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

NN := ideal<ZF | {25, 25, w^4 - 2*w^3 - 4*w^2 + 6*w + 1}>;

primesArray := [
[5, 5, w^4 - w^3 - 4*w^2 + 2*w + 1],
[5, 5, w^2 - w - 2],
[7, 7, -w^2 + 2],
[11, 11, -w^2 + 3],
[13, 13, -w^4 + w^3 + 4*w^2 - 2*w - 2],
[23, 23, -w^4 + 5*w^2 + w - 4],
[31, 31, -w^4 + w^3 + 5*w^2 - 2*w - 2],
[32, 2, 2],
[37, 37, w^3 - 3*w - 1],
[41, 41, w^4 - w^3 - 3*w^2 + w + 1],
[47, 47, -w^3 + w^2 + 4*w],
[61, 61, -w^4 + 6*w^2 + w - 4],
[67, 67, -w^4 + w^3 + 5*w^2 - 4*w - 3],
[71, 71, w^4 - 6*w^2 + 7],
[73, 73, -w^4 + w^3 + 4*w^2 - 4*w - 2],
[83, 83, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],
[83, 83, -w^4 + 5*w^2 + 2*w - 4],
[83, 83, w^4 - 5*w^2 + 2],
[89, 89, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 4],
[101, 101, -2*w^4 + 2*w^3 + 8*w^2 - 3*w - 3],
[107, 107, w^4 - 6*w^2 - 2*w + 6],
[107, 107, 2*w^4 - 3*w^3 - 9*w^2 + 9*w + 4],
[109, 109, w^4 - 6*w^2 - 2*w + 3],
[125, 5, w^4 - w^3 - 6*w^2 + 4*w + 5],
[127, 127, -2*w - 3],
[131, 131, -w^4 + 2*w^3 + 3*w^2 - 7*w],
[137, 137, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],
[139, 139, -w^3 + 2*w^2 + 3*w - 3],
[139, 139, 2*w^4 - w^3 - 9*w^2 + 2*w + 4],
[149, 149, -w^4 + 6*w^2 - 2],
[151, 151, w^4 - w^3 - 4*w^2 + 2*w - 2],
[151, 151, w^3 - 2*w^2 - 3*w + 2],
[163, 163, -w^4 - w^3 + 6*w^2 + 5*w - 4],
[169, 13, -w^4 + 6*w^2 - w - 3],
[169, 13, w^4 - 2*w^3 - 2*w^2 + 7*w - 3],
[173, 173, w^4 + w^3 - 6*w^2 - 7*w + 3],
[179, 179, w^4 - w^3 - 3*w^2 + 2*w - 2],
[179, 179, w^4 - 2*w^3 - 2*w^2 + 7*w - 2],
[191, 191, w^2 + w - 4],
[199, 199, -w^3 + 2*w^2 + 4*w - 4],
[229, 229, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 9],
[233, 233, -2*w^3 - w^2 + 5*w + 1],
[233, 233, -2*w^4 + 2*w^3 + 8*w^2 - 5*w - 2],
[239, 239, -w^3 + 6*w],
[241, 241, w^3 - w^2 - 5*w - 1],
[243, 3, -3],
[251, 251, 2*w^4 - 3*w^3 - 6*w^2 + 4*w + 2],
[251, 251, -w^4 + w^3 + 5*w^2 - 2*w - 7],
[257, 257, -2*w^2 + 2*w + 5],
[257, 257, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3],
[263, 263, w^3 + w^2 - 4*w - 2],
[263, 263, -2*w + 5],
[263, 263, w^4 - 2*w^3 - 5*w^2 + 5*w + 4],
[271, 271, w^4 - 2*w^3 - 2*w^2 + 8*w - 4],
[271, 271, 2*w^4 - 2*w^3 - 9*w^2 + 5*w + 2],
[271, 271, -5*w^4 + 8*w^3 + 18*w^2 - 25*w + 3],
[277, 277, -w^3 + 2*w^2 + 4*w - 2],
[283, 283, -w^3 + 2*w^2 + 2*w - 5],
[307, 307, -w^4 - w^3 + 7*w^2 + 4*w - 5],
[307, 307, 2*w^4 + w^3 - 11*w^2 - 7*w + 4],
[317, 317, -2*w^3 + 2*w^2 + 7*w - 4],
[331, 331, 2*w^2 - w - 4],
[343, 7, w^4 + w^3 - 7*w^2 - 4*w + 10],
[349, 349, -w^4 + 2*w^3 + 5*w^2 - 7*w - 4],
[349, 349, 2*w^4 - 2*w^3 - 8*w^2 + 5*w + 3],
[353, 353, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 5],
