/*
  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, -4, -1, 9, -2, -3, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {21, 21, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + 3*w - 4}>;

primesArray := [
[3, 3, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 5*w + 1],
[7, 7, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 3*w + 5],
[25, 5, -w^3 + 2*w^2 + 2*w - 3],
[37, 37, w^3 - w^2 - 4*w + 2],
[37, 37, w^3 - 2*w^2 - 3*w + 2],
[49, 7, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 5*w - 3],
[49, 7, -w^5 + 3*w^4 + 3*w^3 - 9*w^2 - 3*w + 4],
[53, 53, w^4 - 2*w^3 - 3*w^2 + 3*w + 1],
[53, 53, w^4 - 2*w^3 - 3*w^2 + 5*w],
[61, 61, -3*w^5 + 6*w^4 + 12*w^3 - 16*w^2 - 12*w + 5],
[61, 61, 2*w^5 - 5*w^4 - 7*w^3 + 16*w^2 + 7*w - 7],
[61, 61, -2*w^5 + 5*w^4 + 7*w^3 - 15*w^2 - 8*w + 6],
[61, 61, -w^3 + 2*w^2 + 4*w - 3],
[64, 2, -2],
[67, 67, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 2*w - 3],
[67, 67, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 4*w + 5],
[71, 71, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 2*w - 4],
[71, 71, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - w],
[81, 3, -w^4 + 2*w^3 + 2*w^2 - 3*w + 2],
[101, 101, w^5 - 3*w^4 - 3*w^3 + 9*w^2 + 4*w - 3],
[101, 101, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 4*w + 3],
[103, 103, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 - 2*w + 3],
[103, 103, 2*w^5 - 5*w^4 - 5*w^3 + 13*w^2 + w - 3],
[107, 107, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 4*w - 3],
[107, 107, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - w - 1],
[113, 113, -2*w^5 + 4*w^4 + 9*w^3 - 12*w^2 - 10*w + 6],
[113, 113, -3*w^5 + 9*w^4 + 7*w^3 - 27*w^2 - 2*w + 12],
[113, 113, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 4*w + 6],
[113, 113, -2*w^5 + 5*w^4 + 5*w^3 - 12*w^2 + 2],
[149, 149, 3*w^5 - 8*w^4 - 8*w^3 + 22*w^2 + 4*w - 7],
[149, 149, 3*w^5 - 7*w^4 - 10*w^3 + 20*w^2 + 7*w - 6],
[151, 151, -w^5 + 4*w^4 - 12*w^2 + 3*w + 3],
[151, 151, -w^5 + w^4 + 6*w^3 - 3*w^2 - 6*w + 3],
[163, 163, 2*w^5 - 5*w^4 - 5*w^3 + 14*w^2 - 4],
[163, 163, 2*w^5 - 5*w^4 - 5*w^3 + 11*w^2 + 3*w - 2],
[167, 167, -w^4 + 3*w^3 - 6*w + 2],
[167, 167, -2*w^5 + 6*w^4 + 5*w^3 - 19*w^2 - w + 7],
[193, 193, 4*w^5 - 10*w^4 - 11*w^3 + 27*w^2 + 4*w - 7],
[193, 193, 4*w^5 - 10*w^4 - 11*w^3 + 26*w^2 + 5*w - 7],
[197, 197, w^5 - 4*w^4 - 2*w^3 + 15*w^2 + 3*w - 8],
[197, 197, -2*w^5 + 5*w^4 + 7*w^3 - 16*w^2 - 4*w + 8],
[197, 197, -w^3 + w^2 + w - 2],
[197, 197, 2*w^5 - 5*w^4 - 5*w^3 + 13*w^2 + 2*w - 4],
[227, 227, -w^4 + 3*w^3 + 2*w^2 - 6*w - 3],
[227, 227, w^5 - 3*w^4 - 3*w^3 + 11*w^2 + 2*w - 4],
[241, 241, -2*w^5 + 6*w^4 + 5*w^3 - 20*w^2 - w + 9],
[241, 241, -w^4 + w^3 + 4*w^2 - 3*w - 1],
[241, 241, -4*w^5 + 11*w^4 + 10*w^3 - 31*w^2 - 2*w + 11],
[241, 241, -2*w^5 + 4*w^4 + 9*w^3 - 11*w^2 - 12*w + 3],
[257, 257, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 3*w - 1],
[257, 257, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - 2*w - 4],
[263, 263, -3*w^5 + 8*w^4 + 8*w^3 - 21*w^2 - 6*w + 5],
[263, 263, -3*w^5 + 7*w^4 + 10*w^3 - 21*w^2 - 7*w + 9],
[271, 271, 2*w^5 - 6*w^4 - 6*w^3 + 21*w^2 + 6*w - 11],
[271, 271, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 6*w - 7],
[277, 277, 4*w^5 - 9*w^4 - 15*w^3 + 27*w^2 + 13*w - 9],
[277, 277, -4*w^5 + 11*w^4 + 11*w^3 - 32*w^2 - 6*w + 11],
[281, 281, -w^5 + 3*w^4 + 3*w^3 - 11*w^2 - 3*w + 5],
[281, 281, -w^5 + 2*w^4 + 5*w^3 - 6*w^2 - 9*w + 4],
[311, 311, -w^4 + w^3 + 3*w^2 + w + 1],
[311, 311, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 5*w + 5],
[353, 353, w^5 - 4*w^4 - w^3 + 14*w^2 - 3*w - 6],
