/*
  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 | {71, 71, -2*w^5 + 2*w^4 + 10*w^3 - 8*w^2 - 11*w + 6}>;

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^7 - 27*x^5 + 8*x^4 + 144*x^3 - 80*x^2 - 192*x + 128;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/4*e^6 + 3/8*e^5 - 25/4*e^4 - 57/8*e^3 + 53/2*e^2 + 15*e - 24, 1/4*e^6 + 3/8*e^5 - 6*e^4 - 57/8*e^3 + 83/4*e^2 + 15*e - 12, 11/32*e^6 + 11/16*e^5 - 265/32*e^4 - 221/16*e^3 + 31*e^2 + 63/2*e - 28, 3/16*e^6 + 3/16*e^5 - 73/16*e^4 - 49/16*e^3 + 17*e^2 + 9/2*e - 14, 3/32*e^6 + 1/16*e^5 - 73/32*e^4 - 15/16*e^3 + 33/4*e^2 + 3/2*e - 6, 1/4*e^6 + 3/8*e^5 - 25/4*e^4 - 57/8*e^3 + 53/2*e^2 + 15*e - 26, -1/16*e^6 + 27/16*e^4 - 1/2*e^3 - 9*e^2 + 4*e + 14, -1/4*e^6 - 5/16*e^5 + 13/2*e^4 + 95/16*e^3 - 127/4*e^2 - 31/2*e + 34, -3/16*e^6 - 3/8*e^5 + 73/16*e^4 + 61/8*e^3 - 37/2*e^2 - 19*e + 24, -e^2 + 8, -1/4*e^6 - 5/16*e^5 + 13/2*e^4 + 95/16*e^3 - 127/4*e^2 - 27/2*e + 36, -1/2*e^6 - 3/4*e^5 + 49/4*e^4 + 57/4*e^3 - 193/4*e^2 - 30*e + 45, -1, -1/8*e^5 - 1/2*e^4 + 19/8*e^3 + 21/2*e^2 - e - 20, 11/32*e^6 + 7/16*e^5 - 273/32*e^4 - 129/16*e^3 + 135/4*e^2 + 31/2*e - 24, -5/16*e^6 - 1/2*e^5 + 123/16*e^4 + 10*e^3 - 123/4*e^2 - 25*e + 34, 3/32*e^6 + 5/16*e^5 - 65/32*e^4 - 107/16*e^3 + 7/2*e^2 + 31/2*e + 2, 1/8*e^6 + 7/16*e^5 - 23/8*e^4 - 149/16*e^3 + 10*e^2 + 45/2*e - 14, -11/16*e^6 - 7/8*e^5 + 273/16*e^4 + 129/8*e^3 - 139/2*e^2 - 33*e + 66, -7/16*e^6 - 3/4*e^5 + 165/16*e^4 + 59/4*e^3 - 65/2*e^2 - 35*e + 16, -1/4*e^5 + 23/4*e^3 - 2*e^2 - 13*e + 16, -1/16*e^6 + 1/8*e^5 + 27/16*e^4 - 23/8*e^3 - 8*e^2 + 4*e + 4, -19/32*e^6 - 17/16*e^5 + 457/32*e^4 + 335/16*e^3 - 207/4*e^2 - 93/2*e + 42, 3/16*e^6 + 1/4*e^5 - 65/16*e^4 - 17/4*e^3 + 6*e^2 + 6*e + 8, 5/16*e^6 + 7/16*e^5 - 127/16*e^4 - 141/16*e^3 + 36*e^2 + 57/2*e - 44, -3/16*e^6 - 1/4*e^5 + 73/16*e^4 + 17/4*e^3 - 35/2*e^2 - 2*e + 20, -5/8*e^6 - 9/8*e^5 + 119/8*e^4 + 179/8*e^3 - 51*e^2 - 57*e + 40, -1/16*e^6 + 1/8*e^5 + 27/16*e^4 - 23/8*e^3 - 8*e^2 + 3*e, -11/32*e^6 - 9/16*e^5 + 265/32*e^4 + 167/16*e^3 - 30*e^2 - 33/2*e + 