/*
  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 | {16, 2, 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^12 - 36*x^10 + 492*x^8 - 3144*x^6 + 9168*x^4 - 9696*x^2 + 3136;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 0, 3/64*e^10 - 43/32*e^8 + 105/8*e^6 - 389/8*e^4 + 95/2*e^2 - 15/2, 3/224*e^11 - 47/112*e^9 + 33/7*e^7 - 633/28*e^5 + 296/7*e^3 - 153/7*e, -3/64*e^10 + 43/32*e^8 - 105/8*e^6 + 389/8*e^4 - 97/2*e^2 + 31/2, 5/896*e^11 - 69/448*e^9 + 157/112*e^7 - 467/112*e^5 - 41/28*e^3 + 263/28*e, 1/4*e^6 - 5*e^4 + 26*e^2 - 18, e, -3/224*e^11 + 47/112*e^9 - 33/7*e^7 + 633/28*e^5 - 296/7*e^3 + 153/7*e, -3/224*e^11 + 47/112*e^9 - 33/7*e^7 + 633/28*e^5 - 296/7*e^3 + 160/7*e, 3/224*e^11 - 47/112*e^9 + 33/7*e^7 - 633/28*e^5 + 303/7*e^3 - 209/7*e, -15/448*e^11 + 207/224*e^9 - 471/56*e^7 + 1457/56*e^5 - 87/14*e^3 - 89/14*e, 29/896*e^11 - 445/448*e^9 + 1213/112*e^7 - 5531/112*e^5 + 2327/28*e^3 - 961/28*e, -1/64*e^10 + 17/32*e^8 - 55/8*e^6 + 327/8*e^4 - 203/2*e^2 + 117/2, 1/16*e^10 - 15/8*e^8 + 79/4*e^6 - 171/2*e^4 + 134*e^2 - 60, 7/64*e^10 - 103/32*e^8 + 261/8*e^6 - 1033/8*e^4 + 311/2*e^2 - 99/2, -1/128*e^11 + 17/64*e^9 - 53/16*e^7 + 295/16*e^5 - 175/4*e^3 + 117/4*e, 5/896*e^11 - 69/448*e^9 + 157/112*e^7 - 467/112*e^5 - 69/28*e^3 + 543/28*e, -1/16*e^10 + 15/8*e^8 - 39/2*e^6 + 161/2*e^4 - 109*e^2 + 56, -13/896*e^11 + 157/448*e^9 - 257/112*e^7 - 85/112*e^5 + 969/28*e^3 - 583/28*e, 1/2*e^6 - 10*e^4 + 52*e^2 - 40, -13/896*e^11 + 157/448*e^9 - 257/112*e^7 - 85/112*e^5 + 997/28*e^3 - 751/28*e, -1/8*e^10 + 15/4*e^8 - 79/2*e^6 + 170*e^4 - 257*e^2 + 114, -11/64*e^10 + 163/32*e^8 - 421/8*e^6 + 1749/8*e^4 - 615/2*e^2 + 271/2, -5/112*e^11 + 83/56*e^9 - 255/14*e^7 + 1405/14*e^5 - 1612/7*e^3 + 937/7*e, 27/448*e^11 - 395/224*e^9 + 999/56*e^7 - 3989/56*e^5 + 1271/14*e^3 - 523/14*e, -5/64*e^10 + 77/32*e^8 - 211/8*e^6 + 971/8*e^4 - 419/2*e^2 + 201/2, 1/64*e^11 - 9/32*e^9 - 3/8*e^7 + 233/8*e^5 - 279/2*e^3 + 191/2*e, -7/64*e^10 + 103/32*e^8 - 265/8*e^6 + 1113/8*e^4 - 421/2*e^2 + 211/2, 1/64*e^10 - 17/32*e^8 + 