[353, 353, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w - 1],
[359, 359, -4*w^5 + 10*w^4 + 11*w^3 - 28*w^2 - 4*w + 11],
[359, 359, -2*w^2 + w + 5],
[361, 19, -3*w^5 + 9*w^4 + 6*w^3 - 25*w^2 + 8],
[361, 19, 3*w^5 - 6*w^4 - 12*w^3 + 17*w^2 + 11*w - 5],
[361, 19, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],
[367, 367, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 6*w - 8],
[367, 367, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - 7],
[383, 383, -2*w^5 + 6*w^4 + 6*w^3 - 20*w^2 - 5*w + 8],
[383, 383, 4*w^5 - 10*w^4 - 11*w^3 + 28*w^2 + 3*w - 12],
[419, 419, -w^4 + 5*w^2 + 3*w - 3],
[419, 419, -2*w^5 + 6*w^4 + 6*w^3 - 20*w^2 - 6*w + 11],
[443, 443, w^5 - 3*w^4 - w^3 + 8*w^2 - 3*w - 4],
[443, 443, -w^5 + 2*w^4 + 3*w^3 - 3*w^2 - 3*w - 2],
[461, 461, -w^5 + 2*w^4 + 4*w^3 - 4*w^2 - 4*w - 3],
[461, 461, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - 3*w - 6],
[463, 463, -3*w^5 + 7*w^4 + 10*w^3 - 21*w^2 - 5*w + 8],
[463, 463, -3*w^5 + 8*w^4 + 8*w^3 - 21*w^2 - 4*w + 4],
[499, 499, w^5 - 2*w^4 - 2*w^3 + 2*w^2 - w + 2],
[499, 499, w^5 - 4*w^4 + 11*w^2 - 4*w - 5],
[509, 509, -2*w^5 + 5*w^4 + 8*w^3 - 18*w^2 - 8*w + 10],
[509, 509, w^5 - w^4 - 6*w^3 + 4*w^2 + 7*w - 3],
[541, 541, -2*w^5 + 6*w^4 + 6*w^3 - 20*w^2 - 5*w + 9],
[541, 541, -2*w^5 + 4*w^4 + 10*w^3 - 14*w^2 - 13*w + 6],
[557, 557, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 9*w + 4],
[557, 557, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 6*w - 7],
[577, 577, 2*w^5 - 5*w^4 - 6*w^3 + 16*w^2 + 2*w - 9],
[577, 577, -3*w^5 + 7*w^4 + 10*w^3 - 19*w^2 - 9*w + 8],
[587, 587, -w^4 + 3*w^3 + w^2 - 6*w - 1],
[587, 587, 2*w^5 - 4*w^4 - 9*w^3 + 14*w^2 + 8*w - 7],
[587, 587, -w^5 + 4*w^4 - 12*w^2 + 6*w + 3],
[587, 587, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 5*w - 7],
[587, 587, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 3*w + 4],
[587, 587, -w^4 + w^3 + 4*w^2 - w - 4],
[625, 5, -2*w^5 + 6*w^4 + 3*w^3 - 16*w^2 + 3*w + 6],
[631, 631, 3*w^5 - 8*w^4 - 8*w^3 + 22*w^2 + 5*w - 4],
[631, 631, -3*w^5 + 7*w^4 + 10*w^3 - 20*w^2 - 8*w + 10],
[641, 641, -3*w^5 + 7*w^4 + 11*w^3 - 21*w^2 - 10*w + 10],
[641, 641, 3*w^5 - 8*w^4 - 9*w^3 + 24*w^2 + 6*w - 6],
[643, 643, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + 6*w - 5],
[643, 643, 3*w^5 - 6*w^4 - 13*w^3 + 16*w^2 + 15*w - 3],
[643, 643, 3*w^5 - 8*w^4 - 8*w^3 + 24*w^2 + 5*w - 11],
[643, 643, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + 6*w - 6],
[653, 653, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w + 4],
[653, 653, 2*w^5 - 6*w^4 - 5*w^3 + 19*w^2 + 2*w - 6],
[653, 653, -2*w^5 + 4*w^4 + 9*w^3 - 12*w^2 - 11*w + 6],
[653, 653, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - 4*w + 6],
[659, 659, 2*w^5 - 4*w^4 - 8*w^3 + 12*w^2 + 8*w - 7],
[659, 659, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 - 2*w + 3],
[661, 661, 2*w^5 - 5*w^4 - 6*w^3 + 13*w^2 + 6*w - 1],
[661, 661, -w^5 + w^4 + 5*w^3 - w^2 - 6*w - 1],
[691, 691, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 8*w + 6],
[691, 691, 4*w^5 - 11*w^4 - 11*w^3 + 32*w^2 + 7*w - 11],
[733, 733, -4*w^5 + 10*w^4 + 13*w^3 - 30*w^2 - 8*w + 14],
[733, 733, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 - w + 5],
[739, 739, -w^2 + 5],
[739, 739, w^2 - 2*w - 4],
[757, 757, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 - 2*w],
[757, 757, -w^4 + w^3 + 5*w^2 - w - 6],
[821, 821, -3*w^5 + 7*w^4 + 9*w^3 - 17*w^2 - 7*w + 2],
[821, 821, -2*w^5 + 3*w^4 + 10*w^3 - 9*w^2 - 11*w + 4],
[821, 821, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],
[821, 821, -3*w^5 + 8*w^4 + 7*w^3 - 22*w^2 - w + 9],
[823, 823, 3*w^5 - 7*w^4 - 10*w^3 + 19*w^2 + 10*w - 7],
