/*
  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![5, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {24, 6, 2*w - 4}>;

primesArray := [
[3, 3, w - 2],
[5, 5, w],
[5, 5, -w + 3],
[8, 2, 2],
[9, 3, w^2 + w - 4],
[13, 13, w + 3],
[17, 17, -w^2 + w + 3],
[23, 23, w^2 - 2],
[23, 23, w^2 - 3],
[23, 23, -w^2 + 8],
[29, 29, w - 4],
[37, 37, w^2 + w - 8],
[41, 41, w^2 + 2*w - 4],
[47, 47, 2*w^2 + w - 8],
[59, 59, -2*w^2 - 3*w + 6],
[61, 61, -2*w - 1],
[67, 67, -2*w - 3],
[79, 79, 2*w^2 - 9],
[109, 109, w^2 + 2*w - 6],
[109, 109, 2*w^2 + w - 14],
[109, 109, -w^2 - w - 1],
[113, 113, 2*w^2 + w - 12],
[127, 127, w^2 - 3*w - 2],
[131, 131, -4*w^2 - 5*w + 9],
[137, 137, w^2 + 2*w - 7],
[137, 137, 3*w + 8],
[137, 137, 2*w^2 + w - 17],
[139, 139, 3*w^2 + 2*w - 11],
[149, 149, w - 6],
[151, 151, w^2 + 2*w - 9],
[157, 157, -w^2 - 2*w + 13],
[157, 157, 2*w^2 - 7],
[163, 163, w^2 + 3*w + 3],
[163, 163, 2*w^2 - 3*w - 3],
[163, 163, 2*w^2 - w - 7],
[167, 167, 3*w^2 + w - 13],
[169, 13, 2*w^2 + 5*w - 1],
[173, 173, w^2 + 2*w + 2],
[179, 179, 4*w^2 + 2*w - 19],
[181, 181, w^2 + 3*w - 6],
[191, 191, 3*w^2 + 2*w - 14],
[193, 193, w^2 - 3*w - 3],
[197, 197, 3*w^2 - 2*w - 13],
[211, 211, -w - 6],
[223, 223, -w^2 + w - 3],
[223, 223, -3*w - 2],
[223, 223, 2*w^2 - 2*w - 3],
[229, 229, 2*w^2 - w - 4],
[233, 233, 2*w + 7],
[239, 239, -3*w - 4],
[239, 239, w^2 - 11],
[239, 239, 4*w^2 - w - 21],
[241, 241, 3*w^2 + w - 19],
[251, 251, -w^2 - 4*w - 1],
[257, 257, w - 7],
[263, 263, 3*w^2 + w - 12],
[269, 269, 3*w^2 - 4*w - 11],
[271, 271, 4*w^2 + 6*w - 11],
[277, 277, -4*w^2 - 3*w + 17],
[283, 283, 5*w^2 + 4*w - 18],
[289, 17, 3*w^2 + 2*w - 9],
[307, 307, -2*w^2 - w + 18],
[307, 307, w^2 + 2*w + 3],
[307, 307, -w^2 - 3],
[313, 313, 3*w^2 - 13],
[317, 317, 3*w^2 - 2*w - 12],
[331, 331, w^2 + 3*w - 9],
[331, 331, w^2 + 3*w - 11],
[331, 331, w^2 + 3*w + 4],
[343, 7, -7],
[347, 347, 2*w^2 + 2*w - 13],
[353, 353, 4*w^2 + 5*w - 13],
[359, 359, 3*w^2 + 6*w - 1],
[379, 379, w^2 - 3*w - 6],
[383, 383, 3*w^2 - 23],
[383, 383, w^2 - 12],
[383, 383, 3*w^2 + 3*w - 13],
[401, 401, -6*w^2 - 7*w + 16],
[409, 409, -4*w^2 + 3*w + 21],
[421, 421, w^2 - 3*w - 9],
[433, 433, w^2 - 3*w - 8],
[433, 433, w^2 + 4*w - 8],
[433, 433, 3*w^2 - w - 12],