[353, 353, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 2],
[353, 353, 2*w^4 - 9*w^2 - 3*w - 1],
[361, 19, w^3 + w^2 - 5*w - 3],
[373, 373, w^2 - 3*w - 2],
[379, 379, -w^4 + 2*w^3 + 4*w^2 - 8*w - 1],
[383, 383, -w^4 + 3*w^3 + 2*w^2 - 9*w + 6],
[389, 389, w^3 - 3*w - 4],
[389, 389, -w^3 + w^2 + 2*w - 4],
[397, 397, -2*w^4 + 4*w^3 + 5*w^2 - 13*w + 7],
[409, 409, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 1],
[409, 409, w^4 - 6*w^2 + 2*w + 1],
[421, 421, -2*w^4 + w^3 + 9*w^2 - 4*w - 5],
[431, 431, -2*w^4 + w^3 + 10*w^2 - 2*w - 3],
[433, 433, -w^4 + 4*w^2 + 2*w - 3],
[439, 439, -3*w^4 + 2*w^3 + 14*w^2 - 4*w - 4],
[443, 443, -3*w^4 + 3*w^3 + 13*w^2 - 6*w - 9],
[449, 449, -4*w^4 + 6*w^3 + 15*w^2 - 17*w + 1],
[449, 449, -w^4 + w^3 + 6*w^2 - 3*w - 3],
[457, 457, -2*w^4 + 12*w^2 + w - 10],
[457, 457, -w^4 + 2*w^3 + 4*w^2 - 7*w - 4],
[463, 463, 2*w^4 - 8*w^2 - 5*w - 1],
[479, 479, -w^4 + w^3 + 6*w^2 - 6*w - 3],
[479, 479, -w^4 + 5*w^2 + 2*w - 6],
[487, 487, 2*w^4 - w^3 - 9*w^2 + 2*w + 2],
[503, 503, 2*w^4 - w^3 - 8*w^2 - 2*w + 4],
[529, 23, w^4 - 3*w^3 - 4*w^2 + 10*w],
[529, 23, w^2 + w - 5],
[541, 541, -2*w^3 + w^2 + 7*w + 1],
[557, 557, w^4 - 2*w^3 - 6*w^2 + 4*w + 7],
[557, 557, w^3 - w^2 - 7*w + 2],
[557, 557, 3*w^4 - 3*w^3 - 13*w^2 + 7*w + 2],
[569, 569, -w^4 + 2*w^3 + 3*w^2 - 7*w - 1],
[577, 577, -4*w^4 + 6*w^3 + 15*w^2 - 19*w + 1],
[593, 593, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3],
[593, 593, 2*w^3 + w^2 - 5*w - 2],
[593, 593, -w^4 + w^3 + 3*w^2 - 4*w + 3],
[599, 599, -2*w^3 + 8*w - 1],
[607, 607, 2*w^3 - w^2 - 7*w + 5],
[607, 607, 3*w^4 - w^3 - 13*w^2 - 2*w + 3],
[607, 607, -w^4 - w^3 + 6*w^2 + 4*w - 4],
[613, 613, w^4 - 6*w^2 - 2*w],
[613, 613, -2*w^4 + 2*w^3 + 10*w^2 - 5*w - 5],
[613, 613, w^4 + w^3 - 5*w^2 - 3*w + 4],
[617, 617, 2*w^4 - 10*w^2 - w + 7],
[643, 643, -2*w^4 + 2*w^3 + 9*w^2 - 3*w - 5],
[647, 647, -2*w^3 + 2*w^2 + 4*w - 3],
[653, 653, w^3 + 3*w^2 - 2*w - 5],
[659, 659, w^3 - 2*w - 3],
[659, 659, -w^4 + 3*w^3 + 2*w^2 - 10*w + 1],
[661, 661, -2*w^4 + 2*w^3 + 7*w^2 - 3*w - 4],
[673, 673, 2*w^4 - 3*w^3 - 10*w^2 + 9*w + 5],
[683, 683, w^2 + w + 2],
[701, 701, -w^4 + 6*w^2 + 3*w - 5],
[709, 709, -w^4 + w^3 + 5*w^2 - 5*w - 3],
[719, 719, 2*w^4 - 3*w^3 - 9*w^2 + 7*w + 2],
[719, 719, 2*w^4 - 2*w^3 - 11*w^2 + 7*w + 8],
[727, 727, -w^4 + 5*w^2 - 7],
[733, 733, -2*w^4 + w^3 + 8*w^2 - 5],
[733, 733, 2*w^4 + w^3 - 11*w^2 - 6*w + 5],
[739, 739, 2*w^4 - 3*w^3 - 7*w^2 + 6*w + 1],
[751, 751, 3*w^4 - 3*w^3 - 15*w^2 + 7*w + 9],
[751, 751, -2*w^4 + 2*w^3 + 9*w^2 - 2*w - 8],
[751, 751, w^3 + w^2 - 6*w - 2],
[769, 769, -w^4 + 8*w^2 - 10],
[769, 769, -2*w^4 + 2*w^3 + 10*w^2 - 7*w - 8],
