/*
  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![3, -4, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[3, 3, w],
[3, 3, w^3 - 2*w^2 - 6*w + 1],
[9, 3, w^3 - 2*w^2 - 5*w + 1],
[16, 2, 2],
[17, 17, w^3 - w^2 - 8*w - 5],
[17, 17, -2*w^3 + 4*w^2 + 11*w - 2],
[17, 17, -w^3 + 3*w^2 + 2*w - 2],
[17, 17, w^3 - 2*w^2 - 4*w + 1],
[25, 5, w^3 - 3*w^2 - 3*w + 1],
[25, 5, w^2 - 2*w - 7],
[29, 29, w^3 - 2*w^2 - 6*w - 1],
[29, 29, w - 2],
[53, 53, w^3 - 3*w^2 - 4*w + 4],
[53, 53, -w^3 + 3*w^2 + 4*w - 5],
[61, 61, -w^3 + 2*w^2 + 7*w - 4],
[61, 61, -4*w^3 + 11*w^2 + 14*w - 8],
[101, 101, -w^3 + 2*w^2 + 7*w - 1],
[103, 103, -w^3 + 3*w^2 + 2*w - 5],
[103, 103, -w^3 + w^2 + 8*w + 2],
[113, 113, w^3 - 3*w^2 - 3*w + 7],
[113, 113, w^2 - 2*w - 1],
[127, 127, w^3 - 2*w^2 - 4*w - 2],
[127, 127, -2*w^3 + 4*w^2 + 11*w + 1],
[131, 131, -3*w^3 + 7*w^2 + 14*w - 8],
[131, 131, -w^2 + w + 8],
[131, 131, 2*w^3 - 5*w^2 - 9*w + 1],
[131, 131, -3*w^3 + 6*w^2 + 16*w + 1],
[139, 139, -2*w^3 + 5*w^2 + 9*w - 2],
[139, 139, -w^2 + w + 7],
[157, 157, -2*w^3 + 5*w^2 + 7*w - 5],
[157, 157, -2*w^3 + 3*w^2 + 13*w + 2],
[157, 157, -2*w^3 + 5*w^2 + 6*w - 2],
[157, 157, -3*w^3 + 5*w^2 + 19*w + 4],
[169, 13, -2*w^3 + 4*w^2 + 10*w - 1],
[173, 173, 3*w^3 - 8*w^2 - 10*w + 7],
[173, 173, 2*w^3 - 2*w^2 - 15*w - 8],
[179, 179, -3*w^2 + 6*w + 17],
[179, 179, -2*w^3 + 3*w^2 + 12*w + 1],
[179, 179, -3*w^3 + 7*w^2 + 13*w - 7],
[179, 179, -2*w^3 + 3*w^2 + 14*w + 4],
[191, 191, w^3 - 3*w^2 - w + 4],
[191, 191, -2*w^3 + 3*w^2 + 14*w + 2],
[199, 199, -w^3 + 8*w + 10],
[199, 199, 5*w^3 - 15*w^2 - 14*w + 14],
[211, 211, -w - 4],
[211, 211, -w^3 + 2*w^2 + 6*w - 5],
[211, 211, 2*w^3 - 3*w^2 - 12*w - 4],
[211, 211, -3*w^3 + 7*w^2 + 13*w - 4],
[257, 257, -3*w^3 + 8*w^2 + 10*w - 2],
[257, 257, 2*w^3 - 2*w^2 - 16*w - 7],
[263, 263, 3*w^3 - 6*w^2 - 17*w + 1],
[263, 263, w^3 - 2*w^2 - 3*w - 1],
[269, 269, 5*w^3 - 13*w^2 - 19*w + 7],
[269, 269, -3*w^3 + 6*w^2 + 18*w - 1],
[277, 277, w^3 - 4*w^2 - w + 5],
[277, 277, -w^3 + 4*w^2 + w - 11],
[283, 283, -w^3 + 5*w^2 - w - 14],
