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

NN := ideal<ZF | {13, 13, w^3 - 2*w^2 - 4*w + 2}>;

primesArray := [
[3, 3, w],
[4, 2, -w^3 + 3*w^2 + w - 2],
[11, 11, -w^3 + 3*w^2 + 2*w - 5],
[13, 13, w^3 - 2*w^2 - 4*w + 2],
[13, 13, -w + 2],
[17, 17, w^3 - 3*w^2 - 2*w + 2],
[19, 19, -w^2 + w + 4],
[19, 19, w^3 - 2*w^2 - 3*w + 2],
[23, 23, w^2 - 2*w - 1],
[27, 3, w^3 - 2*w^2 - 5*w - 1],
[29, 29, -w^3 + 2*w^2 + 5*w - 1],
[37, 37, -w^3 + 2*w^2 + 5*w - 4],
[41, 41, 2*w^3 - 5*w^2 - 6*w + 4],
[53, 53, w^3 - 2*w^2 - 3*w - 2],
[59, 59, w - 4],
[67, 67, 2*w^2 - 3*w - 8],
[73, 73, 2*w^3 - 5*w^2 - 6*w + 5],
[89, 89, -2*w^3 + 6*w^2 + 5*w - 7],
[97, 97, -2*w^3 + 6*w^2 + 3*w - 7],
[97, 97, -w^3 + 3*w^2 + 3*w - 1],
[103, 103, w^2 - 4*w - 4],
[113, 113, 2*w^3 - 5*w^2 - 4*w + 1],
[113, 113, -w^3 + 3*w^2 + w - 5],
[139, 139, 2*w^2 - 3*w - 4],
[139, 139, w^3 - w^2 - 6*w - 1],
[149, 149, 2*w^3 - 5*w^2 - 5*w + 2],
[149, 149, w^3 - 4*w^2 + 10],
[151, 151, -3*w - 1],
[157, 157, w^2 - w - 8],
[157, 157, w^3 - 7*w - 8],
[163, 163, -3*w^3 + 8*w^2 + 8*w - 10],
[163, 163, w^3 - 3*w^2 - 4*w + 7],
[163, 163, -w^3 + 4*w^2 + w - 8],
[163, 163, -2*w^3 + 5*w^2 + 7*w - 4],
[167, 167, -2*w^3 + 5*w^2 + 8*w - 4],
[169, 13, -2*w^3 + 4*w^2 + 7*w - 5],
[173, 173, 2*w^3 - 4*w^2 - 9*w + 4],
[173, 173, -w^3 + 5*w^2 - 2*w - 5],
[181, 181, 3*w^3 - 7*w^2 - 12*w + 7],
[191, 191, w^3 - 3*w^2 - 3*w - 1],
[199, 199, 2*w^3 - 3*w^2 - 10*w - 1],
[199, 199, 2*w^3 - 6*w^2 - 6*w + 13],
[223, 223, -2*w^3 + 6*w^2 + 4*w - 11],
[223, 223, 3*w^3 - 7*w^2 - 12*w + 8],
[223, 223, -2*w^3 + 5*w^2 + 6*w - 1],
[229, 229, -w^3 + 3*w^2 + 2*w + 1],
[229, 229, 3*w^3 - 4*w^2 - 15*w - 5],
[233, 233, -3*w^3 + 7*w^2 + 11*w - 5],
[233, 233, 2*w^3 - 5*w^2 - 4*w + 5],
[241, 241, w^3 - 3*w^2 - 5*w + 7],
[257, 257, w^3 - 3*w^2 - 4*w + 8],
[263, 263, w^2 - 4*w - 5],
[271, 271, -2*w^3 + 6*w^2 + 5*w - 13],
[277, 277, -2*w^3 + 6*w^2 + 3*w - 8],
[277, 277, w^3 - 2*w^2 - 2*w - 2],
[281, 281, 2*w^3 - 5*w^2 - 4*w + 2],
[283, 283, -3*w^2 + 7*w + 10],
[283, 283, -w^3 + 4*w^2 - w - 7],
[293, 293, 2*w^3 - 5*w^2 - 3*w + 4],
[293, 293, w^3 - 8*w - 4],
[307, 307, -w^3 + 2*w^2 + 6*w - 2],
[311, 311, w^2 - 5],
