/*
  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![9, 3, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {61, 61, 1/3*w^3 - 4/3*w^2 + 2/3*w + 4}>;

primesArray := [
[5, 5, -1/3*w^3 - 2/3*w^2 + 7/3*w + 4],
[5, 5, 1/3*w^3 - 4/3*w^2 - 1/3*w + 3],
[9, 3, -w],
[9, 3, 1/3*w^3 - 1/3*w^2 - 7/3*w + 1],
[16, 2, 2],
[19, 19, 2/3*w^3 - 2/3*w^2 - 11/3*w],
[19, 19, -1/3*w^3 + 1/3*w^2 + 1/3*w + 1],
[31, 31, 2/3*w^3 + 1/3*w^2 - 11/3*w - 2],
[31, 31, 2/3*w^3 - 5/3*w^2 - 5/3*w + 5],
[41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 6],
[41, 41, -1/3*w^3 + 4/3*w^2 + 7/3*w - 3],
[61, 61, 1/3*w^3 - 4/3*w^2 + 2/3*w + 4],
[61, 61, 2/3*w^3 + 1/3*w^2 - 14/3*w - 2],
[71, 71, -1/3*w^3 + 4/3*w^2 + 1/3*w - 7],
[71, 71, 1/3*w^3 + 2/3*w^2 - 7/3*w],
[89, 89, -1/3*w^3 + 1/3*w^2 + 10/3*w - 3],
[89, 89, 1/3*w^3 - 1/3*w^2 - 10/3*w - 1],
[101, 101, 1/3*w^3 - 1/3*w^2 - 7/3*w - 3],
[101, 101, -2/3*w^3 + 5/3*w^2 + 11/3*w - 4],
[101, 101, 2/3*w^3 - 5/3*w^2 - 8/3*w + 3],
[101, 101, w - 4],
[109, 109, 2/3*w^3 + 1/3*w^2 - 8/3*w - 4],
[109, 109, -w^3 + 2*w^2 + 4*w - 4],
[121, 11, w^3 - w^2 - 4*w + 1],
[121, 11, -w^3 + w^2 + 4*w - 2],
[131, 131, 4/3*w^3 - 1/3*w^2 - 22/3*w - 1],
[131, 131, -w^3 + 7*w + 2],
[139, 139, 2/3*w^3 - 2/3*w^2 - 14/3*w + 1],
[139, 139, 2*w - 1],
[139, 139, 1/3*w^3 + 5/3*w^2 - 7/3*w - 10],
[139, 139, -2/3*w^3 + 8/3*w^2 + 5/3*w - 5],
[149, 149, w^3 + w^2 - 7*w - 11],
[149, 149, -w^3 - w^2 + 5*w + 4],
[151, 151, -2/3*w^3 - 4/3*w^2 + 14/3*w + 5],
[151, 151, 2/3*w^3 - 8/3*w^2 - 2/3*w + 9],
[169, 13, -2/3*w^3 - 1/3*w^2 + 11/3*w],
[169, 13, -2/3*w^3 + 5/3*w^2 + 5/3*w - 7],
[179, 179, 4/3*w^3 - 7/3*w^2 - 13/3*w + 6],
[179, 179, -4/3*w^3 + 1/3*w^2 + 19/3*w + 1],
[181, 181, 5/3*w^3 - 8/3*w^2 - 20/3*w + 5],
[181, 181, -1/3*w^3 + 1/3*w^2 + 10/3*w - 1],
[181, 181, -4/3*w^3 + 1/3*w^2 + 16/3*w + 3],
[191, 191, w^2 - 3*w - 1],
[191, 191, -1/3*w^3 + 4/3*w^2 + 10/3*w - 3],
[199, 199, -w^3 + 3*w^2 + w - 7],
[199, 199, -4/3*w^3 + 1/3*w^2 + 19/3*w - 1],