[823, 823, 3*w^5 - 8*w^4 - 8*w^3 + 23*w^2 + 5*w - 8],
[827, 827, w^5 - w^4 - 7*w^3 + 4*w^2 + 9*w - 2],
[827, 827, -w^5 + 4*w^4 + w^3 - 13*w^2 + 3*w + 4],
[853, 853, -5*w^5 + 13*w^4 + 12*w^3 - 33*w^2 - 2*w + 9],
[853, 853, 2*w^4 - 6*w^3 - 4*w^2 + 16*w - 1],
[857, 857, 3*w^5 - 6*w^4 - 11*w^3 + 16*w^2 + 8*w - 8],
[857, 857, 2*w^5 - 7*w^4 - 3*w^3 + 21*w^2 - 2*w - 6],
[859, 859, -3*w^5 + 7*w^4 + 11*w^3 - 20*w^2 - 12*w + 8],
[859, 859, -3*w^5 + 7*w^4 + 12*w^3 - 23*w^2 - 12*w + 9],
[859, 859, 3*w^5 - 8*w^4 - 10*w^3 + 25*w^2 + 9*w - 10],
[859, 859, -2*w^5 + 7*w^4 + 2*w^3 - 20*w^2 + 5*w + 7],
[863, 863, w^3 - 6*w],
[863, 863, 3*w^5 - 7*w^4 - 11*w^3 + 21*w^2 + 9*w - 8],
[863, 863, -3*w^5 + 8*w^4 + 9*w^3 - 24*w^2 - 5*w + 7],
[863, 863, -w^3 + 3*w^2 + 3*w - 5],
[877, 877, 2*w^5 - 6*w^4 - 4*w^3 + 15*w^2 + 2*w - 1],
[877, 877, -2*w^5 + 4*w^4 + 8*w^3 - 13*w^2 - 6*w + 8],
[881, 881, -4*w^5 + 10*w^4 + 13*w^3 - 29*w^2 - 9*w + 8],
[881, 881, -4*w^5 + 10*w^4 + 13*w^3 - 30*w^2 - 8*w + 11],
[883, 883, -4*w^5 + 9*w^4 + 16*w^3 - 30*w^2 - 14*w + 12],
[883, 883, w^5 - w^4 - 8*w^3 + 7*w^2 + 10*w - 5],
[907, 907, -2*w^5 + 4*w^4 + 9*w^3 - 14*w^2 - 9*w + 7],
[907, 907, -w^4 + 3*w^3 + w^2 - 7*w],
[907, 907, w^5 - 4*w^4 - w^3 + 12*w^2 - 2*w - 2],
[907, 907, -w^4 + 3*w^3 + 3*w^2 - 9*w - 1],
[907, 907, -w^4 + w^3 + 4*w^2 - 4],
[907, 907, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 4*w + 5],
[919, 919, -4*w^5 + 10*w^4 + 12*w^3 - 28*w^2 - 6*w + 9],
[919, 919, 4*w^5 - 10*w^4 - 12*w^3 + 28*w^2 + 6*w - 7],
[929, 929, -w^5 + 4*w^4 + 2*w^3 - 14*w^2 - 2*w + 8],
[929, 929, -2*w^5 + 5*w^4 + 5*w^3 - 14*w^2 + w + 5],
[937, 937, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 8*w + 5],
[937, 937, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 5*w - 5],
[941, 941, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 5*w + 5],
[941, 941, -w^5 + 3*w^4 + 2*w^3 - 9*w^2],
[947, 947, 3*w^5 - 6*w^4 - 12*w^3 + 16*w^2 + 12*w - 6],
[947, 947, -2*w^5 + 5*w^4 + 6*w^3 - 13*w^2 - 5*w + 6],
[947, 947, 2*w^4 - 4*w^3 - 5*w^2 + 6*w + 1],
[947, 947, -3*w^5 + 9*w^4 + 6*w^3 - 26*w^2 + w + 7],
[953, 953, -3*w^5 + 6*w^4 + 12*w^3 - 18*w^2 - 10*w + 10],
[953, 953, 2*w^5 - 7*w^4 - 3*w^3 + 22*w^2 - w - 7],
[971, 971, -w^5 + w^4 + 5*w^3 - 7*w - 3],
[971, 971, -w^5 + 4*w^4 - w^3 - 11*w^2 + 7*w + 5],
[997, 997, -4*w^5 + 9*w^4 + 14*w^3 - 25*w^2 - 12*w + 10],
[997, 997, -3*w^5 + 8*w^4 + 9*w^3 - 23*w^2 - 6*w + 5],
[997, 997, -2*w^4 + 4*w^3 + 6*w^2 - 9*w - 1],
[997, 997, 4*w^5 - 11*w^4 - 10*w^3 + 31*w^2 + 4*w - 8],
[1021, 1021, 3*w^5 - 7*w^4 - 11*w^3 + 23*w^2 + 10*w - 15],
[1021, 1021, -4*w^5 + 11*w^4 + 10*w^3 - 32*w^2 - 4*w + 12],
[1031, 1031, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 - 2*w + 6],
[1031, 1031, 2*w^5 - 4*w^4 - 8*w^3 + 11*w^2 + 10*w - 5],
[1039, 1039, -w^4 + 7*w^2 + 2*w - 6],
[1039, 1039, w^5 - 4*w^4 + w^3 + 10*w^2 - 7*w - 4],
[1039, 1039, w^5 - 2*w^4 - 3*w^3 + 5*w^2 + 2*w - 4],
[1039, 1039, -3*w^5 + 8*w^4 + 7*w^3 - 21*w^2 + w + 7],
[1051, 1051, -w^4 + 3*w^3 + 2*w^2 - 5*w - 2],
[1051, 1051, w^4 - 2*w^3 - 2*w^2 + 4*w - 4],
[1061, 1061, 4*w^5 - 10*w^4 - 11*w^3 + 26*w^2 + 4*w - 6],
[1061, 1061, 4*w^5 - 10*w^4 - 11*w^3 + 27*w^2 + 3*w - 7],
[1087, 1087, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + w - 4],
[1087, 1087, 2*w^5 - 5*w^4 - 6*w^3 + 14*w^2 + w - 2],
[1097, 1097, -2*w^5 + 6*w^4 + 4*w^3 - 17*w^2 - w + 4],
[1097, 1097, -2*w^5 + 4*w^4 + 8*w^3 - 11*w^2 - 9*w + 6],
[1103, 1103, 4*w^5 - 10*w^4 - 13*w^3 + 30*w^2 + 8*w - 10],
[1103, 1103, 4*w^5 - 10*w^4 - 13*w^3 + 29*w^2 + 9*w - 9],
[1129, 1129, 3*w^5 - 9*w^4 - 4*w^3 + 24*w^2 - 8*w - 7],
[1129, 1129, 3*w^5 - 6*w^4 - 10*w^3 + 12*w^2 + 7*w + 1],
