/*
  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![-7, -14, 7, 13, -4, -3, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {7, 7, w}>;

primesArray := [
[7, 7, w],
[7, 7, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 + 5],
[13, 13, w^2 - 3],
[13, 13, -w^2 + 2*w + 2],
[29, 29, w^4 - 2*w^3 - 4*w^2 + 4*w + 6],
[29, 29, -w^4 + 2*w^3 + 4*w^2 - 6*w - 5],
[41, 41, -w^2 + 4],
[41, 41, -w^2 + 2*w + 3],
[43, 43, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 - w + 3],
[43, 43, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 4*w + 2],
[64, 2, -2],
[71, 71, w^5 - 3*w^4 - w^3 + 6*w^2 - 2*w + 2],
[71, 71, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 - w + 5],
[71, 71, w^4 - 3*w^3 - 2*w^2 + 6*w + 2],
[71, 71, 2*w^5 - 6*w^4 - 4*w^3 + 19*w^2 - w - 12],
[83, 83, -w^4 + w^3 + 5*w^2 - w - 6],
[83, 83, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 4*w - 1],
[97, 97, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 5*w + 4],
[97, 97, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + 2*w - 2],
[113, 113, -2*w^4 + 3*w^3 + 8*w^2 - 6*w - 6],
[113, 113, -w^5 + 4*w^4 - w^3 - 12*w^2 + 9*w + 10],
[113, 113, -w^5 + w^4 + 5*w^3 + w^2 - 7*w - 9],
[113, 113, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 6*w - 1],
[125, 5, -w^5 + 4*w^4 - 12*w^2 + 4*w + 4],
[125, 5, w^5 - w^4 - 6*w^3 + 2*w^2 + 9*w - 1],
[139, 139, -w^4 + w^3 + 5*w^2 - 2*w - 8],
[139, 139, w^5 - 4*w^4 + 13*w^2 - 4*w - 10],
[167, 167, -w^5 + 3*w^4 + 2*w^3 - 11*w^2 + 2*w + 9],
[167, 167, -w^5 + 7*w^3 + 2*w^2 - 9*w - 3],
[167, 167, 2*w^5 - 6*w^4 - 3*w^3 + 15*w^2 - w - 5],
[167, 167, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - 8],
[167, 167, -w^5 + 5*w^4 - 3*w^3 - 13*w^2 + 11*w + 4],
[167, 167, 2*w^4 - 5*w^3 - 4*w^2 + 11*w + 2],
[169, 13, -w^4 + 2*w^3 + 2*w^2 - 3*w + 1],
[169, 13, 2*w^4 - 4*w^3 - 7*w^2 + 9*w + 8],
[181, 181, -w^5 + 4*w^4 - 14*w^2 + 5*w + 12],
[181, 181, -w^3 + 2*w^2 + 3*w - 2],
[181, 181, -w^3 + w^2 + 4*w - 2],
[181, 181, 3*w^4 - 7*w^3 - 9*w^2 + 15*w + 9],
[197, 197, w^5 - 2*w^4 - 5*w^3 + 5*w^2 + 8*w + 3],
[197, 197, -w^5 + 8*w^3 + w^2 - 14*w - 4],
[197, 197, w^4 - w^3 - 6*w^2 + 5*w + 6],
[197, 197, w^5 - 3*w^4 - 3*w^3 + 12*w^2 - 10],
[211, 211, -w^5 + 4*w^4 + w^3 - 13*w^2 + 2*w + 8],
[211, 211, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 3*w - 3],
[211, 211, -w^5 + 3*w^4 + w^3 - 8*w^2 + 3*w + 5],
