/*
  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^2 + 3*x - 2;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e + 4, e + 4, e + 6, -4*e - 8, -3*e - 4, -1, 2*e + 2, 2, -e + 8, 2*e - 3, 12, -4*e - 4, 4*e + 14, -14, -2*e - 12, 3*e + 2, -2*e + 4, -4*e - 8, 4*e + 14, 8*e + 10, 20, -2*e - 10, -3*e - 6, 4*e, 6*e + 18, -8*e - 8, 6*e, -3*e - 6, -2*e - 12, -5*e - 14, 3*e - 8, 7*e + 4, 6*e + 16, -2*e, 3*e - 2, 6*e + 8, 26, -4*e - 18, 3*e + 4, 8*e + 18, 3*e + 20, 6*e + 18, -12*e - 20, -4, -8*e, e + 24, 6*e + 20, -4*e - 24, 3*e + 18, -10*e - 14, e - 14, -3*e - 6, 2*e - 22, -6*e + 2, -13*e - 18, -4*e - 2, 2*e + 6, 11*e + 8, 4*e - 14, -4*e + 12, -13*e - 18, 3*e + 2, e + 30, 2*e, -11*e - 20, -3*e - 12, -6*e - 10, -10*e - 26, -4*e - 6, -2, 5*e + 2, -7*e - 22, 4*e + 26, 2*e + 18, -4*e - 36, -9*e - 18, -2*e, 11*e + 14, -12*e - 30, 4*e - 18, -7*e, -4*e - 18, -12*e - 18, 16*e + 26, 7*e - 10, 4*e - 4, -2*e - 22, -14*e - 34, -10*e, -6*e - 4, e + 22, -2*e - 8, -2*e - 6, 13*e + 34, -22, 6*e + 24, -3*e + 28, 6*e + 30, -9*e - 24, 13*e + 32, -7*e - 36, 6*e + 18, -20, -10*e + 12, -40, -11*e + 10, -5*e - 14, -8*e, -4*e - 4, 10*e + 40, 18*e + 24, -4*e - 12, -13*e - 24, 2*e + 30, 8*e + 6, e + 20, 12*e + 16, -3*e + 38, 8*e + 40, -2*e + 32, -13*e - 22, 4*e - 4, 2*e + 46, 15*e + 32, -6*e - 24, 7*e + 44, -11*e - 14, -3*e - 16, -4*e + 10, 13*e + 32, 6*e + 18, -3*e - 12, 5*e + 16, -11*e - 8, -8*e - 22, -3*e, -16*e - 18, -3*e + 6, -8*e - 12, -e + 38, -14*e - 44, 15*e + 22, -5*e - 14, -17*e - 26, 8*e + 48, -7*e - 18, 20*e + 40, -4*e + 36, -3*e - 46, 12*e - 2, 10*e + 30, 8*e + 6, 18, 18*e + 18, 18*e + 28, 3*e + 10, -20*e - 36, -8*e - 28, 3*e + 2, 6*e + 20, 2*e - 32, -8*e + 12, 14*e + 32, 12*e + 42, -18*e - 22, -13*e - 10, 11*e + 34, 5*e - 10, 4*e + 8, 16*e + 8, 20*e + 30, -26, -16*e - 48, 12*e + 40, -e - 6, 8*e, -17*e - 38, -6*e - 12, -6*e - 18, -8*e + 18, -22*e - 50, -6*e - 22, 2*e - 8, -19*e - 14, -10*e + 26, -7*e - 36, 16*e + 26, 14*e - 6, 32, 4*e + 56, -6*e - 10, e + 32, -6*e - 4, -14*e - 44, -20*e - 32, -13*e - 22, 5*e - 24, 13*e + 36, e + 62, -6*e + 4, -2*e - 46, -14*e - 34, -11*e - 18, -5*e - 6, -4*e - 12, -3*e - 14, -3*e, -5*e, -8*e - 18, -14, -17*e - 18, -31*e - 46, 14*e + 26, 11*e + 12, -3*e + 22, -15*e - 36, 2*e + 36, -54, -18*e - 2, -e + 32, -14, e - 42, 2*e - 52, 19*e, -5*e - 34, 13*e + 10, -8*e - 44, -36, -21*e - 10, 18*e + 8, 6*e + 4, 17*e + 46, -24*e - 40, 22*e + 44, -8*e - 24, 28*e + 48, -74, -22*e - 10, 4*e + 54, 6*e + 26, -15*e - 4, 24*e + 18, 17*e - 14, -10*e + 32, -9*e + 38, e + 10, 3*e - 12, -16*e - 18, -8*e - 6, 14*e + 60, 12*e + 24, 2*e - 14, 3*e + 40, -4*e - 62, -23*e - 46, -24*e - 20, 16*e + 48, 5*e + 46, -31*e - 48, -9*e + 12, -9*e + 40, 14*e + 54, -24*e - 12, -24*e - 28, -16*e - 38, -13*e + 24, -e + 12, -21*e - 28, -26*e - 42, 30, -12*e - 6, 8*e + 62, 6*e + 26, 11*e + 44, 7*e - 16, -20*e - 34, -6*e - 40, -8*e - 20];
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;