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

NN := ideal<ZF | {49, 7, w^3 - w^2 - 6*w - 1}>;

primesArray := [
[5, 5, -w^3 + w^2 + 5*w],
[7, 7, -w^3 + 2*w^2 + 3*w - 3],
[7, 7, w^3 - 2*w^2 - 3*w],
[13, 13, -w^2 + w + 3],
[13, 13, -w^2 + w + 2],
[16, 2, 2],
[23, 23, -w^2 + 3*w + 1],
[23, 23, -2*w^3 + 3*w^2 + 9*w - 2],
[25, 5, w^3 - 2*w^2 - 2*w + 2],
[49, 7, w^3 - w^2 - 6*w - 1],
[53, 53, 2*w^3 - 2*w^2 - 8*w - 3],
[53, 53, 2*w^3 - 2*w^2 - 8*w - 1],
[67, 67, 2*w^3 - 4*w^2 - 7*w + 2],
[67, 67, -2*w^3 + 4*w^2 + 6*w - 1],
[81, 3, -3],
[83, 83, 2*w^3 - 3*w^2 - 6*w - 1],
[83, 83, 3*w^3 - 4*w^2 - 12*w + 1],
[103, 103, 3*w^3 - 4*w^2 - 12*w - 1],
[103, 103, 2*w^3 - 3*w^2 - 6*w + 1],
[107, 107, w^3 - w^2 - 3*w - 3],
[107, 107, 2*w^3 - 2*w^2 - 9*w],
[109, 109, -3*w^3 + 5*w^2 + 10*w - 5],
[109, 109, -2*w^3 + 3*w^2 + 9*w + 1],
[109, 109, -2*w^3 + 2*w^2 + 7*w],
[109, 109, w^2 - 3*w - 4],
[121, 11, 3*w^3 - 4*w^2 - 12*w],
[121, 11, 2*w^2 - 3*w - 6],
[139, 139, w^3 - 7*w - 1],
[139, 139, -w^3 + 2*w^2 + 5*w - 4],
[139, 139, 3*w^3 - 3*w^2 - 14*w - 2],
[139, 139, -3*w^3 + 5*w^2 + 13*w - 2],
[149, 149, -w^2 + 4*w + 1],
[149, 149, -3*w^3 + 4*w^2 + 14*w - 1],
[149, 149, 2*w^2 - 3*w - 5],
[149, 149, -w^3 + 3*w^2 + 3*w - 4],
[167, 167, -w^3 - w^2 + 7*w + 4],
[167, 167, -4*w^3 + 6*w^2 + 15*w - 4],
[169, 13, w^3 - w^2 - 6*w + 4],
[173, 173, w^2 - 3*w - 5],
[173, 173, 2*w^3 - 2*w^2 - 11*w],
[179, 179, 2*w^3 - 2*w^2 - 9*w + 1],
[179, 179, w^3 - w^2 - 3*w - 4],
[179, 179, w^2 - 5],
[179, 179, w^3 - 6*w - 1],
[197, 197, w^3 + w^2 - 7*w - 6],
[197, 197, 2*w^3 - 2*w^2 - 9*w + 2],
[223, 223, -w^2 + 7],
[223, 223, 2*w^2 - w - 4],
[227, 227, 2*w^3 - 2*w^2 - 11*w - 2],
[227, 227, w^3 - w^2 - 7*w - 1],
[233, 233, w^3 - w^2 - 7*w],
[233, 233, -2*w^3 + 2*w^2 + 11*w + 1],
[257, 257, -2*w^3 + 4*w^2 + 8*w - 1],
[257, 257, 2*w^2 - 4*w - 7],
[277, 277, -4*w^3 + 6*w^2 + 15*w - 3],
[277, 277, -3*w^3 + 5*w^2 + 9*w],
[281, 281, -2*w^3 + 5*w^2 + 3*w - 5],
[281, 281, -2*w^3 + 5*w^2 + 5*w - 7],
[281, 281, 2*w^3 - 5*w^2 - 5*w + 4],
[281, 281, -4*w^3 + 7*w^2 + 15*w - 4],