[317, 317, -w^3 + 5*w^2 - 2*w - 11],
[331, 331, -w^3 + 3*w^2 - 5],
[349, 349, -2*w^2 + 5*w + 2],
[353, 353, -2*w^3 + 5*w^2 + 4*w - 4],
[353, 353, 3*w^3 - 7*w^2 - 10*w + 2],
[353, 353, -2*w^3 + 7*w^2 + 2*w - 10],
[353, 353, w^3 + w^2 - 9*w - 13],
[361, 19, -w^3 + w^2 + 5*w - 1],
[373, 373, -w^3 + 3*w^2 + 4*w - 11],
[373, 373, 2*w^3 - 5*w^2 - 5*w - 2],
[383, 383, w^3 - 5*w^2 + w + 13],
[389, 389, 3*w^3 - 10*w^2 - 4*w + 16],
[397, 397, -w^3 + 2*w^2 + 3*w + 5],
[397, 397, -3*w^3 + 7*w^2 + 10*w - 7],
[419, 419, -w^3 + 2*w^2 + 4*w + 4],
[419, 419, 3*w^3 - 7*w^2 - 12*w + 5],
[421, 421, -w^2 + 5*w - 2],
[421, 421, 3*w^3 - 7*w^2 - 10*w + 5],
[431, 431, w^2 - 10],
[439, 439, w^3 - w^2 - 7*w - 7],
[461, 461, -2*w^3 + 7*w^2 + 5*w - 10],
[467, 467, -3*w^3 + 6*w^2 + 14*w - 4],
[479, 479, 3*w^2 - 6*w - 5],
[479, 479, -4*w^3 + 11*w^2 + 10*w - 10],
[491, 491, -w^3 + w^2 + 6*w - 1],
[499, 499, -4*w^3 + 9*w^2 + 15*w - 10],
[499, 499, 2*w^3 - 5*w^2 - 7*w + 2],
[503, 503, -2*w^3 + 7*w^2 - 8],
[521, 521, 2*w^2 - w - 7],
[521, 521, -3*w^3 + 7*w^2 + 11*w - 11],
[569, 569, 2*w^2 - 5*w - 1],
[569, 569, -4*w - 1],
[571, 571, -w^3 + 3*w^2 - 7],
[577, 577, -w^3 + w^2 + 8*w - 4],
[587, 587, -3*w^3 + 8*w^2 + 7*w - 8],
[593, 593, 3*w^2 - 4*w - 11],
[593, 593, -3*w^3 + 9*w^2 + 6*w - 14],
[601, 601, -3*w^3 + 9*w^2 + 4*w - 10],
[601, 601, 5*w^3 - 9*w^2 - 24*w - 2],
[607, 607, 2*w^2 - 7*w - 5],
[613, 613, w^3 - 6*w - 2],
[613, 613, -w^2 + 3*w + 8],
[617, 617, -w^2 + 4*w - 5],
[619, 619, 3*w^3 - 6*w^2 - 13*w + 5],
[625, 5, -5],
[631, 631, -2*w^3 + 4*w^2 + 8*w - 7],
[647, 647, -2*w^3 + 4*w^2 + 5*w - 1],
[647, 647, w^3 - 3*w^2 - 2*w - 2],
[653, 653, w^3 - 4*w^2 - w + 13],
[653, 653, w^3 - 2*w^2 - 7*w + 5],
[659, 659, -4*w^3 + 10*w^2 + 15*w - 14],
[659, 659, 5*w^3 - 12*w^2 - 17*w + 14],
[661, 661, w^2 - 5*w - 2],
[661, 661, -w^3 + 5*w^2 - 2*w - 7],
[673, 673, -w^3 + w^2 + 6*w - 2],
[683, 683, 2*w^3 - 8*w^2 + w + 8],
[683, 683, 5*w^3 - 12*w^2 - 19*w + 14],
[701, 701, -2*w^3 + 6*w^2 + 5*w - 4],
[709, 709, -w^2 + 3*w - 4],
[709, 709, -2*w^3 + 5*w^2 + 9*w - 8],
[719, 719, w^2 - 7],
[719, 719, -2*w^3 + 3*w^2 + 8*w + 5],
[733, 733, -2*w^3 + 7*w^2 + 3*w - 10],
[739, 739, 4*w^3 - 11*w^2 - 13*w + 14],
