/*
  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, -9, 2, 11, -5, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {43,43,w^5 - 7*w^3 - w^2 + 10*w + 3}>;

primesArray := [
[13, 13, 3*w^5 - 2*w^4 - 17*w^3 + 10*w^2 + 16*w - 3],
[13, 13, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 3],
[13, 13, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 13*w],
[13, 13, -w^2 + 3],
[41, 41, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 1],
[41, 41, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 4],
[41, 41, w^5 - 6*w^3 + w^2 + 7*w - 2],
[41, 41, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 20*w - 1],
[43, 43, 3*w^5 - 2*w^4 - 17*w^3 + 11*w^2 + 16*w - 6],
[43, 43, -2*w^5 + w^4 + 12*w^3 - 6*w^2 - 14*w + 5],
[49, 7, 2*w^5 - w^4 - 12*w^3 + 5*w^2 + 13*w - 4],
[64, 2, -2],
[71, 71, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 1],
[71, 71, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 15*w],
[83, 83, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 26*w - 1],
[83, 83, -4*w^5 + 2*w^4 + 23*w^3 - 9*w^2 - 23*w],
[97, 97, -w^5 + 7*w^3 + w^2 - 11*w - 3],
[97, 97, -2*w^5 + w^4 + 11*w^3 - 6*w^2 - 9*w + 3],
[113, 113, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 27*w + 2],
[113, 113, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 6],
[113, 113, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 12*w + 3],
[113, 113, 3*w^5 - w^4 - 18*w^3 + 4*w^2 + 20*w],
[127, 127, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w],
[127, 127, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 2],
[127, 127, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 5],
[127, 127, -8*w^5 + 5*w^4 + 46*w^3 - 23*w^2 - 45*w + 4],
[139, 139, w^3 + w^2 - 4*w - 1],
[139, 139, 5*w^5 - 2*w^4 - 29*w^3 + 9*w^2 + 29*w + 1],
[139, 139, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 5],
[139, 139, w^5 - 6*w^3 + 5*w + 2],
[167, 167, w^4 - 4*w^2 + 1],
[167, 167, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 21*w - 1],
[169, 13, -3*w^5 + w^4 + 18*w^3 - 5*w^2 - 20*w],
[181, 181, -w^4 + w^3 + 5*w^2 - 3*w - 3],
[181, 181, -5*w^5 + 3*w^4 + 29*w^3 - 15*w^2 - 30*w + 6],
[181, 181, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 4],
[181, 181, w^3 + w^2 - 4*w - 2],
[197, 197, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 2],
[197, 197, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 26*w + 4],
[197, 197, -2*w^5 + 2*w^4 + 12*w^3 - 11*w^2 - 12*w + 5],
[197, 197, 2*w^5 - w^4 - 12*w^3 + 6*w^2 + 13*w - 7],
