/*
  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, 9, 2, -6, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17, 17, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 3*w - 3}>;

primesArray := [
[5, 5, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w],
[11, 11, w^5 - 2*w^4 - 3*w^3 + 4*w^2 + w + 1],
[11, 11, w^2 - w - 2],
[17, 17, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 3*w - 3],
[27, 3, w^5 - w^4 - 5*w^3 + w^2 + 6*w + 1],
[27, 3, w^4 - 2*w^3 - 3*w^2 + 5*w],
[37, 37, -w^2 + 2*w + 2],
[47, 47, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 2*w - 1],
[53, 53, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 + 3],
[59, 59, w^4 - 2*w^3 - 2*w^2 + 3*w - 2],
[64, 2, -2],
[67, 67, w^5 - 2*w^4 - 3*w^3 + 4*w^2 + w - 2],
[67, 67, -w^5 + 3*w^4 + w^3 - 7*w^2 + 3*w + 2],
[71, 71, w^4 - w^3 - 4*w^2 + 2*w + 1],
[71, 71, w^4 - w^3 - 5*w^2 + 2*w + 4],
[73, 73, 2*w^5 - 4*w^4 - 7*w^3 + 10*w^2 + 5*w - 3],
[83, 83, w^4 - 2*w^3 - 4*w^2 + 5*w + 2],
[83, 83, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - 3],
[89, 89, -w^5 + w^4 + 6*w^3 - 2*w^2 - 8*w + 1],
[97, 97, 2*w^5 - 4*w^4 - 7*w^3 + 8*w^2 + 7*w],
[103, 103, -w^5 + w^4 + 6*w^3 - 2*w^2 - 9*w + 1],
[107, 107, w^5 - 2*w^4 - 3*w^3 + 3*w^2 + 2*w + 2],
[113, 113, w^5 - w^4 - 5*w^3 + w^2 + 4*w + 2],
[127, 127, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 12*w],
[127, 127, -2*w^5 + 3*w^4 + 9*w^3 - 6*w^2 - 9*w - 1],
[131, 131, w^4 - 3*w^3 - 2*w^2 + 7*w + 1],
[137, 137, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - 2*w - 2],
[137, 137, w^2 - 2*w - 3],
[149, 149, -w^4 + 3*w^3 + w^2 - 7*w + 1],
[151, 151, -w^5 + w^4 + 6*w^3 - 3*w^2 - 8*w + 4],
[163, 163, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w + 5],
[173, 173, -2*w^5 + 3*w^4 + 9*w^3 - 7*w^2 - 10*w + 1],
[179, 179, w^5 - 2*w^4 - 3*w^3 + 5*w^2 + w - 3],
[191, 191, -w^5 + w^4 + 6*w^3 - 2*w^2 - 7*w - 1],
[193, 193, 2*w^5 - 4*w^4 - 8*w^3 + 10*w^2 + 9*w],
[193, 193, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 7*w],
[211, 211, -w^5 + w^4 + 6*w^3 - 2*w^2 - 8*w - 4],
[211, 211, -3*w^5 + 5*w^4 + 13*w^3 - 13*w^2 - 12*w + 3],
[223, 223, 3*w^5 - 4*w^4 - 15*w^3 + 9*w^2 + 17*w - 1],
[227, 227, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 5*w - 3],
[229, 229, 2*w^5 - 4*w^4 - 9*w^3 + 12*w^2 + 11*w - 5],
[229, 229, -w^5 + w^4 + 6*w^3 - 2*w^2 - 10*w],
[233, 233, -w^5 + w^4 + 7*w^3 - 3*w^2 - 12*w + 1],
[239, 239, 2*w^5 - 4*w^4 - 8*w^3 + 11*w^2 + 9*w - 2],
[239, 239, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 2*w - 6],
[241, 241, -2*w^5 + 3*w^4 + 10*w^3 - 6*w^2 - 14*w - 3],
[251, 251, -2*w^5 + 3*w^4 + 11*w^3 - 8*w^2 - 17*w],
[251, 251, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 - 2*w],
[251, 251, -3*w^5 + 6*w^4 + 11*w^3 - 15*w^2 - 10*w + 2],
[257, 257, 2*w^5 - 3*w^4 - 9*w^3 + 8*w^2 + 9*w - 1],
[257, 257, -w^5 + 9*w^3 - 16*w - 3],
[263, 263, -2*w^5 + 3*w^4 + 9*w^3 - 8*w^2 - 11*w + 2],
[269, 269, 2*w^5 - 4*w^4 - 8*w^3 + 10*w^2 + 9*w - 1],
[271, 271, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 11*w - 2],
[271, 271, 3*w^5 - 6*w^4 - 11*w^3 + 14*w^2 + 11*w],
[277, 277, -w^5 + 3*w^4 + w^3 - 7*w^2 + 4*w + 3],
[293, 293, w^5 - w^4 - 7*w^3 + 3*w^2 + 12*w - 2],
[293, 293, -w^5 + w^4 + 6*w^3 - 2*w^2 - 10*w - 1],
[307, 307, w^5 - 7*w^3 - 2*w^2 + 9*w + 5],
[313, 313, -3*w^5 + 4*w^4 + 15*w^3 - 9*w^2 - 17*w - 1],
[313, 313, -w^3 + 6*w + 1],
[317, 317, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 8*w - 1],
[317, 317, -3*w^5 + 4*w^4 + 15*w^3 - 9*w^2 - 19*w],
[317, 317, 2*w^4 - 3*w^3 - 8*w^2 + 5*w + 5],
[337, 337, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 5*w - 3],
[337, 337, 3*w^5 - 5*w^4 - 14*w^3 + 14*w^2 + 17*w - 4],
[347, 347, 2*w^5 - 5*w^4 - 5*w^3 + 13*w^2 + 2*w - 3],
[347, 347, w^5 - 7*w^3 - w^2 + 9*w],
[353, 353, 3*w^5 - 5*w^4 - 13*w^3 + 12*w^2 + 14*w - 2],
[361, 19, -w^5 + w^4 + 7*w^3 - 3*w^2 - 10*w - 1],
[367, 367, 2*w^5 - 4*w^4 - 7*w^3 + 11*w^2 + 3*w - 3],
[389, 389, -2*w^5 + 3*w^4 + 9*w^3 - 5*w^2 - 10*w - 2],
[389, 389, 2*w^5 - 4*w^4 - 9*w^3 + 13*w^2 + 10*w - 6],
[397, 397, 2*w^5 - 4*w^4 - 8*w^3 + 9*w^2 + 9*w + 1],
[401, 401, -3*w^5 + 4*w^4 + 15*w^3 - 9*w^2 - 18*w],
[401, 401, -2*w^4 + 4*w^3 + 6*w^2 - 8*w - 1],
[401, 401, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 6*w - 3],
[401, 401, -2*w^3 + 3*w^2 + 6*w - 4],
[419, 419, w^5 - 2*w^4 - 3*w^3 + 6*w^2 - 3],
[433, 433, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + w + 1],
[443, 443, 2*w^5 - 3*w^4 - 11*w^3 + 8*w^2 + 16*w + 1],
[443, 443, w^3 - w^2 - 4*w + 3],
[461, 461, -w^5 + 3*w^4 - 6*w^2 + 6*w + 2],
[479, 479, 3*w^5 - 5*w^4 - 13*w^3 + 12*w^2 + 12*w],
[479, 479, 2*w^5 - 4*w^4 - 9*w^3 + 11*w^2 + 12*w],