[739, 739, -5*w^3 + 10*w^2 + 21*w - 4],
[739, 739, 3*w^3 - 8*w^2 - 6*w + 8],
[739, 739, -w^3 + w^2 + 8*w - 5],
[743, 743, -w - 5],
[743, 743, w^3 - w^2 - 8*w - 8],
[751, 751, -2*w^3 + 5*w^2 + 8*w - 2],
[751, 751, -2*w^3 + 6*w^2 - 1],
[757, 757, -2*w^3 + 6*w^2 + 3*w - 10],
[757, 757, 5*w^2 - 9*w - 22],
[761, 761, -2*w^3 + 4*w^2 + 9*w - 7],
[761, 761, 2*w^3 - 3*w^2 - 12*w - 1],
[769, 769, 5*w^3 - 13*w^2 - 14*w + 10],
[773, 773, -3*w^3 + 5*w^2 + 16*w + 1],
[773, 773, w^3 - 11*w + 2],
[787, 787, -w^3 + 4*w^2 - w - 11],
[797, 797, -5*w^3 + 12*w^2 + 17*w - 11],
[809, 809, -2*w^3 + 7*w^2 + 4*w - 11],
[809, 809, 4*w^3 - 9*w^2 - 14*w + 4],
[821, 821, -w^3 + 5*w^2 - 2*w - 17],
[827, 827, -3*w^3 + 8*w^2 + 9*w - 7],
[829, 829, 2*w^3 - 6*w^2 - 5*w + 2],
[829, 829, -w^3 + 4*w^2 - w - 10],
[839, 839, w^2 - 8],
[853, 853, -5*w^3 + 12*w^2 + 18*w - 16],
[853, 853, w^3 - 6*w - 8],
[863, 863, 5*w^3 - 11*w^2 - 19*w + 7],
[881, 881, -3*w^3 + 9*w^2 + 9*w - 11],
[907, 907, -w^3 + 2*w^2 + 7*w - 4],
[907, 907, -2*w^3 + 3*w^2 + 7*w + 4],
[929, 929, -w^3 + 5*w^2 - 3*w - 11],
[937, 937, 2*w^3 - 5*w^2 - 6*w + 11],
[937, 937, -2*w^3 + 6*w^2 + 3*w - 11],
[941, 941, -3*w^3 + 8*w^2 + 11*w - 8],
[953, 953, -4*w^3 + 10*w^2 + 10*w - 13],
[953, 953, w^3 - 5*w^2 - w + 13],
[967, 967, 3*w^3 - 7*w^2 - 9*w + 7],
[967, 967, -2*w^3 + 5*w^2 + 8*w + 1],
[967, 967, 4*w^3 - 13*w^2 - 6*w + 20],
[967, 967, -2*w^2 + 3*w + 13],
[971, 971, -6*w^3 + 13*w^2 + 25*w - 8],
[977, 977, w^3 - w^2 - 4*w - 10],
[977, 977, 2*w^3 - 6*w^2 - 7*w + 13],
[983, 983, -4*w^3 + 10*w^2 + 11*w - 13],
[983, 983, -w^3 + 5*w^2 - w - 7],
[983, 983, 3*w^3 - 9*w^2 - 7*w + 19],
[983, 983, -2*w^3 + 3*w^2 + 10*w - 1],
[997, 997, 5*w^3 - 12*w^2 - 17*w + 13],
[997, 997, -2*w^3 + 3*w^2 + 13*w - 4],
[997, 997, w^3 - 7*w - 2],
[997, 997, w^3 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 - 3*x^8 - 13*x^7 + 36*x^6 + 47*x^5 - 110*x^4 - 44*x^3 + 100*x^2 + 10*x - 25;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -4/35*e^8 + 16/105*e^7 + 206/105*e^6 - 26/15*e^5 - 1124/105*e^4 + 97/21*e^3 + 671/35*e^2 - 29/7*e - 169/21, 34/105*e^8 - 19/35*e^7 - 179/35*e^6 + 77/15*e^5 + 871/35*e^4 - 122/21*e^3 - 1207/35*e^2 - 58/21*e + 263/21, -1, -4/35*e^8 + 16/105*e^7 + 206/105*e^6 - 26/15*e^5 - 1124/105*e^4 + 97/21*e^3 + 671/35*e^2 - 22/7*e - 211/21, 46/105*e^8 - 73/105*e^7 - 743/105*e^6 + 103/15*e^5 + 3737/105*e^4 - 73/7*e^3 - 1843/35*e^2 + 8/21*e + 137/7, 16/105*e^8 - 11/35*e^7 - 76/35*e^6 + 53/15*e^5 + 309/35*e^4 - 176/21*e^3 - 323/35*e^2 + 11/21*e + 41/21, 16/105*e^8 - 11/35*e^7 - 76/35*e^6 + 53/15*e^5 + 309/35*e^4 - 197/21*e^3 - 288/35*e^2 + 116/21*e + 41/21, -88/105*e^8 + 199/105*e^7 + 1289/105*e^6 - 319/15*e^5 - 5711/105*e^4 + 374/7*e^3 + 2494/35*e^2 - 638/21*e - 165/7, 53/105*e^8 - 43/35*e^7 - 243/35*e^6 + 199/15*e^5 + 982/35*e^4 - 625/21*e^3 - 1164/35*e^2 + 253/21*e + 292/21, 94/105*e^8 - 242/105*e^7 - 1252/105*e^6 + 129/5*e^5 + 4663/105*e^4 - 1363/21*e^3 - 1447/35*e^2 + 734/21*e + 268/21, 6/35*e^8 - 8/35*e^7 - 103/35*e^6 + 13/5*e^5 + 562/35*e^4 - 45/7*e^3 - 1024/35*e^2 + 12/7*e + 116/7, -8/15*e^8 + 14/15*e^7 + 124/15*e^6 - 143/15*e^5 - 601/15*e^4 + 18*e^3 + 309/5*e^2 - 22/3*e - 26, 22/105*e^8 - 37/35*e^7 - 52/35*e^6 + 191/15*e^5 - 157/35*e^4 - 809/21*e^3 + 1074/35*e^2 + 506/21*e - 361/21, -62/105*e^8 + 47/35*e^7 + 312/35*e^6 - 226/15*e^5 - 1473/35*e^4 + 766/21*e^3 + 2271/35*e^2 - 292/21*e - 613/21, -118/105*e^8 + 169/105*e^7 + 1979/105*e^6 - 244/15*e^5 - 10376/105*e^4 + 197/7*e^3 + 5414/35*e^2 - 278/21*e - 405/7, 13/21*e^8 - 12/7*e^7 - 60/7*e^6 + 59/3*e^5 + 241/7*e^4 - 1093/21*e^3 - 241/7*e^2 + 739/21*e + 181/21, -89/105*e^8 + 212/105*e^7 + 1207/105*e^6 - 317/15*e^5 - 4603/105*e^4 + 296/7*e^3 + 1357/35*e^2 - 283/21*e - 75/7, 11/105*e^8 - 73/105*e^7 - 8/105*e^6 + 41/5*e^5 - 988/105*e^4 - 485/21*e^3 + 992/35*e^2 + 169/21*e - 205/21, 1/15*e^8 - 1/5*e^7 - 1/5*e^6 + 26/15*e^5 - 31/5*e^4 - 2/3*e^3 + 147/5*e^2 - 13/3*e - 46/3, -12/35*e^8 + 118/105*e^7 + 443/105*e^6 - 188/15*e^5 - 1412/105*e^4 + 634/21*e^3 + 228/35*e^2 - 101/7*e - 10/21, 17/35*e^8 - 103/105*e^7 - 788/105*e^6 + 158/15*e^5 + 3902/105*e^4 - 505/21*e^3 - 2213/35*e^2 + 111/7*e + 706/21, -8/105*e^8 + 34/105*e^7 + 44/105*e^6 - 18/5*e^5 + 499/105*e^4 + 158/21*e^3 - 1011/35*e^2 + 68/21*e + 361/21, -68/105*e^8 + 184/105*e^7 + 899/105*e^6 - 103/5*e^5 - 3371/105*e^4 + 1280/21*e^3 + 