/*
  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, 2, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17, 17, w^2 - 2}>;

primesArray := [
[7, 7, w^4 - 6*w^2 - 2*w + 4],
[9, 3, -w^4 + w^3 + 5*w^2 - w - 4],
[11, 11, -2*w^4 + w^3 + 10*w^2 + w - 3],
[11, 11, w^4 - 6*w^2 - 3*w + 3],
[17, 17, w^2 - 2],
[23, 23, -w^3 + w^2 + 3*w],
[27, 3, w^4 - w^3 - 4*w^2 + 2*w - 1],
[29, 29, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 3],
[32, 2, -2],
[47, 47, w^4 - 2*w^3 - 3*w^2 + 5*w],
[47, 47, w^3 - w^2 - 4*w - 1],
[53, 53, -w^4 + 7*w^2 - 3],
[53, 53, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 2],
[53, 53, 3*w^4 - 2*w^3 - 16*w^2 + 8],
[67, 67, -w^4 + 6*w^2 + 4*w - 3],
[73, 73, 2*w^4 - 12*w^2 - 4*w + 5],
[83, 83, w^4 - 5*w^2 - 3*w + 3],
[97, 97, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[103, 103, 2*w^3 - 3*w^2 - 7*w + 3],
[109, 109, -3*w^4 + 2*w^3 + 14*w^2 + w - 6],
[113, 113, -2*w^4 + 2*w^3 + 8*w^2 - w - 2],
[137, 137, -2*w^4 + 3*w^3 + 9*w^2 - 7*w - 4],
[139, 139, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 1],
[157, 157, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],
[157, 157, w^4 - 6*w^2 - 4*w + 6],
[163, 163, 2*w^4 - 2*w^3 - 8*w^2 + 2*w + 3],
[169, 13, w^4 - w^3 - 6*w^2 + w + 6],
[173, 173, -w^4 + w^3 + 4*w^2 - 2*w + 3],
[173, 173, -3*w^4 + w^3 + 17*w^2 + 3*w - 10],
[173, 173, -3*w^4 + 3*w^3 + 14*w^2 - 5*w - 7],
[179, 179, -2*w^4 + w^3 + 10*w^2 + 2*w - 7],
[179, 179, 2*w^4 - 12*w^2 - 3*w + 6],
[181, 181, 3*w^4 - 2*w^3 - 14*w^2 - 2*w + 4],
[181, 181, w^4 - 2*w^3 - 4*w^2 + 5*w + 3],
[191, 191, 3*w^4 - w^3 - 16*w^2 - 5*w + 7],
[191, 191, 4*w^4 - 3*w^3 - 20*w^2 + 2*w + 8],
[191, 191, -2*w^4 + 2*w^3 + 10*w^2 - 3*w - 3],
[193, 193, -2*w^4 + 2*w^3 + 9*w^2 - 3*w - 6],
[197, 197, -2*w^4 + w^3 + 9*w^2 + 2*w - 4],
[197, 197, -2*w^4 + 2*w^3 + 8*w^2 - w + 1],
[197, 197, -w^4 + w^3 + 4*w^2 - 4],
[199, 199, w^4 - 7*w^2 - 3*w + 5],
[199, 199, -2*w^4 + 2*w^3 + 9*w^2 - 4*w - 5],
[211, 211, w^2 - 5],
[211, 211, 2*w^4 - 12*w^2 - 5*w + 7],
[211, 211, w^4 - 2*w^3 - 4*w^2 + 7*w + 1],
[227, 227, -3*w^4 + 3*w^3 + 14*w^2 - 3*w - 6],
[227, 227, w^4 + w^3 - 6*w^2 - 7*w + 4],
[229, 229, w^3 - 6*w - 2],
