/*
  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![9, 6, -7, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 3, w}>;

primesArray := [
[4, 2, -1/3*w^3 + 2/3*w^2 + 4/3*w - 1],
[7, 7, 1/3*w^3 - 2/3*w^2 - 7/3*w + 1],
[7, 7, w - 2],
[7, 7, w - 1],
[9, 3, w],
[9, 3, -1/3*w^3 + 2/3*w^2 + 7/3*w - 2],
[23, 23, -1/3*w^3 + 2/3*w^2 + 1/3*w - 2],
[23, 23, -2/3*w^3 + 4/3*w^2 + 11/3*w - 4],
[31, 31, -2/3*w^3 + 1/3*w^2 + 14/3*w + 2],
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 7],
[41, 41, -1/3*w^3 + 5/3*w^2 + 4/3*w - 4],
[47, 47, 1/3*w^3 - 5/3*w^2 - 1/3*w + 5],
[47, 47, w^2 - w - 4],
[73, 73, -w^3 + 3*w^2 + 4*w - 8],
[73, 73, -2/3*w^3 + 7/3*w^2 - 1/3*w - 4],
[73, 73, -w^3 + w^2 + 7*w + 1],
[73, 73, 2/3*w^3 - 4/3*w^2 - 14/3*w + 5],
[79, 79, w^2 - 5],
[79, 79, -2/3*w^3 + 7/3*w^2 + 8/3*w - 6],
[97, 97, -2/3*w^3 + 7/3*w^2 + 5/3*w - 4],
[97, 97, 1/3*w^3 + 1/3*w^2 - 7/3*w - 5],
[103, 103, w^3 - 3*w^2 - 2*w + 5],
[103, 103, -4/3*w^3 + 11/3*w^2 + 13/3*w - 10],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 3],
[113, 113, w^3 - 2*w^2 - 5*w + 1],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 1],
[113, 113, -1/3*w^3 + 5/3*w^2 + 4/3*w - 10],
[127, 127, -2/3*w^3 + 1/3*w^2 + 14/3*w + 1],
[127, 127, -2/3*w^3 + 7/3*w^2 + 2/3*w - 6],
[137, 137, -1/3*w^3 + 5/3*w^2 - 2/3*w - 6],
[137, 137, 1/3*w^3 + 1/3*w^2 - 10/3*w - 1],
[151, 151, -w^3 + 2*w^2 + 3*w - 5],
[151, 151, 1/3*w^3 + 1/3*w^2 - 10/3*w - 7],
[169, 13, -2/3*w^3 + 7/3*w^2 + 5/3*w - 2],
[169, 13, 1/3*w^3 + 1/3*w^2 - 7/3*w - 7],
[191, 191, 1/3*w^3 - 2/3*w^2 + 2/3*w - 2],
[191, 191, 2*w - 5],
[191, 191, -w^3 + 2*w^2 + 6*w - 2],
[191, 191, 2/3*w^3 - 4/3*w^2 - 14/3*w - 1],
[239, 239, 1/3*w^3 - 2/3*w^2 - 7/3*w - 3],
[239, 239, w - 5],
[241, 241, 1/3*w^3 + 1/3*w^2 - 10/3*w],
[241, 241, 4/3*w^3 - 17/3*w^2 + 2/3*w + 9],
[241, 241, -1/3*w^3 + 5/3*w^2 - 2/3*w - 7],
[241, 241, -4/3*w^3 + 14/3*w^2 + 13/3*w - 14],
[257, 257, -w - 4],
[257, 257, -1/3*w^3 + 2/3*w^2 + 7/3*w - 6],
[263, 263, -4/3*w^3 + 11/3*w^2 + 16/3*w - 8],
[263, 263, 2/3*w^3 - 1/3*w^2 - 8/3*w - 3],
[281, 281, -1/3*w^3 - 1/3*w^2 + 10/3*w - 2],
[281, 281, -1/3*w^3 + 5/3*w^2 - 5/3*w - 5],
