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

NN := ideal<ZF | {53, 53, -w^4 + w^3 + 4*w^2 - w - 4}>;

primesArray := [
[3, 3, -w^4 + w^3 + 4*w^2 - 2*w - 2],
[5, 5, w^2 - w - 2],
[19, 19, w^4 - 2*w^3 - 4*w^2 + 5*w + 4],
[23, 23, -w^3 + w^2 + 3*w - 1],
[29, 29, 2*w^4 - 3*w^3 - 8*w^2 + 7*w + 4],
[32, 2, 2],
[37, 37, w^3 - 2*w^2 - 2*w + 2],
[41, 41, -2*w^4 + 3*w^3 + 9*w^2 - 8*w - 6],
[43, 43, -2*w^4 + 3*w^3 + 8*w^2 - 8*w - 6],
[47, 47, w^4 - 2*w^3 - 5*w^2 + 6*w + 5],
[53, 53, -w^4 + w^3 + 4*w^2 - w - 4],
[61, 61, w^2 - 2*w - 3],
[67, 67, w^4 - w^3 - 4*w^2 + 3*w],
[67, 67, -w^4 + w^3 + 5*w^2 - 2*w - 2],
[71, 71, w^4 - w^3 - 4*w^2 + 5],
[71, 71, w^4 - 2*w^3 - 3*w^2 + 5*w + 3],
[71, 71, 2*w^4 - 2*w^3 - 8*w^2 + 5*w + 4],
[73, 73, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6],
[81, 3, -2*w^4 + 3*w^3 + 10*w^2 - 9*w - 10],
[97, 97, -2*w^4 + 3*w^3 + 7*w^2 - 5*w - 4],
[101, 101, -w^4 + 2*w^3 + 2*w^2 - 4*w + 2],
[103, 103, -w^4 + w^3 + 3*w^2 - 2],
[103, 103, -2*w^4 + 2*w^3 + 10*w^2 - 5*w - 7],
[107, 107, 3*w^4 - 4*w^3 - 12*w^2 + 10*w + 5],
[107, 107, -w^4 + 6*w^2 + w - 4],
[107, 107, w^4 - 2*w^3 - 5*w^2 + 8*w + 3],
[109, 109, 3*w^4 - 3*w^3 - 13*w^2 + 6*w + 8],
[127, 127, -w^4 + 6*w^2 + 2*w - 5],
[131, 131, -2*w^4 + 3*w^3 + 7*w^2 - 7*w - 5],
[137, 137, w^4 - w^3 - 3*w^2 - 2],
[137, 137, 3*w^4 - 5*w^3 - 12*w^2 + 12*w + 9],
[151, 151, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 6],
[151, 151, 2*w^4 - 4*w^3 - 8*w^2 + 11*w + 6],
[157, 157, -w^4 + 2*w^3 + 2*w^2 - 5*w],
[167, 167, -w^4 + 6*w^2 - w - 5],
[167, 167, w^4 - w^3 - 6*w^2 + 3*w + 4],
[167, 167, w^4 - 5*w^2 - 3*w + 3],
[169, 13, -w^3 + 4*w - 1],
[173, 173, w^4 - 2*w^3 - 5*w^2 + 7*w + 4],
[181, 181, 2*w^4 - 4*w^3 - 5*w^2 + 6*w + 6],
[181, 181, w^3 - 2*w^2 - 3*w + 3],
[193, 193, -w^4 + w^3 + 4*w^2 - 3*w + 1],
[197, 197, 2*w^4 - 3*w^3 - 9*w^2 + 7*w + 5],
[211, 211, 2*w^4 - 3*w^3 - 8*w^2 + 6*w + 5],
[229, 229, 2*w^2 - 2*w - 5],
[239, 239, w^2 - 2*w - 4],
[257, 257, -w^3 + 4*w - 2],
[257, 257, w^4 - 2*w^3 - 6*w^2 + 7*w + 7],
[263, 263, -2*w^4 + 4*w^3 + 7*w^2 - 10*w - 7],
[271, 271, -w^4 + 2*w^3 + 5*w^2 - 8*w - 6],
[271, 271, 2*w^4 - 2*w^3 - 9*w^2 + 4*w + 3],
[271, 271, -w^3 + 2*w^2 + 3*w - 2],
[277, 277, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],
[277, 277, 2*w^4 - 2*w^3 - 7*w^2 + 3*w + 2],
