/*
  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, 3, 4, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {27, 3, -w^3 + w^2 + 3*w - 2}>;

primesArray := [
[3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3],
[9, 3, -w^4 + 5*w^2 - 3],
[9, 3, -w^4 + w^3 + 5*w^2 - 3*w - 2],
[13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1],
[17, 17, w^4 - w^3 - 5*w^2 + 3*w + 1],
[19, 19, -w^3 + w^2 + 4*w - 2],
[23, 23, -w^2 + 3],
[31, 31, w^3 - 4*w + 2],
[32, 2, 2],
[37, 37, w^3 - 3*w - 1],
[53, 53, -2*w^4 + w^3 + 9*w^2 - 3*w - 2],
[59, 59, -w^4 + 5*w^2 + w - 4],
[61, 61, -w^4 + w^3 + 5*w^2 - 4*w],
[67, 67, -w^4 + 6*w^2 + 2*w - 4],
[71, 71, 2*w^4 - w^3 - 9*w^2 + 4*w + 5],
[79, 79, 2*w^4 - w^3 - 10*w^2 + 2*w + 7],
[83, 83, -w^4 + 2*w^3 + 5*w^2 - 7*w - 2],
[83, 83, -w^4 + w^3 + 4*w^2 - 3*w + 3],
[83, 83, w^4 - w^3 - 5*w^2 + 4*w - 1],
[83, 83, -w^4 + w^3 + 4*w^2 - 4*w - 2],
[83, 83, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],
[97, 97, -w^4 - w^3 + 6*w^2 + 4*w - 4],
[101, 101, -w^4 + 4*w^2 + 2*w - 2],
[107, 107, -2*w^4 + w^3 + 11*w^2 - 3*w - 7],
[127, 127, w^3 - w^2 - 4*w],
[131, 131, w^4 - w^3 - 4*w^2 + 2*w - 1],
[137, 137, w^4 - w^3 - 6*w^2 + 3*w + 3],
[139, 139, -2*w^4 + w^3 + 9*w^2 - 3*w + 1],
[157, 157, 2*w^4 - w^3 - 9*w^2 + 4*w + 2],
[163, 163, -w^2 - 2*w + 3],
[167, 167, w^4 - 5*w^2 + 2*w + 1],
[169, 13, -2*w^4 + 2*w^3 + 9*w^2 - 7*w - 5],
[169, 13, w^4 - 2*w^3 - 5*w^2 + 8*w + 2],
[191, 191, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],
[193, 193, 2*w^4 - 2*w^3 - 8*w^2 + 7*w],
[199, 199, -w^4 + 2*w^3 + 5*w^2 - 9*w - 3],
[211, 211, -3*w^4 + w^3 + 13*w^2 - 2*w - 1],
[227, 227, 2*w^3 - w^2 - 9*w + 1],
[233, 233, -w^3 + w^2 + 4*w + 1],
[251, 251, -3*w^4 + 14*w^2 + w - 2],
[257, 257, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 5],
[257, 257, -w^4 + w^3 + 5*w^2 - 6*w - 1],
[257, 257, 2*w^4 - 2*w^3 - 11*w^2 + 9*w + 9],
[257, 257, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 3],
[257, 257, 2*w^4 - w^3 - 10*w^2 + 5*w + 5],
[269, 269, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],
[271, 271, -w^4 + 4*w^2 + w - 3],
[277, 277, -2*w^4 + 9*w^2 + w - 4],
[281, 281, w^4 - w^3 - 3*w^2 + 3*w - 4],
[283, 283, 2*w^4 - 3*w^3 - 8*w^2 + 13*w],
[289, 17, -w^4 + w^3 + 7*w^2 - 5*w - 5],