[281, 281, -1/3*w^3 + 5/3*w^2 - 2/3*w - 9],
[281, 281, -2/3*w^3 + 1/3*w^2 + 17/3*w],
[289, 17, -w^3 + 2*w^2 + 4*w - 4],
[289, 17, w^3 - 2*w^2 - 4*w + 2],
[311, 311, -2/3*w^3 + 10/3*w^2 - 4/3*w - 9],
[311, 311, -2/3*w^3 - 2/3*w^2 + 20/3*w + 5],
[313, 313, -1/3*w^3 + 5/3*w^2 - 2/3*w - 8],
[313, 313, -1/3*w^3 - 1/3*w^2 + 10/3*w - 1],
[313, 313, w^3 - 4*w^2 - 2*w + 10],
[313, 313, 1/3*w^3 + 4/3*w^2 - 10/3*w - 8],
[337, 337, -4/3*w^3 + 5/3*w^2 + 19/3*w + 4],
[337, 337, 5/3*w^3 - 13/3*w^2 - 17/3*w + 5],
[353, 353, -4/3*w^3 + 8/3*w^2 + 25/3*w - 3],
[353, 353, -1/3*w^3 - 1/3*w^2 + 16/3*w + 6],
[353, 353, -2/3*w^3 + 10/3*w^2 + 2/3*w - 13],
[353, 353, -5/3*w^3 + 7/3*w^2 + 29/3*w - 3],
[367, 367, -5/3*w^3 + 4/3*w^2 + 35/3*w + 4],
[367, 367, 2/3*w^3 - 7/3*w^2 - 8/3*w + 4],
[367, 367, -4/3*w^3 + 14/3*w^2 + 1/3*w - 8],
[367, 367, w^2 - 7],
[383, 383, 4/3*w^3 - 14/3*w^2 - 10/3*w + 15],
[383, 383, -5/3*w^3 + 13/3*w^2 + 20/3*w - 12],
[383, 383, 2*w^2 - 2*w - 13],
[383, 383, w^3 - w^2 - 8*w - 1],
[401, 401, -4/3*w^3 + 11/3*w^2 + 22/3*w - 11],
[401, 401, 5/3*w^3 - 10/3*w^2 - 29/3*w + 5],
[409, 409, -w^3 + 2*w^2 + 3*w - 2],
[409, 409, -1/3*w^3 + 8/3*w^2 - 2/3*w - 6],
[409, 409, 1/3*w^3 - 8/3*w^2 + 2/3*w + 12],
[409, 409, 4/3*w^3 - 8/3*w^2 - 19/3*w + 4],
[433, 433, -w^3 + 3*w^2 - 1],
[433, 433, -1/3*w^3 + 5/3*w^2 - 8/3*w - 5],
[433, 433, -w^3 + w^2 + 8*w - 2],
[433, 433, 5/3*w^3 - 7/3*w^2 - 32/3*w - 2],
[439, 439, w^2 - 10],
[439, 439, -2/3*w^3 + 4/3*w^2 + 17/3*w - 7],
[449, 449, -2/3*w^3 + 4/3*w^2 + 11/3*w - 8],
[449, 449, -4/3*w^3 + 5/3*w^2 + 22/3*w + 2],
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 6],
[457, 457, 2/3*w^3 - 7/3*w^2 - 14/3*w + 9],
[463, 463, w^3 - 3*w^2 - 6*w + 11],
[463, 463, -1/3*w^3 + 5/3*w^2 + 10/3*w - 4],
[479, 479, -1/3*w^3 + 5/3*w^2 - 2/3*w + 1],
[479, 479, 5/3*w^3 - 10/3*w^2 - 23/3*w + 9],
[487, 487, -2/3*w^3 + 7/3*w^2 + 2/3*w - 8],
[487, 487, -2/3*w^3 + 1/3*w^2 + 14/3*w - 1],
[503, 503, w^3 - 4*w^2 - 2*w + 16],
[503, 503, -1/3*w^3 + 5/3*w^2 - 2/3*w - 11],
[521, 521, -4/3*w^3 + 11/3*w^2 + 10/3*w - 7],
[521, 521, 4/3*w^3 - 5/3*w^2 - 22/3*w],
[529, 23, -1/3*w^3 + 2/3*w^2 + 4/3*w - 6],
