/*
  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 | {49, 7, w^4 - 2*w^3 - 3*w^2 + 4*w + 2}>;

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^2 - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 0, -8, -3/2*e, -3/2*e, -3/2*e, -3/2*e, -9/2*e, -9/2*e, 8, 9/2*e, 9/2*e, 8, -7, -3*e, -3*e, -12, -12, -16, -9/2*e, -9/2*e, 4, 4, 12, 12, 9/2*e, 0, 0, 9/2*e, 9/2*e, 9/2*e, -6*e, -6*e, 3*e, 3*e, 0, 0, -15/2*e, -15/2*e, -24, 9/2*e, 9/2*e, -24, -24, -24, -8, 15/2*e, 15/2*e, -8, 6, 6, 0, 0, 6*e, 6*e, -8, -8, -9/2*e, -9/2*e, -12, -12, -9/2*e, -9/2*e, -12, -12, -2, -2, -38, 28, 28, 12, 12, 9*e, 9*e, -24, -24, -30, -30, 0, 0, 4, 4, -30, -30, -16, -16, -9/2*e, -9/2*e, -2, -2, 9*e, 12, -9*e, -9*e, 12, 9*e, -32, -6*e, -6*e, -24, -24, -32, -4, -4, -32, -24, 48, 48, -24, -9*e, -9*e, 21/2*e, 21/2*e, -44, -44, 21/2*e, 21/2*e, 3*e, 3*e, 27/2*e, 27/2*e, -24, 0, 0, -24, 18*e, 18*e, -9*e, -9*e, 9/2*e, 9/2*e, -9/2*e, -9/2*e, -4, -4, -4, -4, 0, 0, 0, 0, 40, 40, 24, 24, -9*e, -9*e, 28, 9*e, -3*e, -3*e, 9*e, 28, -16, -16, 9/2*e, 9/2*e, 2, 2, -18, -18, -48, 24, 24, -48, 42, 42, 9*e, 9*e, 3/2*e, 33/2*e, 33/2*e, 3/2*e, 27/2*e, 27/2*e, 18*e, 18*e, -12*e, 18*e, 18*e, -12*e, -15*e, -15*e, 27/2*e, 27/2*e, 44, 44, 27/2*e, 27/2*e, 60, 60, -9/2*e, -9/2*e, -45/2*e, 18, 9/2*e, 9/2*e, 18, -45/2*e, -9*e, 9*e, 9*e, -9*e, 27/2*e, 27/2*e, 0, 0, -28, -28, 22, 22, 56, 56, -3*e, -3*e, -15/2*e, -15/2*e, -8, -8, -12, -12, 21/2*e, 21/2*e, 45/2*e, -48, -48, 45/2*e, -9/2*e, -9/2*e, -18*e, -18*e, 0, 0, 27/2*e, 27/2*e, 18*e, 18*e, -12, -12, -18*e, -18*e, 40, 40, -27*e, -27*e, -60, -60, 34, 34, -9*e, -9*e, -36, -36, -9/2*e, -9/2*e, 27*e, 27*e, -44, -44, 27/2*e, 27/2*e, -2, -2, -82, 2, 2, 27/2*e, 27/2*e, -9/2*e, -9/2*e, 48, 48, 32, 32, 6*e, 6*e, 40, 21/2*e, 21/2*e, 40, 6*e, 20, 20, 6*e, -57/2*e, -57/2*e, -27*e, -27*e, 36, 36, 12, 12, -14, 27/2*e, 27/2*e, -66, 45/2*e, 45/2*e, -66, 48, 48, -30, -30, 6, 6, 48, 48, -21*e, -21*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
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;