/*
  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, -2, 4, 5, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {64, 2, -2}>;

primesArray := [
[13, 13, -w^5 + 3*w^4 + 2*w^3 - 9*w^2 + w + 4],
[13, 13, -w^2 + w + 2],
[27, 3, 2*w^5 - 4*w^4 - 7*w^3 + 9*w^2 + 4*w - 2],
[27, 3, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 - w + 5],
[29, 29, w^3 - 2*w^2 - 2*w + 3],
[29, 29, 2*w^5 - 4*w^4 - 7*w^3 + 8*w^2 + 4*w - 2],
[43, 43, -w^5 + 3*w^4 + w^3 - 6*w^2 + 3*w + 1],
[43, 43, -w^4 + w^3 + 5*w^2 - 4],
[49, 7, w^5 - 4*w^4 + 11*w^2 - 3*w - 4],
[64, 2, -2],
[71, 71, 2*w^5 - 6*w^4 - 4*w^3 + 17*w^2 - w - 6],
[71, 71, 2*w^4 - 4*w^3 - 6*w^2 + 7*w + 2],
[71, 71, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - w],
[71, 71, -2*w^5 + 5*w^4 + 6*w^3 - 14*w^2 - 3*w + 5],
[83, 83, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 5*w + 5],
[83, 83, 3*w^5 - 6*w^4 - 10*w^3 + 12*w^2 + 5*w - 2],
[83, 83, -2*w^5 + 5*w^4 + 5*w^3 - 11*w^2 - 3*w + 3],
[83, 83, 3*w^5 - 7*w^4 - 8*w^3 + 15*w^2 + 2*w - 4],
[97, 97, -3*w^5 + 6*w^4 + 10*w^3 - 12*w^2 - 5*w + 3],
[97, 97, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 3],
[97, 97, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 5*w - 1],
[97, 97, -3*w^5 + 7*w^4 + 8*w^3 - 15*w^2 - 2*w + 3],
[113, 113, -3*w^5 + 6*w^4 + 11*w^3 - 14*w^2 - 7*w + 3],
[113, 113, 2*w^5 - 5*w^4 - 5*w^3 + 12*w^2 - 6],
[113, 113, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - 1],
[113, 113, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 2*w + 5],
[125, 5, 3*w^5 - 6*w^4 - 10*w^3 + 11*w^2 + 6*w - 1],
[125, 5, -2*w^5 + 4*w^4 + 6*w^3 - 6*w^2 - 3*w - 1],
[127, 127, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 2*w + 5],
[127, 127, 2*w^5 - 3*w^4 - 9*w^3 + 6*w^2 + 7*w - 2],
[127, 127, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 4*w + 4],
[127, 127, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - w + 3],
[139, 139, 3*w^5 - 7*w^4 - 9*w^3 + 16*w^2 + 6*w - 4],
[139, 139, -w^5 + w^4 + 5*w^3 - 6*w - 1],
[167, 167, -2*w^5 + 5*w^4 + 6*w^3 - 12*w^2 - 5*w + 4],
[167, 167, w^5 - 4*w^4 - w^3 + 13*w^2 - 2*w - 4],
[169, 13, -w^5 + 4*w^4 - 11*w^2 + 3*w + 5],
[169, 13, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 3*w - 3],
[181, 181, 2*w^5 - 4*w^4 - 6*w^3 + 6*w^2 + 2*w + 1],
[197, 197, -w^4 + 3*w^3 + 2*w^2 - 6*w - 1],
[197, 197, -3*w^5 + 6*w^4 + 11*w^3 - 14*w^2 - 6*w + 4],
[211, 211, 2*w^5 - 5*w^4 - 6*w^3 + 12*w^2 + 6*w - 1],
[211, 211, -2*w^5 + 4*w^4 + 8*w^3 - 9*w^2 - 8*w + 2],
[211, 211, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 3*w + 3],
[211, 211, -2*w^4 + 5*w^3 + 5*w^2 - 9*w - 3],
[223, 223, -w^5 + 8*w^3 + w^2 - 11*w + 1],