[289, 17, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 6],
[293, 293, -2*w^4 + 11*w^2 - 7],
[317, 317, w^3 - 6*w],
[337, 337, -w^4 + 2*w^3 + 5*w^2 - 9*w - 1],
[337, 337, w^4 - 2*w^3 - 5*w^2 + 6*w + 3],
[337, 337, -w^4 + w^3 + 5*w^2 - 4*w - 6],
[337, 337, -2*w^3 + 8*w + 1],
[337, 337, w^2 + w - 5],
[347, 347, -2*w^4 + w^3 + 9*w^2 - 5*w - 3],
[349, 349, 3*w^4 - 2*w^3 - 13*w^2 + 8*w + 2],
[353, 353, w^4 - 3*w^3 - 4*w^2 + 12*w + 2],
[359, 359, 2*w^3 - w^2 - 7*w + 3],
[361, 19, w^4 - 2*w^3 - 4*w^2 + 7*w + 3],
[361, 19, w^4 - w^3 - 3*w^2 + 3*w - 3],
[367, 367, -2*w^4 + w^3 + 9*w^2 - w - 2],
[379, 379, -2*w^3 + w^2 + 7*w + 2],
[379, 379, -2*w^4 + w^3 + 10*w^2 - 4*w],
[379, 379, 2*w^3 - w^2 - 6*w - 2],
[379, 379, -2*w^3 + 2*w^2 + 6*w - 3],
[379, 379, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6],
[383, 383, 2*w^4 + w^3 - 10*w^2 - 7*w + 4],
[383, 383, -2*w^3 + w^2 + 10*w],
[383, 383, w^3 + w^2 - 5*w - 1],
[383, 383, 3*w^4 + 3*w^3 - 15*w^2 - 15*w + 4],
[383, 383, -2*w^4 + 3*w^3 + 10*w^2 - 10*w - 2],
[389, 389, 3*w^4 - w^3 - 14*w^2 + 3*w + 3],
[397, 397, w^4 - 6*w^2 - 2*w + 5],
[397, 397, w^4 - 2*w^3 - 2*w^2 + 6*w - 5],
[397, 397, 2*w^4 - 2*w^3 - 11*w^2 + 10*w + 6],
[397, 397, w^4 - w^3 - 3*w^2 + 4*w - 6],
[397, 397, w^3 + w^2 - 3*w - 4],
[401, 401, w^3 - w^2 - 6*w + 3],
[401, 401, -w^4 + 4*w^2 + 2*w - 3],
[401, 401, -w^4 + 6*w^2 - w - 7],
[421, 421, -w^4 - w^3 + 5*w^2 + 5*w - 4],
[421, 421, -w^4 + w^3 + 6*w^2 - 3*w - 2],
[421, 421, -w^4 - w^3 + 6*w^2 + 6*w - 5],
[421, 421, 3*w^4 - w^3 - 15*w^2 + w + 10],
[421, 421, 2*w^4 - 9*w^2 + 2*w + 4],
[431, 431, 2*w^4 - 11*w^2 - 2*w + 11],
[439, 439, 2*w^4 - 3*w^3 - 9*w^2 + 10*w + 6],
[443, 443, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 2],
[449, 449, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],
[461, 461, 3*w^4 - 4*w^3 - 14*w^2 + 14*w + 5],
[463, 463, w^3 - 6*w - 1],
[467, 467, 3*w^4 - 3*w^3 - 13*w^2 + 10*w],
[487, 487, 3*w^4 - w^3 - 14*w^2 + w + 3],
[487, 487, 2*w^2 - w - 4],
[487, 487, -2*w^4 + 2*w^3 + 9*w^2 - 8*w - 5],
[487, 487, 2*w^4 - 10*w^2 - 3*w + 4],
[487, 487, -2*w^4 + w^3 + 8*w^2 - 3*w - 1],
[499, 499, -w^4 + 2*w^3 + 4*w^2 - 8*w - 2],
[499, 499, -2*w^4 + w^3 + 10*w^2 - w - 9],
[499, 499, -w^3 - 2*w^2 + 2*w + 5],
[499, 499, -w^4 + 4*w^2 + 4*w],