[1181, 1181, 4*w^5 - 10*w^4 - 14*w^3 + 32*w^2 + 9*w - 13],
[1181, 1181, 3*w^5 - 7*w^4 - 12*w^3 + 22*w^2 + 13*w - 7],
[1181, 1181, -4*w^5 + 10*w^4 + 12*w^3 - 29*w^2 - 5*w + 9],
[1181, 1181, 4*w^5 - 10*w^4 - 12*w^3 + 27*w^2 + 7*w - 7],
[1181, 1181, 3*w^5 - 8*w^4 - 10*w^3 + 26*w^2 + 8*w - 12],
[1181, 1181, -5*w^5 + 13*w^4 + 16*w^3 - 40*w^2 - 11*w + 19],
[1187, 1187, 2*w^5 - 7*w^4 - 4*w^3 + 22*w^2 + w - 11],
[1187, 1187, 3*w^5 - 7*w^4 - 10*w^3 + 21*w^2 + 7*w - 12],
[1187, 1187, 2*w^5 - 4*w^4 - 7*w^3 + 11*w^2 + 4*w - 7],
[1187, 1187, 6*w^5 - 14*w^4 - 22*w^3 + 41*w^2 + 21*w - 14],
[1217, 1217, 3*w^5 - 6*w^4 - 12*w^3 + 17*w^2 + 12*w - 7],
[1217, 1217, 3*w^5 - 9*w^4 - 6*w^3 + 25*w^2 + w - 7],
[1229, 1229, -2*w^5 + 5*w^4 + 8*w^3 - 17*w^2 - 8*w + 8],
[1229, 1229, -2*w^5 + 5*w^4 + 8*w^3 - 17*w^2 - 8*w + 6],
[1231, 1231, -6*w^5 + 16*w^4 + 17*w^3 - 47*w^2 - 8*w + 18],
[1231, 1231, -6*w^5 + 14*w^4 + 21*w^3 - 40*w^2 - 17*w + 10],
[1237, 1237, -3*w^5 + 8*w^4 + 8*w^3 - 22*w^2 - 2*w + 6],
[1237, 1237, 3*w^5 - 7*w^4 - 10*w^3 + 20*w^2 + 5*w - 5],
[1279, 1279, 3*w^5 - 8*w^4 - 8*w^3 + 23*w^2 + w - 6],
[1279, 1279, 2*w^5 - 5*w^4 - 7*w^3 + 16*w^2 + 8*w - 9],
[1291, 1291, -2*w^5 + 4*w^4 + 9*w^3 - 13*w^2 - 9*w + 8],
[1291, 1291, -2*w^5 + 6*w^4 + 5*w^3 - 18*w^2 - 2*w + 3],
[1297, 1297, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + w - 6],
[1297, 1297, -w^5 + 3*w^4 + 2*w^3 - 7*w^2 - 3],
[1303, 1303, -w^4 + w^3 + 6*w^2 - 3*w - 3],
[1303, 1303, -w^4 + 3*w^3 + 3*w^2 - 8*w],
[1319, 1319, -2*w^5 + 5*w^4 + 7*w^3 - 16*w^2 - 8*w + 7],
[1319, 1319, 2*w^5 - 5*w^4 - 7*w^3 + 15*w^2 + 9*w - 7],
[1369, 37, 2*w^5 - 6*w^4 - 5*w^3 + 19*w^2 + 4*w - 13],
[1369, 37, 2*w^5 - 4*w^4 - 9*w^3 + 12*w^2 + 13*w - 1],
[1373, 1373, w^5 - 3*w^4 - 4*w^3 + 10*w^2 + 6*w - 6],
[1373, 1373, 2*w^5 - 5*w^4 - 8*w^3 + 17*w^2 + 9*w - 8],
[1373, 1373, 2*w^5 - 5*w^4 - 8*w^3 + 17*w^2 + 9*w - 7],
[1373, 1373, 4*w^5 - 10*w^4 - 13*w^3 + 31*w^2 + 8*w - 17],
[1381, 1381, -w^4 + 3*w^3 + 2*w^2 - 5*w - 1],
[1381, 1381, w^5 - 2*w^4 - 5*w^3 + 5*w^2 + 9*w + 1],
[1423, 1423, -3*w^5 + 8*w^4 + 9*w^3 - 24*w^2 - 5*w + 6],
[1423, 1423, 3*w^5 - 7*w^4 - 11*w^3 + 21*w^2 + 9*w - 9],
[1427, 1427, -w^5 + 3*w^4 + 2*w^3 - 8*w^2 - 2],
[1427, 1427, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 3*w - 6],
[1433, 1433, w^5 - w^4 - 7*w^3 + 4*w^2 + 9*w - 1],
[1433, 1433, w^5 - 4*w^4 - w^3 + 13*w^2 - 3*w - 5],
[1439, 1439, 3*w^5 - 7*w^4 - 9*w^3 + 17*w^2 + 5*w - 4],
[1439, 1439, 4*w^5 - 11*w^4 - 11*w^3 + 32*w^2 + 6*w - 9],
[1451, 1451, -w^5 + w^4 + 6*w^3 - w^2 - 9*w - 2],
[1451, 1451, -w^5 + 4*w^4 - 13*w^2 + 6*w + 6],
[1471, 1471, 5*w^5 - 14*w^4 - 15*w^3 + 46*w^2 + 11*w - 23],
[1471, 1471, -5*w^5 + 14*w^4 + 11*w^3 - 38*w^2 + w + 11],
[1489, 1489, 4*w^5 - 10*w^4 - 13*w^3 + 30*w^2 + 10*w - 9],
[1489, 1489, 4*w^5 - 10*w^4 - 13*w^3 + 29*w^2 + 11*w - 12],
[1499, 1499, w^5 - 3*w^4 - w^3 + 7*w^2 - 4*w - 3],
[1499, 1499, -w^5 + 2*w^4 + 3*w^3 - 4*w^2 - 3],
[1511, 1511, 4*w^5 - 12*w^4 - 9*w^3 + 35*w^2 + 2*w - 13],
[1511, 1511, -2*w^5 + 6*w^4 + 3*w^3 - 18*w^2 + 4*w + 10],
[1549, 1549, -2*w^5 + 6*w^4 + 4*w^3 - 16*w^2 + 3*w + 1],
[1549, 1549, 2*w^5 - 4*w^4 - 8*w^3 + 12*w^2 + 3*w - 4],
[1579, 1579, 3*w^5 - 9*w^4 - 6*w^3 + 24*w^2 - w - 6],
[1579, 1579, 5*w^5 - 12*w^4 - 15*w^3 + 32*w^2 + 7*w - 13],
[1607, 1607, -w^5 + 4*w^4 - 3*w^3 - 7*w^2 + 12*w - 2],
[1607, 1607, -w^5 + w^4 + 3*w^3 + 2*w^2 - 3],
[1613, 1613, -w^4 + 3*w^3 + w^2 - 6*w - 2],
[1613, 1613, -w^4 + w^3 + 4*w^2 - w - 5],
[1619, 1619, 4*w^5 - 10*w^4 - 12*w^3 + 29*w^2 + 6*w - 8],
[1619, 1619, -2*w^5 + 6*w^4 + 3*w^3 - 17*w^2 + 5*w + 6],
[1627, 1627, 4*w^5 - 11*w^4 - 9*w^3 + 29*w^2 - w - 4],