[569, 569, -1/3*w^3 + 8/3*w^2 + 1/3*w - 10],
[569, 569, -2/3*w^3 + 10/3*w^2 + 5/3*w - 10],
[577, 577, -2/3*w^3 + 10/3*w^2 + 5/3*w - 12],
[577, 577, -1/3*w^3 + 8/3*w^2 + 1/3*w - 8],
[593, 593, w^3 - w^2 - 5*w - 5],
[593, 593, 4/3*w^3 - 11/3*w^2 - 13/3*w + 4],
[599, 599, 2*w^2 - 3*w - 7],
[599, 599, 2/3*w^3 - 1/3*w^2 - 8/3*w - 4],
[599, 599, -1/3*w^3 + 8/3*w^2 - 5/3*w - 9],
[599, 599, -4/3*w^3 + 11/3*w^2 + 16/3*w - 7],
[607, 607, 4/3*w^3 - 8/3*w^2 - 22/3*w + 9],
[607, 607, -2/3*w^3 + 4/3*w^2 + 2/3*w - 5],
[625, 5, -5],
[631, 631, -1/3*w^3 + 2/3*w^2 + 13/3*w - 5],
[631, 631, -2/3*w^3 + 4/3*w^2 + 17/3*w - 1],
[641, 641, -2/3*w^3 + 7/3*w^2 + 14/3*w - 7],
[641, 641, 2/3*w^3 - 7/3*w^2 - 14/3*w + 8],
[647, 647, -2/3*w^3 - 5/3*w^2 + 26/3*w + 8],
[647, 647, -1/3*w^3 + 8/3*w^2 + 1/3*w - 9],
[647, 647, -2/3*w^3 + 10/3*w^2 + 5/3*w - 11],
[647, 647, -2/3*w^3 + 13/3*w^2 - 10/3*w - 13],
[673, 673, -4/3*w^3 + 5/3*w^2 + 25/3*w - 2],
[673, 673, -w^3 + 3*w^2 + w - 7],
[727, 727, -2/3*w^3 + 7/3*w^2 + 2/3*w - 11],
[727, 727, -2/3*w^3 + 1/3*w^2 + 14/3*w - 4],
[743, 743, w^2 - w - 10],
[743, 743, 1/3*w^3 - 5/3*w^2 - 1/3*w - 1],
[769, 769, -w^3 + 2*w^2 + 7*w - 4],
[769, 769, 3*w - 2],
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 8],
[809, 809, -1/3*w^3 + 8/3*w^2 - 2/3*w - 10],
[823, 823, -w^3 + 2*w^2 + 7*w - 5],
[823, 823, 3*w - 1],
[839, 839, 4/3*w^3 - 2/3*w^2 - 25/3*w - 8],
[839, 839, 2/3*w^3 - 7/3*w^2 - 14/3*w + 12],
[857, 857, -1/3*w^3 + 5/3*w^2 - 5/3*w - 7],
[857, 857, -2/3*w^3 + 1/3*w^2 + 17/3*w - 2],
[863, 863, -2/3*w^3 + 7/3*w^2 - 7/3*w - 2],
[863, 863, 5/3*w^3 - 7/3*w^2 - 35/3*w + 1],
[881, 881, -2/3*w^3 + 1/3*w^2 + 14/3*w + 8],
[881, 881, -2/3*w^3 + 13/3*w^2 - 4/3*w - 18],
[881, 881, -5/3*w^3 + 7/3*w^2 + 35/3*w + 3],
[881, 881, -2/3*w^3 + 7/3*w^2 + 2/3*w + 1],
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 7],
[887, 887, -1/3*w^3 + 11/3*w^2 - 8/3*w - 18],
[911, 911, 5/3*w^3 - 4/3*w^2 - 29/3*w - 3],
[911, 911, 2*w^3 - 6*w^2 - 5*w + 13],
[929, 929, 2*w^3 - 4*w^2 - 9*w + 2],
[929, 929, 5/3*w^3 - 10/3*w^2 - 17/3*w],
[929, 929, -4/3*w^3 + 14/3*w^2 + 13/3*w - 11],
[929, 929, 1/3*w^3 + 4/3*w^2 - 7/3*w - 9],