[233, 233, w^4 - 5*w^2 - 4*w + 4],
[233, 233, w^4 - 2*w^3 - 3*w^2 + 5*w + 1],
[239, 239, -w^4 + w^3 + 5*w^2 - 5],
[239, 239, w^4 - w^3 - 6*w^2 + 3*w + 3],
[241, 241, -w^3 + w^2 + 6*w - 1],
[251, 251, -3*w^4 + 2*w^3 + 16*w^2 - 2*w - 7],
[269, 269, 3*w^4 - 3*w^3 - 13*w^2 + 4*w + 5],
[271, 271, w^4 - 2*w^3 - 2*w^2 + 4*w - 3],
[271, 271, w^2 - 2*w - 4],
[277, 277, -w^4 + 2*w^3 + 4*w^2 - 4*w - 2],
[277, 277, w^4 + w^3 - 7*w^2 - 5*w + 4],
[281, 281, 2*w^3 - 3*w^2 - 8*w + 4],
[281, 281, -3*w^4 + 2*w^3 + 15*w^2 - 4],
[293, 293, -3*w^4 + 2*w^3 + 14*w^2 - w - 3],
[307, 307, -w^4 - w^3 + 7*w^2 + 8*w - 5],
[311, 311, 3*w^4 - 3*w^3 - 13*w^2 + 4*w + 3],
[317, 317, -2*w^4 + 12*w^2 + 5*w - 5],
[331, 331, -3*w^4 + 3*w^3 + 13*w^2 - 2*w - 4],
[331, 331, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],
[331, 331, w^4 + w^3 - 8*w^2 - 5*w + 6],
[337, 337, -4*w^4 + 3*w^3 + 19*w^2 - 2*w - 7],
[347, 347, -w^4 + 2*w^3 + 4*w^2 - 4*w - 3],
[349, 349, -3*w^4 + 2*w^3 + 14*w^2 - 5],
[367, 367, 2*w^4 - 11*w^2 - 5*w + 7],
[367, 367, w^4 - 2*w^3 - 2*w^2 + 4*w - 4],
[373, 373, -2*w^4 + 3*w^3 + 9*w^2 - 7*w - 5],
[379, 379, 3*w^4 - w^3 - 16*w^2 - 5*w + 8],
[383, 383, w^4 - w^3 - 3*w^2 - 2],
[383, 383, 3*w^4 - w^3 - 16*w^2 - 5*w + 9],
[389, 389, -2*w^4 + 3*w^3 + 8*w^2 - 7*w - 3],
[397, 397, -w^4 + 3*w^3 + 3*w^2 - 7*w + 1],
[409, 409, 2*w^4 - 12*w^2 - 3*w + 7],
[421, 421, 4*w^4 - 3*w^3 - 20*w^2 + 4*w + 11],
[439, 439, 2*w^4 + w^3 - 12*w^2 - 8*w + 9],
[449, 449, -w^4 + w^3 + 5*w^2 - 4*w - 2],
[457, 457, w^4 + w^3 - 7*w^2 - 5*w + 5],
[461, 461, -2*w^4 + 2*w^3 + 10*w^2 - 3*w - 2],
[461, 461, -w^4 - w^3 + 7*w^2 + 5*w - 6],
[463, 463, w^2 + w - 4],
[463, 463, 3*w^4 - 3*w^3 - 14*w^2 + 3*w + 5],
[467, 467, w^4 - w^3 - 3*w^2 - 3],
[467, 467, -4*w^4 + 3*w^3 + 19*w^2 - 7],
[479, 479, 3*w^4 - 2*w^3 - 14*w^2 - 3*w + 6],
[487, 487, w^4 - 2*w^3 - 3*w^2 + 6*w + 1],
[487, 487, -2*w^4 + w^3 + 10*w^2 + 2*w],
[491, 491, -w^4 + w^3 + 5*w^2 - 6],
[491, 491, w - 4],
[499, 499, 2*w^4 - w^3 - 11*w^2 + 2*w + 4],
[503, 503, 5*w^4 - 3*w^3 - 24*w^2 - w + 7],
[509, 509, -2*w^4 + 3*w^3 + 6*w^2 - 6*w + 2],
[521, 521, 2*w^4 - w^3 - 11*w^2 - 3*w + 6],
[557, 557, -2*w^4 + 2*w^3 + 7*w^2 + 4],
