/*
  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![14, -4, -10, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {18,6,w - 2}>;

primesArray := [
[2, 2, -w - 2],
[7, 7, -2/3*w^3 + 2/3*w^2 + 5*w - 7/3],
[7, 7, -1/3*w^3 + 1/3*w^2 + w - 5/3],
[9, 3, -1/3*w^3 + 1/3*w^2 + 3*w - 5/3],
[9, 3, w + 1],
[17, 17, w + 3],
[17, 17, -1/3*w^3 + 1/3*w^2 + 3*w - 11/3],
[31, 31, 1/3*w^3 - 1/3*w^2 - w - 1/3],
[31, 31, -2/3*w^3 + 2/3*w^2 + 5*w - 1/3],
[41, 41, -2/3*w^3 + 5/3*w^2 + 4*w - 19/3],
[41, 41, w^2 - 5],
[41, 41, 2*w + 3],
[41, 41, 2/3*w^3 + 4/3*w^2 - 5*w - 29/3],
[47, 47, -2/3*w^3 + 5/3*w^2 + 3*w - 19/3],
[47, 47, 1/3*w^3 + 2/3*w^2 - 3*w - 13/3],
[49, 7, -1/3*w^3 + 1/3*w^2 + 2*w - 11/3],
[73, 73, -1/3*w^3 + 1/3*w^2 + 5*w + 19/3],
[73, 73, -2/3*w^3 - 1/3*w^2 + 4*w + 11/3],
[73, 73, -1/3*w^3 + 4/3*w^2 + 2*w - 17/3],
[103, 103, -1/3*w^3 + 4/3*w^2 + w - 23/3],
[103, 103, w^2 - w - 3],
[113, 113, -2/3*w^3 + 2/3*w^2 + 5*w + 11/3],
[113, 113, 1/3*w^3 - 1/3*w^2 - w - 13/3],
[127, 127, 2/3*w^3 + 1/3*w^2 - 9*w - 41/3],
[127, 127, -2/3*w^3 + 5/3*w^2 + 5*w - 31/3],
[127, 127, -1/3*w^3 + 1/3*w^2 - 11/3],
[127, 127, 4/3*w^3 + 5/3*w^2 - 10*w - 43/3],
[151, 151, -1/3*w^3 + 4/3*w^2 + w - 29/3],
[151, 151, 2/3*w^3 - 2/3*w^2 - 6*w - 5/3],
[167, 167, -2/3*w^3 + 5/3*w^2 + 2*w - 19/3],
[167, 167, 2/3*w^3 + 1/3*w^2 - 6*w - 11/3],
[191, 191, -w^3 + 2*w^2 + 6*w - 5],
[191, 191, 1/3*w^3 + 2/3*w^2 - 2*w - 19/3],
[193, 193, -1/3*w^3 + 1/3*w^2 + 5*w + 13/3],
[193, 193, 2/3*w^3 + 7/3*w^2 - 2*w - 23/3],
[199, 199, 1/3*w^3 + 2/3*w^2 + 2*w + 11/3],
[199, 199, 1/3*w^3 + 2/3*w^2 - 5*w - 31/3],
[223, 223, -2/3*w^3 + 5/3*w^2 + 5*w - 13/3],
[223, 223, 4/3*w^3 - 7/3*w^2 - 8*w + 17/3],
[223, 223, 2/3*w^3 + 1/3*w^2 - 4*w - 17/3],
[223, 223, w^3 - w^2 - 7*w - 1],
[233, 233, -1/3*w^3 + 1/3*w^2 + w - 17/3],
[233, 233, -2/3*w^3 + 2/3*w^2 + 5*w - 19/3],
[239, 239, 5/3*w^3 + 7/3*w^2 - 10*w - 41/3],
[239, 239, -5/3*w^3 - 7/3*w^2 + 12*w + 59/3],
[241, 241, -1/3*w^3 + 7/3*w^2 + w - 23/3],
[241, 241, 5/3*w^3 + 10/3*w^2 - 10*w - 59/3],
[257, 257, -1/3*w^3 + 1/3*w^2 + 4*w - 5/3],
[257, 257, 1/3*w^3 - 1/3*w^2 - 4*w - 1/3],
[257, 257, 2/3*w^3 + 1/3*w^2 - 7*w - 23/3],
[257, 257, -4/3*w^3 + 1/3*w^2 + 10*w + 13/3],
[263, 263, -4/3*w^3 + 7/3*w^2 + 9*w - 23/3],
[263, 263, 1/3*w^3 + 2/3*w^2 - w - 13/3],
[271, 271, 5/3*w^3 - 5/3*w^2 - 12*w + 19/3],
[271, 271, w^3 - w^2 - 4*w + 5],
[281, 281, w^2 + 5*w + 5],
[281, 281, -1/3*w^3 - 5/3*w^2 + 2*w + 25/3],
[289, 17, w^3 - w^2 - 6*w + 3],
[311, 311, 1/3*w^3 - 7/3*w^2 + 41/3],
[311, 311, -2*w + 5],
[311, 311, -w^3 - 2*w^2 + 7*w + 15],
[311, 311, 2/3*w^3 - 2/3*w^2 - 6*w - 11/3],
[313, 313, w^3 - w^2 - 5*w + 1],
[313, 313, -4/3*w^3 + 4/3*w^2 + 9*w - 5/3],
[337, 337, 4/3*w^3 - 4/3*w^2 - 11*w + 5/3],
[337, 337, 2/3*w^3 + 1/3*w^2 - 6*w - 29/3],
[359, 359, -w^3 - w^2 + 12*w + 19],
[359, 359, -1/3*w^3 + 4/3*w^2 + 5*w + 1/3],
[359, 359, -1/3*w^3 - 8/3*w^2 - 4*w + 1/3],
[359, 359, w^2 - 3*w - 9],
[361, 19, 2*w^2 - w - 13],
[361, 19, -w^3 + 3*w^2 + 5*w - 9],
[409, 409, -w - 5],
[409, 409, -1/3*w^3 + 1/3*w^2 + 3*w - 17/3],
[431, 431, -w^3 - 3*w^2 + 2*w + 9],
[431, 431, 2/3*w^3 + 7/3*w^2 - w - 23/3],
[433, 433, -1/3*w^3 + 1/3*w^2 + 6*w + 19/3],
[433, 433, 2/3*w^3 + 7/3*w^2 - 3*w - 29/3],
[439, 439, 2/3*w^3 + 4/3*w^2 - 8*w - 47/3],
[439, 439, 4/3*w^3 - 10/3*w^2 - 8*w + 29/3],
[439, 439, 2*w^2 - 13],
[439, 439, 5*w + 9],
[449, 449, w^2 - 11],
[449, 449, 2/3*w^3 - 5/3*w^2 - 4*w + 1/3],
[457, 457, -w^3 - w^2 + 10*w + 17],
[457, 457, -4/3*w^3 + 4/3*w^2 + 7*w + 1/3],
[457, 457, 4/3*w^3 + 5/3*w^2 - 5*w - 19/3],
[457, 457, -w^3 + 2*w^2 + 7*w - 13],
[463, 463, w^3 - 2*w^2 - 5*w + 9],
[463, 463, 2/3*w^3 + 1/3*w^2 - 5*w - 5/3],
[463, 463, 5/3*w^3 - 11/3*w^2 - 9*w + 43/3],
[463, 463, 2/3*w^3 + 4/3*w^2 - 5*w - 23/3],
[487, 487, -4/3*w^3 - 8/3*w^2 + 7*w + 37/3],