[439, 439, -4*w - 11],
[449, 449, 2*w^2 + 3*w - 19],
[457, 457, w^2 - 4*w - 3],
[461, 461, 5*w^2 + 6*w - 16],
[461, 461, -2*w^2 + 3*w + 16],
[461, 461, 2*w^2 + 4*w - 9],
[467, 467, 3*w^2 - 4*w - 12],
[479, 479, 3*w^2 + w - 9],
[487, 487, 5*w^2 + 2*w - 22],
[491, 491, 3*w^2 - 11],
[499, 499, 5*w^2 + w - 24],
[503, 503, 3*w^2 + 2*w - 17],
[503, 503, 3*w^2 + w - 7],
[503, 503, 2*w^2 - 4*w - 7],
[509, 509, 3*w - 11],
[521, 521, 3*w^2 - 3*w - 16],
[523, 523, -w - 8],
[541, 541, 6*w^2 + 7*w - 19],
[547, 547, 3*w^2 + 4*w - 12],
[557, 557, 2*w^2 - 3*w - 12],
[563, 563, 4*w^2 + 3*w - 12],
[563, 563, 4*w^2 + w - 17],
[563, 563, -w^2 - 5*w - 3],
[569, 569, w^2 - 4*w - 4],
[577, 577, w^2 - w - 13],
[587, 587, 4*w^2 + 3*w - 8],
[587, 587, 2*w^2 - 3*w - 13],
[587, 587, 2*w^2 + 3*w - 18],
[593, 593, 4*w - 13],
[599, 599, w - 9],
[613, 613, -w^2 + 3*w - 7],
[613, 613, w^2 + 5*w + 2],
[613, 613, 3*w^2 - 3*w - 17],
[617, 617, w^2 + 5*w + 8],
[641, 641, 4*w^2 + 3*w - 11],
[641, 641, w^2 + w - 14],
[641, 641, 3*w^2 - 2*w - 9],
[647, 647, 2*w^2 + 3*w - 13],
[653, 653, 3*w^2 - 8],
[659, 659, 5*w^2 + 4*w - 21],
[659, 659, 2*w^2 - 4*w - 19],
[659, 659, -2*w^2 - 6*w + 9],
[677, 677, -6*w - 13],
[691, 691, 6*w - 11],
[701, 701, 3*w^2 - 3*w - 19],
[719, 719, 2*w^2 + 3*w - 16],
[727, 727, 3*w^2 + 4*w - 13],
[733, 733, 3*w^2 - w - 8],
[739, 739, 2*w^2 + 4*w - 11],
[739, 739, 4*w^2 - 3*w - 24],
[739, 739, 3*w^2 - w - 6],
[743, 743, 4*w^2 - 2*w - 17],
[751, 751, -w - 9],
[761, 761, 5*w^2 + 5*w - 19],
[769, 769, w^2 - 4*w - 6],
[769, 769, -5*w - 1],
[769, 769, 2*w^2 - 19],
[773, 773, 2*w^2 - 6*w - 3],
[797, 797, 4*w^2 - 17],
[809, 809, -7*w^2 - 4*w + 29],
[811, 811, 6*w^2 + 3*w - 29],
[823, 823, 4*w^2 + 2*w - 13],
[823, 823, -w^2 - 4*w - 7],
[823, 823, 4*w^2 + 4*w - 17],
[827, 827, -2*w^2 - 3*w - 2],
[829, 829, -w^2 + w - 6],
[829, 829, -w^2 + 4*w - 9],
[829, 829, -2*w^2 + 3*w + 21],
[839, 839, 6*w^2 - 2*w - 31],
[841, 29, w^2 + 3*w + 6],
[853, 853, -7*w^2 - 6*w + 27],
[859, 859, 6*w - 1],
[881, 881, w^2 + 5*w - 16],
[907, 907, 2*w^2 + 7*w + 7],
[919, 919, w^2 - 4*w - 9],
[953, 953, -4*w^2 - w + 32],
[967, 967, 5*w^2 - 5*w - 18],
[971, 971, 3*w^2 - 4*w - 24],
[977, 977, 2*w^2 + 4*w + 3],
[997, 997, w^2 + 5*w - 12],
