/*
  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, 3, 4, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1}>;

primesArray := [
[3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3],
[9, 3, -w^4 + 5*w^2 - 3],
[9, 3, -w^4 + w^3 + 5*w^2 - 3*w - 2],
[13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1],
[17, 17, w^4 - w^3 - 5*w^2 + 3*w + 1],
[19, 19, -w^3 + w^2 + 4*w - 2],
[23, 23, -w^2 + 3],
[31, 31, w^3 - 4*w + 2],
[32, 2, 2],
[37, 37, w^3 - 3*w - 1],
[53, 53, -2*w^4 + w^3 + 9*w^2 - 3*w - 2],
[59, 59, -w^4 + 5*w^2 + w - 4],
[61, 61, -w^4 + w^3 + 5*w^2 - 4*w],
[67, 67, -w^4 + 6*w^2 + 2*w - 4],
[71, 71, 2*w^4 - w^3 - 9*w^2 + 4*w + 5],
[79, 79, 2*w^4 - w^3 - 10*w^2 + 2*w + 7],
[83, 83, -w^4 + 2*w^3 + 5*w^2 - 7*w - 2],
[83, 83, -w^4 + w^3 + 4*w^2 - 3*w + 3],
[83, 83, w^4 - w^3 - 5*w^2 + 4*w - 1],
[83, 83, -w^4 + w^3 + 4*w^2 - 4*w - 2],
[83, 83, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],
[97, 97, -w^4 - w^3 + 6*w^2 + 4*w - 4],
[101, 101, -w^4 + 4*w^2 + 2*w - 2],
[107, 107, -2*w^4 + w^3 + 11*w^2 - 3*w - 7],
[127, 127, w^3 - w^2 - 4*w],
[131, 131, w^4 - w^3 - 4*w^2 + 2*w - 1],
[137, 137, w^4 - w^3 - 6*w^2 + 3*w + 3],
[139, 139, -2*w^4 + w^3 + 9*w^2 - 3*w + 1],
[157, 157, 2*w^4 - w^3 - 9*w^2 + 4*w + 2],
[163, 163, -w^2 - 2*w + 3],
[167, 167, w^4 - 5*w^2 + 2*w + 1],
[169, 13, -2*w^4 + 2*w^3 + 9*w^2 - 7*w - 5],
[169, 13, w^4 - 2*w^3 - 5*w^2 + 8*w + 2],
[191, 191, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],
[193, 193, 2*w^4 - 2*w^3 - 8*w^2 + 7*w],
[199, 199, -w^4 + 2*w^3 + 5*w^2 - 9*w - 3],
[211, 211, -3*w^4 + w^3 + 13*w^2 - 2*w - 1],
[227, 227, 2*w^3 - w^2 - 9*w + 1],
[233, 233, -w^3 + w^2 + 4*w + 1],
[251, 251, -3*w^4 + 14*w^2 + w - 2],
[257, 257, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 5],
[257, 257, -w^4 + w^3 + 5*w^2 - 6*w - 1],
[257, 257, 2*w^4 - 2*w^3 - 11*w^2 + 9*w + 9],
[257, 257, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 3],
[257, 257, 2*w^4 - w^3 - 10*w^2 + 5*w + 5],
[269, 269, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],
[271, 271, -w^4 + 4*w^2 + w - 3],
[277, 277, -2*w^4 + 9*w^2 + w - 4],
[281, 281, w^4 - w^3 - 3*w^2 + 3*w - 4],
[283, 283, 2*w^4 - 3*w^3 - 8*w^2 + 13*w],
[289, 17, -w^4 + w^3 + 7*w^2 - 5*w - 5],
[289, 17, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 6],
[293, 293, -2*w^4 + 11*w^2 - 7],
[317, 317, w^3 - 6*w],
[337, 337, -w^4 + 2*w^3 + 5*w^2 - 9*w - 1],
[337, 337, w^4 - 2*w^3 - 5*w^2 + 6*w + 3],
[337, 337, -w^4 + w^3 + 5*w^2 - 4*w - 6],
[337, 337, -2*w^3 + 8*w + 1],
[337, 337, w^2 + w - 5],
