/*
  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, 4, 5, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[3, 3, w - 1],
[5, 5, w^2 - w - 2],
[7, 7, w^4 - 2*w^3 - 3*w^2 + 4*w + 2],
[13, 13, w^3 - 2*w^2 - 2*w + 2],
[29, 29, -w^4 + 3*w^3 + 2*w^2 - 7*w - 1],
[29, 29, -w^2 + 2*w + 3],
[31, 31, w^4 - 2*w^3 - 3*w^2 + 5*w],
[31, 31, w^3 - 2*w^2 - 3*w + 2],
[32, 2, 2],
[43, 43, -w^2 - w + 4],
[53, 53, -w^4 + w^3 + 6*w^2 - 2*w - 5],
[53, 53, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],
[73, 73, w^4 - w^3 - 6*w^2 + 2*w + 6],
[73, 73, w^3 - w^2 - 4*w + 2],
[81, 3, 2*w^4 - 5*w^3 - 4*w^2 + 9*w + 2],
[83, 83, w^3 - w^2 - 5*w],
[89, 89, -w^4 + 2*w^3 + 4*w^2 - 4*w - 2],
[97, 97, w^4 - 3*w^3 - 2*w^2 + 6*w + 2],
[101, 101, -w^4 + 3*w^3 + 3*w^2 - 7*w - 3],
[103, 103, -w^4 + w^3 + 6*w^2 - w - 7],
[103, 103, 2*w^4 - 4*w^3 - 6*w^2 + 8*w + 1],
[103, 103, w^4 - w^3 - 6*w^2 + 3*w + 4],
[107, 107, -w^4 + 2*w^3 + 3*w^2 - 5*w + 2],
[109, 109, w^3 - 4*w - 1],
[109, 109, -w^2 + 2*w + 4],
[131, 131, w^4 - w^3 - 6*w^2 + w + 6],
[137, 137, w^3 - w^2 - 5*w + 3],
[149, 149, w^4 - 3*w^3 - 3*w^2 + 9*w + 6],
[149, 149, -w^4 + 6*w^2 + 3*w],
[163, 163, 2*w^2 - w - 5],
[167, 167, -2*w^4 + 3*w^3 + 8*w^2 - 4*w - 3],
[169, 13, w^4 - 3*w^3 + 6*w - 3],
[169, 13, w^3 - 4*w - 4],
[173, 173, w^4 - w^3 - 4*w^2 + 2*w + 1],
[173, 173, -w^4 + 3*w^3 + 3*w^2 - 7*w - 5],
[173, 173, w^4 - w^3 - 4*w^2 + w - 1],
[179, 179, w^2 - 3*w - 2],
[191, 191, -w^4 + 3*w^3 + w^2 - 6*w - 1],
[193, 193, -w^3 + 2*w^2 + w - 4],
[193, 193, -w^4 + 3*w^3 + 3*w^2 - 8*w - 1],
[199, 199, w^4 - 3*w^3 - w^2 + 7*w],
[199, 199, -2*w^4 + 5*w^3 + 5*w^2 - 10*w - 2],
[199, 199, w^4 - 3*w^3 + 5*w - 4],
[211, 211, 2*w^3 - 2*w^2 - 7*w - 1],
[211, 211, -2*w^3 + 3*w^2 + 5*w - 2],
[223, 223, -2*w^4 + 3*w^3 + 10*w^2 - 7*w - 9],
[227, 227, w^3 - w^2 - 4*w + 3],
[229, 229, w^4 - 4*w^3 + 10*w - 5],
[229, 229, -2*w^4 + 4*w^3 + 7*w^2 - 9*w - 8],
[239, 239, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 3],
[241, 241, w^3 - w^2 - 2*w - 2],
[257, 257, -w^4 + w^3 + 6*w^2 - 4*w - 4],
[263, 263, 2*w^3 - 4*w^2 - 5*w + 6],
[269, 269, w^3 - 2*w^2 - 4*w + 4],
[269, 269, -2*w^4 + 2*w^3 + 11*w^2 - 3*w - 10],
[271, 271, w^3 - 5*w],
[281, 281, w^4 - w^3 - 5*w^2 + 3*w + 3],
[281, 281, -2*w^4 + 4*w^3 + 6*w^2 - 6*w - 3],
[281, 281, w^3 - 5*w - 1],
[283, 283, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 3],
[289, 17, -2*w^4 + 3*w^3 + 8*w^2 - 5*w - 2],
[293, 293, 2*w^4 - 5*w^3 - 7*w^2 + 14*w + 9],
[311, 311, -w^4 + 2*w^3 + 2*w^2 - 4*w + 3],
[313, 313, w^3 - 6*w],
[331, 331, w^4 - 2*w^3 - 2*w^2 + w - 2],
[359, 359, w^4 - w^3 - 5*w^2 + 2*w - 1],
[373, 373, -w^4 + 3*w^3 + 2*w^2 - 8*w - 3],