[211, 211, -1/3*w^3 - 5/3*w^2 + 4/3*w + 8],
[211, 211, w^3 - 3*w^2 - 4*w + 8],
[229, 229, -4/3*w^3 + 4/3*w^2 + 13/3*w - 4],
[229, 229, -5/3*w^3 + 5/3*w^2 + 23/3*w - 5],
[229, 229, 1/3*w^3 - 7/3*w^2 + 2/3*w + 10],
[229, 229, -2/3*w^3 + 2/3*w^2 + 2/3*w - 3],
[239, 239, 4/3*w^3 - 1/3*w^2 - 19/3*w - 2],
[239, 239, -2/3*w^3 + 2/3*w^2 + 11/3*w - 6],
[251, 251, 2*w^2 - w - 8],
[251, 251, -1/3*w^3 - 2/3*w^2 + 7/3*w - 2],
[251, 251, 1/3*w^3 - 4/3*w^2 - 1/3*w + 9],
[251, 251, -1/3*w^3 + 7/3*w^2 + 1/3*w - 7],
[269, 269, -5/3*w^3 + 8/3*w^2 + 17/3*w - 8],
[269, 269, -5/3*w^3 + 11/3*w^2 + 17/3*w - 12],
[271, 271, 1/3*w^3 + 2/3*w^2 - 13/3*w - 3],
[271, 271, 1/3*w^3 + 2/3*w^2 - 13/3*w - 2],
[311, 311, -1/3*w^3 + 7/3*w^2 - 5/3*w - 8],
[311, 311, -w^3 + 5*w - 1],
[331, 331, -2/3*w^3 - 1/3*w^2 + 8/3*w + 6],
[331, 331, -w^3 + 2*w^2 + 4*w - 2],
[349, 349, -w^3 + 2*w^2 + w + 1],
[349, 349, 1/3*w^3 - 1/3*w^2 - 7/3*w - 4],
[349, 349, -4/3*w^3 + 13/3*w^2 + 10/3*w - 11],
[349, 349, w - 5],
[361, 19, -4/3*w^3 + 4/3*w^2 + 16/3*w - 3],
[379, 379, 7/3*w^3 - 1/3*w^2 - 43/3*w - 9],
[379, 379, 2*w^3 - 3*w^2 - 9*w + 8],
[401, 401, 1/3*w^3 + 2/3*w^2 - 7/3*w - 8],
[401, 401, -1/3*w^3 + 10/3*w^2 - 2/3*w - 15],
[401, 401, -4/3*w^3 + 4/3*w^2 + 13/3*w - 3],
[401, 401, -1/3*w^3 + 4/3*w^2 - 5/3*w + 3],
[409, 409, -5/3*w^3 + 8/3*w^2 + 20/3*w - 7],
[409, 409, 4/3*w^3 - 1/3*w^2 - 16/3*w - 1],
[419, 419, 4/3*w^3 - 16/3*w^2 - 1/3*w + 10],
[419, 419, 5/3*w^3 + 7/3*w^2 - 35/3*w - 17],
[421, 421, -2*w^3 + 4*w^2 + 7*w - 10],
[421, 421, -5/3*w^3 + 8/3*w^2 + 29/3*w - 7],
[421, 421, 1/3*w^3 + 2/3*w^2 + 5/3*w - 4],
[421, 421, -5/3*w^3 - 1/3*w^2 + 23/3*w + 5],
[431, 431, -2/3*w^3 + 5/3*w^2 + 11/3*w - 1],
[431, 431, -w^2 - w + 8],
[439, 439, -2/3*w^3 + 11/3*w^2 - 1/3*w - 8],
[439, 439, -2/3*w^3 + 11/3*w^2 + 8/3*w - 15],
[479, 479, w^3 + 2*w^2 - 6*w - 10],
[479, 479, -5/3*w^3 + 5/3*w^2 + 14/3*w],
[479, 479, -5/3*w^3 + 5/3*w^2 + 23/3*w - 3],
[479, 479, -4/3*w^3 + 4/3*w^2 + 13/3*w - 2],
[521, 521, -w^3 + 2*w^2 + 5*w + 1],