[487, 487, 2*w^5 - 2*w^4 - 12*w^3 + 6*w^2 + 15*w - 2],
[491, 491, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 4],
[491, 491, -2*w^5 + 4*w^4 + 8*w^3 - 10*w^2 - 10*w - 1],
[499, 499, w^4 - w^3 - 4*w^2 + 2*w - 1],
[499, 499, w^5 - 8*w^3 + 10*w],
[499, 499, -2*w^5 + 3*w^4 + 8*w^3 - 4*w^2 - 8*w - 4],
[503, 503, -2*w^4 + 4*w^3 + 5*w^2 - 7*w - 2],
[509, 509, 2*w^5 - 4*w^4 - 9*w^3 + 11*w^2 + 13*w - 2],
[509, 509, w^5 - 9*w^3 + 15*w + 2],
[509, 509, -w^5 + 9*w^3 - 16*w - 1],
[521, 521, w^5 - w^4 - 5*w^3 + 5*w + 6],
[523, 523, -2*w^4 + 5*w^3 + 4*w^2 - 10*w + 1],
[523, 523, -w^5 + 7*w^3 + 2*w^2 - 11*w - 3],
[563, 563, -w^5 + w^4 + 5*w^3 - w^2 - 7*w],
[571, 571, -3*w^5 + 4*w^4 + 15*w^3 - 8*w^2 - 18*w - 2],
[577, 577, w^5 - 3*w^4 - w^3 + 7*w^2 - 4*w - 2],
[587, 587, w^5 - w^4 - 5*w^3 - w^2 + 7*w + 5],
[593, 593, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - w - 2],
[593, 593, -w^5 + 9*w^3 - w^2 - 16*w + 1],
[607, 607, w^5 - 2*w^4 - 3*w^3 + 5*w^2 + 2*w - 4],
[607, 607, -2*w^5 + 3*w^4 + 9*w^3 - 5*w^2 - 11*w - 1],
[613, 613, w^5 - 2*w^4 - 6*w^3 + 8*w^2 + 9*w - 4],
[619, 619, -w^5 + 4*w^4 - 11*w^2 + 4*w + 3],
[619, 619, w^2 - 5],
[631, 631, -w^3 + 2*w^2 + w - 5],
[631, 631, -2*w^3 + 3*w^2 + 6*w - 5],
[641, 641, w^5 - 8*w^3 + 13*w + 2],
[643, 643, -w^5 + w^4 + 4*w^3 - 2*w - 4],
[647, 647, -w^5 + w^4 + 6*w^3 - 3*w^2 - 10*w + 1],
[647, 647, -w^4 + 6*w^2 + w - 5],
[653, 653, 2*w^4 - 3*w^3 - 8*w^2 + 6*w + 7],
[659, 659, w^5 - 7*w^3 + 7*w - 4],
[659, 659, -2*w^5 + 4*w^4 + 8*w^3 - 11*w^2 - 8*w + 6],
[673, 673, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - 5],
[673, 673, w^5 - 2*w^4 - 2*w^3 + 3*w^2 - 2*w + 3],
[677, 677, -3*w^5 + 5*w^4 + 13*w^3 - 12*w^2 - 13*w + 3],
[691, 691, -2*w^5 + 3*w^4 + 10*w^3 - 9*w^2 - 12*w + 2],
[691, 691, -w^5 + 2*w^4 + 5*w^3 - 5*w^2 - 11*w - 3],
[709, 709, w^4 - 3*w^3 - 2*w^2 + 9*w + 2],
[719, 719, -2*w^5 + 3*w^4 + 9*w^3 - 7*w^2 - 7*w + 1],
[719, 719, -2*w^5 + 3*w^4 + 11*w^3 - 9*w^2 - 14*w + 2],
[719, 719, 2*w^5 - 2*w^4 - 12*w^3 + 5*w^2 + 15*w - 1],
[727, 727, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 2*w - 4],
[727, 727, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + 5*w - 4],
[733, 733, w^5 - w^4 - 5*w^3 + 2*w^2 + 5*w - 4],
[733, 733, 2*w^5 - 4*w^4 - 6*w^3 + 8*w^2 + 3*w + 1],
[739, 739, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + w - 5],
[739, 739, -2*w^5 + 4*w^4 + 6*w^3 - 7*w^2 - 4*w - 3],
[751, 751, -2*w^5 + 5*w^4 + 5*w^3 - 11*w^2 - 3*w - 1],
[751, 751, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 3*w - 7],
[757, 757, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 7*w],
[761, 761, 3*w^5 - 4*w^4 - 15*w^3 + 10*w^2 + 17*w - 2],
[769, 769, -w^5 + 2*w^4 + 4*w^3 - 3*w^2 - 6*w - 4],
[773, 773, 2*w^4 - 4*w^3 - 5*w^2 + 9*w],
[787, 787, -2*w^5 + 2*w^4 + 11*w^3 - 5*w^2 - 12*w + 2],
[787, 787, 2*w^5 - 4*w^4 - 7*w^3 + 10*w^2 + 7*w - 1],
[787, 787, 3*w^5 - 5*w^4 - 12*w^3 + 11*w^2 + 10*w - 1],
[797, 797, -w^4 + 2*w^3 + 4*w^2 - 6*w],
[809, 809, -3*w^5 + 5*w^4 + 13*w^3 - 13*w^2 - 12*w + 2],
[809, 809, 3*w^5 - 4*w^4 - 14*w^3 + 8*w^2 + 14*w],
[809, 809, -3*w^5 + 6*w^4 + 11*w^3 - 14*w^2 - 10*w + 2],
[821, 821, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + w + 4],
[823, 823, -w^4 + 2*w^3 + 4*w^2 - 3*w - 5],
[823, 823, -2*w^5 + 4*w^4 + 7*w^3 - 9*w^2 - 6*w - 3],
[827, 827, -w^3 + 3*w^2 + 3*w - 4],
[839, 839, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 7*w],
[839, 839, -2*w^5 + 4*w^4 + 9*w^3 - 11*w^2 - 13*w + 1],
[839, 839, -2*w^5 + 3*w^4 + 10*w^3 - 7*w^2 - 11*w - 1],
[841, 29, -w^3 + 3*w^2 + w - 2],
[857, 857, -2*w^5 + 4*w^4 + 8*w^3 - 9*w^2 - 10*w - 3],
[859, 859, 3*w^5 - 4*w^4 - 15*w^3 + 9*w^2 + 17*w + 2],
[859, 859, -2*w^5 + 2*w^4 + 11*w^3 - 3*w^2 - 15*w],
[863, 863, 2*w^5 - 4*w^4 - 7*w^3 + 8*w^2 + 6*w + 4],
[863, 863, 3*w^5 - 5*w^4 - 14*w^3 + 12*w^2 + 18*w - 1],
[881, 881, w^5 - 3*w^4 - w^3 + 6*w^2 - 3*w - 1],
[887, 887, 2*w^5 - 5*w^4 - 5*w^3 + 11*w^2 + 2*w + 1],
[907, 907, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 8*w - 2],
[919, 919, w^5 - w^4 - 7*w^3 + 3*w^2 + 10*w + 2],
[929, 929, 2*w^5 - 4*w^4 - 7*w^3 + 10*w^2 + 4*w - 4],
[937, 937, 3*w^5 - 5*w^4 - 12*w^3 + 10*w^2 + 11*w],
[953, 953, -3*w^5 + 5*w^4 + 14*w^3 - 12*w^2 - 17*w - 1],
[953, 953, -2*w^5 + 2*w^4 + 11*w^3 - 4*w^2 - 13*w - 2],
[967, 967, -w^5 + 3*w^4 - 6*w^2 + 6*w],
[967, 967, 3*w^5 - 6*w^4 - 11*w^3 + 14*w^2 + 9*w - 1],
[977, 977, -w^5 + 3*w^4 - 6*w^2 + 5*w + 2],
[991, 991, -w^3 + 3*w^2 + 2*w - 3],
[997, 997, w^5 - 2*w^4 - 6*w^3 + 7*w^2 + 10*w - 1],
[1009, 1009, w^5 - 2*w^4 - 2*w^3 + 3*w^2 - w + 2],
[1013, 1013, 3*w^5 - 5*w^4 - 14*w^3 + 14*w^2 + 14*w - 4],
[1019, 1019, -3*w^5 + 5*w^4 + 14*w^3 - 13*w^2 - 18*w + 2],
[1031, 1031, -2*w^5 + 2*w^4 + 11*w^3 - 3*w^2 - 15*w - 1],