[373, 373, 2*w^3 - 4*w^2 - 3*w + 3],
[379, 379, -w^4 + 4*w^3 - w^2 - 9*w + 3],
[379, 379, -w^4 + 3*w^3 + w^2 - 6*w - 2],
[379, 379, w^4 + w^3 - 7*w^2 - 6*w + 3],
[383, 383, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],
[383, 383, w^4 - 5*w^2 - 3*w + 3],
[383, 383, 2*w^4 - 5*w^3 - 3*w^2 + 8*w],
[389, 389, 3*w^4 - 7*w^3 - 8*w^2 + 16*w + 1],
[397, 397, -2*w^4 + 3*w^3 + 8*w^2 - 3*w - 7],
[397, 397, -w^4 + 2*w^3 + 3*w^2 - 4*w - 5],
[397, 397, -2*w^4 + 3*w^3 + 7*w^2 - 4*w - 3],
[409, 409, -w^4 + w^3 + 7*w^2 - 5*w - 9],
[419, 419, w^4 - 2*w^3 - w^2 + 2*w - 2],
[419, 419, 2*w^4 - 3*w^3 - 7*w^2 + 5*w + 2],
[419, 419, -w^4 + 4*w^3 - 9*w - 2],
[433, 433, -w^4 + 3*w^3 - 8*w + 4],
[433, 433, w^4 - 7*w^2 - w + 6],
[439, 439, 3*w^4 - 6*w^3 - 10*w^2 + 11*w + 7],
[449, 449, -2*w^4 + 5*w^3 + 5*w^2 - 11*w - 4],
[457, 457, -2*w^4 + 5*w^3 + 4*w^2 - 10*w - 2],
[463, 463, 2*w^4 - 5*w^3 - 4*w^2 + 8*w + 1],
[467, 467, 2*w^4 - 4*w^3 - 6*w^2 + 6*w + 1],
[479, 479, 3*w^4 - 4*w^3 - 12*w^2 + 4*w + 7],
[479, 479, 2*w^4 - 6*w^3 - 4*w^2 + 14*w + 1],
[491, 491, w^4 - 6*w^2 - w + 4],
[503, 503, 2*w^4 - w^3 - 11*w^2 - 3*w + 8],
[503, 503, 2*w^3 - 5*w^2 - 2*w + 6],
[509, 509, w^4 - 5*w^2 - 6*w],
[509, 509, w^4 - 3*w^3 - 3*w^2 + 9*w + 1],
[523, 523, 3*w^4 - 7*w^3 - 7*w^2 + 12*w + 3],
[523, 523, 3*w^4 - 8*w^3 - 7*w^2 + 17*w + 3],
[523, 523, w^4 - 7*w^2 - 2*w + 4],
[529, 23, -w^4 + w^3 + 7*w^2 - 2*w - 9],
[541, 541, w^4 - 3*w^3 - 3*w^2 + 5*w + 2],
[547, 547, -w^3 + 3*w^2 + 3*w - 4],
[557, 557, 2*w^3 - 4*w^2 - 6*w + 9],
[577, 577, 2*w^4 - 2*w^3 - 11*w^2 + 5*w + 10],
[577, 577, w^4 - 4*w^3 - w^2 + 9*w],
[577, 577, 2*w^4 - 3*w^3 - 9*w^2 + 4*w + 8],
[577, 577, w^4 - 7*w^2 - 3*w + 7],
[577, 577, -3*w^4 + 3*w^3 + 13*w^2 - w - 4],
[593, 593, 3*w - 2],
[601, 601, 2*w^4 - 4*w^3 - 7*w^2 + 5*w + 6],
[601, 601, -w^4 + 3*w^3 + w^2 - 7*w - 1],
[607, 607, w^3 - w^2 - 2*w - 3],
[625, 5, 2*w^4 - 5*w^3 - 8*w^2 + 14*w + 11],
[631, 631, -w^2 + w - 2],
[631, 631, w^4 - 7*w^2 - w + 8],
[643, 643, -2*w^4 + 4*w^3 + 9*w^2 - 10*w - 8],
[647, 647, 3*w^4 - 6*w^3 - 10*w^2 + 12*w + 8],
[653, 653, -3*w^4 + 6*w^3 + 11*w^2 - 12*w - 9],
[653, 653, w^4 - w^3 - 2*w^2 - 3*w - 3],
[653, 653, -2*w^4 + 4*w^3 + 7*w^2 - 6*w - 8],
[701, 701, w^4 - w^3 - 2*w^2 - w - 5],
[719, 719, 2*w^4 - 5*w^3 - 6*w^2 + 14*w + 2],
[727, 727, 2*w^4 - 5*w^3 - 5*w^2 + 9*w + 3],
[733, 733, 2*w^4 - 5*w^3 - 5*w^2 + 13*w + 3],
[739, 739, -2*w^3 + 4*w^2 + 4*w - 5],
[743, 743, w^4 - 2*w^3 - 3*w^2 + 6*w - 3],
[743, 743, 2*w^4 - 2*w^3 - 11*w^2 + 2*w + 10],
[757, 757, -w^4 + 2*w^3 + 5*w^2 - 6*w - 10],
[757, 757, 2*w^4 - 6*w^3 - 2*w^2 + 11*w - 4],
[761, 761, 3*w^3 - 4*w^2 - 9*w + 3],
[769, 769, 3*w^3 - 5*w^2 - 9*w + 7],
