/*
  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![31, 0, -12, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[4, 2, -1/2*w^3 - 1/2*w^2 + 7/2*w + 11/2],
[5, 5, -1/2*w^3 + w^2 + 5/2*w - 4],
[5, 5, -1/2*w^2 - w + 3/2],
[9, 3, -1/2*w^3 + 1/2*w^2 + 7/2*w - 9/2],
[9, 3, -1/2*w^3 - 1/2*w^2 + 7/2*w + 9/2],
[19, 19, -1/2*w^2 + w + 5/2],
[19, 19, 1/2*w^2 + w - 5/2],
[29, 29, -3/2*w^2 - w + 17/2],
[29, 29, -3/2*w^2 + w + 17/2],
[31, 31, 1/2*w^3 - 7/2*w],
[59, 59, 1/2*w^2 + w - 11/2],
[59, 59, 1/2*w^2 - w - 11/2],
[61, 61, -1/2*w^3 + w^2 + 7/2*w - 5],
[61, 61, 1/2*w^3 + w^2 - 7/2*w - 5],
[71, 71, -1/2*w^3 - 2*w^2 + 11/2*w + 13],
[71, 71, 3/2*w^3 + 2*w^2 - 23/2*w - 18],
[79, 79, 1/2*w^3 + 1/2*w^2 - 7/2*w - 1/2],
[79, 79, -2*w^2 + w + 10],
[79, 79, 2*w^2 + w - 10],
[79, 79, 1/2*w^3 - 1/2*w^2 - 7/2*w + 1/2],
[89, 89, 1/2*w^3 - 2*w^2 - 5/2*w + 11],
[89, 89, -1/2*w^2 + w + 15/2],
[109, 109, -w^3 + 7*w + 3],
[109, 109, 1/2*w^3 - 2*w^2 - 9/2*w + 12],
[109, 109, w^3 + w^2 - 8*w - 11],
[109, 109, w^3 - 7*w + 3],
[121, 11, 3/2*w^2 - 19/2],
[121, 11, -1/2*w^2 + 13/2],
[131, 131, -1/2*w^3 + 3/2*w^2 + 7/2*w - 15/2],
[131, 131, -1/2*w^3 + 9/2*w + 3],
[179, 179, -w^3 - 5/2*w^2 + 9*w + 35/2],
[179, 179, 1/2*w^3 + 2*w^2 - 7/2*w - 13],
[179, 179, -1/2*w^3 + 2*w^2 + 7/2*w - 13],
[179, 179, 3/2*w^3 + 4*w^2 - 27/2*w - 29],
[181, 181, -1/2*w^3 + w^2 + 5/2*w - 7],
[181, 181, 1/2*w^3 + w^2 - 5/2*w - 7],
[191, 191, 2*w^2 + w - 12],
[191, 191, 2*w^2 - w - 12],
[211, 211, w^3 - w^2 - 7*w + 4],
[211, 211, -w^3 - w^2 + 7*w + 4],
[229, 229, 3/2*w^2 + w - 23/2],
[229, 229, 3/2*w^2 - w - 23/2],
[241, 241, 1/2*w^3 + 3/2*w^2 - 9/2*w - 17/2],
[241, 241, -1/2*w^3 + 3/2*w^2 + 9/2*w - 17/2],
[251, 251, w^3 - 5*w + 3],
[251, 251, 3/2*w^3 - 23/2*w + 2],
[251, 251, 5/2*w^2 + 3*w - 23/2],
[251, 251, w^3 + 3*w^2 - 9*w - 24],
[269, 269, 1/2*w^2 + 2*w - 9/2],
[269, 269, 1/2*w^2 - 2*w - 9/2],
