/*
  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![11, 1, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {31,31,1/2*w^3 - w^2 - w + 1/2}>;

primesArray := [
[4, 2, -1/2*w^3 + w^2 + 3*w - 7/2],
[4, 2, w + 1],
[11, 11, w],
[11, 11, 1/2*w^3 - w^2 - 3*w + 5/2],
[11, 11, -w + 2],
[19, 19, -1/2*w^3 + 3*w + 3/2],
[25, 5, -w^3 + 2*w^2 + 4*w - 4],
[31, 31, w^3 - 2*w^2 - 5*w + 3],
[31, 31, -1/2*w^3 + w^2 + w - 1/2],
[59, 59, 1/2*w^3 - w^2 - 4*w + 15/2],
[59, 59, 1/2*w^3 - w^2 - 4*w - 5/2],
[61, 61, -2*w^3 + 5*w^2 + 8*w - 12],
[61, 61, 1/2*w^3 - 2*w - 5/2],
[71, 71, -1/2*w^3 + 2*w^2 - 15/2],
[71, 71, 3/2*w^3 - 4*w^2 - 5*w + 23/2],
[79, 79, -5/2*w^3 + 6*w^2 + 9*w - 31/2],
[79, 79, -4*w^3 + 10*w^2 + 17*w - 28],
[79, 79, -w^3 + w^2 + 4*w + 1],
[79, 79, 5/2*w^3 - 6*w^2 - 9*w + 23/2],
[81, 3, -3],
[89, 89, 2*w^3 - 4*w^2 - 9*w + 6],
[89, 89, -1/2*w^3 + w + 3/2],
[101, 101, 1/2*w^3 - 5*w - 9/2],
[101, 101, -5/2*w^3 + 7*w^2 + 9*w - 37/2],
[131, 131, 1/2*w^3 - 4*w - 1/2],
[131, 131, -w^3 + 3*w^2 + 2*w - 9],
[139, 139, -w^3 + 2*w^2 + 6*w - 6],
[139, 139, -2*w - 1],
[149, 149, w^2 - 6],
[149, 149, -3/2*w^3 + 4*w^2 + 6*w - 17/2],
[151, 151, w^3 - w^2 - 6*w + 1],
[151, 151, 2*w - 1],
[151, 151, -w^3 + 2*w^2 + 6*w - 4],
[151, 151, 3/2*w^3 - 4*w^2 - 4*w + 21/2],
[169, 13, 7/2*w^3 - 9*w^2 - 13*w + 43/2],
[169, 13, -1/2*w^3 + 2*w^2 + 4*w - 3/2],
[179, 179, -1/2*w^3 + w^2 + 4*w - 7/2],
[179, 179, -1/2*w^3 + w^2 + 4*w - 3/2],
[181, 181, -1/2*w^3 + 3*w^2 + w - 17/2],
[181, 181, -2*w^3 + 6*w^2 + 7*w - 18],
[199, 199, 3/2*w^3 - 2*w^2 - 7*w + 1/2],
[199, 199, -1/2*w^3 + 2*w^2 - w - 17/2],
[199, 199, 5/2*w^3 - 6*w^2 - 9*w + 25/2],
[199, 199, -3*w^3 + 8*w^2 + 11*w - 23],
[211, 211, -2*w^3 + 6*w^2 + 7*w - 16],
[211, 211, -1/2*w^3 + 3*w^2 + w - 21/2],
[211, 211, 1/2*w^3 - 4*w - 13/2],
[211, 211, -w^3 + 3*w^2 + 2*w - 3],
[239, 239, 3/2*w^3 - 2*w^2 - 9*w + 7/2],
[239, 239, 5/2*w^3 - 5*w^2 - 11*w + 17/2],
[241, 241, -2*w^3 + 4*w^2 + 10*w - 7],
[241, 241, -2*w^3 + 3*w^2 + 10*w - 4],