[769, 769, -2*w^4 + 4*w^3 + 7*w^2 - 11*w - 2],
[773, 773, 2*w^4 - 5*w^3 - 6*w^2 + 12*w + 1],
[773, 773, 3*w^4 - 4*w^3 - 12*w^2 + 3*w + 5],
[773, 773, 2*w^3 - w^2 - 10*w - 4],
[797, 797, -w^4 + 4*w^3 + 2*w^2 - 12*w - 3],
[811, 811, w^4 - 2*w^3 - 6*w^2 + 8*w + 4],
[823, 823, w^4 - 4*w^3 - w^2 + 10*w + 1],
[823, 823, 2*w^4 - 4*w^3 - 4*w^2 + 4*w - 3],
[827, 827, -w^4 + 2*w^3 + 2*w^2 - w + 3],
[827, 827, 2*w^4 - 4*w^3 - 5*w^2 + 7*w + 2],
[829, 829, -w^4 + 4*w^3 + 2*w^2 - 10*w - 2],
[829, 829, 3*w^3 - 3*w^2 - 11*w - 2],
[829, 829, w^4 - 2*w^3 - 3*w^2 + 2*w - 3],
[839, 839, -w^4 + 5*w^3 - 3*w^2 - 10*w + 5],
[839, 839, -w^4 + w^3 + 5*w^2 - 9],
[853, 853, 2*w^4 - 6*w^3 - 5*w^2 + 15*w + 4],
[859, 859, -3*w^4 + 8*w^3 + 6*w^2 - 18*w + 5],
[863, 863, w^4 - w^3 - 6*w^2 + 5*w + 2],
[877, 877, 3*w^3 - 4*w^2 - 8*w + 1],
[881, 881, 3*w^4 - 7*w^3 - 9*w^2 + 15*w + 3],
[881, 881, -w^3 + 7*w + 1],
[883, 883, -2*w^4 + 6*w^3 + 3*w^2 - 15*w],
[887, 887, w^4 - w^3 - 5*w^2 + w - 1],
[907, 907, 3*w^4 - 5*w^3 - 11*w^2 + 9*w + 5],
[919, 919, w^4 - 2*w^3 - 6*w^2 + 8*w + 7],
[937, 937, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 3],
[937, 937, -3*w^4 + 8*w^3 + 6*w^2 - 16*w + 1],
[937, 937, 3*w^4 - 5*w^3 - 11*w^2 + 7*w + 4],
[947, 947, w^3 - w^2 - 7*w + 2],
[947, 947, -w^4 + 2*w^3 + 4*w^2 - 7*w - 3],
[967, 967, -w^4 + 4*w^3 - 11*w],
[971, 971, 3*w^4 - 6*w^3 - 8*w^2 + 10*w],
[971, 971, w^4 - 8*w^2 + 2*w + 9],
[971, 971, w^4 - 4*w^3 + 2*w^2 + 7*w - 4],
[977, 977, -w^4 + 4*w^3 - 12*w + 5],
[983, 983, -w^4 + w^3 + 4*w^2 + 3*w - 2],
[983, 983, -2*w^4 + 3*w^3 + 8*w^2 - 6*w - 4],
[997, 997, 2*w^4 - 4*w^3 - 5*w^2 + 5*w - 3],
[997, 997, w^4 + w^3 - 5*w^2 - 9*w - 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^11 - 4*x^10 - 11*x^9 + 51*x^8 + 31*x^7 - 195*x^6 - 4*x^5 + 212*x^4 - 12*x^3 - 85*x^2 + 4*x + 11;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 21/25*e^10 - 76/25*e^9 - 254/25*e^8 + 954/25*e^7 + 943/25*e^6 - 3481/25*e^5 - 1172/25*e^4 + 3071/25*e^3 + 781/25*e^2 - 647/25*e - 172/25, -4/5*e^10 + 19/5*e^9 + 31/5*e^8 - 231/5*e^7 + 33/5*e^6 + 799/5*e^5 - 512/5*e^4 - 604/5*e^3 + 381/5*e^2 + 128/5*e - 57/5, 1, 14/25*e^10 - 84/25*e^9 - 36/25*e^8 + 986/25*e^7 - 1013/25*e^6 - 3179/25*e^5 + 5052/25*e^4 + 1739/25*e^3 - 4371/25*e^2 - 173/25*e + 977/25, 16/25*e^10 - 71/25*e^9 - 134/25*e^8 + 834/25*e^7 + 3/25*e^6 - 2626/25*e^5 + 1513/25*e^4 + 1091/25*e^3 - 999/25*e^2 - 62/25*e + 138/25, -26/25*e^10 + 56/25*e^9 + 474/25*e^8 - 799/25*e^7 - 3133/25*e^6 + 3536/25*e^5 + 8382/25*e^4 - 4776/25*e^3 - 6411/25*e^2 + 1357/25*e + 1282/25, 2/25*e^10 - 37/25*e^9 + 102/25*e^8 + 373/25*e^7 - 1434/25*e^6 - 722/25*e^5 + 4986/25*e^4 - 1223/25*e^3 - 