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

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[5, 5, w + 1],
[5, 5, -w + 2],
[7, 7, -w^2 + 2*w + 1],
[7, 7, -w^2 + 2],
[9, 3, w^3 - w^2 - 4*w],
[16, 2, 2],
[17, 17, -w^3 + 2*w^2 + 3*w - 2],
[17, 17, -w^3 + w^2 + 4*w - 2],
[25, 5, w^2 - w - 3],
[37, 37, 2*w - 1],
[43, 43, -w^3 + 2*w^2 + 2*w - 2],
[43, 43, w^3 - w^2 - 3*w + 1],
[59, 59, 2*w - 5],
[59, 59, -2*w - 3],
[79, 79, -w^3 + 2*w^2 + 5*w - 4],
[79, 79, -w^3 + w^2 + 6*w - 2],
[83, 83, -w^3 + 7*w],
[83, 83, -w^3 + 2*w^2 + 4*w - 2],
[83, 83, -w^3 + w^2 + 5*w - 3],
[83, 83, w^2 - 2*w - 6],
[89, 89, w^2 - 2*w - 4],
[89, 89, w^2 - 5],
[101, 101, -2*w^3 + 5*w^2 + 5*w - 11],
[101, 101, w^2 + w - 4],
[101, 101, w^2 - 3*w - 2],
[101, 101, 2*w^3 - w^2 - 9*w - 3],
[109, 109, w^3 - w^2 - 6*w - 3],
[109, 109, -w^3 + 2*w^2 + 5*w - 9],
[121, 11, w^3 - 8*w],
[121, 11, w^3 - 3*w^2 - 5*w + 7],
[131, 131, w^3 - 5*w - 6],
[131, 131, -w^3 + 3*w^2 + 2*w - 10],
[163, 163, 2*w^2 - 3*w - 6],
[163, 163, 2*w^2 - w - 7],
[167, 167, w^3 - w^2 - 3*w - 4],
[167, 167, -w^3 + 2*w^2 + 2*w - 7],
[193, 193, 2*w^3 - 3*w^2 - 7*w + 2],
[193, 193, 2*w^3 - 3*w^2 - 7*w + 6],
[227, 227, 2*w^3 - 3*w^2 - 8*w + 1],
[227, 227, -2*w^3 + 3*w^2 + 8*w - 8],
[251, 251, 2*w^2 - w - 9],
[251, 251, 2*w^2 - 3*w - 8],
[257, 257, w^3 + w^2 - 8*w - 7],
[257, 257, -4*w^3 + 5*w^2 + 17*w - 1],
[269, 269, 2*w^3 - 2*w^2 - 10*w + 3],
[269, 269, 2*w^3 - 3*w^2 - 9*w + 2],
[269, 269, 2*w^3 - 3*w^2 - 9*w + 8],
[269, 269, -2*w^3 + 4*w^2 + 8*w - 7],
[277, 277, w^3 - w^2 - 4*w - 4],
[277, 277, -w^3 + 2*w^2 + 3*w - 8],
[289, 17, 2*w^2 - 2*w - 3],
[293, 293, -w^2 - 2*w + 5],
[293, 293, -w^3 + 3*w^2 + 3*w - 4],
[293, 293, w^3 - 6*w + 1],
[293, 293, -2*w^3 + 4*w^2 + 7*w - 6],
[311, 311, -w^3 + 3*w^2 + w - 7],
[311, 311, w^3 - 4*w - 4],
[331, 331, -2*w^3 + 4*w^2 + 9*w - 10],
[331, 331, 2*w^3 - 2*w^2 - 11*w + 1],
[337, 337, 2*w^2 - 3*w - 4],
[337, 337, -w^3 + 2*w^2 + 6*w - 3],
[337, 337, w^3 - w^2 - 7*w + 4],
[337, 337, 2*w^2 - w - 5],
[353, 353, -3*w^3 + 4*w^2 + 13*w - 2],
[353, 353, 3*w^3 - 5*w^2 - 12*w + 12],
