/*
  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, -11, 10, 10, -6, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {29,29,w^2 - w - 1}>;

primesArray := [
[7, 7, w^4 - 2*w^3 - 3*w^2 + 6*w - 1],
[7, 7, -w^4 + w^3 + 4*w^2 - w - 1],
[13, 13, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 3*w - 3],
[29, 29, -w^4 + w^3 + 4*w^2 - 3*w - 2],
[29, 29, w^4 - 5*w^2 - 2*w + 4],
[29, 29, -w^4 + w^3 + 3*w^2 - w + 2],
[29, 29, -w^2 + w + 1],
[41, 41, -w^5 + w^4 + 4*w^3 - w^2 - 3*w - 1],
[43, 43, 2*w^3 - 2*w^2 - 7*w + 3],
[43, 43, w^3 - w^2 - 2*w + 1],
[64, 2, -2],
[71, 71, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 3],
[71, 71, w^5 - 2*w^4 - 2*w^3 + 5*w^2 - 3*w - 1],
[83, 83, -2*w^4 + 3*w^3 + 6*w^2 - 7*w + 2],
[83, 83, w^4 - w^3 - 5*w^2 + 2*w + 3],
[83, 83, -w^5 + 3*w^4 + w^3 - 9*w^2 + 5*w + 3],
[83, 83, w^5 - 7*w^3 + 10*w - 1],
[97, 97, w^5 - 2*w^4 - 3*w^3 + 5*w^2 - w - 1],
[97, 97, w^5 - 5*w^3 - w^2 + 3*w - 3],
[113, 113, w^4 - w^3 - 4*w^2 + w + 4],
[113, 113, w^4 - 2*w^3 - 3*w^2 + 6*w + 2],
[127, 127, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3],
[127, 127, w^5 - 9*w^3 + w^2 + 18*w - 1],
[127, 127, 2*w^4 - 4*w^3 - 5*w^2 + 10*w - 2],
[127, 127, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 2],
[127, 127, -3*w^4 + 5*w^3 + 9*w^2 - 11*w],
[127, 127, -w^5 + 3*w^4 + w^3 - 8*w^2 + 4*w - 1],
[139, 139, w^4 - 3*w^3 - 2*w^2 + 10*w - 3],
[139, 139, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 4*w - 3],
[167, 167, -w^4 + w^3 + 5*w^2 - 4*w - 3],
[167, 167, -w^5 + w^4 + 6*w^3 - 3*w^2 - 8*w + 3],
[169, 13, w^4 - 3*w^3 - 2*w^2 + 8*w - 2],
[169, 13, 2*w^4 - 3*w^3 - 7*w^2 + 7*w + 3],
[181, 181, -w^5 + 4*w^3 + 3*w^2 - 2],
[181, 181, w^5 - 3*w^4 - 3*w^3 + 11*w^2 - 4],
[181, 181, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 2*w - 4],
[181, 181, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 3*w - 2],
[197, 197, -3*w^4 + 5*w^3 + 10*w^2 - 11*w - 3],
[197, 197, -3*w^4 + 6*w^3 + 9*w^2 - 16*w - 1],
[197, 197, 2*w^3 - 3*w^2 - 5*w + 4],
[197, 197, -w^5 + 6*w^3 - 6*w + 1],
[223, 223, 2*w^4 - 2*w^3 - 9*w^2 + 3*w + 6],
[223, 223, -3*w^4 + 5*w^3 + 9*w^2 - 13*w + 1],
[223, 223, w^5 - w^4 - 6*w^3 + 4*w^2 + 10*w - 4],
[223, 223, w^4 - 4*w^3 - 2*w^2 + 12*w],
