/*
  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 | {5, 5, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w}>;

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;
K := Rationals(); e := 1;

heckeEigenvaluesArray := [1, 0, -3, -3, -10, 2, 4, -9, -9, 9, -11, -2, -5, 0, -6, -2, -3, -9, -9, 4, -14, -3, -6, -10, 8, 15, -9, -6, -3, 16, -5, 24, 15, -24, 14, -14, 8, 5, -1, -21, 10, -4, 18, -12, 12, 10, -9, 21, 12, 21, 18, -9, 0, 20, -29, -16, -24, -27, 7, -20, 5, -18, -27, 18, -8, -20, -3, 12, 0, 26, -13, -18, -36, -29, 15, 12, -12, 24, 9, -2, 36, 36, -30, -24, 12, -16, -12, -30, -31, 22, 10, 6, -18, -27, 33, 24, 8, -31, 36, 31, -2, -27, -42, 42, 16, 8, 11, 1, -38, -40, 20, -18, 4, -3, 3, -18, -12, -15, -1, -8, 48, -10, -2, 14, 36, -30, 45, -1, 25, 13, -25, 32, 16, 4, 2, 38, -21, 22, -12, 22, -8, -41, -12, 0, -21, -27, 6, -40, 14, 48, 54, 12, -48, -16, 30, -13, 4, 48, -48, -6, 27, 26, 56, -57, -46, 18, -6, -23, -58, 18, 56, -19, 17, 30, 36, 0, 20, -6, 4, 50, -14, -12, -14, -23, -16, -58, 32, 9, -15, 60, 14, 1, 42, -12, -39, -56, 58, 37, -10, -47, 41, -54, 54, 66, -50, -15, -24, -27, -18, 6, -39, 9, 36, -13, 50, -26, 67, -18, -6, -9, -18, 10, -4, -39, -44, 53, 48, 12, 36, -44, -5, 23, -16, 15, 61, 0, 18, 12, -12, -27, -24, 45, 24, -40, -36, 9, 36, 18, -12, 12, -37, -28, -6, 6, 69, -11, 64, 52, 59, 0, 45, 36, -78, -71, 74, -15, -80, -44, 37, -35, -26, -71, 20, 18, 0, -3, -42, 22, -30, -64, -22];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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