/*
  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![29, 8, -12, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, 2/5*w^3 - 14/5*w - 3/5}>;

primesArray := [
[5, 5, -w + 2],
[5, 5, -3/5*w^3 - w^2 + 26/5*w + 42/5],
[9, 3, 2/5*w^3 - 19/5*w - 3/5],
[9, 3, -1/5*w^3 + 2/5*w + 4/5],
[11, 11, -2/5*w^3 - w^2 + 14/5*w + 28/5],
[11, 11, 1/5*w^3 - w^2 - 7/5*w + 36/5],
[16, 2, 2],
[29, 29, -w],
[29, 29, 1/5*w^3 - 12/5*w + 1/5],
[41, 41, -2/5*w^3 - w^2 + 14/5*w + 38/5],
[41, 41, -w^2 + 10],
[41, 41, 11/5*w^3 + 3*w^2 - 102/5*w - 149/5],
[41, 41, -1/5*w^3 + w^2 + 7/5*w - 26/5],
[49, 7, 1/5*w^3 + w^2 - 12/5*w - 19/5],
[49, 7, 1/5*w^3 + w^2 - 12/5*w - 44/5],
[59, 59, -4/5*w^3 - w^2 + 33/5*w + 36/5],
[59, 59, -2*w^3 - 3*w^2 + 17*w + 26],
[61, 61, -1/5*w^3 + w^2 + 12/5*w - 51/5],
[61, 61, 4/5*w^3 + w^2 - 28/5*w - 41/5],
[71, 71, 1/5*w^3 + w^2 - 7/5*w - 49/5],
[71, 71, -2*w^3 - 3*w^2 + 18*w + 28],
[89, 89, -3/5*w^3 + 16/5*w + 7/5],
[89, 89, 3/5*w^3 + w^2 - 26/5*w - 57/5],
[121, 11, -3/5*w^3 + 21/5*w + 2/5],
[131, 131, w^2 - w - 8],
[131, 131, 2/5*w^3 + w^2 - 19/5*w - 23/5],
[149, 149, 2/5*w^3 - w^2 - 14/5*w + 22/5],
[149, 149, 3/5*w^3 + w^2 - 21/5*w - 42/5],
[179, 179, -6/5*w^3 - w^2 + 57/5*w + 69/5],
[179, 179, -2/5*w^3 - 2*w^2 + 14/5*w + 73/5],
[179, 179, 2*w^2 - 11],
[179, 179, -3/5*w^3 + 2*w^2 + 16/5*w - 48/5],
[181, 181, -7/5*w^3 - 2*w^2 + 59/5*w + 78/5],
[181, 181, -2/5*w^3 + 24/5*w - 27/5],
[191, 191, -3/5*w^3 + 2*w^2 + 21/5*w - 78/5],
[191, 191, 2/5*w^3 - 2*w^2 - 9/5*w + 57/5],
[199, 199, -1/5*w^3 + 17/5*w - 16/5],
[199, 199, 1/5*w^3 - 17/5*w - 14/5],
[211, 211, 2/5*w^3 - w^2 + 1/5*w + 7/5],
[211, 211, -1/5*w^3 + w^2 - 3/5*w - 26/5],
[211, 211, 4/5*w^3 + w^2 - 38/5*w - 36/5],
[211, 211, -1/5*w^3 - 2*w^2 + 7/5*w + 44/5],
[239, 239, 3/5*w^3 + w^2 - 16/5*w - 37/5],
[239, 239, 2/5*w^3 + w^2 - 14/5*w - 53/5],
[251, 251, 2/5*w^3 - 24/5*w - 13/5],
[251, 251, 2*w - 3],
[251, 251, w^2 + w - 10],