[373, 373, -2*w^3 + 6*w^2 + 3*w - 11],
[373, 373, -w^3 - w^2 + 6*w + 7],
[373, 373, -w^3 + 4*w^2 + w - 11],
[373, 373, 2*w^3 - 9*w - 4],
[379, 379, w^3 - w^2 - 7*w + 1],
[379, 379, 3*w^3 - 7*w^2 - 7*w + 11],
[379, 379, 3*w^3 - 2*w^2 - 12*w],
[379, 379, -w^3 + 2*w^2 + 6*w - 6],
[383, 383, -3*w^3 + 4*w^2 + 12*w - 8],
[383, 383, 3*w^3 - 5*w^2 - 11*w + 5],
[421, 421, 2*w^3 - 2*w^2 - 9*w - 4],
[421, 421, -2*w^3 + 4*w^2 + 7*w - 13],
[457, 457, -w^3 + 2*w^2 + 6*w - 5],
[457, 457, w^3 - w^2 - 7*w + 2],
[461, 461, -w^3 + 3*w^2 + 4*w - 4],
[461, 461, -w^3 + 7*w - 2],
[463, 463, -w^3 + 4*w^2 + 3*w - 10],
[463, 463, w^3 + w^2 - 8*w - 4],
[467, 467, -w^3 + 6*w - 2],
[467, 467, -w^3 + 3*w^2 + 3*w - 3],
[479, 479, -w^3 + w^2 + w - 4],
[479, 479, w^3 - 2*w^2 - 3],
[487, 487, -3*w^3 + 4*w^2 + 12*w - 3],
[487, 487, -3*w^3 + 5*w^2 + 11*w - 10],
[499, 499, 3*w^3 - 4*w^2 - 11*w + 8],
[499, 499, 3*w^3 - 5*w^2 - 10*w + 4],
[503, 503, -w^3 + 2*w^2 + w - 5],
[503, 503, w^3 - w^2 - 2*w - 3],
[521, 521, 2*w^3 - 4*w^2 - 6*w + 3],
[521, 521, -w^3 + 10*w - 8],
[521, 521, w^3 - 3*w^2 - 7*w + 1],
[521, 521, -2*w^3 + 2*w^2 + 8*w - 5],
[541, 541, 2*w - 7],
[541, 541, 2*w + 5],
[547, 547, w^3 - 4*w^2 + 7],
[547, 547, w^3 + w^2 - 5*w - 4],
[563, 563, -2*w^3 + 4*w^2 + 7*w - 5],
[563, 563, -2*w^3 + 2*w^2 + 9*w - 4],
[587, 587, -w^3 - w^2 + 4*w + 6],
[587, 587, w^3 - 4*w^2 + w + 8],
[593, 593, w^2 + w - 7],
[593, 593, 2*w^3 - 5*w^2 - 4*w + 5],
[593, 593, -2*w^3 + w^2 + 8*w - 2],
[593, 593, w^2 - 3*w - 5],
[613, 613, 2*w^3 - 3*w^2 - 11*w + 2],
[613, 613, w^3 + w^2 - 6*w - 3],
[613, 613, 3*w^3 - 4*w^2 - 12*w + 4],
[613, 613, 2*w^3 - 3*w^2 - 11*w + 10],
[631, 631, -w^3 - w^2 + 7*w + 6],
[631, 631, -w^3 + 4*w^2 + 2*w - 11],
[647, 647, w^2 - 4*w - 3],
[647, 647, w^2 + 2*w - 6],
[677, 677, w^2 + w - 8],
[677, 677, -w^3 + 7*w - 3],
[677, 677, -w^3 + 3*w^2 + 4*w - 3],
[677, 677, w^2 - 3*w - 6],
[709, 709, w^3 - 7*w - 8],
[709, 709, w^3 - 3*w^2 - 4*w + 14],
[739, 739, w^3 - w^2 - 4*w - 5],
[739, 739, -3*w^3 + 3*w^2 + 16*w - 3],
[739, 739, -3*w^3 + 6*w^2 + 13*w - 13],
[739, 739, -w^3 + 2*w^2 + 3*w - 9],
[751, 751, -w - 5],