[347, 347, -2*w^4 + w^3 + 9*w^2 - 5*w - 3],
[349, 349, 3*w^4 - 2*w^3 - 13*w^2 + 8*w + 2],
[353, 353, w^4 - 3*w^3 - 4*w^2 + 12*w + 2],
[359, 359, 2*w^3 - w^2 - 7*w + 3],
[361, 19, w^4 - 2*w^3 - 4*w^2 + 7*w + 3],
[361, 19, w^4 - w^3 - 3*w^2 + 3*w - 3],
[367, 367, -2*w^4 + w^3 + 9*w^2 - w - 2],
[379, 379, -2*w^3 + w^2 + 7*w + 2],
[379, 379, -2*w^4 + w^3 + 10*w^2 - 4*w],
[379, 379, 2*w^3 - w^2 - 6*w - 2],
[379, 379, -2*w^3 + 2*w^2 + 6*w - 3],
[379, 379, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6],
[383, 383, 2*w^4 + w^3 - 10*w^2 - 7*w + 4],
[383, 383, -2*w^3 + w^2 + 10*w],
[383, 383, w^3 + w^2 - 5*w - 1],
[383, 383, 3*w^4 + 3*w^3 - 15*w^2 - 15*w + 4],
[383, 383, -2*w^4 + 3*w^3 + 10*w^2 - 10*w - 2],
[389, 389, 3*w^4 - w^3 - 14*w^2 + 3*w + 3],
[397, 397, w^4 - 6*w^2 - 2*w + 5],
[397, 397, w^4 - 2*w^3 - 2*w^2 + 6*w - 5],
[397, 397, 2*w^4 - 2*w^3 - 11*w^2 + 10*w + 6],
[397, 397, w^4 - w^3 - 3*w^2 + 4*w - 6],
[397, 397, w^3 + w^2 - 3*w - 4],
[401, 401, w^3 - w^2 - 6*w + 3],
[401, 401, -w^4 + 4*w^2 + 2*w - 3],
[401, 401, -w^4 + 6*w^2 - w - 7],
[421, 421, -w^4 - w^3 + 5*w^2 + 5*w - 4],
[421, 421, -w^4 + w^3 + 6*w^2 - 3*w - 2],
[421, 421, -w^4 - w^3 + 6*w^2 + 6*w - 5],
[421, 421, 3*w^4 - w^3 - 15*w^2 + w + 10],
[421, 421, 2*w^4 - 9*w^2 + 2*w + 4],
[431, 431, 2*w^4 - 11*w^2 - 2*w + 11],
[439, 439, 2*w^4 - 3*w^3 - 9*w^2 + 10*w + 6],
[443, 443, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 2],
[449, 449, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],
[461, 461, 3*w^4 - 4*w^3 - 14*w^2 + 14*w + 5],
[463, 463, w^3 - 6*w - 1],
[467, 467, 3*w^4 - 3*w^3 - 13*w^2 + 10*w],
[487, 487, 3*w^4 - w^3 - 14*w^2 + w + 3],
[487, 487, 2*w^2 - w - 4],
[487, 487, -2*w^4 + 2*w^3 + 9*w^2 - 8*w - 5],
[487, 487, 2*w^4 - 10*w^2 - 3*w + 4],
[487, 487, -2*w^4 + w^3 + 8*w^2 - 3*w - 1],
[499, 499, -w^4 + 2*w^3 + 4*w^2 - 8*w - 2],
[499, 499, -2*w^4 + w^3 + 10*w^2 - w - 9],
[499, 499, -w^3 - 2*w^2 + 2*w + 5],
[499, 499, -w^4 + 4*w^2 + 4*w],
[499, 499, -2*w^4 + w^3 + 12*w^2 - 4*w - 9],
[509, 509, 3*w^4 - 2*w^3 - 16*w^2 + 6*w + 9],
[521, 521, -2*w^4 + w^3 + 10*w^2 - 5*w - 4],
[523, 523, w - 4],
[529, 23, -w^3 + 2*w^2 + 4*w - 4],
[529, 23, w^4 + 2*w^3 - 6*w^2 - 11*w + 4],
[569, 569, -2*w^4 + 11*w^2 - 10],
[571, 571, -2*w^4 + w^3 + 10*w^2 - 3*w - 2],
[593, 593, -2*w^4 + 3*w^3 + 10*w^2 - 11*w - 8],
[613, 613, -2*w^3 + 2*w^2 + 9*w - 5],
[617, 617, -2*w^4 + 10*w^2 - w - 7],
[631, 631, -2*w^4 + 2*w^3 + 7*w^2 - 8*w + 3],
[641, 641, -2*w^4 + 2*w^3 + 8*w^2 - 9*w + 3],
[643, 643, -3*w^4 + w^3 + 13*w^2 - w + 1],