[251, 251, -4/5*w^3 - w^2 + 28/5*w + 46/5],
[269, 269, 1/5*w^3 - 12/5*w - 24/5],
[269, 269, w - 5],
[271, 271, -3/5*w^3 + 31/5*w - 33/5],
[271, 271, -8/5*w^3 - 2*w^2 + 66/5*w + 77/5],
[281, 281, 4/5*w^3 - w^2 - 38/5*w + 39/5],
[281, 281, -14/5*w^3 - 3*w^2 + 128/5*w + 161/5],
[311, 311, 1/5*w^3 - 2*w^2 - 2/5*w + 51/5],
[311, 311, 1/5*w^3 - w^2 - 17/5*w + 21/5],
[331, 331, 1/5*w^3 + 3/5*w - 4/5],
[331, 331, 3/5*w^3 - 31/5*w - 2/5],
[359, 359, -w^3 + 6*w - 1],
[359, 359, -6/5*w^3 + 47/5*w - 6/5],
[359, 359, 3/5*w^3 + w^2 - 31/5*w - 27/5],
[359, 359, 3/5*w^3 + w^2 - 26/5*w - 22/5],
[361, 19, 4/5*w^3 - 28/5*w - 11/5],
[361, 19, 1/5*w^3 - 7/5*w - 24/5],
[379, 379, 9/5*w^3 + 3*w^2 - 78/5*w - 121/5],
[379, 379, -4/5*w^3 + 3*w^2 + 28/5*w - 99/5],
[389, 389, -3/5*w^3 - 2*w^2 + 26/5*w + 42/5],
[389, 389, -2*w^2 + w + 17],
[401, 401, 4/5*w^3 - w^2 - 23/5*w + 24/5],
[401, 401, 3/5*w^3 + w^2 - 26/5*w - 7/5],
[401, 401, -4/5*w^3 - w^2 + 38/5*w + 66/5],
[401, 401, 6/5*w^3 + w^2 - 52/5*w - 39/5],
[409, 409, -6/5*w^3 + 52/5*w + 19/5],
[409, 409, 1/5*w^3 + 2*w^2 - 7/5*w - 39/5],
[421, 421, -1/5*w^3 - w^2 + 17/5*w + 44/5],
[421, 421, 2/5*w^3 + w^2 - 24/5*w - 18/5],
[431, 431, -3/5*w^3 - w^2 + 36/5*w + 47/5],
[431, 431, -13/5*w^3 - 4*w^2 + 116/5*w + 182/5],
[431, 431, -2/5*w^3 + w^2 + 24/5*w - 47/5],
[431, 431, 2/5*w^3 - w^2 + 6/5*w - 8/5],
[439, 439, 6/5*w^3 + w^2 - 47/5*w - 49/5],
[439, 439, -2/5*w^3 + 2*w^2 + 9/5*w - 42/5],
[439, 439, 4/5*w^3 - w^2 - 23/5*w + 14/5],
[439, 439, -w^3 - 2*w^2 + 8*w + 17],
[461, 461, w^3 + w^2 - 10*w - 9],
[461, 461, -2*w^3 - 3*w^2 + 17*w + 27],
[461, 461, -1/5*w^3 + w^2 - 8/5*w - 16/5],
[461, 461, -3*w^3 - 4*w^2 + 26*w + 35],
[479, 479, 2*w - 1],
[479, 479, 2/5*w^3 - 24/5*w - 3/5],
[491, 491, w^3 - 2*w^2 - 6*w + 11],
[491, 491, w^3 + w^2 - 10*w - 15],
[499, 499, -1/5*w^3 + 2*w^2 + 2/5*w - 36/5],
[499, 499, 4/5*w^3 + 2*w^2 - 33/5*w - 91/5],
[509, 509, 8/5*w^3 + w^2 - 66/5*w - 42/5],
[509, 509, 1/5*w^3 - 2*w^2 + 8/5*w + 16/5],
