/*
  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![-1, 4, 5, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {29, 29, -w^4 + 3*w^3 + 2*w^2 - 7*w - 1}>;

primesArray := [
[3, 3, w - 1],
[5, 5, w^2 - w - 2],
[7, 7, w^4 - 2*w^3 - 3*w^2 + 4*w + 2],
[13, 13, w^3 - 2*w^2 - 2*w + 2],
[29, 29, -w^4 + 3*w^3 + 2*w^2 - 7*w - 1],
[29, 29, -w^2 + 2*w + 3],
[31, 31, w^4 - 2*w^3 - 3*w^2 + 5*w],
[31, 31, w^3 - 2*w^2 - 3*w + 2],
[32, 2, 2],
[43, 43, -w^2 - w + 4],
[53, 53, -w^4 + w^3 + 6*w^2 - 2*w - 5],
[53, 53, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],
[73, 73, w^4 - w^3 - 6*w^2 + 2*w + 6],
[73, 73, w^3 - w^2 - 4*w + 2],
[81, 3, 2*w^4 - 5*w^3 - 4*w^2 + 9*w + 2],
[83, 83, w^3 - w^2 - 5*w],
[89, 89, -w^4 + 2*w^3 + 4*w^2 - 4*w - 2],
[97, 97, w^4 - 3*w^3 - 2*w^2 + 6*w + 2],
[101, 101, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],
[103, 103, -w^4 + w^3 + 6*w^2 - w - 7],
[103, 103, 2*w^4 - 4*w^3 - 6*w^2 + 8*w + 1],
[103, 103, w^4 - w^3 - 6*w^2 + 3*w + 4],
[107, 107, -w^4 + 2*w^3 + 3*w^2 - 5*w + 2],
[109, 109, w^3 - 4*w - 1],
[109, 109, -w^2 + 2*w + 4],
[131, 131, w^4 - w^3 - 6*w^2 + w + 6],
[137, 137, w^3 - w^2 - 5*w + 3],
[149, 149, w^4 - 3*w^3 - 3*w^2 + 9*w + 6],
[149, 149, -w^4 + 6*w^2 + 3*w],
[163, 163, 2*w^2 - w - 5],
[167, 167, -2*w^4 + 3*w^3 + 8*w^2 - 4*w - 3],
[169, 13, w^4 - 3*w^3 + 6*w - 3],
[169, 13, w^3 - 4*w - 4],
[173, 173, w^4 - w^3 - 4*w^2 + 2*w + 1],
[173, 173, -w^4 + 3*w^3 + 3*w^2 - 7*w - 5],
[173, 173, w^4 - w^3 - 4*w^2 + w - 1],
[179, 179, w^2 - 3*w - 2],
[191, 191, -w^4 + 3*w^3 + w^2 - 6*w - 1],
[193, 193, -w^3 + 2*w^2 + w - 4],
[193, 193, -w^4 + 3*w^3 + 3*w^2 - 8*w - 1],
[199, 199, w^4 - 3*w^3 - w^2 + 7*w],
[199, 199, -2*w^4 + 5*w^3 + 5*w^2 - 10*w - 2],
[199, 199, w^4 - 3*w^3 + 5*w - 4],
[211, 211, 2*w^3 - 2*w^2 - 7*w - 1],
[211, 211, -2*w^3 + 3*w^2 + 5*w - 2],
[223, 223, -2*w^4 + 3*w^3 + 10*w^2 - 7*w - 9],
[227, 227, w^3 - w^2 - 4*w + 3],
[229, 229, w^4 - 4*w^3 + 10*w - 5],
[229, 229, -2*w^4 + 4*w^3 + 7*w^2 - 9*w - 8],
[239, 239, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 3],
[241, 241, w^3 - w^2 - 2*w - 2],
[257, 257, -w^4 + w^3 + 6*w^2 - 4*w - 4],
[263, 263, 2*w^3 - 4*w^2 - 5*w + 6],
[269, 269, w^3 - 2*w^2 - 4*w + 4],
[269, 269, -2*w^4 + 2*w^3 + 11*w^2 - 3*w - 10],
[271, 271, w^3 - 5*w],
[281, 281, w^4 - w^3 - 5*w^2 + 3*w + 3],
[281, 281, -2*w^4 + 4*w^3 + 6*w^2 - 6*w - 3],
[281, 281, w^3 - 5*w - 1],
[283, 283, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 3],
[289, 17, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 2],
