/*
  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 | {19,19,-1/2*w^2 - w + 5/2}>;

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 - 3*x^7 - 9*x^6 + 28*x^5 + 21*x^4 - 64*x^3 - 19*x^2 + 28*x - 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/2*e^7 + 1/2*e^6 + 11/2*e^5 - 4*e^4 - 35/2*e^3 + 7*e^2 + 29/2*e - 5, 1/2*e^7 - 1/2*e^6 - 11/2*e^5 + 4*e^4 + 33/2*e^3 - 7*e^2 - 21/2*e + 5, -1/2*e^7 - 1/2*e^6 + 11/2*e^5 + 7*e^4 - 29/2*e^3 - 25*e^2 - 3/2*e + 5, -1/2*e^7 + 3/2*e^6 + 9/2*e^5 - 13*e^4 - 21/2*e^3 + 25*e^2 + 19/2*e - 7, e^6 - e^5 - 10*e^4 + 6*e^3 + 24*e^2 + e - 6, 1, -e^6 + 10*e^4 + 2*e^3 - 23*e^2 - 13*e + 2, -e^7 + e^6 + 11*e^5 - 9*e^4 - 35*e^3 + 19*e^2 + 33*e - 10, e^7 - 2*e^6 - 9*e^5 + 17*e^4 + 20*e^3 - 32*e^2 - 13*e + 10, e^7 + e^6 - 11*e^5 - 13*e^4 + 30*e^3 + 41*e^2 - 4, -e^7 + 2*e^6 + 9*e^5 - 17*e^4 - 19*e^3 + 30*e^2 + 8*e - 6, 1/2*e^7 - 1/2*e^6 - 13/2*e^5 + 5*e^4 + 55/2*e^3 - 13*e^2 - 75/2*e + 3, 3/2*e^7 - 5/2*e^6 - 27/2*e^5 + 17*e^4 + 55/2*e^3 - 11*e^2 - 9/2*e - 5, -2*e^7 + 2*e^6 + 20*e^5 - 15*e^4 - 52*e^3 + 20*e^2 + 29*e - 14, e^7 - e^6 - 10*e^5 + 7*e^4 + 27*e^3 - 5*e^2 - 20*e - 4, 4*e^7 - 3*e^6 - 42*e^5 + 19*e^4 + 116*e^3 - 8*e^2 - 56*e + 16, 2*e^7 - 3*e^6 - 20*e^5 + 23*e^4 + 54*e^3 - 34*e^2 - 40*e + 20, 3*e^7 - 4*e^6 - 29*e^5 + 30*e^4 + 70*e^3 - 37*e^2 - 30*e + 10, e^6 + e^5 - 11*e^4 - 12*e^3 + 32*e^2 + 35*e - 14, e^7 - 2*e^6 - 9*e^5 + 17*e^4 + 23*e^3 - 30*e^2 - 32*e + 2, 2*e^6 - 2*e^5 - 18*e^4 + 13*e^3 + 34*e^2 - 4*e + 2, -3/2*e^7 + 5/2*e^6 + 25/2*e^5 - 19*e^4 - 39/2*e^3 + 24*e^2 - 3/2*e - 3, 5/2*e^7 - 1/2*e^6 - 53/2*e^5 - 4*e^4 + 139/2*e^3 + 44*e^2 - 17/2*e - 17, -1/2*e^7 + 1/2*e^6 + 9/2*e^5 - 3*e^4 - 11/2*e^3 + e^2 - 43/2*e - 3, 3/2*e^7 - 5/2*e^6 - 31/2*e^5 + 23*e^4 + 95/2*e^3 - 52*e^2 - 101/2*e + 17, -e^7 + 2*e^6 + 10*e^5 - 18*e^4 - 26*e^3 + 35*e^2 + 12*e - 6, -3*e^7 + 4*e^6 + 28*e^5 - 29*e^4 - 63*e^3 + 33*e^2 + 21*e - 14, 2*e^7 - 3*e^6 - 18*e^5 + 21*e^4 + 38*e^3 - 18*e^2 - 12*e - 4, e^5 + e^4 - 8*e^3 - 8*e^2 + 15*e + 6, 3*e^7 - 4*e^6 - 29*e^5 + 30*e^4 + 72*e^3 - 39*e^2 - 42*e + 12, 4*e^7 - 3*e^6 - 39*e^5 + 14*e^4 + 88*e^3 + 25*e^2 + 2*e + 2, -2*e^7 + 5*e^6 + 21*e^5 - 46*e^4 - 64*e^3 + 99*e^2 + 68*e - 28, e^7 + 4*e^6 - 14*e^5 - 40*e^4 + 48*e^3 + 95*e^2 - 4*e - 8, -e^7 + 9*e^5 + 3*e^4 - 18*e^3 - 18*e^2 + e + 10, -e^7 + 2*e^6 + 8*e^5 - 15*e^4 - 8*e^3 + 16*e^2 - 24*e - 4, -3*e^7 + 3*e^6 + 30*e^5 - 21*e^4 - 78*e^3 + 16*e^2 + 42*e + 2, -e^6 + 3*e^5 + 10*e^4 - 22*e^3 - 24*e^2 + 22*e + 6, -2*e^6 + 3*e^5 + 17*e^4 - 20*e^3 - 32*e^2 + 12*e + 4, e^7 - 5*e^6 - 9*e^5 + 45*e^4 + 25*e^3 - 90*e^2 - 45*e + 14, 3*e^7 - 4*e^6 - 30*e^5 + 34*e^4 + 79*e^3 - 61*e^2 - 52*e + 18, 5*e^7 - 4*e^6 - 54*e^5 + 26*e^4 + 160*e^3 - 16*e^2 - 100*e + 18, 7/2*e^7 - 13/2*e^6 - 73/2*e^5 + 54*e^4 + 219/2*e^3 - 96*e^2 - 211/2*e + 41, -5/2*e^7 + 1/2*e^6 + 55/2*e^5 - e^4 - 169/2*e^3 - 10*e^2 + 127/2*e - 11, 5*e^7 - 4*e^6 - 50*e^5 + 24*e^4 + 124*e^3 + e^2 - 36*e, -3*e^7 + e^6 + 30*e^5 - 73*e^3 - 33*e^2 + 8*e - 16, -2*e^7 + 5*e^6 + 20*e^5 - 45*e^4 - 61*e^3 + 93*e^2 + 76*e - 16, -5*e^7 + e^6 + 55*e^5 + 3*e^4 - 162*e^3 - 52*e^2 + 78*e + 4, 7/2*e^7 - 3/2*e^6 - 77/2*e^5 + 6*e^4 + 229/2*e^3 + 16*e^2 - 115/2*e - 5, -5/2*e^7 + 19/2*e^6 + 39/2*e^5 - 88*e^4 - 73/2*e^3 + 193*e^2 + 127/2*e - 47, 3*e^7 - 4*e^6 - 31*e^5 + 32*e^4 + 91*e^3 - 47*e^2 - 82*e + 8, -e^7 + 10*e^5 + 8*e^4 - 21*e^3 - 50*e^2 - 17*e + 10, 3*e^7 - 4*e^6 - 29*e^5 + 30*e^4 + 74*e^3 - 38*e^2 - 49*e - 2, -e^7 + 9*e^5 + 9*e^4 - 15*e^3 - 60*e^2 - 16*e + 34, -11/2*e^7 + 13/2*e^6 + 109/2*e^5 - 50*e^4 - 279/2*e^3 + 71*e^2 + 149/2*e - 37, -3/2*e^7 - 1/2*e^6 + 27/2*e^5 + 15*e^4 - 39/2*e^3 - 76*e^2 - 87/2*e + 19, -1/2*e^7 + 1/2*e^6 + 15/2*e^5 - 3*e^4 - 69/2*e^3 + 3*e^2 + 83/2*e - 1, 1/2*e^7 + 9/2*e^6 - 11/2*e^5 - 49*e^4 + 5/2*e^3 + 139*e^2 + 139/2*e - 43, 