[197, 197, w^5 - 6*w^3 - w^2 + 7*w + 1],
[197, 197, -4*w^5 + 3*w^4 + 24*w^3 - 14*w^2 - 25*w + 2],
[211, 211, 2*w^5 - w^4 - 13*w^3 + 5*w^2 + 18*w - 1],
[211, 211, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 4],
[223, 223, 4*w^5 - w^4 - 23*w^3 + 5*w^2 + 24*w - 1],
[223, 223, -3*w^5 + w^4 + 17*w^3 - 4*w^2 - 15*w - 4],
[223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 15*w^2 + 11*w - 5],
[223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 5],
[239, 239, -w^5 + w^4 + 6*w^3 - 6*w^2 - 7*w + 4],
[239, 239, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 12*w + 2],
[251, 251, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 4],
[251, 251, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 28*w - 1],
[251, 251, 6*w^5 - 4*w^4 - 35*w^3 + 19*w^2 + 37*w - 7],
[251, 251, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 8],
[281, 281, -6*w^5 + 4*w^4 + 35*w^3 - 18*w^2 - 35*w + 3],
[281, 281, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 16*w - 10],
[281, 281, -3*w^5 + w^4 + 17*w^3 - 5*w^2 - 18*w + 3],
[281, 281, 5*w^5 - 3*w^4 - 30*w^3 + 14*w^2 + 33*w - 5],
[293, 293, -w^5 + w^4 + 6*w^3 - 5*w^2 - 8*w + 3],
[307, 307, 5*w^5 - 3*w^4 - 30*w^3 + 13*w^2 + 32*w],
[307, 307, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 20*w + 5],
[307, 307, 3*w^5 - 2*w^4 - 17*w^3 + 8*w^2 + 15*w],
[307, 307, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 11*w - 1],
[307, 307, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 10],
[307, 307, -4*w^5 + 2*w^4 + 23*w^3 - 8*w^2 - 22*w - 3],
[337, 337, -w^5 + w^4 + 5*w^3 - 4*w^2 - 3*w + 3],
[337, 337, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 29*w],
[337, 337, 5*w^5 - 3*w^4 - 28*w^3 + 15*w^2 + 27*w - 5],
[337, 337, -4*w^5 + 4*w^4 + 22*w^3 - 20*w^2 - 19*w + 10],
[349, 349, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 45*w + 9],
[349, 349, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3],
[419, 419, 5*w^5 - 3*w^4 - 29*w^3 + 13*w^2 + 28*w - 1],
[419, 419, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 35*w + 10],
[421, 421, -6*w^5 + 3*w^4 + 34*w^3 - 15*w^2 - 32*w + 3],
[421, 421, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 28*w],
[421, 421, w^5 - 5*w^3 - w^2 + 2*w + 3],
[421, 421, -6*w^5 + 3*w^4 + 35*w^3 - 14*w^2 - 37*w + 2],
[433, 433, -4*w^5 + 4*w^4 + 22*w^3 - 19*w^2 - 19*w + 6],
[433, 433, 7*w^5 - 4*w^4 - 40*w^3 + 19*w^2 + 40*w - 6],
[449, 449, 2*w^4 - w^3 - 10*w^2 + 4*w + 6],
[449, 449, -4*w^5 + 4*w^4 + 23*w^3 - 20*w^2 - 21*w + 10],
[461, 461, -6*w^5 + 3*w^4 + 36*w^3 - 15*w^2 - 40*w + 4],