[571, 571, w^2 - 2*w - 5],
[577, 577, 5*w^4 - 4*w^3 - 24*w^2 + 3*w + 12],
[577, 577, w^4 + w^3 - 8*w^2 - 6*w + 7],
[587, 587, 3*w^4 - 3*w^3 - 13*w^2 + 2*w + 3],
[587, 587, 3*w^4 - 2*w^3 - 16*w^2 + 3*w + 5],
[587, 587, w^4 - 7*w^2 + 7],
[599, 599, 2*w^4 - 13*w^2 - 3*w + 5],
[617, 617, 2*w^4 - w^3 - 11*w^2 + 2*w + 3],
[631, 631, 2*w^2 - w - 4],
[641, 641, w^4 - 5*w^2 - w - 1],
[653, 653, -4*w^4 + 3*w^3 + 19*w^2 - 4],
[653, 653, 4*w^4 - w^3 - 23*w^2 - 4*w + 13],
[659, 659, -2*w^4 + 2*w^3 + 10*w^2 - 5*w - 6],
[661, 661, 3*w^3 - 3*w^2 - 13*w + 2],
[683, 683, 5*w^4 - 3*w^3 - 25*w^2 - 3*w + 13],
[683, 683, 2*w^4 - w^3 - 10*w^2 - 4*w + 8],
[683, 683, 3*w^4 - 4*w^3 - 12*w^2 + 6*w + 2],
[691, 691, 2*w^4 - 11*w^2 - 2*w + 4],
[691, 691, -w^4 + w^3 + 5*w^2 - 4*w - 3],
[691, 691, -5*w^4 + 3*w^3 + 25*w^2 + 2*w - 7],
[709, 709, -w^3 + 2*w^2 + w - 4],
[719, 719, -w^4 + 2*w^3 + 5*w^2 - 6*w - 3],
[727, 727, -3*w^4 + 4*w^3 + 12*w^2 - 6*w - 6],
[727, 727, w^4 + w^3 - 7*w^2 - 7*w],
[727, 727, -4*w^4 + 2*w^3 + 22*w^2 + 3*w - 14],
[733, 733, -3*w^4 + 2*w^3 + 15*w^2 - 2*w - 5],
[743, 743, -4*w^4 + 3*w^3 + 21*w^2 - 2*w - 10],
[743, 743, 4*w^4 - 3*w^3 - 18*w^2 - w + 7],
[751, 751, 2*w^4 - 2*w^3 - 7*w^2 - 3],
[751, 751, -3*w^4 + 4*w^3 + 14*w^2 - 9*w - 6],
[751, 751, 4*w^4 - 3*w^3 - 20*w^2 + 2*w + 7],
[757, 757, 4*w^4 - w^3 - 22*w^2 - 8*w + 10],
[773, 773, -4*w^4 + 2*w^3 + 22*w^2 - w - 11],
[787, 787, w^4 - w^3 - 4*w^2 - w - 2],
[797, 797, 3*w^4 - w^3 - 15*w^2 - 2*w + 3],
[809, 809, -2*w^4 + 2*w^3 + 11*w^2 - 4*w - 6],
[811, 811, -4*w^4 + 3*w^3 + 19*w^2 - 6],
[821, 821, 5*w^4 - 5*w^3 - 22*w^2 + 6*w + 5],
[823, 823, 4*w^4 - 4*w^3 - 18*w^2 + 3*w + 8],
[823, 823, 2*w^2 - w - 7],
[829, 829, -2*w^4 + 13*w^2 + 3*w - 12],
[853, 853, 2*w^3 - 2*w^2 - 8*w - 1],
[853, 853, w^3 - 2*w^2 - 4*w - 2],
[857, 857, -4*w^4 + w^3 + 21*w^2 + 5*w - 10],
[859, 859, 4*w^4 - w^3 - 21*w^2 - 5*w + 7],
[859, 859, w^4 - 3*w^3 - 2*w^2 + 7*w + 1],
[863, 863, -5*w^4 + 3*w^3 + 26*w^2 - w - 12],
[877, 877, 2*w^4 - 3*w^3 - 8*w^2 + 3*w + 2],
[887, 887, -3*w^4 + w^3 + 15*w^2 + 5*w - 7],
[907, 907, -w^4 - 2*w^3 + 9*w^2 + 9*w - 10],
[929, 929, -2*w^4 + 3*w^3 + 10*w^2 - 7*w - 8],
