/*
  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![36, 6, -13, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[4, 2, 1/3*w^3 - 4/3*w^2 - 4/3*w + 6],
[4, 2, -1/3*w^3 - 2/3*w^2 + 10/3*w + 7],
[9, 3, -1/2*w^3 + 3/2*w^2 + 7/2*w - 9],
[9, 3, 1/3*w^3 + 2/3*w^2 - 7/3*w - 5],
[11, 11, -1/6*w^3 + 1/6*w^2 + 1/6*w],
[19, 19, w + 1],
[19, 19, 1/6*w^3 - 1/6*w^2 - 13/6*w + 2],
[25, 5, -1/3*w^3 + 1/3*w^2 + 7/3*w - 1],
[29, 29, -1/6*w^3 + 1/6*w^2 + 13/6*w],
[29, 29, w - 1],
[31, 31, 1/3*w^3 - 1/3*w^2 - 10/3*w + 3],
[31, 31, 1/6*w^3 - 1/6*w^2 - 1/6*w + 2],
[41, 41, -1/2*w^3 - 1/2*w^2 + 9/2*w + 5],
[41, 41, -1/2*w^3 + 3/2*w^2 + 5/2*w - 8],
[59, 59, 1/3*w^3 - 1/3*w^2 - 10/3*w - 1],
[59, 59, -1/3*w^3 + 4/3*w^2 + 7/3*w - 7],
[59, 59, 1/2*w^3 - 1/2*w^2 - 5/2*w + 2],
[79, 79, w^2 - 11],
[79, 79, 1/6*w^3 - 7/6*w^2 - 7/6*w + 3],
[89, 89, -1/6*w^3 + 1/6*w^2 + 19/6*w - 5],
[89, 89, 1/6*w^3 - 1/6*w^2 - 19/6*w - 3],
[109, 109, -1/6*w^3 + 7/6*w^2 + 1/6*w - 9],
[109, 109, -7/6*w^3 + 19/6*w^2 + 49/6*w - 20],
[109, 109, -5/6*w^3 + 17/6*w^2 + 23/6*w - 12],
[109, 109, 1/6*w^3 + 5/6*w^2 - 13/6*w - 4],
[121, 11, 1/6*w^3 - 1/6*w^2 - 7/6*w - 3],
[139, 139, 1/3*w^3 - 4/3*w^2 + 5/3*w - 1],
[139, 139, -5/3*w^3 - 7/3*w^2 + 50/3*w + 29],
[169, 13, w^3 + w^2 - 10*w - 13],
[169, 13, -5/6*w^3 + 17/6*w^2 + 17/6*w - 12],
[179, 179, -1/3*w^3 + 4/3*w^2 + 1/3*w - 7],
[179, 179, -2/3*w^3 + 2/3*w^2 + 17/3*w - 5],
[181, 181, -w^3 + 3*w^2 + 4*w - 13],
[181, 181, -1/6*w^3 + 7/6*w^2 + 1/6*w - 11],
[181, 181, 7/6*w^3 + 11/6*w^2 - 79/6*w - 25],
[181, 181, 1/6*w^3 + 5/6*w^2 - 13/6*w - 2],
[191, 191, -5/6*w^3 - 1/6*w^2 + 53/6*w + 7],
[191, 191, 1/6*w^3 + 11/6*w^2 - 19/6*w - 16],
[211, 211, -1/6*w^3 + 1/6*w^2 - 5/6*w - 2],
[211, 211, -1/2*w^3 + 1/2*w^2 + 9/2*w + 2],
[211, 211, 1/3*w^3 - 1/3*w^2 - 4/3*w - 3],
[211, 211, -1/2*w^3 + 1/2*w^2 + 11/2*w - 4],
[229, 229, 5/6*w^3 + 1/6*w^2 - 53/6*w - 9],
[229, 229, 1/2*w^3 - 3/2*w^2 - 1/2*w + 2],
[239, 239, 5/6*w^3 + 7/6*w^2 - 59/6*w - 19],
[239, 239, 1/2*w^3 - 5/2*w^2 + 1/2*w + 5],
[241, 241, 1/6*w^3 - 1/6*w^2 - 13/6*w - 4],
[241, 241, w - 5],
[269, 269, -1/6*w^3 + 7/6*w^2 + 1/6*w - 2],
[269, 269, 1/3*w^3 - 7/3*w^2 - 1/3*w + 7],
[271, 271, 5/3*w^3 + 7/3*w^2 - 47/3*w - 25],