[211, 211, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 3],
[223, 223, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 6*w + 2],
[223, 223, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 - w + 4],
[293, 293, w^5 - 8*w^3 - w^2 + 13*w + 4],
[293, 293, -w^5 + 5*w^4 - 2*w^3 - 15*w^2 + 8*w + 9],
[307, 307, -w^5 + 4*w^4 + w^3 - 13*w^2 + w + 10],
[307, 307, 2*w^5 - 6*w^4 - 4*w^3 + 17*w^2 + w - 6],
[307, 307, -w^5 + 3*w^4 + w^3 - 8*w^2 + 4*w + 6],
[307, 307, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 2*w - 5],
[307, 307, -2*w^5 + 4*w^4 + 8*w^3 - 11*w^2 - 9*w + 4],
[307, 307, 2*w^4 - 4*w^3 - 9*w^2 + 10*w + 11],
[337, 337, w^5 - 4*w^4 - w^3 + 13*w^2 - 2*w - 5],
[337, 337, -w^5 + w^4 + 5*w^3 + w^2 - 9*w - 5],
[337, 337, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 6*w + 2],
[337, 337, -2*w^4 + 4*w^3 + 9*w^2 - 12*w - 11],
[337, 337, w^5 - w^4 - 7*w^3 + 5*w^2 + 9*w - 2],
[337, 337, -w^5 + 5*w^4 - 2*w^3 - 14*w^2 + 7*w + 8],
[349, 349, 2*w^4 - 4*w^3 - 9*w^2 + 10*w + 13],
[349, 349, w^5 - 5*w^4 + 3*w^3 + 15*w^2 - 12*w - 11],
[379, 379, -w^5 + 5*w^4 - 2*w^3 - 15*w^2 + 7*w + 11],
[379, 379, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 1],
[379, 379, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 8*w - 1],
[379, 379, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 4*w + 8],
[379, 379, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 7*w + 2],
[379, 379, 2*w^4 - 3*w^3 - 8*w^2 + 5*w + 10],
[419, 419, -w^5 + w^4 + 6*w^3 - 10*w - 9],
[419, 419, w^4 - 2*w^3 - w^2 + w - 3],
[419, 419, -2*w^5 + 5*w^4 + 6*w^3 - 14*w^2 - 5*w + 6],
[419, 419, w^5 - 4*w^4 + 14*w^2 - 7*w - 13],
[421, 421, 2*w^5 - 4*w^4 - 7*w^3 + 8*w^2 + 8*w + 2],
[421, 421, 3*w^4 - 6*w^3 - 11*w^2 + 15*w + 12],
[421, 421, -3*w^4 + 6*w^3 + 11*w^2 - 13*w - 13],
[421, 421, w^5 - 4*w^4 - w^3 + 15*w^2 - 4*w - 13],
[433, 433, -2*w^3 + 3*w^2 + 6*w - 3],
[433, 433, w^5 - 3*w^4 - w^3 + 5*w^2 + 3],
[433, 433, -w^5 + 2*w^4 + 3*w^3 - 6*w^2 + 5],
[433, 433, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 6*w - 3],
[449, 449, w^5 - 5*w^4 + w^3 + 17*w^2 - 7*w - 11],
[449, 449, 2*w^4 - 4*w^3 - 7*w^2 + 10*w + 9],
[463, 463, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 3],
[463, 463, w^5 - 9*w^3 + 16*w + 3],
[463, 463, -w^5 + 5*w^4 - w^3 - 17*w^2 + 6*w + 11],