[239, 239, 2*w^4 - w^3 - 8*w^2 + 3],
[239, 239, -w^5 + 3*w^4 - 8*w^2 + 9*w + 3],
[239, 239, -w^5 + 2*w^4 + 2*w^3 - 6*w^2 + 3*w + 4],
[239, 239, w^4 - 6*w^2 + 6],
[239, 239, 3*w^4 - 4*w^3 - 11*w^2 + 10*w + 3],
[239, 239, -w^5 + 7*w^3 - 9*w - 1],
[251, 251, w^5 - 2*w^4 - 2*w^3 + 5*w^2 - 3*w - 2],
[251, 251, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 2],
[281, 281, -3*w^4 + 4*w^3 + 11*w^2 - 10*w - 4],
[281, 281, 3*w^4 - 3*w^3 - 12*w^2 + 5*w + 6],
[293, 293, -w^5 + 8*w^3 - 15*w + 1],
[293, 293, w^5 - 4*w^4 + w^3 + 11*w^2 - 9*w + 1],
[293, 293, -2*w^4 + 5*w^3 + 5*w^2 - 14*w + 3],
[293, 293, -w^5 + 8*w^3 - 14*w + 2],
[293, 293, -2*w^4 + 4*w^3 + 6*w^2 - 9*w + 1],
[293, 293, w^5 - 4*w^4 + 12*w^2 - 5*w - 2],
[307, 307, 2*w^4 - 3*w^3 - 7*w^2 + 9*w + 2],
[307, 307, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 8*w + 6],
[307, 307, -w^3 + 6*w],
[307, 307, w^5 - w^4 - 7*w^3 + 5*w^2 + 12*w - 3],
[337, 337, -2*w^4 + 2*w^3 + 9*w^2 - 4*w - 8],
[337, 337, 3*w^4 - 4*w^3 - 10*w^2 + 9*w + 3],
[349, 349, 2*w^5 - 4*w^4 - 7*w^3 + 13*w^2 + 3*w - 6],
[349, 349, 2*w^5 - 2*w^4 - 9*w^3 + 5*w^2 + 7*w - 2],
[349, 349, w^5 - 3*w^4 + 8*w^2 - 7*w - 2],
[349, 349, w^5 - 9*w^3 + 2*w^2 + 17*w - 4],
[379, 379, w^5 - 3*w^4 + 6*w^2 - 8*w + 5],
[379, 379, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 3*w + 1],
[379, 379, w^5 - 4*w^4 + 12*w^2 - 8*w - 2],
[379, 379, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w + 5],
[419, 419, w^5 - 4*w^4 + w^3 + 10*w^2 - 10*w + 3],
[419, 419, 3*w^4 - 3*w^3 - 11*w^2 + 5*w + 4],
[421, 421, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 8*w - 4],
[421, 421, -w^5 + w^4 + 5*w^3 - 3*w^2 - 7*w + 4],
[433, 433, -2*w^2 + w + 5],
[433, 433, -2*w^4 + 3*w^3 + 5*w^2 - 6*w + 3],
[461, 461, 2*w^3 - 2*w^2 - 5*w + 3],
[461, 461, -w^5 + 5*w^3 + 2*w^2 - 4*w + 1],
[461, 461, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + 3*w + 5],
[461, 461, -w^5 + 3*w^4 + 3*w^3 - 11*w^2 - w + 3],
[463, 463, -2*w^4 + 3*w^3 + 7*w^2 - 8*w - 4],
[463, 463, -w^5 + 4*w^4 + w^3 - 12*w^2 + 3*w - 1],
[491, 491, w^5 - 7*w^3 - w^2 + 10*w + 3],
[491, 491, -w^5 + 4*w^4 - w^3 - 11*w^2 + 10*w + 2],
[503, 503, -2*w^5 + 5*w^4 + 5*w^3 - 15*w^2 + w + 3],
[503, 503, 2*w^4 - w^3 - 8*w^2 + w + 1],
