/*
  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, -1, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 5, w^2 + w - 2}>;

primesArray := [
[3, 3, w + 1],
[5, 5, -w^3 + 5*w + 2],
[5, 5, w - 1],
[11, 11, -w^3 + 5*w],
[13, 13, -w^3 + w^2 + 6*w - 3],
[16, 2, 2],
[19, 19, 2*w^3 - w^2 - 11*w + 2],
[25, 5, -w^2 + w + 3],
[27, 3, w^3 - w^2 - 5*w + 4],
[31, 31, -w - 3],
[37, 37, -w^3 - w^2 + 6*w + 4],
[41, 41, w^3 + w^2 - 6*w - 5],
[43, 43, w^3 - 7*w - 2],
[47, 47, -2*w^3 + 11*w + 2],
[61, 61, w^2 - 3],
[67, 67, w^2 + w - 4],
[79, 79, -4*w^3 + 2*w^2 + 22*w - 9],
[97, 97, -3*w^3 + 2*w^2 + 19*w - 6],
[97, 97, -w^3 + 6*w - 3],
[97, 97, -3*w^3 + 16*w],
[97, 97, -w^3 + w^2 + 4*w - 3],
[101, 101, -2*w^3 + w^2 + 11*w - 6],
[103, 103, w^3 - w^2 - 7*w],
[109, 109, -w^3 - w^2 + 6*w + 2],
[121, 11, 2*w^3 - 11*w + 1],
[127, 127, -2*w^3 + w^2 + 13*w - 1],
[127, 127, w^3 - w^2 - 6*w - 2],
[157, 157, -2*w^3 + 2*w^2 + 10*w - 7],
[157, 157, w - 4],
[163, 163, 2*w^3 - w^2 - 13*w + 3],
[173, 173, -2*w^3 + w^2 + 13*w - 6],
[173, 173, 2*w^3 - 12*w + 1],
[173, 173, 2*w^3 - 11*w + 2],
[173, 173, w^2 + w - 5],
[179, 179, -2*w^3 + 10*w + 3],
[181, 181, -w^3 - w^2 + 7*w + 8],
[191, 191, w^3 - 5*w - 5],
[191, 191, 2*w^3 - w^2 - 9*w - 1],
[193, 193, -3*w^3 + 3*w^2 + 19*w - 7],
[197, 197, -w^3 + w^2 + 6*w - 7],
[223, 223, w^3 + w^2 - 6*w + 1],
[227, 227, 6*w^3 - 3*w^2 - 34*w + 9],
[227, 227, w^3 + w^2 - 8*w],
[233, 233, 2*w^3 - 13*w - 3],
[233, 233, -w^3 + 2*w^2 + 4*w - 6],
[239, 239, -2*w^3 + 10*w + 1],
[239, 239, w^3 - 8*w - 2],
[251, 251, -3*w^3 + w^2 + 18*w],
[251, 251, -5*w^3 + 2*w^2 + 30*w - 11],
[263, 263, -2*w^3 + w^2 + 10*w + 2],
[269, 269, 3*w^3 - 18*w - 8],
[271, 271, 2*w^3 + w^2 - 10*w - 2],
[271, 271, -3*w^3 + 19*w],
[277, 277, -w^3 + w^2 + 4*w - 8],
[277, 277, 2*w^3 + w^2 - 14*w - 8],
[281, 281, w^3 - 8*w],
[307, 307, 3*w^3 - w^2 - 16*w + 2],
[307, 307, 2*w^3 - w^2 - 9*w + 1],