[463, 463, -2*w^4 + 5*w^3 + 4*w^2 - 11*w - 1],
[491, 491, 3*w^4 - 7*w^3 - 9*w^2 + 16*w + 10],
[491, 491, -w^5 + w^4 + 7*w^3 - 5*w^2 - 10*w + 4],
[491, 491, w^3 - w^2 - 3*w - 3],
[491, 491, -w^3 + 2*w^2 + 2*w - 6],
[491, 491, -w^5 + 4*w^4 + w^3 - 12*w^2 + 4],
[491, 491, 3*w^4 - 5*w^3 - 12*w^2 + 11*w + 13],
[503, 503, -w^3 + 4*w^2 + w - 8],
[503, 503, -w^5 + 4*w^4 - w^3 - 10*w^2 + 7*w + 6],
[547, 547, w^5 - 3*w^4 - w^3 + 7*w^2 - 3*w - 4],
[547, 547, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - w - 3],
[587, 587, w^5 - 5*w^4 + 3*w^3 + 14*w^2 - 11*w - 11],
[587, 587, w^5 - 7*w^3 - 3*w^2 + 11*w + 9],
[601, 601, w^5 - 5*w^4 + 2*w^3 + 15*w^2 - 7*w - 9],
[601, 601, w^5 - 8*w^3 - w^2 + 14*w + 3],
[631, 631, w^5 - 2*w^4 - 4*w^3 + 4*w^2 + 6*w - 1],
[631, 631, -w^5 + 3*w^4 + 2*w^3 - 6*w^2 - 3*w - 3],
[631, 631, -w^5 + 2*w^4 + 4*w^3 - 8*w^2 - 2*w + 8],
[631, 631, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + w + 4],
[643, 643, w^5 - 4*w^4 + 12*w^2 - 4*w - 10],
[643, 643, -3*w^4 + 6*w^3 + 10*w^2 - 14*w - 10],
[643, 643, -3*w^4 + 6*w^3 + 10*w^2 - 12*w - 11],
[643, 643, -w^5 + 3*w^4 + w^3 - 9*w^2 + 5*w + 6],
[659, 659, -w^5 + w^4 + 6*w^3 - 2*w^2 - 10*w + 1],
[659, 659, 2*w^4 - 5*w^3 - 4*w^2 + 9*w + 1],
[659, 659, -2*w^4 + 4*w^3 + 5*w^2 - 6*w - 2],
[659, 659, 2*w^4 - 4*w^3 - 5*w^2 + 8*w + 1],
[659, 659, -2*w^4 + 3*w^3 + 7*w^2 - 6*w - 3],
[659, 659, -w^5 + 4*w^4 - 12*w^2 + 3*w + 5],
[673, 673, 2*w^4 - 3*w^3 - 10*w^2 + 9*w + 10],
[673, 673, -2*w^4 + 5*w^3 + 7*w^2 - 12*w - 8],
[701, 701, -4*w^4 + 8*w^3 + 12*w^2 - 17*w - 8],
[701, 701, 2*w^5 - 5*w^4 - 5*w^3 + 13*w^2 + 2*w - 6],
[701, 701, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + 5*w - 5],
[701, 701, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 - 3*w + 1],
[701, 701, w^5 - w^4 - 7*w^3 + 6*w^2 + 7*w - 4],
[727, 727, w^3 - w^2 - 2*w - 3],
[727, 727, w^5 - 3*w^4 - 2*w^3 + 11*w^2 - 3*w - 10],
[727, 727, -w^5 + 2*w^4 + 4*w^3 - 3*w^2 - 6*w - 6],
[727, 727, -w^3 + 2*w^2 + w - 5],
[729, 3, -3],
[743, 743, -2*w^5 + 3*w^4 + 9*w^3 - 5*w^2 - 12*w - 5],
[743, 743, 2*w^5 - 7*w^4 - w^3 + 20*w^2 - 7*w - 12],
[757, 757, -2*w^4 + 5*w^3 + 7*w^2 - 12*w - 11],
[757, 757, -w^5 + 4*w^4 + 2*w^3 - 15*w^2 - w + 10],
[757, 757, -w^5 + w^4 + 8*w^3 - 5*w^2 - 14*w + 1],
[757, 757, -2*w^4 + 3*w^3 + 10*w^2 - 9*w - 13],
[769, 769, w^5 - 3*w^4 + 5*w^2 - 4*w - 1],