[1039, 1039, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 3],
[1049, 1049, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 4],
[1063, 1063, -w^4 + 2*w^3 + 3*w^2 - 2*w - 4],
[1069, 1069, 2*w^5 - 3*w^4 - 8*w^3 + 4*w^2 + 7*w + 4],
[1069, 1069, w^5 - w^4 - 5*w^3 + 2*w^2 + 3*w + 2],
[1091, 1091, -2*w^5 + 2*w^4 + 11*w^3 - 4*w^2 - 15*w],
[1117, 1117, -w^4 + w^3 + 6*w^2 - 3*w - 5],
[1117, 1117, -3*w^5 + 4*w^4 + 16*w^3 - 11*w^2 - 20*w + 1],
[1123, 1123, 2*w^5 - 2*w^4 - 14*w^3 + 7*w^2 + 21*w - 3],
[1129, 1129, w^5 - w^4 - 5*w^3 + 3*w^2 + 3*w - 3],
[1129, 1129, w^4 - 3*w^3 - w^2 + 6*w - 4],
[1151, 1151, 2*w^5 - 3*w^4 - 8*w^3 + 5*w^2 + 7*w + 4],
[1151, 1151, -2*w^5 + 3*w^4 + 10*w^3 - 9*w^2 - 10*w + 5],
[1151, 1151, -2*w^5 + 4*w^4 + 8*w^3 - 11*w^2 - 10*w + 2],
[1153, 1153, -w^3 + 2*w^2 - 3],
[1171, 1171, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 13*w + 1],
[1187, 1187, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 4*w + 5],
[1193, 1193, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 - 3*w],
[1193, 1193, -w^3 + 3*w^2 - 6],
[1201, 1201, -w^4 + 3*w^3 + w^2 - 5*w + 3],
[1201, 1201, 2*w^4 - 5*w^3 - 4*w^2 + 9*w],
[1213, 1213, -2*w^5 + 3*w^4 + 11*w^3 - 8*w^2 - 16*w - 2],
[1213, 1213, 2*w^5 - w^4 - 13*w^3 + w^2 + 17*w - 2],
[1213, 1213, 3*w^5 - 5*w^4 - 12*w^3 + 13*w^2 + 10*w - 2],
[1213, 1213, -2*w^5 + 4*w^4 + 7*w^3 - 11*w^2 - 5*w + 5],
[1217, 1217, -w^5 + 4*w^4 + w^3 - 13*w^2 + 3*w + 9],
[1229, 1229, -w^5 + 2*w^4 + 3*w^3 - 6*w^2 + w + 4],
[1229, 1229, -w^5 + w^4 + 4*w^3 + w^2 - 3*w - 6],
[1231, 1231, 2*w^5 - 4*w^4 - 9*w^3 + 13*w^2 + 9*w - 5],
[1277, 1277, -w^5 + 2*w^4 + 5*w^3 - 5*w^2 - 9*w],
[1277, 1277, w^5 - 9*w^3 + 15*w],
[1301, 1301, 2*w^4 - 4*w^3 - 5*w^2 + 8*w + 1],
[1301, 1301, 2*w^5 - 4*w^4 - 8*w^3 + 9*w^2 + 9*w + 3],
[1307, 1307, 2*w^5 - 4*w^4 - 9*w^3 + 13*w^2 + 9*w - 8],
[1307, 1307, 3*w^5 - 6*w^4 - 12*w^3 + 14*w^2 + 13*w - 1],
[1319, 1319, w^3 - 2*w - 3],
[1319, 1319, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 7*w - 8],
[1321, 1321, 3*w^5 - 5*w^4 - 14*w^3 + 12*w^2 + 18*w],
[1327, 1327, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 4*w - 5],
[1327, 1327, -w^5 + 7*w^3 + 3*w^2 - 10*w - 3],
[1327, 1327, -w^4 + 4*w^3 + w^2 - 10*w - 2],
[1361, 1361, -2*w^5 + 3*w^4 + 11*w^3 - 9*w^2 - 16*w],
[1361, 1361, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + 3*w - 6],
[1367, 1367, w^5 - 2*w^4 - 2*w^3 + 2*w^2 + 4],
[1367, 1367, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - w - 3],
[1369, 37, -w^5 + 7*w^3 + w^2 - 10*w + 1],
[1399, 1399, -w^4 + w^3 + 4*w^2 - 4*w - 4],
[1427, 1427, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 + w + 3],
[1429, 1429, -3*w^5 + 6*w^4 + 12*w^3 - 15*w^2 - 14*w],
[1429, 1429, -2*w^5 + 5*w^4 + 7*w^3 - 15*w^2 - 7*w + 5],
[1433, 1433, -w^5 + 4*w^4 - w^3 - 8*w^2 + 4*w - 2],
[1439, 1439, -w^5 + w^4 + 8*w^3 - 6*w^2 - 12*w + 2],
[1439, 1439, -w^4 - w^3 + 5*w^2 + 7*w - 1],
[1447, 1447, 2*w^5 - 2*w^4 - 11*w^3 + 6*w^2 + 11*w - 3],
[1471, 1471, -w^5 + 2*w^4 + 2*w^3 - 4*w^2 + 3],
[1483, 1483, 2*w^5 - 4*w^4 - 9*w^3 + 12*w^2 + 10*w - 2],
[1483, 1483, -w^4 + 2*w^3 + 2*w^2 - w - 4],
[1487, 1487, -3*w^5 + 4*w^4 + 14*w^3 - 7*w^2 - 16*w - 3],
[1489, 1489, -2*w^3 + 3*w^2 + 5*w - 5],
[1493, 1493, -3*w^5 + 3*w^4 + 18*w^3 - 6*w^2 - 25*w - 1],
[1499, 1499, 3*w^5 - 5*w^4 - 16*w^3 + 15*w^2 + 23*w - 3],
[1511, 1511, 2*w^5 - 4*w^4 - 8*w^3 + 10*w^2 + 11*w],
[1511, 1511, -2*w^5 + 2*w^4 + 11*w^3 - 2*w^2 - 15*w - 3],
[1511, 1511, -2*w^5 + 2*w^4 + 11*w^3 - 3*w^2 - 14*w],
[1571, 1571, -2*w^5 + 4*w^4 + 6*w^3 - 9*w^2 - 2*w + 4],
[1571, 1571, 2*w^5 - w^4 - 12*w^3 + w^2 + 14*w - 1],
[1571, 1571, -3*w^5 + 5*w^4 + 13*w^3 - 13*w^2 - 13*w],
[1579, 1579, -2*w^5 + 3*w^4 + 10*w^3 - 9*w^2 - 10*w - 1],
[1583, 1583, -2*w^5 + 4*w^4 + 6*w^3 - 6*w^2 - 6*w - 5],
[1607, 1607, -w^4 + 4*w^3 + w^2 - 11*w],
[1607, 1607, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 10*w],
[1613, 1613, 2*w^5 - 5*w^4 - 5*w^3 + 10*w^2 + 2*w + 2],
[1619, 1619, -w^5 + w^4 + 8*w^3 - 4*w^2 - 17*w],
[1637, 1637, 2*w^4 - 5*w^3 - 5*w^2 + 11*w],
[1663, 1663, w^5 - 4*w^4 + 2*w^3 + 9*w^2 - 8*w - 2],
[1669, 1669, -w^5 + 2*w^4 + 7*w^3 - 10*w^2 - 13*w + 7],
[1697, 1697, -2*w^5 + 5*w^4 + 3*w^3 - 9*w^2 + 4*w - 2],
[1709, 1709, 2*w^4 - 4*w^3 - 4*w^2 + 10*w - 1],
[1733, 1733, 2*w^5 - 2*w^4 - 14*w^3 + 8*w^2 + 23*w - 4],
[1753, 1753, -w^5 + w^4 + 5*w^3 - 4*w - 4],
[1777, 1777, -4*w^5 + 9*w^4 + 12*w^3 - 22*w^2 - 7*w + 6],
[1783, 1783, w^5 - 2*w^4 - 6*w^3 + 6*w^2 + 11*w - 1],
[1801, 1801, -2*w^5 + 4*w^4 + 7*w^3 - 11*w^2 - 5*w + 8],
[1811, 1811, 2*w^5 - 15*w^3 - 3*w^2 + 22*w + 5],
[1811, 1811, w^2 + w - 5],
[1811, 1811, w^2 - 3*w - 4],