[223, 223, 2*w^5 - 5*w^4 - 5*w^3 + 12*w^2 + w - 1],
[239, 239, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 2*w + 3],
[239, 239, 2*w^5 - 5*w^4 - 4*w^3 + 9*w^2 + w],
[239, 239, -3*w^5 + 7*w^4 + 10*w^3 - 18*w^2 - 7*w + 5],
[239, 239, w^5 - w^4 - 6*w^3 + 2*w^2 + 5*w - 2],
[251, 251, w^5 - w^4 - 6*w^3 + 3*w^2 + 6*w - 2],
[251, 251, -w^5 + 4*w^4 - 12*w^2 + 3*w + 6],
[281, 281, -5*w^5 + 11*w^4 + 17*w^3 - 27*w^2 - 10*w + 9],
[281, 281, 2*w^5 - 5*w^4 - 5*w^3 + 10*w^2 + 3*w + 1],
[293, 293, -2*w^5 + 4*w^4 + 7*w^3 - 7*w^2 - 7*w + 2],
[293, 293, -w^4 + 2*w^3 + 2*w^2 - 3*w + 3],
[293, 293, -2*w^5 + 7*w^4 + w^3 - 19*w^2 + 7*w + 9],
[293, 293, -2*w^5 + 3*w^4 + 9*w^3 - 5*w^2 - 7*w + 2],
[307, 307, 2*w^2 - 3*w - 4],
[307, 307, -w^5 + 4*w^4 - 13*w^2 + 6*w + 6],
[337, 337, 4*w^5 - 9*w^4 - 12*w^3 + 20*w^2 + 6*w - 5],
[337, 337, -2*w^5 + 3*w^4 + 10*w^3 - 6*w^2 - 12*w + 2],
[349, 349, 4*w^5 - 9*w^4 - 11*w^3 + 19*w^2 + 2*w - 4],
[349, 349, 5*w^5 - 11*w^4 - 15*w^3 + 24*w^2 + 5*w - 7],
[379, 379, -2*w^5 + 5*w^4 + 4*w^3 - 10*w^2 + w + 5],
[379, 379, -2*w^5 + 5*w^4 + 6*w^3 - 12*w^2 - 5*w + 5],
[379, 379, -4*w^5 + 8*w^4 + 14*w^3 - 17*w^2 - 10*w + 4],
[379, 379, -5*w^5 + 12*w^4 + 13*w^3 - 27*w^2 - 2*w + 7],
[421, 421, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + w - 3],
[421, 421, 3*w^5 - 9*w^4 - 6*w^3 + 25*w^2 - w - 9],
[433, 433, -5*w^5 + 10*w^4 + 18*w^3 - 21*w^2 - 13*w + 4],
[433, 433, 5*w^5 - 12*w^4 - 15*w^3 + 30*w^2 + 7*w - 9],
[433, 433, -3*w^5 + 8*w^4 + 8*w^3 - 22*w^2 - w + 8],
[433, 433, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 9*w - 3],
[449, 449, 3*w^5 - 9*w^4 - 6*w^3 + 24*w^2 + w - 9],
[449, 449, 3*w^5 - 6*w^4 - 11*w^3 + 15*w^2 + 6*w - 5],
[449, 449, 2*w^5 - 3*w^4 - 8*w^3 + 3*w^2 + 7*w],
[449, 449, 4*w^5 - 8*w^4 - 14*w^3 + 17*w^2 + 8*w - 2],
[449, 449, -3*w^4 + 5*w^3 + 12*w^2 - 9*w - 6],
[449, 449, -3*w^5 + 8*w^4 + 7*w^3 - 20*w^2 - 2*w + 6],
[461, 461, 2*w^4 - 4*w^3 - 7*w^2 + 7*w + 2],
[461, 461, -4*w^5 + 10*w^4 + 11*w^3 - 24*w^2 - 6*w + 8],
[463, 463, 4*w^5 - 8*w^4 - 14*w^3 + 18*w^2 + 6*w - 5],
[463, 463, w^5 - w^4 - 5*w^3 + w^2 + 4*w - 3],
[491, 491, -3*w^5 + 7*w^4 + 8*w^3 - 15*w^2 + 5],
[491, 491, -2*w^5 + 6*w^4 + 3*w^3 - 14*w^2 + 3],
[547, 547, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 10*w - 5],
[547, 547, w^5 - 4*w^4 + 12*w^2 - 2*w - 5],
[547, 547, -3*w^5 + 5*w^4 + 13*w^3 - 11*w^2 - 11*w + 3],
[547, 547, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + w + 6],
[547, 547, -2*w^5 + 6*w^4 + 4*w^3 - 15*w^2 - w + 4],
[547, 547, 4*w^5 - 8*w^4 - 15*w^3 + 18*w^2 + 12*w - 6],
[587, 587, -5*w^5 + 12*w^4 + 14*w^3 - 28*w^2 - 6*w + 7],
[587, 587, 5*w^5 - 12*w^4 - 14*w^3 + 27*w^2 + 6*w - 6],