[997, 997, 5*w^2 - 3*w - 22],
[997, 997, 3*w^2 + 8*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 - 4*x^4 - 8*x^3 + 26*x^2 + 19*x - 9;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -4/19*e^4 + 18/19*e^3 + 23/19*e^2 - 106/19*e - 42/19, -1, -2/19*e^4 + 9/19*e^3 + 21/19*e^2 - 72/19*e - 59/19, 1/19*e^4 + 5/19*e^3 - 39/19*e^2 - 21/19*e + 134/19, -1/19*e^4 - 5/19*e^3 + 20/19*e^2 + 40/19*e + 18/19, 4/19*e^4 - 18/19*e^3 - 23/19*e^2 + 87/19*e + 99/19, 4/19*e^4 - 18/19*e^3 - 4/19*e^2 + 68/19*e - 72/19, 4/19*e^4 - 18/19*e^3 - 23/19*e^2 + 87/19*e + 99/19, -11/19*e^4 + 40/19*e^3 + 87/19*e^2 - 263/19*e - 87/19, 5/19*e^4 - 32/19*e^3 - 5/19*e^2 + 199/19*e + 5/19, -15/19*e^4 + 58/19*e^3 + 91/19*e^2 - 255/19*e - 129/19, 10/19*e^4 - 26/19*e^3 - 86/19*e^2 + 94/19*e + 162/19, 4/19*e^4 - 18/19*e^3 - 23/19*e^2 + 163/19*e - 15/19, -10/19*e^4 + 45/19*e^3 + 67/19*e^2 - 284/19*e - 67/19, 3/19*e^4 - 4/19*e^3 - 79/19*e^2 + 108/19*e + 269/19, -2/19*e^4 - 10/19*e^3 + 78/19*e^2 + 42/19*e - 154/19, -6/19*e^4 + 8/19*e^3 + 82/19*e^2 + 12/19*e - 196/19, -2/19*e^4 + 9/19*e^3 + 21/19*e^2 - 110/19*e + 131/19, -10/19*e^4 + 45/19*e^3 + 48/19*e^2 - 227/19*e - 124/19, 7/19*e^4 - 22/19*e^3 - 83/19*e^2 + 233/19*e + 159/19, 2*e^2 - 4*e - 16, -2/19*e^4 + 28/19*e^3 - 36/19*e^2 - 186/19*e + 150/19, e^4 - 5*e^3 - 2*e^2 + 26*e, 6/19*e^4 - 46/19*e^3 - 25/19*e^2 + 406/19*e + 6/19, 17/19*e^4 - 86/19*e^3 - 93/19*e^2 + 517/19*e + 207/19, -e^2 + 5*e + 11, -6/19*e^4 + 27/19*e^3 - 13/19*e^2 - 26/19*e + 393/19, -15/19*e^4 + 58/19*e^3 + 91/19*e^2 - 312/19*e + 23/19, -7/19*e^4 + 22/19*e^3 + 7/19*e^2 + 33/19*e + 107/19, -e^3 + 3*e^2 + 6*e - 13, 10/19*e^4 - 45/19*e^3 + 28/19*e^2 + 132/19*e - 427/19, 16/19*e^4 - 72/19*e^3 - 92/19*e^2 + 500/19*e + 92/19, -22/19*e^4 + 61/19*e^3 + 212/19*e^2 - 374/19*e - 193/19, 20/19*e^4 - 90/19*e^3 - 134/19*e^2 + 568/19*e + 96/19, 21/19*e^4 - 104/19*e^3 - 97/19*e^2 + 623/19*e + 287/19, 10/19*e^4 - 45/19*e^3 - 48/19*e^2 + 170/19*e - 9/19, -4/19*e^4 + 18/19*e^3 + 4/19*e^2 - 182/19*e + 186/19, 8/19*e^4 + 2/19*e^3 - 122/19*e^2 - 73/19*e + 236/19, 4/19*e^4 - 37/19*e^3 + 34/19*e^2 + 258/19*e - 243/19, 2/19*e^4 - 9/19*e^3 - 21/19*e^2 + 72/19*e - 55/19, 3/19*e^4 - 4/19*e^3 - 60/19*e^2 + 70/19*e + 60/19, e^4 - 4*e^3 - 5*e^2 + 14*e + 