4*e^7 - 5*e^6 - 37*e^5 + 37*e^4 + 80*e^3 - 41*e^2 - 28*e + 12, 5*e^7 - 6*e^6 - 53*e^5 + 47*e^4 + 156*e^3 - 72*e^2 - 114*e + 30, 2*e^6 + 4*e^5 - 24*e^4 - 37*e^3 + 72*e^2 + 76*e - 18, -5*e^7 + 4*e^6 + 54*e^5 - 25*e^4 - 155*e^3 + 6*e^2 + 78*e - 8, -8*e^7 + 7*e^6 + 86*e^5 - 52*e^4 - 251*e^3 + 68*e^2 + 155*e - 58, -4*e^6 - e^5 + 42*e^4 + 21*e^3 - 110*e^2 - 88*e + 30, -7*e^7 + 10*e^6 + 71*e^5 - 81*e^4 - 191*e^3 + 129*e^2 + 119*e - 46, e^6 - 5*e^5 - 6*e^4 + 39*e^3 - 2*e^2 - 60*e + 10, -6*e^7 + 2*e^6 + 63*e^5 - 173*e^3 - 72*e^2 + 76*e + 6, -3*e^7 + 6*e^6 + 30*e^5 - 51*e^4 - 84*e^3 + 96*e^2 + 73*e - 30, -4*e^7 + 3*e^6 + 42*e^5 - 20*e^4 - 120*e^3 + 14*e^2 + 81*e - 18, 9*e^7 - 10*e^6 - 90*e^5 + 72*e^4 + 238*e^3 - 77*e^2 - 145*e + 26, -4*e^7 + 2*e^6 + 41*e^5 - 7*e^4 - 103*e^3 - 31*e^2 + 14*e, 2*e^7 + 5*e^6 - 26*e^5 - 58*e^4 + 83*e^3 + 172*e^2 - 2*e - 34, -4*e^7 + 47*e^5 + 8*e^4 - 151*e^3 - 47*e^2 + 86*e - 6, -7/2*e^7 - 1/2*e^6 + 85/2*e^5 + 12*e^4 - 277/2*e^3 - 50*e^2 + 139/2*e - 17, 7/2*e^7 - 17/2*e^6 - 67/2*e^5 + 72*e^4 + 169/2*e^3 - 126*e^2 - 139/2*e + 13, e^7 - 5*e^5 - 10*e^4 - 18*e^3 + 61*e^2 + 76*e - 24, -e^6 - e^5 + 15*e^4 + 10*e^3 - 60*e^2 - 26*e + 40, -1/2*e^7 + 7/2*e^6 + 9/2*e^5 - 36*e^4 - 33/2*e^3 + 97*e^2 + 111/2*e - 49, -7/2*e^7 + 15/2*e^6 + 61/2*e^5 - 62*e^4 - 131/2*e^3 + 107*e^2 + 97/2*e - 31, -e^7 + 12*e^5 - 2*e^4 - 42*e^3 + 7*e^2 + 42*e + 12, -7*e^7 + 12*e^6 + 67*e^5 - 97*e^4 - 164*e^3 + 160*e^2 + 96*e - 56, -4*e^6 + 4*e^5 + 39*e^4 - 26*e^3 - 95*e^2 + 2*e + 32, e^7 + 3*e^6 - 11*e^5 - 35*e^4 + 22*e^3 + 108*e^2 + 39*e - 26, -5*e^7 + 2*e^6 + 57*e^5 - 6*e^4 - 178*e^3 - 30*e^2 + 102*e - 22, -4*e^7 + 9*e^6 + 37*e^5 - 76*e^4 - 87*e^3 + 132*e^2 + 62*e - 20, 3/2*e^7 + 5/2*e^6 - 35/2*e^5 - 30*e^4 + 93/2*e^3 + 98*e^2 + 37/2*e - 43, -7/2*e^7 + 5/2*e^6 + 77/2*e^5 - 16*e^4 - 219/2*e^3 + 13*e^2 + 75/2*e - 41, -5*e^7 + 4*e^6 + 52*e^5 - 24*e^4 - 141*e^3 + 