[769, 769, -w^5 + 2*w^4 + 2*w^3 - 3*w^2 + w - 2],
[797, 797, -w^5 + 5*w^4 - w^3 - 17*w^2 + 5*w + 12],
[797, 797, w^5 - 9*w^3 + 17*w + 3],
[811, 811, -w^4 + w^3 + 4*w^2 - 6],
[811, 811, w^4 - 3*w^3 - w^2 + 7*w + 2],
[827, 827, w^5 + w^4 - 9*w^3 - 5*w^2 + 16*w + 6],
[827, 827, -2*w^5 + 4*w^4 + 8*w^3 - 11*w^2 - 10*w + 2],
[827, 827, 2*w^5 - 6*w^4 - 4*w^3 + 17*w^2 + 2*w - 9],
[827, 827, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 7*w + 6],
[839, 839, -2*w^5 + 6*w^4 + 4*w^3 - 18*w^2 - w + 10],
[839, 839, -w^5 + 4*w^4 - w^3 - 12*w^2 + 7*w + 12],
[839, 839, -w^5 + w^4 + 5*w^3 + w^2 - 9*w - 9],
[839, 839, -2*w^5 + 4*w^4 + 8*w^3 - 10*w^2 - 11*w + 1],
[841, 29, -3*w^4 + 6*w^3 + 11*w^2 - 14*w - 12],
[841, 29, -2*w^4 + 4*w^3 + 9*w^2 - 11*w - 11],
[853, 853, -w^4 + 6*w^2 + w - 4],
[853, 853, -w^5 + 5*w^4 - 2*w^3 - 17*w^2 + 9*w + 15],
[853, 853, -w^5 + 8*w^3 + 3*w^2 - 16*w - 9],
[853, 853, w^4 - 4*w^3 + 9*w - 2],
[881, 881, -w - 3],
[881, 881, w - 4],
[883, 883, w^2 + w - 5],
[883, 883, w^5 - 7*w^3 - 5*w^2 + 13*w + 11],
[883, 883, w^5 - 5*w^4 + 3*w^3 + 16*w^2 - 13*w - 13],
[883, 883, w^2 - 3*w - 3],
[911, 911, -w^5 + 2*w^4 + 4*w^3 - 4*w^2 - 4*w - 3],
[911, 911, -w^4 + 3*w^3 + 2*w^2 - 5*w - 5],
[937, 937, -2*w^5 + 4*w^4 + 6*w^3 - 10*w^2 - 2*w + 3],
[937, 937, -w^5 + 4*w^4 - 10*w^2 + 2*w + 1],
[937, 937, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 4],
[937, 937, -2*w^4 + 5*w^3 + 4*w^2 - 7*w - 2],
[953, 953, w^5 - 4*w^4 + 2*w^3 + 9*w^2 - 9*w - 4],
[953, 953, w^5 - w^4 - 4*w^3 - w^2 + 4*w + 5],
[967, 967, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 7*w + 2],
[967, 967, w^4 - 3*w^3 - 3*w^2 + 10*w + 3],
[967, 967, -w^4 + w^3 + 6*w^2 - w - 8],
[967, 967, -2*w^5 + 5*w^4 + 6*w^3 - 15*w^2 - 5*w + 9],
[1021, 1021, w^5 - 3*w^4 - 2*w^3 + 11*w^2 - 3*w - 8],
[1021, 1021, -w^5 + 2*w^4 + 4*w^3 - 3*w^2 - 6*w - 4],
[1049, 1049, -w^5 + w^4 + 5*w^3 - 2*w^2 - 4*w - 2],
[1049, 1049, 2*w^5 - 6*w^4 - 4*w^3 + 17*w^2 + 2*w - 8],
[1049, 1049, 2*w^5 - 4*w^4 - 8*w^3 + 11*w^2 + 10*w - 3],
[1049, 1049, w^5 - 4*w^4 + w^3 + 9*w^2 - 6*w - 3],
[1051, 1051, -w^5 + 7*w^3 + 3*w^2 - 10*w - 9],
[1051, 1051, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w + 3],
[1051, 1051, -w^5 + 4*w^4 + w^3 - 12*w^2 - w + 6],
[1051, 1051, w^5 - 5*w^4 + 3*w^3 + 14*w^2 - 12*w - 10],
[1063, 1063, 2*w^5 - 5*w^4 - 6*w^3 + 16*w^2 + 3*w - 9],