[293, 293, 2*w^4 - 5*w^3 - 7*w^2 + 14*w + 9],
[311, 311, -w^4 + 2*w^3 + 2*w^2 - 4*w + 3],
[313, 313, w^3 - 6*w],
[331, 331, w^4 - 2*w^3 - 2*w^2 + w - 2],
[359, 359, w^4 - w^3 - 5*w^2 + 2*w - 1],
[373, 373, -w^4 + 3*w^3 + 2*w^2 - 8*w - 3],
[373, 373, 2*w^3 - 4*w^2 - 3*w + 3],
[379, 379, -w^4 + 4*w^3 - w^2 - 9*w + 3],
[379, 379, -w^4 + 3*w^3 + w^2 - 6*w - 2],
[379, 379, w^4 + w^3 - 7*w^2 - 6*w + 3],
[383, 383, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],
[383, 383, w^4 - 5*w^2 - 3*w + 3],
[383, 383, 2*w^4 - 5*w^3 - 3*w^2 + 8*w],
[389, 389, 3*w^4 - 7*w^3 - 8*w^2 + 16*w + 1],
[397, 397, -2*w^4 + 3*w^3 + 8*w^2 - 3*w - 7],
[397, 397, -w^4 + 2*w^3 + 3*w^2 - 4*w - 5],
[397, 397, -2*w^4 + 3*w^3 + 7*w^2 - 4*w - 3],
[409, 409, -w^4 + w^3 + 7*w^2 - 5*w - 9],
[419, 419, w^4 - 2*w^3 - w^2 + 2*w - 2],
[419, 419, 2*w^4 - 3*w^3 - 7*w^2 + 5*w + 2],
[419, 419, -w^4 + 4*w^3 - 9*w - 2],
[433, 433, -w^4 + 3*w^3 - 8*w + 4],
[433, 433, w^4 - 7*w^2 - w + 6],
[439, 439, 3*w^4 - 6*w^3 - 10*w^2 + 11*w + 7],
[449, 449, -2*w^4 + 5*w^3 + 5*w^2 - 11*w - 4],
[457, 457, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 2],
[463, 463, 2*w^4 - 5*w^3 - 4*w^2 + 8*w + 1],
[467, 467, 2*w^4 - 4*w^3 - 6*w^2 + 6*w + 1],
[479, 479, 3*w^4 - 4*w^3 - 12*w^2 + 4*w + 7],
[479, 479, 2*w^4 - 6*w^3 - 4*w^2 + 14*w + 1],
[491, 491, w^4 - 6*w^2 - w + 4],
[503, 503, 2*w^4 - w^3 - 11*w^2 - 3*w + 8],
[503, 503, 2*w^3 - 5*w^2 - 2*w + 6],
[509, 509, w^4 - 5*w^2 - 6*w],
[509, 509, w^4 - 3*w^3 - 3*w^2 + 9*w + 1],
[523, 523, 3*w^4 - 7*w^3 - 7*w^2 + 12*w + 3],
[523, 523, 3*w^4 - 8*w^3 - 7*w^2 + 17*w + 3],
[523, 523, w^4 - 7*w^2 - 2*w + 4],
[529, 23, -w^4 + w^3 + 7*w^2 - 2*w - 9],
[541, 541, w^4 - 3*w^3 - 3*w^2 + 5*w + 2],
[547, 547, -w^3 + 3*w^2 + 3*w - 4],
[557, 557, 2*w^3 - 4*w^2 - 6*w + 9],
[577, 577, 2*w^4 - 2*w^3 - 11*w^2 + 5*w + 10],
[577, 577, w^4 - 4*w^3 - w^2 + 9*w],
[577, 577, 2*w^4 - 3*w^3 - 9*w^2 + 4*w + 8],
[577, 577, w^4 - 7*w^2 - 3*w + 7],
[577, 577, -3*w^4 + 3*w^3 + 13*w^2 - w - 4],
[593, 593, 3*w - 2],
[601, 601, 2*w^4 - 4*w^3 - 7*w^2 + 5*w + 6],
[601, 601, -w^4 + 3*w^3 + w^2 - 7*w - 1],
[607, 607, w^3 - w^2 - 2*w - 3],
[625, 5, 2*w^4 - 5*w^3 - 8*w^2 + 14*w + 11],
[631, 631, -w^2 + w - 2],
[631, 631, w^4 - 7*w^2 - w + 8],
[643, 643, -2*w^4 + 4*w^3 + 9*w^2 - 10*w - 8],
[647, 647, 3*w^4 - 6*w^3 - 10*w^2 + 12*w + 8],
[653, 653, -3*w^4 + 6*w^3 + 11*w^2 - 12*w - 9],
[653, 653, w^4 - w^3 - 2*w^2 - 3*w - 3],
[653, 653, -2*w^4 + 4*w^3 + 7*w^2 - 6*w - 8],