[283, 283, 2*w^3 - 7*w^2 - 4*w + 10],
[283, 283, -w^3 + w^2 + 9*w + 1],
[283, 283, w^2 - 4*w - 5],
[311, 311, -w^3 + w^2 + 9*w + 5],
[311, 311, w^2 - 4*w - 1],
[313, 313, -4*w^3 + 9*w^2 + 19*w - 10],
[313, 313, -2*w^2 + 5*w + 11],
[337, 337, -3*w^3 + 6*w^2 + 14*w + 2],
[337, 337, -4*w^3 + 8*w^2 + 21*w + 1],
[347, 347, -2*w^3 + 6*w^2 + 7*w - 10],
[347, 347, -w^3 + 4*w^2 + 2*w - 7],
[361, 19, -4*w^3 + 11*w^2 + 13*w - 10],
[361, 19, -2*w^3 + 5*w^2 + 8*w - 10],
[373, 373, 2*w^3 - 5*w^2 - 10*w + 2],
[373, 373, -w^3 + 2*w^2 + 8*w - 2],
[373, 373, -2*w^3 + 4*w^2 + 13*w - 1],
[373, 373, -w^3 + 3*w^2 + 5*w - 8],
[389, 389, -w^3 + 4*w^2 + 2*w - 10],
[389, 389, -2*w^3 + 6*w^2 + 7*w - 7],
[419, 419, -2*w^3 + 5*w^2 + 6*w - 1],
[419, 419, -3*w^3 + 5*w^2 + 19*w + 5],
[433, 433, w^2 - 4*w - 2],
[433, 433, w^3 - w^2 - 9*w - 4],
[439, 439, -w^3 + 4*w^2 + w - 2],
[439, 439, w^3 - 4*w^2 - w + 14],
[443, 443, -2*w^3 + 5*w^2 + 9*w + 1],
[443, 443, -w^2 + w + 10],
[467, 467, w^3 - w^2 - 8*w + 4],
[467, 467, w^3 - 4*w^2 + 2],
[491, 491, -4*w^3 + 12*w^2 + 10*w - 11],
[491, 491, 2*w^3 - w^2 - 18*w - 11],
[491, 491, 3*w^3 - 9*w^2 - 7*w + 11],
[491, 491, 2*w^3 - 20*w - 19],
[503, 503, w^3 - 3*w^2 - 4*w + 10],
[503, 503, -w^3 + w^2 + 10*w - 2],
[523, 523, -4*w^3 + 9*w^2 + 22*w - 7],
[523, 523, w^3 - 3*w^2 - 7*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w + 5],
[529, 23, -2*w^3 + 4*w^2 + 10*w - 7],
[547, 547, 2*w^3 - w^2 - 18*w - 14],
[547, 547, 3*w^3 - 8*w^2 - 14*w + 8],
[571, 571, 5*w^3 - 11*w^2 - 25*w + 5],
[571, 571, -4*w^3 + 9*w^2 + 19*w - 11],
[599, 599, -5*w^3 + 12*w^2 + 20*w - 4],
[599, 599, 4*w^3 - 6*w^2 - 25*w - 11],
[601, 601, -4*w^3 + 9*w^2 + 20*w - 5],
[601, 601, w^3 - 9*w - 14],
[601, 601, 3*w^3 - 8*w^2 - 11*w + 2],
[601, 601, -2*w^3 + 3*w^2 + 14*w - 1],
[641, 641, -2*w^3 + 3*w^2 + 15*w - 5],
[641, 641, -4*w^3 + 8*w^2 + 23*w - 2],
[647, 647, -3*w^3 + 8*w^2 + 13*w - 11],
[647, 647, -w^3 + 4*w^2 + 3*w - 7],
[659, 659, 3*w^3 - 7*w^2 - 14*w - 2],
[659, 659, w^3 - w^2 - 6*w - 11],
[673, 673, -3*w^3 + 7*w^2 + 11*w - 1],