[271, 271, 1/2*w^3 - 1/2*w^2 - 3/2*w - 3/2],
[271, 271, w^3 + 3/2*w^2 - 6*w - 15/2],
[271, 271, 1/2*w^3 + 3*w^2 - 9/2*w - 16],
[271, 271, -1/2*w^3 - 2*w^2 + 5/2*w + 13],
[281, 281, w^3 - 5/2*w^2 - 8*w + 33/2],
[281, 281, w^2 + 2*w - 6],
[281, 281, w^2 - 2*w - 6],
[281, 281, w^3 + 5/2*w^2 - 8*w - 33/2],
[311, 311, -5/2*w^3 - 3*w^2 + 37/2*w + 30],
[311, 311, 2*w^2 - 2*w - 13],
[331, 331, -1/2*w^3 + 7/2*w - 5],
[331, 331, -w^3 + 5/2*w^2 + 7*w - 29/2],
[331, 331, w^3 + 5/2*w^2 - 7*w - 29/2],
[331, 331, 1/2*w^3 - 7/2*w - 5],
[349, 349, 1/2*w^3 - 2*w^2 - 11/2*w + 16],
[349, 349, w^3 + 3*w^2 - 6*w - 19],
[349, 349, 1/2*w^3 - w^2 - 1/2*w - 1],
[349, 349, 1/2*w^3 + 2*w^2 - 11/2*w - 16],
[359, 359, 1/2*w^3 + 3/2*w^2 - 11/2*w - 15/2],
[359, 359, w^3 - 4*w^2 - 2*w + 16],
[361, 19, -1/2*w^2 + 15/2],
[379, 379, 1/2*w^3 + 5/2*w^2 - 11/2*w - 37/2],
[379, 379, -w^3 + 1/2*w^2 + 6*w - 15/2],
[389, 389, w^3 - 1/2*w^2 - 7*w + 5/2],
[389, 389, w^3 + 1/2*w^2 - 7*w - 5/2],
[401, 401, 1/2*w^3 + 3/2*w^2 - 5/2*w - 21/2],
[401, 401, -1/2*w^3 + 3/2*w^2 + 5/2*w - 21/2],
[409, 409, -2*w^3 - 5*w^2 + 18*w + 38],
[409, 409, 1/2*w^3 - w^2 - 13/2*w + 1],
[419, 419, -1/2*w^3 + 1/2*w^2 + 5/2*w - 19/2],
[419, 419, 1/2*w^3 + 1/2*w^2 - 5/2*w - 19/2],
[431, 431, 1/2*w^3 - 3/2*w^2 - 7/2*w + 11/2],
[431, 431, 1/2*w^3 + 3/2*w^2 - 7/2*w - 11/2],
[439, 439, -1/2*w^3 + 5/2*w^2 + 5/2*w - 27/2],
[439, 439, -1/2*w^3 - 5/2*w^2 + 5/2*w + 27/2],
[449, 449, w^3 - 1/2*w^2 - 7*w + 7/2],
[449, 449, w^3 + 1/2*w^2 - 7*w - 7/2],
[461, 461, w^3 - 7/2*w^2 - 5*w + 37/2],
[461, 461, -1/2*w^2 + 2*w + 21/2],
[479, 479, -w^3 + 4*w^2 + 4*w - 20],
[479, 479, 1/2*w^3 - 3/2*w + 8],
[491, 491, 1/2*w^3 - 5/2*w^2 - 3/2*w + 29/2],
[491, 491, 1/2*w^3 + 1/2*w^2 - 1/2*w - 7/2],
[491, 491, 1/2*w^3 + w^2 - 11/2*w - 8],
[491, 491, -1/2*w^3 + w^2 + 5/2*w - 12],
[499, 499, 3*w^2 - w - 21],
[499, 499, 1/2*w^3 + 5/2*w^2 - 11/2*w - 29/2],
[499, 499, w^3 + 5*w^2 - 11*w - 32],