[773, 773, -3*w^4 + 2*w^3 + 16*w^2 - 5*w - 11],
[773, 773, -w^4 - w^3 + 7*w^2 + 4*w - 9],
[787, 787, 2*w^4 - w^3 - 10*w^2 + 3*w + 3],
[787, 787, -w^4 + 6*w^2 + 2*w - 10],
[787, 787, 3*w^4 - 3*w^3 - 13*w^2 + 7*w + 7],
[797, 797, -2*w^4 + w^3 + 9*w^2 - 3*w - 4],
[797, 797, -3*w^4 + 2*w^3 + 13*w^2 - 2*w - 5],
[797, 797, -2*w^3 + 2*w^2 + 5*w - 6],
[797, 797, 2*w^4 - 3*w^3 - 6*w^2 + 8*w - 3],
[797, 797, w^4 - 4*w^2 - w - 4],
[809, 809, 3*w^4 - 2*w^3 - 15*w^2 + 2*w + 8],
[811, 811, 3*w^3 - 2*w^2 - 11*w + 4],
[811, 811, 2*w^4 - 2*w^3 - 10*w^2 + 3*w + 6],
[821, 821, -w^3 + 2*w^2 + 2*w - 6],
[821, 821, -w^4 + 2*w^3 + w^2 - 6*w + 5],
[823, 823, w^4 + w^3 - 8*w^2 - 2*w + 8],
[827, 827, -w^4 + 2*w^2 + 2],
[839, 839, w^4 - 5*w^2 + 2*w + 6],
[841, 29, w^4 - 3*w^3 - w^2 + 11*w - 6],
[853, 853, -2*w^3 + w^2 + 9*w + 2],
[853, 853, -w^4 + 4*w^3 + 2*w^2 - 13*w + 5],
[853, 853, -w^4 - w^3 + 6*w^2 + 4*w - 5],
[863, 863, -w^4 + 2*w^3 + 6*w^2 - 5*w - 8],
[863, 863, 2*w^4 - 11*w^2 + 8],
[877, 877, w^4 + w^3 - 7*w^2 - 5*w + 3],
[877, 877, -w^4 - w^3 + 6*w^2 + 3*w - 4],
[887, 887, 2*w^4 - 2*w^3 - 7*w^2 + 4*w + 1],
[907, 907, 3*w^4 - w^3 - 14*w^2 + 4],
[907, 907, -2*w^4 + 2*w^3 + 8*w^2 - 5*w - 7],
[907, 907, -2*w^4 + w^3 + 12*w^2 - w - 11],
[919, 919, -w^4 + w^3 + 5*w^2 - 5*w - 5],
[929, 929, -w^2 - 3*w + 4],
[937, 937, -3*w^4 + 2*w^3 + 16*w^2 - 4*w - 11],
[937, 937, w^4 - w^3 - 4*w^2 - w - 1],
[937, 937, -w^4 + w^3 + 3*w^2 - 2*w - 4],
[941, 941, 2*w^4 - 3*w^3 - 8*w^2 + 7*w + 1],
[953, 953, -w^3 + 3*w^2 + 4*w - 6],
[967, 967, 2*w^3 - 6*w - 3],
[971, 971, -w^4 + w^3 + 5*w^2 - 5*w - 4],
[977, 977, -2*w^4 + w^3 + 11*w^2 - 11],
[977, 977, w^4 - w^3 - 4*w^2 - w + 4],
[997, 997, w^4 - 2*w^3 - 5*w^2 + 9*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 3*x^5 - 20*x^4 - 65*x^3 + 58*x^2 + 281*x + 183;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, -18/163*e^5 - 42/163*e^4 + 388/163*e^3 + 857/163*e^2 - 1724/163*e - 3148/163, 9/163*e^5 + 21/163*e^4 - 194/163*e^3 - 347/163*e^2 + 862/163*e + 759/163, 21/163*e^5 + 49/163*e^4 - 507/163*e^3 - 1027/163*e^2 + 2609/163*e + 3727/163, 43/163*e^5 + 46/163*e^4 - 945/163*e^3 - 1024/163*e^2 + 4209/163*e + 4224/163, 4/163*e^5 - 45/163*e^4 - 50/163*e^3 + 697/163*e^2 - 124/163*e - 1673/163, 27/163*e^5 + 63/163*e^4 - 582/163*e^3 - 1367/163*e^2 + 2749/163*e + 5211/163, 67/163*e^5 + 102/163*e^4 - 1408/163*e^3 - 2058/163*e^2 + 5910/163*e + 7552/163, -64/163*e^5 - 95/163*e^4 + 1452/163*e^3 + 2051/163*e^2 - 7144/163*e - 9255/163, -39/163*e^5 - 91/163*e^4 + 895/163*e^3 + 1721/163*e^2 - 4496/163*e - 6549/163, -12/163*e^5 - 28/163*e^4 + 150/163*e^3 + 517/163*e^2 + 46/163*e - 