[601, 601, -3*w^5 + 9*w^4 + 5*w^3 - 24*w^2 + 5*w + 10],
[601, 601, -3*w^5 + 8*w^4 + 8*w^3 - 23*w^2 - w + 9],
[601, 601, -w^5 + 8*w^3 + 2*w^2 - 11*w - 1],
[601, 601, 2*w^5 - 3*w^4 - 10*w^3 + 8*w^2 + 10*w - 4],
[617, 617, w^5 - w^4 - 7*w^3 + 4*w^2 + 10*w - 3],
[617, 617, -5*w^5 + 11*w^4 + 16*w^3 - 26*w^2 - 7*w + 8],
[617, 617, 3*w^5 - 7*w^4 - 9*w^3 + 15*w^2 + 7*w - 2],
[617, 617, -2*w^5 + 2*w^4 + 11*w^3 - w^2 - 13*w - 2],
[631, 631, 2*w^4 - 4*w^3 - 7*w^2 + 8*w + 2],
[631, 631, w^5 - w^4 - 5*w^3 - w^2 + 4*w + 5],
[631, 631, 4*w^5 - 9*w^4 - 11*w^3 + 19*w^2 + 2*w - 6],
[631, 631, -3*w^5 + 7*w^4 + 9*w^3 - 15*w^2 - 7*w + 3],
[643, 643, -2*w^5 + 3*w^4 + 9*w^3 - 6*w^2 - 8*w + 4],
[643, 643, 3*w^5 - 8*w^4 - 7*w^3 + 20*w^2 + 2*w - 5],
[659, 659, 3*w^5 - 9*w^4 - 5*w^3 + 23*w^2 - 3*w - 10],
[659, 659, -2*w^4 + 3*w^3 + 7*w^2 - 3*w - 5],
[673, 673, 3*w^5 - 6*w^4 - 11*w^3 + 14*w^2 + 8*w - 2],
[673, 673, 3*w^5 - 8*w^4 - 6*w^3 + 19*w^2 - 3*w - 8],
[673, 673, 3*w^5 - 6*w^4 - 10*w^3 + 13*w^2 + 3*w - 2],
[673, 673, 2*w^5 - 3*w^4 - 10*w^3 + 6*w^2 + 11*w - 2],
[701, 701, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 6*w - 4],
[701, 701, 2*w^5 - 5*w^4 - 5*w^3 + 11*w^2 + 2*w - 6],
[701, 701, -w^5 + w^4 + 5*w^3 - 5*w - 5],
[701, 701, -w^5 + w^4 + 6*w^3 - 2*w^2 - 9*w + 2],
[727, 727, -w^5 + 5*w^4 - w^3 - 16*w^2 + 4*w + 8],
[727, 727, w^5 - w^4 - 4*w^3 - 2*w^2 + 4*w + 4],
[727, 727, -3*w^5 + 9*w^4 + 5*w^3 - 22*w^2 + 2*w + 7],
[727, 727, -w^5 + 2*w^4 + 3*w^3 - 5*w^2 + 5],
[727, 727, -5*w^5 + 11*w^4 + 16*w^3 - 24*w^2 - 10*w + 6],
[727, 727, 4*w^5 - 8*w^4 - 14*w^3 + 17*w^2 + 7*w - 4],
[743, 743, 5*w^5 - 11*w^4 - 16*w^3 + 26*w^2 + 6*w - 8],
[743, 743, -4*w^5 + 7*w^4 + 16*w^3 - 14*w^2 - 12*w + 1],
[743, 743, -4*w^5 + 9*w^4 + 14*w^3 - 23*w^2 - 10*w + 7],
[743, 743, w^5 - 3*w^4 + 6*w^2 - 6*w - 2],
[757, 757, -w^5 + 7*w^3 + 3*w^2 - 7*w - 6],
[757, 757, 5*w^5 - 11*w^4 - 16*w^3 + 24*w^2 + 11*w - 5],
[769, 769, -3*w^5 + 4*w^4 + 14*w^3 - 6*w^2 - 12*w + 1],
[769, 769, 4*w^5 - 7*w^4 - 17*w^3 + 16*w^2 + 14*w - 5],
[797, 797, -5*w^5 + 12*w^4 + 14*w^3 - 27*w^2 - 7*w + 7],
[797, 797, 2*w^5 - 7*w^4 - w^3 + 17*w^2 - 4*w - 7],
[811, 811, 5*w^5 - 13*w^4 - 13*w^3 + 33*w^2 + 3*w - 10],
[811, 811, -4*w^5 + 10*w^4 + 9*w^3 - 22*w^2 + w + 6],
[841, 29, 3*w^5 - 6*w^4 - 9*w^3 + 10*w^2 + 4*w - 1],
[841, 29, -w^5 + w^4 + 4*w^3 + 3*w^2 - 3*w - 7],
[853, 853, -4*w^5 + 9*w^4 + 12*w^3 - 21*w^2 - 4*w + 5],
[853, 853, -3*w^5 + 6*w^4 + 10*w^3 - 13*w^2 - 4*w + 2],
[853, 853, -w^5 + w^4 + 5*w^3 + w^2 - 4*w - 6],
[853, 853, -3*w^5 + 7*w^4 + 8*w^3 - 16*w^2 - w + 7],
[853, 853, -4*w^5 + 10*w^4 + 10*w^3 - 24*w^2 - w + 10],
[853, 853, -w^4 + 3*w^3 + 3*w^2 - 8*w - 2],