[751, 751, w^3 + w^2 - 7*w - 4],
[751, 751, -w^3 + 4*w^2 + 2*w - 9],
[751, 751, w - 6],
[757, 757, -w^3 + 4*w^2 + 2*w - 10],
[757, 757, w^3 + w^2 - 7*w - 5],
[761, 761, -3*w^3 + 2*w^2 + 17*w - 1],
[761, 761, -3*w^3 + 6*w^2 + 9*w - 17],
[761, 761, 2*w^3 - 5*w^2 - 9*w + 9],
[761, 761, 2*w^3 - 2*w^2 - 7*w - 6],
[773, 773, -3*w^3 + 8*w^2 + 6*w - 12],
[773, 773, w^3 - 4*w - 6],
[773, 773, -w^3 + 3*w^2 + w - 9],
[773, 773, -3*w^3 + w^2 + 13*w + 1],
[797, 797, 3*w^3 - 2*w^2 - 13*w + 1],
[797, 797, 3*w^3 - 7*w^2 - 8*w + 11],
[823, 823, 2*w^3 - 3*w^2 - 6*w - 4],
[823, 823, -2*w^3 + 3*w^2 + 7*w + 3],
[823, 823, w^3 - 2*w^2 - 7*w - 3],
[823, 823, -3*w^3 + 5*w^2 + 9*w - 2],
[857, 857, 2*w^3 - w^2 - 10*w + 1],
[857, 857, -2*w^3 + 5*w^2 + 6*w - 8],
[883, 883, 3*w^3 - 3*w^2 - 12*w + 1],
[883, 883, 3*w^3 - 6*w^2 - 9*w + 11],
[887, 887, -2*w^3 + 5*w^2 + 7*w - 9],
[887, 887, -4*w^3 + 6*w^2 + 17*w - 7],
[887, 887, -2*w^3 + 5*w^2 + 9*w - 16],
[887, 887, 2*w^3 - w^2 - 11*w + 1],
[907, 907, -3*w^3 + 3*w^2 + 16*w - 2],
[907, 907, -3*w^3 + 6*w^2 + 13*w - 14],
[919, 919, 2*w^3 - 2*w^2 - 7*w + 9],
[919, 919, -2*w^3 + 2*w^2 + 12*w - 5],
[929, 929, -2*w^3 + 4*w^2 + 8*w - 5],
[929, 929, 3*w^3 - 4*w^2 - 11*w + 12],
[929, 929, 3*w^3 - 5*w^2 - 10*w],
[929, 929, -2*w^3 + 2*w^2 + 10*w - 5],
[967, 967, 3*w^3 - 5*w^2 - 9*w + 11],
[967, 967, -3*w^3 + 7*w^2 + 6*w - 10],
[971, 971, 2*w^3 - 2*w^2 - 7*w - 5],
[971, 971, -2*w^3 + w^2 + 8*w - 3],
[971, 971, 2*w^3 - 5*w^2 - 4*w + 4],
[971, 971, -2*w^3 + 4*w^2 + 5*w - 12],
[983, 983, 4*w - 7],
[983, 983, 2*w^2 - 4*w - 7],
[983, 983, 2*w^2 - 9],
[983, 983, 4*w + 3],
[991, 991, 2*w^3 - 6*w^2 - 5*w + 14],
[991, 991, -2*w^3 + 11*w + 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 - 33*x^8 + 368*x^6 - 1722*x^4 + 3500*x^2 - 2500;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e, 61/50*e^8 - 1813/50*e^6 + 8249/25*e^4 - 25421/25*e^2 + 926, 61/50*e^8 - 1813/50*e^6 + 8249/25*e^4 - 25421/25*e^2 + 926, 8/25*e^8 - 239/25*e^6 + 2194/25*e^4 - 6851/25*e^2 + 250, -1, 42/25*e^9 - 2497/50*e^7 + 22737/50*e^5 - 35099/25*e^3 + 1282*e, -42/25*e^9 + 2497/50*e^7 - 