[487, 487, w^3 + w^2 - 8*w - 9],
[487, 487, -2/3*w^3 - 4/3*w^2 + 4*w + 17/3],
[487, 487, 5/3*w^3 - 11/3*w^2 - 8*w + 37/3],
[503, 503, 2/3*w^3 + 4/3*w^2 - 4*w - 29/3],
[503, 503, 2/3*w^3 + 1/3*w^2 - 10*w - 47/3],
[503, 503, 1/3*w^3 - 1/3*w^2 + w + 17/3],
[503, 503, -4/3*w^3 - 8/3*w^2 + 5*w + 31/3],
[521, 521, -1/3*w^3 + 4/3*w^2 + 9*w + 31/3],
[521, 521, w^3 - 6*w - 3],
[529, 23, -1/3*w^3 + 1/3*w^2 + 2*w - 17/3],
[529, 23, 1/3*w^3 - 1/3*w^2 - 2*w - 13/3],
[569, 569, -4/3*w^3 - 5/3*w^2 + 7*w + 25/3],
[569, 569, -1/3*w^3 - 5/3*w^2 - 2*w - 5/3],
[577, 577, -1/3*w^3 + 7/3*w^2 + 3*w - 23/3],
[577, 577, -4/3*w^3 + 10/3*w^2 + 9*w - 47/3],
[599, 599, -4/3*w^3 - 8/3*w^2 + 6*w + 37/3],
[599, 599, 2/3*w^3 + 4/3*w^2 - 3*w - 23/3],
[601, 601, -2/3*w^3 + 8/3*w^2 + 4*w - 43/3],
[601, 601, -2/3*w^3 + 8/3*w^2 + 4*w - 25/3],
[617, 617, -1/3*w^3 + 7/3*w^2 + 2*w - 23/3],
[617, 617, w^3 - 3*w^2 - 6*w + 15],
[625, 5, -5],
[631, 631, -2*w^3 + 2*w^2 + 14*w - 1],
[631, 631, -w^3 + 10*w - 1],
[641, 641, -w^2 - 1],
[641, 641, -2/3*w^3 + 5/3*w^2 + 4*w - 37/3],
[641, 641, 2/3*w^3 - 8/3*w^2 - 5*w + 37/3],
[641, 641, w^3 - 3*w^2 - 7*w + 11],
[647, 647, -1/3*w^3 - 2/3*w^2 - 3*w - 17/3],
[647, 647, 1/3*w^3 + 2/3*w^2 - 6*w - 37/3],
[673, 673, 2/3*w^3 - 2/3*w^2 - 7*w - 5/3],
[673, 673, 2/3*w^3 - 5/3*w^2 - 3*w + 1/3],
[673, 673, -1/3*w^3 + 1/3*w^2 + 5*w - 11/3],
[673, 673, 1/3*w^3 + 2/3*w^2 - 3*w - 31/3],
[719, 719, -w^3 - 2*w^2 + 4*w + 9],
[719, 719, -w^3 - 2*w^2 + 5*w + 11],
[727, 727, -w^3 + 3*w^2 + 4*w - 13],
[727, 727, -1/3*w^3 + 4/3*w^2 + 4*w - 23/3],
[727, 727, 1/3*w^3 + 5/3*w^2 - 4*w - 25/3],
[727, 727, -2/3*w^3 - 1/3*w^2 + 4*w + 23/3],
[743, 743, 1/3*w^3 + 2/3*w^2 + 3*w + 23/3],
[743, 743, -1/3*w^3 + 7/3*w^2 + 2*w - 53/3],
[751, 751, -2/3*w^3 + 5/3*w^2 + 2*w - 25/3],
[751, 751, -2/3*w^3 - 1/3*w^2 + 6*w + 5/3],
[761, 761, -1/3*w^3 + 7/3*w^2 + w - 29/3],
[761, 761, -2/3*w^3 + 8/3*w^2 + 3*w - 37/3],
[769, 769, -4/3*w^3 + 13/3*w^2 + 6*w - 59/3],
[769, 769, -7/3*w^3 + 7/3*w^2 + 16*w - 5/3],
[823, 823, -4/3*w^3 + 1/3*w^2 + 11*w + 25/3],
[823, 823, w^3 - 2*w^2 - 3*w + 1],
[857, 857, w^2 - 3*w - 5],
[857, 857, -2/3*w^3 - 1/3*w^2 + 8*w - 1/3],
[857, 857, 1/3*w^3 + 2/3*w^2 - 5*w - 13/3],
[857, 857, w^2 - 4*w - 9],
[863, 863, -4/3*w^3 + 7/3*w^2 + 8*w - 5/3],
[863, 863, 2/3*w^3 + 1/3*w^2 - 4*w - 29/3],
[881, 881, -5/3*w^3 - 4/3*w^2 + 11*w + 41/3],
[881, 881, -4/3*w^3 - 5/3*w^2 + 6*w + 19/3],
[887, 887, -4/3*w^3 + 4/3*w^2 + 10*w + 1/3],
[887, 887, 2/3*w^3 - 2/3*w^2 - 2*w - 5/3],
[911, 911, w^3 - 9*w - 1],
[911, 911, 5/3*w^3 + 10/3*w^2 - 12*w - 65/3],
[911, 911, 1/3*w^3 - 1/3*w^2 + w + 23/3],
[911, 911, -2/3*w^3 + 5/3*w^2 + w - 25/3],
[919, 919, -2/3*w^3 + 8/3*w^2 - 37/3],
[919, 919, 2/3*w^3 + 4/3*w^2 - 8*w - 23/3],
[929, 929, -2*w^3 - 3*w^2 + 11*w + 17],
[929, 929, 1/3*w^3 - 1/3*w^2 - 4*w - 19/3],
[937, 937, 4/3*w^3 - 1/3*w^2 - 9*w - 13/3],
[937, 937, 3*w - 1],
[937, 937, -w^3 + w^2 + 7*w - 9],
[937, 937, w^3 - w^2 - 9*w + 1],
[961, 31, 4/3*w^3 - 4/3*w^2 - 8*w + 11/3],
[967, 967, w^2 - 3*w - 11],
[967, 967, 1/3*w^3 + 2/3*w^2 - w - 19/3],
[967, 967, 1/3*w^3 + 5/3*w^2 + 6*w + 23/3],
[967, 967, 4/3*w^3 - 7/3*w^2 - 9*w + 17/3],
[977, 977, w^3 - 4*w^2 - 5*w + 23],
[977, 977, 2/3*w^3 - 11/3*w^2 - 3*w + 31/3],
[983, 983, 4/3*w^3 - 4/3*w^2 - 6*w + 23/3],
[983, 983, -2*w^3 + 2*w^2 + 14*w - 9],
[991, 991, 2/3*w^3 + 1/3*w^2 - 5*w + 1/3],
[991, 991, -w^3 + 2*w^2 + 5*w - 11],
[1009, 1009, -1/3*w^3 + 1/3*w^2 + 9*w + 37/3],
[1009, 1009, 2/3*w^3 + 7/3*w^2 - 6*w - 47/3],
[1031, 1031, -4/3*w^3 + 10/3*w^2 + 7*w - 47/3],
[1031, 1031, w^3 - w^2 - 9*w - 3],
[1031, 1031, 1/3*w^3 - 7/3*w^2 + 53/3],
[1031, 1031, 1/3*w^3 + 5/3*w^2 - 3*w - 19/3],
[1033, 1033, -4/3*w^3 - 5/3*w^2 + 5*w + 13/3],
[1033, 1033, 2*w^3 + 2*w^2 - 14*w - 19],
[1039, 1039, -4/3*w^3 + 10/3*w^2 + 10*w - 53/3],
[1039, 1039, -2/3*w^3 + 8/3*w^2 + 6*w - 19/3],