[521, 521, 1/3*w^3 + 2/3*w^2 - 1/3*w - 10],
[541, 541, 1/3*w^3 + 5/3*w^2 + 2/3*w - 7],
[541, 541, 5/3*w^3 - 5/3*w^2 - 26/3*w],
[569, 569, -w - 5],
[569, 569, 1/3*w^3 - 1/3*w^2 - 7/3*w + 6],
[571, 571, w^3 - 4*w - 8],
[571, 571, -1/3*w^3 + 7/3*w^2 + 7/3*w - 4],
[599, 599, 5/3*w^3 - 5/3*w^2 - 14/3*w + 4],
[599, 599, -4/3*w^3 + 13/3*w^2 + 13/3*w - 15],
[599, 599, 2*w^3 - 3*w^2 - 8*w + 5],
[599, 599, 7/3*w^3 - 7/3*w^2 - 34/3*w + 6],
[601, 601, -4/3*w^3 + 13/3*w^2 + 13/3*w - 13],
[601, 601, -w^3 + 8*w + 5],
[619, 619, -5/3*w^3 + 11/3*w^2 + 17/3*w - 13],
[619, 619, 4/3*w^3 + 2/3*w^2 - 19/3*w - 2],
[631, 631, w^3 - w^2 - 7*w + 1],
[631, 631, 3*w - 2],
[641, 641, -2/3*w^3 + 5/3*w^2 + 2/3*w - 7],
[641, 641, -1/3*w^3 + 7/3*w^2 - 5/3*w - 9],
[661, 661, -5/3*w^3 + 11/3*w^2 + 20/3*w - 9],
[661, 661, w^3 + w^2 - 4*w - 7],
[691, 691, -5/3*w^3 + 8/3*w^2 + 17/3*w - 6],
[691, 691, 2*w^2 - 3*w - 10],
[691, 691, 5/3*w^3 - 2/3*w^2 - 23/3*w - 1],
[691, 691, 1/3*w^3 + 5/3*w^2 - 13/3*w - 3],
[701, 701, 2/3*w^3 + 4/3*w^2 - 20/3*w - 11],
[701, 701, -2/3*w^3 - 1/3*w^2 + 17/3*w + 9],
[709, 709, -2/3*w^3 - 7/3*w^2 + 17/3*w + 11],
[709, 709, -1/3*w^3 + 10/3*w^2 - 2/3*w - 8],
[709, 709, -2/3*w^3 + 11/3*w^2 - 1/3*w - 10],
[709, 709, 3*w^2 - 2*w - 14],
[719, 719, 7/3*w^3 + 8/3*w^2 - 46/3*w - 21],
[719, 719, -2*w^3 + 7*w^2 + 2*w - 13],
[719, 719, 2/3*w^3 + 7/3*w^2 - 14/3*w - 9],
[719, 719, -w^3 + 4*w^2 + 2*w - 13],
[761, 761, 1/3*w^3 + 5/3*w^2 - 1/3*w - 8],
[761, 761, 5/3*w^3 - 2/3*w^2 - 14/3*w + 1],
[761, 761, -8/3*w^3 + 11/3*w^2 + 38/3*w - 11],
[761, 761, -4/3*w^3 + 10/3*w^2 + 19/3*w - 9],
[809, 809, -2/3*w^3 + 11/3*w^2 + 5/3*w - 15],
[809, 809, 3*w^2 - w - 8],
[811, 811, 2/3*w^3 + 7/3*w^2 - 14/3*w - 11],
[811, 811, 1/3*w^3 + 8/3*w^2 - 13/3*w - 10],
[811, 811, 1/3*w^3 - 10/3*w^2 + 5/3*w + 11],
[811, 811, -w^3 + 4*w^2 + 2*w - 11],
[821, 821, 2*w^3 - w^2 - 12*w - 5],
[821, 821, 1/3*w^3 + 5/3*w^2 - 7/3*w - 14],
[829, 829, -2/3*w^3 + 11/3*w^2 + 5/3*w - 10],