22737/50*e^5 + 35099/25*e^3 - 1282*e, 107/50*e^8 - 3181/50*e^6 + 14488/25*e^4 - 44802/25*e^2 + 1638, -23/25*e^8 + 684/25*e^6 - 6239/25*e^4 + 19356/25*e^2 - 710, 53/25*e^8 - 1574/25*e^6 + 14304/25*e^4 - 43991/25*e^2 + 1600, 53/25*e^8 - 1574/25*e^6 + 14304/25*e^4 - 43991/25*e^2 + 1600, 34/25*e^9 - 2019/50*e^7 + 18349/50*e^5 - 28248/25*e^3 + 1031*e, -34/25*e^9 + 2019/50*e^7 - 18349/50*e^5 + 28248/25*e^3 - 1031*e, 307/50*e^8 - 9131/50*e^6 + 41613/25*e^4 - 128727/25*e^2 + 4724, 307/50*e^8 - 9131/50*e^6 + 41613/25*e^4 - 128727/25*e^2 + 4724, -67/50*e^9 + 993/25*e^7 - 17981/50*e^5 + 27412/25*e^3 - 981*e, 39/50*e^9 - 581/25*e^7 + 10627/50*e^5 - 16554/25*e^3 + 613*e, -39/50*e^9 + 581/25*e^7 - 10627/50*e^5 + 16554/25*e^3 - 613*e, 67/50*e^9 - 993/25*e^7 + 17981/50*e^5 - 27412/25*e^3 + 981*e, -39/50*e^9 + 581/25*e^7 - 10627/50*e^5 + 16554/25*e^3 - 614*e, 39/50*e^9 - 581/25*e^7 + 10627/50*e^5 - 16554/25*e^3 + 614*e, 169/50*e^9 - 5027/50*e^7 + 22921/25*e^5 - 71034/25*e^3 + 2613*e, -9/10*e^9 + 267/10*e^7 - 1211/5*e^5 + 3709/5*e^3 - 667*e, 9/10*e^9 - 267/10*e^7 + 1211/5*e^5 - 3709/5*e^3 + 667*e, -169/50*e^9 + 5027/50*e^7 - 22921/25*e^5 + 71034/25*e^3 - 2613*e, -37/10*e^8 + 1101/10*e^6 - 5023/5*e^4 + 15582/5*e^2 - 2868, -37/10*e^8 + 1101/10*e^6 - 5023/5*e^4 + 15582/5*e^2 - 2868, 261/50*e^8 - 7763/50*e^6 + 35374/25*e^4 - 109371/25*e^2 + 4002, 261/50*e^8 - 7763/50*e^6 + 35374/25*e^4 - 109371/25*e^2 + 4002, -161/50*e^9 + 2394/25*e^7 - 43623/50*e^5 + 67371/25*e^3 - 2457*e, 161/50*e^9 - 2394/25*e^7 + 43623/50*e^5 - 67371/25*e^3 + 2457*e, -14/25*e^8 + 412/25*e^6 - 3677/25*e^4 + 10883/25*e^2 - 380, -14/25*e^8 + 412/25*e^6 - 3677/25*e^4 + 10883/25*e^2 - 380, 23/25*e^9 - 684/25*e^7 + 6239/25*e^5 - 19406/25*e^3 + 728*e, -23/25*e^9 + 684/25*e^7 - 6239/25*e^5 + 19406/25*e^3 - 728*e, -7*e^8 + 208*e^6 - 1893*e^4 + 5843*e^2 - 5338, -7*e^8 + 208*e^6 - 1893*e^4 + 5843*e^2 - 5338, -191/50*e^9 + 2839/25*e^7 - 51713/50*e^5 + 79901/25*e^3 - 2925*e, 191/50*e^9 - 2839/25*e^7 + 51713/50*e^5 - 79901/25*e^3 + 2925*e, 129/50*e^9 - 1916/25*e^7 + 34847/50*e^5 - 53644/25*e^3 + 1943*e, -129/50*e^9 + 1916/25*e^7 - 