[277, 277, -2*w^4 + 4*w^3 + 7*w^2 - 12*w - 5],
[281, 281, -w^4 + 3*w^3 + 4*w^2 - 9*w - 5],
[281, 281, -3*w^4 + 4*w^3 + 12*w^2 - 8*w - 7],
[281, 281, -2*w^4 + 3*w^3 + 8*w^2 - 6*w - 4],
[289, 17, 3*w^4 - 5*w^3 - 13*w^2 + 14*w + 9],
[331, 331, -w^4 + 2*w^3 + 2*w^2 - 3*w + 2],
[337, 337, 2*w^4 - w^3 - 9*w^2 + w + 2],
[337, 337, -w^4 + 2*w^3 + 3*w^2 - 4*w + 2],
[337, 337, -3*w^4 + 3*w^3 + 13*w^2 - 7*w - 8],
[347, 347, -w^4 + 2*w^3 + w^2 - 2*w + 2],
[347, 347, 3*w^4 - 4*w^3 - 11*w^2 + 9*w + 4],
[349, 349, 3*w^4 - 5*w^3 - 12*w^2 + 12*w + 7],
[353, 353, w^4 - w^3 - 5*w^2 + 3*w],
[353, 353, 2*w^2 - w - 5],
[359, 359, 2*w^4 - 2*w^3 - 9*w^2 + 6*w + 5],
[373, 373, w^3 - 2*w - 3],
[373, 373, -3*w^4 + 5*w^3 + 11*w^2 - 12*w - 8],
[379, 379, -3*w^4 + 5*w^3 + 13*w^2 - 14*w - 14],
[383, 383, w^4 - 7*w^2 + 2*w + 2],
[389, 389, w^4 - 3*w^2 - 5*w],
[389, 389, w^4 - w^3 - 3*w^2 + 2*w - 3],
[389, 389, 2*w^3 - 2*w^2 - 6*w + 1],
[397, 397, -w^3 + 4*w^2 + w - 6],
[419, 419, 2*w^4 - 3*w^3 - 10*w^2 + 10*w + 8],
[439, 439, -2*w^4 + 3*w^3 + 9*w^2 - 8*w - 4],
[439, 439, 2*w^4 - 3*w^3 - 7*w^2 + 8*w + 5],
[443, 443, w^4 + w^3 - 7*w^2 - 4*w + 4],
[449, 449, -2*w^4 + 2*w^3 + 7*w^2 - 3*w - 3],
[457, 457, 4*w^4 - 6*w^3 - 15*w^2 + 13*w + 9],
[461, 461, 3*w^4 - 5*w^3 - 11*w^2 + 11*w + 6],
[463, 463, -w^4 + 4*w^2 + 3*w - 2],
[467, 467, 3*w^4 - 5*w^3 - 11*w^2 + 13*w + 7],
[467, 467, 2*w^4 - 2*w^3 - 9*w^2 + 6*w + 4],
[479, 479, -3*w^4 + 3*w^3 + 12*w^2 - 4*w - 7],
[487, 487, w^4 - w^3 - 7*w^2 + 3*w + 11],
[503, 503, 2*w^4 - 4*w^3 - 7*w^2 + 13*w + 1],
[509, 509, w^4 - 3*w^3 - 4*w^2 + 8*w + 5],
[521, 521, -w^4 + 2*w^3 + 2*w^2 - 7*w + 2],
[523, 523, 2*w^4 - 3*w^3 - 6*w^2 + 6*w],
[523, 523, -3*w^4 + 4*w^3 + 15*w^2 - 12*w - 11],
[523, 523, -w^4 + 2*w^3 + 4*w^2 - 8*w - 4],
[547, 547, w^3 - 3*w^2 - w + 7],
[557, 557, -w^4 + 4*w^3 + 4*w^2 - 14*w - 6],
[557, 557, w^4 - 2*w^3 - 6*w^2 + 8*w + 7],
[563, 563, -w^4 + w^3 + 6*w^2 - 4*w - 10],
[563, 563, w^4 - w^3 - 4*w^2 + 4*w - 1],
[569, 569, -3*w^4 + 6*w^3 + 10*w^2 - 14*w - 6],
[569, 569, -w^4 + w^3 + 5*w^2 - 4],
[571, 571, 5*w^4 - 8*w^3 - 19*w^2 + 20*w + 12],
[571, 571, -3*w^4 + 6*w^3 + 12*w^2 - 19*w - 7],
[587, 587, -2*w^4 + 2*w^3 + 7*w^2 - 5*w],
[599, 599, -2*w^4 + 3*w^3 + 9*w^2 - 10*w - 7],
[601, 601, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 5],