[503, 503, -2*w^5 + 3*w^4 + 7*w^3 - 7*w^2 - 3*w + 1],
[503, 503, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 11*w + 2],
[547, 547, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 10*w - 5],
[547, 547, -w^4 + 2*w^3 + 2*w^2 - 4*w + 5],
[547, 547, -2*w^5 + 4*w^4 + 7*w^3 - 12*w^2 - 3*w + 4],
[547, 547, -w^5 + w^4 + 3*w^3 - 2*w^2 + 3*w + 2],
[587, 587, -w^5 - w^4 + 8*w^3 + 4*w^2 - 13*w + 1],
[587, 587, w^5 - w^4 - 5*w^3 + 2*w^2 + 7*w - 2],
[617, 617, w^5 - w^4 - 4*w^3 + 3*w^2 - 1],
[617, 617, w^4 - 2*w^3 - 5*w^2 + 5*w + 6],
[617, 617, -w^5 + w^4 + 4*w^3 - 3*w^2 + 4],
[617, 617, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 11*w + 8],
[631, 631, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 11*w + 1],
[631, 631, -w^5 + w^4 + 4*w^3 - 2*w - 4],
[631, 631, -2*w^4 + 5*w^3 + 4*w^2 - 12*w + 4],
[631, 631, -2*w^5 + 5*w^4 + 5*w^3 - 15*w^2 + w + 4],
[643, 643, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 2],
[643, 643, -w^5 + 2*w^4 + w^3 - 3*w^2 + 6*w - 3],
[659, 659, -2*w^4 + 4*w^3 + 7*w^2 - 10*w - 2],
[659, 659, w^4 - 3*w^3 - 3*w^2 + 10*w - 1],
[659, 659, w^5 - 4*w^4 + 13*w^2 - 6*w - 6],
[659, 659, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 7*w - 5],
[673, 673, 2*w^5 - 3*w^4 - 7*w^3 + 8*w^2 + 2*w - 3],
[673, 673, w^5 - 4*w^4 - w^3 + 12*w^2 - 3*w - 2],
[673, 673, -2*w^5 + 3*w^4 + 9*w^3 - 10*w^2 - 8*w + 5],
[673, 673, w^5 - w^4 - 4*w^3 + 3*w + 6],
[701, 701, -2*w^4 + 4*w^3 + 6*w^2 - 11*w - 3],
[701, 701, 2*w^4 - 3*w^3 - 7*w^2 + 6*w + 5],
[727, 727, -w^5 + 2*w^4 + w^3 - 3*w^2 + 7*w - 2],
[727, 727, 2*w^5 - 3*w^4 - 7*w^3 + 8*w^2 + 3*w - 4],
[727, 727, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 2*w + 1],
[727, 727, w^5 + 2*w^4 - 9*w^3 - 8*w^2 + 16*w + 2],
[729, 3, -3],
[743, 743, w^5 - 4*w^4 + 13*w^2 - 8*w - 5],
[743, 743, -w^3 + w^2 + w - 2],
[743, 743, w^4 - 4*w^3 - w^2 + 10*w - 2],
[743, 743, 3*w^3 - 3*w^2 - 11*w + 6],
[757, 757, -w^5 + 4*w^4 - 11*w^2 + 5*w - 2],
[757, 757, -3*w^4 + 3*w^3 + 11*w^2 - 5*w - 2],
[757, 757, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - 4],
[757, 757, w^5 - 8*w^3 + w^2 + 15*w - 4],
[769, 769, w^5 - 5*w^3 - 3*w^2 + 4*w + 6],
[769, 769, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 4*w + 3],