34847/50*e^5 + 53644/25*e^3 - 1943*e, 26/25*e^9 - 1541/50*e^7 + 13961/50*e^5 - 21397/25*e^3 + 786*e, -26/25*e^9 + 1541/50*e^7 - 13961/50*e^5 + 21397/25*e^3 - 786*e, -23/25*e^9 + 684/25*e^7 - 6239/25*e^5 + 19381/25*e^3 - 717*e, 76/25*e^9 - 2258/25*e^7 + 20543/25*e^5 - 63322/25*e^3 + 2301*e, -76/25*e^9 + 2258/25*e^7 - 20543/25*e^5 + 63322/25*e^3 - 2301*e, 23/25*e^9 - 684/25*e^7 + 6239/25*e^5 - 19381/25*e^3 + 717*e, 109/50*e^8 - 3247/50*e^6 + 14856/25*e^4 - 46474/25*e^2 + 1736, 109/50*e^8 - 3247/50*e^6 + 14856/25*e^4 - 46474/25*e^2 + 1736, 1/25*e^8 - 33/25*e^6 + 368/25*e^4 - 1622/25*e^2 + 66, 106/25*e^9 - 3148/25*e^7 + 28608/25*e^5 - 87932/25*e^3 + 3189*e, -231/50*e^9 + 6873/50*e^7 - 31354/25*e^5 + 97266/25*e^3 - 3585*e, 231/50*e^9 - 6873/50*e^7 + 31354/25*e^5 - 97266/25*e^3 + 3585*e, -106/25*e^9 + 3148/25*e^7 - 28608/25*e^5 + 87932/25*e^3 - 3189*e, 8/25*e^9 - 239/25*e^7 + 2194/25*e^5 - 6901/25*e^3 + 274*e, -8/25*e^9 + 239/25*e^7 - 2194/25*e^5 + 6901/25*e^3 - 274*e, 133/25*e^8 - 3964/25*e^6 + 36269/25*e^4 - 113026/25*e^2 + 4188, 133/25*e^8 - 3964/25*e^6 + 36269/25*e^4 - 113026/25*e^2 + 4188, -149/50*e^8 + 4417/50*e^6 - 20016/25*e^4 + 61314/25*e^2 - 2216, 3/2*e^8 - 89/2*e^6 + 404*e^4 - 1244*e^2 + 1146, 3/2*e^8 - 89/2*e^6 + 404*e^4 - 1244*e^2 + 1146, -149/50*e^8 + 4417/50*e^6 - 20016/25*e^4 + 61314/25*e^2 - 2216, -27/5*e^9 + 1607/10*e^7 - 14667/10*e^5 + 22764/5*e^3 - 4200*e, 27/5*e^9 - 1607/10*e^7 + 14667/10*e^5 - 22764/5*e^3 + 4200*e, -39/25*e^8 + 1162/25*e^6 - 10627/25*e^4 + 33058/25*e^2 - 1202, 127/50*e^8 - 3791/50*e^6 + 17393/25*e^4 - 54422/25*e^2 + 2022, 127/50*e^8 - 3791/50*e^6 + 17393/25*e^4 - 54422/25*e^2 + 2022, -39/25*e^8 + 1162/25*e^6 - 10627/25*e^4 + 33058/25*e^2 - 1202, -24/25*e^8 + 717/25*e^6 - 6582/25*e^4 + 20603/25*e^2 - 774, -61/5*e^8 + 1813/5*e^6 - 16508/5*e^4 + 51002/5*e^2 - 9330, -61/5*e^8 + 1813/5*e^6 - 16508/5*e^4 + 51002/5*e^2 - 9330, -24/25*e^8 + 717/25*e^6 - 6582/25*e^4 + 20603/25*e^2 - 774, -e^9 + 30*e^7 - 278*e^5 + 889*e^3 - 850*e, e^9 - 30*e^7 + 278*e^5 - 889*e^3 + 850*e, 139/50*e^8 - 4137/50*e^6 + 18876/25*e^4 - 58554/25*e^2 + 2160, 139/50*e^8 - 4137/50*e^6 + 18876/25*e^4 - 58554/25*e^2 + 2160, 163/25*e^8 - 4854/25*e^6 + 44359/25*e^4 - 138011/25*e^2 + 5110, 163/25*e^8 - 4854/25*e^6 + 44359/25*e^4 - 138011/25*e^2 + 5110, -44/25*e^9 + 1302/25*e^7 - 11742/25*e^5 + 35443/25*e^3 - 1249*e, 44/25*e^9 - 1302/25*e^7 + 11742/25*e^5 - 35443/25*e^3 + 1249*e, -187/50*e^8 + 5571/50*e^6 - 25458/25*e^4 + 79107/25*e^2 - 2918, -187/50*e^8 + 5571/50*e^6 - 25458/25*e^4 + 79107/25*e^2 - 2918, 157/50*e^9 - 2328/25*e^7 + 42201/50*e^5 - 64527/25*e^3 + 2317*e, -157/50*e^9 + 2328/25*e^7 - 42201/50*e^5 + 64527/25*e^3 - 2317*e, -199/25*e^9 + 5917/25*e^7 - 53907/25*e^5 + 166653/25*e^3 - 6104*e, 199/25*e^9 - 5917/25*e^7 + 53907/25*e^5 - 166653/25*e^3 + 6104*e, -e^8 + 30*e^6 - 278*e^4 + 888*e^2 - 838, -e^8 + 30*e^6 - 278*e^4 + 888*e^2 - 838, -298/25*e^8 + 8859/25*e^6 - 80689/25*e^4 + 249406/25*e^2 - 9130, -298/25*e^8 + 8859/25*e^6 - 80689/25*e^4 + 249406/25*e^2 - 9130, 8/25*e^9 - 239/25*e^7 + 2194/25*e^5 - 6851/25*e^3 + 244*e, -8/25*e^9 + 239/25*e^7 - 2194/25*e^5 + 6851/25*e^3 - 244*e, 2*e^9 - 119/2*e^7 + 1085/2*e^5 - 1678*e^3 + 1528*e, 23/10*e^9 - 342/5*e^7 + 6229/10*e^5 - 9598/5*e^3 + 1738*e, -23/10*e^9 + 342/5*e^7 - 6229/10*e^5 + 9598/5*e^3 - 1738*e, -2*e^9 + 119/2*e^7 - 1085/2*e^5 + 1678*e^3 - 1528*e, 291/25*e^8 - 8653/25*e^6 + 78838/25*e^4 - 243802/25*e^2 + 8940, 291/25*e^8 - 8653/25*e^6 + 78838/25*e^4 - 243802/25*e^2 + 8940, 21/5*e^8 - 623/5*e^6 + 5653/5*e^4 - 17357/5*e^2 + 3168, 21/5*e^8 - 623/5*e^6 + 5653/5*e^4 - 17357/5*e^2 + 3168, -19/5*e^9 + 1129/10*e^7 - 10269/10*e^5 + 15803/5*e^3 - 2851*e, 19/5*e^9 - 1129/10*e^7 + 10269/10*e^5 - 15803/5*e^3 + 2851*e, 83/50*e^9 - 1232/25*e^7 + 22369/50*e^5 - 34238/25*e^3 + 1213*e, -83/50*e^9 + 1232/25*e^7 - 22369/50*e^5 + 34238/25*e^3 - 1213*e, -191/50*e^9 + 2839/25*e^7 - 51713/50*e^5 + 79901/25*e^3 - 2920*e, 2/5*e^9 - 117/10*e^7 + 1027/10*e^5 - 1444/5*e^3 + 218*e, -2/5*e^9 + 117/10*e^7 - 1027/10*e^5 + 1444/5*e^3 - 218*e, 191/50*e^9 - 2839/25*e^7 + 51713/50*e^5 - 79901/25*e^3 + 2920*e, -337/50*e^8 + 10021/50*e^6 - 45683/25*e^4 + 141632/25*e^2 - 5212, -271/50*e^8 + 8043/50*e^6 - 