[673, 673, -4*w^3 + 7*w^2 + 24*w + 5],
[677, 677, 5*w^3 - 12*w^2 - 21*w + 11],
[677, 677, -2*w^3 + 6*w^2 + 5*w - 1],
[677, 677, -2*w^3 + 6*w^2 + 8*w - 7],
[677, 677, 2*w^3 - 6*w^2 - 8*w + 11],
[719, 719, -2*w^3 + 6*w^2 + 10*w - 7],
[719, 719, -4*w^3 + 10*w^2 + 20*w - 13],
[727, 727, 3*w^3 - 10*w^2 - 8*w + 10],
[727, 727, 3*w^3 - 8*w^2 - 13*w + 10],
[727, 727, 2*w^3 - 8*w^2 - 3*w + 23],
[727, 727, -w^3 + 4*w^2 + 3*w - 8],
[751, 751, -7*w^3 + 18*w^2 + 25*w - 7],
[751, 751, w^3 - 7*w^2 + 5*w + 26],
[757, 757, -4*w^3 + 9*w^2 + 18*w - 7],
[757, 757, -3*w^3 + 5*w^2 + 17*w + 1],
[797, 797, 2*w^2 - 4*w - 5],
[797, 797, 2*w^3 - 6*w^2 - 6*w + 11],
[823, 823, -2*w^3 + 2*w^2 + 15*w + 14],
[823, 823, -3*w^3 + 8*w^2 + 10*w - 1],
[829, 829, -6*w^3 + 16*w^2 + 22*w - 7],
[829, 829, 5*w^3 - 13*w^2 - 20*w + 8],
[829, 829, -w^3 + 7*w + 4],
[829, 829, -5*w^3 + 12*w^2 + 23*w - 14],
[841, 29, -3*w^3 + 6*w^2 + 15*w - 2],
[859, 859, -4*w^3 + 7*w^2 + 24*w - 1],
[859, 859, -5*w^3 + 15*w^2 + 14*w - 13],
[881, 881, 3*w^3 - 7*w^2 - 16*w + 4],
[881, 881, -w^3 + 3*w^2 + 6*w - 7],
[883, 883, -2*w^3 + 3*w^2 + 12*w + 10],
[883, 883, -3*w^3 + 7*w^2 + 13*w + 2],
[887, 887, 4*w^3 - 11*w^2 - 11*w + 11],
[887, 887, 2*w^3 - 7*w^2 - 3*w + 7],
[887, 887, 4*w^3 - 5*w^2 - 29*w - 10],
[887, 887, 3*w^2 - 7*w - 16],
[907, 907, -w^3 + 3*w^2 + 8*w - 5],
[907, 907, -5*w^3 + 11*w^2 + 28*w - 8],
[911, 911, -4*w^3 + 10*w^2 + 18*w - 17],
[911, 911, 2*w^2 - 2*w - 1],
[919, 919, 4*w^3 - 7*w^2 - 25*w - 4],
[919, 919, 2*w^3 - 5*w^2 - 5*w + 1],
[937, 937, -w^3 + 3*w^2 + w - 8],
[937, 937, -2*w^3 + 3*w^2 + 14*w - 2],
[953, 953, -2*w^3 + w^2 + 17*w + 19],
[953, 953, 4*w^3 - 11*w^2 - 13*w + 4],
[961, 31, 2*w^3 - 3*w^2 - 11*w - 8],
[961, 31, w^3 - 4*w^2 + 2*w + 11],
[971, 971, 2*w^2 - 5*w - 4],
[971, 971, -w^3 + 4*w^2 - 11],
[991, 991, -2*w^3 + 6*w^2 + 9*w - 8],
[991, 991, -2*w^3 + 3*w^2 + 13*w - 1],
[991, 991, 3*w^3 - 8*w^2 - 14*w + 11],
[991, 991, -2*w^3 + 5*w^2 + 7*w - 8],
[997, 997, -2*w^3 + 5*w^2 + 6*w + 1],
[997, 997, -3*w^3 + 5*w^2 + 19*w + 7],
[997, 997, -6*w^3 + 12*w^2 + 33*w - 4],
