/*
  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^6 + 4*x^5 - 5*x^4 - 25*x^3 - 5*x^2 + 17*x - 3;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -1/2*e^5 - 5/2*e^4 + e^3 + 31/2*e^2 + 10*e - 15/2, 1/2*e^5 + 5/2*e^4 - e^3 - 31/2*e^2 - 10*e + 13/2, -1, -2*e - 3, -e^5 - 4*e^4 + 4*e^3 + 23*e^2 + 12*e - 9, e^5 + 3*e^4 - 7*e^3 - 15*e^2 + 2*e, 2*e^5 + 7*e^4 - 11*e^3 - 38*e^2 - 9*e + 9, e^3 + 2*e^2 - 5*e - 6, -e^5 - 3*e^4 + 7*e^3 + 16*e^2 - 2*e - 9, -e^5 - 4*e^4 + 4*e^3 + 23*e^2 + 15*e - 9, e^5 + 6*e^4 + e^3 - 39*e^2 - 36*e + 21, -2*e^5 - 7*e^4 + 11*e^3 + 38*e^2 + 9*e - 9, -1/2*e^5 - 3/2*e^4 + 4*e^3 + 15/2*e^2 - 7*e - 3/2, 1/2*e^5 + 7/2*e^4 + 2*e^3 - 47/2*e^2 - 26*e + 25/2, 1/2*e^5 + 7/2*e^4 + 3*e^3 - 43/2*e^2 - 31*e + 17/2, 5/2*e^5 + 17/2*e^4 - 15*e^3 - 91/2*e^2 - 4*e + 17/2, 3/2*e^5 + 9/2*e^4 - 11*e^3 - 51/2*e^2 + 5*e + 17/2, -e^5 - 6*e^4 - e^3 + 38*e^2 + 36*e - 18, -3/2*e^5 - 17/2*e^4 + 109/2*e^2 + 41*e - 57/2, e^5 + 5*e^4 - 2*e^3 - 32*e^2 - 21*e + 18, -1/2*e^5 - 1/2*e^4 + 6*e^3 - 3/2*e^2 - 10*e + 21/2, -2*e^5 - 6*e^4 + 14*e^3 + 31*e^2 - 2*e - 3, -e^4 - 4*e^3 + 4*e^2 + 21*e + 6, e^5 + 2*e^4 - 11*e^3 - 12*e^2 + 21*e + 9, -3/2*e^5 - 11/2*e^4 + 9*e^3 + 71/2*e^2 + 2*e - 45/2, 5/2*e^5 + 21/2*e^4 - 10*e^3 - 127/2*e^2 - 28*e + 33/2, 3/2*e^5 + 7/2*e^4 - 12*e^3 - 27/2*e^2 + 10*e - 15/2, 1/2*e^5 - 1/2*e^4 - 9*e^3 + 17/2*e^2 + 35*e - 21/2, 1/2*e^5 + 1/2*e^4 - 6*e^3 + 5/2*e^2 + 16*e - 39/2, 1/2*e^5 + 1/2*e^4 - 7*e^3 - 5/2*e^2 + 18*e - 21/2, -e^4 - e^3 + 8*e^2 - 5*e - 9, -1/2*e^5 - 1/2*e^4 + 6*e^3 - 1/2*e^2 - 14*e + 7/2, -2*e^5 - 10*e^4 + 2*e^3 + 60*e^2 + 55*e - 28, -2*e^5 - 6*e^4 + 14*e^3 + 30*e^2 - 12*e, 2*e^5 + 8*e^4 - 10*e^3 - 48*e^2 - 13*e + 15, 5/2*e^5 + 19/2*e^4 - 13*e^3 - 111/2*e^2 - 13*e + 33/2, 5/2*e^5 + 11/2*e^4 - 23*e^3 - 49/2*e^2 + 31*e - 21/2, -e^5 - 5*e^4 + 25*e^2 + 29*e, 3/2*e^5 + 17/2*e^4 + e^3 - 103/2*e^2 - 47*e + 33/2, -2*e^5 - 4*e^4 + 20*e^3 + 17*e^2 - 