[701, 701, w^4 - w^3 - 2*w^2 - w - 5],
[719, 719, 2*w^4 - 5*w^3 - 6*w^2 + 14*w + 2],
[727, 727, 2*w^4 - 5*w^3 - 5*w^2 + 9*w + 3],
[733, 733, 2*w^4 - 5*w^3 - 5*w^2 + 13*w + 3],
[739, 739, -2*w^3 + 4*w^2 + 4*w - 5],
[743, 743, w^4 - 2*w^3 - 3*w^2 + 6*w - 3],
[743, 743, 2*w^4 - 2*w^3 - 11*w^2 + 2*w + 10],
[757, 757, -w^4 + 2*w^3 + 5*w^2 - 6*w - 10],
[757, 757, 2*w^4 - 6*w^3 - 2*w^2 + 11*w - 4],
[761, 761, 3*w^3 - 4*w^2 - 9*w + 3],
[769, 769, 3*w^3 - 5*w^2 - 9*w + 7],
[769, 769, -2*w^4 + 4*w^3 + 7*w^2 - 11*w - 2],
[773, 773, 2*w^4 - 5*w^3 - 6*w^2 + 12*w + 1],
[773, 773, 3*w^4 - 4*w^3 - 12*w^2 + 3*w + 5],
[773, 773, 2*w^3 - w^2 - 10*w - 4],
[797, 797, -w^4 + 4*w^3 + 2*w^2 - 12*w - 3],
[811, 811, w^4 - 2*w^3 - 6*w^2 + 8*w + 4],
[823, 823, w^4 - 4*w^3 - w^2 + 10*w + 1],
[823, 823, 2*w^4 - 4*w^3 - 4*w^2 + 4*w - 3],
[827, 827, -w^4 + 2*w^3 + 2*w^2 - w + 3],
[827, 827, 2*w^4 - 4*w^3 - 5*w^2 + 7*w + 2],
[829, 829, -w^4 + 4*w^3 + 2*w^2 - 10*w - 2],
[829, 829, 3*w^3 - 3*w^2 - 11*w - 2],
[829, 829, w^4 - 2*w^3 - 3*w^2 + 2*w - 3],
[839, 839, -w^4 + 5*w^3 - 3*w^2 - 10*w + 5],
[839, 839, -w^4 + w^3 + 5*w^2 - 9],
[853, 853, 2*w^4 - 6*w^3 - 5*w^2 + 15*w + 4],
[859, 859, -3*w^4 + 8*w^3 + 6*w^2 - 18*w + 5],
[863, 863, w^4 - w^3 - 6*w^2 + 5*w + 2],
[877, 877, 3*w^3 - 4*w^2 - 8*w + 1],
[881, 881, 3*w^4 - 7*w^3 - 9*w^2 + 15*w + 3],
[881, 881, -w^3 + 7*w + 1],
[883, 883, -2*w^4 + 6*w^3 + 3*w^2 - 15*w],
[887, 887, w^4 - w^3 - 5*w^2 + w - 1],
[907, 907, 3*w^4 - 5*w^3 - 11*w^2 + 9*w + 5],
[919, 919, w^4 - 2*w^3 - 6*w^2 + 8*w + 7],
[937, 937, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 3],
[937, 937, -3*w^4 + 8*w^3 + 6*w^2 - 16*w + 1],
[937, 937, 3*w^4 - 5*w^3 - 11*w^2 + 7*w + 4],
[947, 947, w^3 - w^2 - 7*w + 2],
[947, 947, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],
[967, 967, -w^4 + 4*w^3 - 11*w],
[971, 971, 3*w^4 - 6*w^3 - 8*w^2 + 10*w],
[971, 971, w^4 - 8*w^2 + 2*w + 9],
[971, 971, w^4 - 4*w^3 + 2*w^2 + 7*w - 4],
[977, 977, -w^4 + 4*w^3 - 12*w + 5],
[983, 983, -w^4 + w^3 + 4*w^2 + 3*w - 2],
[983, 983, -2*w^4 + 3*w^3 + 8*w^2 - 6*w - 4],
[997, 997, 2*w^4 - 4*w^3 - 5*w^2 + 5*w - 3],
[997, 997, w^4 + w^3 - 5*w^2 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 6*x^7 + 2*x^6 + 44*x^5 - 60*x^4 - 73*x^3 + 147*x^2 - 16*x - 38;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/3*e^5 - 2/3*e^4 - 8/3*e^3 + 11/3*e^2 + 14/3*e - 5/3, 1/3*e^7 - 4/3*e^6 - 5/3*e^5 + 31/3*e^4 - 2*e^3 - 20*e^2 + 37/3*e + 19/3, 2/3*e^7 - 3*e^6 - 3*e^5 + 73/3*e^4 - 4*e^3 - 54*e^2 + 18*e + 71/3, 1, -1/3*e^7 + 