[307, 307, -2*w^3 + 2*w^2 + 13*w - 7],
[307, 307, -w^3 + w^2 + 7*w - 6],
[313, 313, -3*w^3 + w^2 + 15*w + 4],
[317, 317, -4*w^3 + w^2 + 22*w - 3],
[317, 317, 2*w^3 - w^2 - 10*w - 3],
[337, 337, w^3 + w^2 - 4*w - 5],
[349, 349, -2*w^3 - w^2 + 11*w + 3],
[349, 349, -4*w^3 + 2*w^2 + 24*w - 11],
[353, 353, 3*w^3 - w^2 - 17*w + 7],
[353, 353, -3*w^3 - 2*w^2 + 17*w + 11],
[367, 367, 2*w^3 - 11*w - 7],
[373, 373, -w^3 + w^2 + 4*w - 5],
[383, 383, -4*w^3 + 3*w^2 + 23*w - 9],
[383, 383, 4*w^3 - w^2 - 22*w],
[389, 389, w^2 - 2*w - 5],
[401, 401, -3*w^3 + 2*w^2 + 16*w - 9],
[401, 401, 2*w^3 - w^2 - 14*w - 1],
[409, 409, 2*w^3 + 2*w^2 - 13*w - 9],
[419, 419, -4*w^3 + 24*w + 7],
[419, 419, 2*w^2 + 2*w - 7],
[419, 419, -w^3 + 2*w^2 + 8*w - 6],
[419, 419, 6*w^3 - 3*w^2 - 33*w + 13],
[431, 431, 2*w^3 - w^2 - 12*w - 5],
[439, 439, -3*w^3 + 2*w^2 + 18*w - 4],
[439, 439, w^3 + 2*w^2 - 7*w - 3],
[443, 443, 6*w^3 - 3*w^2 - 33*w + 7],
[449, 449, -3*w^3 + w^2 + 14*w - 4],
[457, 457, 2*w^3 - 9*w],
[457, 457, 3*w^3 - w^2 - 16*w - 2],
[479, 479, 3*w^3 - w^2 - 19*w + 4],
[487, 487, -w^3 + w^2 + 4*w - 6],
[491, 491, 3*w^3 + w^2 - 17*w - 5],
[499, 499, 5*w^3 - 3*w^2 - 28*w + 8],
[499, 499, 2*w^3 - w^2 - 12*w - 1],
[503, 503, -2*w^3 - w^2 + 12*w + 3],
[503, 503, w^3 - 3*w - 4],
[509, 509, -3*w - 5],
[509, 509, -4*w^3 + 3*w^2 + 25*w - 10],
[523, 523, -2*w^3 + w^2 + 14*w - 6],
[529, 23, 4*w^3 - 2*w^2 - 21*w + 5],
[529, 23, -3*w^3 - w^2 + 15*w + 6],
[557, 557, -2*w^3 + w^2 + 13*w + 2],
[557, 557, 3*w^3 - 2*w^2 - 16*w + 3],
[569, 569, 2*w^2 - w - 8],
[587, 587, w^3 + 2*w^2 - 5*w - 16],
[587, 587, -4*w^3 + w^2 + 25*w - 5],
[599, 599, w^3 + 2*w^2 - 6*w - 5],
[599, 599, -5*w^3 + 3*w^2 + 30*w - 15],
[601, 601, w^3 - 6*w - 6],
[607, 607, -w^3 + w^2 + 3*w - 4],
[607, 607, -w^3 + 2*w - 3],
[613, 613, 6*w^3 - 2*w^2 - 31*w + 11],
[613, 613, w^3 + w^2 - 4*w - 6],
[617, 617, 5*w^3 - w^2 - 30*w + 2],