5, -16/19*e^4 + 72/19*e^3 + 73/19*e^2 - 481/19*e - 73/19, 10/19*e^4 - 45/19*e^3 - 86/19*e^2 + 284/19*e + 143/19, 3/19*e^4 - 42/19*e^3 + 73/19*e^2 + 184/19*e - 187/19, 21/19*e^4 - 85/19*e^3 - 97/19*e^2 + 357/19*e - 112/19, 6/19*e^4 + 11/19*e^3 - 139/19*e^2 - 88/19*e + 291/19, -21/19*e^4 + 66/19*e^3 + 173/19*e^2 - 300/19*e - 249/19, -12/19*e^4 + 73/19*e^3 + 50/19*e^2 - 584/19*e - 69/19, 22/19*e^4 - 80/19*e^3 - 136/19*e^2 + 298/19*e + 174/19, 2/19*e^4 - 9/19*e^3 - 21/19*e^2 - 4/19*e + 287/19, -4/19*e^4 + 18/19*e^3 + 23/19*e^2 - 49/19*e - 213/19, -13/19*e^4 + 30/19*e^3 + 127/19*e^2 - 107/19*e - 51/19, -14/19*e^4 + 82/19*e^3 + 14/19*e^2 - 542/19*e + 24/19, -9/19*e^4 + 12/19*e^3 + 47/19*e^2 + 113/19*e + 105/19, 18/19*e^4 - 62/19*e^3 - 132/19*e^2 + 268/19*e + 170/19, -1/19*e^4 + 14/19*e^3 + 1/19*e^2 - 131/19*e - 1/19, 6/19*e^4 - 8/19*e^3 - 63/19*e^2 - 31/19*e + 101/19, -18/19*e^4 + 43/19*e^3 + 189/19*e^2 - 192/19*e - 151/19, 22/19*e^4 - 99/19*e^3 - 60/19*e^2 + 488/19*e - 73/19, -12/19*e^4 + 54/19*e^3 + 12/19*e^2 - 280/19*e + 254/19, -17/19*e^4 + 86/19*e^3 + 55/19*e^2 - 498/19*e - 169/19, 12/19*e^4 - 16/19*e^3 - 145/19*e^2 - 5/19*e + 449/19, 17/19*e^4 - 67/19*e^3 - 150/19*e^2 + 422/19*e + 492/19, -4/19*e^4 + 18/19*e^3 + 42/19*e^2 - 220/19*e - 118/19, -9/19*e^4 + 12/19*e^3 + 85/19*e^2 - 20/19*e + 29/19, -28/19*e^4 + 126/19*e^3 + 142/19*e^2 - 742/19*e - 256/19, -16/19*e^4 + 72/19*e^3 + 149/19*e^2 - 519/19*e - 529/19, 2*e^2 - 8*e - 6, -5/19*e^4 + 32/19*e^3 - 14/19*e^2 - 218/19*e + 204/19, -8/19*e^4 - 2/19*e^3 + 160/19*e^2 + 92/19*e - 540/19, 22/19*e^4 - 42/19*e^3 - 212/19*e^2 + 32/19*e + 440/19, 7/19*e^4 + 16/19*e^3 - 197/19*e^2 - 52/19*e + 615/19, 22/19*e^4 - 118/19*e^3 - 98/19*e^2 + 754/19*e + 402/19, 3/19*e^4 - 42/19*e^3 + 73/19*e^2 + 298/19*e - 111/19, 49/19*e^4 - 173/19*e^3 - 334/19*e^2 + 890/19*e + 258/19, 26/19*e^4 - 117/19*e^3 - 102/19*e^2 + 556/19*e - 31/19, 1/19*e^4 - 14/19*e^3 - 1/19*e^2 + 169/19*e + 305/19, -13/19*e^4 + 30/19*e^3 + 89/19*e^2 - 31/19*e - 13/19, 3/19*e^4 + 34/19*e^3 - 174/19*e^2 - 234/19*e + 668/19, 2/19*e^4 + 48/19*e^3 - 154/19*e^2 - 308/19*e + 344/19, 11/19*e^4 - 78/19*e^3 - 11/19*e^2 + 510/19*e + 239/19, -18/19*e^4 + 81/19*e^3 + 75/19*e^2 - 344/19*e - 75/19, -14/19*e^4 + 63/19*e^3 + 109/19*e^2 - 504/19*e - 451/19, -18/19*e^4 + 81/19*e^3 - 20/19*e^2 - 249/19*e + 552/19, 2/19*e^4 + 10/19*e^3 + 55/19*e^2 - 346/19*e - 264/19, 29/19*e^4 - 121/19*e^3 - 143/19*e^2 + 493/19*e + 48/19, 14/19*e^4 - 44/19*e^3 - 90/19*e^2 + 86/19*e + 90/19, -8/19*e^4 + 36/19*e^3 - 11/19*e^2 - 41/19*e + 543/19, 12/19*e^4 - 54/19*e^3 - 126/19*e^2 + 508/19*e + 164/19, 2/19*e^4 + 48/19*e^3 - 230/19*e^2 - 194/19*e + 648/19, 20/19*e^4 - 90/19*e^3 - 77/19*e^2 + 283/19*e - 37/19, 12/19*e^4 - 16/19*e^3 - 164/19*e^2 - 24/19*e + 240/19, -6/19*e^4 + 46/19*e^3 - 32/19*e^2 - 178/19*e + 108/19, 8/19*e^4 - 93/19*e^3 + 30/19*e^2 + 744/19*e - 87/19, 5/19*e^4 + 25/19*e^3 - 138/19*e^2 - 162/19*e + 252/19, -7/19*e^4 + 60/19*e^3 - 107/19*e^2 - 252/19*e + 411/19, 4/19*e^4 - 37/19*e^3 - 4/19*e^2 + 448/19*e - 205/19, -9/19*e^4 + 50/19*e^3 + 66/19*e^2 - 400/19*e - 142/19, -11/19*e^4 + 78/19*e^3 - 27/19*e^2 - 396/19*e + 521/19, 28/19*e^4 - 88/19*e^3 - 256/19*e^2 + 400/19*e + 522/19, 2/19*e^4 + 10/19*e^3 - 2/19*e^2 - 232/19*e - 264/19, 29/19*e^4 - 102/19*e^3 - 219/19*e^2 + 664/19*e - 9/19, 4/19*e^4 - 18/19*e^3 - 4/19*e^2 + 144/19*e + 156/19, -2/19*e^4 - 29/19*e^3 + 40/19*e^2 + 251/19*e + 378/19, -14/19*e^4 + 63/19*e^3 + 90/19*e^2 - 485/19*e - 280/19, 8/19*e^4 - 55/19*e^3 - 8/19*e^2 + 402/19*e - 315/19, 10/19*e^4 - 64/19*e^3 - 67/19*e^2 + 607/19*e + 105/19, 2/19*e^4 - 9/19*e^3 + 36/19*e^2 + 34/19*e - 435/19, 32/19*e^4 - 125/19*e^3 - 279/19*e^2 + 848/19*e + 279/19, 16/19*e^4 - 53/19*e^3 - 16/19*e^2 + 120/19*e - 459/19, -28/19*e^4 + 107/19*e^3 + 256/19*e^2 - 647/19*e - 712/19, -2/19*e^4 - 29/19*e^3 + 135/19*e^2 + 42/19*e - 439/19, 42/19*e^4 - 170/19*e^3 - 308/19*e^2 + 942/19*e + 308/19, 16/19*e^4 - 15/19*e^3 - 168/19*e^2 - 127/19*e + 510/19, -1/19*e^4 + 14/19*e^3 + 20/19*e^2 - 150/19*e + 246/19, -e^3 + e^2 + 4*e - 3, 7/19*e^4 + 35/19*e^3 - 235/19*e^2 - 261/19*e + 672/19, 28/19*e^4 - 126/19*e^3 - 47/19*e^2 + 381/19*e - 561/19, 18/19*e^4 - 43/19*e^3 - 246/19*e^2 + 325/19*e + 474/19, 4/19*e^4 - 37/19*e^3 + 72/19*e^2 + 68/19*e - 357/19, -10/19*e^4 + 26/19*e^3 + 86/19*e^2 - 170/19*e + 636/19, -12/19*e^4 + 16/19*e^3 + 126/19*e^2 - 14/19*e - 240/19, -17/19*e^4 + 67/19*e^3 + 150/19*e^2 - 460/19*e + 192/19, 6/19*e^4 - 84/19*e^3 + 222/19*e^2 + 406/19*e - 754/19, -2*e^4 + 7*e^3 + 17*e^2 - 44*e - 3, 22/19*e^4 - 80/19*e^3 - 174/19*e^2 + 526/19*e + 402/19, -47/19*e^4 + 202/19*e^3 + 199/19*e^2 - 932/19*e + 29/19, -22/19*e^4 + 61/19*e^3 + 193/19*e^2 - 70/19*e - 649/19, 14/19*e^4 - 82/19*e^3 - 14/19*e^2 + 390/19*e + 128/19, 10/19*e^4 - 26/19*e^3 - 200/19*e^2 + 170/19*e + 656/19, -4/19*e^4 + 18/19*e^3 + 137/19*e^2 - 277/19*e - 631/19, -32/19*e^4 + 144/19*e^3 + 108/19*e^2 - 734/19*e - 222/19, 12/19*e^4 - 16/19*e^3 - 145/19*e^2 + 223/19*e - 7/19, 3/19*e^4 + 53/19*e^3 - 155/19*e^2 - 329/19*e + 174/19, -34/19*e^4 + 153/19*e^3 + 205/19*e^2 - 692/19*e - 661/19, 35/19*e^4 - 110/19*e^3 - 263/19*e^2 + 443/19*e + 149/19, -14/19*e^4 + 6/19*e^3 + 223/19*e^2 + 180/19*e - 850/19, -9/19*e^4 + 88/19*e^3 + 9/19*e^2 - 685/19*e - 351/19, -21/19*e^4 + 104/19*e^3 + 59/19*e^2 - 433/19*e + 207/19, -33/19*e^4 + 158/19*e^3 + 33/19*e^2 - 751/19*e + 195/19, 26/19*e^4 - 98/19*e^3 - 178/19*e^2 + 252/19*e + 482/19, -24/19*e^4 + 146/19*e^3 + 62/19*e^2 - 978/19*e - 328/19, -8/19*e^4 - 2/19*e^3 + 46/19*e^2 + 282/19*e + 68/19, -24/19*e^4 + 108/19*e^3 + 81/19*e^2 - 693/19*e + 71/19, 4/19*e^4 - 56/19*e^3 + 110/19*e^2 + 486/19*e - 414/19, 12/19*e^4 - 35/19*e^3 - 183/19*e^2 + 166/19*e + 791/19, 16/19*e^4 - 91/19*e^3 + 79/19*e^2 + 424/19*e - 535/19, 5/19*e^4 - 32/19*e^3 - 5/19*e^2 + 66/19*e - 451/19, -14/19*e^4 + 82/19*e^3 + 71/19*e^2 - 485/19*e - 375/19, -13/19*e^4 - 8/19*e^3 + 279/19*e^2 + 83/19*e - 697/19, -24/19*e^4 + 70/19*e^3 + 290/19*e^2 - 693/19*e - 784/19, 2*e^4 - 8*e^3 - 12*e^2 + 50*e + 8, 14/19*e^4 - 82/19*e^3 - 14/19*e^2 + 447/19*e - 24/19, 54/19*e^4 - 224/19*e^3 - 263/19*e^2 + 1089/19*e + 377/19, 69/19*e^4 - 244/19*e^3 - 487/19*e^2 + 1268/19*e + 677/19, -e^4 + 6*e^3 - 3*e^2 - 26*e + 45, -36/19*e^4 + 124/19*e^3 + 188/19*e^2 - 612/19*e + 116/19, 10/19*e^4 - 64/19*e^3 + 66/19*e^2 + 94/19*e - 294/19, -15/19*e^4 - 18/19*e^3 + 357/19*e^2 + 11/19*e - 699/19, -4/19*e^4 + 18/19*e^3 - 15/19*e^2 + 65/19*e + 53/19, -32/19*e^4 + 182/19*e^3 + 146/19*e^2 - 1285/19*e - 526/19, -43/19*e^4 + 146/19*e^3 + 385/19*e^2 - 769/19*e - 385/19];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w - 2}>] := -1;
ALEigenvalues[ideal<ZF | {8, 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;