[929, 929, w^4 + w^3 - 5*w^2 - 8*w],
[929, 929, 3*w^4 - w^3 - 15*w^2 - 3*w + 1],
[941, 941, -4*w^4 + 3*w^3 + 18*w^2 + 2*w - 5],
[941, 941, -w^4 - w^3 + 8*w^2 + 6*w - 6],
[947, 947, -4*w^4 + w^3 + 22*w^2 + 5*w - 11],
[967, 967, w^4 + 2*w^3 - 8*w^2 - 9*w + 5],
[971, 971, 2*w^4 - w^3 - 11*w^2 - 4*w + 5],
[971, 971, 4*w^4 - 2*w^3 - 21*w^2 - 3*w + 7],
[977, 977, -3*w^4 + 2*w^3 + 16*w^2 - 4*w - 6],
[991, 991, 2*w^3 - 4*w^2 - 7*w + 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - x^5 - 18*x^4 + 18*x^3 + 61*x^2 - 33*x - 64;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/2*e^5 + 3/2*e^4 + 15/2*e^3 - 49/2*e^2 - 5*e + 38, -1/3*e^5 + e^4 + 5*e^3 - 16*e^2 - 10/3*e + 68/3, 1/3*e^5 - 1/2*e^4 - 11/2*e^3 + 17/2*e^2 + 65/6*e - 32/3, 1, -1/2*e^5 + e^4 + 8*e^3 - 16*e^2 - 25/2*e + 20, -e^5 + 2*e^4 + 15*e^3 - 34*e^2 - 14*e + 52, -2/3*e^5 + 1/2*e^4 + 21/2*e^3 - 17/2*e^2 - 109/6*e + 22/3, -2/3*e^5 + 11*e^3 - e^2 - 74/3*e - 17/3, 1/3*e^5 - e^4 - 5*e^3 + 15*e^2 + 10/3*e - 56/3, -2/3*e^5 + e^4 + 10*e^3 - 17*e^2 - 26/3*e + 64/3, e^5 - 2*e^4 - 15*e^3 + 34*e^2 + 14*e - 50, -1/3*e^5 + 2*e^4 + 5*e^3 - 32*e^2 - 4/3*e + 170/3, 2/3*e^5 - 11*e^3 + 2*e^2 + 77/3*e + 2/3, -e - 4, -5/3*e^5 + 2*e^4 + 26*e^3 - 34*e^2 - 125/3*e + 118/3, 1/3*e^5 - 5*e^3 + 22/3*e + 28/3, 3/2*e^5 - 4*e^4 - 23*e^3 + 65*e^2 + 49/2*e - 90, -3/2*e^5 + e^4 + 24*e^3 - 16*e^2 - 87/2*e + 4, 2/3*e^5 - 3*e^4 - 10*e^3 + 48*e^2 + 2/3*e - 214/3, e^5 - e^4 - 16*e^3 + 17*e^2 + 25*e - 14, -2/3*e^5 + 3*e^4 + 9*e^3 - 49*e^2 + 19/3*e + 238/3, e^5 - e^4 - 16*e^3 + 17*e^2 + 33*e - 12, 5/6*e^5 - 13*e^3 + e^2 + 131/6*e + 46/3, -e^5 + 2*e^4 + 15*e^3 - 34*e^2 - 12*e + 46, -4/3*e^5 + 2*e^4 + 20*e^3 - 34*e^2 - 70/3*e + 116/3, 1/6*e^5 - 3*e^4 - 2*e^3 + 48*e^2 - 83/6*e - 262/3, 1/3*e^5 - 2*e^4 - 5*e^3 + 32*e^2 + 16/3*e - 182/3, -7/6*e^5 + 3/2*e^4 + 37/2*e^3 - 53/2*e^2 - 86/3*e + 118/3, 7/6*e^5 - 1/2*e^4 - 37/2*e^3 + 15/2*e^2 + 92/3*e + 38/3, 13/6*e^5 - 5/2*e^4 - 69/2*e^3 + 87/2*e^2 + 173/3*e - 136/3, -1/3*e^5 + 2*e^4 + 5*e^3 - 32*e^2 - 16/3*e + 164/3, -7/6*e^5 + 7/2*e^4 + 35/2*e^3 - 117/2*e^2 - 41/3*e + 286/3, 2*e^5 - 4*e^4 - 31*e^3 + 67*e^2 + 35*e - 106, -3/2*e^5 + 3/2*e^4 + 49/2*e^3 - 