[499, 499, -2*w^4 + w^3 + 12*w^2 - 4*w - 9],
[509, 509, 3*w^4 - 2*w^3 - 16*w^2 + 6*w + 9],
[521, 521, -2*w^4 + w^3 + 10*w^2 - 5*w - 4],
[523, 523, w - 4],
[529, 23, -w^3 + 2*w^2 + 4*w - 4],
[529, 23, w^4 + 2*w^3 - 6*w^2 - 11*w + 4],
[569, 569, -2*w^4 + 11*w^2 - 10],
[571, 571, -2*w^4 + w^3 + 10*w^2 - 3*w - 2],
[593, 593, -2*w^4 + 3*w^3 + 10*w^2 - 11*w - 8],
[613, 613, -2*w^3 + 2*w^2 + 9*w - 5],
[617, 617, -2*w^4 + 10*w^2 - w - 7],
[631, 631, -2*w^4 + 2*w^3 + 7*w^2 - 8*w + 3],
[641, 641, -2*w^4 + 2*w^3 + 8*w^2 - 9*w + 3],
[643, 643, -3*w^4 + w^3 + 13*w^2 - w + 1],
[643, 643, w^4 + w^3 - 3*w^2 - 6*w - 2],
[643, 643, -w^4 - w^3 + 6*w^2 + 3*w - 4],
[643, 643, 2*w^4 - w^3 - 10*w^2 + 4*w + 1],
[643, 643, 2*w^4 - 2*w^3 - 8*w^2 + 6*w + 5],
[647, 647, -2*w^3 + w^2 + 6*w - 4],
[659, 659, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 10],
[661, 661, 2*w^4 + 3*w^3 - 11*w^2 - 14*w + 6],
[673, 673, w^4 - 3*w^3 - 3*w^2 + 10*w - 1],
[683, 683, w^4 - 2*w^3 - 4*w^2 + 6*w - 4],
[701, 701, w^3 - w^2 - 2*w + 4],
[727, 727, -w^4 + 6*w^2 + 2*w - 7],
[743, 743, w^4 - w^3 - 7*w^2 + 5*w + 9],
[769, 769, w^4 + 2*w^3 - 5*w^2 - 9*w + 4],
[787, 787, w^4 + w^3 - 7*w^2 - 3*w + 5],
[797, 797, -2*w^4 - w^3 + 10*w^2 + 4*w - 4],
[797, 797, w^4 - 3*w^3 - 5*w^2 + 12*w + 5],
[797, 797, -3*w^4 + 2*w^3 + 13*w^2 - 8*w],
[797, 797, -w^4 + w^3 + 3*w^2 - 5*w - 1],
[797, 797, 2*w^4 - 11*w^2 + 2*w + 7],
[809, 809, -4*w^4 + 2*w^3 + 18*w^2 - 6*w + 1],
[809, 809, 3*w^4 - 3*w^3 - 14*w^2 + 11*w + 8],
[809, 809, w^4 - 2*w^3 - 2*w^2 + 8*w - 4],
[809, 809, 2*w^4 - 8*w^2 - 2*w + 3],
[809, 809, 2*w^4 - 3*w^3 - 9*w^2 + 13*w + 4],
[821, 821, -2*w^4 + 2*w^3 + 8*w^2 - 6*w + 3],
[823, 823, -2*w^4 + 3*w^3 + 8*w^2 - 12*w - 1],
[829, 829, 2*w^4 - 3*w^3 - 10*w^2 + 13*w + 4],
[839, 839, -2*w^4 + 12*w^2 - w - 10],
[863, 863, 2*w^4 - w^3 - 9*w^2 + 5],
[877, 877, -w^4 + w^3 + 5*w^2 - 5*w - 7],
[881, 881, 2*w^4 - w^3 - 9*w^2 + w - 1],
[887, 887, 3*w^4 - 3*w^3 - 15*w^2 + 11*w + 6],
[907, 907, -3*w^4 + w^3 + 13*w^2 - w - 5],
[919, 919, w^4 + w^3 - 3*w^2 - 5*w - 3],
[929, 929, 2*w^4 - 3*w^3 - 10*w^2 + 9*w + 4],
[937, 937, -2*w^4 + 10*w^2 + 3*w - 7],
[941, 941, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],
[961, 31, 2*w^4 + 2*w^3 - 10*w^2 - 12*w + 5],
[961, 31, -w^4 + 5*w^2 - w - 7],