24, -21/32*e^6 - 13/16*e^5 + 519/32*e^4 + 235/16*e^3 - 127/2*e^2 - 53/2*e + 52, 3/16*e^6 - 73/16*e^4 + 3/2*e^3 + 31/2*e^2 - 9*e - 10, -1/4*e^6 - 5/16*e^5 + 25/4*e^4 + 95/16*e^3 - 26*e^2 - 35/2*e + 24, -1/2*e^6 - e^5 + 12*e^4 + 20*e^3 - 87/2*e^2 - 44*e + 40, -3/8*e^6 - 13/16*e^5 + 71/8*e^4 + 263/16*e^3 - 123/4*e^2 - 75/2*e + 26, -1/16*e^6 - 3/16*e^5 + 23/16*e^4 + 65/16*e^3 - 19/4*e^2 - 25/2*e + 16, -7/8*e^6 - 23/16*e^5 + 171/8*e^4 + 453/16*e^3 - 327/4*e^2 - 137/2*e + 78, 1/2*e^5 + 1/2*e^4 - 23/2*e^3 - 15/2*e^2 + 28*e + 6, 1/8*e^6 + 1/2*e^5 - 25/8*e^4 - 23/2*e^3 + 61/4*e^2 + 35*e - 20, -11/32*e^6 - 11/16*e^5 + 265/32*e^4 + 221/16*e^3 - 31*e^2 - 67/2*e + 30, 21/32*e^6 + 17/16*e^5 - 511/32*e^4 - 327/16*e^3 + 243/4*e^2 + 79/2*e - 50, -17/32*e^6 - 15/16*e^5 + 427/32*e^4 + 305/16*e^3 - 58*e^2 - 97/2*e + 60, 3/16*e^6 + 1/4*e^5 - 77/16*e^4 - 21/4*e^3 + 89/4*e^2 + 23*e - 30, -7/16*e^6 - 15/16*e^5 + 169/16*e^4 + 309/16*e^3 - 159/4*e^2 - 105/2*e + 34, 1/16*e^6 + 3/8*e^5 - 27/16*e^4 - 69/8*e^3 + 12*e^2 + 28*e - 26, 1/8*e^6 + 1/8*e^5 - 23/8*e^4 - 11/8*e^3 + 15/2*e^2 - 10*e + 2, -5/32*e^6 - 5/16*e^5 + 127/32*e^4 + 99/16*e^3 - 69/4*e^2 - 19/2*e + 14, -17/32*e^6 - 9/16*e^5 + 411/32*e^4 + 159/16*e^3 - 91/2*e^2 - 41/2*e + 28, -3/16*e^6 - 5/8*e^5 + 65/16*e^4 + 107/8*e^3 - 8*e^2 - 35*e + 6, 7/16*e^6 + 5/16*e^5 - 181/16*e^4 - 87/16*e^3 + 53*e^2 + 39/2*e - 58, -1/8*e^6 - 3/8*e^5 + 23/8*e^4 + 65/8*e^3 - 19/2*e^2 - 26*e + 10, 3/16*e^6 + 1/4*e^5 - 73/16*e^4 - 17/4*e^3 + 35/2*e^2 + 6*e - 18, 1/4*e^6 + 1/4*e^5 - 27/4*e^4 - 19/4*e^3 + 38*e^2 + 16*e - 50, -1/16*e^6 + 23/16*e^4 - 1/2*e^3 - 17/4*e^2 + e + 6, 11/16*e^6 + 13/16*e^5 - 281/16*e^4 - 239/16*e^3 + 159/2*e^2 + 67/2*e - 84, 1/4*e^6 + 3/8*e^5 - 27/4*e^4 - 65/8*e^3 + 39*e^2 + 32*e - 62, 1/32*e^6 - 1/16*e^5 - 35/32*e^4 + 31/16*e^3 + 39/4*e^2 - 25/2*e - 4, -9/16*e^6 - 1/2*e^5 + 227/16*e^4 + 8*e^3 - 60*e^2 - 14*e + 54, -5/8*e^6 - 15/16*e^5 + 121/8*e^4 + 269/16*e^3 - 221/4*e^2 - 45/2*e + 34, 3/4*e^6 + 3/4*e^5 - 73/4*e^4 - 49/4*e^3 + 66*e^2 + 18*e - 36, -3/16*e^6 - 3/4*e^5 + 81/16*e^4 + 67/4*e^3 - 31*e^2 - 50*e + 54, -5/16*e^6 - 1/8*e^5 + 119/16*e^4 + 7/8*e^3 - 22*e^2 + 3*e + 2, -1/8*e^6 + 27/8*e^4 - e^3 - 18*e^2 + 12*e + 32, 5/16*e^6 + 7/16*e^5 - 127/16*e^4 - 141/16*e^3 + 36*e^2 + 49/2*e - 38, 7/32*e^6 + 11/16*e^5 - 149/32*e^4 - 221/16*e^3 + 33/4*e^2 + 43/2*e + 10, 1/4*e^6 - 25/4*e^4 + 2*e^3 + 49/2*e^2 - 16*e - 16, -1/16*e^6 - 1/4*e^5 + 19/16*e^4 + 21/4*e^3 + 1/2*e^2 - 16*e + 4, 25/16*e^6 + 17/8*e^5 - 615/16*e^4 - 319/8*e^3 + 603/4*e^2 + 85*e - 130, -7/32*e^6 - 5/16*e^5 + 181/32*e^4 + 107/16*e^3 - 105/4*e^2 - 51/2*e + 36, 7/16*e^6 + 1/2*e^5 - 173/16*e^4 - 9*e^3 + 44*e^2 + 20*e - 54, -15/16*e^6 - 11/8*e^5 + 369/16*e^4 + 205/8*e^3 - 357/4*e^2 - 45*e + 74, -1/8*e^6 + 1/8*e^5 + 27/8*e^4 - 35/8*e^3 - 17*e^2 + 21*e + 10, 1/16*e^6 - 27/16*e^4 + 1/2*e^3 + 9*e^2 - 3*e - 12, -19/16*e^6 - 7/4*e^5 + 465/16*e^4 + 131/4*e^3 - 112*e^2 - 62*e + 98, -3/8*e^6 - 3/4*e^5 + 73/8*e^4 + 61/4*e^3 - 37*e^2 - 34*e + 58, 3/8*e^6 + 3/4*e^5 - 73/8*e^4 - 57/4*e^3 + 37*e^2 + 23*e - 50, -9/16*e^6 - e^5 + 211/16*e^4 + 39/2*e^3 - 42*e^2 - 48*e + 32, -9/32*e^6 - 3/16*e^5 + 251/32*e^4 + 45/16*e^3 - 183/4*e^2 - 3/2*e + 56, -1/2*e^6 - 3/4*e^5 + 25/2*e^4 + 61/4*e^3 - 53*e^2 - 45*e + 56, 5/8*e^6 + 17/16*e^5 - 121/8*e^4 - 339/16*e^3 + 217/4*e^2 + 107/2*e - 50, 3/8*e^6 + 3/8*e^5 - 77/8*e^4 - 49/8*e^3 + 91/2*e^2 + 3*e - 56, e^6 + 2*e^5 - 49/2*e^4 - 41*e^3 + 197/2*e^2 + 103*e - 106, -45/32*e^6 - 31/16*e^5 + 1103/32*e^4 + 577/16*e^3 - 265/2*e^2 - 143/2*e + 114, 29/32*e^6 + 23/16*e^5 - 711/32*e^4 - 441/16*e^3 + 345/4*e^2 + 103/2*e - 62, 17/16*e^6 + 2*e^5 - 411/16*e^4 - 81/2*e^3 + 97*e^2 + 100*e - 90, -1/8*e^6 - 1/8*e^5 + 23/8*e^4 + 11/8*e^3 - 17/2*e^2 + 5*e + 4, 5/16*e^6 + 11/16*e^5 - 123/16*e^4 - 217/16*e^3 + 133/4*e^2 + 45/2*e - 34, 9/8*e^6 + 15/8*e^5 - 219/8*e^4 - 293/8*e^3 + 104*e^2 + 83*e - 84, 1/4*e^6 - 1/8*e^5 - 25/4*e^4 + 43/8*e^3 + 47/2*e^2 - 30*e - 10, -19/32*e^6 - 9/16*e^5 + 473/32*e^4 + 151/16*e^3 - 245/4*e^2 - 35/2*e + 66, -5/8*e^6 - 5/8*e^5 + 117/8*e^4 + 79/8*e^3 - 169/4*e^2 - 9*e + 8, -7/16*e^6 - e^5 + 165/16*e^4 + 41/2*e^3 - 71/2*e^2 - 50*e + 34, -15/16*e^6 - 3/4*e^5 + 373/16*e^4 + 43/4*e^3 - 93*e^2 - 6*e + 86, -13/16*e^6 - 7/8*e^5 + 319/16*e^4 + 121/8*e^3 - 76*e^2 - 31*e + 58, 1/4*e^6 + 7/8*e^5 - 21/4*e^4 - 149/8*e^3 + 15/2*e^2 + 43*e - 6, 1/8*e^6 + 5/8*e^5 - 23/8*e^4 - 111/8*e^3 + 23/2*e^2 + 35*e - 30, -9/16*e^6 - 7/8*e^5 + 219/16*e^4 + 137/8*e^3 - 101/2*e^2 - 39*e + 38, 9/16*e^6 + 3/8*e^5 - 243/16*e^4 - 45/8*e^3 + 82*e^2 + 11*e - 90, -3/4*e^6 - 7/8*e^5 + 37/2*e^4 + 125/8*e^3 - 295/4*e^2 - 29*e + 88, -3/2*e^6 - 21/8*e^5 + 145/4*e^4 + 415/8*e^3 - 545/4*e^2 - 119*e + 132, 11/8*e^6 + 7/4*e^5 - 273/8*e^4 - 129/4*e^3 + 138*e^2 + 66*e - 126, -7/16*e^6 - 11/16*e^5 + 173/16*e^4 + 201/16*e^3 - 89/2*e^2 - 33/2*e + 36, 1/8*e^6 + 3/8*e^5 - 27/8*e^4 - 65/8*e^3 + 19*e^2 + 27*e - 4, 17/16*e^6 + e^5 - 435/16*e^4 - 35/2*e^3 + 245/2*e^2 + 40*e - 116, -7/16*e^6 + 5/16*e^5 + 181/16*e^4 - 167/16*e^3 - 50*e^2 + 79/2*e + 40, 5/16*e^6 + 9/16*e^5 - 123/16*e^4 - 179/16*e^3 + 125/4*e^2 + 63/2*e - 42, -19/16*e^6 - 13/8*e^5 + 473/16*e^4 + 243/8*e^3 - 243/2*e^2 - 61*e + 108, -23/16*e^6 - 3*e^5 + 549/16*e^4 + 121/2*e^3 - 245/2*e^2 - 146*e + 108, 25/32*e^6 + 9/16*e^5 - 619/32*e^4 - 127/16*e^3 + 299/4*e^2 + 17/2*e - 62, -11/16*e^6 - 13/8*e^5 + 269/16*e^4 + 275/8*e^3 - 275/4*e^2 - 87*e + 86, 5/8*e^6 + 9/8*e^5 - 121/8*e^4 - 179/8*e^3 + 215/4*e^2 + 47*e - 20, -5/16*e^6 - 1/4*e^5 + 127/16*e^4 + 13/4*e^3 - 71/2*e^2 + 2*e + 50, -5/16*e^5 + 1/4*e^4 + 127/16*e^3 - 33/4*e^2 - 47/2*e + 44, -17/16*e^6 - 7/4*e^5 + 419/16*e^4 + 135/4*e^3 - 207/2*e^2 - 68*e + 80, 5/4*e^6 + 11/8*e^5 - 125/4*e^4 - 193/8*e^3 + 259/2*e^2 + 45*e - 116, 5/8*e^6 + 15/16*e^5 - 133/8*e^4 - 301/16*e^3 + 351/4*e^2 + 97/2*e - 106, -3/16*e^6 + 73/16*e^4 - 3/2*e^3 - 27/2*e^2 + 5*e - 12, -31/32*e^6 - 21/16*e^5 + 749/32*e^4 + 379/16*e^3 - 339/4*e^2 - 83/2*e + 60, 17/32*e^6 + 21/16*e^5 - 379/32*e^4 - 435/16*e^3 + 57/2*e^2 + 129/2*e - 16, 7/4*e^6 + 21/8*e^5 - 171/4*e^4 - 399/8*e^3 + 327/2*e^2 + 99*e - 138, -23/16*e^6 - 25/16*e^5 + 577/16*e^4 + 435/16*e^3 - 609/4*e^2 - 99/2*e + 144, 49/32*e^6 + 33/16*e^5 - 1227/32*e^4 - 