[1063, 1063, -2*w^5 + 5*w^4 + 6*w^3 - 12*w^2 - 7*w + 1],
[1091, 1091, -w^4 + 2*w^3 + 4*w^2 - 3*w - 8],
[1091, 1091, -w^4 + 2*w^3 + 4*w^2 - 7*w - 6],
[1093, 1093, 3*w^5 - 10*w^4 - 4*w^3 + 30*w^2 - 5*w - 16],
[1093, 1093, 3*w^4 - 5*w^3 - 11*w^2 + 10*w + 9],
[1093, 1093, 3*w^4 - 7*w^3 - 8*w^2 + 15*w + 6],
[1093, 1093, 2*w^5 - 2*w^4 - 11*w^3 + 4*w^2 + 14*w + 4],
[1217, 1217, -w^5 + 4*w^4 + w^3 - 12*w^2 - w + 5],
[1217, 1217, -w^5 + 6*w^4 - 4*w^3 - 19*w^2 + 13*w + 15],
[1217, 1217, -w^5 - w^4 + 10*w^3 + 5*w^2 - 18*w - 10],
[1217, 1217, w^5 - w^4 - 7*w^3 + 5*w^2 + 11*w - 4],
[1231, 1231, -w^4 + w^3 + 7*w^2 - 3*w - 8],
[1231, 1231, w^4 - 3*w^3 - 4*w^2 + 10*w + 4],
[1259, 1259, -w^5 - w^4 + 9*w^3 + 9*w^2 - 17*w - 19],
[1259, 1259, w^5 - 2*w^4 - 4*w^3 + 8*w^2 + 4*w - 10],
[1259, 1259, -w^5 + 7*w^4 - 6*w^3 - 22*w^2 + 17*w + 16],
[1259, 1259, 2*w^5 - 6*w^4 - 3*w^3 + 17*w^2 - w - 13],
[1289, 1289, -w^5 + 3*w^4 + 2*w^3 - 11*w^2 + 3*w + 9],
[1289, 1289, w^5 - 8*w^3 - w^2 + 14*w + 2],
[1289, 1289, w^5 - 5*w^4 + 2*w^3 + 15*w^2 - 7*w - 8],
[1289, 1289, w^5 - 2*w^4 - 4*w^3 + 3*w^2 + 6*w + 5],
[1301, 1301, w^5 - w^4 - 7*w^3 + 2*w^2 + 14*w + 2],
[1301, 1301, w^5 - 4*w^4 - 2*w^3 + 13*w^2 + w - 4],
[1301, 1301, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 - w + 9],
[1301, 1301, 2*w^5 - 4*w^4 - 8*w^3 + 11*w^2 + 9*w - 1],
[1301, 1301, 2*w^5 - w^4 - 15*w^3 + 26*w + 10],
[1301, 1301, w^5 - 8*w^3 + 13*w + 2],
[1303, 1303, -w^5 + 3*w^4 + w^3 - 5*w^2 + w - 2],
[1303, 1303, w^5 + w^4 - 8*w^3 - 7*w^2 + 13*w + 9],
[1303, 1303, -2*w^5 + 2*w^4 + 13*w^3 - 5*w^2 - 21*w - 6],
[1303, 1303, w^5 - 2*w^4 - 3*w^3 + 6*w^2 - w - 3],
[1373, 1373, 2*w^5 - 5*w^4 - 5*w^3 + 13*w^2 + 3*w - 4],
[1373, 1373, -2*w^5 + 7*w^4 + 3*w^3 - 21*w^2 - 2*w + 12],
[1373, 1373, -w^5 + w^4 + 6*w^3 - 2*w^2 - 11*w - 1],
[1373, 1373, w^5 - 4*w^4 + 12*w^2 - 2*w - 8],
[1373, 1373, -2*w^5 + 3*w^4 + 11*w^3 - 10*w^2 - 17*w + 3],
[1373, 1373, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 5],
[1399, 1399, -2*w^5 + 5*w^4 + 6*w^3 - 15*w^2 - 5*w + 8],
[1399, 1399, -2*w^5 + 4*w^4 + 8*w^3 - 11*w^2 - 8*w + 1],
[1399, 1399, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 + 8],
[1399, 1399, w^5 - 3*w^4 + 6*w^2 - 6*w - 2],
[1427, 1427, -2*w^4 + 3*w^3 + 10*w^2 - 9*w - 11],
[1427, 1427, -2*w^4 + 5*w^3 + 7*w^2 - 12*w - 9],
[1429, 1429, 3*w^5 - 7*w^4 - 10*w^3 + 19*w^2 + 10*w - 2],