[461, 461, -7*w^5 + 5*w^4 + 40*w^3 - 23*w^2 - 39*w + 6],
[461, 461, 4*w^5 - 2*w^4 - 22*w^3 + 9*w^2 + 19*w + 1],
[461, 461, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 32*w - 1],
[463, 463, 4*w^5 - 2*w^4 - 23*w^3 + 10*w^2 + 23*w],
[463, 463, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 34*w + 7],
[491, 491, 7*w^5 - 5*w^4 - 41*w^3 + 24*w^2 + 41*w - 8],
[491, 491, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 15*w - 2],
[491, 491, -w^5 + 6*w^3 + w^2 - 7*w],
[491, 491, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 3],
[491, 491, 5*w^5 - 3*w^4 - 29*w^3 + 16*w^2 + 30*w - 9],
[491, 491, -3*w^5 + 2*w^4 + 18*w^3 - 8*w^2 - 20*w - 2],
[503, 503, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 7],
[503, 503, 6*w^5 - 3*w^4 - 35*w^3 + 14*w^2 + 37*w - 3],
[547, 547, 2*w^5 - 12*w^3 - w^2 + 15*w + 3],
[547, 547, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 6*w + 3],
[547, 547, 7*w^5 - 4*w^4 - 41*w^3 + 19*w^2 + 41*w - 3],
[547, 547, 2*w^5 - w^4 - 13*w^3 + 6*w^2 + 17*w - 3],
[547, 547, -4*w^5 + 3*w^4 + 23*w^3 - 13*w^2 - 21*w],
[547, 547, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 7],
[587, 587, 2*w^4 - w^3 - 10*w^2 + 4*w + 7],
[587, 587, 5*w^5 - 2*w^4 - 29*w^3 + 10*w^2 + 31*w - 2],
[601, 601, w^3 + 2*w^2 - 4*w - 4],
[601, 601, w^5 - 7*w^3 - 2*w^2 + 11*w + 5],
[617, 617, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 9],
[617, 617, -2*w^5 + 13*w^3 - 16*w],
[643, 643, -2*w^5 + 2*w^4 + 12*w^3 - 8*w^2 - 13*w],
[643, 643, -3*w^5 + w^4 + 18*w^3 - 3*w^2 - 21*w - 3],
[659, 659, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 33*w + 2],
[659, 659, w^5 - 6*w^3 + 2*w^2 + 7*w - 4],
[673, 673, -2*w^5 + 2*w^4 + 10*w^3 - 11*w^2 - 5*w + 6],
[673, 673, 6*w^5 - 4*w^4 - 34*w^3 + 20*w^2 + 33*w - 9],
[673, 673, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 16*w + 5],
[673, 673, 7*w^5 - 4*w^4 - 40*w^3 + 20*w^2 + 38*w - 6],
[673, 673, -3*w^5 + 2*w^4 + 16*w^3 - 11*w^2 - 12*w + 6],
[673, 673, w^5 + w^4 - 6*w^3 - 6*w^2 + 8*w + 5],
[701, 701, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 9],
[701, 701, 4*w^5 - 3*w^4 - 22*w^3 + 15*w^2 + 20*w - 6],
[727, 727, -2*w^5 + 2*w^4 + 11*w^3 - 8*w^2 - 9*w - 2],
[727, 727, -3*w^5 + 3*w^4 + 18*w^3 - 15*w^2 - 20*w + 9],
[729, 3, -3],
[743, 743, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 22*w],
[743, 743, -5*w^5 + 2*w^4 + 29*w^3 - 9*w^2 - 31*w - 1],
[743, 743, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 31*w + 6],
[743, 743, -3*w^5 + 3*w^4 + 17*w^3 - 14*w^2 - 17*w + 3],