3*e^2 + 62*e - 8, -2*e^7 - 7*e^6 + 22*e^5 + 77*e^4 - 48*e^3 - 216*e^2 - 60*e + 44, -2*e^7 + 23*e^5 + 5*e^4 - 73*e^3 - 30*e^2 + 42*e + 8, 7*e^7 - 3*e^6 - 75*e^5 + 14*e^4 + 209*e^3 + 20*e^2 - 78*e + 22, -3*e^7 + 26*e^5 + 18*e^4 - 31*e^3 - 117*e^2 - 88*e + 36, e^6 - 6*e^5 - 5*e^4 + 47*e^3 - 6*e^2 - 70*e - 8, -5*e^7 + 9*e^6 + 51*e^5 - 80*e^4 - 149*e^3 + 162*e^2 + 144*e - 60, -8*e^7 + 8*e^6 + 86*e^5 - 64*e^4 - 255*e^3 + 101*e^2 + 176*e - 56, 2*e^7 - 3*e^6 - 22*e^5 + 27*e^4 + 71*e^3 - 56*e^2 - 66*e + 10, -7*e^7 + 8*e^6 + 71*e^5 - 64*e^4 - 190*e^3 + 103*e^2 + 109*e - 42, e^6 + 3*e^5 - 13*e^4 - 26*e^3 + 45*e^2 + 40*e - 8, 6*e^7 - 8*e^6 - 59*e^5 + 60*e^4 + 149*e^3 - 74*e^2 - 78*e, -6*e^7 + 63*e^5 + 18*e^4 - 164*e^3 - 108*e^2 + 16*e + 24, e^7 + e^6 - 13*e^5 - 8*e^4 + 46*e^3 + 11*e^2 - 31*e + 22, -13/2*e^7 + 15/2*e^6 + 131/2*e^5 - 54*e^4 - 343/2*e^3 + 54*e^2 + 189/2*e - 17, -13/2*e^7 + 5/2*e^6 + 143/2*e^5 - 6*e^4 - 413/2*e^3 - 48*e^2 + 163/2*e - 19, 11/2*e^7 - 7/2*e^6 - 109/2*e^5 + 11*e^4 + 251/2*e^3 + 68*e^2 + 3/2*e - 35, -1/2*e^7 + 5/2*e^6 + 9/2*e^5 - 20*e^4 - 37/2*e^3 + 29*e^2 + 109/2*e + 11, -e^7 - 4*e^6 + 17*e^5 + 42*e^4 - 72*e^3 - 110*e^2 + 42*e + 16, -e^7 + 8*e^6 + 8*e^5 - 86*e^4 - 28*e^3 + 235*e^2 + 101*e - 62, -6*e^7 - 3*e^6 + 68*e^5 + 47*e^4 - 196*e^3 - 173*e^2 + 32*e + 16, 6*e^7 + e^6 - 64*e^5 - 28*e^4 + 170*e^3 + 129*e^2 - 16*e - 10, 2*e^7 - 6*e^6 - 13*e^5 + 46*e^4 - 5*e^3 - 65*e^2 + 62*e + 32, 2*e^7 + 7*e^6 - 25*e^5 - 77*e^4 + 70*e^3 + 209*e^2 + 43*e - 22, -5/2*e^7 + 7/2*e^6 + 41/2*e^5 - 23*e^4 - 45/2*e^3 + 15*e^2 - 107/2*e - 5, -3/2*e^7 - 9/2*e^6 + 43/2*e^5 + 43*e^4 - 161/2*e^3 - 101*e^2 + 79/2*e + 29, 3*e^7 - e^6 - 33*e^5 + 4*e^4 + 97*e^3 + 15*e^2 - 59*e - 6, -2*e^7 + 4*e^6 + 19*e^5 - 33*e^4 - 52*e^3 + 55*e^2 + 62*e - 6, 7*e^7 - 15*e^6 - 63*e^5 + 126*e^4 + 139*e^3 - 220*e^2 - 104*e + 48, -3*e^6 + 3*e^5 + 27*e^4 - 13*e^3 - 57*e^2 - 32*e, -3*e^7 + 6*e^6 + 32*e^5 - 52*e^4 - 103*e^3 + 98*e^2 + 111*e - 22, -3*e^7 + 2*e^6 + 30*e^5 - 11*e^4 - 78*e^3 - 2*e^2 + 44*e - 4, e^7 + 3*e^6 - 16*e^5 - 34*e^4 + 59*e^3 + 97*e^2 + 4*e - 22, 2*e^7 - 7*e^6 - 14*e^5 + 59*e^4 + 10*e^3 - 111*e^2 + 6*e + 32, 3/2*e^7 - 7/2*e^6 - 35/2*e^5 + 39*e^4 + 133/2*e^3 - 116*e^2 - 187/2*e + 31, 13/2*e^7 - 5/2*e^6 - 143/2*e^5 + 7*e^4 + 431/2*e^3 + 47*e^2 - 265/2*e - 5, 3*e^7 - 5*e^6 - 28*e^5 + 42*e^4 + 57*e^3 - 77*e^2 + 9*e + 42, -4*e^7 - 8*e^6 + 45*e^5 + 95*e^4 - 120*e^3 - 282*e^2 - 34*e + 32, 5/2*e^7 - 19/2*e^6 - 45/2*e^5 + 90*e^4 + 115/2*e^3 - 198*e^2 - 155/2*e + 37, -3/2*e^7 - 11/2*e^6 + 41/2*e^5 + 61*e^4 - 133/2*e^3 - 166*e^2 - 37/2*e + 33, -13/2*e^7 + 15/2*e^6 + 141/2*e^5 - 65*e^4 - 443/2*e^3 + 131*e^2 + 413/2*e - 59, -7/2*e^7 + 3/2*e^6 + 69/2*e^5 - 3*e^4 - 153/2*e^3 - 31*e^2 - 37/2*e - 21, 8*e^7 - 11*e^6 - 81*e^5 + 85*e^4 + 218*e^3 - 123*e^2 - 136*e + 72, 2*e^7 + e^6 - 16*e^5 - 23*e^4 + 9*e^3 + 114*e^2 + 74*e - 66, 6*e^7 - 4*e^6 - 61*e^5 + 23*e^4 + 156*e^3 + 5*e^2 - 40*e + 6, -2*e^7 + 4*e^6 + 14*e^5 - 22*e^4 + e^3 - 13*e^2 - 72*e + 14, 5*e^7 - 53*e^5 - 14*e^4 + 137*e^3 + 87*e^2 - 6*e - 32, -2*e^7 + 12*e^6 + 16*e^5 - 120*e^4 - 41*e^3 + 294*e^2 + 99*e - 70, 5*e^6 - 2*e^5 - 54*e^4 + e^3 + 151*e^2 + 68*e - 58, 2*e^5 - 4*e^4 - 16*e^3 + 24*e^2 + 32*e - 26, -6*e^7 + 11*e^6 + 58*e^5 - 88*e^4 - 146*e^3 + 130*e^2 + 102*e - 16, -4*e^7 + 6*e^6 + 41*e^5 - 55*e^4 - 119*e^3 + 127*e^2 + 96*e - 66, -e^7 + e^6 + 13*e^5 - 6*e^4 - 55*e^3 + 5*e^2 + 80*e - 10, -12*e^7 + 18*e^6 + 118*e^5 - 141*e^4 - 300*e^3 + 201*e^2 + 175*e - 50, 2*e^7 + 5*e^6 - 23*e^5 - 57*e^4 + 52*e^3 + 164*e^2 + 73*e - 26, -10*e^7 + 13*e^6 + 99*e^5 - 95*e^4 - 248*e^3 + 114*e^2 + 95*e - 54, 2*e^7 - 13*e^6 - 10*e^5 + 120*e^4 - 4*e^3 - 265*e^2 - 44*e + 72, -3*e^7 + 9*e^6 + 37*e^5 - 89*e^4 - 149*e^3 + 220*e^2 + 208*e - 82, 13/2*e^7 - 31/2*e^6 - 125/2*e^5 + 147*e^4 + 