3003/25*e^2 + 761/25*e + 411/25, 17/25*e^10 - 102/25*e^9 - 33/25*e^8 + 1158/25*e^7 - 1339/25*e^6 - 3387/25*e^5 + 6281/25*e^4 + 592/25*e^3 - 4513/25*e^2 + 481/25*e + 831/25, -13/25*e^10 - 22/25*e^9 + 437/25*e^8 + 138/25*e^7 - 4054/25*e^6 + 343/25*e^5 + 13141/25*e^4 - 2463/25*e^3 - 10843/25*e^2 + 1041/25*e + 2316/25, -4/5*e^10 + 19/5*e^9 + 31/5*e^8 - 231/5*e^7 + 28/5*e^6 + 809/5*e^5 - 462/5*e^4 - 674/5*e^3 + 251/5*e^2 + 198/5*e - 2/5, -36/25*e^10 + 216/25*e^9 + 89/25*e^8 - 2514/25*e^7 + 2612/25*e^6 + 7921/25*e^5 - 12673/25*e^4 - 3711/25*e^3 + 9779/25*e^2 + 227/25*e - 1923/25, 133/25*e^10 - 548/25*e^9 - 1342/25*e^8 + 6717/25*e^7 + 2739/25*e^6 - 23438/25*e^5 + 3669/25*e^4 + 17758/25*e^3 - 3162/25*e^2 - 3331/25*e + 569/25, 39/25*e^10 - 184/25*e^9 - 286/25*e^8 + 2161/25*e^7 - 488/25*e^6 - 6854/25*e^5 + 5427/25*e^4 + 3039/25*e^3 - 3796/25*e^2 + 152/25*e + 552/25, 11/5*e^10 - 46/5*e^9 - 109/5*e^8 + 564/5*e^7 + 208/5*e^6 - 1981/5*e^5 + 328/5*e^4 + 1581/5*e^3 - 154/5*e^2 - 387/5*e - 7/5, -38/25*e^10 + 103/25*e^9 + 612/25*e^8 - 1412/25*e^7 - 3554/25*e^6 + 5968/25*e^5 + 8416/25*e^4 - 7613/25*e^3 - 5668/25*e^2 + 2316/25*e + 1016/25, 162/25*e^10 - 622/25*e^9 - 1813/25*e^8 + 7688/25*e^7 + 5571/25*e^6 - 27257/25*e^5 - 3709/25*e^4 + 21887/25*e^3 + 3782/25*e^2 - 4409/25*e - 1434/25, 29/25*e^10 - 199/25*e^9 + 29/25*e^8 + 2271/25*e^7 - 3293/25*e^6 - 6844/25*e^5 + 14047/25*e^4 + 2179/25*e^3 - 9831/25*e^2 + 497/25*e + 1597/25, -67/25*e^10 + 402/25*e^9 + 158/25*e^8 - 4658/25*e^7 + 4964/25*e^6 + 14487/25*e^5 - 23981/25*e^4 - 6067/25*e^3 + 18438/25*e^2 + 19/25*e - 3506/25, 44/25*e^10 - 89/25*e^9 - 831/25*e^8 + 1331/25*e^7 + 5602/25*e^6 - 6334/25*e^5 - 14808/25*e^4 + 9869/25*e^3 + 10009/25*e^2 - 3183/25*e - 1758/25, 8/25*e^10 + 52/25*e^9 - 417/25*e^8 - 508/25*e^7 + 4314/25*e^6 + 987/25*e^5 - 14381/25*e^4 + 1258/25*e^3 + 11038/25*e^2 - 756/25*e - 2081/25, -3*e^10 + 14*e^9 + 23*e^8 - 166*e^7 + 25*e^6 + 541*e^5 - 370*e^4 - 299*e^3 + 242*e^2 + 31*e - 32, -68/25*e^10 + 158/25*e^9 + 1182/25*e^8 - 2182/25*e^7 - 7469/25*e^6 + 9223/25*e^5 + 19251/25*e^4 - 11543/25*e^3 - 14323/25*e^2 + 3651/25*e + 2826/25, 38/25*e^10 - 53/25*e^9 - 812/25*e^8 + 887/25*e^7 + 6004/25*e^6 - 4693/25*e^5 - 16916/25*e^4 + 8138/25*e^3 + 11918/25*e^2 - 2816/25*e - 2216/25, -242/25*e^10 + 977/25*e^9 + 2508/25*e^8 - 11958/25*e^7 - 5861/25*e^6 + 41612/25*e^5 - 3181/25*e^4 - 31242/25*e^3 + 1713/25*e^2 + 5969/25*e + 394/25, 2/25*e^10 + 13/25*e^9 - 98/25*e^8 - 152/25*e^7 + 1016/25*e^6 + 553/25*e^5 - 3489/25*e^4 - 748/25*e^3 + 3047/25*e^2 + 561/25*e - 539/25, 83/25*e^10 - 373/25*e^9 - 717/25*e^8 + 4517/25*e^7 + 289/25*e^6 - 15463/25*e^5 + 6769/25*e^4 + 11108/25*e^3 - 3637/25*e^2 - 2256/25*e + 169/25, 152/25*e^10 - 612/25*e^9 - 1598/25*e^8 + 7548/25*e^7 + 3941/25*e^6 - 26722/25*e^5 + 1186/25*e^4 + 21602/25*e^3 - 728/25*e^2 - 4614/25*e - 189/25, 102/25*e^10 - 362/25*e^9 - 1248/25*e^8 + 4498/25*e^7 + 4791/25*e^6 - 16022/25*e^5 - 6614/25*e^4 + 12877/25*e^3 + 4822/25*e^2 - 2689/25*e - 989/25, -152/25*e^10 + 512/25*e^9 + 1998/25*e^8 - 6498/25*e^7 - 8866/25*e^6 + 24197/25*e^5 + 16139/25*e^4 - 22877/25*e^3 - 12847/25*e^2 + 6039/25*e + 2789/25, 16/5*e^10 - 76/5*e^9 - 119/5*e^8 + 904/5*e^7 - 177/5*e^6 - 2976/5*e^5 + 2113/5*e^4 + 1776/5*e^3 - 1344/5*e^2 - 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36191/25*e^3 - 11151/25*e^2 + 7762/25*e + 3412/25, 21/25*e^10 + 249/25*e^9 - 1579/25*e^8 - 2371/25*e^7 + 17168/25*e^6 + 3794/25*e^5 - 57622/25*e^4 + 9721/25*e^3 + 43056/25*e^2 - 5097/25*e - 8097/25, -46/25*e^10 + 226/25*e^9 + 304/25*e^8 - 2629/25*e^7 + 932/25*e^6 + 8156/25*e^5 - 7278/25*e^4 - 3046/25*e^3 + 4044/25*e^2 - 303/25*e - 428/25, 36/5*e^10 - 181/5*e^9 - 234/5*e^8 + 2164/5*e^7 - 837/5*e^6 - 7231/5*e^5 + 6483/5*e^4 + 4686/5*e^3 - 5019/5*e^2 - 687/5*e + 998/5, 31/5*e^10 - 166/5*e^9 - 154/5*e^8 + 1954/5*e^7 - 1317/5*e^6 - 6296/5*e^5 + 7743/5*e^4 + 3341/5*e^3 - 6144/5*e^2 - 247/5*e + 1193/5, 127/25*e^10 - 837/25*e^9 + 27/25*e^8 + 9498/25*e^7 - 13309/25*e^6 - 27997/25*e^5 + 58361/25*e^4 + 6402/25*e^3 - 42678/25*e^2 + 2561/25*e + 7361/25, -297/25*e^10 + 1082/25*e^9 + 3578/25*e^8 - 13553/25*e^7 - 13301/25*e^6 + 49317/25*e^5 + 17329/25*e^4 - 43522/25*e^3 - 13842/25*e^2 + 10579/25*e + 2879/25, -463/25*e^10 + 2003/25*e^9 + 4287/25*e^8 - 24362/25*e^7 - 4879/25*e^6 + 83918/25*e^5 - 28734/25*e^4 - 61013/25*e^3 + 22432/25*e^2 + 11216/25*e - 3584/25, -411/25*e^10 + 1591/25*e^9 + 4539/25*e^8 - 19589/25*e^7 - 13438/25*e^6 + 68846/25*e^5 + 7227/25*e^4 - 53311/25*e^3 - 8221/25*e^2 + 10327/25*e + 3452/25, -138/25*e^10 + 303/25*e^9 + 2462/25*e^8 - 4212/25*e^7 - 15979/25*e^6 + 17968/25*e^5 + 42241/25*e^4 - 22913/25*e^3 - 32393/25*e^2 + 7466/25*e + 6516/25, -256/25*e^10 + 1061/25*e^9 + 2519/25*e^8 - 12869/25*e^7 - 4548/25*e^6 + 43891/25*e^5 - 9108/25*e^4 - 30156/25*e^3 + 5959/25*e^2 + 5017/25*e - 608/25, 321/25*e^10 - 1101/25*e^9 - 4129/25*e^8 + 13904/25*e^7 + 17518/25*e^6 - 51181/25*e^5 - 29022/25*e^4 + 46096/25*e^3 + 20331/25*e^2 - 10197/25*e - 4622/25, 159/25*e^10 - 304/25*e^9 - 3066/25*e^8 + 4566/25*e^7 + 21097/25*e^6 - 21699/25*e^5 - 57063/25*e^4 + 33359/25*e^3 + 40524/25*e^2 - 11163/25*e - 7463/25, -214/25*e^10 + 1134/25*e^9 + 1111/25*e^8 - 13336/25*e^7 + 8438/25*e^6 + 42829/25*e^5 - 50777/25*e^4 - 22064/25*e^3 + 39396/25*e^2 + 1198/25*e - 7577/25, 24/5*e^10 - 149/5*e^9 - 36/5*e^8 + 1721/5*e^7 - 