[1627, 1627, 4*w^5 - 11*w^4 - 12*w^3 + 35*w^2 + 10*w - 18],
[1637, 1637, w^5 - 4*w^4 + w^3 + 11*w^2 - 7*w - 4],
[1637, 1637, w^5 - w^4 - 5*w^3 + 7*w + 2],
[1657, 1657, 4*w^5 - 10*w^4 - 14*w^3 + 31*w^2 + 15*w - 15],
[1657, 1657, -4*w^5 + 10*w^4 + 14*w^3 - 31*w^2 - 15*w + 11],
[1681, 41, 2*w^4 - 4*w^3 - 7*w^2 + 9*w + 3],
[1693, 1693, w^5 - 4*w^4 - w^3 + 15*w^2 - w - 10],
[1693, 1693, -5*w^5 + 12*w^4 + 17*w^3 - 35*w^2 - 16*w + 11],
[1697, 1697, -7*w^5 + 17*w^4 + 21*w^3 - 48*w^2 - 9*w + 19],
[1697, 1697, 2*w^5 - 5*w^4 - 5*w^3 + 13*w^2 - w - 7],
[1709, 1709, -3*w^5 + 9*w^4 + 7*w^3 - 27*w^2 - 3*w + 14],
[1709, 1709, -4*w^5 + 10*w^4 + 11*w^3 - 30*w^2 - 3*w + 14],
[1721, 1721, -w^5 + 9*w^3 - 17*w + 2],
[1721, 1721, -w^5 + 5*w^4 - w^3 - 17*w^2 + 5*w + 7],
[1723, 1723, -w^5 + 2*w^4 + 2*w^3 - 2*w^2 + w - 3],
[1723, 1723, w^5 - 3*w^4 + 6*w^2 - 6*w - 1],
[1759, 1759, w^4 - 4*w^3 - w^2 + 8*w - 1],
[1759, 1759, -3*w^5 + 9*w^4 + 7*w^3 - 27*w^2 - 4*w + 14],
[1777, 1777, 3*w^5 - 9*w^4 - 7*w^3 + 29*w^2 - 13],
[1777, 1777, 3*w^5 - 8*w^4 - 8*w^3 + 26*w^2 + 3*w - 13],
[1777, 1777, -3*w^5 + 9*w^4 + 6*w^3 - 25*w^2 + 11],
[1777, 1777, 3*w^5 - 6*w^4 - 13*w^3 + 16*w^2 + 16*w - 3],
[1783, 1783, 3*w^5 - 7*w^4 - 10*w^3 + 19*w^2 + 10*w - 8],
[1783, 1783, -6*w^5 + 15*w^4 + 17*w^3 - 42*w^2 - 7*w + 15],
[1783, 1783, 7*w^5 - 18*w^4 - 20*w^3 + 52*w^2 + 8*w - 20],
[1783, 1783, -3*w^5 + 8*w^4 + 8*w^3 - 23*w^2 - 5*w + 7],
[1801, 1801, -2*w^5 + 6*w^4 + 3*w^3 - 15*w^2 + 4*w + 4],
[1801, 1801, -2*w^5 + 4*w^4 + 7*w^3 - 10*w^2 - 3*w],
[1811, 1811, -2*w^5 + 4*w^4 + 8*w^3 - 12*w^2 - 9*w + 7],
[1811, 1811, 3*w^5 - 10*w^4 - 7*w^3 + 33*w^2 + 3*w - 14],
[1823, 1823, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 9*w - 5],
[1823, 1823, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 5*w + 5],
[1847, 1847, w^5 - 3*w^4 - 2*w^3 + 8*w^2 + w - 6],
[1847, 1847, -w^5 + 2*w^4 + 4*w^3 - 6*w^2 - 4*w - 1],
[1849, 43, -2*w^4 + 4*w^3 + 6*w^2 - 8*w - 3],
[1901, 1901, -5*w^5 + 13*w^4 + 13*w^3 - 37*w^2 - 3*w + 13],
[1901, 1901, -5*w^5 + 12*w^4 + 15*w^3 - 30*w^2 - 11*w + 6],
[1913, 1913, 3*w^5 - 9*w^4 - 7*w^3 + 29*w^2 + w - 13],
[1913, 1913, -w^5 + 4*w^4 + w^3 - 12*w^2 + 3*w + 1],
[1913, 1913, 2*w^5 - 3*w^4 - 10*w^3 + 7*w^2 + 14*w + 1],
[1913, 1913, -2*w^4 + 2*w^3 + 9*w^2 + w - 5],
[1931, 1931, -3*w^5 + 6*w^4 + 13*w^3 - 16*w^2 - 17*w + 5],
[1931, 1931, 4*w^5 - 9*w^4 - 15*w^3 + 26*w^2 + 17*w - 10],
[1949, 1949, -3*w^5 + 8*w^4 + 9*w^3 - 23*w^2 - 10*w + 10],
[1949, 1949, 2*w^5 - 7*w^4 - 3*w^3 + 20*w^2 - w - 8],
[1973, 1973, -2*w^5 + 6*w^4 + 5*w^3 - 17*w^2 - 3*w + 9],
[1973, 1973, -5*w^5 + 13*w^4 + 13*w^3 - 34*w^2 - 7*w + 11],
[1979, 1979, 6*w^5 - 15*w^4 - 18*w^3 + 43*w^2 + 8*w - 13],
[1979, 1979, -2*w^5 + 3*w^4 + 11*w^3 - 11*w^2 - 12*w + 7],
[1987, 1987, -5*w^5 + 13*w^4 + 13*w^3 - 35*w^2 - 3*w + 7],
[1987, 1987, -w^5 + 2*w^4 + 6*w^3 - 6*w^2 - 11*w]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 6*x^5 - 81*x^4 + 404*x^3 + 444*x^2 - 2016*x - 2144;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 1, 2, 3/136*e^5 - 11/102*e^4 - 725/408*e^3 + 439/68*e^2 + 637/102*e - 607/51, e, 5/918*e^5 - 16/459*e^4 - 331/918*e^3 + 1015/459*e^2 - 1591/459*e - 1160/459, -5/918*e^5 + 16/459*e^4 + 331/918*e^3 - 1015/459*e^2 + 1591/459*e + 2996/459, -11/3672*e^5 - 13/918*e^4 + 1279/3672*e^3 + 2357/1836*e^2 - 5395/918*e - 739/459, 1/1836*e^5 - 95/918*e^4 + 607/1836*e^3 + 4003/459*e^2 - 8926/459*e - 20924/459, 101/3672*e^5 - 131/918*e^4 - 7849/3672*e^3 + 15913/1836*e^2 + 2551/918*e - 7541/459, 10/459*e^5 - 103/1836*e^4 - 1783/918*e^3 + 3847/1836*e^2 + 7100/459*e + 4081/459, -35/3672*e^5 + 28/459*e^4 + 2623/3672*e^3 - 7105/1836*e^2 + 749/918*e + 2183/459, -5/918*e^5 + 16/459*e^4 + 331/918*e^3 - 1015/459*e^2 + 2050/459*e + 2078/459, -5/408*e^5 - 1/204*e^4 + 167/136*e^3 + 181/102*e^2 - 1661/102*e - 283/17, -10/459*e^5 + 103/1836*e^4 + 1783/918*e^3 - 3847/1836*e^2 - 6641/459*e - 5917/459, 29/918*e^5 - 155/918*e^4 - 2287/918*e^3 + 9479/918*e^2 + 2492/459*e - 9482/459, -23/1836*e^5 + 43/918*e^4 + 1951/1836*e^3 - 1187/459*e^2 - 2323/459*e + 3280/459, 41/1836*e^5 - 293/1836*e^4 - 2959/1836*e^3 + 19859/1836*e^2 - 1826/459*e - 15007/459, -1/102*e^5 + 23/204*e^4 + 28/51*e^3 - 1679/204*e^2 + 461/51*e + 1813/51, -10/459*e^5 + 103/1836*e^4 + 1783/918*e^3 - 3847/1836*e^2 - 8018/459*e - 4081/459, -127/3672*e^5 + 71/459*e^4 + 10427/3672*e^3 - 16601/1836*e^2 - 12215/918*e + 8743/459, -1/102*e^5 + 23/204*e^4 + 28/51*e^3 - 1679/204*e^2 + 461/51*e + 1711/51, -1/102*e^5 + 23/204*e^4 + 28/51*e^3 - 1679/204*e^2 + 461/51*e + 1711/51, 5/918*e^5 - 16/459*e^4 - 331/918*e^3 + 1015/459*e^2 - 2050/459*e - 1160/459, -101/3672*e^5 + 131/918*e^4 + 7849/3672*e^3 - 15913/1836*e^2 - 2551/918*e + 8459/459, 1/27*e^5 - 11/54*e^4 - 77/27*e^3 + 677/54*e^2 + 53/27*e - 734/27, 35/1836*e^5 - 56/459*e^4 - 2623/1836*e^3 + 7105/918*e^2 + 169/459*e - 7120/459, 1/1836*e^5 - 95/918*e^4 + 607/1836*e^3 + 4003/459*e^2 - 8467/459*e - 21842/459, -25/918*e^5 + 167/1836*e^4 + 1057/459*e^3 - 7907/1836*e^2 - 5050/459*e - 4757/459, -11/1836*e^5 + 127/918*e^4 + 55/1836*e^3 - 5018/459*e^2 + 10058/459*e + 23920/459, -25/1836*e^5 + 40/459*e^4 + 1961/1836*e^3 - 5075/918*e^2 - 1760/459*e + 5042/459, -11/459*e^5 + 49/918*e^4 + 973/459*e^3 - 1129/918*e^2 - 7504/459*e - 13562/459, 7/3672*e^5 + 25/459*e^4 - 1259/3672*e^3 - 9595/1836*e^2 + 8357/918*e + 17189/459, -1/36*e^5 + 7/36*e^4 + 71/36*e^3 - 469/36*e^2 + 76/9*e + 389/9, 49/1224*e^5 - 29/153*e^4 - 3917/1224*e^3 + 6887/612*e^2 + 2399/306*e - 2689/153, 7/612*e^5 - 55/612*e^4 - 545/612*e^3 + 3913/612*e^2 + 452/153*e - 3617/153, -1/612*e^5 - 7/306*e^4 + 209/612*e^3 + 281/153*e^2 - 1835/153*e - 292/153, -1/24*e^5 + 1/3*e^4 + 23/8*e^3 - 275/12*e^2 + 83/6*e + 75, 5/204*e^5 + 1/102*e^4 - 167/68*e^3 - 181/51*e^2 + 1508/51*e + 668/17, 1/306*e^5 - 23/612*e^4 - 28/153*e^3 + 1883/612*e^2 - 665/153*e - 2629/153, -1/36*e^5 + 7/36*e^4 + 71/36*e^3 - 469/36*e^2 + 85/9*e + 353/9, 19/306*e^5 - 91/306*e^4 - 1523/306*e^3 + 5419/306*e^2 + 2155/153*e - 5122/153, -2/153*e^5 + 23/153*e^4 + 112/153*e^3 - 1730/153*e^2 + 2048/153*e + 8374/153, 1/51*e^5 + 5/204*e^4 - 197/102*e^3 - 977/204*e^2 + 1169/51*e + 2885/51, 1/204*e^5 - 1/68*e^4 - 107/204*e^3 + 253/204*e^2 + 164/17*e - 677/51, 2/459*e^5 + 79/459*e^4 - 469/459*e^3 - 6991/459*e^2 + 16720/459*e + 39770/459, -25/1836*e^5 + 40/459*e^4 + 1961/1836*e^3 - 5075/918*e^2 - 1760/459*e + 3206/459, -11/1836*e^5 + 127/918*e^4 + 55/1836*e^3 - 5018/459*e^2 + 10058/459*e + 22084/459, 83/3672*e^5 - 41/918*e^4 - 7759/3672*e^3 + 3079/1836*e^2 + 19705/918*e - 1907/459, -13/918*e^5 + 11/459*e^4 + 1289/918*e^3 - 344/459*e^2 - 8746/459*e + 1486/459, -19/1836*e^5 - 31/918*e^4 + 1931/1836*e^3 + 1973/459*e^2 - 6203/459*e - 15850/459, -241/3672*e^5 + 101/459*e^4 + 20789/3672*e^3 - 19547/1836*e^2 - 33215/918*e - 863/459, 151/3672*e^5 - 211/918*e^4 - 11771/3672*e^3 + 26063/1836*e^2 + 3317/918*e - 7993/459, -13/918*e^5 + 11/459*e^4 + 1289/918*e^3 - 344/459*e^2 - 9664/459*e - 1268/459, -25/459*e^5 + 167/918*e^4 + 2114/459*e^3 - 7907/918*e^2 - 11936/459*e - 7678/459, -25/1836*e^5 + 40/459*e^4 + 1961/1836*e^3 - 5075/918*e^2 - 2219/459*e + 4124/459, -103/3672*e^5 + 113/459*e^4 + 6635/3672*e^3 - 31925/1836*e^2 + 14383/918*e + 28465/459, 7/918*e^5 + 247/1836*e^4 - 553/459*e^3 - 22315/1836*e^2 + 15184/459*e + 30047/459, 