1154/35*e^2 - 1039/21*e - 113/21, 29/21*e^8 - 55/21*e^7 - 443/21*e^6 + 27*e^5 + 2042/21*e^4 - 1063/21*e^3 - 851/7*e^2 + 437/21*e + 757/21, -302/105*e^8 + 706/105*e^7 + 4286/105*e^6 - 367/5*e^5 - 17939/105*e^4 + 3539/21*e^3 + 6906/35*e^2 - 1654/21*e - 1340/21, 17/15*e^8 - 31/15*e^7 - 266/15*e^6 + 109/5*e^5 + 1244/15*e^4 - 131/3*e^3 - 481/5*e^2 + 70/3*e + 53/3, -13/21*e^8 + 29/21*e^7 + 187/21*e^6 - 16*e^5 - 793/21*e^4 + 953/21*e^3 + 318/7*e^2 - 802/21*e - 209/21, -13/105*e^8 + 134/105*e^7 + 19/105*e^6 - 83/5*e^5 + 1034/105*e^4 + 1144/21*e^3 - 1236/35*e^2 - 614/21*e + 416/21, -28/15*e^8 + 23/5*e^7 + 128/5*e^6 - 758/15*e^5 - 507/5*e^4 + 359/3*e^3 + 529/5*e^2 - 173/3*e - 68/3, 17/21*e^8 - 25/21*e^7 - 272/21*e^6 + 11*e^5 + 1352/21*e^4 - 214/21*e^3 - 684/7*e^2 - 250/21*e + 934/21, -72/35*e^8 + 201/35*e^7 + 921/35*e^6 - 326/5*e^5 - 3174/35*e^4 + 1198/7*e^3 + 2418/35*e^2 - 676/7*e - 111/7, -31/105*e^8 + 6/35*e^7 + 191/35*e^6 - 23/15*e^5 - 1104/35*e^4 + 26/21*e^3 + 1993/35*e^2 + 22/21*e - 611/21, -32/105*e^8 + 241/105*e^7 + 71/105*e^6 - 142/5*e^5 + 2206/105*e^4 + 1871/21*e^3 - 2329/35*e^2 - 1114/21*e + 562/21, 2/15*e^8 - 1/15*e^7 - 41/15*e^6 + 4/5*e^5 + 254/15*e^4 - 5/3*e^3 - 146/5*e^2 + 10/3*e + 44/3, -43/35*e^8 + 242/105*e^7 + 1987/105*e^6 - 352/15*e^5 - 9493/105*e^4 + 908/21*e^3 + 4527/35*e^2 - 135/7*e - 1010/21, -59/35*e^8 + 137/35*e^7 + 832/35*e^6 - 212/5*e^5 - 3403/35*e^4 + 663/7*e^3 + 3536/35*e^2 - 272/7*e - 149/7, -4/5*e^8 + 31/15*e^7 + 161/15*e^6 - 332/15*e^5 - 614/15*e^4 + 139/3*e^3 + 201/5*e^2 - 12*e - 37/3, -5/21*e^8 + 2/21*e^7 + 73/21*e^6 + 1/3*e^5 - 256/21*e^4 - 67/7*e^3 - 29/7*e^2 + 160/21*e + 150/7, 73/105*e^8 - 74/105*e^7 - 1294/105*e^6 + 28/5*e^5 + 7036/105*e^4 + 65/21*e^3 - 3414/35*e^2 - 148/21*e + 583/21, 64/105*e^8 - 97/105*e^7 - 1052/105*e^6 + 127/15*e^5 + 5318/105*e^4 - 34/7*e^3 - 2447/35*e^2 - 355/21*e + 162/7, 62/35*e^8 - 141/35*e^7 - 866/35*e^6 + 221/5*e^5 + 3509/35*e^4 - 752/7*e^3 - 3733/35*e^2 + 467/7*e + 193/7, -211/105*e^8 + 538/105*e^7 + 2858/105*e^6 - 868/15*e^5 - 11072/105*e^4 + 1042/7*e^3 + 3868/35*e^2 - 1808/21*e - 239/7, 34/105*e^8 - 22/105*e^7 - 572/105*e^6 + 7/15*e^5 + 2963/105*e^4 + 83/7*e^3 - 1557/35*e^2 - 