[643, 643, w^4 + w^3 - 3*w^2 - 6*w - 2],
[643, 643, -w^4 - w^3 + 6*w^2 + 3*w - 4],
[643, 643, 2*w^4 - w^3 - 10*w^2 + 4*w + 1],
[643, 643, 2*w^4 - 2*w^3 - 8*w^2 + 6*w + 5],
[647, 647, -2*w^3 + w^2 + 6*w - 4],
[659, 659, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 10],
[661, 661, 2*w^4 + 3*w^3 - 11*w^2 - 14*w + 6],
[673, 673, w^4 - 3*w^3 - 3*w^2 + 10*w - 1],
[683, 683, w^4 - 2*w^3 - 4*w^2 + 6*w - 4],
[701, 701, w^3 - w^2 - 2*w + 4],
[727, 727, -w^4 + 6*w^2 + 2*w - 7],
[743, 743, w^4 - w^3 - 7*w^2 + 5*w + 9],
[769, 769, w^4 + 2*w^3 - 5*w^2 - 9*w + 4],
[787, 787, w^4 + w^3 - 7*w^2 - 3*w + 5],
[797, 797, -2*w^4 - w^3 + 10*w^2 + 4*w - 4],
[797, 797, w^4 - 3*w^3 - 5*w^2 + 12*w + 5],
[797, 797, -3*w^4 + 2*w^3 + 13*w^2 - 8*w],
[797, 797, -w^4 + w^3 + 3*w^2 - 5*w - 1],
[797, 797, 2*w^4 - 11*w^2 + 2*w + 7],
[809, 809, -4*w^4 + 2*w^3 + 18*w^2 - 6*w + 1],
[809, 809, 3*w^4 - 3*w^3 - 14*w^2 + 11*w + 8],
[809, 809, w^4 - 2*w^3 - 2*w^2 + 8*w - 4],
[809, 809, 2*w^4 - 8*w^2 - 2*w + 3],
[809, 809, 2*w^4 - 3*w^3 - 9*w^2 + 13*w + 4],
[821, 821, -2*w^4 + 2*w^3 + 8*w^2 - 6*w + 3],
[823, 823, -2*w^4 + 3*w^3 + 8*w^2 - 12*w - 1],
[829, 829, 2*w^4 - 3*w^3 - 10*w^2 + 13*w + 4],
[839, 839, -2*w^4 + 12*w^2 - w - 10],
[863, 863, 2*w^4 - w^3 - 9*w^2 + 5],
[877, 877, -w^4 + w^3 + 5*w^2 - 5*w - 7],
[881, 881, 2*w^4 - w^3 - 9*w^2 + w - 1],
[887, 887, 3*w^4 - 3*w^3 - 15*w^2 + 11*w + 6],
[907, 907, -3*w^4 + w^3 + 13*w^2 - w - 5],
[919, 919, w^4 + w^3 - 3*w^2 - 5*w - 3],
[929, 929, 2*w^4 - 3*w^3 - 10*w^2 + 9*w + 4],
[937, 937, -2*w^4 + 10*w^2 + 3*w - 7],
[941, 941, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],
[961, 31, 2*w^4 + 2*w^3 - 10*w^2 - 12*w + 5],
[961, 31, -w^4 + 5*w^2 - w - 7],
[967, 967, -2*w^4 + w^3 + 9*w^2 - 3*w + 2],
[991, 991, -3*w^4 + w^3 + 15*w^2 - 4*w - 5],
[997, 997, -3*w^4 + 15*w^2 - 5],
[997, 997, 2*w^4 - w^3 - 9*w^2 + 6*w + 2],
[997, 997, 2*w^4 - w^3 - 11*w^2 + w + 7],
[997, 997, 2*w^3 - 5*w - 1],
[997, 997, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 + x^11 - 26*x^10 - 19*x^9 + 250*x^8 + 123*x^7 - 1095*x^6 - 282*x^5 + 2152*x^4 + 34*x^3 - 1603*x^2 + 175*x + 305;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -857/3218*e^11 - 1583/3218*e^10 + 20297/3218*e^9 + 16152/1609*e^8 - 174333/3218*e^7 - 235177/3218*e^6 + 329084/1609*e^5 + 719279/3218*e^4 - 517564/1609*e^3 - 812571/3218*e^2 + 505287/3218*e + 130214/1609, -3013/6436*e^11 - 5235/6436*e^10 + 74213/6436*e^9 + 28228/1609*e^8 - 661407/6436*e^7 - 876491/6436*e^6 + 1282433/3218*e^5 + 2853611/6436*e^4 - 