1501/163, -7/163*e^5 + 38/163*e^4 + 169/163*e^3 - 690/163*e^2 - 1087/163*e + 1420/163, -22/163*e^5 + 3/163*e^4 + 438/163*e^3 + 160/163*e^2 - 1926/163*e - 1638/163, 80/163*e^5 + 78/163*e^4 - 1652/163*e^3 - 1871/163*e^2 + 6974/163*e + 9083/163, -86/163*e^5 - 92/163*e^4 + 1890/163*e^3 + 2211/163*e^2 - 8744/163*e - 10404/163, 104/163*e^5 + 134/163*e^4 - 2278/163*e^3 - 2905/163*e^2 + 10468/163*e + 12411/163, -25/163*e^5 - 4/163*e^4 + 557/163*e^3 + 4/163*e^2 - 2811/163*e + 228/163, -43/163*e^5 - 46/163*e^4 + 945/163*e^3 + 1024/163*e^2 - 4535/163*e - 4224/163, 28/163*e^5 + 11/163*e^4 - 676/163*e^3 - 337/163*e^2 + 3370/163*e + 2307/163, 98/163*e^5 + 120/163*e^4 - 2040/163*e^3 - 2891/163*e^2 + 8372/163*e + 12720/163, -39/163*e^5 - 91/163*e^4 + 895/163*e^3 + 1884/163*e^2 - 4659/163*e - 8505/163, -126/163*e^5 - 131/163*e^4 + 2716/163*e^3 + 3065/163*e^2 - 11905/163*e - 13397/163, -110/163*e^5 - 148/163*e^4 + 2353/163*e^3 + 3082/163*e^2 - 10771/163*e - 12102/163, -68/163*e^5 - 50/163*e^4 + 1502/163*e^3 + 1354/163*e^2 - 7346/163*e - 7582/163, 37/163*e^5 + 32/163*e^4 - 707/163*e^3 - 684/163*e^2 + 2276/163*e + 3066/163, -73/163*e^5 - 116/163*e^4 + 1483/163*e^3 + 2398/163*e^2 - 5887/163*e - 9036/163, 90/163*e^5 + 210/163*e^4 - 1940/163*e^3 - 3959/163*e^2 + 8620/163*e + 12806/163, -74/163*e^5 - 64/163*e^4 + 1577/163*e^3 + 1694/163*e^2 - 6997/163*e - 8414/163, 76/163*e^5 + 123/163*e^4 - 1602/163*e^3 - 2405/163*e^2 + 7098/163*e + 8637/163, -118/163*e^5 - 221/163*e^4 + 2616/163*e^3 + 4459/163*e^2 - 11664/163*e - 17069/163, -4/163*e^5 + 45/163*e^4 + 50/163*e^3 - 1023/163*e^2 - 202/163*e + 3629/163, -136/163*e^5 - 263/163*e^4 + 3004/163*e^3 + 5479/163*e^2 - 14366/163*e - 21521/163, -109/163*e^5 - 200/163*e^4 + 2585/163*e^3 + 4275/163*e^2 - 13573/163*e - 17614/163, -223/163*e^5 - 303/163*e^4 + 4825/163*e^3 + 6660/163*e^2 - 21449/163*e - 27880/163, 27/163*e^5 + 63/163*e^4 - 582/163*e^3 - 1041/163*e^2 + 2912/163*e + 2277/163, -229/163*e^5 - 317/163*e^4 + 5063/163*e^3 + 7163/163*e^2 - 23545/163*e - 30831/163, -82/163*e^5 - 137/163*e^4 + 1840/163*e^3 + 2908/163*e^2 - 8216/163*e - 11262/163, 107/163*e^5 + 141/163*e^4 - 2397/163*e^3 - 3401/163*e^2 + 12005/163*e + 16413/163, 83/163*e^5 + 85/163*e^4 - 1771/163*e^3 - 2041/163*e^2 + 7859/163*e + 10477/163, 95/163*e^5 + 113/163*e^4 - 2247/163*e^3 - 2721/163*e^2 + 11725/163*e + 13445/163, 101/163*e^5 + 127/163*e^4 - 2159/163*e^3 - 2735/163*e^2 + 9746/163*e + 11343/163, 27/163*e^5 + 63/163*e^4 - 745/163*e^3 - 1693/163*e^2 + 5031/163*e + 8145/163, 113/163*e^5 + 155/163*e^4 - 2635/163*e^3 - 3415/163*e^2 + 12960/163*e + 14637/163, 120/163*e^5 + 117/163*e^4 - 2478/163*e^3 - 2725/163*e^2 + 