[1063, 1063, -1/3*w^3 - 8/3*w^2 + 4*w + 37/3],
[1063, 1063, -2/3*w^3 + 5/3*w^2 + 2*w - 43/3],
[1063, 1063, -5/3*w^3 + 14/3*w^2 + 8*w - 61/3],
[1063, 1063, 2/3*w^3 + 4/3*w^2 - 5*w - 41/3],
[1097, 1097, -4/3*w^3 + 7/3*w^2 + 8*w - 47/3],
[1097, 1097, 5/3*w^3 - 8/3*w^2 - 8*w + 19/3],
[1097, 1097, -2/3*w^3 + 8/3*w^2 + 4*w - 31/3],
[1097, 1097, -2/3*w^3 + 8/3*w^2 + 4*w - 37/3],
[1129, 1129, 2/3*w^3 + 10/3*w^2 - w - 29/3],
[1129, 1129, -2/3*w^3 + 2/3*w^2 + 10*w + 29/3],
[1151, 1151, -w^3 + w^2 + 15*w + 17],
[1151, 1151, -2/3*w^3 + 5/3*w^2 + 6*w - 1/3],
[1153, 1153, w^3 - w^2 - 9*w + 3],
[1153, 1153, -3*w - 1],
[1201, 1201, 1/3*w^3 + 8/3*w^2 - 4*w - 43/3],
[1201, 1201, -5/3*w^3 - 13/3*w^2 + 7*w + 53/3],
[1231, 1231, 2/3*w^3 + 1/3*w^2 - 3*w - 17/3],
[1231, 1231, -5/3*w^3 + 8/3*w^2 + 11*w - 19/3],
[1279, 1279, 4/3*w^3 - 1/3*w^2 - 10*w - 1/3],
[1279, 1279, -4/3*w^3 + 7/3*w^2 + 6*w - 29/3],
[1279, 1279, 2/3*w^3 + 1/3*w^2 - 7*w + 1/3],
[1279, 1279, -1/3*w^3 + 4/3*w^2 - w - 29/3],
[1297, 1297, -2*w^3 - 3*w^2 + 9*w + 11],
[1297, 1297, w^3 + 3*w^2 - 1],
[1303, 1303, 8/3*w^3 - 11/3*w^2 - 17*w + 19/3],
[1303, 1303, 4/3*w^3 + 2/3*w^2 - 13*w - 19/3],
[1319, 1319, 5/3*w^3 - 8/3*w^2 - 9*w + 37/3],
[1319, 1319, 4/3*w^3 - 1/3*w^2 - 9*w + 5/3],
[1321, 1321, -1/3*w^3 + 1/3*w^2 + 5*w - 5/3],
[1321, 1321, 2/3*w^3 - 2/3*w^2 - 7*w + 1/3],
[1327, 1327, 2/3*w^3 + 1/3*w^2 - 11*w - 53/3],
[1327, 1327, -1/3*w^3 + 1/3*w^2 - 2*w - 23/3],
[1361, 1361, 2/3*w^3 + 1/3*w^2 - 8*w - 41/3],
[1361, 1361, -5/3*w^3 - 7/3*w^2 + 7*w + 29/3],
[1367, 1367, -1/3*w^3 + 7/3*w^2 + 3*w - 41/3],
[1367, 1367, 4/3*w^3 - 10/3*w^2 - 9*w + 29/3],
[1369, 37, -4/3*w^3 + 10/3*w^2 + 9*w - 65/3],
[1369, 37, 1/3*w^3 - 7/3*w^2 - 3*w + 5/3],
[1409, 1409, -2/3*w^3 - 13/3*w^2 - w + 29/3],
[1409, 1409, -4/3*w^3 - 2/3*w^2 + 5*w + 1/3],
[1423, 1423, 2/3*w^3 + 1/3*w^2 + 19/3],
[1423, 1423, 7/3*w^3 - 16/3*w^2 - 12*w + 59/3],
[1423, 1423, -4/3*w^3 - 2/3*w^2 + 9*w + 37/3],
[1423, 1423, 7/3*w^3 - 13/3*w^2 - 13*w + 29/3],
[1433, 1433, 1/3*w^3 - 1/3*w^2 - w - 19/3],
[1433, 1433, -2*w^3 - 2*w^2 + 13*w + 15],
[1433, 1433, 2/3*w^3 - 2/3*w^2 - 5*w - 17/3],
[1433, 1433, -2/3*w^3 + 11/3*w^2 + 2*w - 61/3],
[1439, 1439, 2/3*w^3 + 4/3*w^2 - 6*w - 23/3],
[1439, 1439, -4/3*w^3 + 10/3*w^2 + 6*w - 41/3],
[1447, 1447, -5/3*w^3 + 11/3*w^2 + 7*w - 37/3],
[1447, 1447, 2/3*w^3 - 5/3*w^2 - 8*w + 31/3],
[1471, 1471, 5/3*w^3 - 2/3*w^2 - 14*w + 1/3],
[1471, 1471, -w^3 + 2*w^2 + 2*w - 9],
[1481, 1481, -1/3*w^3 - 5/3*w^2 + 7*w + 43/3],
[1481, 1481, -8/3*w^3 + 17/3*w^2 + 17*w - 73/3],
[1489, 1489, 4*w + 5],
[1489, 1489, 4/3*w^3 - 4/3*w^2 - 12*w + 23/3],
[1511, 1511, w^3 - 4*w^2 - 3*w + 9],
[1511, 1511, 3*w^2 - 3*w - 23],
[1543, 1543, 2/3*w^3 + 1/3*w^2 - 6*w + 1/3],
[1543, 1543, -2/3*w^3 + 5/3*w^2 + 2*w - 31/3],
[1553, 1553, -7/3*w^3 + 10/3*w^2 + 13*w - 41/3],
[1553, 1553, 2*w^3 - w^2 - 13*w + 3],
[1559, 1559, 2*w^2 - 15],
[1559, 1559, -4/3*w^3 + 10/3*w^2 + 8*w - 23/3],
[1567, 1567, -1/3*w^3 - 2/3*w^2 - 4*w - 23/3],
[1567, 1567, -1/3*w^3 - 2/3*w^2 + 7*w + 43/3],
[1601, 1601, -2/3*w^3 + 2/3*w^2 + 5*w - 25/3],
[1601, 1601, -1/3*w^3 + 1/3*w^2 + w - 23/3],
[1607, 1607, -7/3*w^3 + 10/3*w^2 + 18*w - 29/3],
[1607, 1607, -2*w^3 + 4*w^2 + 12*w - 19],
[1607, 1607, -2/3*w^3 - 4/3*w^2 + 4*w + 11/3],
[1607, 1607, -4/3*w^3 - 5/3*w^2 + 11*w + 43/3],
[1657, 1657, -4/3*w^3 + 1/3*w^2 + 6*w - 5/3],
[1657, 1657, 8/3*w^3 - 11/3*w^2 - 18*w + 43/3],
[1697, 1697, 2/3*w^3 - 2/3*w^2 - 8*w - 11/3],
[1697, 1697, -2/3*w^3 - 1/3*w^2 + 10*w + 41/3],
[1697, 1697, -w^3 - 3*w^2 + 7*w + 17],
[1697, 1697, -2/3*w^3 - 1/3*w^2 + 13*w + 59/3],
[1721, 1721, -2/3*w^3 - 7/3*w^2 - 4*w - 19/3],
[1721, 1721, -5/3*w^3 - 7/3*w^2 + 5*w + 11/3],
[1753, 1753, -7/3*w^3 - 14/3*w^2 + 15*w + 85/3],
[1753, 1753, -5/3*w^3 - 13/3*w^2 + 11*w + 71/3],
[1753, 1753, 1/3*w^3 + 2/3*w^2 - 6*w - 25/3],