[283, 283, 3*w^3 - 4*w^2 - 13*w - 3],
[283, 283, -w^3 + 2*w^2 + w - 4],
[313, 313, -2*w^3 + 4*w^2 + 5*w - 5],
[313, 313, 3*w^3 - 5*w^2 - 11*w],
[347, 347, -w^3 + 2*w^2 + w - 5],
[347, 347, 3*w^3 - 4*w^2 - 13*w - 4],
[353, 353, 3*w^3 - 5*w^2 - 12*w + 1],
[353, 353, -w^3 + 3*w^2 - 5],
[361, 19, 3*w^3 - 3*w^2 - 13*w - 9],
[361, 19, 2*w^3 - 2*w^2 - 7*w + 4],
[373, 373, -3*w^3 + 4*w^2 + 11*w + 1],
[373, 373, -3*w^3 + 4*w^2 + 11*w],
[383, 383, 3*w^3 - 5*w^2 - 13*w + 1],
[383, 383, 2*w^2 - 5*w - 6],
[397, 397, 4*w^3 - 5*w^2 - 19*w - 1],
[397, 397, 2*w^3 - 2*w^2 - 9*w - 7],
[431, 431, 2*w^3 - 2*w^2 - 7*w - 5],
[431, 431, 3*w^3 - 6*w^2 - 11*w + 3],
[431, 431, 3*w^3 - 3*w^2 - 13*w],
[431, 431, w^3 - 4*w^2 + w + 8],
[457, 457, 4*w^3 - 7*w^2 - 13*w],
[457, 457, -3*w^2 + 4*w + 5],
[463, 463, -3*w^3 + 5*w^2 + 11*w + 1],
[463, 463, 2*w^3 - 4*w^2 - 5*w + 6],
[487, 487, -w^3 + 4*w^2 - 7],
[487, 487, -2*w^3 + 5*w^2 + 6*w - 5],
[499, 499, -6*w^3 + 8*w^2 + 25*w - 1],
[499, 499, -w^3 + 4*w^2 - 5],
[499, 499, -2*w^3 + 5*w^2 + 6*w - 7],
[499, 499, 3*w^3 - 5*w^2 - 7*w],
[509, 509, -4*w^3 + 8*w^2 + 15*w - 8],
[509, 509, 3*w^3 - 4*w^2 - 10*w - 2],
[509, 509, -3*w^3 + 3*w^2 + 11*w + 6],
[509, 509, 4*w^3 - 5*w^2 - 16*w],
[521, 521, -4*w^3 + 5*w^2 + 15*w - 2],
[521, 521, w^3 - w^2 - 8*w + 1],
[521, 521, -4*w^3 + 7*w^2 + 12*w - 6],
[521, 521, 3*w^3 - 3*w^2 - 16*w - 1],
[523, 523, 3*w^3 - w^2 - 17*w - 8],
[523, 523, -5*w^3 + 5*w^2 + 23*w + 4],
[529, 23, 3*w^3 - 6*w^2 - 10*w + 2],
[547, 547, -2*w^3 + 5*w^2 + 6*w - 6],
[547, 547, -w^3 + 4*w^2 - 6],
[557, 557, -3*w^3 + 6*w^2 + 9*w - 10],
[557, 557, 3*w^3 - 6*w^2 - 9*w - 1],
[571, 571, 4*w^3 - 4*w^2 - 17*w - 6],
[571, 571, -2*w^3 + 6*w^2 + 3*w - 10],
[571, 571, 4*w^3 - 5*w^2 - 18*w + 3],
[571, 571, -3*w^3 + 7*w^2 + 9*w - 5],
[587, 587, 3*w^3 - 3*w^2 - 14*w],
[587, 587, w^3 - w^2 - 2*w - 4],
[593, 593, -w^3 + 3*w^2 + 2*w - 10],
[593, 593, -5*w^3 + 7*w^2 + 19*w - 4],
[613, 613, 3*w^3 - 5*w^2 - 12*w - 4],
[613, 613, -w^3 + 3*w^2 - 10],
[631, 631, 4*w^3 - 8*w^2 - 13*w + 4],
[631, 631, 3*w^3 - 4*w^2 - 15*w + 2],