41*e, 3*e^5 + 12*e^4 - 12*e^3 - 67*e^2 - 29*e + 21, 2*e^5 + 9*e^4 - 4*e^3 - 53*e^2 - 49*e + 21, -4*e^5 - 13*e^4 + 26*e^3 + 71*e^2 - 2*e - 24, -5/2*e^5 - 9/2*e^4 + 25*e^3 + 23/2*e^2 - 47*e + 35/2, -5*e^5 - 17*e^4 + 31*e^3 + 95*e^2 + 2*e - 29, e^5 + 3*e^4 - 6*e^3 - 11*e^2 - 4*e - 12, 1/2*e^5 + 9/2*e^4 + 3*e^3 - 73/2*e^2 - 25*e + 63/2, e^3 - 2*e^2 - 17*e + 9, -7/2*e^5 - 29/2*e^4 + 15*e^3 + 173/2*e^2 + 35*e - 69/2, 4*e^4 + 10*e^3 - 33*e^2 - 46*e + 24, -3/2*e^5 - 11/2*e^4 + 9*e^3 + 73/2*e^2 + 4*e - 57/2, -2*e^5 - 8*e^4 + 8*e^3 + 44*e^2 + 25*e - 15, 2*e^5 + 10*e^4 - 4*e^3 - 65*e^2 - 44*e + 27, 5/2*e^5 + 23/2*e^4 - 6*e^3 - 133/2*e^2 - 45*e + 45/2, e^4 + 2*e^3 - 9*e^2 - 9*e + 9, 7/2*e^5 + 21/2*e^4 - 25*e^3 - 109/2*e^2 + 16*e + 37/2, 9/2*e^5 + 31/2*e^4 - 26*e^3 - 167/2*e^2 - 13*e + 39/2, 2*e^5 + 6*e^4 - 14*e^3 - 27*e^2 + 8*e - 18, -1/2*e^5 - 13/2*e^4 - 10*e^3 + 91/2*e^2 + 64*e - 63/2, -15/2*e^5 - 51/2*e^4 + 46*e^3 + 285/2*e^2 + 13*e - 81/2, 9/2*e^5 + 31/2*e^4 - 28*e^3 - 175/2*e^2 - 2*e + 53/2, -4*e^5 - 17*e^4 + 15*e^3 + 100*e^2 + 47*e - 29, e^5 - 16*e^3 + 4*e^2 + 43*e - 11, e^4 + 5*e^3 - 3*e^2 - 30*e - 14, -e^5 - 8*e^4 - 7*e^3 + 51*e^2 + 69*e - 21, -2*e^5 - 9*e^4 + 7*e^3 + 59*e^2 + 30*e - 39, e^5 + 5*e^4 - 2*e^3 - 30*e^2 - 11*e - 3, 2*e^5 + 10*e^4 - 4*e^3 - 61*e^2 - 31*e + 27, -3*e^5 - 7*e^4 + 28*e^3 + 32*e^2 - 43*e - 1, -3*e^5 - 11*e^4 + 16*e^3 + 60*e^2 + 8*e - 10, -e^5 - 10*e^4 - 12*e^3 + 67*e^2 + 80*e - 44, -e^5 - 4*e^4 + 4*e^3 + 24*e^2 + 23*e - 8, -4*e^5 - 13*e^4 + 24*e^3 + 69*e^2 + 5*e - 33, 1/2*e^5 + 11/2*e^4 + 6*e^3 - 81/2*e^2 - 41*e + 57/2, 5/2*e^5 + 15/2*e^4 - 15*e^3 - 57/2*e^2 - 6*e - 49/2, -1/2*e^5 - 7/2*e^4 - 2*e^3 + 45/2*e^2 + 22*e - 25/2, -3/2*e^5 - 3/2*e^4 + 16*e^3 - 17/2*e^2 - 33*e + 69/2, -5/2*e^5 - 21/2*e^4 + 11*e^3 + 133/2*e^2 + 23*e - 87/2, 1/2*e^5 + 5/2*e^4 - 23/2*e^2 - 7*e + 3/2, 3/2*e^5 + 9/2*e^4 - 10*e^3 - 35/2*e^2 + 6*e - 21/2, 1/2*e^5 - 3/2*e^4 - 13*e^3 + 19/2*e^2 + 41*e + 3/2, 4*e^5 + 18*e^4 - 14*e^3 - 109*e^2 - 48*e + 48, e^5 - 13*e^3 + 16*e^2 + 29*e - 47, -9/2*e^5 - 31/2*e^4 + 26*e^3 + 169/2*e^2 + 27*e - 29/2, -4*e^5 - 14*e^4 + 23*e^3 + 80*e^2 + 18*e - 17, 4*e^5 + 12*e^4 - 28*e^3 - 67*e^2 + 4*e + 29, -2*e^5 - 8*e^4 + 8*e^3 + 40*e^2 + 20*e + 6, -1/2*e^5 + 7/2*e^4 + 19*e^3 - 51/2*e^2 - 77*e + 15/2, -2*e^5 - 9*e^4 + 7*e^3 + 59*e^2 + 29*e - 30, -1/2*e^5 + 5/2*e^4 + 13*e^3 - 47/2*e^2 - 47*e + 33/2, 5/2*e^5 + 15/2*e^4 - 19*e^3 - 89/2*e^2 + 18*e + 75/2, -3*e^5 - 12*e^4 + 12*e^3 + 70*e^2 + 44*e - 21, -3/2*e^5 - 15/2*e^4 + 3*e^3 + 89/2*e^2 + 28*e - 9/2, -7/2*e^5 - 19/2*e^4 + 28*e^3 + 99/2*e^2 - 22*e - 15/2, 3/2*e^5 + 5/2*e^4 - 17*e^3 - 17/2*e^2 + 35*e - 33/2, 5/2*e^5 + 11/2*e^4 - 22*e^3 - 39/2*e^2 + 23*e - 39/2, e^5 + 4*e^4 - 5*e^3 - 21*e^2 + 12, -1/2*e^5 - 7/2*e^4 - 3*e^3 + 35/2*e^2 + 26*e - 3/2, 7/2*e^5 + 21/2*e^4 - 23*e^3 - 99/2*e^2 + 3/2, -4*e^5 - 11*e^4 + 28*e^3 + 47*e^2 - 9*e + 27, 2*e^5 + 4*e^4 - 18*e^3 - 13*e^2 + 17*e - 20, -1/2*e^5 - 7/2*e^4 - e^3 + 49/2*e^2 + 24*e - 61/2, 2*e^5 + 9*e^4 - 9*e^3 - 57*e^2 - 12*e + 21, 1/2*e^5 - 5/2*e^4 - 13*e^3 + 49/2*e^2 + 41*e - 27/2, 1/2*e^5 + 13/2*e^4 + 12*e^3 - 79/2*e^2 - 72*e + 13/2, 5/2*e^5 + 13/2*e^4 - 21*e^3 - 65/2*e^2 + 35*e + 7/2, 7*e^4 + 18*e^3 - 56*e^2 - 90*e + 33, -3/2*e^5 - 7/2*e^4 + 12*e^3 + 29/2*e^2 - 17*e - 15/2, 7*e^5 + 25*e^4 - 41*e^3 - 142*e^2 - 13*e + 50, -4*e^5 - 18*e^4 + 11*e^3 + 105*e^2 + 69*e - 37, 1/2*e^5 + 9/2*e^4 + 5*e^3 - 69/2*e^2 - 47*e + 21/2, e^5 + 12*e^4 + 16*e^3 - 87*e^2 - 106*e + 72, -17/2*e^5 - 63/2*e^4 + 43*e^3 + 349/2*e^2 + 59*e - 113/2, -3/2*e^5 - 23/2*e^4 - 9*e^3 + 161/2*e^2 + 93*e - 129/2, -7/2*e^5 - 21/2*e^4 + 24*e^3 + 105/2*e^2 + 5*e + 3/2, 6*e^5 + 15*e^4 - 50*e^3 - 67*e^2 + 62*e + 3, e^5 + 8*e^4 + 7*e^3 - 50*e^2 - 67*e + 12, -3*e^5 - 9*e^4 + 21*e^3 + 46*e^2 - 17*e - 6, -2*e^4 - 7*e^3 + 17*e^2 + 51*e - 21, e^5 - 15*e^3 + 12*e^2 + 47*e - 