[241, 241, -w^3 + 2*w^2 + 2*w - 2],
[241, 241, -5/2*w^3 + 6*w^2 + 8*w - 27/2],
[251, 251, -1/2*w^3 + w^2 + w - 13/2],
[251, 251, 5/2*w^3 - 5*w^2 - 11*w + 25/2],
[251, 251, -2*w^3 + 4*w^2 + 7*w - 10],
[251, 251, -w^3 + 2*w^2 + 5*w - 9],
[269, 269, 1/2*w^3 - 3*w^2 + 13/2],
[269, 269, -3/2*w^3 + 5*w^2 + 4*w - 35/2],
[271, 271, -3/2*w^3 + 4*w^2 + 6*w - 13/2],
[271, 271, -5/2*w^3 + 6*w^2 + 11*w - 29/2],
[281, 281, 7/2*w^3 - 10*w^2 - 11*w + 53/2],
[281, 281, 1/2*w^3 - 5*w - 5/2],
[281, 281, -1/2*w^3 - 2*w^2 + 5*w + 19/2],
[281, 281, 3/2*w^3 - 5*w^2 - 5*w + 27/2],
[331, 331, 3/2*w^3 - 4*w^2 - 7*w + 17/2],
[331, 331, 1/2*w^3 - 2*w^2 - 3*w + 17/2],
[349, 349, 1/2*w^3 + w^2 - 4*w - 9/2],
[349, 349, 5/2*w^3 - 7*w^2 - 8*w + 39/2],
[359, 359, 1/2*w^3 - w^2 - 3/2],
[359, 359, 3/2*w^3 - 3*w^2 - 8*w + 7/2],
[361, 19, -1/2*w^3 + w^2 + 2*w - 13/2],
[379, 379, 5/2*w^3 - 4*w^2 - 12*w + 15/2],
[379, 379, -3*w^3 + 7*w^2 + 10*w - 17],
[389, 389, -7/2*w^3 + 9*w^2 + 13*w - 51/2],
[389, 389, w^3 - 5*w - 1],
[389, 389, -9/2*w^3 + 11*w^2 + 18*w - 59/2],
[389, 389, -7/2*w^3 + 8*w^2 + 13*w - 43/2],
[401, 401, -1/2*w^3 + 3*w^2 + 4*w - 9/2],
[401, 401, 1/2*w^3 + 3*w^2 - 4*w - 23/2],
[421, 421, -7/2*w^3 + 9*w^2 + 16*w - 61/2],
[421, 421, 1/2*w^3 - 4*w^2 + w + 35/2],
[449, 449, -7/2*w^3 + 8*w^2 + 17*w - 47/2],
[449, 449, 3*w^3 - 7*w^2 - 14*w + 17],
[461, 461, -5/2*w^3 + 5*w^2 + 11*w - 21/2],
[461, 461, 2*w^3 - 4*w^2 - 7*w + 8],
[479, 479, 3/2*w^3 - 2*w^2 - 10*w + 9/2],
[479, 479, 5/2*w^3 - 5*w^2 - 12*w + 17/2],
[491, 491, 2*w^2 - 2*w - 5],
[491, 491, 2*w^3 - 6*w^2 - 6*w + 19],
[509, 509, 3/2*w^3 - 4*w^2 - 4*w + 25/2],
[509, 509, w^3 - w^2 - 6*w + 3],
[521, 521, 3/2*w^3 - 3*w^2 - 8*w + 5/2],
[521, 521, 3*w^3 - 8*w^2 - 12*w + 20],
[529, 23, 3/2*w^3 - 2*w^2 - 8*w + 9/2],
[529, 23, -2*w^3 + 5*w^2 + 6*w - 14],
[541, 541, w^3 - w^2 - 4*w - 3],
[541, 541, 5/2*w^3 - 6*w^2 - 10*w + 23/2],
[569, 569, 1/2*w^3 - w - 7/2],
[569, 569, 7/2*w^3 - 7*w^2 - 16*w + 27/2],
[569, 569, 5/2*w^3 - 6*w^2 - 11*w + 27/2],