[757, 757, -6*w^5 + 3*w^4 + 35*w^3 - 13*w^2 - 37*w + 2],
[757, 757, 3*w^5 - 3*w^4 - 17*w^3 + 13*w^2 + 16*w - 1],
[757, 757, -8*w^5 + 5*w^4 + 47*w^3 - 25*w^2 - 48*w + 10],
[757, 757, 4*w^5 - 2*w^4 - 25*w^3 + 10*w^2 + 29*w - 4],
[797, 797, -6*w^5 + 4*w^4 + 35*w^3 - 19*w^2 - 34*w + 3],
[797, 797, -w^5 + w^4 + 7*w^3 - 6*w^2 - 10*w + 3],
[811, 811, 5*w^5 - 4*w^4 - 28*w^3 + 18*w^2 + 25*w - 4],
[811, 811, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 17*w + 7],
[827, 827, w^5 - w^4 - 5*w^3 + 5*w^2 + w - 4],
[827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 3],
[827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 2],
[827, 827, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 4],
[839, 839, -2*w^5 + 2*w^4 + 10*w^3 - 10*w^2 - 6*w + 7],
[839, 839, -6*w^5 + 4*w^4 + 34*w^3 - 20*w^2 - 32*w + 5],
[841, 29, -4*w^5 + 3*w^4 + 24*w^3 - 15*w^2 - 25*w + 7],
[841, 29, 5*w^5 - 3*w^4 - 30*w^3 + 15*w^2 + 32*w - 6],
[841, 29, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 33*w - 2],
[853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 29*w + 9],
[853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 30*w + 7],
[881, 881, -3*w^5 + 3*w^4 + 16*w^3 - 14*w^2 - 13*w + 3],
[881, 881, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 25*w + 2],
[881, 881, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 9],
[881, 881, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 34*w + 2],
[883, 883, 3*w^5 - w^4 - 17*w^3 + 6*w^2 + 16*w - 5],
[883, 883, 7*w^5 - 5*w^4 - 40*w^3 + 24*w^2 + 37*w - 8],
[911, 911, w^5 - 8*w^3 + 14*w],
[911, 911, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 17*w - 1],
[911, 911, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 44*w + 9],
[911, 911, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 11*w],
[937, 937, 5*w^5 - 2*w^4 - 30*w^3 + 9*w^2 + 32*w - 1],
[937, 937, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 15*w - 4],
[967, 967, -w^5 + 7*w^3 - 9*w + 2],
[967, 967, 6*w^5 - 4*w^4 - 35*w^3 + 20*w^2 + 36*w - 11]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 8*x^3 + 11*x^2 - 42*x - 89;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/2*e^3 - 3*e^2 + e + 35/2, -2*e^3 - 9*e^2 + 9*e + 50, 3/2*e^3 + 7*e^2 - 7*e - 91/2, e, -2*e^3 - 11*e^2 + 5*e + 64, 1/2*e^3 + 3/2*e^2 - 7/2*e - 10, -2*e^3 - 17/2*e^2 + 21/2*e + 95/2, 5/2*e^3 + 27/2*e^2 - 19/2*e - 85, 5/2*e^3 + 14*e^2 - 8*e - 169/2, 1, 3*e^3 + 29/2*e^2 - 21/2*e - 167/2, 2*e^3 + 17/2*e^2 - 21/2*e - 101/2, -9/2*e^3 - 22*e^2 + 18*e + 265/2, e^3 + 3*e^2 - 8*e - 16, 3/2*e^3 + 15/2*e^2 - 7/2*e - 44, -5*e^3 - 45/2*e^2 + 49/2*e + 259/2, 9/2*e^3 + 22*e^2 - 19*e - 267/2, -11/2*e^3 - 26*e^2 + 21*e + 297/2, -3/2*e^3 - 7*e^2 + 7*e + 77/2, 3*e^3 + 27/2*e^2 - 25/2*e - 149/2, -e^3 - 5/2*e^2 + 11/2*e + 5/2, 15/2*e^3 + 34*e^2 - 31*e - 385/2, -7*e^3 - 65/2*e^2 + 65/2*e + 389/2, 1/2*e^3 + 9/2*e^2 - 3/2*e - 47, 13/2*e^3 + 29*e^2 - 28*e - 327/2, 9/2*e^3 + 18*e^2 - 24*e - 201/2, -13/2*e^3 - 31*e^2 + 27*e + 353/2, -5/2*e^3 - 31/2*e^2 + 15/2*e + 103, 10*e^3 + 48*e^2 - 43*e - 287, -1/2*e^3 + e^2 + 10*e - 1/2, -5/2*e^3 - 21/2*e^2 + 21/2*e + 49, 3/2*e^3 + 3*e^2 - 16*e - 43/2, e^3 + 7*e^2 + e - 45, -3/2*e^3 - 11/2*e^2 + 9/2*e + 23, -2*e^3 - 17/2*e^2 + 23/2*e + 81/2, -5/2*e^3 - 25/2*e^2 + 29/2*e + 85, 7*e^3 + 30*e^2 - 30*e - 152, -6*e^3 - 27*e^2 + 26*e + 145, -3*e^2 - e + 31, 7*e^3 + 32*e^2 - 34*e - 187, -7*e^3 - 73/2*e^2 + 47/2*e + 415/2, -2*e^3 - 19/2*e^2 + 21/2*e + 125/2, -9/2*e^3 - 21*e^2 + 16*e + 219/2, -6*e^3 - 51/2*e^2 + 63/2*e + 291/2, 8*e^3 + 75/2*e^2 - 71/2*e - 437/2, 10*e^3 + 48*e^2 - 44*e - 291, -5/2*e^2 - 17/2*e + 29/2, 6*e^3 + 29*e^2 - 25*e - 164, -3*e^3 - 33/2*e^2 + 19/2*e + 201/2, 7*e^3 + 69/2*e^2 - 45/2*e - 403/2, -9/2*e^2 - 11/2*e + 63/2, 9/2*e^3 + 21*e^2 - 26*e - 269/2, 4*e^3 + 33/2*e^2 - 39/2*e - 201/2, -3*e^3 - 31/2*e^2 + 25/2*e + 197/2, -5*e^3 - 27*e^2 + 16*e + 160, -11/2*e^3 - 41/2*e^2 + 59/2*e + 102, -5*e^3 - 41/2*e^2 + 51/2*e + 189/2, -27/2*e^3 - 125/2*e^2 + 117/2*e + 363, -3/2*e^3 - 7*e^2 + 7*e + 81/2, 5*e^3 + 41/2*e^2 - 55/2*e - 227/2, -15/2*e^3 - 57/2*e^2 + 87/2*e + 156, -11/2*e^3 - 20*e^2 + 33*e + 197/2, 2*e^3 + 27/2*e^2 + 1/2*e - 135/2, -7*e^3 - 63/2*e^2 + 71/2*e + 357/2, -2*e^3 - 23/2*e^2 + 15/2*e + 173/2, -1/2*e^2 + 3/2*e + 3/2, -e^3 - 9/2*e^2 - 3/2*e + 27/2, -17/2*e^3 - 41*e^2 + 39*e + 505/2, -25/2*e^3 - 107/2*e^2 + 121/2*e + 305, 6*e^3 + 57/2*e^2 - 41/2*e - 335/2, 19/2*e^3 + 46*e^2 - 34*e - 537/2, 7*e^3 + 69/2*e^2 - 61/2*e - 417/2, -4*e^3 - 13*e^2 + 32*e + 68, -5/2*e^3 - 21/2*e^2 + 11/2*e + 53, 5*e^3 + 51/2*e^2 - 31/2*e - 311/2, -1/2*e^3 - 11/2*e^2 - 11/2*e + 30, 7*e^3 + 61/2*e^2 - 77/2*e - 383/2, 1/2*e^3 + 7/2*e^2 + 7/2*e - 11, -6*e^3 - 61/2*e^2 + 43/2*e + 395/2, -7/2*e^3 - 22*e^2 + 14*e + 319/2, 5*e^3 + 24*e^2 - 14*e - 148, -4*e^3 - 23/2*e^2 + 49/2*e + 61/2, e^3 + 21/2*e^2 + 21/2*e - 121/2, 6*e^3 + 55/2*e^2 - 71/2*e - 325/2, -6*e^3 - 27*e^2 + 33*e + 164, 3/2*e^3 + 13/2*e^2 - 37/2*e - 40, -4*e^3 - 25/2*e^2 + 57/2*e + 139/2, 4*e^3 + 45/2*e^2 - 5/2*e - 245/2, -7*e^3 - 33*e^2 + 