[883, 883, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 4*w + 6],
[883, 883, -w^5 + 5*w^4 - 3*w^3 - 12*w^2 + 9*w + 3],
[883, 883, 5*w^5 - 12*w^4 - 14*w^3 + 27*w^2 + 7*w - 6],
[883, 883, 2*w^5 - 6*w^4 - 3*w^3 + 16*w^2 - 3*w - 9],
[937, 937, 5*w^5 - 11*w^4 - 16*w^3 + 26*w^2 + 8*w - 7],
[937, 937, -w^5 + 2*w^4 + 4*w^3 - 3*w^2 - 6*w],
[937, 937, 2*w^5 - 6*w^4 - 4*w^3 + 18*w^2 - 3*w - 10],
[937, 937, 2*w^5 - 7*w^4 - 3*w^3 + 22*w^2 - 2*w - 10],
[953, 953, 4*w^5 - 7*w^4 - 16*w^3 + 14*w^2 + 13*w - 3],
[953, 953, 5*w^5 - 11*w^4 - 15*w^3 + 23*w^2 + 7*w - 3],
[953, 953, w^5 + w^4 - 9*w^3 - 6*w^2 + 12*w + 3],
[953, 953, -3*w^5 + 7*w^4 + 10*w^3 - 17*w^2 - 9*w + 5],
[953, 953, w^5 - 4*w^4 + w^3 + 8*w^2 - 5*w + 1],
[953, 953, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 4*w + 7],
[967, 967, 4*w^5 - 10*w^4 - 11*w^3 + 26*w^2 + w - 9],
[967, 967, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 3*w - 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - x^7 - 73*x^6 + 105*x^5 + 1154*x^4 - 3116*x^3 + 1968*x^2 - 240*x - 32;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, e, -73819/4112952*e^7 + 5393/685492*e^6 + 2720603/2056476*e^5 - 1176551/1028238*e^4 - 89670745/4112952*e^3 + 22427675/514119*e^2 - 3892525/514119*e - 869167/514119, -73819/4112952*e^7 + 5393/685492*e^6 + 2720603/2056476*e^5 - 1176551/1028238*e^4 - 89670745/4112952*e^3 + 22427675/514119*e^2 - 3892525/514119*e - 869167/514119, 13871/4112952*e^7 - 2043/1370984*e^6 - 1065077/4112952*e^5 + 829225/4112952*e^4 + 4941689/1028238*e^3 - 3916054/514119*e^2 - 4259488/514119*e + 3861734/514119, 13871/4112952*e^7 - 2043/1370984*e^6 - 1065077/4112952*e^5 + 829225/4112952*e^4 + 4941689/1028238*e^3 - 3916054/514119*e^2 - 4259488/514119*e + 3861734/514119, -172735/8225904*e^7 + 15607/2741968*e^6 + 12721777/8225904*e^5 - 9002813/8225904*e^4 - 26324519/1028238*e^3 + 24431785/514119*e^2 + 126335/1028238*e - 2536517/514119, -172735/8225904*e^7 + 15607/2741968*e^6 + 12721777/8225904*e^5 - 9002813/8225904*e^4 - 26324519/1028238*e^3 + 24431785/514119*e^2 + 126335/1028238*e - 2536517/514119, -93843/2741968*e^7 + 122043/2741968*e^6 + 6886707/2741968*e^5 - 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22766587/2056476*e^5 + 8036036/514119*e^4 + 725891507/4112952*e^3 - 480987989/1028238*e^2 + 135441239/514119*e - 2340547/514119, 25883/1370984*e^7 + 558/171373*e^6 - 471677/342746*e^5 + 357593/685492*e^4 + 30458755/1370984*e^3 - 6819658/171373*e^2 + 83857/342746*e + 2262600/171373, 622349/4112952*e^7 - 101323/685492*e^6 - 22766587/2056476*e^5 + 8036036/514119*e^4 + 725891507/4112952*e^3 - 480987989/1028238*e^2 + 