[829, 829, 3*w^2 - w - 13],
[839, 839, 4/3*w^3 - 1/3*w^2 - 25/3*w + 1],
[839, 839, -2/3*w^3 + 5/3*w^2 - 1/3*w - 6],
[841, 29, 5/3*w^3 - 5/3*w^2 - 20/3*w + 1],
[841, 29, -5/3*w^3 + 5/3*w^2 + 20/3*w - 4],
[859, 859, -1/3*w^3 - 8/3*w^2 + 13/3*w + 11],
[859, 859, -1/3*w^3 + 10/3*w^2 - 5/3*w - 10],
[859, 859, 2/3*w^3 - 8/3*w^2 + 1/3*w + 9],
[859, 859, -2/3*w^3 + 5/3*w^2 + 2/3*w - 8],
[881, 881, -4/3*w^3 + 16/3*w^2 + 4/3*w - 11],
[881, 881, 1/3*w^3 - 13/3*w^2 - 4/3*w + 18],
[911, 911, -1/3*w^3 + 10/3*w^2 + 1/3*w - 14],
[911, 911, -1/3*w^3 + 10/3*w^2 + 1/3*w - 9],
[929, 929, 4/3*w^3 - 7/3*w^2 - 22/3*w + 4],
[929, 929, 1/3*w^3 + 2/3*w^2 + 2/3*w - 6],
[961, 31, 5/3*w^3 - 5/3*w^2 - 20/3*w + 2],
[971, 971, -2/3*w^3 + 11/3*w^2 + 2/3*w - 13],
[971, 971, 1/3*w^3 + 8/3*w^2 - 10/3*w - 9],
[991, 991, -4/3*w^3 + 10/3*w^2 + 16/3*w - 17],
[991, 991, -2*w^3 + 4*w^2 + 11*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 2*x^5 - 15*x^4 - 40*x^3 - 20*x^2 + 12*x + 7;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1/2*e^5 + 8*e^3 + 4*e^2 - 13/2*e + 1/2, 4*e^5 + 3*e^4 - 127/2*e^3 - 161/2*e^2 + 16*e + 43/2, 3/2*e^5 + 1/2*e^4 - 24*e^3 - 41/2*e^2 + 15*e + 4, 3/2*e^5 + e^4 - 24*e^3 - 28*e^2 + 17/2*e + 13/2, -11/2*e^5 - 9/2*e^4 + 87*e^3 + 233/2*e^2 - 15*e - 33, -9/2*e^5 - 7/2*e^4 + 143/2*e^3 + 93*e^2 - 20*e - 61/2, -e^5 - e^4 + 16*e^3 + 24*e^2 - 5*e - 11, -1/2*e^5 + e^4 + 17/2*e^3 - 23/2*e^2 - 41/2*e + 4, -21/2*e^5 - 7*e^4 + 167*e^3 + 197*e^2 - 111/2*e - 107/2, 11/2*e^5 + 5/2*e^4 - 175/2*e^3 - 85*e^2 + 41*e + 39/2, 1, -1/2*e^5 - e^4 + 8*e^3 + 20*e^2 + 3/2*e - 33/2, 2*e^5 + e^4 - 31*e^3 - 33*e^2 + 3*e + 4, -19/2*e^5 - 6*e^4 + 303/2*e^3 + 349/2*e^2 - 121/2*e - 57, e^5 + 1/2*e^4 - 15*e^3 - 35/2*e^2 - 11/2*e + 3/2, 6*e^5 + 3*e^4 - 191/2*e^3 - 193/2*e^2 + 38*e + 35/2, 13*e^5 + 10*e^4 - 207*e^3 - 265*e^2 + 61*e + 82, 11/2*e^5 + 7/2*e^4 - 88*e^3 - 201/2*e^2 + 40*e + 31, -3*e^5 - 3*e^4 + 97/2*e^3 + 143/2*e^2 - 19*e - 63/2, -8*e^5 - 13/2*e^4 + 127*e^3 + 337/2*e^2 - 63/2*e - 89/2, -11/2*e^5 - 5*e^4 + 88*e^3 + 124*e^2 - 47/2*e - 87/2, 17/2*e^5 + 7*e^4 - 135*e^3 - 181*e^2 + 67/2*e + 107/2, -23/2*e^5 - 8*e^4 + 365/2*e^3 + 443/2*e^2 - 109/2*e - 64, 15/2*e^5 + 6*e^4 - 237/2*e^3 - 315/2*e^2 + 39/2*e + 44, 9/2*e^5 + 5*e^4 - 143/2*e^3 - 233/2*e^2 + 7/2*e + 36, -3/2*e^5 - e^4 + 47/2*e^3 + 59/2*e^2 - 9/2*e - 16, -2*e^5 + 65/2*e^3 + 33/2*e^2 - 31*e - 31/2, -10*e^5 - 13/2*e^4 + 317/2*e^3 + 186*e^2 - 89/2*e - 49, 12*e^5 + 21/2*e^4 - 191*e^3 - 529/2*e^2 + 87/2*e + 169/2, -12*e^5 - 7*e^4 + 192*e^3 + 209*e^2 - 86*e - 53, -9*e^5 - 5*e^4 + 285/2*e^3 + 307/2*e^2 - 47*e - 59/2, 11*e^5 + 15/2*e^4 - 353/2*e^3 - 209*e^2 + 153/2*e + 67, -19*e^5 - 27/2*e^4 + 605/2*e^3 + 371*e^2 - 203/2*e - 121, -1/2*e^5 - 1/2*e^4 + 15/2*e^3 + 11*e^2 + 9*e + 5/2, -e^4 - 3/2*e^3 + 31/2*e^2 + 34*e - 13/2, 9/2*e^5 + 1/2*e^4 - 145/2*e^3 - 46*e^2 + 57*e + 17/2, -e^5 + 16*e^3 + 6*e^2 - 11*e + 9, 37/2*e^5 + 27/2*e^4 - 294*e^3 - 733/2*e^2 + 84*e + 107, e^5 + e^4 - 17*e^3 - 24*e^2 + 18*e + 21, -10*e^5 - 8*e^4 + 317/2*e^3 + 421/2*e^2 - 33*e - 139/2, 4*e^5 + e^4 - 129/2*e^3 - 95/2*e^2 + 47*e + 9/2, -11/2*e^5 - 9/2*e^4 + 173/2*e^3 + 118*e^2 - 11*e - 89/2, 25/2*e^5 + 8*e^4 - 395/2*e^3 - 461/2*e^2 + 109/2*e + 54, -15*e^5 - 12*e^4 + 239*e^3 + 314*e^2 - 74*e - 113, -13/2*e^5 - 3*e^4 + 104*e^3 + 100*e^2 - 109/2*e - 33/2, 21*e^5 + 14*e^4 - 669/2*e^3 - 791/2*e^2 + 116*e + 237/2, 7*e^5 + 3*e^4 - 223/2*e^3 - 209/2*e^2 + 56*e + 41/2, -4*e^5 - 7/2*e^4 + 65*e^3 + 175/2*e^2 - 61/2*e - 75/2, 19/2*e^5 + 8*e^4 - 299/2*e^3 - 407/2*e^2 + 19/2*e + 52, 20*e^5 + 29/2*e^4 - 318*e^3 - 787/2*e^2 + 179/2*e + 203/2, 3*e^5 + 3*e^4 - 95/2*e^3 - 139/2*e^2 + 4*e + 7/2, 21/2*e^5 + 10*e^4 - 335/2*e^3 - 489/2*e^2 + 79/2*e + 90, -21/2*e^5 - 10*e^4 + 165*e^3 + 246*e^2 - 1/2*e - 141/2, 27/2*e^5 + 8*e^4 - 431/2*e^3 - 475/2*e^2 + 183/2*e + 56, 21/2*e^5 + 13/2*e^4 - 335/2*e^3 - 190*e^2 + 67*e + 99/2, -39/2*e^5 - 14*e^4 + 311*e^3 + 382*e^2 - 219/2*e - 243/2, -31/2*e^5 - 9*e^4 + 247*e^3 + 271*e^2 - 201/2*e - 121/2, 6*e^5 + 7*e^4 - 96*e^3 - 158*e^2 + 10*e + 48, 27/2*e^5 + 12*e^4 - 214*e^3 - 301*e^2 + 75/2*e + 181/2, -3/2*e^5 + 3/2*e^4 + 47/2*e^3 - 12*e^2 - 20*e + 39/2, -7*e^5 - 13/2*e^4 + 112*e^3 + 319/2*e^2 - 57/2*e - 113/2, 19*e^5 + 25/2*e^4 - 605/2*e^3 - 356*e^2 + 215/2*e + 98, -17/2*e^5 - 11/2*e^4 + 267/2*e^3 + 157*e^2 - 24*e - 73/2, -10*e^5 - 15/2*e^4 + 160*e^3 + 397/2*e^2 - 113/2*e - 85/2, 16*e^5 + 12*e^4 - 511/2*e^3 - 643/2*e^2 + 91*e + 233/2, 11/2*e^5 + 5/2*e^4 - 177/2*e^3 - 86*e^2 + 51*e + 53/2, -9*e^5 - 15/2*e^4 + 143*e^3 + 385/2*e^2 - 65/2*e - 95/2, -9/2*e^5 - 4*e^4 + 141/2*e^3 + 203/2*e^2 - 11/2*e - 41, 53/2*e^5 + 18*e^4 - 843/2*e^3 - 1009/2*e^2 + 285/2*e + 150, -63/2*e^5 - 23*e^4 + 999/2*e^3 + 1243/2*e^2 - 255/2*e - 158, -32*e^5 - 47/2*e^4 + 1015/2*e^3 + 634*e^2 - 251/2*e - 159, 7/2*e^5 + 9/2*e^4 - 109/2*e^3 - 98*e^2 - 18*e + 47/2, 31/2*e^5 + 13*e^4 - 493/2*e^3 - 669/2*e^2 + 115/2*e + 109, -23/2*e^5 - 10*e^4 + 361/2*e^3 + 511/2*e^2 - 21/2*e - 74, 24*e^5 + 33/2*e^4 - 382*e^3 - 919/2*e^2 + 247/2*e + 239/2, 2*e^5 + 2*e^4 - 61/2*e^3 - 97/2*e^2 - 12*e + 3/2, 11/2*e^5 + 3*e^4 - 87*e^3 - 95*e^2 + 57/2*e + 41/2, 11/2*e^5 + 9/2*e^4 - 90*e^3 - 231/2*e^2 + 57*e + 50, -11*e^5 - 19/2*e^4 + 175*e^3 + 479/2*e^2 - 87/2*e - 133/2, 4*e^5 + 5*e^4 - 129/2*e^3 - 227/2*e^2 + 18*e + 89/2, -30*e^5 - 43/2*e^4 + 955/2*e^3 + 586*e^2 - 299/2*e - 169, -1/2*e^5 + 3/2*e^4 + 9*e^3 - 37/2*e^2 - 30*e - 10, 39*e^5 + 51/2*e^4 - 1239/2*e^3 - 726*e^2 + 417/2*e + 196, 9*e^5 + 13/2*e^4 - 145*e^3 - 355/2*e^2 + 129/2*e + 151/2, -7/2*e^5 - 5/2*e^4 + 111/2*e^3 + 69*e^2 - 13*e - 31/2, 65/2*e^5 + 21*e^4 - 518*e^3 - 602*e^2 + 383/2*e + 357/2, -30*e^5 - 22*e^4 + 478*e^3 + 596*e^2 - 152*e - 182, 20*e^5 + 13*e^4 - 635/2*e^3 - 741/2*e^2 + 103*e + 183/2, -3*e^5 + e^4 + 47*e^3 + 10*e^2 - 33*e + 7, -6*e^5 - 9/2*e^4 + 195/2*e^3 + 119*e^2 - 101/2*e - 34, 9/2*e^5 + 2*e^4 - 141/2*e^3 - 141/2*e^2 + 37/2*e + 24, 33/2*e^5 + 14*e^4 - 259*e^3 - 358*e^2 + 29/2*e + 189/2, 20*e^5 + 15*e^4 - 317*e^3 - 402*e^2 + 