615/16*e^3 + 163*e^2 + 155/2*e - 152, -9/16*e^6 - 1/2*e^5 + 219/16*e^4 + 8*e^3 - 99/2*e^2 - 10*e + 20, -7/8*e^6 - 11/8*e^5 + 169/8*e^4 + 209/8*e^3 - 151/2*e^2 - 54*e + 48, -1/2*e^6 - 3/8*e^5 + 23/2*e^4 + 41/8*e^3 - 27*e^2 - 3*e - 22, e^6 + 19/16*e^5 - 99/4*e^4 - 329/16*e^3 + 391/4*e^2 + 49/2*e - 72, -19/16*e^6 - 21/8*e^5 + 449/16*e^4 + 427/8*e^3 - 97*e^2 - 123*e + 70, 17/32*e^6 + 9/16*e^5 - 427/32*e^4 - 159/16*e^3 + 55*e^2 + 49/2*e - 44, -5/16*e^6 - e^5 + 119/16*e^4 + 43/2*e^3 - 28*e^2 - 53*e + 40, -19/16*e^6 - 2*e^5 + 477/16*e^4 + 81/2*e^3 - 521/4*e^2 - 101*e + 146, 7/16*e^6 + 1/8*e^5 - 173/16*e^4 + 1/8*e^3 + 39*e^2 - 5*e - 16, 7/8*e^6 + 15/8*e^5 - 173/8*e^4 - 309/8*e^3 + 92*e^2 + 97*e - 108, -3/16*e^6 - 3/8*e^5 + 81/16*e^4 + 61/8*e^3 - 26*e^2 - 14*e + 14, -7/32*e^6 - 9/16*e^5 + 141/32*e^4 + 183/16*e^3 - 5/2*e^2 - 61/2*e - 4, 9/8*e^6 + 13/8*e^5 - 227/8*e^4 - 255/8*e^3 + 128*e^2 + 81*e - 158, 19/16*e^6 + 31/16*e^5 - 457/16*e^4 - 613/16*e^3 + 103*e^2 + 201/2*e - 88, 3/16*e^6 + 1/4*e^5 - 65/16*e^4 - 17/4*e^3 + 6*e^2 + 6*e + 16, -1/2*e^6 - 9/8*e^5 + 12*e^4 + 179/8*e^3 - 85/2*e^2 - 42*e + 28, -15/16*e^6 - e^5 + 389/16*e^4 + 37/2*e^3 - 117*e^2 - 60*e + 142, 31/16*e^6 + 35/16*e^5 - 769/16*e^4 - 625/16*e^3 + 771/4*e^2 + 153/2*e - 166, 27/16*e^6 + 11/4*e^5 - 649/16*e^4 - 211/4*e^3 + 146*e^2 + 112*e - 118, -1/2*e^6 - 1/2*e^5 + 25/2*e^4 + 15/2*e^3 - 53*e^2 + 9*e + 58, 9/16*e^6 + 11/8*e^5 - 227/16*e^4 - 229/8*e^3 + 67*e^2 + 65*e - 92, -5/16*e^6 + 7/16*e^5 + 151/16*e^4 - 189/16*e^3 - 127/2*e^2 + 69/2*e + 58, -3/4*e^6 - 19/16*e^5 + 71/4*e^4 + 361/16*e^3 - 59*e^2 - 97/2*e + 54, 29/32*e^6 + 29/16*e^5 - 695/32*e^4 - 587/16*e^3 + 319/4*e^2 + 171/2*e - 96, -17/16*e^6 - 3*e^5 + 403/16*e^4 + 127/2*e^3 - 181/2*e^2 - 156*e + 92, 2*e^6 + 21/8*e^5 - 99/2*e^4 - 391/8*e^3 + 399/2*e^2 + 102*e - 182, -17/16*e^6 - 33/16*e^5 + 427/16*e^4 + 667/16*e^3 - 237/2*e^2 - 219/2*e + 120];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {71, 71, -2*w^5 + 2*w^4 + 10*w^3 - 8*w^2 - 11*w + 6}>] := 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;