[1429, 1429, -w^4 + 7*w^2 - 2*w - 3],
[1429, 1429, -w^5 + 2*w^4 + 4*w^3 - 2*w^2 - 7*w - 4],
[1429, 1429, -3*w^5 + 8*w^4 + 8*w^3 - 23*w^2 - 5*w + 13],
[1471, 1471, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 8*w + 3],
[1471, 1471, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 2*w - 5],
[1483, 1483, w^4 - 3*w^3 + 4*w - 4],
[1483, 1483, w^4 - w^3 - 3*w^2 + w - 2],
[1499, 1499, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 - 2*w + 4],
[1499, 1499, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 7*w + 3],
[1511, 1511, -2*w^5 + 5*w^4 + 6*w^3 - 15*w^2 - 4*w + 5],
[1511, 1511, -2*w^5 + 5*w^4 + 5*w^3 - 15*w^2 + 8],
[1511, 1511, 3*w^5 - 10*w^4 - 4*w^3 + 31*w^2 - 5*w - 20],
[1511, 1511, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 6*w - 5],
[1553, 1553, w^5 - 3*w^4 - 2*w^3 + 7*w^2 + 3*w + 2],
[1553, 1553, -w^5 + 2*w^4 + 4*w^3 - 7*w^2 - 4*w + 8],
[1567, 1567, -3*w^4 + 7*w^3 + 7*w^2 - 15*w - 1],
[1567, 1567, -2*w^5 + 6*w^4 + 4*w^3 - 19*w^2 - w + 13],
[1583, 1583, -3*w^5 + 6*w^4 + 13*w^3 - 18*w^2 - 15*w + 6],
[1583, 1583, 2*w^5 - 3*w^4 - 10*w^3 + 9*w^2 + 10*w - 3],
[1583, 1583, 2*w^5 - 4*w^4 - 7*w^3 + 12*w^2 + 5*w - 9],
[1583, 1583, -w^5 + w^4 + 5*w^3 + 2*w^2 - 10*w - 9],
[1597, 1597, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 7*w - 4],
[1597, 1597, w^5 - 4*w^4 + 12*w^2 - 5*w - 9],
[1597, 1597, -w^5 + w^4 + 6*w^3 - 2*w^2 - 8*w - 5],
[1597, 1597, 2*w^5 - 5*w^4 - 7*w^3 + 16*w^2 + 6*w - 8],
[1609, 1609, w^5 - 7*w^3 - 4*w^2 + 13*w + 10],
[1609, 1609, w^4 - 4*w^3 + 8*w - 1],
[1609, 1609, -w^4 + 6*w^2 - 4],
[1609, 1609, w^5 - 5*w^4 + 3*w^3 + 15*w^2 - 11*w - 13],
[1667, 1667, -w^5 + 9*w^3 - 17*w - 4],
[1667, 1667, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 4*w - 8],
[1667, 1667, 2*w^5 - 5*w^4 - 5*w^3 + 12*w^2 + 4*w - 3],
[1667, 1667, w^5 - 5*w^4 + w^3 + 17*w^2 - 5*w - 13],
[1681, 41, 3*w^2 - 3*w - 5],
[1681, 41, -3*w^4 + 6*w^3 + 12*w^2 - 15*w - 16],
[1693, 1693, w^5 - w^4 - 6*w^3 + w^2 + 8*w + 6],
[1693, 1693, -w^5 + 4*w^4 - 12*w^2 + 7*w + 3],
[1693, 1693, w^5 - 6*w^4 + 4*w^3 + 22*w^2 - 17*w - 20],
[1693, 1693, w^5 - 4*w^4 + w^3 + 13*w^2 - 9*w - 13],
[1709, 1709, -w^5 - w^4 + 10*w^3 + 6*w^2 - 20*w - 13],
[1709, 1709, -w^5 + 4*w^4 - w^3 - 11*w^2 + 7*w + 10],
[1709, 1709, 2*w^4 - 4*w^3 - 6*w^2 + 5*w + 8],
[1709, 1709, 3*w^5 - 7*w^4 - 10*w^3 + 19*w^2 + 10*w - 3],