51/8*e^6 - 255/8*e^4 + 123/2*e^2 - 77/2, -1/56*e^11 + 9/14*e^9 - 123/14*e^7 + 386/7*e^5 - 1013/7*e^3 + 561/7*e, -1/8*e^10 + 7/2*e^8 - 131/4*e^6 + 109*e^4 - 58*e^2 - 6, -1/8*e^10 + 15/4*e^8 - 159/4*e^6 + 174*e^4 - 272*e^2 + 114, -15/224*e^11 + 207/112*e^9 - 116/7*e^7 + 1317/28*e^5 + 109/7*e^3 - 348/7*e, -5/448*e^11 + 69/224*e^9 - 157/56*e^7 + 467/56*e^5 + 27/14*e^3 - 123/14*e, 1/896*e^11 + 31/448*e^9 - 299/112*e^7 + 3177/112*e^5 - 2937/28*e^3 + 2091/28*e, 1/64*e^11 - 17/32*e^9 + 53/8*e^7 - 295/8*e^5 + 175/2*e^3 - 121/2*e, 9/64*e^10 - 129/32*e^8 + 311/8*e^6 - 1087/8*e^4 + 183/2*e^2 + 19/2, 1/16*e^11 - 15/8*e^9 + 79/4*e^7 - 171/2*e^5 + 136*e^3 - 80*e, -1/32*e^10 + 17/16*e^8 - 51/4*e^6 + 255/4*e^4 - 119*e^2 + 49, -3/64*e^11 + 43/32*e^9 - 103/8*e^7 + 349/8*e^5 - 39/2*e^3 - 61/2*e, 1/16*e^10 - 15/8*e^8 + 20*e^6 - 179/2*e^4 + 148*e^2 - 66, 1/8*e^10 - 15/4*e^8 + 40*e^6 - 180*e^4 + 312*e^2 - 156, -53/448*e^11 + 821/224*e^9 - 2283/56*e^7 + 10875/56*e^5 - 5087/14*e^3 + 2689/14*e, 15/224*e^11 - 235/112*e^9 + 165/7*e^7 - 3165/28*e^5 + 1494/7*e^3 - 877/7*e, -1/64*e^10 + 9/32*e^8 - 3/8*e^6 - 113/8*e^4 + 117/2*e^2 - 43/2, 1/4*e^7 - 5*e^5 + 29*e^3 - 46*e, 25/896*e^11 - 345/448*e^9 + 757/112*e^7 - 1887/112*e^5 - 485/28*e^3 + 251/28*e, -37/896*e^11 + 533/448*e^9 - 1285/112*e^7 + 4419/112*e^5 - 615/28*e^3 - 423/28*e, -1/2*e^6 + 10*e^4 - 51*e^2 + 42, 1/16*e^10 - 15/8*e^8 + 20*e^6 - 179/2*e^4 + 148*e^2 - 52, -1/8*e^10 + 15/4*e^8 - 79/2*e^6 + 171*e^4 - 268*e^2 + 124, 3/16*e^10 - 45/8*e^8 + 59*e^6 - 501/2*e^4 + 364*e^2 - 142, 9/64*e^10 - 137/32*e^8 + 367/8*e^6 - 1607/8*e^4 + 615/2*e^2 - 245/2, 5/64*e^10 - 77/32*e^8 + 211/8*e^6 - 963/8*e^4 + 395/2*e^2 - 169/2, -43/896*e^11 + 683/448*e^9 - 1983/112*e^7 + 10221/112*e^5 - 5533/28*e^3 + 3607/28*e, -1/16*e^10 + 15/8*e^8 - 77/4*e^6 + 151/2*e^4 - 84*e^2 + 32, 1/16*e^10 - 15/8*e^8 + 77/4*e^6 - 151/2*e^4 + 86*e^2 - 48, 13/448*e^11 - 213/224*e^9 + 649/56*e^7 - 3611/56*e^5 + 2181/14*e^3 - 1461/14*e, -29/448*e^11 + 445/224*e^9 - 1213/56*e^7 + 5531/56*e^5 - 2327/14*e^3 + 933/14*e, 11/224*e^11 - 163/112*e^9 + 107/7*e^7 - 1901/28*e^5 + 852/7*e^3 - 624/7*e, -3/32*e^10 + 47/16*e^8 - 131/4*e^6 + 609/4*e^4 - 259*e^2 + 115, -7/64*e^10 + 103/32*e^8 - 261/8*e^6 + 1033/8*e^4 - 313/2*e^2 + 115/2, -7/64*e^10 + 95/32*e^8 - 213/8*e^6 + 657/8*e^4 - 51/2*e^2 - 29/2, -11/448*e^11 + 163/224*e^9 - 435/56*e^7 + 2013/56*e^5 - 901/14*e^3 + 155/14*e, 1/128*e^11 - 17/64*e^9 + 57/16*e^7 - 375/16*e^5 + 287/4*e^3 - 269/4*e, -23/896*e^11 + 295/448*e^9 - 543/112*e^7 + 289/112*e^5 + 1807/28*e^3 - 1893/28*e, 1/16*e^10 - 15/8*e^8 + 39/2*e^6 - 163/2*e^4 + 119*e^2 - 52, 3/16*e^10 - 45/8*e^8 + 59*e^6 - 501/2*e^4 + 364*e^2 - 140, 11/896*e^11 - 219/448*e^9 + 799/112*e^7 - 5149/112*e^5 + 3337/28*e^3 - 1919/28*e, 1/64*e^10 - 17/32*e^8 + 51/8*e^6 - 255/8*e^4 + 123/2*e^2 - 93/2, -1/64*e^10 + 25/32*e^8 - 103/8*e^6 + 703/8*e^4 - 467/2*e^2 + 285/2, 1/16*e^10 - 15/8*e^8 + 39/2*e^6 - 163/2*e^4 + 120*e^2 - 74, -7/64*e^10 + 95/32*e^8 - 209/8*e^6 + 593/8*e^4 + 15/2*e^2 - 77/2, -1/64*e^10 + 17/32*e^8 - 51/8*e^6 + 247/8*e^4 - 99/2*e^2 + 29/2, -1/112*e^11 + 25/56*e^9 - 221/28*e^7 + 841/14*e^5 - 1259/7*e^3 + 788/7*e, 3/64*e^10 - 43/32*e^8 + 109/8*e^6 - 461/8*e^4 + 183/2*e^2 - 103/2, 3/32*e^11 - 47/16*e^9 + 133/4*e^7 - 653/4*e^5 + 322*e^3 - 170*e, -15/224*e^11 + 235/112*e^9 - 165/7*e^7 + 3165/28*e^5 - 1494/7*e^3 + 905/7*e, -1/4*e^7 + 5*e^5 - 26*e^3 + 22*e, 5/32*e^10 - 73/16*e^8 + 181/4*e^6 - 675/4*e^4 + 165*e^2 - 45, 1/32*e^10 - 13/16*e^8 + 27/4*e^6 - 71/4*e^4 + e^2 - 9, 5/64*e^10 - 77/32*e^8 + 207/8*e^6 - 899/8*e^4 + 331/2*e^2 - 129/2, 17/224*e^11 - 229/112*e^9 + 489/28*e^7 - 1179/28*e^5 - 327/7*e^3 + 288/7*e, -25/448*e^11 + 401/224*e^9 - 1177/56*e^7 + 6143/56*e^5 - 3365/14*e^3 + 2017/14*e, 13/448*e^11 - 157/224*e^9 + 257/56*e^7 + 85/56*e^5 - 1011/14*e^3 + 891/14*e, 1/4*e^7 - 5*e^5 + 27*e^3 - 31*e, -11/224*e^11 + 163/112*e^9 - 421/28*e^7 + 1733/28*e^5 - 544/7*e^3 - 48/7*e, 1/112*e^11 + 3/56*e^9 - 171/28*e^7 + 993/14*e^5 - 1828/7*e^3 + 1249/7*e, 7/32*e^10 - 103/16*e^8 + 265/4*e^6 - 1105/4*e^4 + 393*e^2 - 151, 1/16*e^10 - 17/8*e^8 + 51/2*e^6 - 253/2*e^4 + 229*e^2 - 116, 5/896*e^11 - 69/448*e^9 + 129/112*e^7 + 93/112*e^5 - 797/28*e^3 + 1047/28*e, -41/896*e^11 + 633/448*e^9 - 1741/112*e^7 + 8063/112*e^5 - 3539/28*e^3 + 1685/28*e, 11/896*e^11 - 219/448*e^9 + 827/112*e^7 - 5709/112*e^5 + 4009/28*e^3 - 1863/28*e, -15/64*e^10 + 223/32*e^8 - 573/8*e^6 + 2321/8*e^4 - 751/2*e^2 + 315/2, -155/896*e^11 + 2363/448*e^9 - 6435/112*e^7 + 29821/112*e^5 - 13457/28*e^3 + 7079/28*e, -1/32*e^11 + 13/16*e^9 - 6*e^7 + 7/4*e^5 + 91*e^3 - 86*e, -3/16*e^10 + 43/8*e^8 - 103/2*e^6 + 351/2*e^4 - 103*e^2 + 4, -1/8*e^10 + 15/4*e^8 - 40*e^6 + 178*e^4 - 284*e^2 + 110, 1/4*e^8 - 15/2*e^6 + 75*e^4 - 266*e^2 + 178, -1/128*e^11 + 33/64*e^9 - 161/16*e^7 + 1271/16*e^5 - 971/4*e^3 + 613/4*e, -1/14*e^11 + 65/28*e^9 - 781/28*e^7 + 1054/7*e^5 - 2379/7*e^3 + 1306/7*e, 5/64*e^11 - 77/32*e^9 + 209/8*e^7 - 931/8*e^5 + 365/2*e^3 - 149/2*e, -9/64*e^10 + 129/32*e^8 - 315/8*e^6 + 1159/8*e^4 - 263/2*e^2 + 37/2, -3/16*e^10 + 41/8*e^8 - 46*e^6 + 279/2*e^4 - 32*e^2 - 28, 9/112*e^11 - 141/56*e^9 + 403/14*e^7 - 2039/14*e^5 + 2154/7*e^3 - 1303/7*e, 1/16*e^10 - 17/8*e^8 + 26*e^6 - 273/2*e^4 + 283*e^2 - 152, 37/448*e^11 - 533/224*e^9 + 1299/56*e^7 - 4699/56*e^5 + 1007/14*e^3 - 249/14*e, 43/448*e^11 - 627/224*e^9 + 1577/56*e^7 - 6189/56*e^5 + 1753/14*e^3 - 135/14*e, -1/8*e^10 + 15/4*e^8 - 81/2*e^6 + 190*e^4 - 364*e^2 + 196, 17/64*e^10 - 249/32*e^8 + 635/8*e^6 - 2615/8*e^4 + 931/2*e^2 - 421/2, -5/56*e^11 + 19/7*e^9 - 817/28*e^7 + 915/7*e^5 - 1558/7*e^3 + 922/7*e, 3/224*e^11 - 47/112*e^9 + 125/28*e^7 - 493/28*e^5 + 93/7*e^3 + 162/7*e, 15/224*e^11 - 207/112*e^9 + 116/7*e^7 - 1345/28*e^5 + 3/7*e^3 - 86/7*e, -39/448*e^11 + 527/224*e^9 - 1135/56*e^7 + 2825/56*e^5 + 737/14*e^3 - 1161/14*e, 1/64*e^10 - 9/32*e^8 - 1/8*e^6 + 201/8*e^4 - 245/2*e^2 + 195/2, -87/896*e^11 + 1335/448*e^9 - 3667/112*e^7 + 17265/112*e^5 - 8185/28*e^3 + 5235/28*e, 3/64*e^10 - 35/32*e^8 + 53/8*e^6 + 59/8*e^4 - 253/2*e^2 + 201/2, 5/112*e^11 - 83/56*e^9 + 503/28*e^7 - 1335/14*e^5 + 1409/7*e^3 - 643/7*e, 41/448*e^11 - 633/224*e^9 + 