135441239/514119*e - 2340547/514119, -99073/1370984*e^7 + 57303/685492*e^6 + 3591705/685492*e^5 - 2890327/342746*e^4 - 110791691/1370984*e^3 + 82391833/342746*e^2 - 32345089/171373*e + 2884387/171373, -269869/8225904*e^7 + 18127/2741968*e^6 + 19510405/8225904*e^5 - 13347413/8225904*e^4 - 152946629/4112952*e^3 + 156771247/2056476*e^2 - 17603192/514119*e - 6491513/514119, -99073/1370984*e^7 + 57303/685492*e^6 + 3591705/685492*e^5 - 2890327/342746*e^4 - 110791691/1370984*e^3 + 82391833/342746*e^2 - 32345089/171373*e + 2884387/171373, -269869/8225904*e^7 + 18127/2741968*e^6 + 19510405/8225904*e^5 - 13347413/8225904*e^4 - 152946629/4112952*e^3 + 156771247/2056476*e^2 - 17603192/514119*e - 6491513/514119, 90525/685492*e^7 - 6525/171373*e^6 - 1660594/171373*e^5 + 2382899/342746*e^4 + 109056909/685492*e^3 - 51210612/171373*e^2 + 2866683/171373*e + 5506006/171373, 176585/1370984*e^7 - 195025/1370984*e^6 - 12978529/1370984*e^5 + 19835265/1370984*e^4 + 104297033/685492*e^3 - 142049381/342746*e^2 + 40236386/171373*e - 2711876/171373, 176585/1370984*e^7 - 195025/1370984*e^6 - 12978529/1370984*e^5 + 19835265/1370984*e^4 + 104297033/685492*e^3 - 142049381/342746*e^2 + 40236386/171373*e - 2711876/171373, 90525/685492*e^7 - 6525/171373*e^6 - 1660594/171373*e^5 + 2382899/342746*e^4 + 109056909/685492*e^3 - 51210612/171373*e^2 + 2866683/171373*e + 5506006/171373, -152371/4112952*e^7 + 14563/685492*e^6 + 5659295/2056476*e^5 - 2854757/1028238*e^4 - 191585185/4112952*e^3 + 101514781/1028238*e^2 + 5991494/514119*e - 16964065/514119, -152371/4112952*e^7 + 14563/685492*e^6 + 5659295/2056476*e^5 - 2854757/1028238*e^4 - 191585185/4112952*e^3 + 101514781/1028238*e^2 + 5991494/514119*e - 16964065/514119, -397043/8225904*e^7 + 15583/2741968*e^6 + 28971833/8225904*e^5 - 16991485/8225904*e^4 - 116606225/2056476*e^3 + 53783273/514119*e^2 - 27010703/1028238*e - 608851/514119, -397043/8225904*e^7 + 15583/2741968*e^6 + 28971833/8225904*e^5 - 16991485/8225904*e^4 - 116606225/2056476*e^3 + 53783273/514119*e^2 - 27010703/1028238*e - 608851/514119, 5987/2056476*e^7 + 695/1370984*e^6 - 784021/4112952*e^5 + 384947/4112952*e^4 + 7145599/4112952*e^3 - 13104965/2056476*e^2 + 13793746/514119*e - 15132428/514119, -42517/4112952*e^7 + 18939/685492*e^6 + 1544387/2056476*e^5 - 1130521/514119*e^4 - 46192099/4112952*e^3 + 46869097/1028238*e^2 - 29043541/514119*e + 13459103/514119, -42517/4112952*e^7 + 18939/685492*e^6 + 1544387/2056476*e^5 - 1130521/514119*e^4 - 46192099/4112952*e^3 + 46869097/1028238*e^2 - 29043541/514119*e + 13459103/514119, 5987/2056476*e^7 + 695/1370984*e^6 - 784021/4112952*e^5 + 384947/4112952*e^4 + 7145599/4112952*e^3 - 