[1823, 1823, -4*w^5 + 7*w^4 + 18*w^3 - 20*w^2 - 20*w + 5],
[1831, 1831, 3*w^5 - 5*w^4 - 14*w^3 + 11*w^2 + 19*w + 4],
[1831, 1831, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 + w - 2],
[1847, 1847, w^5 - 2*w^4 - w^3 + 3*w^2 - 7*w - 2],
[1849, 43, w^3 - 3*w - 4],
[1861, 1861, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - 6*w + 10],
[1861, 1861, w^4 - 4*w^3 - 3*w^2 + 10*w + 3],
[1861, 1861, -w^5 + 4*w^4 - 2*w^3 - 10*w^2 + 9*w + 3],
[1861, 1861, w^5 - 2*w^4 - 2*w^3 + 5*w^2 - 4*w - 2],
[1867, 1867, -w^5 + 7*w^3 - w^2 - 7*w + 3],
[1873, 1873, -3*w^5 + 5*w^4 + 12*w^3 - 11*w^2 - 11*w],
[1877, 1877, 4*w^5 - 8*w^4 - 13*w^3 + 17*w^2 + 8*w + 1],
[1889, 1889, -w^4 + 5*w^2 + 3*w - 4],
[1901, 1901, -2*w^5 + 3*w^4 + 10*w^3 - 6*w^2 - 15*w - 1],
[1949, 1949, -w^5 + 2*w^4 + 2*w^3 - 2*w^2 + w - 5],
[1951, 1951, -2*w^5 + 4*w^4 + 6*w^3 - 7*w^2 - 3*w - 2],
[1973, 1973, w^5 - 9*w^3 + w^2 + 16*w],
[1987, 1987, -3*w^5 + 6*w^4 + 12*w^3 - 16*w^2 - 14*w + 3],
[1999, 1999, -2*w^5 + 5*w^4 + 6*w^3 - 15*w^2 - 4*w + 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 4*x^5 - 13*x^4 - 41*x^3 + 58*x^2 + 64*x - 22;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 2/173*e^5 + 29/173*e^4 + 19/173*e^3 - 315/173*e^2 - 164/173*e + 136/173, -18/173*e^5 - 88/173*e^4 + 175/173*e^3 + 759/173*e^2 - 773/173*e - 532/173, 1, 13/173*e^5 + 102/173*e^4 + 37/173*e^3 - 750/173*e^2 - 201/173*e + 192/173, 9/173*e^5 + 44/173*e^4 - 174/173*e^3 - 639/173*e^2 + 992/173*e + 958/173, 25/173*e^5 + 103/173*e^4 - 368/173*e^3 - 1083/173*e^2 + 1583/173*e + 662/173, -4/173*e^5 - 58/173*e^4 - 211/173*e^3 + 111/173*e^2 + 1539/173*e + 766/173, -16/173*e^5 - 59/173*e^4 + 194/173*e^3 + 617/173*e^2 - 245/173*e - 1434/173, e^2 + e - 10, 25/173*e^5 + 103/173*e^4 - 195/173*e^3 - 1083/173*e^2 - 320/173*e + 2219/173, 25/173*e^5 + 103/173*e^4 - 22/173*e^3 - 218/173*e^2 - 839/173*e - 2106/173, -10/173*e^5 + 28/173*e^4 + 251/173*e^3 - 328/173*e^2 - 910/173*e - 680/173, 21/173*e^5 + 45/173*e^4 - 406/173*e^3 - 453/173*e^2 + 2084/173*e + 44/173, -28/173*e^5 - 60/173*e^4 + 426/173*e^3 + 431/173*e^2 - 1856/173*e - 866/173, -33/173*e^5 - 219/173*e^4 + 119/173*e^3 + 1651/173*e^2 - 754/173*e - 514/173, -37/173*e^5 - 277/173*e^4 - 92/173*e^3 + 2281/173*e^2 + 1304/173*e - 2516/173, 27/173*e^5 + 132/173*e^4 - 176/173*e^3 - 879/173*e^2 + 727/173*e - 586/173, 24/173*e^5 + 175/173*e^4 + 228/173*e^3 - 1012/173*e^2 - 2314/173*e + 1286/173, 49/173*e^5 + 105/173*e^4 - 832/173*e^3 - 1057/173*e^2 + 3421/173*e + 1256/173, 30/173*e^5 + 89/173*e^4 - 407/173*e^3 - 746/173*e^2 + 1346/173*e - 728/173, -28/173*e^5 - 60/173*e^4 + 599/173*e^3 + 777/173*e^2 - 3240/173*e - 520/173, -40/173*e^5 - 61/173*e^4 + 1004/173*e^3 + 1110/173*e^2 - 5370/173*e - 2028/173, 51/173*e^5 + 134/173*e^4 - 640/173*e^3 - 853/173*e^2 + 1700/173*e - 1722/173, 40/173*e^5 + 234/173*e^4 - 139/173*e^3 - 1975/173*e^2 - 512/173*e + 2028/173, -20/173*e^5 - 117/173*e^4 + 156/173*e^3 + 901/173*e^2 - 1128/173*e - 668/173, 15/173*e^5 + 131/173*e^4 + 229/173*e^3 - 546/173*e^2 - 1922/173*e - 18/173, -9/173*e^5 - 44/173*e^4 + 174/173*e^3 + 812/173*e^2 - 819/173*e - 1650/173, 21/173*e^5 + 45/173*e^4 - 233/173*e^3 - 453/173*e^2 + 8/173*e + 1082/173, -52/173*e^5 - 235/173*e^4 + 371/173*e^3 + 1616/173*e^2 - 926/173*e - 422/173, -78/173*e^5 - 439/173*e^4 + 297/173*e^3 + 3462/173*e^2 + 168/173*e - 3920/173, 33/173*e^5 + 46/173*e^4 - 465/173*e^3 - 94/173*e^2 + 927/173*e - 870/173, -29/173*e^5 + 12/173*e^4 + 676/173*e^3 + 156/173*e^2 - 2466/173*e - 1280/173, 30/173*e^5 + 89/173*e^4 - 407/173*e^3 - 227/173*e^2 + 2038/173*e - 2458/173, 106/173*e^5 + 499/173*e^4 - 896/173*e^3 - 4066/173*e^2 + 3418/173*e + 3056/173, 19/173*e^5 + 16/173*e^4 - 252/173*e^3 + 208/173*e^2 + 518/173*e - 1130/173, -27/173*e^5 - 132/173*e^4 + 176/173*e^3 + 1052/173*e^2 - 900/173*e + 240/173, -64/173*e^5 - 236/173*e^4 + 430/173*e^3 + 1430/173*e^2 + 58/173*e - 1238/173, -49/173*e^5 - 278/173*e^4 + 140/173*e^3 + 1749/173*e^2 + 385/173*e - 1256/173, 35/173*e^5 + 248/173*e^4 - 100/173*e^3 - 2312/173*e^2 + 71/173*e + 3072/173, -27/173*e^5 - 132/173*e^4 + 176/173*e^3 + 1225/173*e^2 - 381/173*e - 3566/173, -45/173*e^5 - 220/173*e^4 + 178/173*e^3 + 1984/173*e^2 + 922/173*e - 5136/173, 32/173*e^5 + 118/173*e^4 - 561/173*e^3 - 1234/173*e^2 + 3950/173*e + 1484/173, 1/173*e^5 - 72/173*e^4 - 77/173*e^3 + 794/173*e^2 - 947/173*e - 3046/173, 23/173*e^5 + 74/173*e^4 - 560/173*e^3 - 1460/173*e^2 + 2612/173*e + 2602/173, 38/173*e^5 + 32/173*e^4 - 504/173*e^3 + 589/173*e^2 + 1382/173*e - 3990/173, -62/173*e^5 - 380/173*e^4 + 103/173*e^3 + 2845/173*e^2 + 759/173*e - 3178/173, 39/173*e^5 + 133/173*e^4 - 754/173*e^3 - 1558/173*e^2 + 4068/173*e + 1614/173, -68/173*e^5 - 