[569, 569, -5/2*w^3 + 5*w^2 + 8*w - 17/2],
[571, 571, 3/2*w^3 - 3*w^2 - 8*w + 3/2],
[571, 571, 7/2*w^3 - 9*w^2 - 14*w + 45/2],
[599, 599, -7/2*w^3 + 10*w^2 + 13*w - 53/2],
[599, 599, -1/2*w^3 + 4*w^2 + w - 29/2],
[619, 619, -3/2*w^3 + 6*w^2 + 3*w - 31/2],
[619, 619, -3/2*w^3 + 4*w^2 + 7*w - 9/2],
[619, 619, 1/2*w^3 - 2*w^2 - 3*w + 25/2],
[619, 619, -3/2*w^3 + 6*w^2 + 3*w - 41/2],
[641, 641, -3/2*w^3 + 5*w^2 + 6*w - 39/2],
[641, 641, -7/2*w^3 + 8*w^2 + 17*w - 45/2],
[661, 661, 5/2*w^3 - 7*w^2 - 8*w + 41/2],
[661, 661, -w^3 + 4*w^2 - 14],
[701, 701, -1/2*w^3 + 3*w^2 + 5*w - 5/2],
[701, 701, 11/2*w^3 - 14*w^2 - 21*w + 69/2],
[719, 719, -3/2*w^3 + 4*w^2 + 7*w - 13/2],
[719, 719, -3*w^3 + 7*w^2 + 12*w - 13],
[719, 719, -w^2 - 4*w - 2],
[719, 719, 5*w^3 - 14*w^2 - 17*w + 37],
[739, 739, w^3 - 7*w - 3],
[739, 739, -5/2*w^3 + 7*w^2 + 7*w - 37/2],
[751, 751, 5*w^3 - 14*w^2 - 16*w + 36],
[751, 751, w^3 + 2*w^2 - 8*w - 12],
[761, 761, 1/2*w^3 - 2*w^2 + 2*w + 15/2],
[761, 761, w^3 - w^2 - 8*w + 3],
[769, 769, -1/2*w^3 + 4*w^2 - w - 29/2],
[769, 769, -5/2*w^3 + 8*w^2 + 7*w - 43/2],
[809, 809, 1/2*w^3 - w^2 - 3*w - 7/2],
[809, 809, w - 6],
[811, 811, -5*w^3 + 12*w^2 + 24*w - 38],
[811, 811, -2*w^3 + 6*w^2 + 6*w - 21],
[811, 811, 2*w^2 - 2*w - 3],
[811, 811, 2*w^2 + 4*w - 1],
[821, 821, -1/2*w^3 + 5*w + 17/2],
[821, 821, 2*w^3 - 2*w^2 - 10*w - 1],
[829, 829, -3*w^3 + 9*w^2 + 10*w - 23],
[829, 829, 1/2*w^3 - 4*w^2 + 31/2],
[839, 839, -3*w^3 + 8*w^2 + 12*w - 28],
[839, 839, -3*w^3 + 6*w^2 + 15*w - 17],
[841, 29, -5/2*w^3 + 5*w^2 + 10*w - 17/2],
[841, 29, 5/2*w^3 - 5*w^2 - 10*w + 23/2],
[911, 911, -1/2*w^3 + 6*w - 3/2],
[911, 911, w^2 - 4*w - 6],
[919, 919, 7/2*w^3 - 8*w^2 - 14*w + 37/2],
[919, 919, 2*w^3 - 3*w^2 - 8*w + 4],
[929, 929, 1/2*w^3 - 2*w^2 - 1/2],
[929, 929, 1/2*w^3 - 2*w^2 - 3*w + 29/2],
[929, 929, 2*w^3 - 2*w^2 - 7*w + 2],
[929, 929, w^2 - 2*w - 10],
[941, 941, 5/2*w^3 - 6*w^2 - 13*w + 25/2],
[941, 941, -3/2*w^3 + 8*w + 7/2],
[941, 941, 7/2*w^3 - 10*w^2 - 12*w + 45/2],