9972/163*e + 10283/163, 109/163*e^5 + 200/163*e^4 - 2585/163*e^3 - 4438/163*e^2 + 13736/163*e + 19244/163, 43/163*e^5 + 46/163*e^4 - 945/163*e^3 - 1187/163*e^2 + 4209/163*e + 3735/163, -5/163*e^5 + 97/163*e^4 - 19/163*e^3 - 1727/163*e^2 + 481/163*e + 3273/163, 19/163*e^5 - 10/163*e^4 - 319/163*e^3 + 10/163*e^2 + 1041/163*e + 2526/163, 4/163*e^5 - 45/163*e^4 - 50/163*e^3 + 860/163*e^2 + 202/163*e - 2814/163, 85/163*e^5 + 144/163*e^4 - 1796/163*e^3 - 3078/163*e^2 + 6982/163*e + 11352/163, -77/163*e^5 - 71/163*e^4 + 1859/163*e^3 + 1701/163*e^2 - 9675/163*e - 9645/163, 71/163*e^5 + 220/163*e^4 - 1621/163*e^3 - 4132/163*e^2 + 8231/163*e + 13866/163, 7/163*e^5 - 38/163*e^4 - 169/163*e^3 + 690/163*e^2 + 924/163*e - 3050/163, 128/163*e^5 + 190/163*e^4 - 2904/163*e^3 - 4102/163*e^2 + 14614/163*e + 16880/163, -91/163*e^5 - 158/163*e^4 + 1871/163*e^3 + 3418/163*e^2 - 7611/163*e - 13814/163, 27/163*e^5 + 63/163*e^4 - 745/163*e^3 - 1041/163*e^2 + 5031/163*e + 3907/163, -130/163*e^5 - 86/163*e^4 + 2766/163*e^3 + 2368/163*e^2 - 11944/163*e - 12050/163, -105/163*e^5 - 82/163*e^4 + 2209/163*e^3 + 2038/163*e^2 - 9459/163*e - 8692/163, 145/163*e^5 + 284/163*e^4 - 3361/163*e^3 - 5826/163*e^2 + 17021/163*e + 23258/163, 106/163*e^5 + 193/163*e^4 - 2466/163*e^3 - 4105/163*e^2 + 12036/163*e + 16383/163, -49/163*e^5 - 60/163*e^4 + 1020/163*e^3 + 1038/163*e^2 - 4512/163*e - 4730/163, -13/163*e^5 + 24/163*e^4 + 407/163*e^3 - 676/163*e^2 - 2205/163*e + 3196/163, 133/163*e^5 + 256/163*e^4 - 3048/163*e^3 - 5472/163*e^2 + 14622/163*e + 23224/163, 60/163*e^5 + 140/163*e^4 - 1402/163*e^3 - 2748/163*e^2 + 6942/163*e + 8320/163, -166/163*e^5 - 333/163*e^4 + 3868/163*e^3 + 6853/163*e^2 - 20282/163*e - 27963/163, -176/163*e^5 - 302/163*e^4 + 3830/163*e^3 + 6170/163*e^2 - 17364/163*e - 23862/163, 49/163*e^5 + 60/163*e^4 - 1183/163*e^3 - 1038/163*e^2 + 5979/163*e + 3426/163, -80/163*e^5 - 241/163*e^4 + 1978/163*e^3 + 4805/163*e^2 - 10397/163*e - 16907/163, 13/163*e^5 + 139/163*e^4 - 407/163*e^3 - 2258/163*e^2 + 3183/163*e + 4628/163, -54/163*e^5 - 126/163*e^4 + 1327/163*e^3 + 3060/163*e^2 - 7291/163*e - 15638/163, -124/163*e^5 - 235/163*e^4 + 2854/163*e^3 + 5125/163*e^2 - 13434/163*e - 21813/163, -118/163*e^5 - 58/163*e^4 + 2453/163*e^3 + 1362/163*e^2 - 11175/163*e - 6474/163, -8/163*e^5 + 90/163*e^4 + 100/163*e^3 - 2046/163*e^2 + 900/163*e + 8562/163, -171/163*e^5 - 236/163*e^4 + 3849/163*e^3 + 5126/163*e^2 - 18497/163*e - 23060/163, -209/163*e^5 - 216/163*e^4 + 4487/163*e^3 + 5106/163*e^2 - 20253/163*e - 22570/163, 136/163*e^5 + 263/163*e^4 - 3004/163*e^3 - 5479/163*e^2 + 14040/163*e + 21847/163, 35/163*e^5 - 27/163*e^4 - 519/163*e^3 + 1005/163*e^2 + 382/163*e - 5633/163, -145/163*e^5 - 284/163*e^4 + 3361/163*e^3 + 5989/163*e^2 - 17347/163*e - 26844/163, -17/163*e^5 - 94/163*e^4 + 457/163*e^3 + 1398/163*e^2 - 2896/163*e - 836/163, 31/163*e^5 + 18/163*e^4 - 795/163*e^3 - 507/163*e^2 + 5233/163*e + 2234/163, -158/163*e^5 - 260/163*e^4 + 3442/163*e^3 + 5313/163*e^2 - 15966/163*e - 17943/163, 174/163*e^5 + 406/163*e^4 - 3968/163*e^3 - 8230/163*e^2 + 19219/163*e + 29670/163, -289/163*e^5 - 457/163*e^4 + 6139/163*e^3 + 9585/163*e^2 - 26249/163*e - 36543/163, 232/163*e^5 + 324/163*e^4 - 5182/163*e^3 - 7170/163*e^2 + 24430/163*e + 30758/163, 21/163*e^5 + 49/163*e^4 - 507/163*e^3 - 1353/163*e^2 + 3261/163*e + 8617/163, -137/163*e^5 - 211/163*e^4 + 3098/163*e^3 + 4449/163*e^2 - 14824/163*e - 19595/163, 131/163*e^5 + 34/163*e^4 - 2697/163*e^3 - 1338/163*e^2 + 10935/163*e + 6864/163, 155/163*e^5 + 253/163*e^4 - 3323/163*e^3 - 4980/163*e^2 + 14755/163*e + 15408/163, -8/163*e^5 - 73/163*e^4 + 426/163*e^3 + 1051/163*e^2 - 3990/163*e - 2359/163, -62/163*e^5 - 36/163*e^4 + 1264/163*e^3 + 525/163*e^2 - 5250/163*e - 2349/163, -73/163*e^5 - 279/163*e^4 + 1483/163*e^3 + 5495/163*e^2 - 6213/163*e - 18001/163, 25/163*e^5 + 167/163*e^4 - 557/163*e^3 - 3101/163*e^2 + 1833/163*e + 11345/163, 187/163*e^5 + 382/163*e^4 - 4375/163*e^3 - 8206/163*e^2 + 22239/163*e + 31690/163, -105/163*e^5 - 82/163*e^4 + 2209/163*e^3 + 2201/163*e^2 - 10763/163*e - 12441/163, 8/163*e^5 - 90/163*e^4 - 100/163*e^3 + 1068/163*e^2 - 85/163*e + 4152/163, 165/163*e^5 + 222/163*e^4 - 3611/163*e^3 - 4786/163*e^2 + 16401/163*e + 19620/163, -189/163*e^5 - 278/163*e^4 + 4237/163*e^3 + 5820/163*e^2 - 19243/163*e - 22296/163, 10/163*e^5 + 132/163*e^4 - 288/163*e^3 - 2088/163*e^2 + 1646/163*e + 1604/163, -60/163*e^5 - 303/163*e^4 + 1402/163*e^3 + 5519/163*e^2 - 6942/163*e - 17937/163, -96/163*e^5 - 224/163*e^4 + 1852/163*e^3 + 4136/163*e^2 - 6478/163*e - 12660/163, -206/163*e^5 - 372/163*e^4 + 4694/163*e^3 + 7870/163*e^2 - 23606/163*e - 30630/163, 216/163*e^5 + 341/163*e^4 - 4656/163*e^3 - 7187/163*e^2 + 19710/163*e + 29463/163, -153/163*e^5 - 194/163*e^4 + 3135/163*e^3 + 4269/163*e^2 - 13513/163*e - 18445/163, -194/163*e^5 - 181/163*e^4 + 4218/163*e^3 + 4093/163*e^2 - 19740/163*e - 18371/163, -197/163*e^5 - 351/163*e^4 + 4337/163*e^3 + 7034/163*e^2 - 19321/163*e - 24166/163, 311/163*e^5 + 454/163*e^4 - 6577/163*e^3 - 9256/163*e^2 + 28175/163*e + 35084/163, 153/163*e^5 + 194/163*e^4 - 3461/163*e^3 - 4758/163*e^2 + 18077/163*e + 22846/163, 128/163*e^5 + 190/163*e^4 - 2904/163*e^3 - 4102/163*e^2 + 13636/163*e + 17206/163, -141/163*e^5 - 166/163*e^4 + 3311/163*e^3 + 3915/163*e^2 - 