30, -1/2*e^5 - 3/2*e^4 + 4*e^3 + 15/2*e^2 - 12*e - 3/2, -11/2*e^5 - 43/2*e^4 + 26*e^3 + 259/2*e^2 + 52*e - 109/2, -6*e^5 - 23*e^4 + 29*e^3 + 133*e^2 + 46*e - 59, e^5 + 2*e^4 - 7*e^3 + e^2 + 4*e - 32, 3/2*e^5 + 13/2*e^4 - 5*e^3 - 77/2*e^2 - 25*e + 23/2, 4*e^5 + 15*e^4 - 22*e^3 - 90*e^2 - 15*e + 24, 13/2*e^5 + 47/2*e^4 - 36*e^3 - 263/2*e^2 - 33*e + 75/2, 1/2*e^5 + 19/2*e^4 + 18*e^3 - 143/2*e^2 - 114*e + 99/2, -9/2*e^5 - 29/2*e^4 + 30*e^3 + 157/2*e^2 - 15*e - 57/2, 4*e^5 + 19*e^4 - 9*e^3 - 110*e^2 - 72*e + 45, 3/2*e^5 + 13/2*e^4 - 2*e^3 - 65/2*e^2 - 39*e + 15/2, 4*e^5 + 15*e^4 - 19*e^3 - 77*e^2 - 25*e - 9, -13/2*e^5 - 45/2*e^4 + 40*e^3 + 255/2*e^2 - 5*e - 99/2, 2*e^5 + 4*e^4 - 19*e^3 - 20*e^2 + 28*e + 21, -e^5 - 5*e^4 - 3*e^3 + 23*e^2 + 54*e - 12, -2*e^5 - 8*e^4 + 9*e^3 + 47*e^2 + 15*e - 27, 5*e^5 + 15*e^4 - 37*e^3 - 83*e^2 + 20*e + 42, -5/2*e^5 - 17/2*e^4 + 14*e^3 + 95/2*e^2 + 19*e - 47/2, -3/2*e^5 - 23/2*e^4 - 9*e^3 + 145/2*e^2 + 88*e - 65/2, 1/2*e^5 + 11/2*e^4 + 10*e^3 - 73/2*e^2 - 74*e + 21/2, 2*e^5 + e^4 - 28*e^3 + 8*e^2 + 75*e - 33, 3*e^5 + 8*e^4 - 22*e^3 - 33*e^2 + 17*e + 9, -4*e^5 - 8*e^4 + 39*e^3 + 35*e^2 - 51*e + 15, e^5 + 3*e^4 - 5*e^3 - 8*e^2 - 8*e - 45, -e^5 + e^4 + 16*e^3 - 18*e^2 - 39*e + 21, -3*e^5 - 20*e^4 - 8*e^3 + 131*e^2 + 120*e - 78, -4*e^5 - 15*e^4 + 22*e^3 + 94*e^2 + 20*e - 57, 17/2*e^5 + 59/2*e^4 - 49*e^3 - 327/2*e^2 - 16*e + 129/2, 3/2*e^5 + 17/2*e^4 - 2*e^3 - 121/2*e^2 - 41*e + 87/2, -15/2*e^5 - 63/2*e^4 + 27*e^3 + 369/2*e^2 + 110*e - 165/2, -e^5 - e^4 + 14*e^3 + 7*e^2 - 24*e - 12, 3*e^5 + 12*e^4 - 14*e^3 - 67*e^2 - 19*e + 9, -4*e^5 - 11*e^4 + 31*e^3 + 60*e^2 - 30*e - 48, -1/2*e^5 - 3/2*e^4 + 6*e^3 + 21/2*e^2 - 14*e - 15/2, 2*e^5 + 2*e^4 - 28*e^3 - 5*e^2 + 82*e + 9, 9/2*e^5 + 37/2*e^4 - 20*e^3 - 223/2*e^2 - 38*e + 117/2, -1/2*e^5 - 9/2*e^4 - 5*e^3 + 67/2*e^2 + 50*e - 75/2, 5/2*e^5 + 15/2*e^4 - 17*e^3 - 75/2*e^2 + 5*e + 3/2, -3*e^5 - 7*e^4 + 29*e^3 + 39*e^2 - 58*e - 21, -e^5 - 8*e^4 - 7*e^3 + 54*e^2 + 58*e - 22, 3/2*e^5 + 7/2*e^4 - 15*e^3 - 45/2*e^2 + 26*e + 71/2, -e^5 + e^4 + 17*e^3 - 16*e^2 - 43*e + 44, 5/2*e^5 + 17/2*e^4 - 15*e^3 - 105/2*e^2 - 11*e + 101/2, -5/2*e^5 - 17/2*e^4 + 16*e^3 + 95/2*e^2 - 4*e - 17/2, -3/2*e^5 - 21/2*e^4 - 7*e^3 + 127/2*e^2 + 80*e - 21/2, -9/2*e^5 - 47/2*e^4 + 8*e^3 + 305/2*e^2 + 102*e - 171/2, 7/2*e^5 + 35/2*e^4 - 5*e^3 - 221/2*e^2 - 95*e + 87/2, 3*e^5 + 11*e^4 - 17*e^3 - 62*e^2 - 17*e - 12, 3*e^5 + 8*e^4 - 21*e^3 - 36*e^2 + 6*e + 6, -3/2*e^5 - 3/2*e^4 + 20*e^3 + 21/2*e^2 - 47*e - 75/2, -2*e^5 - 2*e^4 + 24*e^3 - 11*e^2 - 59*e + 63, -3*e^5 - 17*e^4 - 2*e^3 + 110*e^2 + 108*e - 75, -7*e^4 - 19*e^3 + 56*e^2 + 90*e - 18, 11/2*e^5 + 39/2*e^4 - 34*e^3 - 235/2*e^2 + 3*e + 105/2, 2*e^5 + 8*e^4 - 4*e^3 - 40*e^2 - 52*e + 4, 2*e^5 + 10*e^4 - 2*e^3 - 60*e^2 - 71*e + 1, 19/2*e^5 + 67/2*e^4 - 52*e^3 - 361/2*e^2 - 44*e + 63/2, 3/2*e^5 + 7/2*e^4 - 15*e^3 - 43/2*e^2 + 32*e + 87/2, 3/2*e^5 + 5/2*e^4 - 16*e^3 - 13/2*e^2 + 23*e - 81/2, -7*e^5 - 19*e^4 + 57*e^3 + 101*e^2 - 65*e - 42, 9/2*e^5 + 23/2*e^4 - 38*e^3 - 107/2*e^2 + 51*e + 7/2, -4*e^5 - 20*e^4 + 8*e^3 + 119*e^2 + 76*e - 25, -e^5 - 8*e^4 - 11*e^3 + 45*e^2 + 91*e - 12, -11/2*e^5 - 37/2*e^4 + 33*e^3 + 203/2*e^2 + 4*e - 87/2, 7*e^5 + 31*e^4 - 19*e^3 - 183*e^2 - 133*e + 84, e^5 - 2*e^4 - 25*e^3 + 12*e^2 + 92*e - 6, 11/2*e^5 + 53/2*e^4 - 11*e^3 - 319/2*e^2 - 112*e + 177/2, 7/2*e^5 + 25/2*e^4 - 19*e^3 - 141/2*e^2 - 14*e + 45/2, -6*e^5 - 17*e^4 + 46*e^3 + 88*e^2 - 36*e - 15, -1/2*e^5 - 13/2*e^4 - 10*e^3 + 99/2*e^2 + 80*e - 63/2, -15/2*e^5 - 65/2*e^4 + 23*e^3 + 375/2*e^2 + 130*e - 147/2, 3*e^5 + 16*e^4 - 4*e^3 - 103*e^2 - 76*e + 66, -5*e^5 - 11*e^4 + 49*e^3 + 61*e^2 - 80*e - 28, 5*e^5 + 23*e^4 - 17*e^3 - 144*e^2 - 57*e + 68, -7/2*e^5 - 39/2*e^4 - 2*e^3 + 237/2*e^2 + 117*e - 135/2, 3*e^5 + 6*e^4 - 30*e^3 - 25*e^2 + 59*e + 21, 9/2*e^5 + 35/2*e^4 - 20*e^3 - 