[1753, 1753, 1/3*w^3 + 8/3*w^2 - 2*w - 43/3],
[1759, 1759, -1/3*w^3 + 4/3*w^2 + 4*w - 29/3],
[1759, 1759, -w^3 + 2*w^2 + 8*w - 3],
[1777, 1777, -2/3*w^3 + 11/3*w^2 + 4*w - 31/3],
[1777, 1777, -5/3*w^3 + 14/3*w^2 + 12*w - 67/3],
[1777, 1777, -4/3*w^3 + 13/3*w^2 + 8*w - 71/3],
[1777, 1777, w^3 - 4*w^2 - 8*w + 13],
[1783, 1783, -5/3*w^3 + 2/3*w^2 + 13*w + 29/3],
[1783, 1783, 4/3*w^3 - 7/3*w^2 - 5*w - 1/3],
[1831, 1831, -1/3*w^3 - 5/3*w^2 + 25/3],
[1831, 1831, 4/3*w^3 + 2/3*w^2 - 7*w - 25/3],
[1831, 1831, -7/3*w^3 + 13/3*w^2 + 16*w - 47/3],
[1831, 1831, 3*w^3 - 5*w^2 - 19*w + 15],
[1847, 1847, -4/3*w^3 - 2/3*w^2 + 8*w + 31/3],
[1847, 1847, -8/3*w^3 - 10/3*w^2 + 19*w + 89/3],
[1889, 1889, -2/3*w^3 - 1/3*w^2 + 10*w + 53/3],
[1889, 1889, 2/3*w^3 + 7/3*w^2 + 3*w + 13/3],
[1889, 1889, 3*w^3 - 2*w^2 - 24*w + 1],
[1889, 1889, -5/3*w^3 - 7/3*w^2 + 5*w + 17/3],
[1913, 1913, 5/3*w^3 - 5/3*w^2 - 9*w + 1/3],
[1913, 1913, -4/3*w^3 + 7/3*w^2 + 9*w - 47/3],
[1951, 1951, -7/3*w^3 - 8/3*w^2 + 17*w + 67/3],
[1951, 1951, 4/3*w^3 - 10/3*w^2 - 10*w + 59/3],
[1993, 1993, -8/3*w^3 + 8/3*w^2 + 19*w - 19/3],
[1993, 1993, 5/3*w^3 - 5/3*w^2 - 7*w + 13/3],
[1999, 1999, 2/3*w^3 + 1/3*w^2 - 3*w - 41/3],
[1999, 1999, -5/3*w^3 + 8/3*w^2 + 11*w + 5/3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 + 4*x^7 - 27*x^6 - 117*x^5 + 133*x^4 + 843*x^3 + 673*x^2 + 6*x - 76;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, -4052/86739*e^7 - 27575/173478*e^6 + 231031/173478*e^5 + 13981/2991*e^4 - 491295/57826*e^3 - 996537/28913*e^2 - 2279659/173478*e + 539263/86739, 5042/86739*e^7 + 30245/173478*e^6 - 294199/173478*e^5 - 15178/2991*e^4 + 690377/57826*e^3 + 1065552/28913*e^2 + 755797/173478*e - 661813/86739, 1, 683/173478*e^7 + 3308/86739*e^6 - 21965/173478*e^5 - 6835/5982*e^4 + 51121/57826*e^3 + 508031/57826*e^2 + 267875/173478*e - 484292/86739, 430/28913*e^7 + 1456/28913*e^6 - 11966/28913*e^5 - 1245/997*e^4 + 78012/28913*e^3 + 213466/28913*e^2 + 60700/28913*e - 60238/28913, -9151/173478*e^7 - 17779/86739*e^6 + 268639/173478*e^5 + 34367/5982*e^4 - 623465/57826*e^3 - 2288167/57826*e^2 - 1305901/173478*e + 326764/86739, 2363/86739*e^7 + 6253/86739*e^6 - 80819/86739*e^5 - 6691/2991*e^4 + 242819/28913*e^3 + 501610/28913*e^2 - 725455/86739*e - 635086/86739, 2365/57826*e^7 + 4004/28913*e^6 - 65813/57826*e^5 - 8343/1994*e^4 + 342327/57826*e^3 + 1810149/57826*e^2 + 1519283/57826*e - 35546/28913, -6859/86739*e^7 - 19325/86739*e^6 + 204454/86739*e^5 + 19523/2991*e^4 - 495347/28913*e^3 - 1395048/28913*e^2 - 387286/86739*e + 1132058/86739, 7823/86739*e^7 + 26758/86739*e^6 - 228725/86739*e^5 - 27160/2991*e^4 + 504425/28913*e^3 + 1904707/28913*e^2 + 2161322/86739*e - 375238/86739, 18271/173478*e^7 + 29185/86739*e^6 - 538567/173478*e^5 - 60077/5982*e^4 + 1218025/57826*e^3 + 4343307/57826*e^2 + 4190917/173478*e - 1366987/86739, -35093/173478*e^7 - 51479/86739*e^6 + 1019867/173478*e^5 + 104725/5982*e^4 - 2297455/57826*e^3 - 7451423/57826*e^2 - 6170189/173478*e + 1599479/86739, 2098/86739*e^7 + 10039/173478*e^6 - 119321/173478*e^5 - 5147/2991*e^4 + 274057/57826*e^3 + 388950/28913*e^2 - 91507/173478*e - 739838/86739, 7445/173478*e^7 + 42019/173478*e^6 - 101236/86739*e^5 - 40951/5982*e^4 + 170652/28913*e^3 + 2764145/57826*e^2 + 2504338/86739*e - 747740/86739, -1257/28913*e^7 - 8647/57826*e^6 + 75473/57826*e^5 + 4393/997*e^4 - 517555/57826*e^3 - 965996/28913*e^2 - 680995/57826*e + 436846/28913, -6734/86739*e^7 - 50579/173478*e^6 + 405313/173478*e^5 + 24946/2991*e^4 - 974903/57826*e^3 - 1698682/28913*e^2 - 1939507/173478*e + 587389/86739, 3475/173478*e^7 - 1009/173478*e^6 - 46670/86739*e^5 + 13/5982*e^4 + 90954/28913*e^3 + 99875/57826*e^2 + 469178/86739*e + 487154/86739, 9173/57826*e^7 + 12975/28913*e^6 - 277051/57826*e^5 - 26617/1994*e^4 + 2017741/57826*e^3 + 