[937, 937, -w^3 + 4*w^2 + w - 13],
[937, 937, 2/3*w^3 + 2/3*w^2 - 17/3*w - 3],
[953, 953, -2*w^3 + w^2 + 15*w + 8],
[953, 953, -5/3*w^3 + 19/3*w^2 - 1/3*w - 11],
[961, 31, 4/3*w^3 - 8/3*w^2 - 16/3*w + 5],
[967, 967, -4/3*w^3 + 2/3*w^2 + 31/3*w + 11],
[967, 967, 7/3*w^3 - 5/3*w^2 - 46/3*w - 9],
[977, 977, 2*w^3 - 5*w^2 - 8*w + 10],
[977, 977, -w^3 + 3*w^2 + 2*w - 14],
[977, 977, -4/3*w^3 + 8/3*w^2 + 25/3*w - 11],
[977, 977, -w^3 + w^2 + 6*w - 7],
[983, 983, -5/3*w^3 + 16/3*w^2 + 8/3*w - 12],
[983, 983, -1/3*w^3 + 11/3*w^2 - 11/3*w - 15],
[983, 983, 3*w^2 - 2*w - 16],
[983, 983, -5/3*w^3 + 16/3*w^2 + 11/3*w - 14],
[991, 991, 4/3*w^3 - 8/3*w^2 - 22/3*w + 1],
[991, 991, 2/3*w^3 - 4/3*w^2 - 2/3*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 + 6*x^5 + 5*x^4 - 24*x^3 - 35*x^2 + 12*x + 25;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -2*e^5 - 9*e^4 + 3*e^3 + 43*e^2 + 9*e - 38, -e^5 - 5*e^4 + e^3 + 25*e^2 + 5*e - 23, e^5 + 4*e^4 - 3*e^3 - 20*e^2 - e + 17, 1, e^5 + 5*e^4 - 24*e^2 - 10*e + 20, -e^3 - 2*e^2 + 3*e + 4, 2*e^5 + 8*e^4 - 4*e^3 - 34*e^2 - 6*e + 24, 2*e^5 + 9*e^4 - 4*e^3 - 45*e^2 - 4*e + 42, -e^5 - 3*e^4 + 6*e^3 + 16*e^2 - 9*e - 17, -3*e^5 - 14*e^4 + 3*e^3 + 67*e^2 + 22*e - 57, 2*e^5 + 10*e^4 - 2*e^3 - 50*e^2 - 12*e + 42, e^5 + 5*e^4 + e^3 - 19*e^2 - 11*e + 7, -e^5 - 7*e^4 - 6*e^3 + 28*e^2 + 25*e - 19, 2*e^5 + 10*e^4 - e^3 - 46*e^2 - 11*e + 34, -e^5 - 4*e^4 + e^3 + 12*e^2 + 6*e + 4, e^4 + e^3 - 6*e^2 + 2*e + 6, 3*e^5 + 13*e^4 - 6*e^3 - 62*e^2 - 9*e + 55, 4*e^5 + 18*e^4 - 6*e^3 - 86*e^2 - 19*e + 65, -6*e^5 - 28*e^4 + 7*e^3 + 131*e^2 + 33*e - 103, 6*e^5 + 28*e^4 - 7*e^3 - 135*e^2 - 36*e + 112, -e^5 - 4*e^4 + 5*e^3 + 22*e^2 - 12*e - 26, 7*e^5 + 32*e^4 - 9*e^3 - 150*e^2 - 34*e + 124, -e^5 - 3*e^4 + 4*e^3 + 12*e^2 + 2*e - 4, -2*e^4 - 2*e^3 + 19*e^2 + 9*e - 34, 2*e^5 + 10*e^4 - 2*e^3 - 49*e^2 - 7*e + 36, -4*e^5 - 18*e^4 + 6*e^3 + 84*e^2 + 15*e - 69, 5*e^5 + 24*e^4 - 7*e^3 - 120*e^2 - 22*e + 108, 7*e^5 + 32*e^4 - 10*e^3 - 152*e^2 - 31*e + 128, -3*e^5 - 12*e^4 + 7*e^3 + 52*e^2 + 2*e - 42, -2*e^5 - 11*e^4 - 2*e^3 + 52*e^2 + 18*e - 47, 7*e^5 + 32*e^4 - 11*e^3 - 154*e^2 - 28*e + 132, -8*e^5 - 34*e^4 + 15*e^3 + 