[997, 997, 4*w^3 - 11*w^2 - 15*w + 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 2*x^3 - 9*x^2 + 10*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [2/5*e^3 - 3/5*e^2 - 17/5*e + 9/5, -2/5*e^3 + 3/5*e^2 + 17/5*e - 9/5, 2/5*e^3 - 3/5*e^2 - 22/5*e + 9/5, 1, -e + 1, -3/5*e^3 + 7/5*e^2 + 28/5*e - 36/5, 4/5*e^3 - 6/5*e^2 - 39/5*e + 23/5, -1/5*e^3 - 1/5*e^2 + 16/5*e - 2/5, 2/5*e^3 - 3/5*e^2 - 12/5*e + 19/5, -6/5*e^3 + 9/5*e^2 + 56/5*e - 17/5, 9/5*e^3 - 16/5*e^2 - 79/5*e + 38/5, -e^3 + 2*e^2 + 7*e - 10, e^3 - 3*e^2 - 8*e + 15, -1/5*e^3 + 9/5*e^2 - 4/5*e - 27/5, -11/5*e^3 + 19/5*e^2 + 106/5*e - 77/5, -1/5*e^3 - 1/5*e^2 + 26/5*e - 7/5, -6, -7/5*e^3 + 13/5*e^2 + 57/5*e - 54/5, 7/5*e^3 - 13/5*e^2 - 57/5*e + 34/5, 11/5*e^3 - 9/5*e^2 - 96/5*e - 8/5, -3*e^3 + 3*e^2 + 28*e - 10, -e^3 + 3*e^2 + 11*e - 10, -11/5*e^3 + 9/5*e^2 + 121/5*e - 2/5, 3*e^3 - 3*e^2 - 27*e + 6, -16/5*e^3 + 29/5*e^2 + 161/5*e - 97/5, -8/5*e^3 + 7/5*e^2 + 103/5*e - 11/5, -3*e^3 + 3*e^2 + 27*e - 6, -3/5*e^3 - 3/5*e^2 + 33/5*e + 4/5, 3/5*e^3 + 3/5*e^2 - 33/5*e - 44/5, -8/5*e^3 + 12/5*e^2 + 103/5*e - 66/5, -4*e^3 + 6*e^2 + 41*e - 24, -3/5*e^3 + 2/5*e^2 + 23/5*e - 6/5, 7/5*e^3 - 8/5*e^2 - 67/5*e + 14/5, 8/5*e^3 - 12/5*e^2 - 88/5*e + 36/5, 19/5*e^3 - 31/5*e^2 - 204/5*e + 143/5, 17/5*e^3 - 23/5*e^2 - 192/5*e + 109/5, e^3 - 4*e^2 - 6*e + 17, 17/5*e^3 - 13/5*e^2 - 187/5*e + 44/5, 7/5*e^3 - 23/5*e^2 - 77/5*e + 124/5, -e^3 + 4*e^2 + 6*e - 17, -8/5*e^3 + 17/5*e^2 + 93/5*e - 91/5, -16/5*e^3 + 19/5*e^2 + 171/5*e - 77/5, -2/5*e^3 - 2/5*e^2 + 22/5*e + 16/5, 2/5*e^3 + 2/5*e^2 - 22/5*e - 16/5, 2*e^3 - 4*e^2 - 20*e + 30, 6/5*e^3 - 4/5*e^2 - 76/5*e + 82/5, 2/5*e^3 + 12/5*e^2 - 52/5*e - 106/5, 14/5*e^3 - 36/5*e^2 - 124/5*e + 98/5, -6/5*e^3 + 9/5*e^2 + 76/5*e - 82/5, -14/5*e^3 + 21/5*e^2 + 144/5*e - 118/5, -3/5*e^3 + 7/5*e^2 + 13/5*e - 66/5, 11/5*e^3 - 19/5*e^2 - 101/5*e + 22/5, -8/5*e^3 + 2/5*e^2 + 83/5*e - 66/5, 4/5*e^3 + 4/5*e^2 - 39/5*e - 112/5, -13/5*e^3 + 12/5*e^2 + 143/5*e - 1/5, -7/5*e^3 + 18/5*e^2 + 77/5*e - 49/5, 28/5*e^3 - 47/5*e^2 - 293/5*e + 151/5, 4*e^3 - 5*e^2 - 47*e + 13, 8/5*e^3 - 22/5*e^2 - 58/5*e + 86/5, -8/5*e^3 + 22/5*e^2 + 58/5*e - 86/5, 2*e^3 - 5*e^2 - 21*e + 21, 14/5*e^3 - 11/5*e^2 - 159/5*e + 23/5, 7/5*e^3 - 3/5*e^2 - 72/5*e - 76/5, -3/5*e^3 - 3/5*e^2 + 28/5*e - 46/5, -3*e^3 + 5*e^2 + 26*e - 14, 11/5*e^3 - 19/5*e^2 - 86/5*e + 72/5, -21/5*e^3 + 24/5*e^2 + 186/5*e - 117/5, 21/5*e^3 - 24/5*e^2 - 186/5*e - 3/5, 7/5*e^3 - 33/5*e^2 - 32/5*e + 144/5, -11/5*e^3 + 39/5*e^2 + 76/5*e - 162/5, 18/5*e^3 - 12/5*e^2 - 198/5*e + 66/5, -2*e^3 + 4*e^2 + 22*e - 26, -14/5*e^3 + 16/5*e^2 + 154/5*e - 98/5, 6/5*e^3 - 24/5*e^2 - 66/5*e + 162/5, -14/5*e^3 + 36/5*e^2 + 99/5*e - 88/5, 18/5*e^3 - 42/5*e^2 - 143/5*e + 206/5, -14/5*e^3 + 1/5*e^2 + 129/5*e + 77/5, 22/5*e^3 - 13/5*e^2 - 217/5*e + 39/5, -4/5*e^3 + 11/5*e^2 + 34/5*e - 108/5, -e^2 + 2*e - 8, 8/5*e^3 - 12/5*e^2 - 88/5*e + 56/5, 8/5*e^3 - 12/5*e^2 - 88/5*e + 56/5, 26/5*e^3 - 44/5*e^2 - 216/5*e + 202/5, -26/5*e^3 + 44/5*e^2 + 216/5*e - 82/5, -21/5*e^3 + 44/5*e^2 + 206/5*e - 227/5, -11/5*e^3 + 4/5*e^2 + 146/5*e - 57/5, -2/5*e^3 - 2/5*e^2 + 2/5*e + 56/5, 21/5*e^3 - 14/5*e^2 - 196/5*e + 7/5, -21/5*e^3 + 14/5*e^2 + 196/5*e - 7/5, 18/5*e^3 - 22/5*e^2 - 178/5*e + 96/5, -23/5*e^3 + 22/5*e^2 + 238/5*e - 101/5, -1/5*e^3 + 14/5*e^2 + 26/5*e - 127/5, 11/5*e^3 - 39/5*e^2 - 61/5*e + 192/5, -19/5*e^3 + 51/5*e^2 + 149/5*e - 168/5, 14/5*e^3 - 21/5*e^2 - 154/5*e + 18/5, -22/5*e^3 + 33/5*e^2 + 242/5*e - 144/5, -7*e^3 + 8*e^2 + 70*e - 17, 3/5*e^3 + 8/5*e^2 + 2/5*e - 39/5, -21/5*e^3 + 19/5*e^2 + 181/5*e + 38/5, 29/5*e^3 - 31/5*e^2 - 269/5*e + 138/5, -18/5*e^3 + 37/5*e^2 + 173/5*e - 101/5, -6/5*e^3 - 1/5*e^2 + 91/5*e + 53/5, -2*e^3 + 4*e^2 + 15*e - 25, -4/5*e^3 + 6/5*e^2 - 1/5*e - 13/5, 32/5*e^3 - 48/5*e^2 - 307/5*e + 149/5, 14/5*e^3 - 26/5*e^2 - 119/5*e + 33/5, 24/5*e^3 - 36/5*e^2 - 224/5*e + 38/5, -8/5*e^3 + 12/5*e^2 + 48/5*e - 106/5, -4*e^3 + 4*e^2 + 48*e - 20, -28/5*e^3 + 52/5*e^2 + 288/5*e - 236/5, 1/5*e^3 - 14/5*e^2 + 4/5*e + 127/5, -1/5*e^3 + 14/5*e^2 - 4/5*e - 7/5, 6/5*e^3 - 24/5*e^2 + 9/5*e + 87/5, -42/5*e^3 + 78/5*e^2 + 387/5*e - 