/*
  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 | {9, 3, -w^4 + w^3 + 5*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^7 - 13*x^5 - 3*x^4 + 35*x^3 - 4*x^2 - 13*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -2/3*e^5 + 1/3*e^4 + 20/3*e^3 + 4/3*e^2 - 26/3*e - 7/3, 1, -4/3*e^6 + 1/3*e^5 + 16*e^4 + 2*e^3 - 110/3*e^2 + 5*e + 22/3, -2/3*e^6 + 22/3*e^4 + 11/3*e^3 - 12*e^2 - 8/3*e + 10/3, -1/3*e^6 + 1/3*e^5 + 4*e^4 - 2*e^3 - 29/3*e^2 + 3*e - 2/3, 1/3*e^6 + 1/3*e^5 - 16/3*e^4 - 8/3*e^3 + 52/3*e^2 - 19/3*e - 6, 2/3*e^6 - 31/3*e^4 + 4/3*e^3 + 36*e^2 - 58/3*e - 31/3, 1/3*e^6 - 14/3*e^4 - 4/3*e^3 + 14*e^2 + 10/3*e - 14/3, -1/3*e^6 + e^5 + 8/3*e^4 - 23/3*e^3 - 3*e^2 + 32/3*e + 8/3, -5/3*e^6 - 2/3*e^5 + 68/3*e^4 + 28/3*e^3 - 182/3*e^2 + 32/3*e + 14, -4/3*e^6 - 4/3*e^5 + 52/3*e^4 + 53/3*e^3 - 112/3*e^2 - 29/3*e + 4, e^6 - 12*e^4 - 4*e^3 + 25*e^2 - 4, -2*e^6 - e^5 + 24*e^4 + 19*e^3 - 48*e^2 - 13*e + 8, -1/3*e^6 - e^5 + 17/3*e^4 + 25/3*e^3 - 15*e^2 + 11/3*e - 1/3, e^6 + 1/3*e^5 - 38/3*e^4 - 22/3*e^3 + 91/3*e^2 + 7/3*e - 19/3, 5/3*e^6 - 61/3*e^4 - 23/3*e^3 + 47*e^2 + 11/3*e - 25/3, -1/3*e^6 + 5/3*e^5 + 10/3*e^4 - 52/3*e^3 - 31/3*e^2 + 112/3*e + 2, -e^6 + 5/3*e^5 + 35/3*e^4 - 38/3*e^3 - 103/3*e^2 + 59/3*e + 43/3, 2*e^6 + 1/3*e^5 - 80/3*e^4 - 22/3*e^3 + 214/3*e^2 - 47/3*e - 46/3, -4/3*e^6 + 47/3*e^4 + 22/3*e^3 - 32*e^2 - 34/3*e + 23/3, e^6 - e^5 - 10*e^4 + 4*e^3 + 15*e^2 - 5*e + 3, -1/3*e^6 - 1/3*e^5 + 19/3*e^4 + 11/3*e^3 - 85/3*e^2 - 17/3*e + 17, -5/3*e^5 + 7/3*e^4 + 44/3*e^3 - 26/3*e^2 - 35/3*e + 5/3, 1/3*e^6 + e^5 - 5/3*e^4 - 46/3*e^3 - 17*e^2 + 79/3*e + 49/3, 1/3*e^6 - 5/3*e^5 - 1/3*e^4 + 37/3*e^3 - 41/3*e^2 - 52/3*e + 13, 1/3*e^6 - e^5 - 14/3*e^4 + 38/3*e^3 + 17*e^2 - 95/3*e - 8/3, 2*e^6 + 2/3*e^5 - 76/3*e^4 - 35/3*e^3 + 170/3*e^2 - 46/3*e - 14/3, 1/3*e^6 - e^5 - 8/3*e^4 + 20/3*e^3 + 3*e^2 - 17/3*e + 28/3, e^5 + 2*e^4 - 14*e^3 - 24*e^2 + 30*e + 16, 4/3*e^6 - 7/3*e^5 - 13*e^4 + 16*e^3 + 65/3*e^2 - 27*e + 11/3, -e^6 - 2/3*e^5 + 43/3*e^4 + 26/3*e^3 - 131/3*e^2 - 8/3*e + 65/3, -4/3*e^6 + 41/3*e^4 + 25/3*e^3 - 14*e^2 - 13/3*e - 7/3, e^6 + 2/3*e^5 - 31/3*e^4 - 38/3*e^3 + 29/3*e^2 + 38/3*e + 1/3, -4/3*e^6 - 1/3*e^5 + 49/3*e^4 + 29/3*e^3 - 112/3*e^2 - 32/3*e + 19, -5/3*e^6 + 73/3*e^4 + 2/3*e^3 - 81*e^2 + 79/3*e + 79/3, 4/3*e^6 + 2/3*e^5 - 23*e^4 + 254/3*e^2 - 52*e - 76/3, -1/3*e^6 - e^5 + 11/3*e^4 + 34/3*e^3 + 3*e^2 - 37/3*e - 43/3, -1/3*e^6 - 2/3*e^5 + 4*e^4 + 6*e^3 - 11/3*e^2 + 8*e + 4/3, 2/3*e^6 + 3*e^5 - 31/3*e^4 - 92/3*e^3 + 20*e^2 + 77/3*e + 8/3, e^6 - 1/3*e^5 - 25/3*e^4 - 14/3*e^3 - 7/3*e^2 + 23/3*e + 37/3, -14/3*e^6 - 1/3*e^5 + 58*e^4 + 20*e^3 - 406/3*e^2 + 13*e + 74/3, -10/3*e^6 - e^5 + 128/3*e^4 + 64/3*e^3 - 104*e^2 + 5/3*e + 89/3, -4/3*e^6 - 1/3*e^5 + 70/3*e^4 - 10/3*e^3 - 268/3*e^2 + 157/3*e + 26, 8/3*e^6 - 10/3*e^5 - 98/3*e^4 + 80/3*e^3 + 278/3*e^2 - 206/3*e - 30, 2*e^6 - 7/3*e^5 - 61/3*e^4 + 31/3*e^3 + 122/3*e^2 - 31/3*e - 101/3, 2/3*e^6 - 7/3*e^5 - 35/3*e^4 + 80/3*e^3 + 182/3*e^2 - 167/3*e - 25, 10/3*e^6 - 2/3*e^5 - 121/3*e^4 - 14/3*e^3 + 268/3*e^2 - 79/3*e - 17, 1/3*e^6 + 5/3*e^5 - 7*e^4 - 12*e^3 + 65/3*e^2 - 13*e - 16/3, -2*e^6 - 2/3*e^5 + 67/3*e^4 + 50/3*e^3 - 92/3*e^2 - 35/3*e - 61/3, 4*e^6 - e^5 - 46*e^4 - 12*e^3 + 100*e^2 + 21*e - 30, e^6 - 5/3*e^5 - 41/3*e^4 + 50/3*e^3 + 139/3*e^2 - 113/3*e + 5/3, e^6 - e^5 - 9*e^4 + 3*e^3 + 7*e^2 - e + 5, -1/3*e^6 - 1/3*e^5 + 19/3*e^4 + 2/3*e^3 - 73/3*e^2 + 67/3*e + 2, -1/3*e^6 - 1/3*e^5 + 1/3*e^4 + 38/3*e^3 + 53/3*e^2 - 119/3*e + 5, 11/3*e^5 + 2/3*e^4 - 128/3*e^3 - 70/3*e^2 + 221/3*e + 22/3, 11/3*e^6 + 13/3*e^5 - 50*e^4 - 51*e^3 + 355/3*e^2 + 16*e - 62/3, -2*e^6 - 2*e^5 + 21*e^4 + 34*e^3 - 16*e^2 - 52*e - 6, -4*e^6 + 2*e^5 + 46*e^4 - 4*e^3 - 98*e^2 + 40*e + 21, 4*e^6 - 2/3*e^5 - 143/3*e^4 - 28/3*e^3 + 322/3*e^2 - 50/3*e - 91/3, 10/3*e^6 - e^5 - 110/3*e^4 - 22/3*e^3 + 65*e^2 - 23/3*e - 38/3, -5/3*e^6 - 1/3*e^5 + 21*e^4 + 12*e^3 - 169/3*e^2 - 15*e + 83/3, 1/3*e^6 - 17/3*e^4 + 2/3*e^3 + 21*e^2 - 26/3*e + 19/3, 4*e^6 + 3*e^5 - 56*e^4 - 36*e^3 + 150*e^2 - 9*e - 30, 7/3*e^6 - 86/3*e^4 - 37/3*e^3 + 69*e^2 + 73/3*e - 80/3, 2/3*e^6 - 5/3*e^5 - 5*e^4 + 8*e^3 + 28/3*e^2 + 12*e - 47/3, 3*e^6 + 1/3*e^5 - 119/3*e^4 - 28/3*e^3 + 316/3*e^2 - 59/3*e - 97/3, -2/3*e^6 + 2*e^5 + 13/3*e^4 - 52/3*e^3 + 4*e^2 + 112/3*e - 17/3, -4/3*e^6 + 13/3*e^5 + 14*e^4 - 37*e^3 - 140/3*e^2 + 50*e + 76/3, 2/3*e^6 - 1/3*e^5 - 11/3*e^4 - 16/3*e^3 - 52/3*e^2 + 49/3*e + 21, -13/3*e^5 + 2/3*e^4 + 130/3*e^3 + 65/3*e^2 - 157/3*e - 74/3, -4/3*e^6 + 3*e^5 + 29/3*e^4 - 50/3*e^3 - 4*e^2 + 20/3*e + 47/3, 7/3*e^6 - 2/3*e^5 - 91/3*e^4 - 2/3*e^3 + 259/3*e^2 - 16/3*e - 35, 1/3*e^6 - 5/3*e^5 - 7/3*e^4 + 46/3*e^3 + 4/3*e^2 - 91/3*e + 17, 7/3*e^6 - 1/3*e^5 - 28*e^4 - 5*e^3 + 179/3*e^2 - 15*e - 28/3, 5/3*e^6 - 1/3*e^5 - 50/3*e^4 - 28/3*e^3 + 65/3*e^2 + 49/3*e - 9, -4/3*e^6 + 2/3*e^5 + 64/3*e^4 - 34/3*e^3 - 232/3*e^2 + 166/3*e + 30, 2/3*e^6 + 10/3*e^5 - 14*e^4 - 32*e^3 + 154/3*e^2 + 39*e - 44/3, -5*e^6 + 60*e^4 + 20*e^3 - 129*e^2 + 3*e + 20, 14/3*e^6 - e^5 - 166/3*e^4 - 26/3*e^3 + 120*e^2 - 91/3*e - 64/3, -10/3*e^6 + 1/3*e^5 + 44*e^4 + 2*e^3 - 350/3*e^2 + 46*e + 70/3, 2/3*e^6 - 5/3*e^5 - 9*e^4 + 16*e^3 + 112/3*e^2 - 29*e - 71/3, -4/3*e^6 + 50/3*e^4 + 4/3*e^3 - 36*e^2 + 80/3*e - 43/3, -2/3*e^6 - 3*e^5 + 34/3*e^4 + 92/3*e^3 - 28*e^2 - 113/3*e + 4/3, -3*e^6 - 8/3*e^5 + 130/3*e^4 + 77/3*e^3 - 353/3*e^2 + 115/3*e + 56/3, -e^6 - 3*e^5 + 16*e^4 + 33*e^3 - 47*e^2 - 43*e + 26, 1/3*e^6 - 7/3*e^5 - 2*e^4 + 18*e^3 + 23/3*e^2 - 3*e - 4/3, 1/3*e^6 - 5/3*e^4 - 16/3*e^3 - 9*e^2 + 61/3*e + 61/3, 3*e^6 - e^5 - 33*e^4 - 6*e^3 + 59*e^2 + 5*e + 9, -2*e^6 + 11/3*e^5 + 83/3*e^4 - 110/3*e^3 - 310/3*e^2 + 239/3*e + 118/3, -4/3*e^6 - 8/3*e^5 + 19*e^4 + 26*e^3 - 122/3*e^2 + 13/3, 2/3*e^6 + 4/3*e^5 - 6*e^4 - 22*e^3 - 20/3*e^2 + 44*e + 31/3, -1/3*e^5 + 2/3*e^4 + 10/3*e^3 - 7/3*e^2 - 19/3*e - 14/3, -1/3*e^6 - 8/3*e^5 + 4*e^4 + 30*e^3 + 25/3*e^2 - 48*e - 53/3, 1/3*e^6 + 7/3*e^5 + 2/3*e^4 - 98/3*e^3 - 137/3*e^2 + 179/3*e + 44, -17/3*e^6 - 1/3*e^5 + 71*e^4 + 22*e^3 - 511/3*e^2 + 24*e + 113/3, 10/3*e^6 - 5/3*e^5 - 103/3*e^4 - 14/3*e^3 + 160/3*e^2 + 5/3*e - 27, e^6 + 3*e^5 - 13*e^4 - 34*e^3 + 17*e^2 + 31*e + 5, 10/3*e^6 - e^5 - 113/3*e^4 - 10/3*e^3 + 70*e^2 - 101/3*e - 8/3, -2/3*e^5 - 14/3*e^4 + 44/3*e^3 + 112/3*e^2 - 134/3*e - 16/3, 8/3*e^6 + 5/3*e^5 - 116/3*e^4 - 43/3*e^3 + 332/3*e^2 - 149/3*e - 22, -11/3*e^6 + 2/3*e^5 + 45*e^4 + 2*e^3 - 313/3*e^2 + 46*e + 95/3, 4/3*e^6 + 5/3*e^5 - 23*e^4 - 16*e^3 + 254/3*e^2 + 5*e - 67/3, -2*e^6 - 11/3*e^5 + 82/3*e^4 + 122/3*e^3 - 170/3*e^2 - 83/3*e + 2/3, -e^5 + e^4 + 14*e^3 - 7*e^2 - 47*e + 7, -4*e^6 + 5/3*e^5 + 158/3*e^4 - 32/3*e^3 - 448/3*e^2 + 203/3*e + 112/3, 3*e^6 + 2*e^5 - 44*e^4 - 22*e^3 + 127*e^2 - 16*e - 16, 14/3*e^6 + 11/3*e^5 - 185/3*e^4 - 157/3*e^3 + 452/3*e^2 + 82/3*e - 37, -8/3*e^6 + 4/3*e^5 + 98/3*e^4 - 14/3*e^3 - 242/3*e^2 + 80/3*e + 20, 14/3*e^6 - 11/3*e^5 - 59*e^4 + 26*e^3 + 490/3*e^2 - 85*e - 140/3, -7/3*e^6 - 1/3*e^5 + 91/3*e^4 + 38/3*e^3 - 256/3*e^2 - 23/3*e + 25, 22/3*e^6 + 2/3*e^5 - 91*e^4 - 32*e^3 + 638/3*e^2 - 18*e - 115/3, 5*e^6 - e^5 - 58*e^4 - 10*e^3 + 115*e^2 - 35*e - 6, e^6 - e^5 - 13*e^4 + 8*e^3 + 37*e^2 - 13*e + 4, 4/3*e^6 + 13/3*e^5 - 46/3*e^4 - 164/3*e^3 + 16/3*e^2 + 239/3*e + 8, 4/3*e^6 + 2/3*e^5 - 21*e^4 - 8*e^3 + 230/3*e^2 + 8*e - 118/3, -8/3*e^6 + 2/3*e^5 + 33*e^4 - 2*e^3 - 244/3*e^2 + 54*e + 59/3, -8/3*e^6 + 8/3*e^5 + 29*e^4 - 14*e^3 - 184/3*e^2 + 34*e + 95/3, 5/3*e^6 + 2*e^5 - 58/3*e^4 - 89/3*e^3 + 23*e^2 + 131/3*e + 32/3, -13/3*e^6 - e^5 + 143/3*e^4 + 94/3*e^3 - 65*e^2 - 43/3*e - 61/3, -10/3*e^6 - 11/3*e^5 + 44*e^4 + 48*e^3 - 296/3*e^2 - 41*e + 55/3, 17/3*e^6 + 7/3*e^5 - 67*e^4 - 46*e^3 + 367/3*e^2 + 13*e + 7/3, -11/3*e^5 + 7/3*e^4 + 119/3*e^3 - 8/3*e^2 - 194/3*e - 1/3, -19/3*e^6 + 13/3*e^5 + 73*e^4 - 14*e^3 - 497/3*e^2 + 43*e + 79/3, -e^6 - 10/3*e^5 + 59/3*e^4 + 88/3*e^3 - 211/3*e^2 - 52/3*e + 100/3, 10/3*e^6 - 2/3*e^5 - 115/3*e^4 - 38/3*e^3 + 244/3*e^2 + 77/3*e - 19, -11/3*e^5 - 14/3*e^4 + 146/3*e^3 + 163/3*e^2 - 317/3*e - 82/3, -7/3*e^6 + 10/3*e^5 + 27*e^4 - 28*e^3 - 203/3*e^2 + 66*e + 7/3, 7/3*e^6 - 20/3*e^5 - 79/3*e^4 + 184/3*e^3 + 253/3*e^2 - 340/3*e - 38, 5*e^6 + e^5 - 60*e^4 - 28*e^3 + 123*e^2 - 13*e - 18, -e^6 + 16/3*e^5 + 34/3*e^4 - 166/3*e^3 - 137/3*e^2 + 292/3*e + 44/3, 8/3*e^6 - 94/3*e^4 - 38/3*e^3 + 66*e^2 + 14/3*e - 82/3, -10/3*e^6 - 10/3*e^5 + 121/3*e^4 + 140/3*e^3 - 184/3*e^2 - 92/3*e - 27, -2*e^6 + 3*e^5 + 26*e^4 - 26*e^3 - 93*e^2 + 57*e + 48, -14/3*e^6 + 7/3*e^5 + 173/3*e^4 - 14/3*e^3 - 458/3*e^2 + 41/3*e + 45, 1/3*e^6 + 16/3*e^5 - 37/3*e^4 - 140/3*e^3 + 130/3*e^2 + 122/3*e + 13, 4/3*e^6 + 7/3*e^5 - 49/3*e^4 - 83/3*e^3 + 64/3*e^2 + 44/3*e + 13, 2*e^6 + 4/3*e^5 - 62/3*e^4 - 73/3*e^3 + 34/3*e^2 + 43/3*e + 86/3, -2/3*e^6 - 10/3*e^5 + 11*e^4 + 40*e^3 - 94/3*e^2 - 74*e + 41/3, 2*e^6 - 1/3*e^5 - 79/3*e^4 + 16/3*e^3 + 194/3*e^2 - 196/3*e - 5/3, 1/3*e^6 - 7/3*e^5 + 2*e^4 + 16*e^3 - 73/3*e^2 - 7*e - 58/3, -4*e^6 - 5/3*e^5 + 130/3*e^4 + 122/3*e^3 - 164/3*e^2 - 155/3*e - 28/3, -20/3*e^6 + 2/3*e^5 + 81*e^4 + 20*e^3 - 562/3*e^2 + 20*e + 173/3, -1/3*e^6 - 17/3*e^5 + 10*e^4 + 56*e^3 - 65/3*e^2 - 76*e - 32/3, 2/3*e^6 + 10/3*e^5 - 9*e^4 - 31*e^3 + 4/3*e^2 + 2*e + 97/3, 1/3*e^6 - 35/3*e^4 + 50/3*e^3 + 65*e^2 - 266/3*e - 29/3, -17/3*e^6 - 11/3*e^5 + 221/3*e^4 + 160/3*e^3 - 515/3*e^2 - 70/3*e + 35, 16/3*e^6 + 2*e^5 - 173/3*e^4 - 172/3*e^3 + 85*e^2 + 280/3*e - 83/3, -3*e^6 - 14/3*e^5 + 139/3*e^4 + 146/3*e^3 - 413/3*e^2 - 80/3*e + 155/3, -4/3*e^6 + 13/3*e^5 + 10*e^4 - 28*e^3 - 44/3*e^2 + 7*e + 70/3, -10/3*e^6 - 2*e^5 + 143/3*e^4 + 76/3*e^3 - 140*e^2 + 56/3*e + 125/3, -1/3*e^6 - e^5 + 23/3*e^4 + 22/3*e^3 - 38*e^2 + 35/3*e + 173/3, -8/3*e^6 + 2/3*e^5 + 33*e^4 + 2*e^3 - 244/3*e^2 + 17*e + 35/3, -8/3*e^6 + 5/3*e^5 + 40*e^4 - 24*e^3 - 418/3*e^2 + 109*e + 176/3, -1/3*e^6 + 5*e^5 - 10/3*e^4 - 128/3*e^3 + 17*e^2 + 149/3*e + 68/3, -4/3*e^6 + 4/3*e^5 + 14*e^4 - e^3 - 98/3*e^2 - 26*e + 76/3, 6*e^6 + e^5 - 73*e^4 - 32*e^3 + 155*e^2 - 7*e - 9, -22/3*e^6 - 14/3*e^5 + 100*e^4 + 61*e^3 - 776/3*e^2 + 18*e + 166/3, -11/3*e^6 + 2*e^5 + 118/3*e^4 - 4/3*e^3 - 69*e^2 + 82/3*e + 10/3, -2*e^6 - 14/3*e^5 + 79/3*e^4 + 158/3*e^3 - 116/3*e^2 - 152/3*e - 97/3, e^6 + 5*e^5 - 19*e^4 - 48*e^3 + 57*e^2 + 41*e - 15, 19/3*e^6 + 5/3*e^5 - 89*e^4 - 23*e^3 + 773/3*e^2 - 55*e - 175/3, 7/3*e^6 - 3*e^5 - 101/3*e^4 + 104/3*e^3 + 115*e^2 - 347/3*e - 83/3, 8/3*e^6 - 2/3*e^5 - 33*e^4 - e^3 + 232/3*e^2 - 33*e - 53/3, -2*e^6 + e^5 + 22*e^4 - 4*e^3 - 38*e^2 + 47*e - 20, 29/3*e^6 - 14/3*e^5 - 117*e^4 + 12*e^3 + 853/3*e^2 - 84*e - 170/3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
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;