[271, 271, -5/3*w^3 + 17/3*w^2 + 23/3*w - 27],
[281, 281, 1/6*w^3 + 5/6*w^2 - 1/6*w - 7],
[281, 281, -1/2*w^3 + 3/2*w^2 + 9/2*w - 8],
[311, 311, 11/6*w^3 + 7/6*w^2 - 113/6*w - 23],
[311, 311, -4/3*w^3 + 13/3*w^2 + 10/3*w - 13],
[311, 311, -2/3*w^3 + 5/3*w^2 + 20/3*w - 17],
[311, 311, 1/6*w^3 + 5/6*w^2 + 5/6*w + 1],
[349, 349, -1/2*w^3 + 1/2*w^2 + 11/2*w - 1],
[349, 349, -5/6*w^3 + 17/6*w^2 + 29/6*w - 14],
[349, 349, 1/6*w^3 - 1/6*w^2 + 5/6*w - 1],
[349, 349, -2/3*w^3 - 4/3*w^2 + 17/3*w + 13],
[359, 359, -1/6*w^3 + 13/6*w^2 + 7/6*w - 18],
[359, 359, -1/6*w^3 - 11/6*w^2 + 13/6*w + 18],
[361, 19, 2/3*w^3 - 2/3*w^2 - 14/3*w + 1],
[379, 379, 1/2*w^3 + 1/2*w^2 - 7/2*w - 7],
[379, 379, -5/6*w^3 + 11/6*w^2 + 23/6*w - 7],
[379, 379, w^3 - 9*w - 5],
[379, 379, -2/3*w^3 + 5/3*w^2 + 14/3*w - 7],
[401, 401, -1/3*w^3 + 7/3*w^2 + 4/3*w - 17],
[401, 401, -4/3*w^3 - 8/3*w^2 + 43/3*w + 31],
[419, 419, 1/6*w^3 - 1/6*w^2 + 5/6*w],
[419, 419, 1/2*w^3 - 1/2*w^2 - 11/2*w + 2],
[421, 421, -5/6*w^3 - 7/6*w^2 + 47/6*w + 11],
[421, 421, 1/3*w^3 - 1/3*w^2 - 19/3*w - 7],
[431, 431, 5/6*w^3 + 1/6*w^2 - 41/6*w - 6],
[431, 431, 2/3*w^3 - 8/3*w^2 - 8/3*w + 15],
[431, 431, 5/6*w^3 - 11/6*w^2 - 29/6*w + 7],
[431, 431, 1/2*w^3 - 5/2*w^2 + 5/2*w + 2],
[439, 439, 5/6*w^3 + 1/6*w^2 - 41/6*w - 4],
[439, 439, 1/2*w^3 - 1/2*w^2 - 9/2*w + 8],
[449, 449, 5/6*w^3 - 17/6*w^2 - 29/6*w + 18],
[449, 449, 11/6*w^3 - 41/6*w^2 - 41/6*w + 29],
[449, 449, -7/6*w^3 + 31/6*w^2 + 7/6*w - 16],
[449, 449, 5/3*w^3 + 7/3*w^2 - 56/3*w - 33],
[461, 461, -1/6*w^3 + 1/6*w^2 + 19/6*w - 3],
[461, 461, 1/6*w^3 - 1/6*w^2 - 19/6*w - 1],
[479, 479, -2*w - 1],
[479, 479, 1/3*w^3 - 1/3*w^2 - 13/3*w + 3],
[491, 491, 2/3*w^3 + 1/3*w^2 - 14/3*w - 7],
[491, 491, 5/6*w^3 - 11/6*w^2 - 35/6*w + 7],
[509, 509, 5/6*w^3 - 11/6*w^2 - 29/6*w + 8],
[509, 509, 5/6*w^3 + 1/6*w^2 - 41/6*w - 5],
[521, 521, -1/6*w^3 + 7/6*w^2 - 17/6*w + 1],
[521, 521, -1/6*w^3 + 7/6*w^2 - 11/6*w - 3],
[521, 521, -1/6*w^3 + 7/6*w^2 + 19/6*w - 5],
[521, 521, 2/3*w^3 + 1/3*w^2 - 26/3*w - 11],
[541, 541, 1/6*w^3 + 5/6*w^2 - 31/6*w - 17],
[541, 541, 5/6*w^3 - 17/6*w^2 - 47/6*w + 23],
[541, 541, 3*w^3 + 3*w^2 - 32*w - 47],
[541, 541, 2/3*w^3 + 4/3*w^2 - 14/3*w - 11],