24*e + 184, 13/2*e^3 + 23*e^2 - 40*e - 255/2, 6*e^3 + 31*e^2 - 29*e - 198, -4*e^3 - 25*e^2 + 11*e + 161, 3*e^3 + 43/2*e^2 - 7/2*e - 279/2, 9/2*e^3 + 13*e^2 - 38*e - 145/2, 33/2*e^3 + 71*e^2 - 76*e - 777/2, 17/2*e^3 + 35*e^2 - 50*e - 423/2, 29/2*e^3 + 71*e^2 - 55*e - 837/2, -e^3 - 19/2*e^2 - 25/2*e + 107/2, -11*e^3 - 46*e^2 + 49*e + 228, -22*e^3 - 104*e^2 + 85*e + 588, -e^3 - 9/2*e^2 + 9/2*e + 19/2, 15*e^3 + 127/2*e^2 - 159/2*e - 749/2, -13*e^3 - 117/2*e^2 + 107/2*e + 623/2, 7/2*e^3 + 33/2*e^2 - 45/2*e - 112, -23/2*e^3 - 99/2*e^2 + 103/2*e + 264, 13/2*e^3 + 27*e^2 - 40*e - 351/2, 9/2*e^3 + 22*e^2 - 20*e - 221/2, 9/2*e^3 + 35/2*e^2 - 41/2*e - 98, -5/2*e^3 - 5*e^2 + 25*e + 49/2, -5/2*e^3 - 13/2*e^2 + 27/2*e + 12, -4*e^3 - 19*e^2 + 26*e + 116, -5*e^3 - 20*e^2 + 33*e + 117, -3*e^3 - 13*e^2 + 20*e + 102, 9/2*e^3 + 33/2*e^2 - 57/2*e - 90, 8*e^3 + 40*e^2 - 40*e - 247, -9/2*e^3 - 19/2*e^2 + 73/2*e + 31, -23/2*e^3 - 56*e^2 + 44*e + 661/2, -29/2*e^3 - 69*e^2 + 53*e + 805/2, 4*e^3 + 21/2*e^2 - 47/2*e - 55/2, -3/2*e^3 - 5*e^2 + 6*e + 45/2, -7*e^3 - 33*e^2 + 43*e + 205, 13*e^3 + 121/2*e^2 - 111/2*e - 655/2, -13/2*e^3 - 30*e^2 + 25*e + 325/2, -14*e^3 - 69*e^2 + 66*e + 434, 13/2*e^3 + 75/2*e^2 - 31/2*e - 225, -e^3 - 11/2*e^2 + 5/2*e + 31/2, -5*e^3 - 59/2*e^2 + 35/2*e + 381/2, -6*e^3 - 32*e^2 + 14*e + 194, -5/2*e^3 - 7*e^2 + 18*e + 17/2, -5/2*e^3 - 17/2*e^2 + 15/2*e + 27, 7*e^3 + 24*e^2 - 45*e - 141, 2*e^3 + 14*e^2 + 6*e - 77, -7*e^3 - 35*e^2 + 28*e + 217, -5/2*e^3 - 11*e^2 + 15*e + 197/2, 15*e^3 + 127/2*e^2 - 147/2*e - 707/2, 17*e^3 + 80*e^2 - 78*e - 502, 15/2*e^3 + 65/2*e^2 - 55/2*e - 188, -11/2*e^3 - 55/2*e^2 + 67/2*e + 187, 9*e^3 + 71/2*e^2 - 101/2*e - 423/2, -13/2*e^3 - 53/2*e^2 + 67/2*e + 143, 6*e^3 + 21*e^2 - 39*e - 119, 10*e^3 + 95/2*e^2 - 95/2*e - 625/2, -8*e^3 - 73/2*e^2 + 65/2*e + 401/2, -6*e^3 - 37*e^2 + 19*e + 250, e^3 - 9/2*e^2 - 15/2*e + 149/2, -5*e^3 - 41/2*e^2 + 51/2*e + 169/2, 6*e^3 + 43/2*e^2 - 67/2*e - 215/2, 29/2*e^3 + 60*e^2 - 70*e - 641/2, -4*e^3 - 13*e^2 + 29*e + 98, 23/2*e^3 + 53*e^2 - 49*e - 639/2, -7*e^3 - 33*e^2 + 26*e + 170, 13*e^3 + 65*e^2 - 42*e - 369, -3*e^3 - 7*e^2 + 21*e + 37, 5*e^3 + 21*e^2 - 28*e - 103, -9/2*e^3 - 22*e^2 + 13*e + 237/2, 2*e^3 + 17*e^2 + 5*e - 117, -1/2*e^3 + 7*e + 31/2, -1/2*e^3 - 10*e^2 - 8*e + 123/2, -23/2*e^3 - 111/2*e^2 + 119/2*e + 358, -21/2*e^3 - 50*e^2 + 38*e + 521/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {43,43,w^5 - 7*w^3 - w^2 + 10*w + 3}>] := -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;