[617, 617, -2*w^3 + 2*w^2 + 9*w + 3],
[619, 619, -w^3 + 2*w^2 + 6*w - 5],
[619, 619, 2*w^3 - w^2 - 11*w - 3],
[631, 631, -w^3 + 2*w^2 + 5*w - 5],
[631, 631, -3*w^3 - w^2 + 18*w + 5],
[647, 647, 6*w^3 - w^2 - 36*w + 2],
[659, 659, 4*w^3 + w^2 - 24*w - 14],
[661, 661, 2*w^2 - w - 4],
[673, 673, 2*w^3 + w^2 - 14*w - 12],
[677, 677, 3*w^3 + 2*w^2 - 16*w - 7],
[677, 677, -3*w^3 + w^2 + 15*w + 1],
[683, 683, 2*w^2 - 2*w - 9],
[683, 683, 2*w^3 - 14*w + 1],
[691, 691, 5*w^3 - 2*w^2 - 30*w + 6],
[701, 701, -2*w^3 + w^2 + 12*w + 2],
[701, 701, -5*w^3 + 2*w^2 + 26*w],
[719, 719, 3*w^3 - 18*w + 1],
[719, 719, 3*w^3 - w^2 - 19*w + 1],
[727, 727, -3*w^2 - 2*w + 11],
[727, 727, 2*w^3 + w^2 - 13*w - 2],
[733, 733, 2*w^3 - 3*w^2 - 8*w + 5],
[739, 739, -3*w^3 + 16*w - 6],
[743, 743, -4*w^3 - w^2 + 25*w + 8],
[751, 751, w^2 - 2*w - 7],
[751, 751, -3*w^3 + 15*w + 5],
[757, 757, 3*w^3 - 19*w - 2],
[761, 761, 2*w^2 + w - 5],
[769, 769, -2*w^3 + w^2 + 14*w - 9],
[769, 769, 2*w^2 - w - 7],
[773, 773, w^3 - w^2 - 5*w - 4],
[773, 773, w^3 - 3*w - 6],
[787, 787, 3*w^3 - 19*w - 3],
[797, 797, 8*w^3 - 3*w^2 - 47*w + 10],
[809, 809, 4*w^3 - 25*w - 1],
[811, 811, w^3 - 9*w - 1],
[821, 821, 6*w^3 - 3*w^2 - 35*w + 8],
[823, 823, 2*w^3 + w^2 - 10*w + 1],
[823, 823, -2*w^3 + 2*w^2 + 7*w - 4],
[829, 829, -5*w^3 + w^2 + 27*w - 4],
[829, 829, 5*w^3 - 2*w^2 - 26*w + 10],
[839, 839, w^3 + 2*w^2 - 5*w - 7],
[839, 839, 3*w^3 - w^2 - 15*w + 2],
[841, 29, 3*w^3 + w^2 - 14*w - 2],
[841, 29, 4*w^3 - 2*w^2 - 21*w + 10],
[857, 857, -8*w^3 + 4*w^2 + 44*w - 11],
[857, 857, -3*w^3 + 3*w^2 + 20*w - 9],
[859, 859, w^3 + 2*w^2 - 6*w - 6],
[877, 877, w^3 - 3*w^2 - 4*w + 14],
[883, 883, 2*w^3 + w^2 - 10*w - 11],
[887, 887, 3*w^3 + w^2 - 20*w - 2],
[919, 919, 7*w^3 - 3*w^2 - 39*w + 9],
[929, 929, 3*w^3 - 17*w + 2],
[929, 929, -6*w^3 + 2*w^2 + 35*w - 10],
[937, 937, w^3 + w^2 - 3*w - 5],
[937, 937, -2*w^3 + 3*w^2 + 12*w - 12],