17797/163*e - 17433/163, 197/163*e^5 + 188/163*e^4 - 4337/163*e^3 - 3937/163*e^2 + 19321/163*e + 16505/163, -69/163*e^5 - 161/163*e^4 + 1596/163*e^3 + 3095/163*e^2 - 7478/163*e - 14295/163, 228/163*e^5 + 206/163*e^4 - 4806/163*e^3 - 5422/163*e^2 + 20316/163*e + 25422/163, -210/163*e^5 - 327/163*e^4 + 4744/163*e^3 + 7173/163*e^2 - 21526/163*e - 30261/163, -216/163*e^5 - 178/163*e^4 + 4656/163*e^3 + 4416/163*e^2 - 20362/163*e - 21150/163, -141/163*e^5 - 166/163*e^4 + 2985/163*e^3 + 4078/163*e^2 - 12581/163*e - 17596/163, 37/163*e^5 + 32/163*e^4 - 1033/163*e^3 - 1336/163*e^2 + 6351/163*e + 8608/163, 198/163*e^5 + 462/163*e^4 - 4594/163*e^3 - 9264/163*e^2 + 21572/163*e + 31368/163, 93/163*e^5 + 217/163*e^4 - 2059/163*e^3 - 4781/163*e^2 + 9505/163*e + 19905/163, 194/163*e^5 + 181/163*e^4 - 4218/163*e^3 - 4256/163*e^2 + 19414/163*e + 18371/163, 49/163*e^5 + 60/163*e^4 - 1020/163*e^3 - 1364/163*e^2 + 4512/163*e + 2448/163, 280/163*e^5 + 436/163*e^4 - 6108/163*e^3 - 9564/163*e^2 + 27995/163*e + 40674/163, 87/163*e^5 + 40/163*e^4 - 1821/163*e^3 - 1018/163*e^2 + 8061/163*e + 7174/163, 86/163*e^5 - 71/163*e^4 - 1890/163*e^3 + 723/163*e^2 + 10048/163*e + 787/163, -1/163*e^5 - 111/163*e^4 - 232/163*e^3 + 1741/163*e^2 + 3128/163*e - 2801/163, 58/163*e^5 + 81/163*e^4 - 1214/163*e^3 - 1059/163*e^2 + 5700/163*e + 925/163, -66/163*e^5 + 9/163*e^4 + 988/163*e^3 - 9/163*e^2 - 1214/163*e + 139/163, 155/163*e^5 + 90/163*e^4 - 3323/163*e^3 - 3350/163*e^2 + 14755/163*e + 21928/163, -69/163*e^5 - 161/163*e^4 + 1759/163*e^3 + 3095/163*e^2 - 11227/163*e - 12991/163, -45/163*e^5 + 58/163*e^4 + 807/163*e^3 - 1688/163*e^2 - 2191/163*e + 7126/163, -105/163*e^5 + 81/163*e^4 + 2209/163*e^3 - 896/163*e^2 - 9459/163*e - 1357/163, -109/163*e^5 - 200/163*e^4 + 2259/163*e^3 + 3623/163*e^2 - 9335/163*e - 9138/163, 211/163*e^5 + 275/163*e^4 - 4512/163*e^3 - 5491/163*e^2 + 19050/163*e + 17577/163, -254/163*e^5 - 321/163*e^4 + 5620/163*e^3 + 7656/163*e^2 - 25378/163*e - 35167/163, -120/163*e^5 - 117/163*e^4 + 2478/163*e^3 + 2399/163*e^2 - 9972/163*e - 8653/163, 31/163*e^5 + 18/163*e^4 - 469/163*e^3 - 18/163*e^2 + 343/163*e + 278/163, 73/163*e^5 + 116/163*e^4 - 1483/163*e^3 - 2561/163*e^2 + 5887/163*e + 9525/163, -188/163*e^5 - 330/163*e^4 + 4306/163*e^3 + 7502/163*e^2 - 21556/163*e - 34002/163, -92/163*e^5 - 106/163*e^4 + 2128/163*e^3 + 2388/163*e^2 - 11166/163*e - 11562/163, -151/163*e^5 - 298/163*e^4 + 3273/163*e^3 + 6003/163*e^2 - 15205/163*e - 26046/163, 182/163*e^5 + 153/163*e^4 - 3742/163*e^3 - 3576/163*e^2 + 15548/163*e + 17685/163, 141/163*e^5 + 166/163*e^4 - 2985/163*e^3 - 3100/163*e^2 + 13233/163*e + 10098/163, -96/163*e^5 - 