156*e^2 + 29*e - 118, -2*e^3 - 6*e^2 + 8*e + 20, -e^5 - 2*e^4 + 5*e^3 + 7*e^2 - 15, -3*e^5 - 15*e^4 + 3*e^3 + 73*e^2 + 10*e - 62, 2*e^4 + 7*e^3 - e^2 - 18*e - 12, -2*e^5 - 8*e^4 + 4*e^3 + 35*e^2 + 7*e - 32, 6*e^5 + 31*e^4 + e^3 - 146*e^2 - 60*e + 118, -14*e^5 - 64*e^4 + 19*e^3 + 304*e^2 + 71*e - 250, e^4 + 2*e^3 - 6*e^2 - 2*e + 5, 3*e^4 + 6*e^3 - 17*e^2 - 20*e + 12, e^5 + 7*e^4 + 2*e^3 - 37*e^2 - 8*e + 33, 4*e^5 + 17*e^4 - 4*e^3 - 74*e^2 - 36*e + 57, -6*e^5 - 27*e^4 + 7*e^3 + 123*e^2 + 35*e - 102, -2*e^5 - 9*e^4 + 6*e^3 + 52*e^2 - 57, -9*e^5 - 42*e^4 + 11*e^3 + 200*e^2 + 48*e - 172, 3*e^5 + 11*e^4 - 8*e^3 - 46*e^2 - 9*e + 21, e^5 + 4*e^4 + e^3 - 11*e^2 - 8*e + 11, -12*e^5 - 56*e^4 + 14*e^3 + 265*e^2 + 58*e - 223, 7*e^5 + 32*e^4 - 6*e^3 - 144*e^2 - 51*e + 102, 8*e^5 + 38*e^4 - 5*e^3 - 174*e^2 - 57*e + 132, -10*e^5 - 45*e^4 + 14*e^3 + 210*e^2 + 46*e - 163, 2*e^5 + 10*e^4 - 5*e^3 - 56*e^2 + e + 50, -2*e^5 - 12*e^4 - 3*e^3 + 58*e^2 + 17*e - 60, -4*e^5 - 20*e^4 + 94*e^2 + 30*e - 88, -10*e^5 - 48*e^4 + 8*e^3 + 226*e^2 + 64*e - 178, -3*e^5 - 14*e^4 + 4*e^3 + 74*e^2 + 23*e - 76, -2*e^5 - 8*e^4 + 4*e^3 + 38*e^2 + 18*e - 46, -4*e^5 - 20*e^4 - e^3 + 91*e^2 + 41*e - 79, -2*e^5 - 8*e^4 + 8*e^3 + 44*e^2 - 10*e - 44, -9*e^5 - 45*e^4 + 4*e^3 + 216*e^2 + 69*e - 173, 5*e^5 + 22*e^4 - 13*e^3 - 117*e^2 - 6*e + 107, -e^5 - 3*e^4 + 6*e^3 + 22*e^2 - 34, -3*e^5 - 18*e^4 - 9*e^3 + 80*e^2 + 58*e - 66, -3*e^5 - 16*e^4 - e^3 + 80*e^2 + 33*e - 69, -6*e^5 - 30*e^4 + 8*e^3 + 154*e^2 + 14*e - 146, e^5 + 4*e^4 - 4*e^3 - 20*e^2 - e + 8, -5*e^5 - 21*e^4 + 14*e^3 + 106*e^2 + 10*e - 98, 9*e^5 + 42*e^4 - 7*e^3 - 194*e^2 - 60*e + 168, -7*e^5 - 35*e^4 + 2*e^3 + 170*e^2 + 60*e - 148, -5*e^5 - 26*e^4 + 5*e^3 + 138*e^2 + 28*e - 136, -2*e^5 - 3*e^4 + 18*e^3 + 15*e^2 - 34*e - 6, 7*e^5 + 31*e^4 - 13*e^3 - 147*e^2 - 26*e + 106, 7*e^5 + 31*e^4 - 8*e^3 - 141*e^2 - 48*e + 101, 2*e^5 + 10*e^4 - 46*e^2 - 19*e + 33, -8*e^5 - 36*e^4 + 8*e^3 + 158*e^2 + 48*e - 112, -4*e^5 - 18*e^4 + e^3 + 70*e^2 + 31*e - 40, -11*e^5 - 53*e^4 + 10*e^3 + 253*e^2 + 56*e - 215, 3*e^5 + 15*e^4 - 67*e^2 - 30*e + 55, 8*e^5 + 40*e^4 - 7*e^3 - 204*e^2 - 53*e + 190, 4*e^5 + 18*e^4 - 5*e^3 - 79*e^2 - 25*e + 41, -7*e^5 - 36*e^4 - e^3 + 170*e^2 + 60*e - 156, -4*e^5 - 16*e^4 + 12*e^3 + 80*e^2 + 2*e - 96, -6*e^5 - 28*e^4 + 6*e^3 + 132*e^2 + 33*e - 129, 3*e^5 + 15*e^4 - 3*e^3 - 75*e^2 - 11*e + 55, 7*e^5 + 35*e^4 - 2*e^3 - 160*e^2 - 49*e + 115, -8*e^5 - 38*e^4 + 12*e^3 + 196*e^2 + 38*e - 190, -3*e^5 - 12*e^4 + 15*e^3 + 74*e^2 - 18*e - 80, -3*e^5 - 17*e^4 - 4*e^3 + 80*e^2 + 42*e - 68, -7*e^5 - 31*e^4 + 17*e^3 + 159*e^2 + 11*e - 143, -5*e^5 - 21*e^4 + 8*e^3 + 87*e^2 + 20*e - 49, 3*e^5 + 13*e^4 - 4*e^3 - 63*e^2 - 30*e + 41, 3*e^5 + 12*e^4 - 13*e^3 - 72*e^2 + 4*e + 80, 14*e^5 + 63*e^4 - 24*e^3 - 308*e^2 - 52*e + 275, 6*e^5 + 25*e^4 - 16*e^3 - 121*e^2 + 6*e + 108, -2*e^5 - 8*e^4 + 4*e^3 + 32*e^2 + 6*e - 22, 5*e^4 + 18*e^3 - 12*e^2 - 56*e - 1, -21*e^5 - 100*e^4 + 19*e^3 + 474*e^2 + 126*e - 396, -7*e^5 - 30*e^4 + 19*e^3 + 150*e^2 + 6*e - 118, 4*e^3 + 5*e^2 - 20*e + 7, 12*e^5 + 62*e^4 - 5*e^3 - 310*e^2 - 95*e + 260, 6*e^5 + 32*e^4 + 7*e^3 - 139*e^2 - 76*e + 100, -4*e^5 - 19*e^4 + 5*e^3 + 90*e^2 + 10*e - 100, 13*e^5 + 59*e^4 - 24*e^3 - 296*e^2 - 41*e + 267, e^5 + 3*e^4 - 8*e^3 - 30*e^2 + 3*e + 57, 7*e^5 + 31*e^4 - 7*e^3 - 135*e^2 - 39*e + 101, 21*e^5 + 96*e^4 - 27*e^3 - 452*e^2 - 103*e + 381, -16*e^5 - 74*e^4 + 21*e^3 + 353*e^2 + 77*e - 295, -10*e^5 - 52*e^4 - e^3 + 253*e^2 + 91*e - 225, 13*e^5 + 59*e^4 - 16*e^3 - 275*e^2 - 68*e + 225, 11*e^5 + 50*e^4 - 15*e^3 - 237*e^2 - 60*e + 205, -6*e^5 - 28*e^4 + 2*e^3 + 122*e^2 + 46*e - 72, -6*e^5 - 26*e^4 + 12*e^3 + 129*e^2 + 26*e - 127, -14*e^5 - 63*e^4 + 24*e^3 + 303*e^2 + 56*e - 244, -4*e^5 - 16*e^4 + 7*e^3 + 69*e^2 + 28*e - 62, -9*e^5 - 39*e^4 + 21*e^3 + 193*e^2 + 27*e - 167, e^5 - e^4 - 11*e^3 + 11*e^2 + 8*e - 42, 8*e^5 + 39*e^4 - 3*e^3 - 178*e^2 - 70*e + 128, -12*e^5 - 56*e^4 + 16*e^3 + 274*e^2 + 60*e - 242, -15*e^5 - 67*e^4 + 20*e^3 + 302*e^2 + 71*e - 233, 8*e^5 + 36*e^4 - 17*e^3 - 183*e^2 - 19*e + 187, -3*e^5 - 14*e^4 - 2*e^3 + 50*e^2 + 33*e - 32, 16*e^5 + 78*e^4 - 12*e^3 - 372*e^2 - 104*e + 304, -12*e^5 - 57*e^4 + 16*e^3 + 290*e^2 + 66*e - 251, -8*e^5 - 42*e^4 + 4*e^3 + 210*e^2 + 46*e - 182, 2*e^5 + 15*e^4 + 10*e^3 - 73*e^2 - 34*e + 78, 10*e^5 + 