29/918*e^5 - 1/459*e^4 - 2899/918*e^3 - 1457/459*e^2 + 18863/459*e + 15916/459, -151/3672*e^5 + 29/459*e^4 + 14219/3672*e^3 - 1277/1836*e^2 - 36977/918*e - 11897/459, 13/918*e^5 - 175/918*e^4 - 677/918*e^3 + 13081/918*e^2 - 8084/459*e - 24130/459, 1/459*e^5 + 5/1836*e^4 - 163/918*e^3 - 1589/1836*e^2 + 404/459*e + 5809/459, 7/918*e^5 - 53/459*e^4 - 341/918*e^3 + 4175/459*e^2 - 4553/459*e - 23044/459, -2/153*e^5 - 5/306*e^4 + 214/153*e^3 + 977/306*e^2 - 3562/153*e - 3458/153, -11/1224*e^5 + 19/153*e^4 + 463/1224*e^3 - 5905/612*e^2 + 4907/306*e + 9563/153, 29/918*e^5 - 155/918*e^4 - 2287/918*e^3 + 9479/918*e^2 + 2492/459*e - 6728/459, -10/459*e^5 + 103/1836*e^4 + 1783/918*e^3 - 3847/1836*e^2 - 6641/459*e - 3163/459, 1/408*e^5 + 2/17*e^4 - 277/408*e^3 - 2041/204*e^2 + 861/34*e + 1897/51, 53/1836*e^5 + 7/459*e^4 - 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214/459*e^4 - 3428/459*e^3 + 12868/459*e^2 + 8957/459*e - 17504/459, -91/1836*e^5 + 115/459*e^4 + 7187/1836*e^3 - 13883/918*e^2 - 2306/459*e + 19430/459, -11/108*e^5 + 47/108*e^4 + 901/108*e^3 - 2657/108*e^2 - 922/27*e + 1357/27, -1/918*e^5 - 58/459*e^4 + 617/918*e^3 + 4846/459*e^2 - 16726/459*e - 14456/459, -19/306*e^5 + 20/153*e^4 + 1727/306*e^3 - 644/153*e^2 - 6694/153*e - 2732/153, 20/153*e^5 - 563/612*e^4 - 2903/306*e^3 + 37835/612*e^2 - 2987/153*e - 28303/153, -53/1224*e^5 - 7/306*e^4 + 5161/1224*e^3 + 4235/612*e^2 - 13615/306*e - 15133/153, 1/18*e^5 - 5/9*e^4 - 59/18*e^3 + 356/9*e^2 - 446/9*e - 1456/9, -157/1836*e^5 + 533/918*e^4 + 11189/1836*e^3 - 17542/459*e^2 + 9847/459*e + 54626/459, 5/108*e^5 - 7/54*e^4 - 421/108*e^3 + 143/27*e^2 + 397/27*e + 950/27, -47/1836*e^5 + 181/918*e^4 + 3295/1836*e^3 - 6377/459*e^2 + 4739/459*e + 18304/459, 32/459*e^5 - 379/918*e^4 - 2455/459*e^3 + 24607/918*e^2 + 1912/459*e - 43918/459, 8/459*e^5 - 143/459*e^4 - 346/459*e^3 + 11051/459*e^2 - 10232/459*e - 49612/459, 28/459*e^5 - 271/459*e^4 - 1670/459*e^3 + 19171/459*e^2 - 22960/459*e - 66236/459, -91/1836*e^5 + 383/918*e^4 + 5963/1836*e^3 - 13138/459*e^2 + 14065/459*e + 43910/459, 113/918*e^5 - 178/459*e^4 - 9745/918*e^3 + 8251/459*e^2 + 30782/459*e + 17542/459, 77/1224*e^5 - 11/306*e^4 - 7729/1224*e^3 - 2423/612*e^2 + 26035/306*e + 8233/153, -7/612*e^5 + 53/306*e^4 + 443/612*e^3 - 2011/153*e^2 + 823/153*e + 5504/153, 59/918*e^5 - 250/459*e^4 - 3967/918*e^3 + 17485/459*e^2 - 13786/459*e - 60812/459, -43/459*e^5 + 275/918*e^4 + 3734/459*e^3 - 13343/918*e^2 - 24410/459*e - 2386/459, -43/459*e^5 + 214/459*e^4 + 3428/459*e^3 - 12868/459*e^2 - 8957/459*e + 28520/459, 91/1836*e^5 - 115/459*e^4 - 7187/1836*e^3 + 13883/918*e^2 + 2306/459*e - 8414/459, -19/306*e^5 - 73/612*e^4 + 991/153*e^3 + 10429/612*e^2 - 13579/153*e - 23489/153, 35/612*e^5 - 107/153*e^4 - 1807/612*e^3 + 15673/306*e^2 - 11051/153*e - 31090/153, 43/918*e^5 + 46/459*e^4 - 4499/918*e^3 - 6418/459*e^2 + 33778/459*e + 54284/459, 157/1836*e^5 - 301/1836*e^4 - 14555/1836*e^3 + 8203/1836*e^2 + 36818/459*e + 18661/459, 101/3672*e^5 - 142/459*e^4 - 5401/3672*e^3 + 40699/1836*e^2 - 24683/918*e - 32021/459, 179/1836*e^5 - 317/459*e^4 - 13135/1836*e^3 + 42763/918*e^2 - 2576/459*e - 65746/459, 25/204*e^5 - 23/51*e^4 - 699/68*e^3 + 2321/102*e^2 + 2695/51*e - 128/17, -13/408*e^5 + 7/51*e^4 + 339/136*e^3 - 1415/204*e^2 + 149/102*e - 695/17, 1/24*e^5 - 1/3*e^4 - 23/8*e^3 + 275/12*e^2 - 65/6*e - 89, -235/1836*e^5 + 599/918*e^4 + 18923/1836*e^3 - 18115/459*e^2 - 18227/459*e + 41642/459, -31/918*e^5 + 611/1836*e^4 + 919/459*e^3 - 43991/1836*e^2 + 10721/459*e + 50185/459, -1/18*e^5 - 1/9*e^4 + 107/18*e^3 + 148/9*e^2 - 811/9*e - 1256/9, 43/612*e^5 - 107/306*e^4 - 3479/612*e^3 + 3064/153*e^2 + 3272/153*e + 928/153, 101/1836*e^5 - 415/918*e^4 - 6625/1836*e^3 + 14153/459*e^2 - 14738/459*e - 60676/459, -155/1836*e^5 + 95/459*e^4 + 13627/1836*e^3 - 6679/918*e^2 - 23458/459*e - 41078/459, -163/3672*e^5 + 8/459*e^4 + 16727/3672*e^3 + 5467/1836*e^2 - 58997/918*e - 16403/459, 1/27*e^5 - 10/27*e^4 - 68/27*e^3 + 730/27*e^2 - 316/27*e - 2858/27, 29/3672*e^5 - 77/918*e^4 - 2593/3672*e^3 + 14149/1836*e^2 + 11191/918*e - 29375/459, 197/1836*e^5 - 101/459*e^4 - 17509/1836*e^3 + 4189/918*e^2 + 34279/459*e + 54566/459, -31/3672*e^5 + 19/918*e^4 + 2603/3672*e^3 - 1703/1836*e^2 - 5885/918*e - 5087/459, -151/1836*e^5 + 269/918*e^4 + 12995/1836*e^3 - 6835/459*e^2 - 21983/459*e - 7576/459, -13/612*e^5 + 11/306*e^4 + 1289/612*e^3 - 19/153*e^2 - 4985/153*e - 6142/153, 79/1836*e^5 + 145/918*e^4 - 8963/1836*e^3 - 8276/459*e^2 + 39956/459*e + 61828/459, 65/1836*e^5 + 49/459*e^4 - 7057/1836*e^3 - 11591/918*e^2 + 28138/459*e + 42950/459, -79/1224*e^5 + 157/306*e^4 + 5291/1224*e^3 - 21851/612*e^2 + 8851/306*e + 18199/153, -23/612*e^5 + 47/153*e^4 + 1747/612*e^3 - 6811/306*e^2 - 997/153*e + 12562/153, -155/3672*e^5 + 95/918*e^4 + 13015/3672*e^3 - 4843/1836*e^2 - 14737/918*e - 25435/459, 37/1836*e^5 + 2/459*e^4 - 3245/1836*e^3 - 3505/918*e^2 + 4043/459*e + 29062/459, -4/153*e^5 + 41/306*e^4 + 275/153*e^3 - 2177/306*e^2 + 1546/153*e - 4774/153, 59/1836*e^5 + 28/459*e^4 - 6415/1836*e^3 - 8219/918*e^2 + 28144/459*e + 31406/459, 305/3672*e^5 - 335/918*e^4 - 24781/3672*e^3 + 38965/1836*e^2 + 23461/918*e - 22739/459, -19/918*e^5 + 122/459*e^4 + 1013/918*e^3 - 9365/459*e^2 + 10238/459*e + 32560/459, -77/3672*e^5 + 31/459*e^4 + 6505/3672*e^3 - 4615/1836*e^2 - 10531/918*e - 23839/459, -167/1836*e^5 + 206/459*e^4 + 13687/1836*e^3 - 24721/918*e^2 - 17785/459*e + 31918/459, -305/3672*e^5 + 244/459*e^4 + 22333/3672*e^3 - 63751/1836*e^2 + 5609/918*e + 53645/459, 73/918*e^5 - 203/459*e^4 - 5567/918*e^3 + 12524/459*e^2 + 670/459*e - 27952/459, -139/3672*e^5 + 50/459*e^4 + 11711/3672*e^3 - 8021/1836*e^2 - 11285/918*e - 13817/459, 3/34*e^5 - 22/51*e^4 - 725/102*e^3 + 439/17*e^2 + 1478/51*e - 1918/51, 71/612*e^5 - 83/153*e^4 - 5863/612*e^3 + 9823/306*e^2 + 7018/153*e - 12826/153, -1/36*e^5 + 1/9*e^4 + 89/36*e^3 - 113/18*e^2 - 152/9*e - 142/9, 35/1224*e^5 - 28/153*e^4 - 2623/1224*e^3 + 7717/612*e^2 - 443/306*e - 14117/153, 2/459*e^5 - 755/1836*e^4 + 1051/918*e^3 + 62459/1836*e^2 - 38054/459*e - 83395/459, -251/1836*e^5 + 218/459*e^4 + 21451/1836*e^3 - 22495/918*e^2 - 32542/459*e + 1270/459, -5/1224*e^5 + 59/306*e^4 - 383/1224*e^3 - 9889/612*e^2 + 10409/306*e + 7889/153, -23/459*e^5 + 86/459*e^4 + 1951/459*e^3 - 4748/459*e^2 - 11128/459*e + 268/459, 1/918*e^5 - 95/459*e^4 + 607/918*e^3 + 8006/459*e^2 - 18770/459*e - 51028/459, 227/3672*e^5 - 304/459*e^4 - 13375/3672*e^3 + 88309/1836*e^2 - 54491/918*e - 81557/459, -677/3672*e^5 + 563/918*e^4 + 58465/3672*e^3 - 55729/1836*e^2 - 94999/918*e - 1279/459, 13/204*e^5 - 107/204*e^4 - 847/204*e^3 + 7369/204*e^2 - 1832/51*e - 6115/51, -1/68*e^5 - 2/51*e^4 + 287/204*e^3 + 171/34*e^2 - 473/51*e - 2236/51, 131/918*e^5 - 563/918*e^4 - 10753/918*e^3 + 31613/918*e^2 + 20648/459*e - 3158/459, -217/1836*e^5 + 286/459*e^4 + 16997/1836*e^3 - 34871/918*e^2 - 5699/459*e + 45062/459, -77/459*e^5 + 649/918*e^4 + 6352/459*e^3 - 36361/918*e^2 - 27130/459*e + 28078/459, 137/1836*e^5 - 469/918*e^4 - 9865/1836*e^3 + 15512/459*e^2 - 9419/459*e - 34864/459];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 5*w + 1}>] := -1;
ALEigenvalues[ideal<ZF | {7, 7, -w^5 + 3*w^4 + 3*w^3 - 10*w^2 - 3*w + 5}>] := -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;