80*e + 102, 35/2*e^5 + 17/2*e^4 - 561/2*e^3 - 279*e^2 + 148*e + 155/2, 2*e^5 + 9/2*e^4 - 33*e^3 - 173/2*e^2 - 1/2*e + 97/2, e^5 - 1/2*e^4 - 31/2*e^3 + 2*e^2 + 9/2*e - 32, -21*e^5 - 13*e^4 + 669/2*e^3 + 759/2*e^2 - 121*e - 215/2, -69/2*e^5 - 51/2*e^4 + 547*e^3 + 1379/2*e^2 - 142*e - 188, -5*e^5 - 6*e^4 + 78*e^3 + 135*e^2 + 20*e - 38, 1/2*e^5 - 1/2*e^4 - 7*e^3 + 7/2*e^2 + 2*e - 19, -19*e^5 - 27/2*e^4 + 302*e^3 + 739/2*e^2 - 175/2*e - 235/2, -25/2*e^5 - 13/2*e^4 + 401/2*e^3 + 206*e^2 - 112*e - 151/2, 4*e^5 - 65*e^3 - 34*e^2 + 62*e + 7, 8*e^5 + 11/2*e^4 - 125*e^3 - 315/2*e^2 + 25/2*e + 119/2, e^5 + 3/2*e^4 - 29/2*e^3 - 32*e^2 - 41/2*e - 6, 37/2*e^5 + 21/2*e^4 - 295*e^3 - 647/2*e^2 + 130*e + 112, -45*e^5 - 30*e^4 + 1431/2*e^3 + 1687/2*e^2 - 232*e - 427/2, 17/2*e^5 + 7*e^4 - 269/2*e^3 - 357/2*e^2 + 31/2*e + 32, -32*e^5 - 49/2*e^4 + 1017/2*e^3 + 653*e^2 - 289/2*e - 191, -45/2*e^5 - 14*e^4 + 713/2*e^3 + 817/2*e^2 - 223/2*e - 79, -13/2*e^5 - 5/2*e^4 + 105*e^3 + 191/2*e^2 - 78*e - 48, -36*e^5 - 55/2*e^4 + 572*e^3 + 1465/2*e^2 - 289/2*e - 449/2, 32*e^5 + 53/2*e^4 - 1013/2*e^3 - 682*e^2 + 193/2*e + 183, 21*e^5 + 15*e^4 - 334*e^3 - 412*e^2 + 99*e + 119, -17*e^5 - 25/2*e^4 + 539/2*e^3 + 337*e^2 - 131/2*e - 91, -15*e^5 - 17/2*e^4 + 237*e^3 + 521/2*e^2 - 155/2*e - 123/2, -9/2*e^4 + e^3 + 139/2*e^2 + 61/2*e - 77/2, -15/2*e^5 - 2*e^4 + 241/2*e^3 + 183/2*e^2 - 167/2*e - 12, 35/2*e^5 + 11*e^4 - 559/2*e^3 - 635/2*e^2 + 243/2*e + 86, 49*e^5 + 71/2*e^4 - 1557/2*e^3 - 965*e^2 + 445/2*e + 265, 12*e^5 + 25/2*e^4 - 385/2*e^3 - 297*e^2 + 91/2*e + 114, -43*e^5 - 55/2*e^4 + 1367/2*e^3 + 789*e^2 - 453/2*e - 190, -53/2*e^5 - 45/2*e^4 + 839/2*e^3 + 576*e^2 - 81*e - 359/2, -33*e^5 - 20*e^4 + 1051/2*e^3 + 1171/2*e^2 - 198*e - 295/2, 9*e^5 + 8*e^4 - 287/2*e^3 - 397/2*e^2 + 34*e + 131/2, 43/2*e^5 + 15*e^4 - 343*e^3 - 415*e^2 + 265/2*e + 277/2, -4*e^5 + e^4 + 65*e^3 + 15*e^2 - 78*e + 11, -e^5 + 2*e^4 + 18*e^3 - 26*e^2 - 57*e + 25, -30*e^5 - 21*e^4 + 476*e^3 + 579*e^2 - 129*e - 139, 