5741495/57826*e^2 + 785883/57826*e - 694357/28913, 5465/173478*e^7 + 39349/173478*e^6 - 69652/86739*e^5 - 38557/5982*e^4 + 71111/28913*e^3 + 2626115/57826*e^2 + 3353008/86739*e - 451712/86739, 4845/28913*e^7 + 15733/28913*e^6 - 136171/28913*e^5 - 15964/997*e^4 + 831928/28913*e^3 + 3380189/28913*e^2 + 1539893/28913*e - 413132/28913, -8522/86739*e^7 - 65915/173478*e^6 + 463675/173478*e^5 + 33253/2991*e^4 - 815425/57826*e^3 - 2301706/28913*e^2 - 9230119/173478*e + 156865/86739, 2727/57826*e^7 + 3810/28913*e^6 - 67549/57826*e^5 - 8255/1994*e^4 + 248779/57826*e^3 + 1877971/57826*e^2 + 2051819/57826*e - 25534/28913, -6685/173478*e^7 - 20059/86739*e^6 + 208891/173478*e^5 + 37313/5982*e^4 - 474873/57826*e^3 - 2313391/57826*e^2 - 3814129/173478*e - 401498/86739, 5798/86739*e^7 + 27970/86739*e^6 - 156236/86739*e^5 - 26479/2991*e^4 + 281105/28913*e^3 + 1688629/28913*e^2 + 2966765/86739*e + 451064/86739, 14447/173478*e^7 + 27805/173478*e^6 - 201889/86739*e^5 - 28045/5982*e^4 + 443023/28913*e^3 + 1966955/57826*e^2 + 646510/86739*e - 132368/86739, -37781/173478*e^7 - 119725/173478*e^6 + 530995/86739*e^5 + 123637/5982*e^4 - 1091021/28913*e^3 - 8979321/57826*e^2 - 5890099/86739*e + 1905158/86739, 9835/173478*e^7 + 28595/173478*e^6 - 127094/86739*e^5 - 31061/5982*e^4 + 175007/28913*e^3 + 2341107/57826*e^2 + 3675569/86739*e + 9398/86739, -50669/173478*e^7 - 152875/173478*e^6 + 755503/86739*e^5 + 157057/5982*e^4 - 1809385/28913*e^3 - 11312907/57826*e^2 - 2321599/86739*e + 3806798/86739, 19235/173478*e^7 + 65803/173478*e^6 - 281419/86739*e^5 - 64723/5982*e^4 + 656921/28913*e^3 + 4419271/57826*e^2 + 857371/86739*e - 1595750/86739, 19091/86739*e^7 + 139103/173478*e^6 - 1118065/173478*e^5 - 68410/2991*e^4 + 2570865/57826*e^3 + 4666821/28913*e^2 + 6239113/173478*e - 2944999/86739, -16023/57826*e^7 - 25917/28913*e^6 + 462293/57826*e^5 + 51667/1994*e^4 - 3043979/57826*e^3 - 10735635/57826*e^2 - 3880307/57826*e + 635571/28913, -8915/57826*e^7 - 30859/57826*e^6 + 137827/28913*e^5 + 30855/1994*e^4 - 1034619/28913*e^3 - 6399849/57826*e^2 - 360275/28913*e + 525938/28913, 6124/86739*e^7 + 24905/86739*e^6 - 167863/86739*e^5 - 25568/2991*e^4 + 292213/28913*e^3 + 1797218/28913*e^2 + 3716782/86739*e + 33964/86739, 159/997*e^7 + 441/997*e^6 - 4420/997*e^5 - 13107/997*e^4 + 27214/997*e^3 + 98103/997*e^2 + 40530/997*e - 18504/997, -18616/86739*e^7 - 57521/86739*e^6 + 552202/86739*e^5 + 57911/2991*e^4 - 1307473/28913*e^3 - 4074285/28913*e^2 - 2123590/86739*e + 1985516/86739, 7391/86739*e^7 + 76007/173478*e^6 - 434617/173478*e^5 - 38221/2991*e^4 + 954045/57826*e^3 + 2676270/28913*e^2 + 4771543/173478*e - 2498122/86739, -2647/86739*e^7 - 19405/173478*e^6 + 207971/173478*e^5 + 9971/2991*e^4 - 738247/57826*e^3 - 699267/28913*e^2 + 4108051/173478*e - 28051/86739, 20506/86739*e^7 + 67955/86739*e^6 - 596728/86739*e^5 - 69713/2991*e^4 + 1284601/28913*e^3 + 4981435/28913*e^2 + 6448966/86739*e - 1241690/86739, 12415/173478*e^7 + 37331/173478*e^6 - 162992/86739*e^5 - 38531/5982*e^4 + 281932/28913*e^3 + 2768039/57826*e^2 + 2383106/86739*e - 691750/86739, -19495/173478*e^7 - 45781/86739*e^6 + 549229/173478*e^5 + 90053/5982*e^4 - 1033039/57826*e^3 - 6142387/57826*e^2 - 11484367/173478*e + 1135216/86739, -19804/86739*e^7 - 59123/86739*e^6 + 572755/86739*e^5 + 58151/2991*e^4 - 1299705/28913*e^3 - 4012538/28913*e^2 - 2666488/86739*e + 2653010/86739, -4415/28913*e^7 - 14277/28913*e^6 + 124205/28913*e^5 + 14719/997*e^4 - 753916/28913*e^3 - 3166723/28913*e^2 - 1479193/28913*e + 410720/28913, 25040/86739*e^7 + 85459/86739*e^6 - 724379/86739*e^5 - 86017/2991*e^4 + 1578157/28913*e^3 + 5987041/28913*e^2 + 6784403/86739*e - 2183194/86739, -53683/173478*e^7 - 166577/173478*e^6 + 782378/86739*e^5 + 172067/5982*e^4 - 1771689/28913*e^3 - 12515699/57826*e^2 - 4894439/86739*e + 4234288/86739, -3685/28913*e^7 - 9788/28913*e^6 + 107925/28913*e^5 + 9939/997*e^4 - 711578/28913*e^3 - 2086209/28913*e^2 - 965983/28913*e - 112464/28913, -6181/86739*e^7 - 31553/86739*e^6 + 173887/86739*e^5 + 30776/2991*e^4 - 324842/28913*e^3 - 2081122/28913*e^2 - 3765172/86739*e + 979790/86739, -8369/86739*e^7 - 57617/173478*e^6 + 504379/173478*e^5 + 29506/2991*e^4 - 1194171/57826*e^3 - 2134647/28913*e^2 - 2679481/173478*e + 2101381/86739, -914/86739*e^7 + 7529/86739*e^6 + 23552/86739*e^5 - 5747/2991*e^4 - 46398/28913*e^3 + 213147/28913*e^2 + 161452/86739*e + 274354/86739, 2707/57826*e^7 + 18983/57826*e^6 - 36522/28913*e^5 - 19651/1994*e^4 + 103412/28913*e^3 + 4202599/57826*e^2 + 2075788/28913*e + 103622/28913, -12649/28913*e^7 - 40275/28913*e^6 + 362888/28913*e^5 + 40391/997*e^4 - 2370400/28913*e^3 - 8463450/28913*e^2 - 2962260/28913*e + 797408/28913, 27514/86739*e^7 + 204751/173478*e^6 - 1592771/173478*e^5 - 103637/2991*e^4 + 3377947/57826*e^3 + 7294125/28913*e^2 + 19368743/173478*e - 2742653/86739, 9056/86739*e^7 + 24478/86739*e^6 - 258599/86739*e^5 - 25687/2991*e^4 + 549808/28913*e^3 + 1892095/28913*e^2 + 2381771/86739*e - 1970890/86739, 269/86739*e^7 - 9713/86739*e^6 - 5603/86739*e^5 + 9110/2991*e^4 + 7392/28913*e^3 - 580097/28913*e^2 + 94454/86739*e + 1117088/86739, -6647/28913*e^7 - 42459/57826*e^6 + 380837/57826*e^5 + 21877/997*e^4 - 2458505/57826*e^3 - 4695411/28913*e^2 - 3313527/57826*e + 429426/28913, 2310/28913*e^7 + 3115/28913*e^6 - 73696/28913*e^5 - 3790/997*e^4 + 638961/28913*e^3 + 974626/28913*e^2 - 939362/28913*e - 228124/28913, -13385/173478*e^7 - 50029/173478*e^6 + 195988/86739*e^5 + 48133/5982*e^4 - 440362/28913*e^3 - 3178235/57826*e^2 - 1779847/86739*e - 272434/86739, 6589/173478*e^7 + 42617/173478*e^6 - 74264/86739*e^5 - 41765/5982*e^4 - 12975/28913*e^3 + 2748279/57826*e^2 + 6018374/86739*e + 743882/86739, -9931/173478*e^7 - 13048/86739*e^6 + 314551/173478*e^5 + 26609/5982*e^4 - 850837/57826*e^3 - 1906219/57826*e^2 + 1218437/173478*e - 360926/86739, -81115/173478*e^7 - 298193/173478*e^6 + 1177379/86739*e^5 + 296705/5982*e^4 - 2573050/28913*e^3 - 20436689/57826*e^2 - 11201870/86739*e + 3708490/86739, 1498/28913*e^7 - 2093/57826*e^6 - 102065/57826*e^5 + 926/997*e^4 + 1080519/57826*e^3 - 104638/28913*e^2 - 3405023/57826*e - 507858/28913, 26107/173478*e^7 + 82079/173478*e^6 - 407159/86739*e^5 - 84017/5982*e^4 + 1064547/28913*e^3 + 5998779/57826*e^2 - 428683/86739*e - 1644340/86739, 1175/28913*e^7 + 4651/28913*e^6 - 31353/28913*e^5 - 4457/997*e^4 + 173501/28913*e^3 + 909420/28913*e^2 + 292949/28913*e - 488026/28913, 202/997*e^7 + 786/997*e^6 - 5816/997*e^5 - 22201/997*e^4 + 37408/997*e^3 + 155541/997*e^2 + 59561/997*e - 39084/997, -2744/28913*e^7 - 8081/28913*e^6 + 84563/28913*e^5 + 8339/997*e^4 - 667538/28913*e^3 - 1827643/28913*e^2 + 234615/28913*e + 548198/28913, 51058/86739*e^7 + 161723/86739*e^6 - 1483639/86739*e^5 - 163319/2991*e^4 + 3305290/28913*e^3 + 11519198/28913*e^2 + 11221690/86739*e - 4923776/86739, -14793/57826*e^7 - 21145/28913*e^6 + 438823/57826*e^5 + 42657/1994*e^4 - 3080373/57826*e^3 - 8961779/57826*e^2 - 2227407/57826*e + 267683/28913, -14029/173478*e^7 - 32896/86739*e^6 + 446341/173478*e^5 + 67619/5982*e^4 - 1041911/57826*e^3 - 4723101/57826*e^2 - 6460543/173478*e - 577778/86739, -11246/28913*e^7 - 37945/28913*e^6 + 336352/28913*e^5 + 38033/997*e^4 - 2383611/28913*e^3 - 7963698/28913*e^2 - 1695100/28913*e + 1458706/28913, -32378/86739*e^7 - 105868/86739*e^6 + 920108/86739*e^5 + 108376/2991*e^4 - 1924152/28913*e^3 - 7715563/28913*e^2 - 9809201/86739*e + 1614358/86739, 7672/28913*e^7 + 24364/28913*e^6 - 234071/28913*e^5 - 25065/997*e^4 + 1753090/28913*e^3 + 5431524/28913*e^2 + 292264/28913*e - 1408804/28913, -16067/86739*e^7 - 61531/86739*e^6 + 479117/86739*e^5 + 62089/2991*e^4 - 1122528/28913*e^3 - 4400814/28913*e^2 - 