[941, 941, 1/2*w^3 - 3*w^2 + 2*w + 29/2],
[961, 31, 5/2*w^3 - 5*w^2 - 10*w + 21/2],
[971, 971, w^3 - w^2 - 4*w - 5],
[971, 971, 5/2*w^3 - 6*w^2 - 10*w + 19/2],
[991, 991, -1/2*w^3 + 4*w^2 - 27/2],
[991, 991, 3*w^3 - 9*w^2 - 10*w + 25]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 4*x^5 - 8*x^4 + 28*x^3 + 32*x^2 - 32*x - 25;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/2*e^3 + e^2 + 2*e + 1/2, e, e^2 - 2*e - 3, -1/2*e^5 + 3*e^4 - e^3 - 29/2*e^2 + 6*e + 13, 1/2*e^5 - 3*e^4 + e^3 + 27/2*e^2 - 4*e - 6, -1/2*e^5 + 2*e^4 + 3/2*e^3 - 17/2*e^2 + 11/2, 1/2*e^5 - 5/2*e^4 - e^3 + 25/2*e^2 + 3/2*e - 9, -1/2*e^5 + 3*e^4 - 33/2*e^2 + e + 15, -1, 1/2*e^5 - 2*e^4 - 3/2*e^3 + 13/2*e^2 + 5*e + 7/2, e^4 - 1/2*e^3 - 10*e^2 - 4*e + 15/2, -e^3 + 3*e^2 + 3*e + 1, 1/2*e^5 - 3*e^4 + 4*e^3 + 15/2*e^2 - 21*e, 1/2*e^5 - 4*e^4 + 3*e^3 + 41/2*e^2 - 7*e - 19, 1/2*e^4 - e^3 - 11/2*e - 8, 1/2*e^5 - 4*e^4 + 7/2*e^3 + 39/2*e^2 - 11*e - 25/2, 3/2*e^5 - 13/2*e^4 - 3*e^3 + 55/2*e^2 - 3/2*e - 18, -e^3 + 4*e^2 + 2*e - 1, -e^5 + 5*e^4 + 3/2*e^3 - 26*e^2 - 2*e + 49/2, -e^5 + 9/2*e^4 + 3/2*e^3 - 20*e^2 + 9/2*e + 25/2, -e^5 + 7/2*e^4 + 11/2*e^3 - 16*e^2 - 27/2*e + 15/2, 1/2*e^5 - 3*e^4 + 3*e^3 + 21/2*e^2 - 18*e - 1, 3/2*e^4 - 11/2*e^3 - 2*e^2 + 27/2*e - 11/2, e^5 - 11/2*e^4 + e^3 + 25*e^2 - 19/2*e - 16, -e^5 + 5*e^4 - 20*e^2 + 4*e + 4, 1/2*e^5 - 5/2*e^4 + 4*e^3 + 3/2*e^2 - 49/2*e + 5, -e^5 + 9/2*e^4 + 2*e^3 - 17*e^2 - 13/2*e + 4, 1/2*e^5 - 3*e^4 - 1/2*e^3 + 31/2*e^2 + 8*e - 17/2, -3/2*e^3 + 3*e^2 + 5*e - 5/2, e^5 - 9/2*e^4 - 1/2*e^3 + 13*e^2 - 9/2*e + 27/2, 1/2*e^5 - 3/2*e^4 - 9/2*e^3 + 19/2*e^2 + 31/2*e - 35/2, 1/2*e^3 - 10*e - 9/2, 3/2*e^3 - 5*e^2 - 6*e + 15/2, 2*e^3 - 5*e^2 - 8*e + 3, -1/2*e^5 + 3*e^4 + 1/2*e^3 - 45/2*e^2 + 7*e + 69/2, -1/2*e^5 + 8*e^3 - 9/2*e^2 - 10*e + 15, 2*e^4 - 15/2*e^3 - 4*e^2 + 23*e - 7/2, 1/2*e^5 - 3/2*e^4 - 1/2*e^3 + 3/2*e^2 - 23/2*e + 11/2, -1/2*e^5 + 3*e^4 + 1/2*e^3 - 29/2*e^2 - 4*e - 5/2, e^5 - 13/2*e^4 + 7/2*e^3 + 31*e^2 - 31/2*e - 31/2, 5/2*e^5 - 11*e^4 - 