279/5, -3*e^3 + 5*e^2 + 34*e - 25, -21/5*e^3 + 29/5*e^2 + 226/5*e - 127/5, 13/5*e^3 - 32/5*e^2 - 73/5*e + 101/5, -33/5*e^3 + 62/5*e^2 + 293/5*e - 231/5, 13/5*e^3 - 32/5*e^2 - 108/5*e + 11/5, -e^3 + 4*e^2 + 4*e - 39, 52/5*e^3 - 83/5*e^2 - 497/5*e + 229/5, 2*e^2 - 6*e + 6, -4/5*e^3 + 11/5*e^2 - 31/5*e - 73/5, 16/5*e^3 - 34/5*e^2 - 146/5*e + 202/5, 7*e^3 - 14*e^2 - 58*e + 51, -27/5*e^3 + 58/5*e^2 + 202/5*e - 199/5, -49/5*e^3 + 71/5*e^2 + 494/5*e - 223/5, -11/5*e^3 + 19/5*e^2 + 166/5*e - 77/5, 3*e^3 - 2*e^2 - 31*e - 14, -3/5*e^3 - 8/5*e^2 + 23/5*e - 26/5, 17/5*e^3 - 48/5*e^2 - 112/5*e + 189/5, -5*e^3 + 12*e^2 + 40*e - 45, -17/5*e^3 + 8/5*e^2 + 157/5*e + 91/5, 21/5*e^3 - 14/5*e^2 - 201/5*e + 87/5, -7/5*e^3 + 8/5*e^2 + 37/5*e + 21/5, 27/5*e^3 - 38/5*e^2 - 257/5*e + 149/5, -12/5*e^3 + 18/5*e^2 + 132/5*e - 54/5, -28/5*e^3 + 57/5*e^2 + 273/5*e - 161/5, -12/5*e^3 + 3/5*e^2 + 167/5*e + 61/5, -2*e^3 + e^2 + 14*e + 21, 6*e^3 - 7*e^2 - 58*e + 37, 2*e^3 + e^2 - 19*e - 7, -18/5*e^3 + 7/5*e^2 + 183/5*e + 39/5, 1/5*e^3 + 21/5*e^2 - 11/5*e - 138/5, 2/5*e^3 - 23/5*e^2 + 43/5*e + 99/5, -17/5*e^3 + 3/5*e^2 + 187/5*e + 6/5, -34/5*e^3 + 71/5*e^2 + 309/5*e - 263/5, -23/5*e^3 + 42/5*e^2 + 208/5*e - 161/5, 7/5*e^3 - 18/5*e^2 - 32/5*e + 49/5, -7/5*e^3 + 3/5*e^2 + 107/5*e + 36/5, -5*e^3 + 9*e^2 + 49*e - 24, 6/5*e^3 - 34/5*e^2 - 46/5*e + 52/5, 2*e^3 + 2*e^2 - 26*e - 36, 4*e^3 - 5*e^2 - 50*e + 25, 8*e^3 - 13*e^2 - 82*e + 53, -4/5*e^3 + 1/5*e^2 - 36/5*e + 92/5, 64/5*e^3 - 91/5*e^2 - 624/5*e + 348/5, -26/5*e^3 + 24/5*e^2 + 271/5*e - 77/5, -2/5*e^3 + 18/5*e^2 + 37/5*e - 119/5, -24/5*e^3 + 41/5*e^2 + 269/5*e - 173/5, -32/5*e^3 + 43/5*e^2 + 347/5*e - 159/5, 18/5*e^3 - 57/5*e^2 - 153/5*e + 241/5, 9/5*e^3 + 9/5*e^2 - 129/5*e - 32/5, 6/5*e^3 + 21/5*e^2 - 111/5*e - 113/5, 3*e^3 - 9*e^2 - 27*e + 44, -7/5*e^3 + 13/5*e^2 + 42/5*e - 49/5, 19/5*e^3 - 31/5*e^2 - 174/5*e + 93/5, -17/5*e^3 + 8/5*e^2 + 207/5*e + 61/5, -19/5*e^3 + 46/5*e^2 + 189/5*e - 123/5];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {16, 2, 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;