[499, 499, 3*w^2 + w - 21],
[509, 509, -w^3 + 3/2*w^2 + 4*w - 3/2],
[509, 509, 1/2*w^3 + 3*w^2 - 3/2*w - 18],
[529, 23, 1/2*w^3 - 2*w^2 - 11/2*w + 11],
[529, 23, 1/2*w^3 + 2*w^2 - 11/2*w - 11],
[569, 569, 1/2*w^3 + w^2 - 11/2*w - 3],
[569, 569, -1/2*w^3 + w^2 + 11/2*w - 3],
[571, 571, 5/2*w^2 + 2*w - 33/2],
[571, 571, 2*w^3 - 5/2*w^2 - 16*w + 49/2],
[571, 571, 2*w^3 + 5/2*w^2 - 16*w - 49/2],
[571, 571, 5/2*w^2 - 2*w - 33/2],
[599, 599, 1/2*w^3 - 1/2*w^2 - 11/2*w - 3/2],
[599, 599, -1/2*w^3 - 1/2*w^2 + 11/2*w - 3/2],
[601, 601, 2*w^3 + 3/2*w^2 - 15*w - 35/2],
[601, 601, 5/2*w^3 + 3*w^2 - 39/2*w - 29],
[619, 619, 5/2*w^3 + 5/2*w^2 - 37/2*w - 51/2],
[619, 619, w^3 - 1/2*w^2 - 5*w + 11/2],
[619, 619, 3/2*w^2 - 2*w - 23/2],
[619, 619, w^3 + 3*w^2 - 10*w - 21],
[631, 631, -5/2*w^3 + 33/2*w + 13],
[631, 631, 3/2*w^3 - 5*w^2 - 13/2*w + 22],
[631, 631, -2*w^3 - 2*w^2 + 14*w + 23],
[631, 631, -5/2*w^2 - 5*w + 13/2],
[641, 641, -1/2*w^3 + 1/2*w^2 + 11/2*w - 5/2],
[641, 641, 1/2*w^3 + 1/2*w^2 - 11/2*w - 5/2],
[659, 659, -3/2*w^3 - w^2 + 23/2*w + 14],
[659, 659, 3/2*w^3 + 5/2*w^2 - 19/2*w - 27/2],
[661, 661, -1/2*w^3 + 1/2*w^2 + 11/2*w - 9/2],
[661, 661, w^3 + w^2 - 8*w - 5],
[661, 661, -1/2*w^3 + 2*w^2 + 7/2*w - 8],
[661, 661, -1/2*w^3 - 1/2*w^2 + 11/2*w + 9/2],
[691, 691, 1/2*w^3 - 3/2*w^2 - 3/2*w + 19/2],
[691, 691, 1/2*w^3 + 3/2*w^2 - 3/2*w - 19/2],
[709, 709, 1/2*w^3 - 2*w^2 - 3/2*w + 12],
[709, 709, 1/2*w^3 + 2*w^2 - 3/2*w - 12],
[751, 751, 3/2*w^3 - w^2 - 21/2*w + 11],
[751, 751, 3/2*w^3 + w^2 - 21/2*w - 11],
[761, 761, 2*w^3 + 11/2*w^2 - 17*w - 85/2],
[761, 761, -3/2*w^3 - 4*w^2 + 17/2*w + 23],
[769, 769, w^3 - 7/2*w^2 - 8*w + 45/2],
[769, 769, -1/2*w^3 - 5/2*w^2 + 5/2*w + 35/2],
[769, 769, -1/2*w^3 + 5/2*w^2 + 5/2*w - 35/2],
[769, 769, w^3 + 7/2*w^2 - 8*w - 45/2],
[809, 809, w^3 - 7/2*w^2 - 7*w + 41/2],
[809, 809, w^3 + 7/2*w^2 - 7*w - 41/2],
[811, 811, -1/2*w^3 - 9/2*w^2 + 13/2*w + 59/2],