[967, 967, -2*w^4 + w^3 + 9*w^2 - 3*w + 2],
[991, 991, -3*w^4 + w^3 + 15*w^2 - 4*w - 5],
[997, 997, -3*w^4 + 15*w^2 - 5],
[997, 997, 2*w^4 - w^3 - 9*w^2 + 6*w + 2],
[997, 997, 2*w^4 - w^3 - 11*w^2 + w + 7],
[997, 997, 2*w^3 - 5*w - 1],
[997, 997, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 - 7*x^4 + 6*x^3 + 35*x^2 - 40*x - 20;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -1, 2/5*e^4 - 7/5*e^3 - 2*e^2 + 5*e + 2, 3/10*e^4 - 13/10*e^3 - 2*e^2 + 15/2*e + 4, 1/5*e^4 - 6/5*e^3 + 7*e - 1, -2/5*e^4 + 7/5*e^3 + 3*e^2 - 8*e - 4, 1/10*e^4 - 1/10*e^3 - e^2 - 3/2*e + 4, 1/10*e^4 - 1/10*e^3 - e^2 - 1/2*e, e^2 - 3*e - 2, 1/2*e^4 - 3/2*e^3 - 5*e^2 + 17/2*e + 14, -1/5*e^4 + 6/5*e^3 - 2*e^2 - 2*e + 11, -1/5*e^4 + 1/5*e^3 + 2*e^2 + 2*e - 2, 3/5*e^4 - 18/5*e^3 + e^2 + 16*e - 5, -e^3 + 4*e^2 + e - 8, 3/5*e^4 - 13/5*e^3 - e^2 + 7*e + 2, 3/5*e^4 - 13/5*e^3 - e^2 + 8*e - 4, -2/5*e^4 + 2/5*e^3 + 5*e^2 - 4, -e^4 + 4*e^3 + 3*e^2 - 16*e + 9, -1/5*e^4 + 6/5*e^3 - 2*e^2 - 2*e + 10, -1/2*e^4 + 5/2*e^3 + e^2 - 21/2*e + 4, 2*e^3 - 6*e^2 - 11*e + 8, 3/5*e^4 - 13/5*e^3 - e^2 + 11*e - 1, -1/5*e^4 + 1/5*e^3 + 2*e^2 - e - 1, -4/5*e^4 + 24/5*e^3 - 22*e + 7, -3*e^2 + 4*e + 22, 3/5*e^4 - 13/5*e^3 + 6*e, 1/5*e^4 - 6/5*e^3 + e^2 + 8*e - 16, 17/10*e^4 - 67/10*e^3 - 8*e^2 + 63/2*e + 2, -2/5*e^4 + 2/5*e^3 + 6*e^2 - 5*e - 4, 1/5*e^4 - 16/5*e^3 + 7*e^2 + 17*e - 18, -1/5*e^4 + 6/5*e^3 - 8*e - 4, -2/5*e^4 + 12/5*e^3 - 15*e + 12, -7/5*e^4 + 32/5*e^3 + 5*e^2 - 31*e - 1, 3/2*e^4 - 11/2*e^3 - 9*e^2 + 41/2*e + 24, -7/5*e^4 + 37/5*e^3 + 2*e^2 - 34*e + 2, 1/5*e^4 + 4/5*e^3 - 8*e^2 + 26, -1/5*e^4 + 6/5*e^3 - 2*e^2 - 7*e + 24, -1/5*e^4 + 1/5*e^3 + 5*e + 15, -3*e^3 + 8*e^2 + 18*e - 18, -2/5*e^4 + 12/5*e^3 - 20*e + 10, -2/5*e^4 + 22/5*e^3 - 7*e^2 - 16*e + 20, -1/5*e^4 + 1/5*e^3 + 3*e^2 - 4*e + 14, -e^4 + 2*e^3 + 10*e^2 - 4*e - 22, 9/10*e^4 - 59/10*e^3 + 4*e^2 + 47/2*e - 14, -2/5*e^4 + 17/5*e^3 - 4*e^2 - 9*e + 12, -6/5*e^4 + 31/5*e^3 + 2*e^2 - 22*e - 12, 1/5*e^4 - 1/5*e^3 - 3*e^2 + 7, -3/10*e^4 + 23/10*e^3 + e^2 - 27/2*e - 14, -1/5*e^4 + 6/5*e^3 + e^2 - 5*e - 20, 6/5*e^4 - 26/5*e^3 - 6*e^2 + 28*e + 14, -7/10*e^4 + 27/10*e^3 + 9*e^2 - 39/2*e - 34, -7/5*e^4 + 17/5*e^3 + 14*e^2 - 16*e - 24, 2/5*e^4 - 7/5*e^3 - 3*e^2 + 7*e + 6, -12/5*e^4 + 57/5*e^3 + 4*e^2 - 48*e, -1/5*e^4 + 