[521, 521, w^3 - 9*w - 4],
[521, 521, 3/5*w^3 + w^2 - 16/5*w - 42/5],
[521, 521, 3/5*w^3 - w^2 - 26/5*w + 23/5],
[521, 521, -3/5*w^3 + 11/5*w + 22/5],
[529, 23, 3/5*w^3 + w^2 - 36/5*w - 62/5],
[529, 23, -4/5*w^3 + 2*w^2 + 23/5*w - 64/5],
[541, 541, -w^3 - w^2 + 9*w + 7],
[541, 541, 1/5*w^3 + 2*w^2 - 7/5*w - 64/5],
[541, 541, 3/5*w^3 + w^2 - 26/5*w - 12/5],
[571, 571, -w^3 - w^2 + 6*w + 9],
[571, 571, -w^3 + w^2 + 8*w - 4],
[619, 619, -w^3 - 2*w^2 + 8*w + 11],
[619, 619, -2/5*w^3 + 2*w^2 + 9/5*w - 72/5],
[619, 619, 2/5*w^3 - 2*w^2 - 4/5*w + 57/5],
[619, 619, -6/5*w^3 - 2*w^2 + 52/5*w + 69/5],
[631, 631, 1/5*w^3 + w^2 + 3/5*w - 24/5],
[631, 631, 2/5*w^3 - w^2 - 24/5*w + 42/5],
[659, 659, -13/5*w^3 - 4*w^2 + 111/5*w + 177/5],
[659, 659, -11/5*w^3 - 3*w^2 + 92/5*w + 119/5],
[659, 659, -8/5*w^3 - w^2 + 61/5*w + 27/5],
[659, 659, -8/5*w^3 + 71/5*w + 2/5],
[661, 661, -w^3 - w^2 + 7*w + 11],
[661, 661, 4/5*w^3 - w^2 - 28/5*w + 9/5],
[691, 691, 1/5*w^3 - w^2 - 17/5*w + 41/5],
[691, 691, 3/5*w^3 + 2*w^2 - 26/5*w - 67/5],
[701, 701, 12/5*w^3 + 4*w^2 - 99/5*w - 158/5],
[701, 701, -17/5*w^3 - 4*w^2 + 149/5*w + 193/5],
[709, 709, -1/5*w^3 + w^2 + 12/5*w - 6/5],
[709, 709, 1/5*w^3 + w^2 - 2/5*w - 59/5],
[719, 719, 2/5*w^3 + w^2 - 29/5*w - 8/5],
[719, 719, -w^3 + w^2 + 7*w - 4],
[719, 719, 6/5*w^3 + w^2 - 42/5*w - 44/5],
[719, 719, -2/5*w^3 - w^2 + 29/5*w + 53/5],
[739, 739, 14/5*w^3 + 4*w^2 - 123/5*w - 171/5],
[739, 739, -1/5*w^3 + 2*w^2 + 12/5*w - 56/5],
[739, 739, -6/5*w^3 + 3*w^2 + 37/5*w - 96/5],
[739, 739, 2/5*w^3 + 2*w^2 - 9/5*w - 73/5],
[751, 751, -9/5*w^3 - 2*w^2 + 73/5*w + 86/5],
[751, 751, -w^3 + 2*w^2 + 5*w - 8],
[761, 761, 1/5*w^3 + w^2 - 22/5*w + 16/5],
[761, 761, -16/5*w^3 - 4*w^2 + 147/5*w + 214/5],
[769, 769, 13/5*w^3 + 3*w^2 - 121/5*w - 172/5],
[769, 769, 3/5*w^3 - w^2 - 41/5*w + 78/5],
[769, 769, -12/5*w^3 - 4*w^2 + 99/5*w + 153/5],
[769, 769, -6/5*w^3 + 3*w^2 + 27/5*w - 61/5],