[769, 769, w^4 - 3*w^3 - w^2 + 9*w - 3],
[769, 769, w^5 - 4*w^4 - w^3 + 13*w^2 - 4*w - 5],
[797, 797, w^5 + w^4 - 8*w^3 - 4*w^2 + 11*w + 2],
[797, 797, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 9*w - 6],
[811, 811, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 4*w - 4],
[811, 811, 2*w^4 - 2*w^3 - 6*w^2 + 3*w - 4],
[811, 811, -2*w^5 + 3*w^4 + 9*w^3 - 8*w^2 - 9*w + 1],
[811, 811, -w^5 + w^4 + 2*w^3 + w^2 + 4*w - 5],
[827, 827, w^5 + w^4 - 9*w^3 - 2*w^2 + 14*w - 3],
[827, 827, w^5 - 3*w^4 + w^3 + 6*w^2 - 11*w + 4],
[827, 827, w^5 - 3*w^4 + 9*w^2 - 9*w - 4],
[827, 827, w^5 - w^4 - 4*w^3 + 4*w + 5],
[827, 827, -w^5 + 4*w^4 + 2*w^3 - 13*w^2 - w + 3],
[827, 827, -w^5 + w^4 + 7*w^3 - 4*w^2 - 11*w + 3],
[839, 839, -w^5 + w^4 + 7*w^3 - 4*w^2 - 13*w + 4],
[839, 839, -w^5 + 3*w^4 + w^3 - 8*w^2 + 3*w - 1],
[841, 29, -2*w^4 + 5*w^3 + 5*w^2 - 13*w + 1],
[853, 853, -2*w^4 + 10*w^2 + 3*w - 6],
[853, 853, 2*w^5 - 4*w^4 - 8*w^3 + 13*w^2 + 8*w - 6],
[853, 853, -2*w^4 + w^3 + 9*w^2 - 2*w - 4],
[853, 853, -2*w^5 + 3*w^4 + 8*w^3 - 8*w^2 - 7*w + 1],
[881, 881, w^5 - 5*w^3 - 2*w^2 + 5*w - 1],
[881, 881, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - 5*w + 5],
[883, 883, 3*w^4 - 4*w^3 - 11*w^2 + 10*w + 5],
[883, 883, -3*w^4 + 3*w^3 + 12*w^2 - 5*w - 7],
[883, 883, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 5*w - 2],
[883, 883, w^5 - w^4 - 6*w^3 + 4*w^2 + 10*w - 6],
[911, 911, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 4],
[911, 911, 2*w^5 - 4*w^4 - 6*w^3 + 10*w^2 + 2*w + 1],
[911, 911, w^5 - w^4 - 3*w^3 + 2*w^2 - 2*w - 3],
[911, 911, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 5*w + 2],
[911, 911, w^5 - 2*w^4 - w^3 + 2*w^2 - 5*w + 6],
[911, 911, -2*w^5 + 4*w^4 + 8*w^3 - 12*w^2 - 8*w + 3],
[937, 937, w^4 - 2*w^3 - 5*w^2 + 7*w + 5],
[937, 937, w^5 + w^4 - 7*w^3 - 5*w^2 + 9*w + 4],
[937, 937, 3*w^4 - 4*w^3 - 9*w^2 + 7*w - 1],
[937, 937, -w^5 + 4*w^4 + w^3 - 14*w^2 + 4*w + 8],
[953, 953, w^5 - 3*w^4 - w^3 + 7*w^2 - 5*w + 4],
[953, 953, w^5 - 2*w^4 - 3*w^3 + 4*w^2 + 2],
[967, 967, w^5 - 6*w^4 + 2*w^3 + 20*w^2 - 12*w - 6],
[967, 967, w^5 - 4*w^4 - 2*w^3 + 14*w^2 + w - 5],
[967, 967, -w^5 + 8*w^3 - 2*w^2 - 14*w + 10],