294/173*e^4 + 565/173*e^3 + 2060/173*e^2 - 2382/173*e - 1510/173, 32/173*e^5 + 118/173*e^4 - 215/173*e^3 - 542/173*e^2 + 1182/173*e + 100/173, 72/173*e^5 + 352/173*e^4 - 873/173*e^3 - 3728/173*e^2 + 4130/173*e + 4204/173, 98/173*e^5 + 383/173*e^4 - 799/173*e^3 - 2460/173*e^2 + 1306/173*e - 602/173, -75/173*e^5 - 309/173*e^4 + 585/173*e^3 + 2384/173*e^2 - 1462/173*e - 3370/173, -116/173*e^5 - 471/173*e^4 + 1147/173*e^3 + 3565/173*e^2 - 4847/173*e - 2352/173, -69/173*e^5 - 222/173*e^4 + 988/173*e^3 + 2650/173*e^2 - 2992/173*e - 4692/173, e^5 + 4*e^4 - 8*e^3 - 29*e^2 + 16*e + 24, 9/173*e^5 + 44/173*e^4 - 347/173*e^3 - 812/173*e^2 + 2030/173*e - 772/173, -66/173*e^5 - 265/173*e^4 + 757/173*e^3 + 2610/173*e^2 - 3065/173*e - 3796/173, -15/173*e^5 + 42/173*e^4 + 463/173*e^3 - 319/173*e^2 - 2230/173*e + 18/173, -85/173*e^5 - 281/173*e^4 + 836/173*e^3 + 2056/173*e^2 - 2372/173*e - 3358/173, -17/173*e^5 + 13/173*e^4 + 617/173*e^3 + 861/173*e^2 - 2412/173*e - 4616/173, -4/173*e^5 - 58/173*e^4 + 135/173*e^3 + 284/173*e^2 - 2440/173*e + 2496/173, -42/173*e^5 - 263/173*e^4 - 53/173*e^3 + 2117/173*e^2 + 2752/173*e - 3548/173, -58/173*e^5 - 322/173*e^4 + 314/173*e^3 + 2388/173*e^2 - 780/173*e - 1522/173, -93/173*e^5 - 397/173*e^4 + 1106/173*e^3 + 4008/173*e^2 - 3100/173*e - 4248/173, -79/173*e^5 - 540/173*e^4 + 374/173*e^3 + 4571/173*e^2 - 2172/173*e - 1912/173, -13/173*e^5 - 102/173*e^4 - 37/173*e^3 + 1442/173*e^2 + 1412/173*e - 1922/173, -35/173*e^5 - 75/173*e^4 + 619/173*e^3 + 409/173*e^2 - 4569/173*e + 734/173, -48/173*e^5 - 4/173*e^4 + 1101/173*e^3 + 294/173*e^2 - 4368/173*e - 496/173, 32/173*e^5 - 55/173*e^4 - 734/173*e^3 + 1015/173*e^2 + 3431/173*e - 3360/173, -40/173*e^5 - 234/173*e^4 + 312/173*e^3 + 2667/173*e^2 + 512/173*e - 3758/173, -67/173*e^5 - 366/173*e^4 + 142/173*e^3 + 2854/173*e^2 + 1688/173*e - 2826/173, 165/173*e^5 + 749/173*e^4 - 1287/173*e^3 - 5833/173*e^2 + 3251/173*e + 5684/173, 56/173*e^5 + 466/173*e^4 + 186/173*e^3 - 3976/173*e^2 - 3208/173*e + 4846/173, 21/173*e^5 + 45/173*e^4 - 579/173*e^3 - 972/173*e^2 + 4506/173*e + 2466/173, -128/173*e^5 - 645/173*e^4 + 860/173*e^3 + 4936/173*e^2 - 2652/173*e - 4206/173, 17/173*e^5 - 13/173*e^4 - 617/173*e^3 - 688/173*e^2 + 2758/173*e + 2886/173, -7/173*e^5 - 15/173*e^4 + 20/173*e^3 - 22/173*e^2 + 228/173*e - 130/173, 55/173*e^5 + 192/173*e^4 - 948/173*e^3 - 2002/173*e^2 + 5697/173*e + 1664/173, -47/173*e^5 - 249/173*e^4 + 159/173*e^3 + 1434/173*e^2 + 48/173*e + 3378/173, 43/173*e^5 + 191/173*e^4 - 24/173*e^3 - 804/173*e^2 - 1796/173*e - 1920/173, 69/173*e^5 + 222/173*e^4 - 815/173*e^3 - 1785/173*e^2 + 2473/173*e + 194/173, 164/173*e^5 + 648/173*e^4 - 1902/173*e^3 - 5935/173*e^2 + 6966/173*e + 5270/173, 22/173*e^5 - 27/173*e^4 - 483/173*e^3 - 5/173*e^2 + 1483/173*e + 4956/173, -82/173*e^5 - 497/173*e^4 + 605/173*e^3 + 4438/173*e^2 - 3137/173*e - 2808/173, 97/173*e^5 + 282/173*e^4 - 1414/173*e^3 - 2389/173*e^2 + 6578/173*e + 1060/173, 12/173*e^5 + 1/173*e^4 - 59/173*e^3 + 532/173*e^2 - 1503/173*e - 914/173, -36/173*e^5 - 3/173*e^4 + 523/173*e^3 - 904/173*e^2 - 2584/173*e + 1012/173, 10/173*e^5 - 28/173*e^4 - 1116/173*e^3 - 1229/173*e^2 + 8522/173*e + 2410/173, -53/173*e^5 + 10/173*e^4 + 967/173*e^3 - 908/173*e^2 - 4304/173*e + 548/173, 66/173*e^5 + 92/173*e^4 - 1276/173*e^3 - 880/173*e^2 + 5314/173*e + 2758/173, 17/173*e^5 - 13/173*e^4 - 617/173*e^3 - 515/173*e^2 + 2931/173*e + 1502/173, -10/173*e^5 - 318/173*e^4 - 787/173*e^3 + 2613/173*e^2 + 4626/173*e - 3794/173, -67/173*e^5 - 366/173*e^4 + 488/173*e^3 + 3892/173*e^2 + 650/173*e - 6286/173, -193/173*e^5 - 809/173*e^4 + 1713/173*e^3 + 6437/173*e^2 - 4588/173*e - 5512/173, 138/173*e^5 + 617/173*e^4 - 1111/173*e^3 - 4781/173*e^2 + 3216/173*e + 1772/173, 6/173*e^5 - 86/173*e^4 - 462/173*e^3 + 93/173*e^2 + 1584/173*e + 3868/173, 42/173*e^5 + 263/173*e^4 - 466/173*e^3 - 2982/173*e^2 + 2438/173*e + 3202/173, -49/173*e^5 - 278/173*e^4 - 33/173*e^3 + 1230/173*e^2 + 1423/173*e + 2550/173, 16/173*e^5 + 232/173*e^4 + 671/173*e^3 - 1655/173*e^2 - 4772/173*e + 1088/173, 112/173*e^5 + 413/173*e^4 - 1012/173*e^3 - 3627/173*e^2 + 850/173*e + 6232/173, 32/173*e^5 + 291/173*e^4 - 42/173*e^3 - 3137/173*e^2 + 490/173*e + 4252/173, 147/173*e^5 + 661/173*e^4 - 1285/173*e^3 - 5074/173*e^2 + 3343/173*e + 2730/173, -43/173*e^5 - 191/173*e^4 + 716/173*e^3 + 2015/173*e^2 - 3567/173*e - 156/173, -64/173*e^5 - 63/173*e^4 + 1295/173*e^3 + 1084/173*e^2 - 3748/173*e - 2622/173, 83/173*e^5 + 425/173*e^4 - 682/173*e^3 - 3644/173*e^2 + 3055/173*e + 2530/173, 112/173*e^5 + 413/173*e^4 - 1185/173*e^3 - 3108/173*e^2 + 5002/173*e + 1388/173, 43/173*e^5 + 191/173*e^4 - 370/173*e^3 - 1323/173*e^2 + 2010/173*e - 