1769/56*e^7 - 8623/56*e^5 + 4267/14*e^3 - 2217/14*e, 9/64*e^10 - 129/32*e^8 + 311/8*e^6 - 1087/8*e^4 + 179/2*e^2 + 43/2, -29/448*e^11 + 445/224*e^9 - 1213/56*e^7 + 5531/56*e^5 - 2355/14*e^3 + 1157/14*e, 1/16*e^10 - 15/8*e^8 + 39/2*e^6 - 159/2*e^4 + 96*e^2 - 2, 3/32*e^10 - 47/16*e^8 + 129/4*e^6 - 569/4*e^4 + 207*e^2 - 83, -23/64*e^10 + 343/32*e^8 - 897/8*e^6 + 3833/8*e^4 - 1445/2*e^2 + 595/2, -1/224*e^11 - 3/112*e^9 + 89/28*e^7 - 1049/28*e^5 + 963/7*e^3 - 628/7*e, -27/224*e^11 + 395/112*e^9 - 999/28*e^7 + 3989/28*e^5 - 1278/7*e^3 + 572/7*e, -11/224*e^11 + 191/112*e^9 - 156/7*e^7 + 3749/28*e^5 - 2413/7*e^3 + 1478/7*e, 1/28*e^11 - 29/28*e^9 + 74/7*e^7 - 310/7*e^5 + 430/7*e^3 + 61/7*e, -5/32*e^10 + 73/16*e^8 - 179/4*e^6 + 631/4*e^4 - 97*e^2 - 27, -1/16*e^10 + 15/8*e^8 - 19*e^6 + 141/2*e^4 - 56*e^2 + 10, 33/224*e^11 - 489/112*e^9 + 635/14*e^7 - 5367/28*e^5 + 1954/7*e^3 - 731/7*e, 3/16*e^10 - 43/8*e^8 + 53*e^6 - 409/2*e^4 + 238*e^2 - 38, -3/64*e^10 + 51/32*e^8 - 165/8*e^6 + 1005/8*e^4 - 669/2*e^2 + 375/2, 5/64*e^10 - 69/32*e^8 + 155/8*e^6 - 443/8*e^4 - 21/2*e^2 + 47/2, -55/896*e^11 + 759/448*e^9 - 1727/112*e^7 + 5361/112*e^5 - 333/28*e^3 - 653/28*e, -9/896*e^11 + 281/448*e^9 - 1397/112*e^7 + 11503/112*e^5 - 9211/28*e^3 + 5765/28*e, 109/896*e^11 - 1661/448*e^9 + 4509/112*e^7 - 20731/112*e^5 + 9315/28*e^3 - 5769/28*e, 45/448*e^11 - 677/224*e^9 + 1805/56*e^7 - 8011/56*e^5 + 3215/14*e^3 - 1189/14*e, -15/112*e^11 + 221/56*e^9 - 1131/28*e^7 + 2311/14*e^5 - 1546/7*e^3 + 536/7*e, -1/16*e^10 + 13/8*e^8 - 55/4*e^6 + 77/2*e^4 - 6*e^2 + 12, -1/56*e^11 + 11/28*e^9 - 43/28*e^7 - 111/7*e^5 + 779/7*e^3 - 902/7*e, 1/8*e^10 - 4*e^8 + 93/2*e^6 - 235*e^4 + 473*e^2 - 254, 41/896*e^11 - 521/448*e^9 + 957/112*e^7 - 671/112*e^5 - 2845/28*e^3 + 2683/28*e, 11/64*e^10 - 171/32*e^8 + 473/8*e^6 - 2189/8*e^4 + 933/2*e^2 - 423/2, 1/14*e^11 - 65/28*e^9 + 767/28*e^7 - 991/7*e^5 + 2113/7*e^3 - 1285/7*e, 33/448*e^11 - 489/224*e^9 + 1249/56*e^7 - 4975/56*e^5 + 1471/14*e^3 - 185/14*e, 11/64*e^10 - 147/32*e^8 + 321/8*e^6 - 933/8*e^4 + 25/2*e^2 + 