2028/5*e^6 - 5319/5*e^5 + 9417/5*e^4 + 2114/5*e^3 - 7046/5*e^2 + 192/5*e + 1312/5, 192/25*e^10 - 677/25*e^9 - 2408/25*e^8 + 8558/25*e^7 + 9736/25*e^6 - 31762/25*e^5 - 14844/25*e^4 + 30017/25*e^3 + 10462/25*e^2 - 7669/25*e - 2094/25, -139/25*e^10 + 384/25*e^9 + 2161/25*e^8 - 5011/25*e^7 - 12337/25*e^6 + 19454/25*e^5 + 30498/25*e^4 - 20164/25*e^3 - 26704/25*e^2 + 5498/25*e + 6348/25, 226/25*e^10 - 1031/25*e^9 - 1824/25*e^8 + 12249/25*e^7 - 842/25*e^6 - 39911/25*e^5 + 24843/25*e^4 + 21426/25*e^3 - 17914/25*e^2 - 1382/25*e + 2818/25, 14/5*e^10 - 9/5*e^9 - 341/5*e^8 + 221/5*e^7 + 2707/5*e^6 - 1509/5*e^5 - 7828/5*e^4 + 3244/5*e^3 + 5189/5*e^2 - 1118/5*e - 823/5, 93/25*e^10 - 658/25*e^9 + 193/25*e^8 + 7457/25*e^7 - 11906/25*e^6 - 21923/25*e^5 + 50399/25*e^4 + 4893/25*e^3 - 37227/25*e^2 + 1874/25*e + 7024/25, -192/25*e^10 + 627/25*e^9 + 2558/25*e^8 - 7833/25*e^7 - 11686/25*e^6 + 27987/25*e^5 + 22694/25*e^4 - 22192/25*e^3 - 20062/25*e^2 + 4569/25*e + 4844/25, 22/25*e^10 - 32/25*e^9 - 428/25*e^8 + 378/25*e^7 + 3076/25*e^6 - 1042/25*e^5 - 9104/25*e^4 - 578/25*e^3 + 8092/25*e^2 + 646/25*e - 1204/25, -237/25*e^10 + 847/25*e^9 + 2913/25*e^8 - 10638/25*e^7 - 11221/25*e^6 + 38782/25*e^5 + 15134/25*e^4 - 33987/25*e^3 - 9282/25*e^2 + 7684/25*e + 1634/25, -278/25*e^10 + 1043/25*e^9 + 3247/25*e^8 - 13122/25*e^7 - 11074/25*e^6 + 48358/25*e^5 + 10546/25*e^4 - 44803/25*e^3 - 6183/25*e^2 + 10446/25*e + 1196/25, -232/25*e^10 + 792/25*e^9 + 2968/25*e^8 - 9868/25*e^7 - 12631/25*e^6 + 35302/25*e^5 + 22124/25*e^4 - 28682/25*e^3 - 19402/25*e^2 + 6124/25*e + 4949/25, 177/25*e^10 - 462/25*e^9 - 2873/25*e^8 + 6223/25*e^7 + 16916/25*e^6 - 25447/25*e^5 - 41139/25*e^4 + 30077/25*e^3 + 30147/25*e^2 - 9164/25*e - 5464/25, -136/25*e^10 + 391/25*e^9 + 2039/25*e^8 - 5039/25*e^7 - 11063/25*e^6 + 19071/25*e^5 + 25677/25*e^4 - 18236/25*e^3 - 20596/25*e^2 + 4302/25*e + 4152/25, -17/5*e^10 + 72/5*e^9 + 163/5*e^8 - 878/5*e^7 - 236/5*e^6 + 3022/5*e^5 - 936/5*e^4 - 2102/5*e^3 + 958/5*e^2 + 79/5*e - 231/5, -491/25*e^10 + 1896/25*e^9 + 5459/25*e^8 - 23484/25*e^7 - 16278/25*e^6 + 83601/25*e^5 + 7937/25*e^4 - 68241/25*e^3 - 5001/25*e^2 + 14587/25*e + 1512/25, 58/5*e^10 - 253/5*e^9 - 532/5*e^8 + 3092/5*e^7 + 539/5*e^6 - 10803/5*e^5 + 3924/5*e^4 + 8458/5*e^3 - 3242/5*e^2 - 1786/5*e + 539/5, -21/25*e^10 + 101/25*e^9 + 129/25*e^8 - 1129/25*e^7 + 582/25*e^6 + 3081/25*e^5 - 4203/25*e^4 + 554/25*e^3 + 3619/25*e^2 - 1178/25*e - 728/25, 556/25*e^10 - 2186/25*e^9 - 6019/25*e^8 + 26944/25*e^7 + 16523/25*e^6 - 94966/25*e^5 - 2942/25*e^4 + 74656/25*e^3 + 2991/25*e^2 - 15292/25*e - 1867/25, -554/25*e^10 + 2224/25*e^9 + 5846/25*e^8 - 27446/25*e^7 - 14582/25*e^6 + 97244/25*e^5 - 