[571, 571, -1/3*w^3 + 1/3*w^2 + 13/3*w - 1],
[571, 571, 2*w - 1],
[571, 571, -1/3*w^3 + 4/3*w^2 + 7/3*w - 1],
[571, 571, 1/6*w^3 + 5/6*w^2 - 7/6*w - 13],
[599, 599, 1/2*w^3 + 1/2*w^2 - 9/2*w - 2],
[599, 599, -1/6*w^3 + 1/6*w^2 + 19/6*w - 2],
[599, 599, 1/6*w^3 - 1/6*w^2 - 19/6*w],
[599, 599, 1/2*w^3 - 3/2*w^2 - 5/2*w + 11],
[601, 601, 1/6*w^3 + 11/6*w^2 - 7/6*w - 15],
[601, 601, -1/3*w^3 + 7/3*w^2 - 5/3*w - 5],
[601, 601, 2/3*w^3 + 4/3*w^2 - 26/3*w - 19],
[601, 601, 1/2*w^3 - 5/2*w^2 - 7/2*w + 13],
[659, 659, w^2 - 13],
[659, 659, -5/6*w^3 + 11/6*w^2 + 29/6*w - 12],
[659, 659, -1/6*w^3 + 7/6*w^2 + 7/6*w - 1],
[659, 659, 5/6*w^3 + 1/6*w^2 - 41/6*w - 1],
[691, 691, 1/2*w^3 - 1/2*w^2 - 11/2*w + 10],
[691, 691, -1/6*w^3 + 1/6*w^2 - 5/6*w - 8],
[701, 701, -2*w^2 + w + 11],
[701, 701, -1/6*w^3 + 13/6*w^2 + 7/6*w - 9],
[701, 701, -1/6*w^3 + 13/6*w^2 + 1/6*w - 16],
[701, 701, -w^3 + w^2 + 9*w - 1],
[709, 709, -2/3*w^3 + 11/3*w^2 + 14/3*w - 21],
[709, 709, 1/3*w^3 + 5/3*w^2 + 2/3*w - 5],
[709, 709, 7/6*w^3 - 25/6*w^2 - 43/6*w + 25],
[709, 709, 2/3*w^3 - 2/3*w^2 - 23/3*w - 1],
[719, 719, 2*w^2 - 2*w - 19],
[719, 719, -2*w^2 + 2*w + 7],
[739, 739, 5/6*w^3 - 5/6*w^2 - 29/6*w + 4],
[739, 739, -5/6*w^3 + 11/6*w^2 + 53/6*w - 17],
[739, 739, -1/6*w^3 - 5/6*w^2 - 11/6*w],
[739, 739, -w^3 + w^2 + 8*w - 5],
[751, 751, -1/6*w^3 + 1/6*w^2 + 25/6*w + 8],
[751, 751, -1/3*w^3 + 1/3*w^2 + 16/3*w - 11],
[761, 761, -2/3*w^3 + 5/3*w^2 + 8/3*w - 9],
[761, 761, 5/6*w^3 + 1/6*w^2 - 47/6*w - 3],
[769, 769, 1/3*w^3 - 7/3*w^2 - 7/3*w + 11],
[769, 769, -4/3*w^3 - 8/3*w^2 + 46/3*w + 33],
[769, 769, -1/2*w^3 + 5/2*w^2 + 5/2*w - 20],
[769, 769, 2*w^2 - 17],
[809, 809, 1/6*w^3 + 11/6*w^2 - 31/6*w - 15],
[809, 809, -5/6*w^3 - 1/6*w^2 + 35/6*w + 3],
[809, 809, 1/6*w^3 + 11/6*w^2 - 31/6*w - 9],
[809, 809, w^3 - 2*w^2 - 7*w + 11],
[811, 811, -1/6*w^3 - 5/6*w^2 + 25/6*w + 7],
[811, 811, 1/6*w^3 + 5/6*w^2 - 25/6*w - 15],
[811, 811, -1/6*w^3 - 5/6*w^2 + 25/6*w - 4],
[811, 811, 1/6*w^3 + 5/6*w^2 - 25/6*w - 4],
[821, 821, -1/2*w^3 - 5/2*w^2 + 13/2*w + 16],
[821, 821, 1/6*w^3 - 13/6*w^2 - 7/6*w + 21],
[821, 821, 3/2*w^3 + 1/2*w^2 - 35/2*w - 20],
[821, 821, -1/6*w^3 + 13/6*w^2 + 7/6*w - 7],