[607, 607, w^4 - 3*w^3 - 3*w^2 + 6*w],
[607, 607, -3*w^4 + 5*w^3 + 14*w^2 - 13*w - 10],
[619, 619, w^4 - w^3 - 2*w^2 - 2],
[619, 619, 3*w^4 - 3*w^3 - 13*w^2 + 4*w + 7],
[619, 619, -w^4 + 4*w^3 + 3*w^2 - 13*w - 6],
[625, 5, -w^4 + 4*w^3 + 3*w^2 - 11*w - 2],
[641, 641, -3*w^4 + 5*w^3 + 14*w^2 - 16*w - 14],
[643, 643, -5*w^4 + 7*w^3 + 19*w^2 - 18*w - 7],
[659, 659, -3*w^4 + 6*w^3 + 12*w^2 - 19*w - 9],
[661, 661, 2*w^3 - 2*w^2 - 5*w + 1],
[673, 673, w^4 - 2*w^3 - 3*w^2 + 3*w - 1],
[677, 677, w^4 - 4*w^2 - 2*w + 3],
[677, 677, -2*w^4 + 3*w^3 + 11*w^2 - 9*w - 14],
[677, 677, -2*w^4 + 5*w^3 + 8*w^2 - 17*w - 7],
[691, 691, 3*w^4 - 4*w^3 - 14*w^2 + 13*w + 6],
[709, 709, w^4 - 2*w^3 - 5*w^2 + 5*w + 8],
[719, 719, w^4 - w^3 - 3*w^2 + 2*w - 4],
[727, 727, -3*w^4 + 4*w^3 + 11*w^2 - 8*w - 5],
[733, 733, w^4 + w^3 - 7*w^2 - 5*w + 6],
[733, 733, -4*w^4 + 6*w^3 + 15*w^2 - 14*w - 8],
[739, 739, 3*w^4 - 3*w^3 - 13*w^2 + 7*w + 7],
[739, 739, -3*w^4 + 4*w^3 + 11*w^2 - 11*w - 2],
[757, 757, w^2 - 4*w + 1],
[757, 757, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 3],
[773, 773, -5*w^4 + 8*w^3 + 20*w^2 - 19*w - 14],
[811, 811, -2*w^4 + 2*w^3 + 10*w^2 - 8*w - 7],
[811, 811, 2*w^4 - 10*w^2 - 3*w + 6],
[811, 811, 3*w^4 - 3*w^3 - 14*w^2 + 9*w + 10],
[821, 821, w^4 - 3*w^3 + 3*w - 2],
[823, 823, 2*w^4 - 4*w^3 - 9*w^2 + 13*w + 8],
[827, 827, -3*w^4 + 5*w^3 + 11*w^2 - 13*w - 1],
[839, 839, 3*w^4 - 5*w^3 - 13*w^2 + 13*w + 7],
[841, 29, -2*w^3 + 3*w^2 + 3*w - 3],
[841, 29, -4*w^4 + 6*w^3 + 14*w^2 - 13*w - 5],
[853, 853, -w^4 + 4*w^2 + 5*w - 1],
[857, 857, -w^4 + 4*w^3 + 2*w^2 - 11*w - 4],
[863, 863, w^4 - 4*w^3 - w^2 + 11*w],
[877, 877, -4*w^4 + 4*w^3 + 17*w^2 - 8*w - 7],
[881, 881, -4*w^4 + 6*w^3 + 17*w^2 - 17*w - 15],
[881, 881, -2*w^3 + 3*w^2 + 5*w - 4],
[907, 907, -3*w^4 + 6*w^3 + 13*w^2 - 18*w - 12],
[907, 907, -w^4 + 5*w^2 + 4*w - 6],
[919, 919, 3*w^4 - 3*w^3 - 13*w^2 + 7*w + 5],
[919, 919, w^4 - 4*w^3 - 3*w^2 + 13*w + 1],
[919, 919, -4*w^4 + 7*w^3 + 17*w^2 - 20*w - 13],
[929, 929, -w^3 + 5*w - 3],
[941, 941, 4*w^4 - 6*w^3 - 15*w^2 + 16*w + 9],
[947, 947, 3*w^4 - 6*w^3 - 9*w^2 + 16*w + 3],
[953, 953, -w^4 + 2*w^3 + 6*w^2 - 8*w - 6],
[953, 953, 3*w^4 - 3*w^3 - 12*w^2 + 8*w + 6],
[961, 31, -2*w^4 + 4*w^3 + 9*w^2 - 14*w - 5],
[967, 967, -2*w^3 + 5*w + 4],