53/2*e^2 - 47*e + 28, -1/6*e^5 + 3*e^4 + 2*e^3 - 48*e^2 + 83/6*e + 268/3, -4/3*e^5 + 3*e^4 + 21*e^3 - 49*e^2 - 103/3*e + 200/3, 4/3*e^5 - e^4 - 21*e^3 + 17*e^2 + 112/3*e - 26/3, -10/3*e^5 + 11/2*e^4 + 103/2*e^3 - 183/2*e^2 - 395/6*e + 374/3, -1/6*e^5 - 3/2*e^4 + 5/2*e^3 + 45/2*e^2 - 23/3*e - 146/3, 8/3*e^5 - 6*e^4 - 40*e^3 + 100*e^2 + 104/3*e - 454/3, 5/6*e^5 - 2*e^4 - 13*e^3 + 31*e^2 + 107/6*e - 116/3, 1/3*e^5 - e^4 - 5*e^3 + 15*e^2 + 22/3*e - 56/3, 2*e^5 - 5*e^4 - 30*e^3 + 83*e^2 + 27*e - 132, 3/2*e^5 - 7/2*e^4 - 47/2*e^3 + 113/2*e^2 + 28*e - 72, -1/3*e^5 + 6*e^3 - 2*e^2 - 49/3*e + 44/3, -2/3*e^5 + 11*e^3 + 2*e^2 - 71/3*e - 92/3, -1/3*e^5 + 5*e^3 - 28/3*e - 4/3, -3*e^4 + e^3 + 47*e^2 - 25*e - 82, -2*e^5 + 4*e^4 + 31*e^3 - 68*e^2 - 39*e + 98, -4/3*e^5 + 3*e^4 + 21*e^3 - 49*e^2 - 73/3*e + 206/3, -7/3*e^5 + 5*e^4 + 36*e^3 - 82*e^2 - 127/3*e + 320/3, 11/3*e^5 - 5*e^4 - 58*e^3 + 83*e^2 + 263/3*e - 280/3, e^5 - 16*e^3 + 4*e^2 + 31*e + 2, -7/6*e^5 + 7/2*e^4 + 35/2*e^3 - 117/2*e^2 - 29/3*e + 280/3, 7/3*e^5 - 7*e^4 - 35*e^3 + 115*e^2 + 64/3*e - 542/3, -2*e^5 + 3*e^4 + 32*e^3 - 49*e^2 - 52*e + 48, -2/3*e^5 - e^4 + 11*e^3 + 17*e^2 - 86/3*e - 152/3, 3*e^5 - 7/2*e^4 - 95/2*e^3 + 119/2*e^2 + 151/2*e - 62, 8/3*e^5 - 5*e^4 - 40*e^3 + 85*e^2 + 116/3*e - 358/3, -8/3*e^5 + 4*e^4 + 42*e^3 - 68*e^2 - 203/3*e + 286/3, 2/3*e^5 + e^4 - 12*e^3 - 14*e^2 + 110/3*e + 62/3, 19/6*e^5 - 7*e^4 - 48*e^3 + 118*e^2 + 283/6*e - 538/3, 2/3*e^5 - 7/2*e^4 - 17/2*e^3 + 115/2*e^2 - 71/6*e - 304/3, 3/2*e^5 - 4*e^4 - 23*e^3 + 65*e^2 + 33/2*e - 92, 1/3*e^5 - 4*e^3 + 2*e^2 - 23/3*e + 10/3, 4/3*e^5 - 21*e^3 + 2*e^2 + 115/3*e + 76/3, 3/2*e^4 - 3/2*e^3 - 51/2*e^2 + 33/2*e + 56, 2/3*e^5 - 10*e^3 + 2*e^2 + 32/3*e + 20/3, 1/3*e^5 - 5/2*e^4 - 9/2*e^3 + 81/2*e^2 - 13/6*e - 206/3, -2*e^5 + e^4 + 33*e^3 - 19*e^2 - 69*e + 20, 5/2*e^5 - 6*e^4 - 39*e^3 + 95*e^2 + 103/2*e - 118, -4*e^5 + 6*e^4 + 63*e^3 - 101*e^2 - 97*e + 128, 2/3*e^5 - 12*e^3 + 104/3*e + 8/3, -e^5 + 4*e^4 + 14*e^3 - 68*e^2 + 3*e + 118, -e^5 + e^4 + 15*e^3 - 15*e^2 - 14*e + 4, 5/3*e^5 - 6*e^4 - 25*e^3 + 98*e^2 + 44/3*e - 448/3, -1/2*e^5 - e^4 + 8*e^3 + 16*e^2 - 45/2*e - 36, -1/6*e^5 - e^4 + 2*e^3 + 16*e^2 + 