[809, 809, 1/5*w^3 + 3*w^2 - 22/5*w - 104/5],
[809, 809, w^3 + 3*w^2 - 10*w - 17],
[811, 811, -5*w + 11],
[811, 811, -16/5*w^3 - 5*w^2 + 137/5*w + 214/5],
[821, 821, -1/5*w^3 + w^2 + 17/5*w - 36/5],
[821, 821, w^2 + 2*w - 6],
[829, 829, 3*w^2 + w - 17],
[829, 829, 2/5*w^3 + 3*w^2 - 9/5*w - 108/5],
[841, 29, w^3 - 7*w - 3],
[859, 859, 6/5*w^3 + w^2 - 37/5*w - 34/5],
[859, 859, 2/5*w^3 + w^2 - 14/5*w - 68/5],
[881, 881, 1/5*w^3 + 2*w^2 - 17/5*w - 79/5],
[881, 881, 3/5*w^3 + 2*w^2 - 31/5*w - 47/5],
[911, 911, 4/5*w^3 - 3*w^2 - 28/5*w + 89/5],
[911, 911, 1/5*w^3 - w^2 - 7/5*w + 61/5],
[911, 911, w^3 - 6*w - 3],
[911, 911, -11/5*w^3 - 3*w^2 + 92/5*w + 139/5],
[919, 919, 7/5*w^3 + 2*w^2 - 54/5*w - 88/5],
[919, 919, 4/5*w^3 - 2*w^2 - 23/5*w + 39/5],
[929, 929, -7/5*w^3 + w^2 + 54/5*w - 17/5],
[929, 929, -1/5*w^3 + w^2 + 22/5*w - 71/5],
[941, 941, 9/5*w^3 + w^2 - 73/5*w - 31/5],
[941, 941, 1/5*w^3 - w^2 - 2/5*w + 66/5],
[941, 941, 2/5*w^3 + 2*w^2 - 19/5*w - 103/5],
[941, 941, -w^3 + w^2 + 5*w - 3],
[961, 31, w^3 - 7*w - 2],
[961, 31, 2/5*w^3 - 14/5*w - 33/5],
[971, 971, -2/5*w^3 - 2*w^2 + 24/5*w + 93/5],
[971, 971, 2/5*w^3 + 2*w^2 - 24/5*w - 33/5],
[991, 991, -w^3 + 2*w^2 + 7*w - 10],
[991, 991, -7/5*w^3 - 2*w^2 + 49/5*w + 78/5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 2*x^5 - 18*x^4 + 30*x^3 + 85*x^2 - 84*x - 100;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, -1, e, e, 1/4*e^5 + 1/4*e^4 - 15/4*e^3 - 11/4*e^2 + 12*e + 5, 1/4*e^5 + 1/4*e^4 - 15/4*e^3 - 11/4*e^2 + 12*e + 5, 3/4*e^5 + 5/4*e^4 - 43/4*e^3 - 63/4*e^2 + 57/2*e + 35, -1/4*e^5 - 1/4*e^4 + 11/4*e^3 + 11/4*e^2 - 2*e - 5, -1/4*e^5 - 1/4*e^4 + 11/4*e^3 + 11/4*e^2 - 2*e - 5, -1/4*e^5 - 3/4*e^4 + 11/4*e^3 + 37/4*e^2 - 3*e - 15, 1/2*e^4 - 11/2*e^2 + 8, 1/2*e^4 - 11/2*e^2 + 8, -1/4*e^5 - 3/4*e^4 + 11/4*e^3 + 37/4*e^2 - 3*e - 15, -e^5 - e^4 + 14*e^3 + 12*e^2 - 35*e - 20, -e^5 - e^4 + 14*e^3 + 12*e^2 - 35*e - 20, -1/2*e^5 - e^4 + 15/2*e^3 + 15*e^2 - 24*e - 