2016783/3218*e^3 - 3325395/6436*e^2 + 1936155/6436*e + 533641/3218, 1, -483/6436*e^11 - 262/1609*e^10 + 5193/3218*e^9 + 20127/6436*e^8 - 79497/6436*e^7 - 34233/1609*e^6 + 266929/6436*e^5 + 399869/6436*e^4 - 394677/6436*e^3 - 483261/6436*e^2 + 99395/3218*e + 160393/6436, 704/1609*e^11 + 1291/1609*e^10 - 17047/1609*e^9 - 54645/3218*e^8 + 299011/3218*e^7 + 414253/3218*e^6 - 1145137/3218*e^5 - 1313695/3218*e^4 + 898993/1609*e^3 + 746894/1609*e^2 - 441117/1609*e - 460365/3218, 314/1609*e^11 + 2029/6436*e^10 - 31675/6436*e^9 - 45153/6436*e^8 + 144487/3218*e^7 + 363965/6436*e^6 - 1148287/6436*e^5 - 617775/3218*e^4 + 1860795/6436*e^3 + 751193/3218*e^2 - 942755/6436*e - 479113/6436, -2891/6436*e^11 - 4897/6436*e^10 + 72589/6436*e^9 + 27022/1609*e^8 - 660655/6436*e^7 - 858993/6436*e^6 + 655578/1609*e^5 + 2847903/6436*e^4 - 1058822/1609*e^3 - 3316079/6436*e^2 + 2093021/6436*e + 530537/3218, -1905/3218*e^11 - 1597/1609*e^10 + 23230/1609*e^9 + 33970/1609*e^8 - 408585/3218*e^7 - 521435/3218*e^6 + 1557221/3218*e^5 + 842523/1609*e^4 - 1197717/1609*e^3 - 978284/1609*e^2 + 1129203/3218*e + 614797/3218, 553/1609*e^11 + 2563/3218*e^10 - 26091/3218*e^9 - 53993/3218*e^8 + 112425/1609*e^7 + 402749/3218*e^6 - 857499/3218*e^5 - 622522/1609*e^4 + 1367355/3218*e^3 + 693002/1609*e^2 - 675027/3218*e - 420467/3218, 1593/6436*e^11 + 1044/1609*e^10 - 17487/3218*e^9 - 83431/6436*e^8 + 279841/6436*e^7 + 292139/3218*e^6 - 1007989/6436*e^5 - 1701125/6436*e^4 + 1610107/6436*e^3 + 1899271/6436*e^2 - 423219/3218*e - 620761/6436, -361/1609*e^11 - 710/1609*e^10 + 8762/1609*e^9 + 15303/1609*e^8 - 77136/1609*e^7 - 117825/1609*e^6 + 297022/1609*e^5 + 376462/1609*e^4 - 469288/1609*e^3 - 424066/1609*e^2 + 231763/1609*e + 131659/1609, -295/1609*e^11 - 237/1609*e^10 + 8622/1609*e^9 + 7286/1609*e^8 - 88889/1609*e^7 - 76258/1609*e^6 + 388028/1609*e^5 + 312997/1609*e^4 - 661504/1609*e^3 - 419712/1609*e^2 + 331950/1609*e + 151486/1609, 131/3218*e^11 - 201/1609*e^10 - 2138/1609*e^9 + 7613/3218*e^8 + 44567/3218*e^7 - 22535/1609*e^6 - 180563/3218*e^5 + 86639/3218*e^4 + 248477/3218*e^3 - 23733/3218*e^2 - 39397/1609*e - 37659/3218, 3745/3218*e^11 + 12917/6436*e^10 - 185613/6436*e^9 - 282103/6436*e^8 + 415823/1609*e^7 + 2225225/6436*e^6 - 6475765/6436*e^5 - 3695577/3218*e^4 + 10204765/6436*e^3 + 2209548/1609*e^2 - 4955305/6436*e - 2873577/6436, 267/6436*e^11 + 1109/6436*e^10 - 3053/6436*e^9 - 4433/1609*e^8 - 3445/6436*e^7 + 82661/6436*e^6 + 29472/1609*e^5 - 96503/6436*e^4 - 73579/1609*e^3 - 1109/6436*e^2 + 194011/6436*e + 16445/3218, -857/3218*e^11 - 1583/3218*e^10 + 20297/3218*e^9 + 16152/1609*e^8 - 174333/3218*e^7 - 235177/3218*e^6 + 329084/1609*e^5 + 722497/3218*e^4 - 515955/1609*e^3 - 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1547015/1609*e^7 - 4148029/3218*e^6 + 12243153/3218*e^5 + 6983277/1609*e^4 - 19534637/3218*e^3 - 8351213/1609*e^2 + 9477885/3218*e + 5448705/3218, 5471/6436*e^11 + 7671/3218*e^10 - 27597/1609*e^9 - 285885/6436*e^8 + 789821/6436*e^7 + 900615/3218*e^6 - 2514647/6436*e^5 - 4482181/6436*e^4 + 3774807/6436*e^3 + 4160013/6436*e^2 - 485676/1609*e - 1156195/6436, -1055/3218*e^11 - 7613/6436*e^10 + 46261/6436*e^9 + 159371/6436*e^8 - 192171/3218*e^7 - 1171885/6436*e^6 + 1528139/6436*e^5 + 1786579/3218*e^4 - 2884767/6436*e^3 - 2029165/3218*e^2 + 1836635/6436*e + 1208003/6436, -3879/3218*e^11 - 4648/1609*e^10 + 45363/1609*e^9 + 193921/3218*e^8 - 779797/3218*e^7 - 716021/1609*e^6 + 3000869/3218*e^5 + 4407327/3218*e^4 - 4923563/3218*e^3 - 5022047/3218*e^2 + 1250056/1609*e + 1668089/3218, -1925/3218*e^11 - 13647/6436*e^10 + 73063/6436*e^9 + 259561/6436*e^8 - 241003/3218*e^7 - 1667439/6436*e^6 + 1398165/6436*e^5 + 2089999/3218*e^4 - 1982809/6436*e^3 - 1837895/3218*e^2 + 776257/6436*e + 902945/6436, 2907/1609*e^11 + 5548/1609*e^10 - 71404/1609*e^9 - 119588/1609*e^8 + 638254/1609*e^7 + 921533/1609*e^6 - 2500882/1609*e^5 - 2953685/1609*e^4 + 4008921/1609*e^3 + 3344390/1609*e^2 - 1930408/1609*e - 1029207/1609, -3969/3218*e^11 - 4166/1609*e^10 + 47360/1609*e^9 + 175745/3218*e^8 - 822133/3218*e^7 - 659012/1609*e^6 + 3134795/3218*e^5 + 4111367/3218*e^4 - 4942879/3218*e^3 - 4620761/3218*e^2 + 1210343/1609*e + 1477373/3218, 2855/3218*e^11 + 7857/3218*e^10 - 58763/3218*e^9 - 74091/1609*e^8 + 433907/3218*e^7 + 954801/3218*e^6 - 722508/1609*e^5 - 2483587/3218*e^4 + 1130360/1609*e^3 + 2518273/3218*e^2 - 1192215/3218*e - 393621/1609, 963/3218*e^11 - 339/3218*e^10 - 27933/3218*e^9 + 58/1609*e^8 + 273109/3218*e^7 + 77505/3218*e^6 - 530021/1609*e^5 - 541973/3218*e^4 + 699636/1609*e^3 + 888507/3218*e^2 - 477245/3218*e - 201544/1609, 11271/6436*e^11 + 4305/1609*e^10 - 143307/3218*e^9 - 387083/6436*e^8 + 2629973/6436*e^7 + 790596/1609*e^6 - 10449273/6436*e^5 - 10894333/6436*e^4 + 16685581/6436*e^3 + 13435629/6436*e^2 - 4059481/3218*e - 4525469/6436, -6638/1609*e^11 - 24885/3218*e^10 + 321243/3218*e^9 + 528507/3218*e^8 - 1409252/1609*e^7 - 4028033/3218*e^6 + 10813931/3218*e^5 + 6450911/1609*e^4 - 17028655/3218*e^3 - 7510437/1609*e^2 + 8298205/3218*e + 4844155/3218, 9883/3218*e^11 + 9536/1609*e^10 - 120089/1609*e^9 - 