50*e^4 - 221*e^2 - 80*e + 159, 15*e^5 + 66*e^4 - 25*e^3 - 306*e^2 - 50*e + 244, 8*e^5 + 36*e^4 - 18*e^3 - 177*e^2 + 3*e + 160, 7*e^5 + 26*e^4 - 19*e^3 - 105*e^2 - 2*e + 45, 12*e^5 + 58*e^4 - 14*e^3 - 281*e^2 - 53*e + 240, 6*e^5 + 24*e^4 - 15*e^3 - 103*e^2 + 4*e + 70, -13*e^5 - 59*e^4 + 26*e^3 + 291*e^2 + 30*e - 259, 5*e^5 + 24*e^4 - 7*e^3 - 118*e^2 - 19*e + 71, 22*e^5 + 100*e^4 - 29*e^3 - 467*e^2 - 108*e + 400, -3*e^5 - 14*e^4 + 5*e^3 + 70*e^2 + e - 75, 2*e^5 + 11*e^4 + 12*e^3 - 34*e^2 - 66*e + 13, 10*e^5 + 42*e^4 - 23*e^3 - 202*e^2 - 17*e + 178, -16*e^5 - 69*e^4 + 31*e^3 + 327*e^2 + 57*e - 284, -5*e^5 - 15*e^4 + 21*e^3 + 49*e^2 - 37*e - 9, 10*e^5 + 48*e^4 - 3*e^3 - 214*e^2 - 83*e + 162, 11*e^5 + 54*e^4 - 15*e^3 - 281*e^2 - 48*e + 273, 7*e^5 + 35*e^4 - 9*e^3 - 183*e^2 - 38*e + 148, -4*e^5 - 22*e^4 - 4*e^3 + 108*e^2 + 40*e - 118, 9*e^5 + 37*e^4 - 15*e^3 - 153*e^2 - 31*e + 87, -17*e^5 - 86*e^4 + 3*e^3 + 408*e^2 + 135*e - 343, -16*e^5 - 74*e^4 + 16*e^3 + 336*e^2 + 90*e - 258, 6*e^5 + 26*e^4 - 13*e^3 - 134*e^2 - 35*e + 132, 7*e^5 + 28*e^4 - 17*e^3 - 130*e^2 - 10*e + 120, -4*e^5 - 25*e^4 - 6*e^3 + 128*e^2 + 60*e - 85, e^5 - e^4 - 16*e^3 + 8*e^2 + 49*e - 15, -21*e^5 - 97*e^4 + 26*e^3 + 453*e^2 + 110*e - 365, 14*e^5 + 67*e^4 - 16*e^3 - 334*e^2 - 90*e + 293, 4*e^5 + 17*e^4 - 10*e^3 - 84*e^2 + 6*e + 93, 2*e^5 + 16*e^4 + 9*e^3 - 88*e^2 - 47*e + 54, 24*e^5 + 109*e^4 - 26*e^3 - 494*e^2 - 134*e + 389, 12*e^5 + 47*e^4 - 30*e^3 - 204*e^2 - 16*e + 127, -14*e^5 - 58*e^4 + 38*e^3 + 286*e^2 + 14*e - 252, 8*e^5 + 41*e^4 - 196*e^2 - 70*e + 183, 9*e^5 + 40*e^4 - 9*e^3 - 178*e^2 - 64*e + 128, -14*e^5 - 71*e^4 - 2*e^3 + 324*e^2 + 114*e - 247, 4*e^5 + 15*e^4 - 10*e^3 - 68*e^2 - 28*e + 43, -9*e^5 - 48*e^4 - 9*e^3 + 222*e^2 + 118*e - 182, -14*e^5 - 66*e^4 + 9*e^3 + 300*e^2 + 85*e - 236, -22*e^5 - 105*e^4 + 21*e^3 + 498*e^2 + 128*e - 414, -19*e^5 - 89*e^4 + 16*e^3 + 410*e^2 + 118*e - 334, -2*e^5 - 12*e^4 - 8*e^3 + 47*e^2 + 46*e - 31, -5*e^5 - 16*e^4 + 19*e^3 + 69*e^2 - 37, -8*e^5 - 28*e^4 + 24*e^3 + 112*e^2 - 6*e - 82];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9, 3, 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;