[937, 937, -3*w^3 - 2*w^2 + 19*w + 8],
[937, 937, 3*w^3 - 2*w^2 - 13*w + 3],
[941, 941, w^3 + 2*w^2 - 6*w - 8],
[941, 941, 5*w^3 - w^2 - 28*w - 3],
[947, 947, -4*w^3 + 2*w^2 + 25*w - 6],
[953, 953, w^3 - 7*w - 7],
[953, 953, -3*w^3 + 2*w^2 + 18*w - 3],
[967, 967, w^3 - 2*w^2 - 5*w - 3],
[971, 971, w^2 + 2*w - 6],
[977, 977, 2*w^3 - 11*w - 8],
[997, 997, -3*w^3 + w^2 + 17*w + 2],
[997, 997, -2*w^3 + 2*w^2 + 14*w - 7]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 12*x^4 + 33*x^2 + 3*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -1, -1, 1/3*e^5 - 1/3*e^4 - 14/3*e^3 + 8/3*e^2 + 43/3*e + 2/3, e^2 - 2, 1/3*e^5 - 1/3*e^4 - 11/3*e^3 + 8/3*e^2 + 28/3*e + 2/3, 1/3*e^5 - 1/3*e^4 - 11/3*e^3 + 8/3*e^2 + 28/3*e + 2/3, 1/3*e^5 - 1/3*e^4 - 14/3*e^3 + 5/3*e^2 + 43/3*e + 11/3, -e^2 + e + 4, 4/3*e^5 - 1/3*e^4 - 44/3*e^3 + 8/3*e^2 + 103/3*e + 2/3, -2/3*e^5 - 1/3*e^4 + 25/3*e^3 + 11/3*e^2 - 74/3*e - 13/3, 1/3*e^5 - 1/3*e^4 - 14/3*e^3 + 8/3*e^2 + 43/3*e - 7/3, 4/3*e^5 - 1/3*e^4 - 53/3*e^3 + 8/3*e^2 + 160/3*e + 8/3, e^5 + e^4 - 11*e^3 - 7*e^2 + 28*e + 6, 5/3*e^5 + 1/3*e^4 - 55/3*e^3 - 2/3*e^2 + 125/3*e - 8/3, 1/3*e^5 + 2/3*e^4 - 14/3*e^3 - 10/3*e^2 + 49/3*e - 19/3, -1/3*e^5 + 1/3*e^4 + 11/3*e^3 - 5/3*e^2 - 25/3*e + 4/3, 2*e^5 - 26*e^3 + 80*e + 3, -2/3*e^5 - 1/3*e^4 + 22/3*e^3 + 8/3*e^2 - 65/3*e - 16/3, e^5 - e^4 - 11*e^3 + 8*e^2 + 28*e + 1, -1/3*e^5 + 1/3*e^4 + 8/3*e^3 - 11/3*e^2 - 10/3*e + 34/3, -7/3*e^5 + 1/3*e^4 + 86/3*e^3 - 8/3*e^2 - 244/3*e - 2/3, e^5 + e^4 - 11*e^3 - 8*e^2 + 25*e + 15, 5/3*e^5 + 1/3*e^4 - 55/3*e^3 - 5/3*e^2 + 134/3*e + 13/3, -e^5 - e^4 + 9*e^3 + 7*e^2 - 12*e - 12, e^5 - 11*e^3 - e^2 + 27*e - 2, -5/3*e^5 - 1/3*e^4 + 55/3*e^3 + 8/3*e^2 - 128/3*e - 1/3, -5/3*e^5 + 2/3*e^4 + 64/3*e^3 - 16/3*e^2 - 203/3*e + 14/3, -7/3*e^5 - 2/3*e^4 + 80/3*e^3 + 13/3*e^2 - 205/3*e - 29/3, 2/3*e^5 - 5/3*e^4 - 22/3*e^3 + 34/3*e^2 + 53/3*e - 5/3, -e^5 + e^4 + 13*e^3 - 7*e^2 - 42*e, -2*e^5 + e^4 + 25*e^3 - 6*e^2 - 68*e - 7, 2*e^5 + e^4 - 25*e^3 - 7*e^2 + 77*e + 9, e^5 + e^4 - 14*e^3 - 8*e^2 + 49*e + 19, -5/3*e^5 - 4/3*e^4 + 49/3*e^3 + 32/3*e^2 - 95/3*e - 46/3, 4/3*e^5 + 2/3*e^4 - 41/3*e^3 - 10/3*e^2 + 82/3*e - 19/3, 2*e^5 - 22*e^3 + 50*e + 6, -7/3*e^5 + 1/3*e^4 + 86/3*e^3 - 14/3*e^2 - 253/3*e - 5/3, e^5 + e^4 - 12*e^3 - 7*e^2 + 37*e + 3, 4*e^5 + e^4 - 46*e^3 - 5*e^2 + 120*e + 10, -5/3*e^5 - 1/3*e^4 + 49/3*e^3 + 11/3*e^2 - 98/3*e - 46/3, 3*e^5 + e^4 - 35*e^3 - 4*e^2 + 89*e - 3, e^5 - 12*e^3 + 3*e^2 + 35*e - 6, -e^4 - e^3 + 8*e^2 + 4*e - 10, -e^2 - 9*e + 4, 2/3*e^5 + 1/3*e^4 - 37/3*e^3 - 11/3*e^2 + 158/3*e + 28/3, -1/3*e^5 + 1/3*e^4 + 8/3*e^3 - 8/3*e^2 - 1/3*e + 34/3, 8/3*e^5 - 2/3*e^4 - 88/3*e^3 + 13/3*e^2 + 203/3*e + 25/3, 4/3*e^5 + 2/3*e^4 - 38/3*e^3 - 7/3*e^2 + 73/3*e - 22/3, 1/3*e^5 + 5/3*e^4 - 14/3*e^3 - 49/3*e^2 + 52/3*e + 71/3, -4/3*e^5 + 1/3*e^4 + 50/3*e^3 - 2/3*e^2 - 139/3*e - 17/3, e^4 + 3*e^3 - 7*e^2 - 16*e + 9, -5/3*e^5 - 7/3*e^4 + 58/3*e^3 + 47/3*e^2 - 149/3*e - 16/3, 8/3*e^5 + 1/3*e^4 - 88/3*e^3 - 14/3*e^2 + 209/3*e + 31/3, 2/3*e^5 - 2/3*e^4 - 22/3*e^3 + 19/3*e^2 + 50/3*e + 19/3, -e^5 + e^4 + 12*e^3 - 10*e^2 - 35*e + 15, 14/3*e^5 + 1/3*e^4 - 160/3*e^3 + 7/3*e^2 + 401/3*e + 1/3, -7/3*e^5 + 1/3*e^4 + 86/3*e^3 - 5/3*e^2 - 262/3*e - 17/3, -5/3*e^5 + 2/3*e^4 + 64/3*e^3 - 16/3*e^2 - 167/3*e - 22/3, -e^3 + e^2 + 13*e + 7, e^5 + e^4 - 14*e^3 - 6*e^2 + 48*e + 14, 4*e^5 - 42*e^3 + 2*e^2 + 95*e + 7, 1/3*e^5 + 2/3*e^4 - 2/3*e^3 - 31/3*e^2 - 41/3*e + 92/3, 14/3*e^5 + 4/3*e^4 - 163/3*e^3 - 26/3*e^2 + 422/3*e + 31/3, -2/3*e^5 - 4/3*e^4 + 25/3*e^3 + 35/3*e^2 - 95/3*e - 49/3, 2*e^3 - 15*e - 12, -5/3*e^5 - 1/3*e^4 + 67/3*e^3 + 20/3*e^2 - 203/3*e - 46/3, -10/3*e^5 - 8/3*e^4 + 110/3*e^3 + 58/3*e^2 - 265/3*e - 59/3, -2/3*e^5 + 2/3*e^4 + 28/3*e^3 - 28/3*e^2 - 95/3*e + 41/3, -1/3*e^5 - 11/3*e^4 + 5/3*e^3 + 85/3*e^2 + 8/3*e - 74/3, -2*e^5 + 2*e^4 + 28*e^3 - 18*e^2 - 83*e + 13, -e^5 + e^4 + 13*e^3 - 8*e^2 - 45*e - 8, 10/3*e^5 + 2/3*e^4 - 122/3*e^3 - 7/3*e^2 + 340/3*e - 4/3, 13/3*e^5 - 1/3*e^4 - 155/3*e^3 + 2/3*e^2 + 424/3*e + 53/3, 2*e^5 - 2*e^4 - 23*e^3 + 16*e^2 + 60*e - 11, -11/3*e^5 - 1/3*e^4 + 118/3*e^3 + 14/3*e^2 - 260/3*e - 49/3, -4/3*e^5 + 1/3*e^4 + 41/3*e^3 - 17/3*e^2 - 76/3*e + 34/3, 4*e^5 - e^4 - 43*e^3 + 12*e^2 + 99*e - 11, -17/3*e^5 + 2/3*e^4 + 199/3*e^3 - 16/3*e^2 - 503/3*e - 40/3, 2*e^5 - 22*e^3 + 7*e^2 + 50*e - 19, 5/3*e^5 - 5/3*e^4 - 67/3*e^3 + 34/3*e^2 + 221/3*e - 2/3, -4/3*e^5 + 10/3*e^4 + 56/3*e^3 - 68/3*e^2 - 181/3*e + 1/3, -2/3*e^5 - 7/3*e^4 + 19/3*e^3 + 44/3*e^2 - 23/3*e + 14/3, -2*e^5 + e^4 + 21*e^3 - 11*e^2 - 43*e + 22, 1/3*e^5 + 2/3*e^4 - 17/3*e^3 - 13/3*e^2 + 55/3*e + 20/3, 2/3*e^5 + 1/3*e^4 - 1/3*e^3 + 4/3*e^2 - 85/3*e - 44/3, 4/3*e^5 + 5/3*e^4 - 44/3*e^3 - 28/3*e^2 + 85/3*e - 22/3, 5/3*e^5 + 1/3*e^4 - 49/3*e^3 - 23/3*e^2 + 68/3*e + 94/3, -6*e^5 - e^4 + 72*e^3 + 11*e^2 - 195*e - 14, -e^5 + 2*e^4 + 9*e^3 - 23*e^2 - 11*e + 36, -6*e^5 - e^4 + 72*e^3 + 9*e^2 - 193*e - 11, -8/3*e^5 - 1/3*e^4 + 82/3*e^3 + 8/3*e^2 - 170/3*e - 1/3, 8/3*e^5 - 2/3*e^4 - 88/3*e^3 + 37/3*e^2 + 212/3*e - 53/3, e^4 + 6*e^3 - e^2 - 45*e - 21, -5/3*e^5 + 2/3*e^4 + 70/3*e^3 - 1/3*e^2 - 233/3*e - 91/3, -2*e^5 + 2*e^4 + 27*e^3 - 15*e^2 - 79*e - 5, -13/3*e^5 - 5/3*e^4 + 161/3*e^3 + 43/3*e^2 - 457/3*e - 59/3, e^5 - 2*e^4 - 10*e^3 + 20*e^2 + 20*e - 36, -8/3*e^5 + 5/3*e^4 + 109/3*e^3 - 28/3*e^2 - 332/3*e - 52/3, -4/3*e^5 - 2/3*e^4 + 29/3*e^3 + 13/3*e^2 - 1/3*e + 7/3, 5*e^5 - 59*e^3 - e^2 + 150*e + 19, -4*e^5 + e^4 + 44*e^3 - 9*e^2 - 109*e + 5, 10/3*e^5 + 2/3*e^4 - 116/3*e^3 - 19/3*e^2 + 298/3*e + 71/3, e^3 - 7*e, 17/3*e^5 + 1/3*e^4 - 193/3*e^3 + 4/3*e^2 + 506/3*e - 5/3, -4/3*e^5 + 1/3*e^4 + 35/3*e^3 - 20/3*e^2 - 43/3*e + 91/3, -3*e^5 - 2*e^4 + 30*e^3 + 13*e^2 - 57*e - 15, -2*e^5 + e^4 + 22*e^3 - 11*e^2 - 41*e + 9, 4/3*e^5 + 5/3*e^4 - 29/3*e^3 - 49/3*e^2 + 25/3*e + 80/3, 10/3*e^5 - 4/3*e^4 - 116/3*e^3 + 35/3*e^2 + 286/3*e - 22/3, 4/3*e^5 - 4/3*e^4 - 68/3*e^3 + 23/3*e^2 + 256/3*e + 32/3, 2/3*e^5 + 10/3*e^4 - 31/3*e^3 - 101/3*e^2 + 98/3*e + 136/3, 1/3*e^5 + 8/3*e^4 - 8/3*e^3 - 64/3*e^2 + 19/3*e + 74/3, -8/3*e^5 + 5/3*e^4 + 91/3*e^3 - 37/3*e^2 - 218/3*e - 4/3, -1/3*e^5 + 1/3*e^4 + 23/3*e^3 - 20/3*e^2 - 79/3*e + 79/3, -e^5 - 2*e^4 + 10*e^3 + 12*e^2 - 27*e + 4, 4/3*e^5 + 2/3*e^4 - 53/3*e^3 - 19/3*e^2 + 160/3*e - 22/3, 14/3*e^5 + 4/3*e^4 - 154/3*e^3 - 26/3*e^2 + 371/3*e + 4/3, -5/3*e^5 - 4/3*e^4 + 64/3*e^3 + 44/3*e^2 - 191/3*e - 76/3, -e^5 - 2*e^4 + 14*e^3 + 10*e^2 - 43*e + 12, -8/3*e^5 + 2/3*e^4 + 106/3*e^3 - 16/3*e^2 - 311/3*e + 2/3, -6*e^5 - e^4 + 72*e^3 + 9*e^2 - 204*e - 20, -13/3*e^5 + 1/3*e^4 + 170/3*e^3 - 5/3*e^2 - 529/3*e - 50/3, -1/3*e^5 + 10/3*e^4 + 8/3*e^3 - 83/3*e^2 + 8/3*e + 82/3, -3*e^5 - e^4 + 38*e^3 + 7*e^2 - 113*e - 12, 5*e^5 + e^4 - 62*e^3 - 5*e^2 + 179*e + 13, -e^4 - 9*e^3 + 3*e^2 + 57*e + 13, 1/3*e^5 - 4/3*e^4 - 11/3*e^3 + 17/3*e^2 + 40/3*e + 104/3, -4*e^5 + e^4 + 49*e^3 - 9*e^2 - 135*e - 19, 2/3*e^5 + 1/3*e^4 - 13/3*e^3 + 1/3*e^2 + 26/3*e - 56/3, 1/3*e^5 - 1/3*e^4 + 10/3*e^3 - 10/3*e^2 - 125/3*e + 44/3, -16/3*e^5 - 5/3*e^4 + 182/3*e^3 + 40/3*e^2 - 424/3*e - 89/3, -2/3*e^5 + 8/3*e^4 + 43/3*e^3 - 58/3*e^2 - 185/3*e + 5/3, -3*e^5 - 3*e^4 + 34*e^3 + 25*e^2 - 94*e - 20, -10/3*e^5 - 5/3*e^4 + 107/3*e^3 + 13/3*e^2 - 262/3*e + 100/3, 14/3*e^5 - 5/3*e^4 - 178/3*e^3 + 40/3*e^2 + 509/3*e + 28/3, 8/3*e^5 + 4/3*e^4 - 100/3*e^3 - 35/3*e^2 + 269/3*e - 26/3, 2*e^5 + 4*e^4 - 19*e^3 - 30*e^2 + 40*e + 14, -2*e^5 - 4*e^4 + 21*e^3 + 23*e^2 - 49*e + 17, -e^5 + 12*e^3 - 2*e^2 - 50*e + 7, -4/3*e^5 - 2/3*e^4 + 44/3*e^3 + 1/3*e^2 - 106/3*e + 97/3, -13/3*e^5 + 1/3*e^4 + 155/3*e^3 - 23/3*e^2 - 397/3*e + 31/3, -8/3*e^5 - 1/3*e^4 + 106/3*e^3 - 7/3*e^2 - 338/3*e + 53/3, -19/3*e^5 + 4/3*e^4 + 230/3*e^3 - 17/3*e^2 - 637/3*e - 53/3, 