13104965/2056476*e^2 + 13793746/514119*e - 15132428/514119, 786667/4112952*e^7 - 160935/1370984*e^6 - 57636121/4112952*e^5 + 59913821/4112952*e^4 + 233319451/1028238*e^3 - 258469211/514119*e^2 + 86920522/514119*e - 5118776/514119, 580289/4112952*e^7 - 139681/1370984*e^6 - 42572003/4112952*e^5 + 49189039/4112952*e^4 + 172081871/1028238*e^3 - 405558581/1028238*e^2 + 82623419/514119*e + 12179288/514119, 580289/4112952*e^7 - 139681/1370984*e^6 - 42572003/4112952*e^5 + 49189039/4112952*e^4 + 172081871/1028238*e^3 - 405558581/1028238*e^2 + 82623419/514119*e + 12179288/514119, 786667/4112952*e^7 - 160935/1370984*e^6 - 57636121/4112952*e^5 + 59913821/4112952*e^4 + 233319451/1028238*e^3 - 258469211/514119*e^2 + 86920522/514119*e - 5118776/514119, 482769/1370984*e^7 - 335553/1370984*e^6 - 35293101/1370984*e^5 + 39955413/1370984*e^4 + 282886751/685492*e^3 - 166514282/171373*e^2 + 73715933/171373*e - 359346/171373, 400747/2056476*e^7 - 36437/342746*e^6 - 7340923/514119*e^5 + 14279761/1028238*e^4 + 476476063/2056476*e^3 - 512082185/1028238*e^2 + 75884030/514119*e + 8949314/514119, 482769/1370984*e^7 - 335553/1370984*e^6 - 35293101/1370984*e^5 + 39955413/1370984*e^4 + 282886751/685492*e^3 - 166514282/171373*e^2 + 73715933/171373*e - 359346/171373, 50096/514119*e^7 - 44673/685492*e^6 - 14624591/2056476*e^5 + 16401301/2056476*e^4 + 232872647/2056476*e^3 - 138871556/514119*e^2 + 67581934/514119*e - 7420586/514119, 400747/2056476*e^7 - 36437/342746*e^6 - 7340923/514119*e^5 + 14279761/1028238*e^4 + 476476063/2056476*e^3 - 512082185/1028238*e^2 + 75884030/514119*e + 8949314/514119, 50096/514119*e^7 - 44673/685492*e^6 - 14624591/2056476*e^5 + 16401301/2056476*e^4 + 232872647/2056476*e^3 - 138871556/514119*e^2 + 67581934/514119*e - 7420586/514119, -56863/4112952*e^7 + 4471/1370984*e^6 + 4109857/4112952*e^5 - 3121949/4112952*e^4 - 7723787/514119*e^3 + 17487182/514119*e^2 - 19878763/514119*e + 3235544/514119, -22923/342746*e^7 - 9357/685492*e^6 + 3371051/685492*e^5 - 759481/685492*e^4 - 56611869/685492*e^3 + 19166188/171373*e^2 + 13225757/171373*e - 6008068/171373, -22923/342746*e^7 - 9357/685492*e^6 + 3371051/685492*e^5 - 759481/685492*e^4 - 56611869/685492*e^3 + 19166188/171373*e^2 + 13225757/171373*e - 6008068/171373, -56863/4112952*e^7 + 4471/1370984*e^6 + 4109857/4112952*e^5 - 3121949/4112952*e^4 - 7723787/514119*e^3 + 17487182/514119*e^2 - 19878763/514119*e + 3235544/514119, -24451/342746*e^7 + 82109/685492*e^6 + 3599601/685492*e^5 - 7406311/685492*e^4 - 57608499/685492*e^3 + 45999284/171373*e^2 - 32021531/171373*e + 5503212/171373, -24451/342746*e^7 + 82109/685492*e^6 + 3599601/685492*e^5 - 7406311/685492*e^4 - 