7/2*e^3 + 85/2*e^2 - 7*e - 31/2, 1/2*e^5 - 5*e^4 + 5*e^3 + 61/2*e^2 - 18*e - 31, -1/2*e^5 + 9/2*e^4 - 3*e^3 - 53/2*e^2 + 15/2*e + 14, 1/2*e^5 - 2*e^4 - 9/2*e^3 + 33/2*e^2 + 10*e - 25/2, -3/2*e^5 + 13/2*e^4 + 7*e^3 - 77/2*e^2 - 25/2*e + 27, 1/2*e^4 - 5/2*e^3 + 3*e^2 - 3/2*e - 31/2, e^5 - 11/2*e^4 + e^3 + 23*e^2 - 11/2*e - 4, e^5 - 6*e^4 - 5/2*e^3 + 39*e^2 + 13*e - 69/2, -1/2*e^5 + 17/2*e^3 - 3/2*e^2 - 24*e + 3/2, 3/2*e^5 - 7*e^4 - 2*e^3 + 57/2*e^2 - 7, -1/2*e^5 + e^4 + 3*e^3 - 1/2*e^2 + 3*e - 14, -e^5 + 7*e^4 - 5*e^3 - 32*e^2 + 17*e + 12, 2*e^5 - 19/2*e^4 - 7/2*e^3 + 41*e^2 + 11/2*e - 27/2, -3*e^5 + 31/2*e^4 + 5/2*e^3 - 76*e^2 + 9/2*e + 109/2, -5/2*e^4 + 17/2*e^3 + 8*e^2 - 49/2*e + 5/2, 3/2*e^5 - 8*e^4 - 2*e^3 + 87/2*e^2 - 6*e - 35, -1/2*e^5 + 5*e^4 - 5/2*e^3 - 73/2*e^2 + 8*e + 69/2, -1/2*e^5 + 3*e^4 + 1/2*e^3 - 35/2*e^2 + 4*e + 13/2, e^5 - 6*e^4 + 5*e^3 + 18*e^2 - 23*e - 1, -1/2*e^5 + 5/2*e^4 + e^3 - 31/2*e^2 + 11/2*e + 25, -3*e^5 + 29/2*e^4 + 9/2*e^3 - 72*e^2 + 13/2*e + 123/2, -3/2*e^4 - 1/2*e^3 + 24*e^2 + 3/2*e - 79/2, 2*e^4 - 7*e^3 - 3*e^2 + 16*e - 2, -e^5 + 4*e^4 + 9/2*e^3 - 25*e^2 + 3*e + 85/2, 3/2*e^5 - 15/2*e^4 + 5/2*e^3 + 49/2*e^2 - 43/2*e - 15/2, e^5 - 11/2*e^4 + 7/2*e^3 + 20*e^2 - 55/2*e - 27/2, -2*e^5 + 11*e^4 - 7/2*e^3 - 49*e^2 + 32*e + 83/2, -3/2*e^5 + 13/2*e^4 - 45/2*e^2 + 33/2*e + 29, -5/2*e^4 + 3*e^3 + 22*e^2 + 3/2*e - 28, -e^5 + 5*e^4 + 6*e^3 - 36*e^2 - 21*e + 29, -1/2*e^5 + 13/2*e^3 + 9/2*e^2 - 16*e - 21/2, -2*e^5 + 12*e^4 - 2*e^3 - 64*e^2 + 21*e + 55, 3/2*e^5 - 11*e^4 + 6*e^3 + 127/2*e^2 - 30*e - 70, -5/2*e^5 + 25/2*e^4 + 5*e^3 - 137/2*e^2 + 1/2*e + 57, -2*e^5 + 12*e^4 - 9/2*e^3 - 54*e^2 + 25*e + 75/2, -9/2*e^3 + 6*e^2 + 31*e - 17/2, -1/2*e^5 + 4*e^4 - 13/2*e^3 - 27/2*e^2 + 36*e + 13/2, 1/2*e^5 - 3/2*e^4 - 4*e^3 + 33/2*e^2 - 7/2*e - 34, -1/2*e^4 + 3/2*e^3 - e^2 - 3/2*e + 7/2, e^5 - 19/2*e^4 + 29/2*e^3 + 38*e^2 - 97/2*e - 19/2, 3*e^5 - 29/2*e^4 - 11/2*e^3 + 69*e^2 + 9/2*e - 85/2, 7/2*e^4 - 17/2*e^3 - 22*e^2 + 45/2*e + 45/2, 3/2*e^5 - 21/2*e^4 + 