335/2*e^3 - 330*e^2 - 351/2*e + 85, -5/2*e^7 - 1/2*e^6 + 49/2*e^5 + 17*e^4 - 113/2*e^3 - 87*e^2 - 5/2*e + 43, -1/2*e^7 + 11/2*e^6 - 3/2*e^5 - 53*e^4 + 49/2*e^3 + 125*e^2 + 19/2*e - 23, -11/2*e^7 + 7/2*e^6 + 113/2*e^5 - 15*e^4 - 299/2*e^3 - 30*e^2 + 133/2*e - 23, -11*e^7 + 17*e^6 + 106*e^5 - 136*e^4 - 262*e^3 + 209*e^2 + 160*e - 60, -8*e^7 + 12*e^6 + 80*e^5 - 100*e^4 - 207*e^3 + 170*e^2 + 112*e - 56, 8*e^7 - 15*e^6 - 80*e^5 + 129*e^4 + 215*e^3 - 248*e^2 - 156*e + 82, -12*e^7 + 20*e^6 + 118*e^5 - 162*e^4 - 314*e^3 + 264*e^2 + 244*e - 76, 4*e^7 - e^6 - 43*e^5 + 6*e^4 + 120*e^3 - 5*e^2 - 47*e + 26, 2*e^7 - 13*e^6 - 13*e^5 + 123*e^4 + 20*e^3 - 269*e^2 - 95*e + 50, 7*e^6 - 3*e^5 - 76*e^4 + 5*e^3 + 210*e^2 + 82*e - 64, 7/2*e^7 - 9/2*e^6 - 67/2*e^5 + 36*e^4 + 169/2*e^3 - 61*e^2 - 123/2*e + 41, -11/2*e^7 + 13/2*e^6 + 121/2*e^5 - 49*e^4 - 379/2*e^3 + 60*e^2 + 331/2*e - 19, -e^7 + 5*e^6 + 13*e^5 - 56*e^4 - 65*e^3 + 165*e^2 + 144*e - 60, -6*e^7 + 7*e^6 + 57*e^5 - 52*e^4 - 134*e^3 + 62*e^2 + 70*e - 36, -13/2*e^7 - 7/2*e^6 + 147/2*e^5 + 51*e^4 - 423/2*e^3 - 181*e^2 + 89/2*e + 13, -31/2*e^7 + 41/2*e^6 + 311/2*e^5 - 160*e^4 - 825/2*e^3 + 235*e^2 + 491/2*e - 85, 5*e^7 + 7*e^6 - 55*e^5 - 94*e^4 + 132*e^3 + 320*e^2 + 92*e - 66, -2*e^7 + 9*e^6 + 17*e^5 - 86*e^4 - 51*e^3 + 198*e^2 + 116*e - 50, 5*e^7 - 7*e^6 - 44*e^5 + 52*e^4 + 79*e^3 - 57*e^2 + 11*e + 6, -6*e^7 + 9*e^6 + 60*e^5 - 71*e^4 - 161*e^3 + 106*e^2 + 114*e - 32, -7*e^7 + 10*e^6 + 76*e^5 - 82*e^4 - 233*e^3 + 141*e^2 + 180*e - 50, e^7 - 7*e^6 - 6*e^5 + 72*e^4 + 4*e^3 - 182*e^2 - 39*e + 38, 12*e^7 - 14*e^6 - 124*e^5 + 106*e^4 + 340*e^3 - 145*e^2 - 182*e + 64, -8*e^7 + 8*e^6 + 85*e^5 - 59*e^4 - 242*e^3 + 67*e^2 + 144*e - 20, 3*e^7 - 37*e^5 - 8*e^4 + 125*e^3 + 40*e^2 - 59*e + 10];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19,19,-1/2*w^2 - w + 5/2}>] := -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;