[829, 829, -1/6*w^3 + 19/6*w^2 - 11/6*w - 23],
[829, 829, 5/6*w^3 - 17/6*w^2 - 5/6*w + 8],
[829, 829, -4/3*w^3 - 2/3*w^2 + 43/3*w + 15],
[829, 829, -1/6*w^3 - 17/6*w^2 + 25/6*w + 16],
[839, 839, 1/3*w^3 - 7/3*w^2 + 2/3*w + 5],
[839, 839, 1/2*w^3 + 3/2*w^2 - 13/2*w - 20],
[841, 29, -5/6*w^3 + 5/6*w^2 + 35/6*w - 4],
[859, 859, 1/6*w^3 + 5/6*w^2 - 37/6*w + 7],
[859, 859, 13/6*w^3 + 17/6*w^2 - 127/6*w - 33],
[881, 881, -2/3*w^3 + 2/3*w^2 + 20/3*w + 5],
[881, 881, 17/6*w^3 + 19/6*w^2 - 167/6*w - 42],
[911, 911, -1/6*w^3 + 13/6*w^2 - 11/6*w - 6],
[911, 911, 3/2*w^3 + 1/2*w^2 - 33/2*w - 17],
[911, 911, 5/6*w^3 - 17/6*w^2 + 1/6*w + 5],
[911, 911, 7/6*w^3 - 1/6*w^2 - 61/6*w - 1],
[929, 929, -7/6*w^3 - 11/6*w^2 + 67/6*w + 22],
[929, 929, -1/6*w^3 + 1/6*w^2 - 11/6*w + 6],
[941, 941, 11/6*w^3 + 19/6*w^2 - 107/6*w - 34],
[941, 941, 4/3*w^3 - 10/3*w^2 - 25/3*w + 19],
[941, 941, -11/6*w^3 + 41/6*w^2 + 47/6*w - 31],
[941, 941, 1/2*w^3 + 1/2*w^2 - 13/2*w - 13],
[961, 31, -1/3*w^3 + 1/3*w^2 + 7/3*w + 5],
[991, 991, -1/6*w^3 - 5/6*w^2 + 25/6*w + 6],
[991, 991, 1/6*w^3 + 5/6*w^2 - 25/6*w - 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 + x^8 - 45*x^7 - 21*x^6 + 674*x^5 - 70*x^4 - 3666*x^3 + 2082*x^2 + 3232*x - 160;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, -1, e, e, 20877/224432*e^8 + 63945/224432*e^7 - 712669/224432*e^6 - 1885261/224432*e^5 + 3466431/112216*e^4 + 6700957/112216*e^3 - 10423657/112216*e^2 - 8178415/112216*e + 174323/28054, 2651/112216*e^8 + 22423/112216*e^7 - 89163/112216*e^6 - 731979/112216*e^5 + 522945/56108*e^4 + 2982203/56108*e^3 - 2939739/56108*e^2 - 3329857/56108*e + 52825/14027, 2651/112216*e^8 + 22423/112216*e^7 - 89163/112216*e^6 - 731979/112216*e^5 + 522945/56108*e^4 + 2982203/56108*e^3 - 2939739/56108*e^2 - 3329857/56108*e + 52825/14027, 28715/224432*e^8 + 110431/224432*e^7 - 973835/224432*e^6 - 3351259/224432*e^5 + 4827473/112216*e^4 + 12404723/112216*e^3 - 16296911/112216*e^2 - 14501665/112216*e + 308749/28054, 12483/224432*e^8 + 41879/224432*e^7 - 428739/224432*e^6 - 1246099/224432*e^5 + 2157537/112216*e^4 + 4447187/112216*e^3 - 7332919/112216*e^2 - 4641409/112216*e + 317845/28054, 12483/224432*e^8 + 41879/224432*e^7 - 428739/224432*e^6 - 1246099/224432*e^5 + 2157537/112216*e^4 + 4447187/112216*e^3 - 7332919/112216*e^2 - 4641409/112216*e + 317845/28054, -11057/112216*e^8 - 31045/112216*e^7 + 372777/112216*e^6 + 902817/112216*e^5 - 1735451/56108*e^4 - 3197329/56108*e^3 + 4490221/56108*e^2 + 4783611/56108*e + 25249/14027, -11057/112216*e^8 - 31045/112216*e^7 + 372777/112216*e^6 + 902817/112216*e^5 - 1735451/56108*e^4 - 3197329/56108*e^3 + 4490221/56108*e^2 + 4783611/56108*e + 25249/14027, -17639/224432*e^8 - 63267/224432*e^7 + 592207/224432*e^6 + 1899471/224432*e^5 - 2842529/112216*e^4 - 6936663/112216*e^3 + 8938547/112216*e^2 + 8574653/112216*e - 300507/28054, -17639/224432*e^8 - 63267/224432*e^7 + 592207/224432*e^6 + 1899471/224432*e^5 - 2842529/112216*e^4 - 6936663/112216*e^3 + 8938547/112216*e^2 + 8574653/112216*e - 300507/28054, 17059/224432*e^8 + 114575/224432*e^7 - 570075/224432*e^6 - 3662955/224432*e^5 + 3087453/112216*e^4 + 14463947/112216*e^3 - 14801575/112216*e^2 - 15302969/112216*e + 279235/28054, 17059/224432*e^8 + 114575/224432*e^7 - 570075/224432*e^6 - 3662955/224432*e^5 + 3087453/112216*e^4 + 14463947/112216*e^3 - 14801575/112216*e^2 - 15302969/112216*e + 279235/28054, 1619/56108*e^8 + 339/56108*e^7 - 60231/56108*e^6 + 7105/56108*e^5 + 311951/28054*e^4 - 117853/28054*e^3 - 798663/28054*e^2 + 142011/28054*e + 182410/14027, 1003/17264*e^8 + 1343/17264*e^7 - 34731/17264*e^6 - 32251/17264*e^5 + 161953/8632*e^4 + 85339/8632*e^3 - 350031/8632*e^2 - 272185/8632*e + 28965/2158, 1003/17264*e^8 + 1343/17264*e^7 - 34731/17264*e^6 - 32251/17264*e^5 + 161953/8632*e^4 + 85339/8632*e^3 - 350031/8632*e^2 - 272185/8632*e + 28965/2158, 14667/112216*e^8 + 63823/112216*e^7 - 483443/112216*e^6 - 1968587/112216*e^5 + 2288941/56108*e^4 + 7542123/56108*e^3 - 7434111/56108*e^2 - 9997669/56108*e - 86325/14027, 14667/112216*e^8 + 63823/112216*e^7 - 483443/112216*e^6 - 1968587/112216*e^5 + 2288941/56108*e^4 + 7542123/56108*e^3 - 7434111/56108*e^2 - 9997669/56108*e - 86325/14027, 2651/112216*e^8 + 22423/112216*e^7 - 89163/112216*e^6 - 731979/112216*e^5 + 522945/56108*e^4 + 2982203/56108*e^3 - 2995847/56108*e^2 - 3329857/56108*e + 193095/14027, -2401/17264*e^8 - 6141/17264*e^7 + 79921/17264*e^6 + 173489/17264*e^5 - 357203/8632*e^4 - 584817/8632*e^3 + 803245/8632*e^2 + 896459/8632*e + 5085/2158, -2401/17264*e^8 - 6141/17264*e^7 + 79921/17264*e^6 + 173489/17264*e^5 - 357203/8632*e^4 - 584817/8632*e^3 + 803245/8632*e^2 + 896459/8632*e + 5085/2158, 2651/112216*e^8 + 22423/112216*e^7 - 89163/112216*e^6 - 731979/112216*e^5 + 522945/56108*e^4 + 2982203/56108*e^3 - 2995847/56108*e^2 - 3329857/56108*e + 193095/14027, 2417/17264*e^8 + 8357/17264*e^7 - 82377/17264*e^6 - 248249/17264*e^5 + 408519/8632*e^4 + 879193/8632*e^3 - 1353557/8632*e^2 - 883907/8632*e + 58441/2158, -9777/224432*e^8 - 43669/224432*e^7 + 331673/224432*e^6 + 1370121/224432*e^5 - 1674263/112216*e^4 - 5422401/112216*e^3 + 6033589/112216*e^2 + 7876715/112216*e - 35185/28054, -9777/224432*e^8 - 43669/224432*e^7 + 331673/224432*e^6 + 1370121/224432*e^5 - 1674263/112216*e^4 - 5422401/112216*e^3 + 6033589/112216*e^2 + 7876715/112216*e - 35185/28054, -21237/112216*e^8 - 109489/112216*e^7 + 703189/112216*e^6 + 3433565/112216*e^5 - 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5389219/14027*e + 36820/14027, 119175/224432*e^8 + 357963/224432*e^7 - 4015495/224432*e^6 - 10466023/224432*e^5 + 18885605/112216*e^4 + 36983207/112216*e^3 - 52363323/112216*e^2 - 49707237/112216*e + 1778753/28054, -1943/17264*e^8 - 12843/17264*e^7 + 64647/17264*e^6 + 412679/17264*e^5 - 340077/8632*e^4 - 1654583/8632*e^3 + 1504235/8632*e^2 + 1966821/8632*e + 45055/2158, -1943/17264*e^8 - 12843/17264*e^7 + 64647/17264*e^6 + 412679/17264*e^5 - 340077/8632*e^4 - 1654583/8632*e^3 + 1504235/8632*e^2 + 1966821/8632*e + 45055/2158, 9819/28054*e^8 + 52723/28054*e^7 - 330567/28054*e^6 - 1666713/28054*e^5 + 1715634/14027*e^4 + 6535023/14027*e^3 - 7053032/14027*e^2 - 7626887/14027*e + 159284/14027, 9819/28054*e^8 + 52723/28054*e^7 - 330567/28054*e^6 - 1666713/28054*e^5 + 1715634/14027*e^4 + 6535023/14027*e^3 - 7053032/14027*e^2 - 7626887/14027*e + 159284/14027, -109535/224432*e^8 - 460051/224432*e^7 + 3670863/224432*e^6 + 14129183/224432*e^5 - 17953125/112216*e^4 - 53621551/112216*e^3 + 61198947/112216*e^2 + 69376197/112216*e - 989877/28054, 21237/56108*e^8 + 109489/56108*e^7 - 703189/56108*e^6 - 3433565/56108*e^5 + 3492407/28054*e^4 + 13376209/28054*e^3 - 13310477/28054*e^2 - 16545259/28054*e - 157006/14027, 21237/56108*e^8 + 109489/56108*e^7 - 703189/56108*e^6 - 3433565/56108*e^5 + 3492407/28054*e^4 + 13376209/28054*e^3 - 13310477/28054*e^2 - 16545259/28054*e - 157006/14027, -109535/224432*e^8 - 460051/224432*e^7 + 3670863/224432*e^6 + 14129183/224432*e^5 - 17953125/112216*e^4 - 53621551/112216*e^3 + 61198947/112216*e^2 + 69376197/112216*e - 989877/28054, 7447/56108*e^8 - 1733/56108*e^7 - 262111/56108*e^6 + 162953/56108*e^5 + 1181961/28054*e^4 - 1175519/28054*e^3 - 1546331/28054*e^2 + 907365/28054*e - 96670/14027, 7447/56108*e^8 - 1733/56108*e^7 - 262111/56108*e^6 + 162953/56108*e^5 + 1181961/28054*e^4 - 1175519/28054*e^3 - 1546331/28054*e^2 + 907365/28054*e - 96670/14027, -5895/56108*e^8 - 16379/56108*e^7 + 209467/56108*e^6 + 483607/56108*e^5 - 1098653/28054*e^4 - 1727175/28054*e^3 + 3738883/28054*e^2 + 1948101/28054*e - 451756/14027, 14197/56108*e^8 + 66705/56108*e^7 - 477117/56108*e^6 - 2076177/56108*e^5 + 2398415/28054*e^4 + 7940085/28054*e^3 - 8894161/28054*e^2 - 8869811/28054*e - 221636/14027, -5895/56108*e^8 - 16379/56108*e^7 + 209467/56108*e^6 + 483607/56108*e^5 - 1098653/28054*e^4 - 1727175/28054*e^3 + 3738883/28054*e^2 + 1948101/28054*e - 451756/14027, 14197/56108*e^8 + 66705/56108*e^7 - 477117/56108*e^6 - 2076177/56108*e^5 + 2398415/28054*e^4 + 7940085/28054*e^3 - 8894161/28054*e^2 - 8869811/28054*e - 221636/14027, 5409/56108*e^8 + 55889/56108*e^7 - 166157/56108*e^6 - 1832817/56108*e^5 + 878429/28054*e^4 + 7536513/28054*e^3 - 5214017/28054*e^2 - 9031037/28054*e + 312404/14027, -62197/112216*e^8 - 266601/112216*e^7 + 2130733/112216*e^6 + 8229853/112216*e^5 - 11005115/56108*e^4 - 31105085/56108*e^3 + 41520065/56108*e^2 + 34438895/56108*e - 1017441/14027, -62197/112216*e^8 - 266601/112216*e^7 + 2130733/112216*e^6 + 8229853/112216*e^5 - 11005115/56108*e^4 - 31105085/56108*e^3 + 41520065/56108*e^2 + 34438895/56108*e - 1017441/14027];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, 1/3*w^3 - 4/3*w^2 - 4/3*w + 6}>] := 1;
ALEigenvalues[ideal<ZF | {4, 2, -1/3*w^3 - 2/3*w^2 + 10/3*w + 7}>] := 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;