3*e^3 + 119/2*e^2 - 13/2*e - 45, 1/2*e^5 - 6*e^4 + 29/2*e^3 + 31/2*e^2 - 48*e + 11/2, 3/2*e^5 - 5*e^4 - 13*e^3 + 69/2*e^2 + 32*e - 14, e^5 - 2*e^4 - 23/2*e^3 + 11*e^2 + 30*e - 1/2, -e^5 + 11/2*e^4 + 1/2*e^3 - 37*e^2 + 35/2*e + 121/2, e^5 - 27/2*e^3 - 9*e^2 + 38*e + 43/2, 3/2*e^5 - 7/2*e^4 - 13*e^3 + 27/2*e^2 + 69/2*e - 9, 2*e^4 - 21/2*e^3 + 43*e + 7/2, e^5 - 5*e^4 + e^3 + 20*e^2 - 10*e - 19, 2*e^5 - 27/2*e^4 + 23/2*e^3 + 59*e^2 - 107/2*e - 87/2, -1/2*e^5 + 7/2*e^4 - 4*e^3 - 37/2*e^2 + 47/2*e + 26, e^5 - 9/2*e^4 - 7/2*e^3 + 28*e^2 - 11/2*e - 51/2, 1/2*e^5 - 7/2*e^4 + 7/2*e^3 + 37/2*e^2 - 53/2*e - 41/2, -3/2*e^5 + 11/2*e^4 + 9*e^3 - 71/2*e^2 - 5/2*e + 51, e^5 - 6*e^4 + 3/2*e^3 + 35*e^2 - 21*e - 85/2, -1/2*e^5 + 9/2*e^4 - 9*e^3 - 27/2*e^2 + 83/2*e + 3, -2*e^5 + 12*e^4 + 1/2*e^3 - 63*e^2 - 9*e + 67/2, 1/2*e^5 - 9/2*e^4 + 8*e^3 + 31/2*e^2 - 57/2*e - 1, 1/2*e^5 + 5/2*e^4 - 17*e^3 - 7/2*e^2 + 85/2*e - 11, 1/2*e^5 - 3/2*e^4 - 17/2*e^3 + 43/2*e^2 + 59/2*e - 27/2, -2*e^4 + 17/2*e^3 + 7*e^2 - 38*e - 23/2, 5/2*e^5 - 8*e^4 - 16*e^3 + 81/2*e^2 + 26*e - 23, -3*e^5 + 31/2*e^4 + 5/2*e^3 - 73*e^2 + 9/2*e + 107/2, 3/2*e^5 - 17/2*e^4 - 3/2*e^3 + 91/2*e^2 + 27/2*e - 65/2, -e^5 + 15/2*e^4 - 11*e^3 - 25*e^2 + 95/2*e + 16, 2*e^5 - 7*e^4 - 13*e^3 + 39*e^2 + 23*e - 16, 3/2*e^5 - 8*e^4 - 17/2*e^3 + 119/2*e^2 + 23*e - 111/2, -1/2*e^5 + e^4 + 4*e^3 + 1/2*e^2 - 27, -3*e^5 + 31/2*e^4 + 11/2*e^3 - 83*e^2 - 17/2*e + 83/2, e^5 - 5*e^4 - 3*e^3 + 21*e^2 + 24*e - 12, -2*e^5 + 21/2*e^4 - 1/2*e^3 - 42*e^2 + 23/2*e + 17/2, -e^5 + 7*e^4 - 4*e^3 - 33*e^2 + 6*e + 3, -e^4 + 15/2*e^3 - 6*e^2 - 19*e + 5/2, 3/2*e^5 - 5*e^4 - 11/2*e^3 + 35/2*e^2 - e + 5/2, 2*e^4 - 9*e^3 + 4*e^2 + 17*e - 24, 1/2*e^5 - 17/2*e^4 + 15*e^3 + 97/2*e^2 - 89/2*e - 59, -2*e^5 + 6*e^4 + 10*e^3 - 12*e^2 - 28*e - 22, 3/2*e^5 - 15/2*e^4 - 3*e^3 + 71/2*e^2 + 19/2*e, -3/2*e^5 + 11/2*e^4 + 8*e^3 - 71/2*e^2 - 5/2*e + 46, -2*e^3 + 9*e^2 - e - 30, -9/2*e^5 + 23*e^4 + 9*e^3 - 247/2*e^2 - 15*e + 81, -1/2*e^5 + 5/2*e^4 - 1/2*e^3 - 5/2*e^2 - 3/2*e - 