3847/25*e^4 - 78654/25*e^3 + 2681/25*e^2 + 16628/25*e + 853/25, 52/5*e^10 - 167/5*e^9 - 718/5*e^8 + 2143/5*e^7 + 3456/5*e^6 - 8132/5*e^5 - 7019/5*e^4 + 8087/5*e^3 + 5672/5*e^2 - 2169/5*e - 1239/5, 133/5*e^10 - 528/5*e^9 - 1422/5*e^8 + 6512/5*e^7 + 3729/5*e^6 - 23023/5*e^5 + 114/5*e^4 + 18433/5*e^3 + 3/5*e^2 - 3891/5*e - 296/5, 186/25*e^10 - 741/25*e^9 - 1964/25*e^8 + 9039/25*e^7 + 5063/25*e^6 - 31246/25*e^5 - 277/25*e^4 + 23011/25*e^3 + 3221/25*e^2 - 5077/25*e - 2202/25, -207/25*e^10 + 942/25*e^9 + 1693/25*e^8 - 11268/25*e^7 + 594/25*e^6 + 37327/25*e^5 - 22826/25*e^4 - 22107/25*e^3 + 18673/25*e^2 + 1749/25*e - 3526/25, 141/25*e^10 - 421/25*e^9 - 2059/25*e^8 + 5384/25*e^7 + 10853/25*e^6 - 20076/25*e^5 - 25037/25*e^4 + 18341/25*e^3 + 22551/25*e^2 - 4662/25*e - 5537/25, -506/25*e^10 + 2111/25*e^9 + 4994/25*e^8 - 25819/25*e^7 - 9098/25*e^6 + 89866/25*e^5 - 18133/25*e^4 - 67981/25*e^3 + 13309/25*e^2 + 13217/25*e - 1183/25, 111/25*e^10 - 616/25*e^9 - 439/25*e^8 + 7064/25*e^7 - 5837/25*e^6 - 21321/25*e^5 + 29848/25*e^4 + 6511/25*e^3 - 18279/25*e^2 + 923/25*e + 1998/25, 438/25*e^10 - 1653/25*e^9 - 5062/25*e^8 + 20637/25*e^7 + 16929/25*e^6 - 74643/25*e^5 - 15766/25*e^4 + 64388/25*e^3 + 11668/25*e^2 - 14291/25*e - 3191/25, 254/25*e^10 - 1099/25*e^9 - 2346/25*e^8 + 13321/25*e^7 + 2757/25*e^6 - 45644/25*e^5 + 14372/25*e^4 + 32679/25*e^3 - 7656/25*e^2 - 5853/25*e + 197/25, -196/25*e^10 + 1001/25*e^9 + 1204/25*e^8 - 11929/25*e^7 + 5457/25*e^6 + 39481/25*e^5 - 38578/25*e^4 - 24246/25*e^3 + 29944/25*e^2 + 3472/25*e - 5753/25, 296/25*e^10 - 1276/25*e^9 - 2754/25*e^8 + 15504/25*e^7 + 3293/25*e^6 - 53181/25*e^5 + 17503/25*e^4 + 37671/25*e^3 - 12444/25*e^2 - 6597/25*e + 1628/25, 186/25*e^10 - 516/25*e^9 - 2889/25*e^8 + 6789/25*e^7 + 16263/25*e^6 - 26646/25*e^5 - 38452/25*e^4 + 28286/25*e^3 + 29521/25*e^2 - 7827/25*e - 6427/25, -227/25*e^10 + 962/25*e^9 + 2198/25*e^8 - 11798/25*e^7 - 3516/25*e^6 + 41322/25*e^5 - 10536/25*e^4 - 31902/25*e^3 + 9453/25*e^2 + 5189/25*e - 2061/25, -131/25*e^10 + 386/25*e^9 + 1944/25*e^8 - 5019/25*e^7 - 10398/25*e^6 + 19441/25*e^5 + 23867/25*e^4 - 20231/25*e^3 - 19991/25*e^2 + 4992/25*e + 4892/25, 273/25*e^10 - 1038/25*e^9 - 3102/25*e^8 + 12827/25*e^7 + 10059/25*e^6 - 45403/25*e^5 - 9236/25*e^4 + 36173/25*e^3 + 10303/25*e^2 - 7886/25*e - 3586/25, 234/25*e^10 - 1029/25*e^9 - 2116/25*e^8 + 12541/25*e^7 + 1872/25*e^6 - 43474/25*e^5 + 16212/25*e^4 + 32709/25*e^3 - 11076/25*e^2 - 6463/25*e + 1362/25, -216/25*e^10 + 421/25*e^9 + 4109/25*e^8 - 6184/25*e^7 - 28028/25*e^6 + 28376/25*e^5 + 76012/25*e^4 - 40891/25*e^3 - 56801/25*e^2 + 13112/25*e + 11162/25, -71/25*e^10 + 201/25*e^9 + 1079/25*e^8 - 2604/25*e^7 - 