35/6*e - 122/3, -5/3*e^5 + 6*e^4 + 25*e^3 - 98*e^2 - 26/3*e + 490/3, e^5 - 3*e^4 - 14*e^3 + 51*e^2 + e - 94, -e^5 + e^4 + 16*e^3 - 17*e^2 - 25*e + 30, -2/3*e^5 + 4*e^4 + 9*e^3 - 66*e^2 + 19/3*e + 400/3, -e^5 + 5*e^4 + 13*e^3 - 81*e^2 + 12*e + 130, e^5 - e^4 - 16*e^3 + 14*e^2 + 29*e + 10, -1/3*e^5 - e^4 + 6*e^3 + 15*e^2 - 79/3*e - 142/3, 5/3*e^5 - 27*e^3 + 152/3*e + 122/3, e^4 - e^3 - 15*e^2 + 17*e + 24, 11/3*e^5 - 6*e^4 - 57*e^3 + 98*e^2 + 218/3*e - 376/3, 8/3*e^5 - 3*e^4 - 42*e^3 + 51*e^2 + 182/3*e - 172/3, 5/3*e^5 - 2*e^4 - 26*e^3 + 34*e^2 + 131/3*e - 100/3, 3*e^5 - 5*e^4 - 46*e^3 + 83*e^2 + 60*e - 104, -8/3*e^5 + 4*e^4 + 41*e^3 - 67*e^2 - 161/3*e + 232/3, 3/2*e^5 - 2*e^4 - 25*e^3 + 33*e^2 + 97/2*e - 52, -10/3*e^5 + 10*e^4 + 50*e^3 - 166*e^2 - 100/3*e + 788/3, -8/3*e^5 + 8*e^4 + 39*e^3 - 132*e^2 - 59/3*e + 628/3, 1/3*e^5 - 1/2*e^4 - 9/2*e^3 + 17/2*e^2 - 49/6*e - 8/3, e^5 - 15*e^3 + 18*e + 32, -2/3*e^5 + 5/2*e^4 + 19/2*e^3 - 81/2*e^2 + 65/6*e + 190/3, 11/3*e^5 - 21/2*e^4 - 109/2*e^3 + 349/2*e^2 + 187/6*e - 838/3, e^5 - e^4 - 14*e^3 + 18*e^2 + 5*e - 18, -1/2*e^4 + 1/2*e^3 + 13/2*e^2 - 11/2*e - 8, -11/3*e^5 + 8*e^4 + 58*e^3 - 130*e^2 - 239/3*e + 502/3, 9/2*e^4 - 3/2*e^3 - 141/2*e^2 + 73/2*e + 134, 2*e^5 - 7*e^4 - 30*e^3 + 114*e^2 + 26*e - 180, -3/2*e^5 + 3/2*e^4 + 45/2*e^3 - 49/2*e^2 - 29*e + 16, -3*e^5 + 7*e^4 + 47*e^3 - 117*e^2 - 57*e + 180, -11/3*e^5 + 4*e^4 + 58*e^3 - 70*e^2 - 266/3*e + 256/3, 9/2*e^5 - 23/2*e^4 - 135/2*e^3 + 377/2*e^2 + 59*e - 290, -e^5 + 3/2*e^4 + 31/2*e^3 - 55/2*e^2 - 51/2*e + 44, -2/3*e^5 + 4*e^4 + 9*e^3 - 62*e^2 + 31/3*e + 262/3, -2*e^5 + 7*e^4 + 30*e^3 - 115*e^2 - 14*e + 190, 10/3*e^5 - 7*e^4 - 51*e^3 + 117*e^2 + 172/3*e - 566/3, -4/3*e^5 + 4*e^4 + 20*e^3 - 68*e^2 - 40/3*e + 356/3, -7/3*e^5 + 8*e^4 + 35*e^3 - 133*e^2 - 70/3*e + 674/3, -11/3*e^5 + 5*e^4 + 57*e^3 - 81*e^2 - 248/3*e + 244/3, -5/3*e^5 + e^4 + 26*e^3 - 17*e^2 - 131/3*e - 20/3, 4*e^5 - 10*e^4 - 62*e^3 + 165*e^2 + 68*e - 228, e^5 + 3*e^4 - 18*e^3 - 47*e^2 + 61*e + 92, 7/2*e^5 - 9/2*e^4 - 109/2*e^3 + 151/2*e^2 + 87*e - 88, -4*e^5 + 4*e^4 + 63*e^3 - 70*e^2 - 105*e + 84, -7/3*e^5 + 6*e^4 + 36*e^3 - 98*e^2 - 157/3*e + 434/3, -5/3*e^5 + 