36489/25*e^4 + 111706/25*e^2 - 4020, -271/50*e^8 + 8043/50*e^6 - 36489/25*e^4 + 111706/25*e^2 - 4020, -337/50*e^8 + 10021/50*e^6 - 45683/25*e^4 + 141632/25*e^2 - 5212, -171/10*e^8 + 5083/10*e^6 - 23139/5*e^4 + 71411/5*e^2 - 13024, -171/10*e^8 + 5083/10*e^6 - 23139/5*e^4 + 71411/5*e^2 - 13024, 6/5*e^9 - 178/5*e^7 + 1613/5*e^5 - 4917/5*e^3 + 876*e, -6/5*e^9 + 178/5*e^7 - 1613/5*e^5 + 4917/5*e^3 - 876*e, -137/25*e^9 + 4071/25*e^7 - 37041/25*e^5 + 114189/25*e^3 - 4167*e, 17/50*e^9 - 511/50*e^7 + 2378/25*e^5 - 7687/25*e^3 + 305*e, -17/50*e^9 + 511/50*e^7 - 2378/25*e^5 + 7687/25*e^3 - 305*e, 137/25*e^9 - 4071/25*e^7 + 37041/25*e^5 - 114189/25*e^3 + 4167*e, 973/50*e^8 - 28909/50*e^6 + 131507/25*e^4 - 405428/25*e^2 + 14766, 973/50*e^8 - 28909/50*e^6 + 131507/25*e^4 - 405428/25*e^2 + 14766, 331/25*e^8 - 9848/25*e^6 + 89833/25*e^4 - 278432/25*e^2 + 10254, 456/25*e^8 - 13548/25*e^6 + 123258/25*e^4 - 380032/25*e^2 + 13856, 456/25*e^8 - 13548/25*e^6 + 123258/25*e^4 - 380032/25*e^2 + 13856, 331/25*e^8 - 9848/25*e^6 + 89833/25*e^4 - 278432/25*e^2 + 10254, -98/25*e^8 + 2909/25*e^6 - 26414/25*e^4 + 81006/25*e^2 - 2902, -23/25*e^8 + 684/25*e^6 - 6214/25*e^4 + 18906/25*e^2 - 680, -23/25*e^8 + 684/25*e^6 - 6214/25*e^4 + 18906/25*e^2 - 680, -98/25*e^8 + 2909/25*e^6 - 26414/25*e^4 + 81006/25*e^2 - 2902, -61/5*e^8 + 1813/5*e^6 - 16498/5*e^4 + 50812/5*e^2 - 9220, -61/5*e^8 + 1813/5*e^6 - 16498/5*e^4 + 50812/5*e^2 - 9220, 1/5*e^9 - 61/10*e^7 + 581/10*e^5 - 962/5*e^3 + 182*e, -313/50*e^9 + 4652/25*e^7 - 84709/50*e^5 + 130693/25*e^3 - 4764*e, 313/50*e^9 - 4652/25*e^7 + 84709/50*e^5 - 130693/25*e^3 + 4764*e, -1/5*e^9 + 61/10*e^7 - 581/10*e^5 + 962/5*e^3 - 182*e, -123/50*e^9 + 3659/50*e^7 - 16682/25*e^5 + 51653/25*e^3 - 1897*e, -199/25*e^9 + 5917/25*e^7 - 53907/25*e^5 + 166653/25*e^3 - 6103*e, 199/25*e^9 - 5917/25*e^7 + 53907/25*e^5 - 166653/25*e^3 + 6103*e, 123/50*e^9 - 3659/50*e^7 + 16682/25*e^5 - 51653/25*e^3 + 1897*e, -123/50*e^9 + 3659/50*e^7 - 16682/25*e^5 + 51603/25*e^3 - 1877*e, 123/50*e^9 - 3659/50*e^7 + 16682/25*e^5 - 51603/25*e^3 + 1877*e, 407/50*e^8 - 12081/50*e^6 + 54888/25*e^4 - 