/2*e^5 + 146*e^4 - 4*e^3 - 1955/4*e^2 + 21*e + 416, -6*e^6 + 4*e^5 + 93*e^4 - 54*e^3 - 164*e^2 + 38*e + 62, -59/2*e^6 + 39/2*e^5 + 487*e^4 - 276*e^3 - 2473/2*e^2 + 699/2*e + 850, -1/4*e^6 - 5/2*e^5 + 9*e^4 + 39*e^3 - 323/4*e^2 - 68*e + 96, 6*e^6 - 108*e^4 - e^3 + 384*e^2 - 27*e - 334, -21/2*e^6 + 8*e^5 + 171*e^4 - 113*e^3 - 803/2*e^2 + 134*e + 222, -1/2*e^6 + 3*e^5 + 2*e^4 - 43*e^3 + 149/2*e^2 + 34*e - 122, 33/4*e^6 - 9/2*e^5 - 137*e^4 + 60*e^3 + 1415/4*e^2 - 43*e - 248, -39/4*e^6 + 7/2*e^5 + 169*e^4 - 46*e^3 - 2109/4*e^2 + 53*e + 430, -17*e^6 + e^5 + 293*e^4 - 4*e^3 - 890*e^2 - 9*e + 690, 19/2*e^6 - 19/2*e^5 - 148*e^4 + 136*e^3 + 543/2*e^2 - 327/2*e - 118, -27/4*e^6 + 15/2*e^5 + 111*e^4 - 108*e^3 - 1121/4*e^2 + 137*e + 218, 29/2*e^6 - 17/2*e^5 - 239*e^4 + 120*e^3 + 1203/2*e^2 - 313/2*e - 410, 5*e^6 - 85*e^4 - 3*e^3 + 245*e^2 + 5*e - 162, -22*e^6 + 8*e^5 + 376*e^4 - 108*e^3 - 1106*e^2 + 146*e + 852, -5/2*e^6 + 3/2*e^5 + 43*e^4 - 21*e^3 - 267/2*e^2 + 45/2*e + 106];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {59, 59, -w^5 + w^4 + 5*w^3 - 3*w^2 - 4*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;