352/21*e + 181/7, -1/3*e^8 + e^7 + 4*e^6 - 35/3*e^5 - 11*e^4 + 97/3*e^3 - 49/3*e + 20/3, 11/15*e^8 - 23/15*e^7 - 163/15*e^6 + 251/15*e^5 + 727/15*e^4 - 39*e^3 - 313/5*e^2 + 70/3*e + 27, -11/105*e^8 + 1/35*e^7 + 96/35*e^6 - 13/15*e^5 - 744/35*e^4 + 79/21*e^3 + 1808/35*e^2 + 125/21*e - 649/21, -37/35*e^8 + 96/35*e^7 + 501/35*e^6 - 156/5*e^5 - 1949/35*e^4 + 561/7*e^3 + 2173/35*e^2 - 270/7*e - 146/7, 1/5*e^8 + 16/15*e^7 - 109/15*e^6 - 227/15*e^5 + 901/15*e^4 + 184/3*e^3 - 589/5*e^2 - 42*e + 128/3, -1/21*e^8 - 22/21*e^7 + 79/21*e^6 + 14*e^5 - 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1016/21*e^3 - 843/7*e^2 + 620/21*e + 944/21, 502/105*e^8 - 1276/105*e^7 - 6821/105*e^6 + 2041/15*e^5 + 26954/105*e^4 - 2403/7*e^3 - 10331/35*e^2 + 4028/21*e + 677/7, 272/105*e^8 - 491/105*e^7 - 4366/105*e^6 + 761/15*e^5 + 22024/105*e^4 - 785/7*e^3 - 11511/35*e^2 + 1174/21*e + 867/7, 136/35*e^8 - 754/105*e^7 - 6374/105*e^6 + 1139/15*e^5 + 30971/105*e^4 - 3277/21*e^3 - 15394/35*e^2 + 391/7*e + 3646/21, 61/35*e^8 - 524/105*e^7 - 2494/105*e^6 + 844/15*e^5 + 9616/105*e^4 - 2888/21*e^3 - 2594/35*e^2 + 451/7*e + 143/21, 74/105*e^8 - 29/35*e^7 - 439/35*e^6 + 157/15*e^5 + 2571/35*e^4 - 709/21*e^3 - 5637/35*e^2 + 232/21*e + 2161/21, -e^8 + e^7 + 19*e^6 - 10*e^5 - 112*e^4 + 17*e^3 + 186*e^2 - 36*e - 69, -314/105*e^8 + 757/105*e^7 + 4562/105*e^6 - 414/5*e^5 - 19868/105*e^4 + 4637/21*e^3 + 8102/35*e^2 - 2938/21*e - 1355/21, -358/105*e^8 + 944/105*e^7 + 4804/105*e^6 - 503/5*e^5 - 18121/105*e^4 + 5275/21*e^3 + 5289/35*e^2 - 2921/21*e - 745/21, 232/105*e^8 - 177/35*e^7 - 1102/35*e^6 + 821/15*e^5 + 4708/35*e^4 - 2552/21*e^3 - 6206/35*e^2 + 821/21*e + 1529/21, 566/105*e^8 - 1093/105*e^7 - 8783/105*e^6 + 556/5*e^5 + 41792/105*e^4 - 4994/21*e^3 - 18938/35*e^2 + 2707/21*e + 3581/21, 2/15*e^8 - 2/5*e^7 - 12/5*e^6 + 82/15*e^5 + 73/5*e^4 - 73/3*e^3 - 146/5*e^2 + 142/3*e + 88/3, 6/35*e^8 - 94/105*e^7 - 134/105*e^6 + 164/15*e^5 - 484/105*e^4 - 730/21*e^3 + 1391/35*e^2 + 194/7*e - 611/21, 18/35*e^8 - 142/105*e^7 - 542/105*e^6 + 212/15*e^5 + 158/105*e^4 - 622/21*e^3 + 2213/35*e^2 + 50/7*e - 1070/21, -2/21*e^8 + 11/7*e^7 - 15/7*e^6 - 58/3*e^5 + 223/7*e^4 + 1286/21*e^3 - 