105/2, -5/32*e^10 + 81/16*e^8 - 235/4*e^6 + 1155/4*e^4 - 541*e^2 + 285, 27/448*e^11 - 395/224*e^9 + 999/56*e^7 - 3989/56*e^5 + 1285/14*e^3 - 635/14*e, 9/64*e^10 - 137/32*e^8 + 363/8*e^6 - 1543/8*e^4 + 551/2*e^2 - 237/2, 1/28*e^11 - 9/7*e^9 + 485/28*e^7 - 737/7*e^5 + 1837/7*e^3 - 982/7*e, -23/128*e^11 + 343/64*e^9 - 891/16*e^7 + 3713/16*e^5 - 1285/4*e^3 + 459/4*e, -1/14*e^11 + 51/28*e^9 - 375/28*e^7 + 67/7*e^5 + 1072/7*e^3 - 913/7*e, 9/64*e^10 - 129/32*e^8 + 315/8*e^6 - 1167/8*e^4 + 293/2*e^2 - 61/2, -3/8*e^10 + 11*e^8 - 223/2*e^6 + 444*e^4 - 547*e^2 + 178, 3/16*e^10 - 43/8*e^8 + 53*e^6 - 409/2*e^4 + 247*e^2 - 84, -1/8*e^10 + 7/2*e^8 - 32*e^6 + 96*e^4 + e^2 - 54, -19/224*e^11 + 279/112*e^9 - 174/7*e^7 + 2581/28*e^5 - 519/7*e^3 - 235/7*e, 51/896*e^11 - 771/448*e^9 + 2083/112*e^7 - 9557/112*e^5 + 4185/28*e^3 - 1775/28*e, 7/16*e^10 - 101/8*e^8 + 125*e^6 - 957/2*e^4 + 529*e^2 - 132, 5/448*e^11 - 125/224*e^9 + 563/56*e^7 - 4443/56*e^5 + 3487/14*e^3 - 2033/14*e, 13/448*e^11 - 213/224*e^9 + 649/56*e^7 - 3555/56*e^5 + 2013/14*e^3 - 1125/14*e, 1/32*e^10 - 13/16*e^8 + 29/4*e^6 - 103/4*e^4 + 25*e^2 + 23, 3/32*e^10 - 47/16*e^8 + 129/4*e^6 - 577/4*e^4 + 231*e^2 - 131, -3/56*e^11 + 47/28*e^9 - 132/7*e^7 + 626/7*e^5 - 1079/7*e^3 + 339/7*e, -3/8*e^10 + 11*e^8 - 223/2*e^6 + 444*e^4 - 554*e^2 + 242, 3/16*e^10 - 41/8*e^8 + 47*e^6 - 315/2*e^4 + 113*e^2 + 4, -1/28*e^11 + 29/28*e^9 - 141/14*e^7 + 247/7*e^5 - 150/7*e^3 - 124/7*e, -11/32*e^10 + 163/16*e^8 - 425/4*e^6 + 1829/4*e^4 - 717*e^2 + 339, -1/16*e^10 + 17/8*e^8 - 53/2*e^6 + 293/2*e^4 - 334*e^2 + 162, -1/64*e^11 + 17/32*e^9 - 55/8*e^7 + 327/8*e^5 - 195/2*e^3 + 53/2*e, 39/448*e^11 - 583/224*e^9 + 1555/56*e^7 - 7081/56*e^5 + 3169/14*e^3 - 1527/14*e, 17/448*e^11 - 201/224*e^9 + 293/56*e^7 + 697/56*e^5 - 2007/14*e^3 + 1639/14*e, -1/224*e^11 + 25/112*e^9 - 25/7*e^7 + 603/28*e^5 - 255/7*e^3 - 145/7*e, 11/112*e^11 - 149/56*e^9 + 639/28*e^7 - 753/14*e^5 - 585/7*e^3 + 964/7*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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