5943/25*e^6 + 10006/25*e^5 + 13772/25*e^4 - 10321/25*e^3 - 10206/25*e^2 + 3422/25*e + 2372/25, 402/25*e^10 - 1587/25*e^9 - 4348/25*e^8 + 19648/25*e^7 + 11866/25*e^6 - 70022/25*e^5 - 1789/25*e^4 + 57877/25*e^3 + 2197/25*e^2 - 13264/25*e - 1539/25, -348/25*e^10 + 1538/25*e^9 + 3102/25*e^8 - 18702/25*e^7 - 2234/25*e^6 + 64578/25*e^5 - 25964/25*e^4 - 48148/25*e^3 + 17247/25*e^2 + 9886/25*e - 1689/25, 226/25*e^10 - 1206/25*e^9 - 1149/25*e^8 + 14174/25*e^7 - 9117/25*e^6 - 45536/25*e^5 + 53668/25*e^4 + 23926/25*e^3 - 39464/25*e^2 - 2632/25*e + 6768/25, 14/5*e^10 - 89/5*e^9 - 6/5*e^8 + 1001/5*e^7 - 1358/5*e^6 - 2884/5*e^5 + 6122/5*e^4 + 409/5*e^3 - 4831/5*e^2 + 447/5*e + 992/5, 602/25*e^10 - 2462/25*e^9 - 6173/25*e^8 + 30323/25*e^7 + 13616/25*e^6 - 107122/25*e^5 + 12186/25*e^4 + 86002/25*e^3 - 9903/25*e^2 - 18389/25*e + 786/25, 17/5*e^10 - 7/5*e^9 - 433/5*e^8 + 228/5*e^7 + 3551/5*e^6 - 1757/5*e^5 - 10654/5*e^4 + 4102/5*e^3 + 7862/5*e^2 - 1499/5*e - 1594/5, -243/25*e^10 + 733/25*e^9 + 3532/25*e^8 - 9482/25*e^7 - 18269/25*e^6 + 36373/25*e^5 + 39626/25*e^4 - 36968/25*e^3 - 29423/25*e^2 + 10251/25*e + 6176/25, 11/5*e^10 - 66/5*e^9 - 24/5*e^8 + 764/5*e^7 - 852/5*e^6 - 2386/5*e^5 + 4208/5*e^4 + 1046/5*e^3 - 3749/5*e^2 + 168/5*e + 833/5, 7/5*e^10 - 57/5*e^9 + 52/5*e^8 + 613/5*e^7 - 1334/5*e^6 - 1562/5*e^5 + 5201/5*e^4 - 453/5*e^3 - 3563/5*e^2 + 401/5*e + 611/5, -243/25*e^10 + 958/25*e^9 + 2657/25*e^8 - 11932/25*e^7 - 7594/25*e^6 + 43273/25*e^5 + 2826/25*e^4 - 38493/25*e^3 - 3048/25*e^2 + 9101/25*e + 1801/25, 34/25*e^10 - 79/25*e^9 - 591/25*e^8 + 1116/25*e^7 + 3672/25*e^6 - 4949/25*e^5 - 8913/25*e^4 + 7134/25*e^3 + 5199/25*e^2 - 3113/25*e - 663/25, -243/25*e^10 + 708/25*e^9 + 3632/25*e^8 - 9232/25*e^7 - 19444/25*e^6 + 35698/25*e^5 + 43651/25*e^4 - 36318/25*e^3 - 32748/25*e^2 + 9126/25*e + 7176/25, 278/25*e^10 - 1118/25*e^9 - 2897/25*e^8 + 13697/25*e^7 + 6924/25*e^6 - 47758/25*e^5 + 2929/25*e^4 + 36178/25*e^3 - 1142/25*e^2 - 7221/25*e - 321/25, 268/25*e^10 - 1358/25*e^9 - 1632/25*e^8 + 16007/25*e^7 - 7531/25*e^6 - 51623/25*e^5 + 52499/25*e^4 + 27093/25*e^3 - 39677/25*e^2 - 2126/25*e + 7024/25, -118/25*e^10 + 558/25*e^9 + 932/25*e^8 - 6782/25*e^7 + 681/25*e^6 + 23448/25*e^5 - 13599/25*e^4 - 17668/25*e^3 + 8952/25*e^2 + 3776/25*e - 1349/25, -246/25*e^10 + 901/25*e^9 + 2929/25*e^8 - 11254/25*e^7 - 10468/25*e^6 + 40606/25*e^5 + 11447/25*e^4 - 34321/25*e^3 - 6056/25*e^2 + 6972/25*e + 697/25, -34/5*e^10 + 114/5*e^9 + 451/5*e^8 - 1456/5*e^7 - 2032/5*e^6 + 5499/5*e^5 + 3768/5*e^4 - 5429/5*e^3 - 2964/5*e^2 + 1303/5*e + 668/5];
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 - 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;