8/3*e^5 + 4/3*e^4 - 103/3*e^3 - 32/3*e^2 + 326/3*e + 37/3, 10/3*e^5 + 5/3*e^4 - 122/3*e^3 - 34/3*e^2 + 322/3*e + 29/3, 10/3*e^5 + 2/3*e^4 - 104/3*e^3 - 19/3*e^2 + 223/3*e + 50/3, 4*e^5 + 2*e^4 - 44*e^3 - 15*e^2 + 91*e + 34, -8/3*e^5 - 1/3*e^4 + 85/3*e^3 - 4/3*e^2 - 176/3*e + 41/3, 5/3*e^5 + 1/3*e^4 - 73/3*e^3 - 23/3*e^2 + 269/3*e + 112/3, -17/3*e^5 - 1/3*e^4 + 211/3*e^3 + 5/3*e^2 - 605/3*e - 61/3, 14/3*e^5 + 1/3*e^4 - 166/3*e^3 + 1/3*e^2 + 482/3*e + 40/3, -e^5 + 2*e^4 + 17*e^3 - 16*e^2 - 73*e + 23, 2*e^5 - 28*e^3 + 4*e^2 + 98*e - 31, -3*e^5 - e^4 + 38*e^3 + 10*e^2 - 116*e - 7, 4*e^5 - 48*e^3 + 136*e - 9, e^5 - 2*e^4 - 8*e^3 + 18*e^2 + 7*e - 22, -e^5 + 6*e^3 - 2*e^2 + e - 1, -5*e^5 - 2*e^4 + 57*e^3 + 20*e^2 - 146*e - 27, 19/3*e^5 - 7/3*e^4 - 227/3*e^3 + 62/3*e^2 + 580/3*e + 23/3, 6*e^5 - 3*e^4 - 74*e^3 + 27*e^2 + 206*e - 4, -1/3*e^5 - 2/3*e^4 - 7/3*e^3 + 25/3*e^2 + 122/3*e - 77/3, -e^5 - 2*e^4 + 9*e^3 + 15*e^2 - 9*e - 7, -2*e^5 + e^4 + 29*e^3 - 8*e^2 - 105*e - 8, -1/3*e^5 - 5/3*e^4 + 26/3*e^3 + 22/3*e^2 - 145/3*e + 19/3, 4/3*e^5 - 10/3*e^4 - 44/3*e^3 + 68/3*e^2 + 88/3*e + 11/3, 5*e^5 + e^4 - 56*e^3 - 3*e^2 + 141*e + 10, -8/3*e^5 - 1/3*e^4 + 88/3*e^3 - 7/3*e^2 - 224/3*e + 26/3, -13/3*e^5 + 1/3*e^4 + 140/3*e^3 - 26/3*e^2 - 313/3*e + 70/3, 2*e^2 + 9*e - 20, 7*e^5 - e^4 - 82*e^3 + 12*e^2 + 222*e + 7, 9*e^5 - e^4 - 105*e^3 + 6*e^2 + 276*e + 7, 4/3*e^5 + 5/3*e^4 - 62/3*e^3 - 52/3*e^2 + 196/3*e + 65/3, -2*e^5 - e^4 + 24*e^3 + 7*e^2 - 76*e - 9, 10/3*e^5 + 2/3*e^4 - 110/3*e^3 + 11/3*e^2 + 256/3*e - 115/3, 4*e^5 - e^4 - 41*e^3 + 7*e^2 + 85*e + 6, -10/3*e^5 + 1/3*e^4 + 140/3*e^3 + 10/3*e^2 - 457/3*e - 65/3, 5/3*e^5 - 8/3*e^4 - 79/3*e^3 + 49/3*e^2 + 314/3*e + 1/3, 16/3*e^5 - 10/3*e^4 - 185/3*e^3 + 86/3*e^2 + 505/3*e - 43/3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, -w^3 + 5*w + 2}>] := 1;
ALEigenvalues[ideal<ZF | {5, 5, 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;