55/2, -2*e^5 + 29/2*e^4 - 19/2*e^3 - 74*e^2 + 79/2*e + 151/2, -2*e^5 + 11*e^4 + 4*e^3 - 69*e^2 - 2*e + 60, 2*e^5 - 19/2*e^4 - 3/2*e^3 + 39*e^2 - 19/2*e - 5/2, e^5 - 5*e^4 - 11/2*e^3 + 34*e^2 + 26*e - 37/2, -3*e^5 + 33/2*e^4 - 5*e^3 - 64*e^2 + 53/2*e + 21, -3/2*e^5 + 8*e^4 - 65/2*e^2 + 6*e - 16, -3/2*e^5 + 8*e^4 + 5/2*e^3 - 95/2*e^2 - 5*e + 107/2, 5/2*e^5 - 9*e^4 - 37/2*e^3 + 113/2*e^2 + 48*e - 87/2, 3/2*e^5 - 13/2*e^4 - 6*e^3 + 73/2*e^2 + 37/2*e - 36, 3*e^5 - 35/2*e^4 + 7*e^3 + 79*e^2 - 95/2*e - 62, 2*e^5 - 15/2*e^4 - 27/2*e^3 + 46*e^2 + 61/2*e - 91/2, e^5 - 13/2*e^4 + 3/2*e^3 + 38*e^2 - 15/2*e - 89/2, -2*e^5 + 27/2*e^4 - 17/2*e^3 - 63*e^2 + 43/2*e + 89/2, -3/2*e^5 + 11/2*e^4 + 7*e^3 - 49/2*e^2 - 7/2*e + 11, -1/2*e^4 + 3/2*e^3 + 21/2*e + 13/2, 5/2*e^5 - 19/2*e^4 - 17*e^3 + 119/2*e^2 + 91/2*e - 49, 1/2*e^5 - e^4 - 19/2*e^3 + 37/2*e^2 + 34*e - 45/2, -3/2*e^5 + 17/2*e^4 + 7/2*e^3 - 101/2*e^2 - 31/2*e + 95/2, -7/2*e^5 + 35/2*e^4 + 4*e^3 - 183/2*e^2 + 41/2*e + 95, -5/2*e^5 + 37/2*e^4 - 14*e^3 - 197/2*e^2 + 127/2*e + 97, -5/2*e^5 + 14*e^4 + 7/2*e^3 - 159/2*e^2 - 5*e + 155/2, 2*e^5 - 23/2*e^4 + 9/2*e^3 + 52*e^2 - 73/2*e - 93/2, -5/2*e^5 + 12*e^4 + 3/2*e^3 - 109/2*e^2 + 20*e + 71/2, 1/2*e^5 - 11/2*e^4 + 9/2*e^3 + 67/2*e^2 + 9/2*e - 83/2, e^5 - 7/2*e^4 - 3*e^3 + 5*e^2 + 25/2*e + 4, 11/2*e^5 - 63/2*e^4 + 13/2*e^3 + 297/2*e^2 - 103/2*e - 235/2, -3*e^5 + 33/2*e^4 + 3/2*e^3 - 89*e^2 + 25/2*e + 159/2, -e^5 + 7/2*e^4 + 3/2*e^3 - 5*e^2 + 23/2*e - 65/2, 3/2*e^5 - 9/2*e^4 - 12*e^3 + 71/2*e^2 + 21/2*e - 53, e^5 - 2*e^4 - 9*e^3 + 2*e^2 + 40*e + 6, -3*e^5 + 31/2*e^4 + 5/2*e^3 - 86*e^2 + 49/2*e + 209/2, -3/2*e^5 + 11*e^4 - 8*e^3 - 125/2*e^2 + 46*e + 67, -11/2*e^5 + 27*e^4 + 17/2*e^3 - 259/2*e^2 + 2*e + 179/2, -9/2*e^4 + 11*e^3 + 24*e^2 - 33/2*e - 12, -4*e^5 + 22*e^4 - e^3 - 104*e^2 + 21*e + 90, 5/2*e^5 - 13*e^4 - 3/2*e^3 + 121/2*e^2 + e - 51/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {31,31,1/2*w^3 - w^2 - w + 1/2}>] := 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;