531/7*e^2 - 797/21*e + 901/21, 157/105*e^8 - 132/35*e^7 - 772/35*e^6 + 626/15*e^5 + 3533/35*e^4 - 1979/21*e^3 - 5101/35*e^2 + 566/21*e + 1199/21, -244/105*e^8 + 512/105*e^7 + 3337/105*e^6 - 249/5*e^5 - 12448/105*e^4 + 1900/21*e^3 + 2747/35*e^2 - 152/21*e + 17/21, -47/105*e^8 + 121/105*e^7 + 731/105*e^6 - 77/5*e^5 - 3539/105*e^4 + 1259/21*e^3 + 1756/35*e^2 - 1585/21*e - 197/21, -197/105*e^8 + 391/105*e^7 + 2921/105*e^6 - 192/5*e^5 - 12689/105*e^4 + 1523/21*e^3 + 4351/35*e^2 - 646/21*e - 584/21, 1/5*e^8 + 11/15*e^7 - 119/15*e^6 - 142/15*e^5 + 1061/15*e^4 + 119/3*e^3 - 824/5*e^2 - 52*e + 235/3, -40/7*e^8 + 251/21*e^7 + 1801/21*e^6 - 388/3*e^5 - 8188/21*e^4 + 6082/21*e^3 + 3518/7*e^2 - 1086/7*e - 3550/21, -29/35*e^8 + 221/105*e^7 + 1441/105*e^6 - 421/15*e^5 - 7939/105*e^4 + 2147/21*e^3 + 5206/35*e^2 - 730/7*e - 1304/21, 304/105*e^8 - 872/105*e^7 - 3877/105*e^6 + 469/5*e^5 + 13273/105*e^4 - 5059/21*e^3 - 3232/35*e^2 + 2771/21*e + 667/21, -208/35*e^8 + 1532/105*e^7 + 8752/105*e^6 - 2407/15*e^5 - 35908/105*e^4 + 7928/21*e^3 + 13017/35*e^2 - 1368/7*e - 2285/21, 76/105*e^8 - 26/35*e^7 - 501/35*e^6 + 143/15*e^5 + 3209/35*e^4 - 626/21*e^3 - 6793/35*e^2 + 383/21*e + 2027/21, -62/15*e^8 + 52/5*e^7 + 287/5*e^6 - 1747/15*e^5 - 1158/5*e^4 + 877/3*e^3 + 1216/5*e^2 - 523/3*e - 229/3, -527/105*e^8 + 1076/105*e^7 + 8026/105*e^6 - 1676/15*e^5 - 37684/105*e^4 + 1797/7*e^3 + 17851/35*e^2 - 2776/21*e - 1311/7, -431/105*e^8 + 351/35*e^7 + 1986/35*e^6 - 1663/15*e^5 - 8094/35*e^4 + 5623/21*e^3 + 9718/35*e^2 - 2563/21*e - 1741/21, -61/105*e^8 + 23/105*e^7 + 1123/105*e^6 + 9/5*e^5 - 6577/105*e^4 - 659/21*e^3 + 3898/35*e^2 + 361/21*e - 1177/21, -25/21*e^8 + 73/21*e^7 + 344/21*e^6 - 115/3*e^5 - 1448/21*e^4 + 603/7*e^3 + 660/7*e^2 - 376/21*e - 174/7, 14/15*e^8 - 14/5*e^7 - 64/5*e^6 + 499/15*e^5 + 251/5*e^4 - 289/3*e^3 - 232/5*e^2 + 244/3*e + 58/3, -62/15*e^8 + 146/15*e^7 + 871/15*e^6 - 1562/15*e^5 - 3589/15*e^4 + 223*e^3 + 1291/5*e^2 - 235/3*e - 64, 10/3*e^8 - 17/3*e^7 - 163/3*e^6 + 61*e^5 + 829/3*e^4 - 392/3*e^3 - 424*e^2 + 169/3*e + 497/3, 71/105*e^8 - 433/105*e^7 - 128/105*e^6 + 246/5*e^5 - 5833/105*e^4 - 3056/21*e^3 + 7017/35*e^2 + 1465/21*e - 2335/21, -32/35*e^8 - 47/105*e^7 + 2033/105*e^6 + 172/15*e^5 - 12947/105*e^4 - 1583/21*e^3 + 7783/35*e^2 + 447/7*e - 1723/21, -113/105*e^8 + 244/105*e^7 + 1724/105*e^6 - 138/5*e^5 - 8426/105*e^4 + 1691/21*e^3 + 4904/35*e^2 - 1108/21*e - 1172/21, 13/105*e^8 + 146/105*e^7 - 614/105*e^6 - 281/15*e^5 + 5336/105*e^4 + 496/7*e^3 - 3594/35*e^2 - 1129/21*e + 230/7, -172/105*e^8 + 241/105*e^7 + 2801/105*e^6 - 346/15*e^5 - 13964/105*e^4 + 290/7*e^3 + 6386/35*e^2 - 428/21*e - 270/7, -37/105*e^8 + 172/35*e^7 - 148/35*e^6 - 926/15*e^5 + 2617/35*e^4 + 4208/21*e^3 - 6019/35*e^2 - 2699/21*e + 1198/21, 47/15*e^8 - 101/15*e^7 - 691/15*e^6 + 1082/15*e^5 + 3064/15*e^4 - 158*e^3 - 1286/5*e^2 + 217/3*e + 99, -572/105*e^8 + 507/35*e^7 + 2542/35*e^6 - 2446/15*e^5 - 9533/35*e^4 + 8644/21*e^3 + 8966/35*e^2 - 4441/21*e - 1093/21, -109/105*e^8 + 29/35*e^7 + 579/35*e^6 - 47/15*e^5 - 2851/35*e^4 - 733/21*e^3 + 4517/35*e^2 + 1441/21*e - 1391/21, -54/35*e^8 + 72/35*e^7 + 892/35*e^6 - 82/5*e^5 - 4638/35*e^4 - 43/7*e^3 + 7151/35*e^2 + 249/7*e - 519/7, -31/105*e^8 + 193/105*e^7 - 22/105*e^6 - 313/15*e^5 + 2953/105*e^4 + 396/7*e^3 - 2312/35*e^2 - 167/21*e + 137/7, -323/105*e^8 + 268/35*e^7 + 1438/35*e^6 - 1264/15*e^5 - 5177/35*e^4 + 4225/21*e^3 + 3014/35*e^2 - 2095/21*e + 326/21, -101/105*e^8 + 263/105*e^7 + 1483/105*e^6 - 458/15*e^5 - 6427/105*e^4 + 655/7*e^3 + 2378/35*e^2 - 1714/21*e + 4/7, -86/35*e^8 + 379/105*e^7 + 4184/105*e^6 - 524/15*e^5 - 21086/105*e^4 + 1039/21*e^3 + 10559/35*e^2 - 123/7*e - 2566/21, -17/105*e^8 - 43/35*e^7 + 177/35*e^6 + 269/15*e^5 - 1293/35*e^4 - 1514/21*e^3 + 1986/35*e^2 + 1247/21*e - 520/21, -199/105*e^8 + 382/105*e^7 + 3002/105*e^6 - 547/15*e^5 - 13658/105*e^4 + 438/7*e^3 + 5367/35*e^2 - 650/21*e - 136/7, -8/105*e^8 - 211/105*e^7 + 814/105*e^6 + 331/15*e^5 - 8251/105*e^4 - 358/7*e^3 + 6899/35*e^2 + 110/21*e - 736/7, 12/35*e^8 + 54/35*e^7 - 416/35*e^6 - 99/5*e^5 + 3574/35*e^4 + 470/7*e^3 - 8663/35*e^2 - 256/7*e + 820/7, -53/105*e^8 + 43/35*e^7 + 278/35*e^6 - 199/15*e^5 - 1507/35*e^4 + 646/21*e^3 + 3089/35*e^2 - 463/21*e - 1048/21];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, w^3 - 2*w^2 - 4*w + 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;