40, -1/2*e^5 - e^4 + 15/2*e^3 + 15*e^2 - 24*e - 40, 5/4*e^5 + 5/4*e^4 - 71/4*e^3 - 59/4*e^2 + 45*e + 25, 5/4*e^5 + 5/4*e^4 - 71/4*e^3 - 59/4*e^2 + 45*e + 25, e^2 - 10, e^2 - 10, 1/2*e^4 - 13/2*e^2 + e + 10, 1/2*e^4 - 13/2*e^2 + e + 10, -1/4*e^5 - 3/4*e^4 + 11/4*e^3 + 37/4*e^2 - 4*e - 3, 7/4*e^5 + 7/4*e^4 - 101/4*e^3 - 93/4*e^2 + 72*e + 55, 7/4*e^5 + 7/4*e^4 - 101/4*e^3 - 93/4*e^2 + 72*e + 55, -1/4*e^5 - 1/4*e^4 + 19/4*e^3 + 19/4*e^2 - 21*e - 15, -1/4*e^5 - 1/4*e^4 + 19/4*e^3 + 19/4*e^2 - 21*e - 15, -5/2*e^5 - 2*e^4 + 71/2*e^3 + 24*e^2 - 91*e - 50, -7/4*e^5 - 5/4*e^4 + 97/4*e^3 + 55/4*e^2 - 59*e - 25, -7/4*e^5 - 5/4*e^4 + 97/4*e^3 + 55/4*e^2 - 59*e - 25, -5/2*e^5 - 2*e^4 + 71/2*e^3 + 24*e^2 - 91*e - 50, -2*e^5 - 2*e^4 + 27*e^3 + 25*e^2 - 59*e - 50, -2*e^5 - 2*e^4 + 27*e^3 + 25*e^2 - 59*e - 50, e^5 + 2*e^4 - 14*e^3 - 23*e^2 + 34*e + 30, e^5 + 2*e^4 - 14*e^3 - 23*e^2 + 34*e + 30, 7/4*e^5 + 13/4*e^4 - 97/4*e^3 - 167/4*e^2 + 59*e + 85, 7/4*e^5 + 13/4*e^4 - 97/4*e^3 - 167/4*e^2 + 59*e + 85, -2*e^5 - 3*e^4 + 29*e^3 + 39*e^2 - 80*e - 88, 1/2*e^5 + 3/2*e^4 - 13/2*e^3 - 39/2*e^2 + 10*e + 38, 1/2*e^5 + 3/2*e^4 - 13/2*e^3 - 39/2*e^2 + 10*e + 38, -2*e^5 - 3*e^4 + 29*e^3 + 39*e^2 - 80*e - 88, -5/4*e^5 - 3/4*e^4 + 67/4*e^3 + 45/4*e^2 - 37*e - 35, -5/4*e^5 - 3/4*e^4 + 67/4*e^3 + 45/4*e^2 - 37*e - 35, -3/4*e^5 - 5/4*e^4 + 45/4*e^3 + 59/4*e^2 - 32*e - 23, -3/4*e^5 - 5/4*e^4 + 45/4*e^3 + 59/4*e^2 - 32*e - 23, -9/4*e^5 - 5/4*e^4 + 123/4*e^3 + 59/4*e^2 - 72*e - 27, -9/4*e^5 - 5/4*e^4 + 123/4*e^3 + 59/4*e^2 - 72*e - 27, -1/4*e^5 - 5/4*e^4 + 11/4*e^3 + 51/4*e^2 - 2*e - 5, -1/4*e^5 - 5/4*e^4 + 11/4*e^3 + 51/4*e^2 - 2*e - 5, 3/2*e^5 + e^4 - 39/2*e^3 - 12*e^2 + 40*e + 28, 3/2*e^5 + e^4 - 39/2*e^3 - 12*e^2 + 40*e + 28, -7/4*e^5 - 7/4*e^4 + 97/4*e^3 + 81/4*e^2 - 56*e - 43, -7/4*e^5 - 7/4*e^4 + 97/4*e^3 + 81/4*e^2 - 56*e - 43, 2*e^5 + 3/2*e^4 - 28*e^3 - 35/2*e^2 + 66*e + 38, 2*e^5 + 3/2*e^4 - 28*e^3 - 35/2*e^2 + 66*e + 38, -9/4*e^5 - 7/4*e^4 + 131/4*e^3 + 89/4*e^2 - 90*e - 57, -9/4*e^5 - 7/4*e^4 + 131/4*e^3 + 89/4*e^2 - 90*e - 57, 2*e^5 + e^4 - 28*e^3 - 14*e^2 + 68*e + 40, 2*e^5 + e^4 - 28*e^3 - 14*e^2 + 68*e + 40, -5/2*e^5 - 3*e^4 + 71/2*e^3 + 38*e^2 - 95*e - 70, -5/2*e^5 - 3*e^4 + 71/2*e^3 + 38*e^2 - 95*e - 70, -e^4 + 15*e^2 - e - 10, -e^5 - 1/2*e^4 + 15*e^3 + 13/2*e^2 - 43*e, -5/4*e^5 - 5/4*e^4 + 71/4*e^3 + 67/4*e^2 - 52*e - 55, -5/4*e^5 - 5/4*e^4 + 71/4*e^3 + 67/4*e^2 - 52*e - 55, 1/2*e^5 - e^4 - 13/2*e^3 + 14*e^2 + 16*e - 30, 1/2*e^5 - e^4 - 13/2*e^3 + 14*e^2 + 16*e - 30, -5/2*e^5 - 3/2*e^4 + 69/2*e^3 + 41/2*e^2 - 85*e - 60, -5/2*e^5 - 4*e^4 + 69/2*e^3 + 51*e^2 - 90*e - 102, -5/2*e^5 - 3/2*e^4 + 69/2*e^3 + 41/2*e^2 - 85*e - 60, -5/2*e^5 - 4*e^4 + 69/2*e^3 + 51*e^2 - 90*e - 102, -e^5 + 14*e^3 - 3*e^2 - 33*e, -e^5 + 14*e^3 - 3*e^2 - 33*e, 3/4*e^5 - 1/4*e^4 - 49/4*e^3 - 1/4*e^2 + 45*e + 15, 3/4*e^5 - 1/4*e^4 - 49/4*e^3 - 1/4*e^2 + 45*e + 15, -3/2*e^5 - e^4 + 43/2*e^3 + 10*e^2 - 54*e - 10, 7/2*e^5 + 5*e^4 - 99/2*e^3 - 64*e^2 + 132*e + 130, -3/2*e^5 - e^4 + 43/2*e^3 + 10*e^2 - 54*e - 10, 7/2*e^5 + 5*e^4 - 99/2*e^3 - 64*e^2 + 132*e + 130, 1/4*e^5 + 5/4*e^4 - 11/4*e^3 - 63/4*e^2 + 9*e + 25, -e^5 - 3/2*e^4 + 14*e^3 + 39/2*e^2 - 32*e - 50, 1/4*e^5 + 5/4*e^4 - 11/4*e^3 - 63/4*e^2 + 9*e + 25, -e^5 - 3/2*e^4 + 14*e^3 + 39/2*e^2 - 32*e - 50, 9/2*e^5 + 11/2*e^4 - 123/2*e^3 - 137/2*e^2 + 143*e + 140, -e^4 - e^3 + 8*e^2 + 9*e + 10, 9/2*e^5 + 11/2*e^4 - 123/2*e^3 - 137/2*e^2 + 143*e + 140, -e^4 - e^3 + 8*e^2 + 9*e + 10, -3/4*e^5 + 3/4*e^4 + 41/4*e^3 - 37/4*e^2 - 28*e + 5, -3/4*e^5 + 3/4*e^4 + 41/4*e^3 - 37/4*e^2 - 28*e + 5, -1/2*e^5 + 9/2*e^3 - 5*e^2 + 6*e + 30, -1/2*e^5 + 9/2*e^3 - 5*e^2 + 6*e + 30, 7/2*e^5 + 9/2*e^4 - 97/2*e^3 - 109/2*e^2 + 121*e + 100, 7/2*e^5 + 9/2*e^4 - 97/2*e^3 - 109/2*e^2 + 121*e + 100, -1/2*e^5 + 13/2*e^3 - e^2 - 12*e + 10, -1/2*e^5 + 13/2*e^3 - e^2 - 12*e + 10, -2*e^5 - 3/2*e^4 + 27*e^3 + 39/2*e^2 - 59*e - 40, 1/2*e^5 - 13/2*e^3 - 3*e^2 + 10*e + 22, 1/2*e^5 - 13/2*e^3 - 3*e^2 + 10*e + 22, -2*e^5 - 3/2*e^4 + 27*e^3 + 39/2*e^2 - 59*e - 40, -e^4 + 9*e^2 - 4*e - 10, -e^4 + 9*e^2 - 4*e - 10, -3/2*e^5 - 2*e^4 + 43/2*e^3 + 29*e^2 - 55*e - 90, 7/2*e^5 + 9/2*e^4 - 105/2*e^3 - 107/2*e^2 + 158*e + 108, -3/2*e^5 - 2*e^4 + 43/2*e^3 + 29*e^2 - 55*e - 90, -1/4*e^5 - 7/4*e^4 + 15/4*e^3 + 89/4*e^2 - 16*e - 53, -1/4*e^5 - 7/4*e^4 + 15/4*e^3 + 89/4*e^2 - 16*e - 53, 1/4*e^5 + 1/4*e^4 - 11/4*e^3 - 23/4*e^2 + 3*e + 25, 1/4*e^5 + 1/4*e^4 - 11/4*e^3 - 23/4*e^2 + 3*e + 25, -3/4*e^5 - 9/4*e^4 + 53/4*e^3 + 127/4*e^2 - 49*e - 85, -3/4*e^5 - 9/4*e^4 + 53/4*e^3 + 127/4*e^2 - 49*e - 85, 3/2*e^5 + 3/2*e^4 - 41/2*e^3 - 37/2*e^2 + 48*e + 18, 3/2*e^5 + 3/2*e^4 - 41/2*e^3 - 37/2*e^2 + 48*e + 18, e^5 + 4*e^4 - 12*e^3 - 50*e^2 + 18*e + 80, e^5 + 4*e^4 - 12*e^3 - 50*e^2 + 18*e + 80, -2*e^5 - 1/2*e^4 + 28*e^3 + 15/2*e^2 - 70*e - 30, -2*e^5 - 1/2*e^4 + 28*e^3 + 15/2*e^2 - 70*e - 30, 7/4*e^5 + 5/4*e^4 - 97/4*e^3 - 55/4*e^2 + 55*e + 5, 7/4*e^5 + 5/4*e^4 - 97/4*e^3 - 55/4*e^2 + 55*e + 5, e^5 - 1/2*e^4 - 14*e^3 + 7/2*e^2 + 32*e + 20, e^5 - 1/2*e^4 - 14*e^3 + 7/2*e^2 + 32*e + 20, -3*e^5 - 9/2*e^4 + 42*e^3 + 121/2*e^2 - 110*e - 148, -3*e^5 - 9/2*e^4 + 42*e^3 + 121/2*e^2 - 110*e - 148, -3/2*e^5 - 3*e^4 + 45/2*e^3 + 35*e^2 - 62*e - 50, -3/2*e^5 - 3*e^4 + 45/2*e^3 + 35*e^2 - 62*e - 50, -9/2*e^5 - 5*e^4 + 125/2*e^3 + 60*e^2 - 150*e - 120, -3/2*e^5 - 3*e^4 + 47/2*e^3 + 39*e^2 - 78*e - 80, -3/2*e^5 - 3*e^4 + 47/2*e^3 + 39*e^2 - 78*e - 80, -9/2*e^5 - 5*e^4 + 125/2*e^3 + 60*e^2 - 150*e - 120, e^5 + e^4 - 12*e^3 - 13*e^2 + 27*e + 30, -e^5 + 1/2*e^4 + 16*e^3 - 9/2*e^2 - 48*e - 10, e^5 + e^4 - 12*e^3 - 13*e^2 + 27*e + 30, -e^5 + 1/2*e^4 + 16*e^3 - 9/2*e^2 - 48*e - 10, 7/4*e^5 + 11/4*e^4 - 93/4*e^3 - 133/4*e^2 + 48*e + 73, 7/4*e^5 + 11/4*e^4 - 93/4*e^3 - 133/4*e^2 + 48*e + 73, 3/4*e^5 - 1/4*e^4 - 45/4*e^3 + 11/4*e^2 + 24*e - 13, 3/4*e^5 - 1/4*e^4 - 45/4*e^3 + 11/4*e^2 + 24*e - 13, -5/2*e^5 - 5/2*e^4 + 67/2*e^3 + 57/2*e^2 - 71*e - 50, -3*e^5 - 4*e^4 + 42*e^3 + 52*e^2 - 113*e - 120, -3*e^5 - 4*e^4 + 42*e^3 + 52*e^2 - 113*e - 120, -5/2*e^5 - 5/2*e^4 + 67/2*e^3 + 57/2*e^2 - 71*e - 50, 5/2*e^5 + 2*e^4 - 77/2*e^3 - 22*e^2 + 119*e + 40, 5/2*e^5 + 2*e^4 - 77/2*e^3 - 22*e^2 + 119*e + 40, -2*e^5 - 2*e^4 + 28*e^3 + 21*e^2 - 66*e - 12, -2*e^5 - 2*e^4 + 28*e^3 + 21*e^2 - 66*e - 12, -1/2*e^4 + 2*e^3 + 17/2*e^2 - 29*e - 20, -1/2*e^4 + 2*e^3 + 17/2*e^2 - 29*e - 20, -5/2*e^5 - 4*e^4 + 73/2*e^3 + 55*e^2 - 107*e - 140, -5/2*e^5 - 4*e^4 + 73/2*e^3 + 55*e^2 - 107*e - 140, -e^3 - 2*e^2 + 3*e + 50, 1/4*e^5 + 9/4*e^4 - 11/4*e^3 - 99/4*e^2 + 7*e + 25, 1/4*e^5 + 9/4*e^4 - 11/4*e^3 - 99/4*e^2 + 7*e + 25, 2*e^4 + 3*e^3 - 24*e^2 - 25*e + 30, 2*e^4 + 3*e^3 - 24*e^2 - 25*e + 30, -3/4*e^5 + 3/4*e^4 + 45/4*e^3 - 57/4*e^2 - 26*e + 37, -7/4*e^5 - 7/4*e^4 + 101/4*e^3 + 85/4*e^2 - 62*e - 37, -7/4*e^5 - 7/4*e^4 + 101/4*e^3 + 85/4*e^2 - 62*e - 37, -3/4*e^5 + 3/4*e^4 + 45/4*e^3 - 57/4*e^2 - 26*e + 37, 7/2*e^5 + 2*e^4 - 97/2*e^3 - 24*e^2 + 118*e + 60, 7/2*e^5 + 2*e^4 - 97/2*e^3 - 24*e^2 + 118*e + 60, -2*e^5 - 5/2*e^4 + 26*e^3 + 51/2*e^2 - 43*e - 30, -2*e^5 - 5/2*e^4 + 26*e^3 + 51/2*e^2 - 43*e - 30, 11/4*e^5 + 15/4*e^4 - 169/4*e^3 - 177/4*e^2 + 132*e + 67, -11/4*e^5 - 17/4*e^4 + 157/4*e^3 + 219/4*e^2 - 106*e - 117, 11/4*e^5 + 15/4*e^4 - 169/4*e^3 - 177/4*e^2 + 132*e + 67, -11/4*e^5 - 17/4*e^4 + 157/4*e^3 + 219/4*e^2 - 106*e - 117, -17/4*e^5 - 25/4*e^4 + 235/4*e^3 + 307/4*e^2 - 150*e - 113, 5*e^5 + 15/2*e^4 - 69*e^3 - 191/2*e^2 + 168*e + 212, -3/4*e^5 - 7/4*e^4 + 37/4*e^3 + 81/4*e^2 - 8*e - 15, -3/4*e^5 - 7/4*e^4 + 37/4*e^3 + 81/4*e^2 - 8*e - 15, 5/2*e^5 + 3*e^4 - 65/2*e^3 - 36*e^2 + 66*e + 70, 5/2*e^5 + 3*e^4 - 65/2*e^3 - 36*e^2 + 66*e + 70];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, -w + 2}>] := 1;
ALEigenvalues[ideal<ZF | {5, 5, -3/5*w^3 - w^2 + 26/5*w + 42/5}>] := 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;