5034/173, 128/173*e^5 + 472/173*e^4 - 1552/173*e^3 - 4071/173*e^2 + 6977/173*e + 2476/173, -76/173*e^5 - 237/173*e^4 + 835/173*e^3 + 2455/173*e^2 - 688/173*e - 4130/173, -19/173*e^5 - 189/173*e^4 - 613/173*e^3 + 484/173*e^2 + 4326/173*e + 4590/173, -15/173*e^5 + 42/173*e^4 + 117/173*e^3 - 1530/173*e^2 - 846/173*e + 6246/173, -14/173*e^5 - 30/173*e^4 - 133/173*e^3 - 909/173*e^2 + 1494/173*e + 3546/173, -11/173*e^5 - 73/173*e^4 - 18/173*e^3 + 954/173*e^2 + 556/173*e - 6284/173, 148/173*e^5 + 589/173*e^4 - 1189/173*e^3 - 3588/173*e^2 + 3261/173*e - 1354/173, -1/173*e^5 - 101/173*e^4 - 96/173*e^3 + 2320/173*e^2 + 2677/173*e - 8026/173, 61/173*e^5 + 106/173*e^4 - 1237/173*e^3 - 1563/173*e^2 + 6070/173*e + 2418/173, -53/173*e^5 + 10/173*e^4 + 1486/173*e^3 + 130/173*e^2 - 7764/173*e + 548/173, -145/173*e^5 - 632/173*e^4 + 1650/173*e^3 + 5797/173*e^2 - 5929/173*e - 4324/173, -67/173*e^5 - 366/173*e^4 - 204/173*e^3 + 1989/173*e^2 + 4456/173*e - 2134/173, 199/173*e^5 + 896/173*e^4 - 1829/173*e^3 - 7901/173*e^2 + 5826/173*e + 5920/173, -91/173*e^5 - 368/173*e^4 + 1125/173*e^3 + 3520/173*e^2 - 3610/173*e - 6188/173, -60/173*e^5 - 178/173*e^4 + 122/173*e^3 - 584/173*e^2 + 1979/173*e + 5954/173, -63/173*e^5 - 135/173*e^4 + 1391/173*e^3 + 2743/173*e^2 - 5906/173*e - 5668/173, 2/173*e^5 + 202/173*e^4 + 884/173*e^3 - 1353/173*e^2 - 4143/173*e + 2558/173, -1/173*e^5 + 72/173*e^4 + 596/173*e^3 + 244/173*e^2 - 5281/173*e + 1316/173, 100/173*e^5 + 412/173*e^4 - 953/173*e^3 - 3121/173*e^2 + 1315/173*e + 226/173, -80/173*e^5 - 295/173*e^4 + 970/173*e^3 + 2566/173*e^2 - 3128/173*e - 3018/173, 82/173*e^5 + 151/173*e^4 - 1643/173*e^3 - 1843/173*e^2 + 7462/173*e + 386/173, -171/173*e^5 - 663/173*e^4 + 1749/173*e^3 + 5394/173*e^2 - 4662/173*e - 2632/173, -75/173*e^5 - 482/173*e^4 - 107/173*e^3 + 2903/173*e^2 + 1479/173*e + 3204/173, 104/173*e^5 + 470/173*e^4 - 396/173*e^3 - 2540/173*e^2 - 1262/173*e - 1232/173, -20/173*e^5 + 56/173*e^4 + 848/173*e^3 - 310/173*e^2 - 6664/173*e + 2100/173, 64/173*e^5 + 236/173*e^4 + 89/173*e^3 + 473/173*e^2 - 3518/173*e - 6720/173, -160/173*e^5 - 763/173*e^4 + 1248/173*e^3 + 5478/173*e^2 - 4180/173*e + 1230/173, 57/173*e^5 + 221/173*e^4 - 756/173*e^3 - 3009/173*e^2 + 1208/173*e + 8720/173, -70/173*e^5 - 323/173*e^4 + 546/173*e^3 + 1337/173*e^2 - 3256/173*e + 3544/173, 41/173*e^5 + 335/173*e^4 + 130/173*e^3 - 2738/173*e^2 - 1978/173*e + 366/173, 38/173*e^5 + 32/173*e^4 - 504/173*e^3 + 762/173*e^2 + 2247/173*e - 2260/173, -135/173*e^5 - 487/173*e^4 + 1226/173*e^3 + 3703/173*e^2 - 2424/173*e - 3644/173, 48/173*e^5 + 177/173*e^4 - 582/173*e^3 - 2197/173*e^2 + 1254/173*e + 5340/173, -30/173*e^5 - 89/173*e^4 + 753/173*e^3 + 1438/173*e^2 - 3768/173*e + 3150/173, -38/173*e^5 - 32/173*e^4 + 1023/173*e^3 + 795/173*e^2 - 4842/173*e - 3622/173, -14/173*e^5 - 30/173*e^4 + 559/173*e^3 + 994/173*e^2 - 3004/173*e - 260/173, 30/173*e^5 - 84/173*e^4 - 1791/173*e^3 - 746/173*e^2 + 12072/173*e + 2040/173, -83/173*e^5 - 252/173*e^4 + 1547/173*e^3 + 2952/173*e^2 - 9802/173*e - 4952/173, 58/173*e^5 + 149/173*e^4 - 1006/173*e^3 - 1350/173*e^2 + 4586/173*e - 208/173, 41/173*e^5 - 11/173*e^4 - 1427/173*e^3 - 662/173*e^2 + 8402/173*e - 326/173, 2/173*e^5 + 29/173*e^4 - 154/173*e^3 - 1526/173*e^2 + 528/173*e + 8440/173, 13/173*e^5 + 275/173*e^4 + 556/173*e^3 - 2307/173*e^2 - 2969/173*e + 2960/173, -195/173*e^5 - 838/173*e^4 + 1521/173*e^3 + 6233/173*e^2 - 2348/173*e - 2880/173, 17/173*e^5 + 333/173*e^4 + 940/173*e^3 - 2072/173*e^2 - 6238/173*e + 1848/173, -62/173*e^5 - 207/173*e^4 + 968/173*e^3 + 1634/173*e^2 - 6334/173*e + 974/173, -10/173*e^5 + 201/173*e^4 + 1289/173*e^3 - 1366/173*e^2 - 8176/173*e + 704/173, 111/173*e^5 + 485/173*e^4 - 589/173*e^3 - 2864/173*e^2 - 452/173*e + 2358/173, 106/173*e^5 + 672/173*e^4 - 377/173*e^3 - 6142/173*e^2 - 561/173*e + 7208/173, -149/173*e^5 - 690/173*e^4 + 920/173*e^3 + 5562/173*e^2 - 65/173*e - 7018/173, 139/173*e^5 + 718/173*e^4 - 842/173*e^3 - 5890/173*e^2 + 366/173*e + 4262/173, 102/173*e^5 + 441/173*e^4 - 761/173*e^3 - 3955/173*e^2 - 579/173*e + 3822/173, -93/173*e^5 - 224/173*e^4 + 1452/173*e^3 + 1932/173*e^2 - 5176/173*e - 788/173, 26/173*e^5 + 204/173*e^4 - 445/173*e^3 - 2192/173*e^2 + 4788/173*e - 654/173, -52/173*e^5 - 408/173*e^4 + 717/173*e^3 + 5076/173*e^2 - 4732/173*e - 3882/173, -94/173*e^5 - 498/173*e^4 + 1010/173*e^3 + 4425/173*e^2 - 6824/173*e - 3970/173, -120/173*e^5 - 356/173*e^4 + 1109/173*e^3 + 1081/173*e^2 - 3654/173*e + 3258/173, -67/173*e^5 - 539/173*e^4 - 723/173*e^3 + 3719/173*e^2 + 7570/173*e - 5594/173, -36/173*e^5 - 176/173*e^4 + 523/173*e^3 + 2037/173*e^2 - 1892/173*e - 1064/173, 24/173*e^5 - 344/173*e^4 - 2021/173*e^3 + 