2*e^4 + 28*e^3 - 34*e^2 - 209/3*e + 136/3, 5/3*e^5 - 27*e^3 + 2*e^2 + 170/3*e + 32/3, -e^5 + 3*e^4 + 17*e^3 - 51*e^2 - 40*e + 88, -3/2*e^5 + 1/2*e^4 + 51/2*e^3 - 27/2*e^2 - 64*e + 12, 5/3*e^5 - e^4 - 28*e^3 + 19*e^2 + 200/3*e - 70/3, -13/3*e^5 + 13/2*e^4 + 135/2*e^3 - 213/2*e^2 - 533/6*e + 404/3, -1/3*e^5 + 2*e^4 + 4*e^3 - 36*e^2 + 23/3*e + 248/3, -3*e^5 + 4*e^4 + 47*e^3 - 70*e^2 - 72*e + 104, -4/3*e^5 + 5*e^4 + 21*e^3 - 79*e^2 - 55/3*e + 320/3, -10/3*e^5 + 8*e^4 + 51*e^3 - 133*e^2 - 127/3*e + 608/3, e^5 - 5*e^4 - 15*e^3 + 82*e^2 + 2*e - 154, 11/3*e^5 - 5*e^4 - 58*e^3 + 85*e^2 + 269/3*e - 310/3, -e^5 + 16*e^3 + 4*e^2 - 31*e - 44, 10/3*e^5 - 6*e^4 - 51*e^3 + 102*e^2 + 187/3*e - 422/3, 11/3*e^5 - 7*e^4 - 56*e^3 + 117*e^2 + 167/3*e - 466/3, -e^5 - 3*e^4 + 18*e^3 + 43*e^2 - 65*e - 60, -e^5 + 3*e^4 + 15*e^3 - 47*e^2 - 11*e + 38, -2/3*e^5 - 2*e^4 + 12*e^3 + 32*e^2 - 110/3*e - 272/3, 11/6*e^5 - 1/2*e^4 - 57/2*e^3 + 23/2*e^2 + 172/3*e + 4/3, -10/3*e^5 + 5*e^4 + 51*e^3 - 82*e^2 - 211/3*e + 266/3, 5*e^5 - 4*e^4 - 81*e^3 + 70*e^2 + 152*e - 66, -e^5 + e^4 + 18*e^3 - 15*e^2 - 51*e - 2, 2/3*e^5 - 6*e^4 - 10*e^3 + 96*e^2 - 13/3*e - 562/3, 3/2*e^5 - 11/2*e^4 - 45/2*e^3 + 185/2*e^2 + 9*e - 176, -1/3*e^5 - e^4 + 5*e^3 + 17*e^2 - 10/3*e - 148/3, -23/6*e^5 + 8*e^4 + 59*e^3 - 129*e^2 - 389/6*e + 500/3, -2/3*e^5 + 8*e^3 - 2*e^2 + 28/3*e - 38/3, 14/3*e^5 - 5*e^4 - 74*e^3 + 87*e^2 + 368/3*e - 352/3, e^5 - 4*e^4 - 14*e^3 + 64*e^2 - 3*e - 100, 3*e^5 - 10*e^4 - 43*e^3 + 164*e^2 + 12*e - 270, -8/3*e^5 + 4*e^4 + 42*e^3 - 68*e^2 - 206/3*e + 334/3, 4/3*e^5 - 7*e^4 - 18*e^3 + 113*e^2 - 44/3*e - 602/3, 7*e^5 - 15*e^4 - 108*e^3 + 251*e^2 + 121*e - 370, 2/3*e^5 - e^4 - 10*e^3 + 25*e^2 + 32/3*e - 238/3, -10/3*e^5 + 5*e^4 + 52*e^3 - 80*e^2 - 238/3*e + 236/3, -17/6*e^5 + 11/2*e^4 + 81/2*e^3 - 189/2*e^2 - 61/3*e + 452/3, -4/3*e^5 + 3/2*e^4 + 39/2*e^3 - 55/2*e^2 - 119/6*e + 200/3, 11/3*e^5 - 3*e^4 - 58*e^3 + 54*e^2 + 287/3*e - 124/3, 11/3*e^5 - 6*e^4 - 58*e^3 + 102*e^2 + 257/3*e - 418/3, 2*e^5 + 1/2*e^4 - 69/2*e^3 - 5/2*e^2 + 183/2*e + 20];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17, 17, w^2 - 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;