[631, 631, w^3 - 9*w - 1],
[631, 631, 3*w^3 - 7*w^2 - 7*w + 9],
[643, 643, 3*w^3 - 3*w^2 - 16*w - 2],
[643, 643, w^3 - w^2 - 8*w],
[647, 647, -w^3 + 4*w^2 + 2*w - 4],
[647, 647, -3*w^2 + 4*w + 10],
[673, 673, w^3 + 2*w^2 - 9*w - 8],
[673, 673, -w^3 + 4*w^2 + 3*w - 7],
[683, 683, 3*w^3 - 3*w^2 - 11*w - 2],
[683, 683, 4*w^3 - 4*w^2 - 17*w - 5],
[709, 709, -5*w^3 + 7*w^2 + 21*w + 1],
[709, 709, -3*w^3 + 2*w^2 + 16*w + 4],
[709, 709, 4*w^3 - 4*w^2 - 19*w - 2],
[709, 709, -2*w^3 + 4*w^2 + 3*w - 4],
[787, 787, 2*w^2 - w - 9],
[787, 787, w^3 + w^2 - 7*w - 2],
[811, 811, 4*w^3 - 7*w^2 - 11*w],
[811, 811, 3*w^3 - 4*w^2 - 9*w - 1],
[811, 811, -6*w^3 + 9*w^2 + 23*w - 5],
[811, 811, 5*w^3 - 6*w^2 - 21*w - 2],
[821, 821, -3*w^3 + 6*w^2 + 11*w - 2],
[821, 821, 2*w^3 - 13*w - 5],
[821, 821, -w^3 + 4*w^2 - w - 9],
[821, 821, -w^3 + 3*w^2 + 5*w - 6],
[841, 29, 2*w^3 - 2*w^2 - 12*w - 1],
[857, 857, w^3 - 4*w^2 + 3*w + 6],
[857, 857, -5*w^3 + 8*w^2 + 21*w - 3],
[863, 863, -w^3 + 4*w^2 + w - 7],
[863, 863, -w^3 + 4*w^2 + w - 6],
[877, 877, -w^3 - w^2 + 8*w + 2],
[877, 877, -w^3 + 3*w^2 + 4*w - 8],
[883, 883, 4*w^3 - 7*w^2 - 14*w + 1],
[883, 883, 3*w^3 - 6*w^2 - 8*w + 7],
[929, 929, 3*w^3 - 5*w^2 - 12*w - 3],
[929, 929, -w^3 + 3*w^2 - 9],
[929, 929, 3*w - 5],
[929, 929, 3*w^3 - 3*w^2 - 15*w + 2],
[937, 937, w^3 - 2*w^2 - 6*w + 6],
[937, 937, 2*w^3 - w^2 - 12*w],
[941, 941, -w^3 + 4*w^2 + 2*w - 7],
[941, 941, -4*w^3 + 6*w^2 + 13*w - 3],
[941, 941, -5*w^3 + 7*w^2 + 19*w + 2],
[941, 941, 3*w^2 - 4*w - 7],
[953, 953, 3*w^3 - 5*w^2 - 12*w - 2],
[953, 953, -w^3 + 3*w^2 - 8],
[961, 31, 4*w^3 - 6*w^2 - 13*w - 2],
[961, 31, 5*w^3 - 7*w^2 - 19*w + 3],
[977, 977, -4*w^3 + 7*w^2 + 19*w - 6],
[977, 977, -3*w^3 + 7*w^2 + 10*w - 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 23*x^4 - 2*x^3 + 112*x^2 - 40*x - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/12*e^5 - 19/12*e^3 - 5/6*e^2 + 5*e + 2, 1/12*e^5 - 19/12*e^3 - 5/6*e^2 + 5*e + 2, -1/6*e^5 + 23/6*e^3 + 2/3*e^2 - 56/3*e + 10/3, -1/6*e^5 + 23/6*e^3 + 2/3*e^2 - 56/3*e + 10/3, 1/6*e^5 + 1/6*e^4 - 23/6*e^3 - 17/6*e^2 + 17*e - 1/3, -1/3*e^3 + 13/3*e + 10/3, -1/3*e^3 + 13/3*e + 10/3, 1/6*e^5 - 23/6*e^3 - 2/3*e^2 + 50/3*e - 4/3, -1, -1/6*e^5 - 1/3*e^4 + 9/2*e^3 + 5*e^2 - 26*e + 8/3, -1/6*e^5 - 1/3*e^4 + 9/2*e^3 + 5*e^2 - 26*e + 8/3, -1/6*e^5 + 19/6*e^3 + 5/3*e^2 - 12*e - 8, -1/6*e^5 + 19/6*e^3 + 5/3*e^2 - 12*e - 8, 1/6*e^4 - 13/6*e^2 - 5/3*e + 4, 1/3*e^5 - 1/3*e^4 - 19/3*e^3 + 3*e^2 + 70/3*e - 6, 1/3*e^5 - 1/3*e^4 - 19/3*e^3 + 3*e^2 + 70/3*e - 6, -1/6*e^4 + 19/6*e^2 + 5/3*e - 4, -1/6*e^4 + 19/6*e^2 + 5/3*e - 4, -1/3*e^4 + 4/3*e^3 + 13/3*e^2 - 14*e - 10/3, -1/3*e^4 + 4/3*e^3 + 13/3*e^2 - 14*e - 10/3, -1/3*e^4 + 2/3*e^3 + 13/3*e^2 - 13/3*e + 4/3, -1/3*e^5 + 1/3*e^4 + 6*e^3 - 3*e^2 - 16*e + 34/3, -1/3*e^4 + 2/3*e^3 + 13/3*e^2 - 13/3*e + 4/3, -1/3*e^5 + 1/3*e^4 + 6*e^3 - 3*e^2 - 16*e + 34/3, 1/6*e^5 + 1/3*e^4 - 31/6*e^3 - 5*e^2 + 98/3*e - 6, 1/6*e^5 + 1/3*e^4 - 31/6*e^3 - 5*e^2 + 98/3*e - 6, 1/12*e^5 + 2/3*e^4 - 13/4*e^3 - 19/2*e^2 + 22*e + 20/3, 1/12*e^5 + 2/3*e^4 - 13/4*e^3 - 19/2*e^2 + 22*e + 20/3, -1/2*e^5 - 1/6*e^4 + 23/2*e^3 + 25/6*e^2 - 175/3*e + 10, -1/2*e^5 - 1/6*e^4 + 23/2*e^3 + 25/6*e^2 - 175/3*e + 10, -1/12*e^5 - 1/3*e^4 + 31/12*e^3 + 31/6*e^2 - 53/3*e - 8, -1/12*e^5 - 1/3*e^4 + 31/12*e^3 + 31/6*e^2 - 53/3*e - 8, 1/12*e^5 + 1/3*e^4 - 35/12*e^3 - 31/6*e^2 + 20*e - 2/3, 1/12*e^5 + 1/3*e^4 - 35/12*e^3 - 31/6*e^2 + 20*e - 2/3, 1/6*e^5 - 1/6*e^4 - 19/6*e^3 + 3/2*e^2 + 35/3*e - 2, 1/6*e^5 - 1/6*e^4 - 19/6*e^3 + 3/2*e^2 + 35/3*e - 2, 1/3*e^5 - 1/6*e^4 - 7*e^3 + 5/6*e^2 + 91/3*e + 32/3, 1/12*e^5 - 5/4*e^3 - 5/6*e^2 - 4/3*e + 2/3, 1/12*e^5 - 5/4*e^3 - 5/6*e^2 - 4/3*e + 2/3, -1/6*e^5 + 1/3*e^4 + 7/2*e^3 - 14/3*e^2 - 41/3*e + 14/3, -1/6*e^5 + 1/3*e^4 + 7/2*e^3 - 14/3*e^2 - 41/3*e + 14/3, -1/12*e^5 - 1/3*e^4 + 23/12*e^3 + 43/6*e^2 - 4*e - 64/3, -1/12*e^5 - 1/3*e^4 + 23/12*e^3 + 43/6*e^2 - 4*e - 64/3, -1/3*e^5 + 1/3*e^4 + 6*e^3 - 3*e^2 - 17*e + 16/3, -1/3*e^5 + 1/3*e^4 + 6*e^3 - 3*e^2 - 17*e + 16/3, 1/3*e^4 - 5/3*e^3 - 7/3*e^2 + 61/3*e - 58/3, 1/3*e^4 - 5/3*e^3 - 7/3*e^2 + 61/3*e - 58/3, -1/3*e^5 - 1/3*e^4 + 23/3*e^3 + 23/3*e^2 - 34*e - 28/3, -1/3*e^5 - 1/3*e^4 + 23/3*e^3 + 23/3*e^2 - 34*e - 28/3, -1/2*e^5 - 1/6*e^4 + 65/6*e^3 + 19/6*e^2 - 131/3*e + 38/3, -1/2*e^5 - 1/6*e^4 + 65/6*e^3 + 19/6*e^2 - 131/3*e + 38/3, 5/12*e^5 - 2/3*e^4 - 33/4*e^3 + 17/2*e^2 + 35*e - 56/3, 5/12*e^5 - 2/3*e^4 - 33/4*e^3 + 17/2*e^2 + 35*e - 56/3, -1/4*e^5 + 1/3*e^4 + 61/12*e^3 - 23/6*e^2 - 62/3*e + 62/3, -1/4*e^5 + 1/3*e^4 + 61/12*e^3 - 23/6*e^2 - 62/3*e + 62/3, -2/3*e^5 - 2/3*e^4 + 16*e^3 + 40/3*e^2 - 221/3*e + 8/3, 1/3*e^5 - 7*e^3 + 2/3*e^2 + 83/3*e - 64/3, 1/3*e^5 - 7*e^3 + 2/3*e^2 + 83/3*e - 64/3, -2/3*e^5 - 2/3*e^4 + 16*e^3 + 40/3*e^2 - 221/3*e + 8/3, -1/6*e^5 - 2/3*e^4 + 29/6*e^3 + 31/3*e^2 - 23*e - 2/3, -1/6*e^5 - 2/3*e^4 + 29/6*e^3 + 31/3*e^2 - 23*e - 2/3, e^4 - 2*e^3 - 14*e^2 + 16*e + 8, e^4 - 2*e^3 - 14*e^2 + 16*e + 8, -1/6*e^5 + 1/3*e^4 + 11/6*e^3 - 14/3*e^2 + 8*e + 58/3, -1/6*e^5 + 1/3*e^4 + 11/6*e^3 - 14/3*e^2 + 8*e + 58/3, -2/3*e^3 + 2*e^2 + 26/3*e - 28/3, -2/3*e^3 + 2*e^2 + 26/3*e - 28/3, -1/12*e^5 - 1/3*e^4 + 35/12*e^3 + 19/6*e^2 - 23*e + 8/3, -1/12*e^5 - 1/3*e^4 + 35/12*e^3 + 19/6*e^2 - 23*e + 8/3, 1/2*e^5 - e^4 - 59/6*e^3 + 12*e^2 + 112/3*e - 50/3, 1/2*e^5 - e^4 - 59/6*e^3 + 12*e^2 + 112/3*e - 50/3, 5/12*e^5 - 103/12*e^3 - 1/6*e^2 + 89/3*e - 22/3, 5/12*e^5 - 103/12*e^3 - 1/6*e^2 + 89/3*e - 22/3, 1/3*e^4 - 1/3*e^3 - 19/3*e^2 - 2*e + 34/3, 1/3*e^4 - 1/3*e^3 - 19/3*e^2 - 2*e + 34/3, 1/6*e^5 - 1/6*e^4 - 19/6*e^3 + 7/2*e^2 + 35/3*e - 10, -1/2*e^5 - 1/3*e^4 + 19/2*e^3 + 34/3*e^2 - 92/3*e - 24, 1/6*e^5 - 1/6*e^4 - 19/6*e^3 + 7/2*e^2 + 35/3*e - 10, -1/2*e^5 - 1/3*e^4 + 19/2*e^3 + 34/3*e^2 - 92/3*e - 24, 1/2*e^5 - 23/2*e^3 - e^2 + 52*e - 22, 1/2*e^5 - 23/2*e^3 - e^2 + 52*e - 22, -5/6*e^5 + 115/6*e^3 + 13/3*e^2 - 268/3*e + 2/3, -5/6*e^5 + 115/6*e^3 + 13/3*e^2 - 268/3*e + 2/3, -2/3*e^4 + 2/3*e^3 + 32/3*e^2 - 6*e - 20/3, -2/3*e^4 + 2/3*e^3 + 32/3*e^2 - 6*e - 20/3, 1/3*e^5 - 19/3*e^3 - 10/3*e^2 + 24*e + 2, -1/6*e^5 + 2/3*e^4 + 11/6*e^3 - 7*e^2 - 10/3*e - 26/3, -1/6*e^5 + 2/3*e^4 + 11/6*e^3 - 7*e^2 - 10/3*e - 26/3, 1/3*e^5 - 19/3*e^3 - 10/3*e^2 + 24*e + 2, 5/12*e^5 - 37/4*e^3 - 13/6*e^2 + 130/3*e + 58/3, -2/3*e^5 - 2/3*e^4 + 46/3*e^3 + 37/3*e^2 - 72*e + 52/3, 5/12*e^5 - 37/4*e^3 - 13/6*e^2 + 130/3*e + 58/3, -2/3*e^5 - 2/3*e^4 + 46/3*e^3 + 37/3*e^2 - 72*e + 52/3, 5/12*e^5 + 2/3*e^4 - 119/12*e^3 - 89/6*e^2 + 136/3*e + 22, -1/3*e^5 - e^4 + 25/3*e^3 + 46/3*e^2 - 38*e - 14, 5/12*e^5 + 2/3*e^4 - 119/12*e^3 - 89/6*e^2 + 136/3*e + 22, -1/3*e^5 - e^4 + 25/3*e^3 + 46/3*e^2 - 38*e - 14, -1/3*e^4 + 2/3*e^3 + 7/3*e^2 - 34/3*e + 88/3, -1/3*e^4 + 2/3*e^3 + 7/3*e^2 - 34/3*e + 88/3, -1/3*e^5 + 1/2*e^4 + 17/3*e^3 - 37/6*e^2 - 25/3*e + 116/3, 1/2*e^5 + 1/3*e^4 - 19/2*e^3 - 34/3*e^2 + 80/3*e + 28, 1/2*e^5 + 1/3*e^4 - 19/2*e^3 - 34/3*e^2 + 80/3*e + 28, -5/6*e^5 + e^4 + 103/6*e^3 - 26/3*e^2 - 220/3*e + 8/3, -5/6*e^5 + e^4 + 103/6*e^3 - 26/3*e^2 - 220/3*e + 8/3, 1/2*e^5 + 1/3*e^4 - 71/6*e^3 - 22/3*e^2 + 51*e - 2/3, 5/4*e^5 - 1/3*e^4 - 317/12*e^3 - 13/6*e^2 + 111*e - 94/3, 1/2*e^5 + 1/3*e^4 - 71/6*e^3 - 22/3*e^2 + 51*e - 2/3, 5/4*e^5 - 1/3*e^4 - 317/12*e^3 - 13/6*e^2 + 111*e - 94/3, -5/6*e^5 - 1/3*e^4 + 115/6*e^3 + 20/3*e^2 - 90*e + 80/3, -5/6*e^5 - 1/3*e^4 + 115/6*e^3 + 20/3*e^2 - 90*e + 80/3, 7/12*e^5 - 1/3*e^4 - 145/12*e^3 + 1/2*e^2 + 145/3*e - 8, 7/12*e^5 - 1/3*e^4 - 145/12*e^3 + 1/2*e^2 + 145/3*e - 8, -2/3*e^5 + 1/2*e^4 + 14*e^3 - 35/6*e^2 - 187/3*e + 80/3, -2/3*e^5 + 1/2*e^4 + 14*e^3 - 35/6*e^2 - 187/3*e + 80/3, -1/12*e^5 + e^4 + 19/12*e^3 - 85/6*e^2 - 11*e + 18, -1/3*e^5 - e^4 + 8*e^3 + 55/3*e^2 - 107/3*e - 62/3, -1/3*e^5 - e^4 + 8*e^3 + 55/3*e^2 - 107/3*e - 62/3, -1/12*e^5 + e^4 + 19/12*e^3 - 85/6*e^2 - 11*e + 18, -1/6*e^5 + 2/3*e^4 + 7/6*e^3 - 3*e^2 + 34/3*e - 36, -1/6*e^5 + 2/3*e^4 + 7/6*e^3 - 3*e^2 + 34/3*e - 36, -1/3*e^5 + 4/3*e^4 + 17/3*e^3 - 16*e^2 - 62/3*e + 14/3, -1/3*e^5 + 4/3*e^4 + 17/3*e^3 - 16*e^2 - 62/3*e + 14/3, 5/6*e^5 + 1/3*e^4 - 109/6*e^3 - 26/3*e^2 + 76*e - 74/3, 5/6*e^5 + 1/3*e^4 - 109/6*e^3 - 26/3*e^2 + 76*e - 74/3, 1/6*e^5 - 2/3*e^4 - 17/6*e^3 + 11*e^2 + 25/3*e - 46/3, 1/6*e^5 - 2/3*e^4 - 17/6*e^3 + 11*e^2 + 25/3*e - 46/3, -1/4*e^5 + 1/3*e^4 + 53/12*e^3 - 23/6*e^2 - 15*e + 88/3, 11/12*e^5 + 1/3*e^4 - 83/4*e^3 - 19/2*e^2 + 102*e - 14/3, 11/12*e^5 + 1/3*e^4 - 83/4*e^3 - 19/2*e^2 + 102*e - 14/3, -1/4*e^5 + 1/3*e^4 + 53/12*e^3 - 23/6*e^2 - 15*e + 88/3, -1/3*e^5 + 25/3*e^3 - 2/3*e^2 - 42*e + 6, -1/3*e^5 + 25/3*e^3 - 2/3*e^2 - 42*e + 6, e^5 - 62/3*e^3 - 6*e^2 + 233/3*e + 50/3, -2/3*e^4 + 4/3*e^3 + 14/3*e^2 - 32/3*e + 152/3, e^5 - 62/3*e^3 - 6*e^2 + 233/3*e + 50/3, -2/3*e^4 + 4/3*e^3 + 14/3*e^2 - 32/3*e + 152/3, -1/3*e^4 - 4/3*e^3 + 34/3*e^2 + 56/3*e - 134/3, 1/3*e^5 - 23/3*e^3 - 1/3*e^2 + 118/3*e - 14/3, -1/3*e^4 - 4/3*e^3 + 34/3*e^2 + 56/3*e - 134/3, 1/3*e^5 - 23/3*e^3 - 1/3*e^2 + 118/3*e - 14/3, 7/6*e^4 - 14/3*e^3 - 85/6*e^2 + 57*e + 68/3, 13/12*e^5 + 2/3*e^4 - 319/12*e^3 - 27/2*e^2 + 406/3*e - 14, 13/12*e^5 + 2/3*e^4 - 319/12*e^3 - 27/2*e^2 + 406/3*e - 14, 1/2*e^5 + 1/3*e^4 - 77/6*e^3 - 22/3*e^2 + 78*e - 14/3, 1/2*e^5 + 1/3*e^4 - 77/6*e^3 - 22/3*e^2 + 78*e - 14/3, 1/6*e^5 - 1/3*e^4 - 5/2*e^3 + 5/3*e^2 - 16/3*e + 28/3, 1/6*e^5 - 1/3*e^4 - 5/2*e^3 + 5/3*e^2 - 16/3*e + 28/3, 1/12*e^5 - e^4 + 29/12*e^3 + 61/6*e^2 - 43*e + 2, 1/12*e^5 - e^4 + 29/12*e^3 + 61/6*e^2 - 43*e + 2, -11/12*e^5 + 5/3*e^4 + 205/12*e^3 - 45/2*e^2 - 196/3*e + 130/3, -11/12*e^5 + 5/3*e^4 + 205/12*e^3 - 45/2*e^2 - 196/3*e + 130/3, -1/2*e^5 + 4/3*e^4 + 61/6*e^3 - 61/3*e^2 - 48*e + 142/3, -1/2*e^5 + 4/3*e^4 + 61/6*e^3 - 61/3*e^2 - 48*e + 142/3, -5/12*e^5 - e^4 + 115/12*e^3 + 91/6*e^2 - 104/3*e - 2/3, -5/12*e^5 - e^4 + 115/12*e^3 + 91/6*e^2 - 104/3*e - 2/3, 1/6*e^5 + 2/3*e^4 - 25/6*e^3 - 31/3*e^2 + 70/3*e + 14, 2/3*e^5 - 14*e^3 - 17/3*e^2 + 148/3*e + 70/3, 2/3*e^5 - 14*e^3 - 17/3*e^2 + 148/3*e + 70/3, 1/6*e^5 + 2/3*e^4 - 25/6*e^3 - 31/3*e^2 + 70/3*e + 14, -1/2*e^5 + 17/2*e^3 + 3*e^2 - 16*e - 6, -1/2*e^5 + 17/2*e^3 + 3*e^2 - 16*e - 6, 11/12*e^5 - 1/3*e^4 - 241/12*e^3 - 17/6*e^2 + 94*e + 26/3, 11/12*e^5 - 1/3*e^4 - 241/12*e^3 - 17/6*e^2 + 94*e + 26/3, 3/4*e^5 + e^4 - 81/4*e^3 - 37/2*e^2 + 114*e + 6, 3/4*e^5 + e^4 - 81/4*e^3 - 37/2*e^2 + 114*e + 6];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {49, 7, w^3 - w^2 - 6*w - 1}>] := 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;