[967, 967, w^5 - 3*w^4 + 7*w^2 - 9*w],
[967, 967, -w^5 + 6*w^3 + 2*w^2 - 9*w - 2],
[967, 967, -w^5 + w^4 + 5*w^3 - 2*w^2 - 4*w - 3],
[1009, 1009, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - 2*w - 4],
[1009, 1009, -2*w^5 + 4*w^4 + 6*w^3 - 11*w^2 - w + 2],
[1009, 1009, 2*w^5 - 3*w^4 - 9*w^3 + 9*w^2 + 9*w - 3],
[1009, 1009, -w^5 + w^4 + 4*w^3 - w^2 - 2*w - 5],
[1021, 1021, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 10],
[1021, 1021, w^5 - 8*w^3 + 3*w^2 + 14*w - 6],
[1021, 1021, 2*w^5 - 4*w^4 - 7*w^3 + 12*w^2 + 4*w - 4],
[1021, 1021, -2*w^5 + 3*w^4 + 8*w^3 - 8*w^2 - 6*w + 1],
[1051, 1051, w^5 - w^4 - 3*w^3 + w^2 - w - 2],
[1051, 1051, 3*w^4 - 6*w^3 - 8*w^2 + 16*w - 3],
[1091, 1091, -w^5 + 4*w^4 - 13*w^2 + 6*w + 5],
[1091, 1091, -w^5 - w^4 + 9*w^3 + 3*w^2 - 16*w + 1],
[1093, 1093, w^2 + w - 5],
[1093, 1093, w^4 - 3*w^3 - w^2 + 9*w - 2],
[1093, 1093, 2*w^5 - 5*w^4 - 5*w^3 + 14*w^2 - w - 2],
[1093, 1093, 2*w^5 - 3*w^4 - 8*w^3 + 7*w^2 + 6*w + 2],
[1163, 1163, -w^3 + 3*w^2 + 2*w - 5],
[1163, 1163, 2*w^4 - 2*w^3 - 6*w^2 + 3*w - 3],
[1217, 1217, w^5 + w^4 - 6*w^3 - 6*w^2 + 3*w + 5],
[1217, 1217, w^5 - w^4 - 5*w^3 + w^2 + 9*w],
[1231, 1231, -w^5 + 4*w^4 - w^3 - 10*w^2 + 10*w - 1],
[1231, 1231, -w^5 + w^4 + 5*w^3 - w^2 - 5*w - 5],
[1259, 1259, w^5 - w^4 - 5*w^3 + 4*w^2 + 6*w - 7],
[1259, 1259, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 4],
[1289, 1289, w^5 + w^4 - 11*w^3 + 23*w - 7],
[1289, 1289, -w^5 + w^4 + 5*w^3 - 2*w^2 - 8*w - 2],
[1289, 1289, 2*w^5 - 13*w^3 - w^2 + 19*w - 1],
[1289, 1289, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 14*w + 9],
[1301, 1301, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 6*w + 4],
[1301, 1301, -w^5 - w^4 + 11*w^3 + 2*w^2 - 23*w + 3],
[1301, 1301, -2*w^5 + w^4 + 11*w^3 - 2*w^2 - 12*w + 1],
[1301, 1301, -2*w^5 + 3*w^4 + 9*w^3 - 10*w^2 - 9*w + 6],
[1303, 1303, w^5 - 4*w^4 - w^3 + 13*w^2 - 2*w - 2],
[1303, 1303, w^5 - 4*w^4 - w^3 + 14*w^2 - 3*w - 4],
[1331, 11, -w^4 + 4*w^3 - 2*w^2 - 10*w + 8],
[1331, 11, w^5 - 5*w^4 + w^3 + 16*w^2 - 10*w - 4],
[1399, 1399, -w^5 + 6*w^4 - w^3 - 21*w^2 + 8*w + 8],
[1399, 1399, 2*w^4 - 4*w^3 - 6*w^2 + 13*w - 4],
[1399, 1399, 2*w^5 - 4*w^4 - 9*w^3 + 15*w^2 + 9*w - 7],