[811, 811, 3*w^3 + 5*w^2 - 24*w - 45],
[821, 821, -2*w^3 - 7/2*w^2 + 12*w + 33/2],
[821, 821, w^3 - 4*w^2 - w + 13],
[829, 829, -w^3 + 1/2*w^2 + 8*w - 1/2],
[829, 829, w^3 + 1/2*w^2 - 8*w - 1/2],
[839, 839, w^3 - 3/2*w^2 - 7*w + 13/2],
[839, 839, w^3 + 3/2*w^2 - 7*w - 13/2],
[841, 29, 5/2*w^2 - 27/2],
[859, 859, 1/2*w^3 - 7/2*w - 6],
[859, 859, -3/2*w^3 + 2*w^2 + 21/2*w - 15],
[859, 859, 3/2*w^3 + 2*w^2 - 21/2*w - 15],
[859, 859, 1/2*w^3 - 7/2*w + 6],
[881, 881, -3/2*w^3 + 21/2*w + 8],
[881, 881, -2*w^3 + 1/2*w^2 + 13*w + 15/2],
[919, 919, -3/2*w^3 + 5/2*w^2 + 15/2*w - 19/2],
[919, 919, 3/2*w^3 + w^2 - 21/2*w - 4],
[929, 929, 1/2*w^3 + w^2 - 9/2*w - 1],
[929, 929, 1/2*w^3 - w^2 - 9/2*w + 1],
[941, 941, 2*w^3 + 7*w^2 - 19*w - 51],
[941, 941, 5/2*w^2 - 3*w - 33/2],
[961, 31, -w^2 + 12],
[971, 971, 3/2*w^2 - 3*w - 25/2],
[971, 971, w^3 + 7/2*w^2 - 6*w - 39/2],
[991, 991, 1/2*w^3 - 3/2*w^2 - 13/2*w + 13/2],
[991, 991, w^3 + 5/2*w^2 - 5*w - 29/2],
[991, 991, w^3 - 5/2*w^2 - 5*w + 29/2],
[991, 991, -3/2*w^3 + 2*w^2 + 17/2*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 30*x^6 + 260*x^4 - 600*x^2 + 400;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, e, 3/50*e^7 - 17/10*e^5 + 64/5*e^3 - 15*e, 3/50*e^7 - 17/10*e^5 + 64/5*e^3 - 15*e, -1/25*e^7 + 11/10*e^5 - 41/5*e^3 + 12*e, -1/25*e^7 + 11/10*e^5 - 41/5*e^3 + 12*e, 1/10*e^6 - 13/5*e^4 + 17*e^2 - 14, 1/10*e^6 - 13/5*e^4 + 17*e^2 - 14, -9/50*e^7 + 26/5*e^5 - 202/5*e^3 + 52*e, -1/5*e^6 + 6*e^4 - 48*e^2 + 60, -1/5*e^6 + 6*e^4 - 48*e^2 + 60, 2/25*e^7 - 12/5*e^5 + 97/5*e^3 - 27*e, 2/25*e^7 - 12/5*e^5 + 97/5*e^3 - 27*e, 1/10*e^7 - 14/5*e^5 + 20*e^3 - 16*e, 1/10*e^7 - 14/5*e^5 + 20*e^3 - 16*e, -9/50*e^7 + 5*e^5 - 182/5*e^3 + 40*e, -1/10*e^6 + 3*e^4 - 24*e^2 + 32, -1/10*e^6 + 3*e^4 - 24*e^2 + 32, -9/50*e^7 + 5*e^5 - 182/5*e^3 + 40*e, -1/5*e^6 + 29/5*e^4 - 44*e^2 + 50, -1/5*e^6 + 29/5*e^4 - 44*e^2 + 50, 3/25*e^7 - 