[983, 983, 2*w^3 - 7*w - 6],
[997, 997, 4*w^4 - 5*w^3 - 15*w^2 + 12*w + 6],
[997, 997, 4*w^4 - 5*w^3 - 16*w^2 + 13*w + 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + 3*x^4 - 13*x^3 - 23*x^2 + 28*x - 3;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, 7/19*e^4 + 22/19*e^3 - 96/19*e^2 - 191/19*e + 166/19, -7/19*e^4 - 22/19*e^3 + 96/19*e^2 + 191/19*e - 147/19, -4/19*e^4 - 18/19*e^3 + 44/19*e^2 + 139/19*e - 141/19, -4/19*e^4 - 18/19*e^3 + 25/19*e^2 + 158/19*e + 30/19, -11/19*e^4 - 40/19*e^3 + 121/19*e^2 + 311/19*e - 212/19, 11/19*e^4 + 40/19*e^3 - 102/19*e^2 - 292/19*e + 60/19, -6/19*e^4 - 8/19*e^3 + 85/19*e^2 + 28/19*e - 164/19, -2/19*e^4 - 9/19*e^3 + 22/19*e^2 + 41/19*e - 42/19, -1, e^4 + 3*e^3 - 12*e^2 - 22*e + 13, 14/19*e^4 + 44/19*e^3 - 173/19*e^2 - 344/19*e + 142/19, -1/19*e^4 - 14/19*e^3 - 8/19*e^2 + 125/19*e - 59/19, 1/19*e^4 + 14/19*e^3 - 11/19*e^2 - 144/19*e + 135/19, 5/19*e^4 + 13/19*e^3 - 74/19*e^2 - 150/19*e + 162/19, -14/19*e^4 - 44/19*e^3 + 173/19*e^2 + 325/19*e - 351/19, 10/19*e^4 + 26/19*e^3 - 110/19*e^2 - 129/19*e + 134/19, -5/19*e^4 - 13/19*e^3 + 55/19*e^2 - 2/19*e - 67/19, 18/19*e^4 + 62/19*e^3 - 217/19*e^2 - 559/19*e + 226/19, -6/19*e^4 - 8/19*e^3 + 104/19*e^2 + 85/19*e - 297/19, 10/19*e^4 + 45/19*e^3 - 91/19*e^2 - 376/19*e + 115/19, 24/19*e^4 + 70/19*e^3 - 283/19*e^2 - 549/19*e + 238/19, -20/19*e^4 - 71/19*e^3 + 220/19*e^2 + 524/19*e - 363/19, -16/19*e^4 - 72/19*e^3 + 138/19*e^2 + 594/19*e - 165/19, 2/19*e^4 + 9/19*e^3 - 60/19*e^2 - 136/19*e + 270/19, 10/19*e^4 + 45/19*e^3 - 91/19*e^2 - 376/19*e + 77/19, -13/19*e^4 - 49/19*e^3 + 105/19*e^2 + 314/19*e + 50/19, 5/19*e^4 + 32/19*e^3 - 17/19*e^2 - 245/19*e - 66/19, -25/19*e^4 - 65/19*e^3 + 313/19*e^2 + 484/19*e - 354/19, 9/19*e^4 + 31/19*e^3 - 99/19*e^2 - 194/19*e + 132/19, -3/19*e^4 - 4/19*e^3 + 14/19*e^2 - 100/19*e + 89/19, -10/19*e^4 - 45/19*e^3 + 91/19*e^2 + 376/19*e - 191/19, 14/19*e^4 + 63/19*e^3 - 135/19*e^2 - 591/19*e + 104/19, 14/19*e^4 + 63/19*e^3 - 154/19*e^2 - 591/19*e + 237/19, 4/19*e^4 + 18/19*e^3 - 44/19*e^2 - 177/19*e - 87/19, 12/19*e^4 + 35/19*e^3 - 132/19*e^2 - 208/19*e + 81/19, 4/19*e^4 + 18/19*e^3 - 44/19*e^2 - 82/19*e + 65/19, 17/19*e^4 + 48/19*e^3 - 206/19*e^2 - 282/19*e + 414/19, -10/19*e^4 - 26/19*e^3 + 129/19*e^2 + 129/19*e - 419/19, -21/19*e^4 - 66/19*e^3 + 231/19*e^2 + 497/19*e - 365/19, 4/19*e^4 - 1/19*e^3 - 101/19*e^2 - 25/19*e + 350/19, -8/19*e^4 - 17/19*e^3 + 145/19*e^2 + 183/19*e - 453/19, -14/19*e^4 - 44/19*e^3 + 173/19*e^2 + 344/19*e - 199/19, 2*e^4 + 7*e^3 - 21*e^2 - 55*e + 28, 15/19*e^4 + 58/19*e^3 - 146/19*e^2 - 450/19*e + 201/19, 7/19*e^4 + 22/19*e^3 - 77/19*e^2 - 39/19*e - 24/19, 9/19*e^4 + 50/19*e^3 - 80/19*e^2 - 441/19*e + 75/19, -5/19*e^4 - 13/19*e^3 + 131/19*e^2 + 169/19*e - 561/19, 22/19*e^4 + 80/19*e^3 - 261/19*e^2 - 717/19*e + 386/19, -42/19*e^4 - 132/19*e^3 + 481/19*e^2 + 937/19*e - 502/19, -8/19*e^4 - 55/19*e^3 + 31/19*e^2 + 525/19*e + 41/19, -10/19*e^4 - 45/19*e^3 + 110/19*e^2 + 414/19*e - 1/19, -6/19*e^4 - 27/19*e^3 + 104/19*e^2 + 256/19*e - 373/19, -22/19*e^4 - 99/19*e^3 + 261/19*e^2 + 964/19*e - 443/19, -20/19*e^4 - 52/19*e^3 + 239/19*e^2 + 334/19*e - 477/19, -12/19*e^4 - 54/19*e^3 + 113/19*e^2 + 569/19*e - 81/19, -30/19*e^4 - 97/19*e^3 + 368/19*e^2 + 786/19*e - 516/19, -4/19*e^4 + 1/19*e^3 + 101/19*e^2 - 108/19*e - 236/19, -29/19*e^4 - 102/19*e^3 + 319/19*e^2 + 870/19*e - 400/19, -30/19*e^4 - 116/19*e^3 + 311/19*e^2 + 881/19*e - 668/19, 27/19*e^4 + 93/19*e^3 - 354/19*e^2 - 791/19*e + 719/19, 4/19*e^4 + 18/19*e^3 - 82/19*e^2 - 291/19*e + 122/19, -22/19*e^4 - 99/19*e^3 + 166/19*e^2 + 755/19*e - 63/19, -27/19*e^4 - 74/19*e^3 + 335/19*e^2 + 487/19*e - 624/19, -10/19*e^4 - 64/19*e^3 + 34/19*e^2 + 528/19*e + 113/19, 3/19*e^4 - 15/19*e^3 - 71/19*e^2 + 252/19*e + 177/19, 9/19*e^4 + 50/19*e^3 - 118/19*e^2 - 498/19*e + 417/19, 49/19*e^4 + 154/19*e^3 - 539/19*e^2 - 1109/19*e + 687/19, -14/19*e^4 - 25/19*e^3 + 211/19*e^2 + 249/19*e - 503/19, 6/19*e^4 + 27/19*e^3 - 47/19*e^2 - 142/19*e - 7/19, -9/19*e^4 - 31/19*e^3 + 118/19*e^2 + 327/19*e - 208/19, 22/19*e^4 + 80/19*e^3 - 242/19*e^2 - 622/19*e + 291/19, 11/19*e^4 + 21/19*e^3 - 178/19*e^2 - 140/19*e + 630/19, 3/19*e^4 + 23/19*e^3 + 5/19*e^2 - 147/19*e - 51/19, -33/19*e^4 - 120/19*e^3 + 344/19*e^2 + 857/19*e - 351/19, -54/19*e^4 - 186/19*e^3 + 651/19*e^2 + 1525/19*e - 887/19, 18/19*e^4 + 81/19*e^3 - 198/19*e^2 - 863/19*e + 264/19, 9/19*e^4 + 31/19*e^3 - 61/19*e^2 - 137/19*e - 77/19, -16/19*e^4 - 53/19*e^3 + 157/19*e^2 + 385/19*e + 25/19, 22/19*e^4 + 80/19*e^3 - 242/19*e^2 - 641/19*e + 462/19, -3/19*e^4 - 23/19*e^3 - 43/19*e^2 + 14/19*e + 507/19, -33/19*e^4 - 120/19*e^3 + 382/19*e^2 + 1066/19*e - 617/19, 2*e^4 + 7*e^3 - 21*e^2 - 57*e + 18, -23/19*e^4 - 37/19*e^3 + 329/19*e^2 + 177/19*e - 749/19, e^4 + 5*e^3 - 7*e^2 - 41*e - 9, -31/19*e^4 - 92/19*e^3 + 360/19*e^2 + 588/19*e - 537/19, 14/19*e^4 + 25/19*e^3 - 230/19*e^2 - 192/19*e + 180/19, -62/19*e^4 - 222/19*e^3 + 644/19*e^2 + 1689/19*e - 865/19, 16/19*e^4 + 72/19*e^3 - 176/19*e^2 - 727/19*e + 279/19, -61/19*e^4 - 189/19*e^3 + 709/19*e^2 + 1317/19*e - 939/19, -7/19*e^4 - 41/19*e^3 + 20/19*e^2 + 419/19*e + 138/19, 14/19*e^4 + 63/19*e^3 - 97/19*e^2 - 496/19*e + 161/19, -14/19*e^4 - 44/19*e^3 + 192/19*e^2 + 211/19*e - 541/19, 60/19*e^4 + 175/19*e^3 - 717/19*e^2 - 1401/19*e + 652/19, 59/19*e^4 + 199/19*e^3 - 687/19*e^2 - 1542/19*e + 802/19, 77/19*e^4 + 242/19*e^3 - 866/19*e^2 - 1683/19*e + 933/19, -8/19*e^4 + 2/19*e^3 + 145/19*e^2 - 235/19*e - 681/19, -7/19*e^4 - 41/19*e^3 + 115/19*e^2 + 476/19*e - 318/19, -7/19*e^4 - 41/19*e^3 + 96/19*e^2 + 533/19*e - 204/19, 27/19*e^4 + 131/19*e^3 - 240/19*e^2 - 1114/19*e + 282/19, 18/19*e^4 + 43/19*e^3 - 217/19*e^2 - 293/19*e + 207/19, 1/19*e^4 + 14/19*e^3 + 103/19*e^2 + 27/19*e - 796/19, 13/19*e^4 + 49/19*e^3 - 143/19*e^2 - 428/19*e - 50/19, -40/19*e^4 - 161/19*e^3 + 459/19*e^2 + 1409/19*e - 783/19, 24/19*e^4 + 89/19*e^3 - 188/19*e^2 - 568/19*e - 351/19, 7/19*e^4 + 22/19*e^3 - 20/19*e^2 - 58/19*e - 347/19, 35/19*e^4 + 110/19*e^3 - 366/19*e^2 - 765/19*e - 82/19, -16/19*e^4 - 34/19*e^3 + 252/19*e^2 + 423/19*e - 469/19, -27/19*e^4 - 74/19*e^3 + 335/19*e^2 + 544/19*e - 130/19, 3/19*e^4 - 15/19*e^3 - 128/19*e^2 + 157/19*e + 310/19, -68/19*e^4 - 230/19*e^3 + 786/19*e^2 + 1907/19*e - 1010/19, 31/19*e^4 + 73/19*e^3 - 417/19*e^2 - 493/19*e + 860/19, -58/19*e^4 - 204/19*e^3 + 676/19*e^2 + 1797/19*e - 933/19, -26/19*e^4 - 41/19*e^3 + 343/19*e^2 + 77/19*e - 584/19, -41/19*e^4 - 118/19*e^3 + 489/19*e^2 + 812/19*e - 348/19, 42/19*e^4 + 132/19*e^3 - 462/19*e^2 - 785/19*e + 635/19, 15/19*e^4 + 77/19*e^3 - 127/19*e^2 - 640/19*e + 182/19, 22/19*e^4 + 80/19*e^3 - 280/19*e^2 - 812/19*e + 177/19, 50/19*e^4 + 149/19*e^3 - 626/19*e^2 - 1234/19*e + 822/19, 5/19*e^4 - 25/19*e^3 - 188/19*e^2 + 211/19*e + 447/19, -12/19*e^4 - 35/19*e^3 + 151/19*e^2 + 417/19*e - 442/19, 9/19*e^4 - 26/19*e^3 - 251/19*e^2 + 224/19*e + 835/19, 20/19*e^4 + 90/19*e^3 - 144/19*e^2 - 676/19*e - 93/19, -4/19*e^4 + 1/19*e^3 + 82/19*e^2 - 89/19*e - 103/19, 51/19*e^4 + 163/19*e^3 - 599/19*e^2 - 1321/19*e + 919/19, -18/19*e^4 - 81/19*e^3 + 179/19*e^2 + 692/19*e - 758/19, -47/19*e^4 - 164/19*e^3 + 479/19*e^2 + 1391/19*e - 341/19, 34/19*e^4 + 115/19*e^3 - 374/19*e^2 - 1115/19*e + 296/19, 54/19*e^4 + 205/19*e^3 - 556/19*e^2 - 1506/19*e + 640/19, 62/19*e^4 + 203/19*e^3 - 739/19*e^2 - 1765/19*e + 998/19, -10/19*e^4 - 26/19*e^3 + 129/19*e^2 + 110/19*e - 210/19, -43/19*e^4 - 127/19*e^3 + 587/19*e^2 + 1100/19*e - 1093/19, -15/19*e^4 - 39/19*e^3 + 184/19*e^2 + 13/19*e - 467/19, -3/19*e^4 + 15/19*e^3 + 128/19*e^2 - 214/19*e - 139/19, 29/19*e^4 + 64/19*e^3 - 414/19*e^2 - 395/19*e + 666/19, -67/19*e^4 - 235/19*e^3 + 794/19*e^2 + 1953/19*e - 913/19, -48/19*e^4 - 159/19*e^3 + 547/19*e^2 + 1155/19*e - 780/19, 39/19*e^4 + 109/19*e^3 - 448/19*e^2 - 714/19*e + 135/19, 23/19*e^4 + 56/19*e^3 - 272/19*e^2 - 272/19*e + 578/19, -25/19*e^4 - 84/19*e^3 + 332/19*e^2 + 807/19*e - 563/19, 27/19*e^4 + 55/19*e^3 - 373/19*e^2 - 259/19*e + 1004/19, 14/19*e^4 + 25/19*e^3 - 249/19*e^2 - 40/19*e + 465/19, 33/19*e^4 + 82/19*e^3 - 344/19*e^2 - 325/19*e + 237/19, 2*e^3 + 4*e^2 - 17*e + 13, -2*e^4 - 6*e^3 + 25*e^2 + 44*e - 33, -40/19*e^4 - 104/19*e^3 + 592/19*e^2 + 725/19*e - 1068/19, 2/19*e^4 + 9/19*e^3 - 98/19*e^2 - 98/19*e + 859/19, -14/19*e^4 - 44/19*e^3 + 192/19*e^2 + 610/19*e - 389/19, -47/19*e^4 - 145/19*e^3 + 593/19*e^2 + 1106/19*e - 1006/19, -25/19*e^4 - 46/19*e^3 + 332/19*e^2 + 408/19*e - 31/19, -8/19*e^4 - 17/19*e^3 + 69/19*e^2 - 102/19*e + 155/19, -33/19*e^4 - 101/19*e^3 + 344/19*e^2 + 705/19*e - 9/19, 23/19*e^4 + 37/19*e^3 - 272/19*e^2 - 63/19*e + 312/19, -61/19*e^4 - 227/19*e^3 + 671/19*e^2 + 2058/19*e - 996/19, 6/19*e^4 + 8/19*e^3 - 85/19*e^2 + 67/19*e + 639/19, 25/19*e^4 + 84/19*e^3 - 237/19*e^2 - 617/19*e + 639/19, 5/19*e^4 + 32/19*e^3 + 21/19*e^2 - 74/19*e - 560/19, 52/19*e^4 + 177/19*e^3 - 572/19*e^2 - 1332/19*e + 256/19, 45/19*e^4 + 174/19*e^3 - 571/19*e^2 - 1654/19*e + 1230/19, -45/19*e^4 - 136/19*e^3 + 476/19*e^2 + 932/19*e - 527/19, -5/19*e^4 - 51/19*e^3 - 21/19*e^2 + 796/19*e + 370/19];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {53, 53, -w^4 + w^3 + 4*w^2 - 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;