1929/173*e^2 + 11007/173*e - 1828/173, 141/173*e^5 + 747/173*e^4 - 823/173*e^3 - 5513/173*e^2 + 1932/173*e + 1976/173, -14/173*e^5 - 203/173*e^4 - 652/173*e^3 + 648/173*e^2 + 5300/173*e - 1298/173, 1/173*e^5 + 274/173*e^4 + 1134/173*e^3 - 1282/173*e^2 - 5964/173*e - 1662/173, -43/173*e^5 - 191/173*e^4 + 24/173*e^3 + 804/173*e^2 + 1796/173*e - 156/173, 45/173*e^5 + 47/173*e^4 - 178/173*e^3 + 1303/173*e^2 - 749/173*e - 746/173, 134/173*e^5 + 732/173*e^4 - 284/173*e^3 - 4324/173*e^2 - 1300/173*e - 230/173, -7/173*e^5 + 331/173*e^4 + 1750/173*e^3 - 714/173*e^2 - 10325/173*e - 4974/173, -71/173*e^5 + 95/173*e^4 + 1834/173*e^3 - 668/173*e^2 - 6634/173*e + 4860/173, -191/173*e^5 - 953/173*e^4 + 1040/173*e^3 + 7333/173*e^2 - 1292/173*e - 7452/173, -83/173*e^5 - 79/173*e^4 + 2239/173*e^3 + 1395/173*e^2 - 11705/173*e - 3222/173, 64/173*e^5 + 409/173*e^4 - 84/173*e^3 - 3160/173*e^2 - 2134/173*e + 5736/173, -160/173*e^5 - 590/173*e^4 + 2286/173*e^3 + 5651/173*e^2 - 8851/173*e - 1884/173, -45/173*e^5 - 220/173*e^4 + 697/173*e^3 + 1811/173*e^2 - 5652/173*e - 3406/173, 131/173*e^5 + 429/173*e^4 - 1783/173*e^3 - 4284/173*e^2 + 6212/173*e + 6486/173, 63/173*e^5 - 38/173*e^4 - 872/173*e^3 + 2620/173*e^2 + 3138/173*e - 9556/173, 110/173*e^5 + 557/173*e^4 - 685/173*e^3 - 5388/173*e^2 + 1014/173*e + 10940/173, -91/173*e^5 - 195/173*e^4 + 1644/173*e^3 + 1963/173*e^2 - 7416/173*e - 1690/173, 96/173*e^5 + 527/173*e^4 - 126/173*e^3 - 4221/173*e^2 - 3201/173*e + 9296/173, -79/173*e^5 - 367/173*e^4 + 1066/173*e^3 + 3360/173*e^2 - 8400/173*e - 3988/173, 116/173*e^5 + 471/173*e^4 - 455/173*e^3 - 3219/173*e^2 - 3630/173*e + 6504/173, 18/173*e^5 + 261/173*e^4 + 344/173*e^3 - 2316/173*e^2 - 2168/173*e + 186/173, 103/173*e^5 + 369/173*e^4 - 1530/173*e^3 - 3680/173*e^2 + 6432/173*e + 2160/173, -151/173*e^5 - 546/173*e^4 + 2285/173*e^3 + 5877/173*e^2 - 11146/173*e - 7500/173, -100/173*e^5 - 585/173*e^4 + 1299/173*e^3 + 7100/173*e^2 - 5986/173*e - 8530/173, 27/173*e^5 + 305/173*e^4 + 170/173*e^3 - 2782/173*e^2 - 138/173*e - 2316/173, -e^5 - 5*e^4 + 10*e^3 + 51*e^2 - 30*e - 52, -125/173*e^5 - 688/173*e^4 - 236/173*e^3 + 4550/173*e^2 + 7309/173*e - 7116/173, 50/173*e^5 + 552/173*e^4 + 475/173*e^3 - 4415/173*e^2 - 1332/173*e + 6168/173, 41/173*e^5 + 335/173*e^4 - 43/173*e^3 - 3257/173*e^2 + 98/173*e + 5902/173, 101/173*e^5 + 167/173*e^4 - 2587/173*e^3 - 3192/173*e^2 + 13516/173*e + 4792/173, 52/173*e^5 - 111/173*e^4 - 1236/173*e^3 + 1152/173*e^2 + 5078/173*e + 3536/173, 147/173*e^5 + 315/173*e^4 - 2496/173*e^3 - 2306/173*e^2 + 12858/173*e + 2384/173, -207/173*e^5 - 839/173*e^4 + 1926/173*e^3 + 6220/173*e^2 - 6208/173*e - 4388/173, -56/173*e^5 - 120/173*e^4 + 1025/173*e^3 + 2246/173*e^2 - 2328/173*e - 5538/173, -124/173*e^5 - 414/173*e^4 + 1590/173*e^3 + 3614/173*e^2 - 5575/173*e - 4280/173, -92/173*e^5 - 296/173*e^4 + 1548/173*e^3 + 3245/173*e^2 - 8545/173*e - 5564/173, -88/173*e^5 - 757/173*e^4 - 317/173*e^3 + 6248/173*e^2 + 2718/173*e - 6676/173, 76/173*e^5 + 583/173*e^4 + 549/173*e^3 - 3320/173*e^2 - 3118/173*e - 5212/173, -56/173*e^5 - 120/173*e^4 + 852/173*e^3 + 689/173*e^2 - 5096/173*e - 1040/173, -99/173*e^5 - 138/173*e^4 + 1395/173*e^3 + 974/173*e^2 - 3127/173*e - 6386/173, 140/173*e^5 + 646/173*e^4 - 746/173*e^3 - 4231/173*e^2 + 284/173*e + 3638/173, 113/173*e^5 + 514/173*e^4 - 224/173*e^3 - 2660/173*e^2 - 3384/173*e - 1658/173, -76/173*e^5 - 64/173*e^4 + 1181/173*e^3 - 486/173*e^2 - 4840/173*e + 714/173, 45/173*e^5 + 47/173*e^4 - 2254/173*e^3 - 3368/173*e^2 + 16205/173*e + 5828/173, 134/173*e^5 + 559/173*e^4 - 1495/173*e^3 - 5016/173*e^2 + 8042/173*e + 4614/173, 32/173*e^5 + 118/173*e^4 - 1253/173*e^3 - 2618/173*e^2 + 8102/173*e + 3214/173, -45/173*e^5 + 126/173*e^4 + 1216/173*e^3 - 784/173*e^2 - 3922/173*e - 1676/173, -39/173*e^5 - 133/173*e^4 + 62/173*e^3 + 866/173*e^2 + 2333/173*e + 1154/173, 150/173*e^5 + 445/173*e^4 - 2208/173*e^3 - 4249/173*e^2 + 9671/173*e + 4318/173, 154/173*e^5 + 849/173*e^4 - 786/173*e^3 - 7647/173*e^2 + 1731/173*e + 9780/173, -5/173*e^5 + 360/173*e^4 + 1423/173*e^3 - 2240/173*e^2 - 6856/173*e + 4158/173, 22/173*e^5 + 319/173*e^4 + 382/173*e^3 - 2254/173*e^2 - 420/173*e - 1618/173, 197/173*e^5 + 1213/173*e^4 - 810/173*e^3 - 10008/173*e^2 + 2011/173*e + 8552/173, 40/173*e^5 + 407/173*e^4 - 485/173*e^3 - 5608/173*e^2 + 3813/173*e + 8256/173, 93/173*e^5 + 397/173*e^4 - 414/173*e^3 - 1586/173*e^2 - 533/173*e - 4056/173, 99/173*e^5 + 138/173*e^4 - 1741/173*e^3 - 1320/173*e^2 + 7452/173*e + 5002/173, -84/173*e^5 - 353/173*e^4 + 1278/173*e^3 + 4753/173*e^2 - 6260/173*e - 9864/173, -203/173*e^5 - 781/173*e^4 + 2483/173*e^3 + 7666/173*e^2 - 7920/173*e - 6192/173, 174/173*e^5 + 620/173*e^4 - 1807/173*e^3 - 4050/173*e^2 + 8568/173*e - 1316/173, -108/173*e^5 - 701/173*e^4 - 507/173*e^3 + 4208/173*e^2 + 5742/173*e + 268/173, -160/173*e^5 - 417/173*e^4 + 2632/173*e^3 + 4786/173*e^2 - 10062/173*e - 6036/173, 95/173*e^5 + 426/173*e^4 - 222/173*e^3 - 2766/173*e^2 - 4330/173*e + 4384/173, 106/173*e^5 + 672/173*e^4 + 661/173*e^3 - 4239/173*e^2 - 8000/173*e + 8938/173, -48/173*e^5 - 4/173*e^4 + 1620/173*e^3 + 813/173*e^2 - 9558/173*e + 4002/173, -158/173*e^5 - 734/173*e^4 + 1094/173*e^3 + 5509/173*e^2 - 2787/173*e - 1402/173, -52/173*e^5 - 581/173*e^4 - 667/173*e^3 + 3865/173*e^2 + 3226/173*e + 3384/173, -15/173*e^5 + 215/173*e^4 + 1501/173*e^3 + 27/173*e^2 - 6382/173*e - 4826/173, 185/173*e^5 + 693/173*e^4 - 1443/173*e^3 - 6388/173*e^2 - 2195/173*e + 12580/173, -30/173*e^5 + 84/173*e^4 + 926/173*e^3 - 1330/173*e^2 - 4460/173*e + 1766/173, 1/173*e^5 + 274/173*e^4 + 615/173*e^3 - 1974/173*e^2 - 1812/173*e - 4776/173, -22/173*e^5 + 27/173*e^4 + 1002/173*e^3 + 1735/173*e^2 - 3213/173*e - 7378/173, 3/173*e^5 - 43/173*e^4 - 750/173*e^3 - 905/173*e^2 + 4252/173*e + 6086/173, -179/173*e^5 - 606/173*e^4 + 1327/173*e^3 + 3021/173*e^2 - 546/173*e + 1322/173, -78/173*e^5 - 439/173*e^4 + 989/173*e^3 + 4154/173*e^2 - 7790/173*e - 5650/173, -4/173*e^5 - 58/173*e^4 + 135/173*e^3 + 2187/173*e^2 + 155/173*e - 7192/173, 144/173*e^5 + 531/173*e^4 - 1919/173*e^3 - 5726/173*e^2 + 8087/173*e + 4948/173, 41/173*e^5 + 162/173*e^4 - 216/173*e^3 - 662/173*e^2 + 790/173*e - 2748/173, -34/173*e^5 - 147/173*e^4 - 150/173*e^3 - 181/173*e^2 + 885/173*e + 6684/173, 33/173*e^5 + 46/173*e^4 + 54/173*e^3 + 425/173*e^2 - 5474/173*e + 860/173, -147/173*e^5 - 488/173*e^4 + 1285/173*e^3 + 4728/173*e^2 + 2712/173*e - 8958/173, 34/173*e^5 + 666/173*e^4 + 1707/173*e^3 - 4144/173*e^2 - 9362/173*e - 1494/173, -10/173*e^5 - 145/173*e^4 - 441/173*e^3 - 328/173*e^2 + 1858/173*e + 6240/173, 1/173*e^5 - 245/173*e^4 - 942/173*e^3 + 1313/173*e^2 + 4762/173*e - 6506/173, 85/173*e^5 + 281/173*e^4 - 1182/173*e^3 - 2402/173*e^2 + 5486/173*e - 2178/173, -37/173*e^5 - 450/173*e^4 - 784/173*e^3 + 3665/173*e^2 + 6667/173*e - 3900/173, 33/173*e^5 + 565/173*e^4 + 746/173*e^3 - 5630/173*e^2 - 5820/173*e + 8126/173, -101/173*e^5 - 686/173*e^4 - 8/173*e^3 + 5441/173*e^2 + 3957/173*e - 5484/173, 47/173*e^5 + 249/173*e^4 - 678/173*e^3 - 2818/173*e^2 + 6180/173*e + 10116/173, -82/173*e^5 - 151/173*e^4 + 2335/173*e^3 + 3919/173*e^2 - 13517/173*e - 6960/173, 238/173*e^5 + 856/173*e^4 - 2756/173*e^3 - 6518/173*e^2 + 12835/173*e + 5112/173, -95/173*e^5 - 599/173*e^4 + 395/173*e^3 + 3977/173*e^2 - 3282/173*e - 1270/173, 30/173*e^5 - 84/173*e^4 - 1099/173*e^3 - 227/173*e^2 + 4806/173*e - 3150/173, -115/173*e^5 - 543/173*e^4 + 551/173*e^3 + 3840/173*e^2 + 434/173*e + 138/173, -58/173*e^5 - 668/173*e^4 - 724/173*e^3 + 4637/173*e^2 + 2334/173*e + 4360/173, -40/173*e^5 + 458/173*e^4 + 2734/173*e^3 - 3042/173*e^2 - 14885/173*e + 2816/173, -162/173*e^5 - 1138/173*e^4 + 537/173*e^3 + 10464/173*e^2 + 482/173*e - 11708/173, -30/173*e^5 - 435/173*e^4 + 61/173*e^3 + 5590/173*e^2 - 2038/173*e - 9998/173, -119/173*e^5 - 428/173*e^4 + 2070/173*e^3 + 4643/173*e^2 - 11867/173*e - 8092/173, 223/173*e^5 + 1244/173*e^4 - 1947/173*e^3 - 11854/173*e^2 + 8010/173*e + 11358/173, -104/173*e^5 - 470/173*e^4 + 742/173*e^3 + 3059/173*e^2 + 224/173*e + 6422/173, -61/173*e^5 - 106/173*e^4 + 1237/173*e^3 + 352/173*e^2 - 8146/173*e + 5886/173, 27/173*e^5 - 214/173*e^4 - 1041/173*e^3 + 851/173*e^2 + 1592/173*e + 4604/173, 28/173*e^5 + 233/173*e^4 + 93/173*e^3 - 2507/173*e^2 - 220/173*e + 6056/173, -54/173*e^5 - 610/173*e^4 - 513/173*e^3 + 6083/173*e^2 + 4428/173*e - 7478/173, -243/173*e^5 - 842/173*e^4 + 2276/173*e^3 + 6008/173*e^2 - 5332/173*e - 5798/173, 298/173*e^5 + 1034/173*e^4 - 3570/173*e^3 - 8010/173*e^2 + 13970/173*e + 8154/173, -172/173*e^5 - 937/173*e^4 + 1307/173*e^3 + 8060/173*e^2 - 4234/173*e - 7198/173, -92/173*e^5 - 815/173*e^4 - 182/173*e^3 + 8262/173*e^2 + 3392/173*e - 13176/173, 138/173*e^5 + 617/173*e^4 - 1284/173*e^3 - 4608/173*e^2 + 6676/173*e + 1426/173, 248/173*e^5 + 1347/173*e^4 - 1623/173*e^3 - 10861/173*e^2 + 7344/173*e + 7522/173, 239/173*e^5 + 1303/173*e^4 - 1622/173*e^3 - 12471/173*e^2 + 2546/173*e + 15560/173, -11/173*e^5 + 273/173*e^4 + 1712/173*e^3 - 1814/173*e^2 - 11208/173*e + 4096/173, 153/173*e^5 + 748/173*e^4 - 190/173*e^3 - 4981/173*e^2 - 5972/173*e + 8328/173, 282/173*e^5 + 1148/173*e^4 - 3030/173*e^3 - 9642/173*e^2 + 13898/173*e + 8450/173];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17, 17, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 3*w - 3}>] := -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;