407955/3218*e^8 + 2119937/3218*e^7 + 1561181/1609*e^6 - 8190971/3218*e^5 - 9975937/3218*e^4 + 12971099/3218*e^3 + 11380693/3218*e^2 - 3180651/1609*e - 3524469/3218, 20313/6436*e^11 + 19289/3218*e^10 - 123914/1609*e^9 - 827331/6436*e^8 + 4390667/6436*e^7 + 3175925/3218*e^6 - 17009485/6436*e^5 - 20375839/6436*e^4 + 26959957/6436*e^3 + 23377199/6436*e^2 - 3277710/1609*e - 7226673/6436, 39191/6436*e^11 + 35179/3218*e^10 - 238120/1609*e^9 - 1497345/6436*e^8 + 8375013/6436*e^7 + 5726109/3218*e^6 - 32084183/6436*e^5 - 36781817/6436*e^4 + 50101987/6436*e^3 + 42605149/6436*e^2 - 6049509/1609*e - 13510859/6436, -10843/3218*e^11 - 11367/1609*e^10 + 126373/1609*e^9 + 467227/3218*e^8 - 2137091/3218*e^7 - 1706097/1609*e^6 + 7949373/3218*e^5 + 10427481/3218*e^4 - 12360837/3218*e^3 - 11766409/3218*e^2 + 3040766/1609*e + 3671969/3218, 27197/6436*e^11 + 48761/6436*e^10 - 657165/6436*e^9 - 512719/3218*e^8 + 5748945/6436*e^7 + 7740945/6436*e^6 - 10979273/3218*e^5 - 24576665/6436*e^4 + 17169203/3218*e^3 + 28369397/6436*e^2 - 16475565/6436*e - 2268631/1609, -15871/6436*e^11 - 8256/1609*e^10 + 191385/3218*e^9 + 703811/6436*e^8 - 3365021/6436*e^7 - 1336368/1609*e^6 + 13023021/6436*e^5 + 16906173/6436*e^4 - 20889413/6436*e^3 - 19167173/6436*e^2 + 5267371/3218*e + 5877717/6436, 1847/1609*e^11 + 6093/3218*e^10 - 94673/3218*e^9 - 139391/3218*e^8 + 437295/1609*e^7 + 1149387/3218*e^6 - 3479125/3218*e^5 - 1976964/1609*e^4 + 5467031/3218*e^3 + 2399995/1609*e^2 - 2498901/3218*e - 1613887/3218, 2505/1609*e^11 + 18713/6436*e^10 - 241395/6436*e^9 - 395805/6436*e^8 + 1051805/3218*e^7 + 2998573/6436*e^6 - 7983683/6436*e^5 - 4750285/3218*e^4 + 12326367/6436*e^3 + 5408951/3218*e^2 - 5831855/6436*e - 3386261/6436, 15197/6436*e^11 + 14365/3218*e^10 - 95652/1609*e^9 - 640127/6436*e^8 + 3505571/6436*e^7 + 2552181/3218*e^6 - 14056629/6436*e^5 - 16901807/6436*e^4 + 22999789/6436*e^3 + 19660603/6436*e^2 - 2874924/1609*e - 6250893/6436, 6447/6436*e^11 + 10001/6436*e^10 - 170041/6436*e^9 - 59800/1609*e^8 + 1616427/6436*e^7 + 2061889/6436*e^6 - 1659780/1609*e^5 - 7382211/6436*e^4 + 2732118/1609*e^3 + 9174171/6436*e^2 - 5509949/6436*e - 1429151/3218, -15239/6436*e^11 - 32249/6436*e^10 + 361195/6436*e^9 + 168182/1609*e^8 - 3121147/6436*e^7 - 4978105/6436*e^6 + 2978619/1609*e^5 + 15278049/6436*e^4 - 4749177/1609*e^3 - 16797891/6436*e^2 + 9394851/6436*e + 1291963/1609, -921/3218*e^11 - 969/3218*e^10 + 28429/3218*e^9 + 16626/1609*e^8 - 300335/3218*e^7 - 359835/3218*e^6 + 647765/1609*e^5 + 1460989/3218*e^4 - 1020576/1609*e^3 - 1830073/3218*e^2 + 887813/3218*e + 294958/1609, 4597/3218*e^11 + 2662/1609*e^10 - 58899/1609*e^9 - 120285/3218*e^8 + 1077641/3218*e^7 + 500206/1609*e^6 - 4208533/3218*e^5 - 3507911/3218*e^4 + 6446885/3218*e^3 + 4250481/3218*e^2 - 1467351/1609*e - 1329965/3218, 8063/1609*e^11 + 14610/1609*e^10 - 398377/3218*e^9 - 317843/1609*e^8 + 3565423/3218*e^7 + 2483053/1609*e^6 - 13896441/3218*e^5 - 16221011/3218*e^4 + 21982483/3218*e^3 + 18878139/3218*e^2 - 5328891/1609*e - 3000936/1609, 13115/3218*e^11 + 48535/6436*e^10 - 640389/6436*e^9 - 1038769/6436*e^8 + 1417724/1609*e^7 + 7966387/6436*e^6 - 21941727/6436*e^5 - 6382389/1609*e^4 + 34706495/6436*e^3 + 7310023/1609*e^2 - 16826981/6436*e - 9239737/6436, 3425/6436*e^11 + 2181/1609*e^10 - 41761/3218*e^9 - 190213/6436*e^8 + 753523/6436*e^7 + 365893/1609*e^6 - 3034963/6436*e^5 - 4620947/6436*e^4 + 5067403/6436*e^3 + 5093415/6436*e^2 - 1212111/3218*e - 1524655/6436, -16657/3218*e^11 - 16915/1609*e^10 + 197777/1609*e^9 + 706403/3218*e^8 - 3410381/3218*e^7 - 2624412/1609*e^6 + 12899649/3218*e^5 + 16247965/3218*e^4 - 20146983/3218*e^3 - 18146261/3218*e^2 + 4855326/1609*e + 5588791/3218, 642/1609*e^11 + 1383/1609*e^10 - 38853/3218*e^9 - 37466/1609*e^8 + 209962/1609*e^7 + 346117/1609*e^6 - 971107/1609*e^5 - 2518811/3218*e^4 + 1788836/1609*e^3 + 1482115/1609*e^2 - 1998077/3218*e - 930583/3218, 2371/1609*e^11 + 5461/1609*e^10 - 53982/1609*e^9 - 110844/1609*e^8 + 446618/1609*e^7 + 795859/1609*e^6 - 1637202/1609*e^5 - 2394683/1609*e^4 + 2553658/1609*e^3 + 2725012/1609*e^2 - 1259721/1609*e - 873107/1609, 2373/1609*e^11 + 3330/1609*e^10 - 60471/1609*e^9 - 74470/1609*e^8 + 555040/1609*e^7 + 606574/1609*e^6 - 2204079/1609*e^5 - 2080072/1609*e^4 + 3517914/1609*e^3 + 2516364/1609*e^2 - 1678048/1609*e - 806976/1609, -25881/6436*e^11 - 23151/3218*e^10 + 160802/1609*e^9 + 1012783/6436*e^8 - 5792163/6436*e^7 - 3979109/3218*e^6 + 22706481/6436*e^5 + 26147147/6436*e^4 - 36061385/6436*e^3 - 30584231/6436*e^2 + 4357955/1609*e + 9710481/6436, 24427/6436*e^11 + 50899/6436*e^10 - 575715/6436*e^9 - 528909/3218*e^8 + 4931855/6436*e^7 + 7815487/6436*e^6 - 9298561/3218*e^5 - 24034159/6436*e^4 + 14588959/3218*e^3 + 26623103/6436*e^2 - 14242579/6436*e - 1995683/1609, -2135/3218*e^11 - 5915/3218*e^10 + 49337/3218*e^9 + 63127/1609*e^8 - 417019/3218*e^7 - 949815/3218*e^6 + 782134/1609*e^5 + 2939775/3218*e^4 - 1234913/1609*e^3 - 3237829/3218*e^2 + 1232293/3218*e + 500013/1609];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1}>] := -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;