168977/25*e^2 + 6142, 469/50*e^8 - 13927/50*e^6 + 63296/25*e^4 - 194909/25*e^2 + 7100, 469/50*e^8 - 13927/50*e^6 + 63296/25*e^4 - 194909/25*e^2 + 7100, 407/50*e^8 - 12081/50*e^6 + 54888/25*e^4 - 168977/25*e^2 + 6142, 297/50*e^9 - 4413/25*e^7 + 80321/50*e^5 - 123892/25*e^3 + 4536*e, -297/50*e^9 + 4413/25*e^7 - 80321/50*e^5 + 123892/25*e^3 - 4536*e, 21/5*e^8 - 623/5*e^6 + 5643/5*e^4 - 17177/5*e^2 + 3058, 21/5*e^8 - 623/5*e^6 + 5643/5*e^4 - 17177/5*e^2 + 3058, 184/25*e^9 - 5472/25*e^7 + 49862/25*e^5 - 154098/25*e^3 + 5626*e, 33/25*e^9 - 989/25*e^7 + 9144/25*e^5 - 29076/25*e^3 + 1104*e, -33/25*e^9 + 989/25*e^7 - 9144/25*e^5 + 29076/25*e^3 - 1104*e, -184/25*e^9 + 5472/25*e^7 - 49862/25*e^5 + 154098/25*e^3 - 5626*e, 169/25*e^8 - 5027/25*e^6 + 45867/25*e^4 - 142493/25*e^2 + 5270, 169/25*e^8 - 5027/25*e^6 + 45867/25*e^4 - 142493/25*e^2 + 5270, 87/50*e^8 - 2571/50*e^6 + 11558/25*e^4 - 34557/25*e^2 + 1190, 87/50*e^8 - 2571/50*e^6 + 11558/25*e^4 - 34557/25*e^2 + 1190, -329/50*e^9 + 4891/25*e^7 - 89097/50*e^5 + 137594/25*e^3 - 5028*e, 81/10*e^9 - 1204/5*e^7 + 21933/10*e^5 - 33901/5*e^3 + 6216*e, -81/10*e^9 + 1204/5*e^7 - 21933/10*e^5 + 33901/5*e^3 - 6216*e, 329/50*e^9 - 4891/25*e^7 + 89097/50*e^5 - 137594/25*e^3 + 5028*e, 157/25*e^8 - 4656/25*e^6 + 42226/25*e^4 - 129454/25*e^2 + 4684, 157/25*e^8 - 4656/25*e^6 + 42226/25*e^4 - 129454/25*e^2 + 4684, -1/5*e^9 + 61/10*e^7 - 581/10*e^5 + 967/5*e^3 - 187*e, 111/50*e^9 - 1644/25*e^7 + 29723/50*e^5 - 45121/25*e^3 + 1597*e, -111/50*e^9 + 1644/25*e^7 - 29723/50*e^5 + 45121/25*e^3 - 1597*e, 1/5*e^9 - 61/10*e^7 + 581/10*e^5 - 967/5*e^3 + 187*e, 231/25*e^9 - 6873/25*e^7 + 62708/25*e^5 - 194582/25*e^3 + 7182*e, 154/25*e^9 - 4582/25*e^7 + 41797/25*e^5 - 129563/25*e^3 + 4784*e, -154/25*e^9 + 4582/25*e^7 - 41797/25*e^5 + 129563/25*e^3 - 4784*e, -231/25*e^9 + 6873/25*e^7 - 62708/25*e^5 + 194582/25*e^3 - 7182*e, 179/50*e^8 - 5307/50*e^6 + 24061/25*e^4 - 73919/25*e^2 + 2708, 179/50*e^8 - 5307/50*e^6 + 24061/25*e^4 - 73919/25*e^2 + 2708];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {16, 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;