[1709, 1709, -w^5 + w^4 + 5*w^3 - 7*w - 8],
[1709, 1709, w^4 - 3*w^3 - w^2 + 3*w - 1],
[1723, 1723, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 5*w + 2],
[1723, 1723, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + w - 4],
[1847, 1847, -2*w^4 + 4*w^3 + 7*w^2 - 8*w - 3],
[1847, 1847, -2*w^5 + 6*w^4 + 5*w^3 - 19*w^2 - 2*w + 13],
[1849, 43, w^4 - 2*w^3 - 6*w^2 + 7*w + 8],
[1849, 43, -3*w^4 + 6*w^3 + 11*w^2 - 14*w - 13],
[1861, 1861, -w^5 + 9*w^3 - 2*w^2 - 14*w + 2],
[1861, 1861, -w^5 + 4*w^4 - w^3 - 10*w^2 + 3*w + 4],
[1861, 1861, -w^5 + w^4 + 5*w^3 - w^2 - 9*w + 1],
[1861, 1861, -w^5 + 5*w^4 - w^3 - 15*w^2 + 4*w + 6],
[1877, 1877, -2*w^5 + 7*w^4 + 2*w^3 - 21*w^2 + 4*w + 15],
[1877, 1877, 2*w^5 - 6*w^4 - 3*w^3 + 17*w^2 - 3*w - 10],
[1877, 1877, w^5 - w^4 - 6*w^3 + w^2 + 12*w + 3],
[1877, 1877, -w^5 + 4*w^4 - 13*w^2 + 3*w + 10],
[1877, 1877, -2*w^5 + 4*w^4 + 7*w^3 - 8*w^2 - 8*w - 3],
[1877, 1877, 2*w^5 - 3*w^4 - 10*w^3 + 7*w^2 + 14*w + 5],
[1889, 1889, -w^5 + 5*w^4 - w^3 - 15*w^2 + 4*w + 4],
[1889, 1889, -2*w^5 + 3*w^4 + 9*w^3 - 7*w^2 - 10*w - 3],
[1931, 1931, -2*w^5 + 5*w^4 + 7*w^3 - 17*w^2 - 4*w + 10],
[1931, 1931, -2*w^5 + 6*w^4 + 3*w^3 - 19*w^2 + 5*w + 12],
[1933, 1933, w^5 - 2*w^4 - 2*w^3 + 5*w^2 - 4*w - 4],
[1933, 1933, w^5 - 6*w^4 + 3*w^3 + 22*w^2 - 13*w - 16],
[1933, 1933, -2*w^5 + 8*w^4 - 23*w^2 + 7*w + 13],
[1933, 1933, -w^5 + 3*w^4 - 3*w^2 + 3*w - 6],
[1987, 1987, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 5*w - 1],
[1987, 1987, 2*w^5 - 5*w^4 - 4*w^3 + 10*w^2 + 2]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 - 2*x - 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, -2*e + 4, 2*e - 2, 2*e, -2*e + 6, 2*e - 10, 3*e - 2, e + 2, -6, 2*e - 1, e + 4, 4*e - 10, -3*e + 10, -4*e + 2, -2*e + 4, 2*e - 2, 3*e - 4, -2*e - 6, -6*e + 10, 10, 10, 2*e - 2, e + 2, -6, 4, -3*e + 18, 20, e + 14, -2*e - 10, -6*e - 4, -4*e + 12, 3*e + 6, -8, -4*e - 2, -2*e - 14, -4*e - 2, -8, -3*e + 16, -e - 4, 5*e - 18, 3*e + 4, -5*e + 2, 12, -2*e + 4, -4*e - 12, 12, -4*e, -4*e, -2*e - 2, 14, 8*e + 2, e + 6, -2*e - 16, -3*e + 14, 4*e - 8, -6*e + 4, 8*e - 6, -8, 8*e + 10, 22, -6*e + 20, 2*e + 22, 4*e + 2, 2*e - 14, -10*e + 4, 6*e - 22, -28, 5*e + 12, 9*e + 2, 8, -2*e - 8, 20, 3*e + 6, -4*e + 14, -7*e + 20, -8*e + 6, 4*e + 26, -5*e - 2, -6*e + 28, -6*e - 2, -10*e + 4, -4*e + 6, 12*e - 6, -2*e + 34, e, -6*e + 32, 4, -2*e + 14, -12*e + 20, -2*e - 32, -2*e - 26, -15*e + 22, 12*e + 4, 2*e - 20, -3*e + 20, 2*e - 16, 10*e - 30, 7*e + 22, -5*e - 12, -6*e - 20, -2*e - 18, 9*e - 6, 2*e - 2, -4*e + 4, 4*e - 8, -3*e - 4, 12*e - 16, -16*e + 24, 3*e + 24, 21*e - 20, -10*e + 32, 8*e - 28, 28, 2*e - 32, -6*e + 12, -7*e + 18, -3*e + 24, -3*e - 14, 16*e - 22, 4*e + 10, -5*e + 30, 12*e - 2, 6*e - 26, 8*e + 14, -17*e + 24, -8*e + 20, 8*e - 24, 13*e - 26, -4*e + 16, 2*e - 12, -5*e + 14, 14*e - 26, -12*e - 6, -3*e + 30, 13*e - 26, -4*e - 10, 13*e - 30, 6*e - 10, 12*e + 8, 23*e - 18, 9*e + 26, 4*e + 24, 8*e - 20, -8, -3*e + 18, -12*e, -18, -4*e + 48, -20*e + 10, -6*e - 30, 5*e - 32, -14*e + 14, -26, -2*e + 26, 4*e + 6, -22*e + 26, -14*e + 26, 2*e - 8, -14*e + 16, -7*e - 32, 18*e - 14, -3*e + 46, -15*e + 20, -2, -4*e + 42, 14*e - 14, -6*e - 8, 8*e + 28, -6*e - 16, 10*e - 28, 9*e - 36, -20*e + 24, 10*e - 38, -4*e + 2, 15*e - 16, 10*e - 18, 10*e + 20, -24*e + 14, -4*e - 40, 10*e - 18, -23*e + 22, 10*e - 4, -6*e - 24, 6*e - 4, 4*e, 3*e - 8, 2*e - 30, 14*e - 2, -10*e + 34, -12*e + 10, -11*e + 44, -2*e - 22, 8*e - 18, -2*e + 2, -4*e - 6, 4*e - 56, -13*e - 12, -17*e - 8, 14*e + 12, -2*e, 20*e - 26, 12*e - 6, 28*e - 30, -12*e + 22, 6*e - 50, -8*e - 12, -2*e + 54, -19*e + 32, -16*e + 38, 17*e, -15*e + 2, e + 42, -16*e + 20, 6*e - 20, -16*e + 10, 24*e - 10, -4*e + 58, 24*e - 22, -2*e + 10, 4*e + 18, 12*e - 48, -6*e - 50, 20*e - 32, -16*e + 32, 16, 5*e - 58, 2*e - 4, -18*e - 6, -12*e + 42, 4*e + 50, 6*e - 36, 12*e + 12, -2*e - 20, 16*e - 28, 18*e - 20, -12*e - 32, -12*e + 24, -20*e + 8, -5*e - 24, -32, -11*e - 22, 28*e - 14, 16*e - 12, 3*e + 36, 12*e - 48, 9*e + 40, 2*e - 16, -18*e + 16, -12*e + 32, -2*e - 28, -6*e + 16, -8*e + 26, 5*e + 18, 4*e - 46, 30*e - 28, 16*e - 46, -8*e - 6, -9*e + 20, 10*e + 20, -21*e + 4, -14*e + 24, -4*e + 28, -11*e + 4, -6*e + 10, -4*e - 50, 26*e - 42, 4*e + 26, -2*e + 14, 12*e + 2, 18*e + 12, 20*e - 38, -6*e - 54, 12*e + 22, -13*e + 12, -14*e - 8, 2*e + 6, 2*e - 26, 30*e - 30, -12*e - 10, 6*e - 46, -21*e + 4, 16*e - 14, -15*e + 8, -14*e + 26, e - 60, -5*e + 6, -16*e + 10, -15*e + 8, 2*e - 78, 10*e + 24, -28, 22*e - 4, -12*e - 22, 20, -12*e - 38, 20*e + 6, 8*e - 52, 8*e + 24];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {7, 7, 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;