[1399, 1399, w^5 - 7*w^3 + w^2 + 10*w],
[1427, 1427, 2*w^5 - 5*w^4 - 7*w^3 + 17*w^2 + 6*w - 6],
[1427, 1427, -w^5 + 3*w^4 + 2*w^3 - 12*w^2 + 3*w + 9],
[1471, 1471, w^5 - 5*w^3 - w^2 + 4*w - 3],
[1471, 1471, w^5 - w^4 - 6*w^3 + 3*w^2 + 7*w - 1],
[1471, 1471, -w^5 + 3*w^4 - w^3 - 6*w^2 + 12*w - 4],
[1471, 1471, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 6],
[1483, 1483, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 6],
[1483, 1483, -w^5 + 4*w^4 - 2*w^3 - 8*w^2 + 12*w - 7],
[1483, 1483, -w^5 + 5*w^4 + w^3 - 18*w^2 + w + 3],
[1483, 1483, w^5 - 4*w^4 - 4*w^3 + 17*w^2 + 7*w - 8],
[1499, 1499, -2*w^5 + 6*w^4 + 6*w^3 - 21*w^2 - 2*w + 8],
[1499, 1499, 2*w^5 - 5*w^4 - 7*w^3 + 17*w^2 + 5*w - 7],
[1567, 1567, 2*w^5 - 7*w^4 - w^3 + 19*w^2 - 11*w],
[1567, 1567, -w^5 + w^4 + 4*w^3 - 2*w^2 - w - 3],
[1567, 1567, w^5 - 2*w^4 - 3*w^3 + 6*w^2 - w - 5],
[1567, 1567, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 5*w - 2],
[1583, 1583, 2*w^4 - 6*w^3 - 4*w^2 + 17*w - 3],
[1583, 1583, -2*w^4 + 5*w^3 + 5*w^2 - 12*w + 1],
[1609, 1609, -4*w^4 + 5*w^3 + 13*w^2 - 11*w + 2],
[1609, 1609, 2*w^5 - 2*w^4 - 10*w^3 + 3*w^2 + 12*w + 5],
[1637, 1637, -w^5 - 2*w^4 + 10*w^3 + 6*w^2 - 18*w],
[1637, 1637, -w^4 + 6*w^2 - 10],
[1637, 1637, -2*w^4 + w^3 + 8*w^2 - 7],
[1637, 1637, -w^5 + 4*w^4 - 3*w^3 - 11*w^2 + 17*w + 2],
[1667, 1667, 2*w^4 - w^3 - 10*w^2 + 2*w + 7],
[1667, 1667, 3*w^4 - 2*w^3 - 12*w^2 + 2*w + 4],
[1667, 1667, w^5 - 7*w^3 + w^2 + 9*w - 2],
[1667, 1667, w^5 - 2*w^4 - 2*w^3 + 6*w^2 - 4*w - 5],
[1681, 41, -w^4 + 4*w^3 + w^2 - 11*w + 4],
[1681, 41, -3*w^4 + 6*w^3 + 9*w^2 - 15*w + 1],
[1693, 1693, w^5 - 3*w^4 - 2*w^3 + 11*w^2 - w - 7],
[1693, 1693, -w^4 - 3*w^3 + 7*w^2 + 12*w - 7],
[1693, 1693, -w^4 + 5*w^2 - w + 1],
[1693, 1693, -w^5 - w^4 + 9*w^3 + 2*w^2 - 16*w + 4],
[1709, 1709, -w^5 + 7*w^3 - 12*w + 3],
[1709, 1709, -w^5 + 3*w^4 + 3*w^3 - 11*w^2 - 3*w + 5],
[1723, 1723, 2*w^5 - w^4 - 10*w^3 + 11*w + 2],
[1723, 1723, 2*w^5 - 6*w^4 - 6*w^3 + 21*w^2 + 4*w - 9],
[1723, 1723, -w^5 + 2*w^4 + 6*w^3 - 11*w^2 - 8*w + 10],
[1723, 1723, -w^5 - w^4 + 5*w^3 + 5*w^2 - 2*w + 2],
[1777, 1777, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 8*w - 2],
[1777, 1777, w^5 - 5*w^4 + 17*w^2 - 5*w - 4],