18/5*e^5 + 148/5*e^3 - 47*e, -3/25*e^7 + 17/5*e^5 - 133/5*e^3 + 45*e, -3/25*e^7 + 17/5*e^5 - 133/5*e^3 + 45*e, 3/25*e^7 - 18/5*e^5 + 148/5*e^3 - 47*e, 1/5*e^4 - 4*e^2 + 26, 1/5*e^6 - 6*e^4 + 49*e^2 - 58, 9/50*e^7 - 51/10*e^5 + 197/5*e^3 - 60*e, 9/50*e^7 - 51/10*e^5 + 197/5*e^3 - 60*e, 11/50*e^7 - 61/10*e^5 + 223/5*e^3 - 48*e, -3/10*e^6 + 41/5*e^4 - 60*e^2 + 84, -3/10*e^6 + 41/5*e^4 - 60*e^2 + 84, 11/50*e^7 - 61/10*e^5 + 223/5*e^3 - 48*e, 1/10*e^6 - 13/5*e^4 + 17*e^2 - 22, 1/10*e^6 - 13/5*e^4 + 17*e^2 - 22, -3/10*e^6 + 41/5*e^4 - 60*e^2 + 72, -3/10*e^6 + 41/5*e^4 - 60*e^2 + 72, 1/25*e^7 - 9/10*e^5 + 21/5*e^3 + 4*e, 1/25*e^7 - 9/10*e^5 + 21/5*e^3 + 4*e, -1/5*e^6 + 29/5*e^4 - 44*e^2 + 46, -1/5*e^6 + 29/5*e^4 - 44*e^2 + 46, -6/25*e^7 + 67/10*e^5 - 251/5*e^3 + 65*e, -6/25*e^7 + 67/10*e^5 - 251/5*e^3 + 65*e, 2/5*e^6 - 56/5*e^4 + 84*e^2 - 116, -19/50*e^7 + 109/10*e^5 - 417/5*e^3 + 104*e, -19/50*e^7 + 109/10*e^5 - 417/5*e^3 + 104*e, 2/5*e^6 - 56/5*e^4 + 84*e^2 - 116, 2/5*e^7 - 57/5*e^5 + 87*e^3 - 111*e, 2/5*e^7 - 57/5*e^5 + 87*e^3 - 111*e, -1/5*e^6 + 24/5*e^4 - 28*e^2 + 16, 2/25*e^7 - 12/5*e^5 + 102/5*e^3 - 40*e, 2/25*e^7 - 12/5*e^5 + 102/5*e^3 - 40*e, -1/5*e^6 + 24/5*e^4 - 28*e^2 + 16, 3/25*e^7 - 7/2*e^5 + 143/5*e^3 - 51*e, -3/25*e^7 + 33/10*e^5 - 113/5*e^3 + 9*e, -3/25*e^7 + 33/10*e^5 - 113/5*e^3 + 9*e, 3/25*e^7 - 7/2*e^5 + 143/5*e^3 - 51*e, 3/10*e^6 - 43/5*e^4 + 64*e^2 - 72, 3/10*e^6 - 43/5*e^4 + 64*e^2 - 72, -1/5*e^6 + 28/5*e^4 - 44*e^2 + 60, -3/50*e^7 + 3/2*e^5 - 39/5*e^3 - 12*e, -3/50*e^7 + 3/2*e^5 - 39/5*e^3 - 12*e, -1/5*e^6 + 28/5*e^4 - 44*e^2 + 60, 3/10*e^6 - 39/5*e^4 + 53*e^2 - 62, -3/5*e^6 + 81/5*e^4 - 116*e^2 + 150, -3/5*e^6 + 81/5*e^4 - 116*e^2 + 150, 3/10*e^6 - 39/5*e^4 + 53*e^2 - 62, 1/25*e^7 - 6/5*e^5 + 46/5*e^3 - 4*e, 1/25*e^7 - 6/5*e^5 + 46/5*e^3 - 4*e, -2/5*e^6 + 12*e^4 - 95*e^2 + 118, 7/10*e^6 - 97/5*e^4 + 144*e^2 - 172, 7/10*e^6 - 97/5*e^4 + 144*e^2 - 172, -1/5*e^7 + 27/5*e^5 - 38*e^3 + 41*e, -1/5*e^7 + 27/5*e^5 - 38*e^3 + 41*e, 2/5*e^4 - 7*e^2 + 6, 2/5*e^4 - 7*e^2 + 6, -16/25*e^7 + 181/10*e^5 - 681/5*e^3 + 165*e, -16/25*e^7 + 181/10*e^5 - 681/5*e^3 + 165*e, -1/10*e^6 + 11/5*e^4 - 8*e^2 - 12, -1/10*e^6 + 11/5*e^4 - 8*e^2 - 12, -23/50*e^7 + 66/5*e^5 - 514/5*e^3 + 140*e, -23/50*e^7 + 66/5*e^5 - 514/5*e^3 + 140*e, 7/10*e^6 - 99/5*e^4 + 148*e^2 - 176, 7/10*e^6 - 99/5*e^4 + 148*e^2 - 176, -7/25*e^7 + 81/10*e^5 - 317/5*e^3 + 81*e, -7/25*e^7 + 81/10*e^5 - 317/5*e^3 + 81*e, -7/10*e^6 + 99/5*e^4 - 151*e^2 + 194, -7/10*e^6 + 99/5*e^4 - 151*e^2 + 194, -1/10*e^6 + 3*e^4 - 24*e^2 + 40, -1/10*e^6 + 3*e^4 - 24*e^2 + 40, 3/10*e^6 - 43/5*e^4 + 68*e^2 - 100, -7/25*e^7 + 81/10*e^5 - 317/5*e^3 + 92*e, -7/25*e^7 + 81/10*e^5 - 317/5*e^3 + 92*e, 3/10*e^6 - 43/5*e^4 + 68*e^2 - 100, -1/2*e^6 + 15*e^4 - 120*e^2 + 156, 1/5*e^7 - 11/2*e^5 + 39*e^3 - 40*e, 1/5*e^7 - 11/2*e^5 + 39*e^3 - 40*e, -1/2*e^6 + 15*e^4 - 120*e^2 + 156, 1/2*e^6 - 69/5*e^4 + 101*e^2 - 110, 1/2*e^6 - 69/5*e^4 + 101*e^2 - 110, 4/25*e^7 - 47/10*e^5 + 179/5*e^3 - 31*e, 4/25*e^7 - 47/10*e^5 + 179/5*e^3 - 31*e, -13/25*e^7 + 149/10*e^5 - 573/5*e^3 + 153*e, -13/25*e^7 + 149/10*e^5 - 573/5*e^3 + 153*e, -4/5*e^6 + 114/5*e^4 - 172*e^2 + 204, 1/2*e^7 - 139/10*e^5 + 103*e^3 - 128*e, 1/2*e^7 - 139/10*e^5 + 103*e^3 - 128*e, -4/5*e^6 + 114/5*e^4 - 172*e^2 + 204, 2/5*e^7 - 58/5*e^5 + 92*e^3 - 132*e, 2/5*e^7 - 58/5*e^5 + 92*e^3 - 132*e, -13/50*e^7 + 15/2*e^5 - 284/5*e^3 + 53*e, -13/50*e^7 + 15/2*e^5 - 284/5*e^3 + 53*e, 9/25*e^7 - 103/10*e^5 + 399/5*e^3 - 124*e, 1/2*e^6 - 69/5*e^4 + 100*e^2 - 108, 1/2*e^6 - 69/5*e^4 + 100*e^2 - 108, 9/25*e^7 - 103/10*e^5 + 399/5*e^3 - 124*e, 19/50*e^7 - 54/5*e^5 + 402/5*e^3 - 88*e, -1/5*e^6 + 6*e^4 - 52*e^2 + 80, -1/5*e^6 + 6*e^4 - 52*e^2 + 80, 19/50*e^7 - 54/5*e^5 + 402/5*e^3 - 88*e, 9/50*e^7 - 49/10*e^5 + 172/5*e^3 - 31*e, 9/50*e^7 - 49/10*e^5 + 172/5*e^3 - 31*e, -8/25*e^7 + 89/10*e^5 - 323/5*e^3 + 64*e, -8/25*e^7 + 89/10*e^5 - 323/5*e^3 + 64*e, 14/25*e^7 - 81/5*e^5 + 634/5*e^3 - 175*e, 8/25*e^7 - 46/5*e^5 + 353/5*e^3 - 95*e, 8/25*e^7 - 46/5*e^5 + 353/5*e^3 - 95*e, 14/25*e^7 - 81/5*e^5 + 634/5*e^3 - 175*e, -3/10*e^6 + 49/5*e^4 - 84*e^2 + 92, -3/10*e^6 + 49/5*e^4 - 84*e^2 + 92, -3/5*e^6 + 17*e^4 - 132*e^2 + 190, -3/5*e^6 + 17*e^4 - 132*e^2 + 190, 17/50*e^7 - 10*e^5 + 396/5*e^3 - 100*e, 17/50*e^7 - 10*e^5 + 396/5*e^3 - 100*e, 3/5*e^6 - 16*e^4 + 112*e^2 - 142, 3/5*e^6 - 16*e^4 + 112*e^2 - 142, e^6 - 148/5*e^4 + 233*e^2 - 282, -e^6 + 144/5*e^4 - 224*e^2 + 282, -e^6 + 144/5*e^4 - 224*e^2 + 282, e^6 - 148/5*e^4 + 233*e^2 - 282, -2/5*e^6 + 54/5*e^4 - 75*e^2 + 94, -2/5*e^6 + 54/5*e^4 - 75*e^2 + 94, 3/5*e^6 - 88/5*e^4 + 140*e^2 - 188, 3/5*e^6 - 88/5*e^4 + 140*e^2 - 188, 1/5*e^7 - 6*e^5 + 49*e^3 - 67*e, 1/5*e^7 - 6*e^5 + 49*e^3 - 67*e, 3/25*e^7 - 16/5*e^5 + 113/5*e^3 - 27*e, 3/25*e^7 - 16/5*e^5 + 113/5*e^3 - 27*e, -1/25*e^7 + 6/5*e^5 - 46/5*e^3 + 4*e, -1/25*e^7 + 6/5*e^5 - 46/5*e^3 + 4*e, -1/5*e^6 + 28/5*e^4 - 40*e^2 + 82, -4/5*e^4 + 16*e^2 - 36, -3/10*e^7 + 87/10*e^5 - 69*e^3 + 108*e, -3/10*e^7 + 87/10*e^5 - 69*e^3 + 108*e, -4/5*e^4 + 16*e^2 - 36, -23/50*e^7 + 133/10*e^5 - 514/5*e^3 + 133*e, -23/50*e^7 + 133/10*e^5 - 514/5*e^3 + 133*e, 11/25*e^7 - 63/5*e^5 + 476/5*e^3 - 112*e, 11/25*e^7 - 63/5*e^5 + 476/5*e^3 - 112*e, -4/25*e^7 + 9/2*e^5 - 169/5*e^3 + 41*e, -4/25*e^7 + 9/2*e^5 - 169/5*e^3 + 41*e, 1/5*e^6 - 31/5*e^4 + 52*e^2 - 74, 1/5*e^6 - 31/5*e^4 + 52*e^2 - 74, 2/5*e^4 - 7*e^2 + 70, 3/5*e^6 - 84/5*e^4 + 124*e^2 - 156, 3/5*e^6 - 84/5*e^4 + 124*e^2 - 156, -7/50*e^7 + 4*e^5 - 156/5*e^3 + 52*e, -3/10*e^6 + 47/5*e^4 - 84*e^2 + 136, -3/10*e^6 + 47/5*e^4 - 84*e^2 + 136, -7/50*e^7 + 4*e^5 - 156/5*e^3 + 52*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4,2,-1/2*w^3-1/2*w^2+7/2*w+11/2}>] := 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;