1/5*e^3 - 2*e^2 + 14*e + 14, -e^3 + 5*e^2 + 6*e - 22, -2/5*e^4 + 2/5*e^3 + e^2 + 10*e + 10, 4/5*e^4 - 4/5*e^3 - 12*e^2 + 2*e + 24, -1/5*e^4 + 16/5*e^3 - 7*e^2 - 18*e + 24, -7/5*e^4 + 32/5*e^3 - 2*e^2 - 13*e + 27, 6/5*e^4 - 26/5*e^3 - 6*e^2 + 26*e + 3, -9/5*e^4 + 44/5*e^3 + 2*e^2 - 35*e + 14, -1/2*e^4 + 5/2*e^3 - 2*e^2 - 5/2*e + 22, -7/5*e^4 + 27/5*e^3 + 7*e^2 - 30*e + 4, 9/10*e^4 - 39/10*e^3 + 9/2*e - 14, 9/5*e^4 - 39/5*e^3 - 6*e^2 + 30*e + 6, -2*e^4 + 12*e^3 - 4*e^2 - 51*e + 20, 13/5*e^4 - 63/5*e^3 - 4*e^2 + 51*e + 2, -e^4 + 4*e^3 + 6*e^2 - 28*e, 4/5*e^4 - 19/5*e^3 - 2*e^2 + 21*e - 4, -8/5*e^4 + 28/5*e^3 + 11*e^2 - 26*e - 18, 1/5*e^4 + 4/5*e^3 - 7*e^2 - e + 18, -1/5*e^4 + 6/5*e^3 - e^2 - 2*e - 15, -1/2*e^4 + 7/2*e^3 - 3*e^2 - 41/2*e + 4, -7/5*e^4 + 27/5*e^3 + 6*e^2 - 20*e - 4, 8/5*e^4 - 33/5*e^3 - 4*e^2 + 20*e + 2, 6/5*e^4 - 31/5*e^3 + 3*e^2 + 19*e - 28, 1/5*e^4 - 1/5*e^3 - 2*e^2 + 2*e + 12, -4/5*e^4 + 19/5*e^3 + 5*e^2 - 25*e - 3, 2/5*e^4 + 8/5*e^3 - 11*e^2 - 14*e + 26, e^4 - 4*e^3 - 6*e^2 + 24*e - 7, 4/5*e^4 - 19/5*e^3 - 2*e^2 + 14*e + 8, -14/5*e^4 + 59/5*e^3 + 10*e^2 - 50*e - 4, 8/5*e^4 - 23/5*e^3 - 12*e^2 + 21*e + 14, 1/5*e^4 - 16/5*e^3 + 5*e^2 + 21*e - 14, -12/5*e^4 + 57/5*e^3 + 8*e^2 - 54*e - 6, 3/10*e^4 - 13/10*e^3 + 2*e^2 + 5/2*e - 12, -7/5*e^4 + 22/5*e^3 + 14*e^2 - 22*e - 36, -14/5*e^4 + 69/5*e^3 + 2*e^2 - 53*e + 16, 6/5*e^4 - 16/5*e^3 - 9*e^2 + 14*e + 6, 6/5*e^4 - 16/5*e^3 - 11*e^2 + 15*e + 4, e^4 - 4*e^3 - 2*e^2 + 4*e + 4, -9/5*e^4 + 24/5*e^3 + 16*e^2 - 16*e - 36, 3/5*e^4 - 13/5*e^3 - 7*e^2 + 22*e + 14, -8/5*e^4 + 18/5*e^3 + 17*e^2 - 14*e - 48, 4/5*e^4 - 24/5*e^3 - e^2 + 29*e + 4, -2/5*e^4 + 22/5*e^3 - 9*e^2 - 9*e + 30, -e^4 + 3*e^3 + 8*e^2 - 13*e - 12, -7/5*e^4 + 22/5*e^3 + 10*e^2 - 24*e, -6/5*e^4 + 16/5*e^3 + 16*e^2 - 26*e - 46, e^4 - 7*e^3 + 7*e^2 + 28*e - 22, -9/5*e^4 + 39/5*e^3 + 9*e^2 - 46*e - 6, -2/5*e^4 + 7/5*e^3 - e^2 + 3*e + 2, -e^4 + 5*e^3 - e^2 - 20*e + 30, 2/5*e^4 - 17/5*e^3 + 3*e^2 + 22*e - 12, 9/10*e^4 - 9/10*e^3 - 10*e^2 - 21/2*e + 28, 1/10*e^4 - 31/10*e^3 + 8*e^2 + 27/2*e - 18, -7/5*e^4 + 22/5*e^3 + 11*e^2 - 14*e - 36, 1/5*e^4 + 4/5*e^3 - 8*e^2 - 4*e + 28, 4/5*e^4 - 14/5*e^3 - 2*e^2 + 3*e + 6, -4/5*e^4 + 4/5*e^3 + 13*e^2 - 8*e - 36, -8/5*e^4 + 38/5*e^3 + 5*e^2 - 31*e + 8, 2/5*e^4 - 7/5*e^3 - 4*e^2 + 17*e - 10, -e^4 + 6*e^3 - 5*e^2 - 19*e + 24, -3/10*e^4 + 33/10*e^3 - 9*e^2 + 5/2*e + 28, -9/5*e^4 + 19/5*e^3 + 19*e^2 - 12*e - 38, -19/5*e^4 + 79/5*e^3 + 15*e^2 - 69*e - 4, e^3 - 3*e^2 - 4*e + 12, -4/5*e^4 + 14/5*e^3 + 7*e^2 - 19*e - 12, 2*e^4 - 8*e^3 - 12*e^2 + 45*e - 6, e^4 - e^3 - 12*e^2 - 7*e + 24, 3*e^4 - 14*e^3 - 10*e^2 + 66*e + 4, -19/10*e^4 + 69/10*e^3 + 15*e^2 - 69/2*e - 44, 11/10*e^4 - 51/10*e^3 - 2*e^2 + 43/2*e - 20, -2/5*e^4 - 3/5*e^3 + 13*e^2 - 4*e - 48, -8/5*e^4 + 38/5*e^3 + 6*e^2 - 39*e - 10, 4/5*e^4 - 4/5*e^3 - 9*e^2 - 2*e + 10, -1/10*e^4 + 1/10*e^3 - 5/2*e + 12, 12/5*e^4 - 57/5*e^3 - 6*e^2 + 63*e - 10, 17/5*e^4 - 67/5*e^3 - 14*e^2 + 51*e + 16, -21/10*e^4 + 61/10*e^3 + 12*e^2 - 31/2*e + 2, -13/10*e^4 + 23/10*e^3 + 12*e^2 - 19/2*e + 4, -13/10*e^4 + 73/10*e^3 + e^2 - 79/2*e + 22, 31/10*e^4 - 131/10*e^3 - 9*e^2 + 99/2*e - 20, -e^4 + 2*e^3 + 10*e^2 - e - 2, 6/5*e^4 - 6/5*e^3 - 14*e^2 - 10*e + 37, 8/5*e^4 - 38/5*e^3 - 6*e^2 + 38*e + 10, -11/10*e^4 + 61/10*e^3 + e^2 - 59/2*e + 16, 5/2*e^4 - 15/2*e^3 - 18*e^2 + 65/2*e + 20, -6/5*e^4 + 21/5*e^3 + 14*e^2 - 26*e - 44, -9/5*e^4 + 39/5*e^3 + 3*e^2 - 28*e + 34, -e^4 + 3*e^3 + 5*e^2 - 7*e + 25, -7/5*e^4 + 47/5*e^3 - 7*e^2 - 42*e + 12, -7/10*e^4 + 17/10*e^3 + 9*e^2 - 9/2*e - 32, 13/10*e^4 - 83/10*e^3 + 3*e^2 + 73/2*e - 10, -7/5*e^4 + 12/5*e^3 + 20*e^2 - 8*e - 58, -7/5*e^4 + 17/5*e^3 + 13*e^2 - 20*e - 3, -33/10*e^4 + 143/10*e^3 + 12*e^2 - 131/2*e - 2, -2*e^4 + 9*e^3 - e^2 - 22*e + 23, 3/5*e^4 - 23/5*e^3 + 28*e + 4, -3/5*e^4 - 12/5*e^3 + 17*e^2 + 15*e - 44, 6/5*e^4 - 26/5*e^3 - 4*e^2 + 17*e + 22, 1/5*e^4 - 16/5*e^3 + 10*e^2 + 11*e - 36, -11/5*e^4 + 46/5*e^3 + 9*e^2 - 26*e - 24, -6/5*e^4 + 46/5*e^3 - 4*e^2 - 52*e + 14, -13/5*e^4 + 53/5*e^3 + 15*e^2 - 49*e - 45, e^4 - 7*e^3 + 4*e^2 + 30*e - 38, 17/10*e^4 - 67/10*e^3 - 5*e^2 + 59/2*e - 4, -8/5*e^4 - 2/5*e^3 + 30*e^2 + e - 59, -11/5*e^4 + 61/5*e^3 - 2*e^2 - 51*e + 8, 6/5*e^4 - 16/5*e^3 - 7*e^2 - 4*e + 32, -4/5*e^4 + 14/5*e^3 + 9*e^2 - 15*e - 18, 6/5*e^4 - 46/5*e^3 + 5*e^2 + 62*e - 28, -e^4 + e^3 + 15*e^2 - 7*e - 22, 4/5*e^4 - 9/5*e^3 - 14*e^2 + 12*e + 39];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3}>] := -1;
ALEigenvalues[ideal<ZF | {9, 3, -w^4 + w^3 + 5*w^2 - 3*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;