[1777, 1777, -2*w^5 + 6*w^4 + 5*w^3 - 20*w^2 + 9],
[1777, 1777, -w^5 + 6*w^4 - 3*w^3 - 17*w^2 + 11*w],
[1847, 1847, 3*w^5 - 4*w^4 - 14*w^3 + 11*w^2 + 14*w - 3],
[1847, 1847, 2*w^3 - 3*w^2 - 8*w + 3],
[1849, 43, 2*w^4 - 5*w^3 - 5*w^2 + 13*w - 2],
[1849, 43, -w^4 - w^3 + 6*w^2 + 4*w - 3],
[1861, 1861, 3*w^4 - 5*w^3 - 8*w^2 + 13*w - 6],
[1861, 1861, 3*w^4 - w^3 - 12*w^2 - w + 2],
[1861, 1861, -w^5 + w^4 + 6*w^3 - 4*w^2 - 11*w + 6],
[1861, 1861, -w^5 + 4*w^4 + 3*w^3 - 13*w^2 - 4*w + 2],
[1933, 1933, -2*w^5 + 4*w^4 + 7*w^3 - 11*w^2 - 4*w + 2],
[1933, 1933, -2*w^5 + 4*w^4 + 6*w^3 - 10*w^2 - w - 2],
[1933, 1933, 2*w^4 - 3*w^3 - 7*w^2 + 8*w + 6],
[1933, 1933, -2*w^4 + 2*w^3 + 8*w^2 - 3*w - 8],
[1973, 1973, w^4 - 4*w^3 + 12*w - 4],
[1973, 1973, w^3 - 2*w - 3],
[1987, 1987, -3*w^5 + 6*w^4 + 10*w^3 - 19*w^2 - 3*w + 9],
[1987, 1987, -w^4 + 4*w^3 + w^2 - 14*w + 1]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [3, -3, -5, 3, 6, 3, -1, 0, 6, -9, 7, 6, -6, -6, 6, 6, 9, 18, 6, 6, 12, 18, 0, -15, -2, -12, 16, 4, -5, 0, -21, -1, -1, 12, -12, -9, -18, 0, 18, -9, 24, 15, -6, -8, 28, 6, 15, -18, -12, -15, -6, -21, -15, 0, 18, -3, -24, 30, -3, -30, 12, -3, 7, -11, -6, -5, 13, -10, -10, -18, 12, 11, 29, 12, -6, -18, 39, -30, -3, -6, -30, 21, 42, -30, 30, -15, 0, -24, 24, 36, -24, 15, 30, 36, -17, -6, -8, 39, -12, -15, 33, -3, 30, 38, -20, 16, 29, 27, -12, 30, -24, 24, -12, 19, 0, 28, -15, 30, 15, 8, -28, -37, 44, 16, 15, 0, -15, -9, -29, 34, 27, 30, -9, -13, 5, -30, 33, -6, -9, 24, 33, 0, 27, 3, -27, -36, -9, -30, -24, -51, -4, 8, 10, -10, -26, -30, -15, 20, -52, -27, 48, -18, 48, -6, 0, -36, 33, -57, -39, 30, 54, -42, -36, 12, -32, -9, -14, -59, -32, -48, -38, 52, -42, -30, 9, 40, 49, 12, 60, -30, -60, -22, 41, -36, 30, 36, 27, 39, 33, -19, 26, 36, -36, -3, 3, -18, -27, -15, 15, 12, -48, -21, 9, 60, 21, -5, -63, -14, 30, -54, -9, -24, 15, -36, -57, 71, 71, 30, -36, 15, 30, 2, 51, 60, 2, 72, 36, 18, -24, 78, -72, 21, -42, 57, 6, -51, -33, 55, -10, 54, 54, -6, 72, 33, -6, -31, 32, 18, -15, 5, -31, 51, 42, 18, -24, -22, -67, -57, -1, 26, -48, -54, 36, -72, -57, 3, -66, -46, 8];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29,29,w^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;