4/3*e^6 + 2*e^5 - 10*e^4 - 8/3*e^3 + 53/3*e^2 + 1/3*e - 2, 2/3*e^7 - 3*e^6 - 11/3*e^5 + 27*e^4 - 8/3*e^3 - 66*e^2 + 24*e + 88/3, -2/3*e^7 + 10/3*e^6 + 2*e^5 - 27*e^4 + 34/3*e^3 + 178/3*e^2 - 31*e - 56/3, -e^7 + 4*e^6 + 19/3*e^5 - 104/3*e^4 - 8/3*e^3 + 242/3*e^2 - 73/3*e - 95/3, -5/3*e^7 + 23/3*e^6 + 7*e^5 - 62*e^4 + 44/3*e^3 + 403/3*e^2 - 178/3*e - 48, 2/3*e^7 - 10/3*e^6 - 8/3*e^5 + 28*e^4 - 6*e^3 - 66*e^2 + 67/3*e + 101/3, 1/3*e^6 - 1/3*e^5 - 14/3*e^4 + 4*e^3 + 15*e^2 - 31/3*e - 2, 2/3*e^7 - 8/3*e^6 - 11/3*e^5 + 61/3*e^4 + 5/3*e^3 - 113/3*e^2 + 6*e + 22/3, 2*e^7 - 9*e^6 - 9*e^5 + 221/3*e^4 - 15*e^3 - 490/3*e^2 + 206/3*e + 185/3, 1/3*e^7 - 2/3*e^6 - 10/3*e^5 + 11/3*e^4 + 12*e^3 - 7/3*e^2 - 38/3*e + 1, 1/3*e^6 - 1/3*e^5 - 14/3*e^4 + 5*e^3 + 16*e^2 - 43/3*e - 17, e^7 - 4*e^6 - 5*e^5 + 91/3*e^4 - 2*e^3 - 188/3*e^2 + 67/3*e + 94/3, 1/3*e^7 - 2/3*e^6 - 10/3*e^5 + 3*e^4 + 14*e^3 + 2*e^2 - 64/3*e - 20/3, -4/3*e^7 + 17/3*e^6 + 20/3*e^5 - 136/3*e^4 + 19/3*e^3 + 94*e^2 - 134/3*e - 80/3, 4/3*e^6 - 11/3*e^5 - 11*e^4 + 77/3*e^3 + 85/3*e^2 - 40*e - 40/3, 5/3*e^7 - 20/3*e^6 - 31/3*e^5 + 56*e^4 + 8*e^3 - 386/3*e^2 + 26*e + 53, -e^7 + 5*e^6 + 7/3*e^5 - 38*e^4 + 55/3*e^3 + 232/3*e^2 - 116/3*e - 24, 1/3*e^7 - 7/3*e^6 + e^5 + 58/3*e^4 - 61/3*e^3 - 139/3*e^2 + 39*e + 67/3, -e^7 + 5*e^6 + 2*e^5 - 113/3*e^4 + 24*e^3 + 226/3*e^2 - 185/3*e - 59/3, -2/3*e^7 + 5/3*e^6 + 19/3*e^5 - 37/3*e^4 - 18*e^3 + 52/3*e^2 + 35/3*e + 26/3, -4/3*e^7 + 19/3*e^6 + 5*e^5 - 155/3*e^4 + 49/3*e^3 + 113*e^2 - 148/3*e - 119/3, -5/3*e^7 + 25/3*e^6 + 5*e^5 - 208/3*e^4 + 103/3*e^3 + 153*e^2 - 286/3*e - 46, -2/3*e^7 + 2*e^6 + 14/3*e^5 - 41/3*e^4 - 25/3*e^3 + 70/3*e^2 - 5/3*e - 6, 2/3*e^7 - 10/3*e^6 - 10/3*e^5 + 30*e^4 - 8/3*e^3 - 230/3*e^2 + 74/3*e + 137/3, 5/3*e^7 - 25/3*e^6 - 4*e^5 + 191/3*e^4 - 106/3*e^3 - 380/3*e^2 + 278/3*e + 106/3, 1/3*e^7 - 7/3*e^6 - 2/3*e^5 + 70/3*e^4 - 12*e^3 - 63*e^2 + 97/3*e + 112/3, -2*e^7 + 29/3*e^6 + 7*e^5 - 232/3*e^4 + 80/3*e^3 + 500/3*e^2 - 86*e - 58, -4/3*e^7 + 6*e^6 + 16/3*e^5 - 140/3*e^4 + 34/3*e^3 + 295/3*e^2 - 119/3*e - 118/3, -4/3*e^7 + 13/3*e^6 + 31/3*e^5 - 103/3*e^4 - 70/3*e^3 + 215/3*e^2 + 49/3*e - 82/3, e^7 - 5*e^6 - 11/3*e^5 + 40*e^4 - 38/3*e^3 - 239/3*e^2 + 139/3*e + 14, 5/3*e^7 - 8*e^6 - 17/3*e^5 + 181/3*e^4 - 53/3*e^3 - 344/3*e^2 + 157/3*e + 65/3, 1/3*e^7 - 1/3*e^6 - 16/3*e^5 + 4*e^4 + 70/3*e^3 - 9*e^2 - 74/3*e - 1, 1/3*e^7 - 5/3*e^6 - 5/3*e^5 + 12*e^4 + 11/3*e^3 - 55/3*e^2 - 17/3*e - 35/3, 2*e^7 - 25/3*e^6 - 34/3*e^5 + 205/3*e^4 + 10/3*e^3 - 147*e^2 + 85/3*e + 