57608499/685492*e^3 + 45999284/171373*e^2 - 32021531/171373*e + 5503212/171373, -1139675/8225904*e^7 + 98597/2741968*e^6 + 82909703/8225904*e^5 - 59409535/8225904*e^4 - 661783717/4112952*e^3 + 656242841/2056476*e^2 - 50969275/514119*e + 3505919/514119, -1139675/8225904*e^7 + 98597/2741968*e^6 + 82909703/8225904*e^5 - 59409535/8225904*e^4 - 661783717/4112952*e^3 + 656242841/2056476*e^2 - 50969275/514119*e + 3505919/514119, 59401/1028238*e^7 + 83/685492*e^6 - 8670665/2056476*e^5 + 3692395/2056476*e^4 + 142039067/2056476*e^3 - 113941357/1028238*e^2 - 11637515/514119*e + 9379408/514119, 59401/1028238*e^7 + 83/685492*e^6 - 8670665/2056476*e^5 + 3692395/2056476*e^4 + 142039067/2056476*e^3 - 113941357/1028238*e^2 - 11637515/514119*e + 9379408/514119, -1205189/2741968*e^7 + 762493/2741968*e^6 + 88149597/2741968*e^5 - 94275725/2741968*e^4 - 709131021/1370984*e^3 + 809390829/685492*e^2 - 80252764/171373*e + 672809/171373, -1205189/2741968*e^7 + 762493/2741968*e^6 + 88149597/2741968*e^5 - 94275725/2741968*e^4 - 709131021/1370984*e^3 + 809390829/685492*e^2 - 80252764/171373*e + 672809/171373, 367001/2056476*e^7 - 176887/1370984*e^6 - 53378815/4112952*e^5 + 62901329/4112952*e^4 + 840927133/4112952*e^3 - 1042304669/2056476*e^2 + 145567996/514119*e + 6447664/514119, 1194877/4112952*e^7 - 77437/685492*e^6 - 43729571/2056476*e^5 + 8990299/514119*e^4 + 1419745963/4112952*e^3 - 356420729/514119*e^2 + 81134656/514119*e + 32664139/514119, 176513/2056476*e^7 - 12334/171373*e^6 - 3202013/514119*e^5 + 8311331/1028238*e^4 + 199285937/2056476*e^3 - 131393213/514119*e^2 + 93524821/514119*e - 9444944/514119, -146689/4112952*e^7 + 101417/1370984*e^6 + 10646683/4112952*e^5 - 26960999/4112952*e^4 - 40493947/1028238*e^3 + 163015741/1028238*e^2 - 70712737/514119*e - 8645776/514119, 105958/514119*e^7 - 31867/342746*e^6 - 7783222/514119*e^5 + 6803345/514119*e^4 + 127543297/514119*e^3 - 516821357/1028238*e^2 + 46373372/514119*e + 22714760/514119, 176513/2056476*e^7 - 12334/171373*e^6 - 3202013/514119*e^5 + 8311331/1028238*e^4 + 199285937/2056476*e^3 - 131393213/514119*e^2 + 93524821/514119*e - 9444944/514119, -146689/4112952*e^7 + 101417/1370984*e^6 + 10646683/4112952*e^5 - 26960999/4112952*e^4 - 40493947/1028238*e^3 + 163015741/1028238*e^2 - 70712737/514119*e - 8645776/514119, 105958/514119*e^7 - 31867/342746*e^6 - 7783222/514119*e^5 + 6803345/514119*e^4 + 127543297/514119*e^3 - 516821357/1028238*e^2 + 46373372/514119*e + 22714760/514119, -771931/4112952*e^7 + 33313/342746*e^6 + 28195697/2056476*e^5 - 6752293/514119*e^4 - 909261241/4112952*e^3 + 989352065/2056476*e^2 - 76273915/514119*e - 20266144/514119, -771931/4112952*e^7 + 33313/342746*e^6 + 28195697/2056476*e^5 - 6752293/514119*e^4 - 909261241/4112952*e^3 + 989352065/2056476*e^2 - 76273915/514119*e - 20266144/514119, 94813/8225904*e^7 - 148681/2741968*e^6 - 6493807/8225904*e^5 + 36012731/8225904*e^4 + 18409261/2056476*e^3 - 46301740/514119*e^2 + 136516777/1028238*e - 1941451/514119, 94813/8225904*e^7 - 148681/2741968*e^6 - 6493807/8225904*e^5 + 36012731/8225904*e^4 + 18409261/2056476*e^3 - 46301740/514119*e^2 + 136516777/1028238*e - 1941451/514119, -1832899/8225904*e^7 + 369391/2741968*e^6 + 133730305/8225904*e^5 - 140274005/8225904*e^4 - 533764039/2056476*e^3 + 612363239/1028238*e^2 - 272362081/1028238*e - 504671/514119, 380179/2056476*e^7 - 122059/1370984*e^6 - 55868963/4112952*e^5 + 50830141/4112952*e^4 + 915195509/4112952*e^3 - 947164321/2056476*e^2 + 48094574/514119*e + 22937480/514119, 380179/2056476*e^7 - 122059/1370984*e^6 - 55868963/4112952*e^5 + 50830141/4112952*e^4 + 915195509/4112952*e^3 - 947164321/2056476*e^2 + 48094574/514119*e + 22937480/514119, -1832899/8225904*e^7 + 369391/2741968*e^6 + 133730305/8225904*e^5 - 140274005/8225904*e^4 - 533764039/2056476*e^3 + 612363239/1028238*e^2 - 272362081/1028238*e - 504671/514119, -342979/4112952*e^7 + 99541/685492*e^6 + 12584945/2056476*e^5 - 13605635/1028238*e^4 - 397457785/4112952*e^3 + 169964177/514119*e^2 - 253944647/1028238*e + 6413108/514119, -46825/178824*e^7 + 3281/29804*e^6 + 1716275/89412*e^5 - 737585/44706*e^4 - 55920835/178824*e^3 + 28685737/44706*e^2 - 5776499/44706*e - 1586248/22353, 85079/2056476*e^7 - 99697/1370984*e^6 - 12525349/4112952*e^5 + 27553667/4112952*e^4 + 197903071/4112952*e^3 - 347610701/2056476*e^2 + 65023432/514119*e - 82640/514119, 85079/2056476*e^7 - 99697/1370984*e^6 - 12525349/4112952*e^5 + 27553667/4112952*e^4 + 197903071/4112952*e^3 - 347610701/2056476*e^2 + 65023432/514119*e - 82640/514119, -342979/4112952*e^7 + 99541/685492*e^6 + 12584945/2056476*e^5 - 13605635/1028238*e^4 - 397457785/4112952*e^3 + 169964177/514119*e^2 - 253944647/1028238*e + 6413108/514119, -46825/178824*e^7 + 3281/29804*e^6 + 1716275/89412*e^5 - 737585/44706*e^4 - 55920835/178824*e^3 + 28685737/44706*e^2 - 5776499/44706*e - 1586248/22353, -542177/2056476*e^7 + 44369/342746*e^6 + 19837975/1028238*e^5 - 9240214/514119*e^4 - 641527703/2056476*e^3 + 343427549/514119*e^2 - 104905651/514119*e - 21746800/514119, -542177/2056476*e^7 + 44369/342746*e^6 + 19837975/1028238*e^5 - 9240214/514119*e^4 - 641527703/2056476*e^3 + 343427549/514119*e^2 - 104905651/514119*e - 21746800/514119];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {64, 2, -2}>] := 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;