224/163*e^4 + 2178/163*e^3 + 4299/163*e^2 - 9412/163*e - 15757/163, -58/163*e^5 - 81/163*e^4 + 1214/163*e^3 + 2363/163*e^2 - 4233/163*e - 11357/163, -229/163*e^5 - 317/163*e^4 + 5063/163*e^3 + 6511/163*e^2 - 23219/163*e - 26919/163, -68/163*e^5 - 213/163*e^4 + 1828/163*e^3 + 4125/163*e^2 - 10280/163*e - 15243/163, 73/163*e^5 + 116/163*e^4 - 1809/163*e^3 - 2398/163*e^2 + 9473/163*e + 8710/163, 95/163*e^5 + 113/163*e^4 - 2084/163*e^3 - 3047/163*e^2 + 10584/163*e + 18009/163, 126/163*e^5 + 294/163*e^4 - 3042/163*e^3 - 6162/163*e^2 + 15980/163*e + 24318/163, 71/163*e^5 + 57/163*e^4 - 1621/163*e^3 - 2013/163*e^2 + 9209/163*e + 12725/163, 31/163*e^5 + 181/163*e^4 - 795/163*e^3 - 3441/163*e^2 + 4092/163*e + 10873/163, 169/163*e^5 + 340/163*e^4 - 3661/163*e^3 - 6860/163*e^2 + 16277/163*e + 22022/163, -16/163*e^5 + 17/163*e^4 + 200/163*e^3 - 17/163*e^2 + 333/163*e - 5207/163, 146/163*e^5 + 395/163*e^4 - 3618/163*e^3 - 8219/163*e^2 + 19924/163*e + 32253/163, -176/163*e^5 - 302/163*e^4 + 3830/163*e^3 + 6496/163*e^2 - 18016/163*e - 25818/163, 109/163*e^5 + 363/163*e^4 - 2585/163*e^3 - 7535/163*e^2 + 13247/163*e + 27557/163, 31/163*e^5 + 181/163*e^4 - 1121/163*e^3 - 3441/163*e^2 + 9145/163*e + 13481/163, 150/163*e^5 + 24/163*e^4 - 3016/163*e^3 - 1328/163*e^2 + 12954/163*e + 11346/163, -261/163*e^5 - 283/163*e^4 + 5300/163*e^3 + 6151/163*e^2 - 21412/163*e - 23315/163, -116/163*e^5 - 162/163*e^4 + 2428/163*e^3 + 3422/163*e^2 - 8140/163*e - 12934/163, 152/163*e^5 + 83/163*e^4 - 3367/163*e^3 - 2691/163*e^2 + 16315/163*e + 18415/163, 97/163*e^5 + 172/163*e^4 - 1783/163*e^3 - 3432/163*e^2 + 6121/163*e + 12364/163, 58/163*e^5 + 81/163*e^4 - 1377/163*e^3 - 2037/163*e^2 + 8471/163*e + 7119/163, -450/163*e^5 - 724/163*e^4 + 9863/163*e^3 + 15068/163*e^2 - 45545/163*e - 60770/163, -44/163*e^5 + 6/163*e^4 + 876/163*e^3 - 332/163*e^2 - 4178/163*e + 3896/163, 383/163*e^5 + 622/163*e^4 - 8618/163*e^3 - 13010/163*e^2 + 42080/163*e + 56152/163, 175/163*e^5 + 354/163*e^4 - 4225/163*e^3 - 7200/163*e^2 + 21959/163*e + 26766/163, -109/163*e^5 - 363/163*e^4 + 2422/163*e^3 + 6231/163*e^2 - 11454/163*e - 17451/163, -40/163*e^5 - 202/163*e^4 + 663/163*e^3 + 4440/163*e^2 - 1531/163*e - 14240/163, 321/163*e^5 + 586/163*e^4 - 6865/163*e^3 - 11670/163*e^2 + 30147/163*e + 41904/163, -122/163*e^5 - 13/163*e^4 + 2666/163*e^3 + 665/163*e^2 - 10888/163*e - 4149/163, 9/163*e^5 - 142/163*e^4 - 31/163*e^3 + 2424/163*e^2 - 116/163*e - 5598/163, -46/163*e^5 - 53/163*e^4 + 738/163*e^3 + 1357/163*e^2 + 448/163*e - 6433/163];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, w^4 - w^3 - 4*w^2 + 2*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;