158/3, -2*e^7 + 9*e^6 + 8*e^5 - 208/3*e^4 + 18*e^3 + 419/3*e^2 - 211/3*e - 121/3, -1/3*e^7 + 4/3*e^6 + 5/3*e^5 - 26/3*e^4 - 3*e^3 + 56/3*e^2 + 10/3*e - 86/3, e^7 - 4*e^6 - 7*e^5 + 100/3*e^4 + 13*e^3 - 221/3*e^2 - 5/3*e + 91/3, 8/3*e^7 - 34/3*e^6 - 40/3*e^5 + 269/3*e^4 - 23/3*e^3 - 185*e^2 + 199/3*e + 172/3, 4/3*e^7 - 7*e^6 - 11/3*e^5 + 60*e^4 - 86/3*e^3 - 424/3*e^2 + 236/3*e + 152/3, -2/3*e^7 + 3*e^6 + 7/3*e^5 - 68/3*e^4 + 34/3*e^3 + 40*e^2 - 46*e + 2, 1/3*e^7 - 7/3*e^6 + 5/3*e^5 + 55/3*e^4 - 80/3*e^3 - 119/3*e^2 + 143/3*e + 79/3, -2/3*e^7 + 3*e^6 + 10/3*e^5 - 80/3*e^4 + 16/3*e^3 + 66*e^2 - 28*e - 29, -1/3*e^7 + e^6 + 11/3*e^5 - 29/3*e^4 - 34/3*e^3 + 74/3*e^2 - 1/3*e - 34/3, -1/3*e^6 + 4/3*e^5 + 1/3*e^4 - 4*e^3 + 2/3*e^2 - 4*e + 59/3, -e^7 + 17/3*e^6 + 4/3*e^5 - 140/3*e^4 + 31*e^3 + 314/3*e^2 - 74*e - 118/3, e^7 - 4*e^6 - 20/3*e^5 + 101/3*e^4 + 22/3*e^3 - 72*e^2 + 13*e + 35/3, 2/3*e^7 - 7/3*e^6 - 10/3*e^5 + 43/3*e^4 + 13/3*e^3 - 70/3*e^2 - 10*e + 14, -1/3*e^7 + 2/3*e^6 + 14/3*e^5 - 23/3*e^4 - 59/3*e^3 + 56/3*e^2 + 34*e + 2, 2/3*e^7 - 2*e^6 - 19/3*e^5 + 20*e^4 + 35/3*e^3 - 155/3*e^2 + 40/3*e + 43/3, -e^7 + 5*e^6 + 3*e^5 - 112/3*e^4 + 10*e^3 + 218/3*e^2 - 67/3*e - 88/3, -3*e^7 + 37/3*e^6 + 46/3*e^5 - 295/3*e^4 + 23/3*e^3 + 214*e^2 - 235/3*e - 260/3, 2*e^7 - 28/3*e^6 - 7*e^5 + 74*e^4 - 91/3*e^3 - 157*e^2 + 304/3*e + 145/3, -7/3*e^7 + 11*e^6 + 32/3*e^5 - 286/3*e^4 + 62/3*e^3 + 228*e^2 - 98*e - 98, -1/3*e^7 + 10/3*e^6 - 13/3*e^5 - 25*e^4 + 40*e^3 + 157/3*e^2 - 61*e - 17, -2/3*e^6 + 1/3*e^5 + 25/3*e^4 - 10/3*e^3 - 79/3*e^2 + 37/3*e + 17, -1/3*e^7 + 11/3*e^6 - 5*e^5 - 85/3*e^4 + 140/3*e^3 + 190/3*e^2 - 220/3*e - 92/3, -2/3*e^7 + 10/3*e^6 + e^5 - 71/3*e^4 + 46/3*e^3 + 134/3*e^2 - 77/3*e - 64/3, 2/3*e^7 - 13/3*e^6 + 2/3*e^5 + 101/3*e^4 - 71/3*e^3 - 73*e^2 + 88/3*e + 106/3, 2/3*e^7 - 4/3*e^6 - 9*e^5 + 44/3*e^4 + 98/3*e^3 - 113/3*e^2 - 76/3*e + 31/3, -7/3*e^7 + 28/3*e^6 + 40/3*e^5 - 221/3*e^4 - 10/3*e^3 + 424/3*e^2 - 44*e - 62/3, -1/3*e^7 + 7/3*e^6 - 8/3*e^5 - 35/3*e^4 + 74/3*e^3 + 31/3*e^2 - 41*e - 50/3, 4/3*e^7 - 6*e^6 - 16/3*e^5 + 137/3*e^4 - 43/3*e^3 - 256/3*e^2 + 167/3*e + 55/3, -7/3*e^7 + 28/3*e^6 + 13*e^5 - 217/3*e^4 - 14/3*e^3 + 412/3*e^2 - 35*e - 52/3, e^7 - 3*e^6 - 19/3*e^5 + 52/3*e^4 + 23/3*e^3 - 10*e^2 + 9*e - 62/3, -1/3*e^7 + 8/3*e^6 - 3*e^5 - 17*e^4 + 86/3*e^3 + 71/3*e^2 - 44*e + 23/3, -4*e^7 + 53/3*e^6 + 55/3*e^5 - 430/3*e^4 + 32*e^3 + 302*e^2 - 446/3*e - 94, 5/3*e^7 - 8*e^6 - 17/3*e^5 + 184/3*e^4 - 62/3*e^3 - 350/3*e^2 + 163/3*e + 62/3, 1/3*e^7 - 7/3*e^6 + 2/3*e^5 + 62/3*e^4 - 68/3*e^3 - 148/3*e^2 + 48*e + 68/3, e^7 - 14/3*e^6 - 14/3*e^5 + 115/3*e^4 - 10/3*e^3 - 253/3*e^2 + 55/3*e + 38, 4/3*e^7 - 20/3*e^6 - 4*e^5 + 151/3*e^4 - 44/3*e^3 - 298/3*e^2 + 103/3*e + 125/3, -e^7 + 16/3*e^6 + 4*e^5 - 46*e^4 + 22/3*e^3 + 112*e^2 - 82/3*e - 202/3, -1/3*e^7 + e^6 + 3*e^5 - 13*e^4 + 137/3*e^2 - 76/3*e - 86/3, 1/3*e^7 - 8/3*e^6 + 2/3*e^5 + 27*e^4 - 26*e^3 - 71*e^2 + 194/3*e + 100/3, 5/3*e^7 - 8*e^6 - 26/3*e^5 + 217/3*e^4 - 26/3*e^3 - 536/3*e^2 + 184/3*e + 242/3, -4*e^7 + 16*e^6 + 23*e^5 - 382/3*e^4 - 8*e^3 + 773/3*e^2 - 220/3*e - 157/3, 4*e^7 - 17*e^6 - 61/3*e^5 + 413/3*e^4 - 34/3*e^3 - 896/3*e^2 + 334/3*e + 299/3, 11/3*e^7 - 43/3*e^6 - 23*e^5 + 359/3*e^4 + 38/3*e^3 - 785/3*e^2 + 227/3*e + 241/3, 7/3*e^7 - 10*e^6 - 31/3*e^5 + 238/3*e^4 - 64/3*e^3 - 506/3*e^2 + 286/3*e + 58, 7/3*e^7 - 9*e^6 - 13*e^5 + 215/3*e^4 - 4*e^3 - 145*e^2 + 63*e + 94/3, -2/3*e^6 + 5/3*e^5 + 5*e^4 - 10*e^3 - 10/3*e^2 + 13/3*e - 61/3, -8/3*e^7 + 12*e^6 + 13*e^5 - 299/3*e^4 + 14*e^3 + 647/3*e^2 - 253/3*e - 67, 2/3*e^7 - 7/3*e^6 - 13/3*e^5 + 20*e^4 + 7/3*e^3 - 149/3*e^2 + 35/3*e + 104/3, -5/3*e^7 + 8*e^6 + 22/3*e^5 - 212/3*e^4 + 61/3*e^3 + 169*e^2 - 80*e - 52, 3*e^7 - 12*e^6 - 58/3*e^5 + 103*e^4 + 50/3*e^3 - 709/3*e^2 + 146/3*e + 78, 4/3*e^7 - 20/3*e^6 - 10/3*e^5 + 51*e^4 - 22*e^3 - 106*e^2 + 140/3*e + 118/3, e^7 - 17/3*e^6 + 124/3*e^4 - 98/3*e^3 - 242/3*e^2 + 61*e + 21, 4/3*e^6 - 3*e^5 - 34/3*e^4 + 73/3*e^3 + 59/3*e^2 - 149/3*e + 25/3, -e^7 + 7*e^6 - 7/3*e^5 - 57*e^4 + 161/3*e^3 + 395/3*e^2 - 283/3*e - 68, -7/3*e^7 + 12*e^6 + 6*e^5 - 292/3*e^4 + 48*e^3 + 643/3*e^2 - 386/3*e - 64, 8/3*e^7 - 12*e^6 - 14*e^5 + 101*e^4 - 9*e^3 - 640/3*e^2 + 248/3*e + 106/3, 5/3*e^6 - 4*e^5 - 40/3*e^4 + 68/3*e^3 + 107/3*e^2 - 21*e - 48, -10/3*e^7 + 14*e^6 + 49/3*e^5 - 109*e^4 + 43/3*e^3 + 211*e^2 - 90*e - 122/3, 1/3*e^7 - 8/3*e^6 + 2*e^5 + 61/3*e^4 - 80/3*e^3 - 106/3*e^2 + 151/3*e + 14/3, -1/3*e^6 - e^5 + 10*e^4 + 2/3*e^3 - 40*e^2 + 25/3*e + 16/3, 2/3*e^6 - 7/3*e^5 - 16/3*e^4 + 52/3*e^3 + 34/3*e^2 - 100/3*e + 4, -1/3*e^7 - e^6 + 7*e^5 + 43/3*e^4 - 44*e^3 - 51*e^2 + 72*e + 47/3, -7/3*e^6 + 20/3*e^5 + 56/3*e^4 - 131/3*e^3 - 152/3*e^2 + 182/3*e + 26, 1/3*e^7 - 2/3*e^6 - 20/3*e^5 + 37/3*e^4 + 95/3*e^3 - 40*e^2 - 109/3*e + 44/3, 11/3*e^7 - 16*e^6 - 17*e^5 + 373/3*e^4 - 13*e^3 - 251*e^2 + 75*e + 221/3, -4/3*e^7 + 14/3*e^6 + 28/3*e^5 - 35*e^4 - 22*e^3 + 63*e^2 + 100/3*e - 34/3, 2*e^7 - 37/3*e^6 + e^5 + 97*e^4 - 241/3*e^3 - 206*e^2 + 487/3*e + 208/3, 2/3*e^7 + 1/3*e^6 - 13*e^5 + 1/3*e^4 + 184/3*e^3 - 4*e^2 - 214/3*e + 16/3, -7/3*e^7 + 34/3*e^6 + 25/3*e^5 - 281/3*e^4 + 119/3*e^3 + 601/3*e^2 - 128*e - 125/3, e^7 - 4*e^6 - 22/3*e^5 + 34*e^4 + 38/3*e^3 - 190/3*e^2 - 4/3*e, 2/3*e^7 - 8/3*e^6 - 14/3*e^5 + 24*e^4 - 4/3*e^3 - 43*e^2 + 119/3*e - 22, e^7 - 13/3*e^6 - 14/3*e^5 + 35*e^4 - 3*e^3 - 242/3*e^2 + 68/3*e + 118/3, 1/3*e^5 - 16/3*e^4 + 22/3*e^3 + 30*e^2 - 34*e - 109/3, -11/3*e^7 + 44/3*e^6 + 22*e^5 - 350/3*e^4 - 58/3*e^3 + 719/3*e^2 - 28*e - 263/3, 14/3*e^7 - 21*e^6 - 65/3*e^5 + 530/3*e^4 - 131/3*e^3 - 1162/3*e^2 + 617/3*e + 113, -4*e^7 + 52/3*e^6 + 55/3*e^5 - 137*e^4 + 83/3*e^3 + 844/3*e^2 - 147*e - 238/3, -2/3*e^7 + 3*e^6 + 4*e^5 - 92/3*e^4 + 4*e^3 + 278/3*e^2 - 103/3*e - 69, 1/3*e^7 - 8/3*e^6 + 2/3*e^5 + 65/3*e^4 - 15*e^3 - 124/3*e^2 + 58/3*e - 1, -e^7 + 16/3*e^6 + 7/3*e^5 - 131/3*e^4 + 74/3*e^3 + 290/3*e^2 - 176/3*e - 55, 8/3*e^7 - 13*e^6 - 6*e^5 + 99*e^4 - 60*e^3 - 604/3*e^2 + 443/3*e + 217/3, -8/3*e^7 + 41/3*e^6 + 23/3*e^5 - 331/3*e^4 + 127/3*e^3 + 752/3*e^2 - 113*e - 331/3, 4/3*e^7 - 10/3*e^6 - 37/3*e^5 + 85/3*e^4 + 67/3*e^3 - 152/3*e^2 + 71/3*e - 8/3, 13/3*e^7 - 18*e^6 - 73/3*e^5 + 439/3*e^4 + 5/3*e^3 - 920/3*e^2 + 286/3*e + 88, -1/3*e^7 + 19/3*e^5 - 11/3*e^4 - 89/3*e^3 + 58/3*e^2 + 112/3*e - 22/3, 5/3*e^7 - 6*e^6 - 13*e^5 + 57*e^4 + 16*e^3 - 433/3*e^2 + 83/3*e + 154/3, -10/3*e^7 + 52/3*e^6 + 28/3*e^5 - 139*e^4 + 164/3*e^3 + 303*e^2 - 436/3*e - 106, -4/3*e^7 + 8*e^6 + 2/3*e^5 - 197/3*e^4 + 131/3*e^3 + 437/3*e^2 - 232/3*e - 172/3, 2*e^7 - 31/3*e^6 - 16/3*e^5 + 260/3*e^4 - 146/3*e^3 - 590/3*e^2 + 434/3*e + 63, -2*e^7 + 25/3*e^6 + 28/3*e^5 - 179/3*e^4 + 26/3*e^3 + 269/3*e^2 - 170/3*e + 16, 1/3*e^7 - 11/3*e^6 + 10/3*e^5 + 33*e^4 - 127/3*e^3 - 235/3*e^2 + 277/3*e + 100/3, e^7 - 20/3*e^6 + 13/3*e^5 + 136/3*e^4 - 169/3*e^3 - 250/3*e^2 + 226/3*e + 26, -4/3*e^6 + 4*e^5 + 9*e^4 - 73/3*e^3 - 22*e^2 + 97/3*e + 46/3, -5*e^7 + 70/3*e^6 + 62/3*e^5 - 551/3*e^4 + 34*e^3 + 384*e^2 - 409/3*e - 141, 8/3*e^7 - 12*e^6 - 32/3*e^5 + 269/3*e^4 - 59/3*e^3 - 484/3*e^2 + 212/3*e + 28, -10/3*e^7 + 44/3*e^6 + 15*e^5 - 353/3*e^4 + 77/3*e^3 + 253*e^2 - 341/3*e - 107, -2/3*e^7 + 13/3*e^6 + 2/3*e^5 - 109/3*e^4 + 19*e^3 + 218/3*e^2 - 92/3*e + 6, -e^7 + 3*e^6 + 23/3*e^5 - 52/3*e^4 - 85/3*e^3 + 58/3*e^2 + 139/3*e - 19/3, 4/3*e^7 - 7*e^6 - 16/3*e^5 + 179/3*e^4 - 43/3*e^3 - 379/3*e^2 + 155/3*e + 76/3, 16/3*e^7 - 24*e^6 - 27*e^5 + 616/3*e^4 - 25*e^3 - 1432/3*e^2 + 524/3*e + 194, -1/3*e^7 + 2*e^6 + 2*e^5 - 40/3*e^4 - 21*e^3 + 76/3*e^2 + 229/3*e - 22, 10/3*e^7 - 