3042662/86739*e + 3666730/86739, -16393/86739*e^7 - 58466/86739*e^6 + 490744/86739*e^5 + 58187/2991*e^4 - 1162549/28913*e^3 - 3988969/28913*e^2 - 2318116/86739*e + 1481660/86739, 12677/173478*e^7 + 17966/86739*e^6 - 413021/173478*e^5 - 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4073087/86739, -36967/86739*e^7 - 227617/173478*e^6 + 2166491/173478*e^5 + 114278/2991*e^4 - 5018311/57826*e^3 - 8006997/28913*e^2 - 8986373/173478*e + 4682957/86739, -35540/86739*e^7 - 104875/86739*e^6 + 1048847/86739*e^5 + 106054/2991*e^4 - 2472683/28913*e^3 - 7486527/28913*e^2 - 3655325/86739*e + 3525022/86739, -50482/86739*e^7 - 176717/86739*e^6 + 1468417/86739*e^5 + 178067/2991*e^4 - 3249478/28913*e^3 - 12480485/28913*e^2 - 12157042/86739*e + 5499074/86739, 44203/86739*e^7 + 262903/173478*e^6 - 2527259/173478*e^5 - 133142/2991*e^4 + 5480915/57826*e^3 + 9460306/28913*e^2 + 20094515/173478*e - 4427423/86739, 3157/57826*e^7 + 4538/28913*e^6 - 79515/57826*e^5 - 10497/1994*e^4 + 326791/57826*e^3 + 2611871/57826*e^2 + 1939041/57826*e - 605000/28913, 49454/86739*e^7 + 170950/86739*e^6 - 1432817/86739*e^5 - 170407/2991*e^4 + 3137822/28913*e^3 + 11778868/28913*e^2 + 12915626/86739*e - 3717646/86739, -4133/173478*e^7 + 937/86739*e^6 + 158315/173478*e^5 - 2861/5982*e^4 - 598645/57826*e^3 + 331757/57826*e^2 + 5659765/173478*e - 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380165/173478*e^6 + 3013663/173478*e^5 + 192754/2991*e^4 - 6364827/57826*e^3 - 13517905/28913*e^2 - 39527785/173478*e + 2764591/86739, -6569/28913*e^7 - 25067/28913*e^6 + 182936/28913*e^5 + 24248/997*e^4 - 1094541/28913*e^3 - 4873604/28913*e^2 - 2311088/28913*e + 1376528/28913, -44456/86739*e^7 - 331049/173478*e^6 + 2489431/173478*e^5 + 169639/2991*e^4 - 4779411/57826*e^3 - 12171671/28913*e^2 - 40805791/173478*e + 4670770/86739, 65479/57826*e^7 + 97813/28913*e^6 - 1909017/57826*e^5 - 200795/1994*e^4 + 12765227/57826*e^3 + 43168215/57826*e^2 + 14588069/57826*e - 2993048/28913, -62683/173478*e^7 - 85414/86739*e^6 + 1780915/173478*e^5 + 176423/5982*e^4 - 3735985/57826*e^3 - 12743583/57826*e^2 - 16858753/173478*e + 1376971/86739, 53253/57826*e^7 + 165121/57826*e^6 - 776395/28913*e^5 - 167831/1994*e^4 + 5160409/28913*e^3 + 35483199/57826*e^2 + 6512130/28913*e - 1226130/28913, 39943/173478*e^7 + 79709/173478*e^6 - 544805/86739*e^5 - 86933/5982*e^4 + 1145770/28913*e^3 + 6900233/57826*e^2 + 2280644/86739*e - 2416594/86739, 104959/173478*e^7 + 158602/86739*e^6 - 3089695/173478*e^5 - 317051/5982*e^4 + 7535995/57826*e^3 + 22319695/57826*e^2 - 1440197/173478*e - 9208453/86739, 49310/86739*e^7 + 349397/173478*e^6 - 2858023/173478*e^5 - 177085/2991*e^4 + 6103173/57826*e^3 + 12575765/28913*e^2 + 32110441/173478*e - 5813098/86739, -13621/86739*e^7 - 54728/86739*e^6 + 384961/86739*e^5 + 53639/2991*e^4 - 785530/28913*e^3 - 3593336/28913*e^2 - 5648521/86739*e - 1637128/86739, 67777/57826*e^7 + 108562/28913*e^6 - 2014923/57826*e^5 - 220595/1994*e^4 + 14130753/57826*e^3 + 47110485/57826*e^2 + 10993741/57826*e - 4828005/28913, -3044/2991*e^7 - 10117/2991*e^6 + 88877/2991*e^5 + 292502/2991*e^4 - 201316/997*e^3 - 701489/997*e^2 - 588779/2991*e + 387022/2991, 151369/173478*e^7 + 267433/86739*e^6 - 4433635/173478*e^5 - 534407/5982*e^4 + 10135515/57826*e^3 + 37122863/57826*e^2 + 26436859/173478*e - 10359313/86739, -62371/57826*e^7 - 215091/57826*e^6 + 918862/28913*e^5 + 215817/1994*e^4 - 6332053/28913*e^3 - 45304381/57826*e^2 - 6051618/28913*e + 3878348/28913, -77713/173478*e^7 - 131032/86739*e^6 + 2291959/173478*e^5 + 268283/5982*e^4 - 5239313/57826*e^3 - 19497731/57826*e^2 - 14288455/173478*e + 8576101/86739, 11812/86739*e^7 + 72002/86739*e^6 - 339865/86739*e^5 - 69860/2991*e^4 + 655422/28913*e^3 + 4625061/28913*e^2 + 7577770/86739*e - 1852946/86739, -28550/86739*e^7 - 172499/173478*e^6 + 1775227/173478*e^5 + 86998/2991*e^4 - 4653315/57826*e^3 - 6147620/28913*e^2 + 4553069/173478*e + 6055708/86739, 6070/86739*e^7 + 41779/173478*e^6 - 365399/173478*e^5 - 22838/2991*e^4 + 855863/57826*e^3 + 1969560/28913*e^2 + 2187755/173478*e - 5739323/86739, -17669/57826*e^7 - 50549/57826*e^6 + 258016/28913*e^5 + 53349/1994*e^4 - 1694364/28913*e^3 - 11771157/57826*e^2 - 2501120/28913*e + 1070386/28913, -119/997*e^7 + 525/1994*e^6 + 7889/1994*e^5 - 6876/997*e^4 - 85011/1994*e^3 + 37244/997*e^2 + 318437/1994*e + 32284/997, 23/173478*e^7 + 34639/173478*e^6 - 14911/86739*e^5 - 29965/5982*e^4 - 31714/28913*e^3 + 1502889/57826*e^2 + 5288668/86739*e + 2563510/86739, -107519/173478*e^7 - 337303/173478*e^6 + 1554580/86739*e^5 + 341899/5982*e^4 - 3347496/28913*e^3 - 24099763/57826*e^2 - 15112945/86739*e + 2812412/86739, 29309/86739*e^7 + 203459/173478*e^6 - 1835221/173478*e^5 - 100777/2991*e^4 + 4804017/57826*e^3 + 6966726/28913*e^2 + 333019/173478*e - 6785749/86739, 7600/86739*e^7 + 19010/86739*e^6 - 282766/86739*e^5 - 18434/2991*e^4 + 1064089/28913*e^3 + 1240371/28913*e^2 - 10346453/86739*e - 3034790/86739, -3594/28913*e^7 - 2218/28913*e^6 + 96786/28913*e^5 + 1572/997*e^4 - 672476/28913*e^3 - 323198/28913*e^2 + 812573/28913*e + 828648/28913, 12043/57826*e^7 + 14472/28913*e^6 - 370365/57826*e^5 - 28365/1994*e^4 + 2935139/57826*e^3 + 6102527/57826*e^2 - 2161543/57826*e - 2227399/28913, -29693/173478*e^7 - 135103/173478*e^6 + 427738/86739*e^5 + 130825/5982*e^4 - 894337/28913*e^3 - 8667821/57826*e^2 - 6428689/86739*e + 815786/86739, 31675/86739*e^7 + 110615/86739*e^6 - 932551/86739*e^5 - 109946/2991*e^4 + 2177089/28913*e^3 + 7635452/28913*e^2 + 3198487/86739*e - 5360498/86739, -26086/86739*e^7 - 107021/86739*e^6 + 711664/86739*e^5 + 108545/2991*e^4 - 1196422/28913*e^3 - 7720267/28913*e^2 - 17467822/86739*e + 1317368/86739, 16489/86739*e^7 + 198673/173478*e^6 - 841997/173478*e^5 - 98600/2991*e^4 + 984791/57826*e^3 + 6705598/28913*e^2 + 33902447/173478*e - 1125560/86739, 25023/57826*e^7 + 42499/28913*e^6 - 765191/57826*e^5 - 85933/1994*e^4 + 5790275/57826*e^3 + 18127103/57826*e^2 + 686067/57826*e - 1710422/28913, -37725/57826*e^7 - 128411/57826*e^6 + 538688/28913*e^5 + 129225/1994*e^4 - 3445630/28913*e^3 - 27352061/57826*e^2 - 5169373/28913*e + 2688476/28913, -15927/57826*e^7 - 54333/57826*e^6 + 200965/28913*e^5 + 54125/1994*e^4 - 944233/28913*e^3 - 11346387/57826*e^2 - 3651684/28913*e + 379926/28913, -34893/57826*e^7 - 50468/28913*e^6 + 1016991/57826*e^5 + 99045/1994*e^4 - 7108901/57826*e^3 - 20157109/57826*e^2 - 3169969/57826*e + 972918/28913, -29228/86739*e^7 - 117391/86739*e^6 + 874811/86739*e^5 + 117619/2991*e^4 - 1962272/28913*e^3 - 8150455/28913*e^2 - 9221318/86739*e + 1871026/86739, 91519/86739*e^7 + 320108/86739*e^6 - 2705602/86739*e^5 - 321194/2991*e^4 + 6233715/28913*e^3 + 22428889/28913*e^2 + 16394689/86739*e - 12410990/86739, 154847/173478*e^7 + 246560/86739*e^6 - 4591205/173478*e^5 - 497989/5982*e^4 + 10673705/57826*e^3 + 35105727/57826*e^2 + 26492111/173478*e - 9058838/86739, 225697/173478*e^7 + 351718/86739*e^6 - 6564157/173478*e^5 - 709547/5982*e^4 + 14865527/57826*e^3 + 49968999/57826*e^2 + 38708365/173478*e - 14452474/86739, 6521/86739*e^7 + 110459/173478*e^6 - 305509/173478*e^5 - 58378/2991*e^4 + 55393/57826*e^3 + 4318859/28913*e^2 + 22559563/173478*e - 4272400/86739, 4895/57826*e^7 + 891/28913*e^6 - 165803/57826*e^5 - 2429/1994*e^4 + 1616271/57826*e^3 + 575571/57826*e^2 - 3463739/57826*e + 1031846/28913, 9487/86739*e^7 + 70181/86739*e^6 - 263062/86739*e^5 - 69632/2991*e^4 + 337979/28913*e^3 + 4669314/28913*e^2 + 11689393/86739*e + 1633690/86739];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2,2,-1/3*w^3 + 1/3*w^2 + 3*w - 8/3}>] := 1;
ALEigenvalues[ideal<ZF | {9,3,-w - 1}>] := -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;