27/2*e^5 + 10*e^4 - 431/2*e^3 - 549/2*e^2 + 153/2*e + 113, -29/2*e^5 - 8*e^4 + 463/2*e^3 + 489/2*e^2 - 207/2*e - 56, -63/2*e^5 - 47/2*e^4 + 498*e^3 + 1263/2*e^2 - 106*e - 157, -37/2*e^5 - 19/2*e^4 + 593/2*e^3 + 301*e^2 - 155*e - 153/2, 8*e^5 + 5*e^4 - 128*e^3 - 141*e^2 + 47*e + 11, 7/2*e^5 - 1/2*e^4 - 109/2*e^3 - 22*e^2 + 23*e - 23/2, -5*e^5 - 15/2*e^4 + 159/2*e^3 + 158*e^2 + 39/2*e - 36, 5*e^5 + 7/2*e^4 - 155/2*e^3 - 96*e^2 - 3/2*e + 18, 71/2*e^5 + 21*e^4 - 567*e^3 - 626*e^2 + 481/2*e + 329/2, -11/2*e^5 - 2*e^4 + 171/2*e^3 + 157/2*e^2 - 17/2*e - 1, -7*e^5 - 9/2*e^4 + 110*e^3 + 259/2*e^2 - 33/2*e - 93/2, -35/2*e^5 - 15*e^4 + 278*e^3 + 381*e^2 - 127/2*e - 221/2, -6*e^5 - 1/2*e^4 + 96*e^3 + 125/2*e^2 - 151/2*e - 75/2, 17*e^5 + 17/2*e^4 - 270*e^3 - 543/2*e^2 + 213/2*e + 105/2, -24*e^5 - 25/2*e^4 + 384*e^3 + 791/2*e^2 - 367/2*e - 219/2, 81/2*e^5 + 61/2*e^4 - 644*e^3 - 1633/2*e^2 + 189*e + 253, 13/2*e^5 + 4*e^4 - 205/2*e^3 - 235/2*e^2 + 55/2*e + 27, 61/2*e^5 + 47/2*e^4 - 965/2*e^3 - 624*e^2 + 100*e + 351/2, -24*e^5 - 39/2*e^4 + 379*e^3 + 1015/2*e^2 - 121/2*e - 235/2, 11*e^5 + 13/2*e^4 - 349/2*e^3 - 192*e^2 + 127/2*e + 22, 21/2*e^5 + 19/2*e^4 - 331/2*e^3 - 236*e^2 + 7*e + 119/2, -85/2*e^5 - 26*e^4 + 1355/2*e^3 + 1523/2*e^2 - 521/2*e - 217, 93/2*e^5 + 35*e^4 - 739*e^3 - 941*e^2 + 417/2*e + 575/2, -41/2*e^5 - 31/2*e^4 + 651/2*e^3 + 415*e^2 - 90*e - 245/2, -36*e^5 - 55/2*e^4 + 1139/2*e^3 + 730*e^2 - 243/2*e - 181, 27/2*e^5 + 25/2*e^4 - 218*e^3 - 613/2*e^2 + 81*e + 114, -25*e^5 - 31/2*e^4 + 797/2*e^3 + 453*e^2 - 301/2*e - 145, 30*e^5 + 41/2*e^4 - 961/2*e^3 - 571*e^2 + 391/2*e + 190, 43/2*e^5 + 18*e^4 - 685/2*e^3 - 925/2*e^2 + 177/2*e + 136, 37/2*e^5 + 11*e^4 - 292*e^3 - 327*e^2 + 165/2*e + 107/2, 43/2*e^5 + 29/2*e^4 - 342*e^3 - 817/2*e^2 + 114*e + 104, 4*e^5 - 1/2*e^4 - 127/2*e^3 - 22*e^2 + 79/2*e - 27];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {61, 61, 1/3*w^3 - 4/3*w^2 + 2/3*w + 4}>] := -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;