13*e^6 - 64/3*e^5 + 314/3*e^4 + 86/3*e^3 - 667/3*e^2 + 20/3*e + 256/3, 7/3*e^7 - 10*e^6 - 13*e^5 + 254/3*e^4 - 11*e^3 - 179*e^2 + 106*e + 112/3, -4/3*e^7 + 6*e^6 + 7*e^5 - 158/3*e^4 + 6*e^3 + 130*e^2 - 54*e - 190/3, -4*e^7 + 18*e^6 + 17*e^5 - 452/3*e^4 + 51*e^3 + 997/3*e^2 - 581/3*e - 296/3, 10/3*e^7 - 43/3*e^6 - 17*e^5 + 114*e^4 + 2/3*e^3 - 752/3*e^2 + 131/3*e + 110, -8/3*e^7 + 32/3*e^6 + 44/3*e^5 - 242/3*e^4 - 17/3*e^3 + 451/3*e^2 - 94/3*e - 92/3, 2/3*e^7 - 4/3*e^6 - 23/3*e^5 + 40/3*e^4 + 25*e^3 - 140/3*e^2 - 31/3*e + 49, -13/3*e^7 + 64/3*e^6 + 38/3*e^5 - 167*e^4 + 71*e^3 + 1045/3*e^2 - 190*e - 122, 3*e^7 - 17*e^6 - 4*e^5 + 138*e^4 - 85*e^3 - 304*e^2 + 180*e + 110, -4/3*e^7 + 4*e^6 + 35/3*e^5 - 97/3*e^4 - 88/3*e^3 + 62*e^2 + 9*e + 22, 1/3*e^7 - 10/3*e^6 + 2*e^5 + 31*e^4 - 73/3*e^3 - 257/3*e^2 + 77/3*e + 43, -5/3*e^7 + 32/3*e^6 - 269/3*e^4 + 203/3*e^3 + 620/3*e^2 - 150*e - 236/3, 2/3*e^7 - 4*e^6 + 4*e^5 + 67/3*e^4 - 54*e^3 - 22*e^2 + 108*e - 16/3, -4*e^7 + 19*e^6 + 47/3*e^5 - 157*e^4 + 167/3*e^3 + 1016/3*e^2 - 598/3*e - 100, -11/3*e^7 + 43/3*e^6 + 71/3*e^5 - 374/3*e^4 - 11*e^3 + 865/3*e^2 - 92*e - 280/3, 5/3*e^7 - 23/3*e^6 - 32/3*e^5 + 71*e^4 + 14/3*e^3 - 174*e^2 + 140/3*e + 62, -1/3*e^7 + 2/3*e^6 + 13/3*e^5 - 32/3*e^4 - 3*e^3 + 88/3*e^2 - 154/3*e - 9, 2/3*e^7 - 8/3*e^6 + e^5 + 12*e^4 - 98/3*e^3 - 13/3*e^2 + 196/3*e + 1, -11/3*e^7 + 56/3*e^6 + 35/3*e^5 - 152*e^4 + 163/3*e^3 + 338*e^2 - 461/3*e - 146, 13/3*e^7 - 22*e^6 - 41/3*e^5 + 550/3*e^4 - 227/3*e^3 - 412*e^2 + 223*e + 152, -22/3*e^7 + 33*e^6 + 94/3*e^5 - 790/3*e^4 + 190/3*e^3 + 1658/3*e^2 - 775/3*e - 175, -10/3*e^7 + 37/3*e^6 + 22*e^5 - 101*e^4 - 59/3*e^3 + 644/3*e^2 - 197/3*e - 52, 22/3*e^7 - 100/3*e^6 - 95/3*e^5 + 824/3*e^4 - 72*e^3 - 1811/3*e^2 + 878/3*e + 632/3, -14/3*e^7 + 64/3*e^6 + 19*e^5 - 170*e^4 + 139/3*e^3 + 1081/3*e^2 - 181*e - 338/3, 5*e^7 - 67/3*e^6 - 65/3*e^5 + 524/3*e^4 - 35*e^3 - 360*e^2 + 463/3*e + 134, -e^7 + 7*e^6 - 2*e^5 - 172/3*e^4 + 50*e^3 + 425/3*e^2 - 304/3*e - 229/3, -e^7 + 14/3*e^6 + 8/3*e^5 - 98/3*e^4 + 58/3*e^3 + 44*e^2 - 137/3*e + 35/3, -5/3*e^7 + 17/3*e^6 + 35/3*e^5 - 139/3*e^4 - 50/3*e^3 + 308/3*e^2 - 82/3, -5/3*e^7 + 19/3*e^6 + 29/3*e^5 - 54*e^4 + e^3 + 135*e^2 - 100/3*e - 254/3, -e^7 + 14/3*e^6 + 14/3*e^5 - 43*e^4 + 31/3*e^3 + 335/3*e^2 - 32*e - 158/3, -16/3*e^7 + 21*e^6 + 32*e^5 - 518/3*e^4 - 11*e^3 + 377*e^2 - 122*e - 391/3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, -w^4 + 3*w^3 + 2*w^2 - 7*w - 1}>] := -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;