197/2*e^2 - 43*e + 79/2, -15/2*e^5 - 47/2*e^4 + 50*e^3 + 237/2*e^2 - 19*e - 29/2, 1/2*e^5 + 13/2*e^4 + 11*e^3 - 89/2*e^2 - 61*e + 31/2, -13/2*e^5 - 35/2*e^4 + 53*e^3 + 173/2*e^2 - 59*e - 29/2, 7/2*e^5 + 27/2*e^4 - 15*e^3 - 159/2*e^2 - 54*e + 93/2, -5*e^5 - 22*e^4 + 14*e^3 + 128*e^2 + 88*e - 63, 1/2*e^5 + 11/2*e^4 + 7*e^3 - 83/2*e^2 - 59*e + 33/2, -7*e^5 - 26*e^4 + 35*e^3 + 143*e^2 + 42*e - 45, 3*e^5 + 11*e^4 - 12*e^3 - 57*e^2 - 58*e - 15, -5/2*e^5 - 21/2*e^4 + 15*e^3 + 143/2*e^2 - 5*e - 63/2, -2*e^5 - 6*e^4 + 10*e^3 + 25*e^2 + 33*e + 16, 21/2*e^5 + 65/2*e^4 - 70*e^3 - 337/2*e^2 + 22*e + 101/2, 5*e^5 + 21*e^4 - 16*e^3 - 122*e^2 - 90*e + 48, 5*e^5 + 19*e^4 - 21*e^3 - 103*e^2 - 59*e + 21, 1/2*e^5 - 7/2*e^4 - 15*e^3 + 63/2*e^2 + 49*e - 81/2, -1/2*e^5 - 17/2*e^4 - 16*e^3 + 113/2*e^2 + 91*e - 45/2, 6*e^5 + 18*e^4 - 45*e^3 - 98*e^2 + 53*e + 50, -13/2*e^5 - 31/2*e^4 + 59*e^3 + 157/2*e^2 - 68*e - 41/2, -7*e^5 - 33*e^4 + 16*e^3 + 201*e^2 + 139*e - 84, 3/2*e^5 + 9/2*e^4 - 13*e^3 - 49/2*e^2 + 28*e - 3/2, -7/2*e^5 - 33/2*e^4 + 12*e^3 + 219/2*e^2 + 50*e - 99/2, -7/2*e^5 - 23/2*e^4 + 24*e^3 + 145/2*e^2 - 9*e - 57/2, 2*e^4 + 6*e^3 - 13*e^2 - 26*e - 24, 4*e^5 + 17*e^4 - 18*e^3 - 107*e^2 - 37*e + 24, -11/2*e^5 - 41/2*e^4 + 31*e^3 + 251/2*e^2 + 11*e - 145/2, -6*e^5 - 23*e^4 + 27*e^3 + 130*e^2 + 51*e - 47, -3*e^5 - 6*e^4 + 32*e^3 + 33*e^2 - 69*e - 12, -4*e^5 - 3*e^4 + 50*e^3 - 15*e^2 - 106*e + 93, -e^5 - 17*e^4 - 31*e^3 + 122*e^2 + 177*e - 90, -1/2*e^5 - 13/2*e^4 - 12*e^3 + 71/2*e^2 + 79*e + 3/2, -3/2*e^5 + 13/2*e^4 + 38*e^3 - 131/2*e^2 - 143*e + 129/2, 3*e^5 + 12*e^4 - 15*e^3 - 74*e^2 - 15*e + 9, 3*e^5 + 13*e^4 - 8*e^3 - 72*e^2 - 45*e + 15, -5*e^5 - 15*e^4 + 32*e^3 + 64*e^2 - e + 21, -2*e^5 - 5*e^4 + 18*e^3 + 20*e^2 - 35*e + 24, -13/2*e^5 - 63/2*e^4 + 9*e^3 + 373/2*e^2 + 159*e - 183/2, -1/2*e^5 - 1/2*e^4 + 10*e^3 + 1/2*e^2 - 52*e + 3/2, 3*e^4 + 8*e^3 - 16*e^2 - 35*e - 42, -5*e^5 - 23*e^4 + 18*e^3 + 149*e^2 + 67*e - 66, 19/2*e^5 + 67/2*e^4 - 56*e^3 - 375/2*e^2 - 13*e + 129/2, -9*e^5 - 34*e^4 + 45*e^3 + 193*e^2 + 73*e - 60, -8*e^5 - 25*e^4 + 56*e^3 + 135*e^2 - 28*e - 42, -6*e^5 - 29*e^4 + 13*e^3 + 169*e^2 + 100*e - 45, -19/2*e^5 - 57/2*e^4 + 68*e^3 + 305/2*e^2 - 31*e - 105/2, -3/2*e^5 - 19/2*e^4 + e^3 + 135/2*e^2 + 29*e - 131/2, 7*e^5 + 27*e^4 - 33*e^3 - 152*e^2 - 55*e + 38, 12*e^5 + 41*e^4 - 75*e^3 - 229*e^2 + 2*e + 84, 23/2*e^5 + 75/2*e^4 - 73*e^3 - 409/2*e^2 - 17*e + 99/2, 3/2*e^5 + 17/2*e^4 - 105/2*e^2 - 42*e + 81/2, -3/2*e^5 - 7/2*e^4 + 10*e^3 + 21/2*e^2 + 3*e + 51/2, 15/2*e^5 + 65/2*e^4 - 25*e^3 - 399/2*e^2 - 122*e + 177/2, 7/2*e^5 + 45/2*e^4 + 10*e^3 - 295/2*e^2 - 171*e + 141/2, 25/2*e^5 + 95/2*e^4 - 60*e^3 - 541/2*e^2 - 97*e + 171/2, -11/2*e^5 - 45/2*e^4 + 24*e^3 + 257/2*e^2 + 40*e - 57/2, -3/2*e^5 - 5/2*e^4 + 16*e^3 - 5/2*e^2 - 33*e + 129/2, -11/2*e^5 - 37/2*e^4 + 27*e^3 + 181/2*e^2 + 53*e - 3/2, 7/2*e^5 + 43/2*e^4 + 9*e^3 - 255/2*e^2 - 148*e + 111/2, 25/2*e^5 + 77/2*e^4 - 85*e^3 - 407/2*e^2 + 15*e + 69/2, 9/2*e^5 + 19/2*e^4 - 43*e^3 - 83/2*e^2 + 74*e - 39/2, -17/2*e^5 - 61/2*e^4 + 49*e^3 + 345/2*e^2 + 25*e - 159/2, -2*e^5 - e^4 + 27*e^3 - 5*e^2 - 74*e - 18, -1/2*e^5 + 13/2*e^4 + 24*e^3 - 115/2*e^2 - 91*e + 129/2, -4*e^5 - 13*e^4 + 21*e^3 + 64*e^2 + 32*e + 22, 3*e^5 + 22*e^4 + 10*e^3 - 150*e^2 - 117*e + 106, -3/2*e^5 - 9/2*e^4 + 10*e^3 + 53/2*e^2 - e - 51/2, -15/2*e^5 - 47/2*e^4 + 52*e^3 + 273/2*e^2 - 12*e - 105/2, e^5 - 4*e^4 - 27*e^3 + 34*e^2 + 106*e - 48, -21/2*e^5 - 83/2*e^4 + 49*e^3 + 485/2*e^2 + 70*e - 183/2, -2*e^5 - 12*e^4 + e^3 + 89*e^2 + 46*e - 72, -7*e^4 - 12*e^3 + 65*e^2 + 48*e - 72, 5*e^5 + 15*e^4 - 41*e^3 - 94*e^2 + 56*e + 61, 5*e^5 + 19*e^4 - 27*e^3 - 107*e^2 - 22*e + 14, 15/2*e^5 + 47/2*e^4 - 50*e^3 - 255/2*e^2 + 7*e + 53/2, -4*e^5 - 13*e^4 + 31*e^3 + 75*e^2 - 35*e - 16, -e^5 - 8*e^4 - 10*e^3 + 45*e^2 + 74*e, 11/2*e^5 + 49/2*e^4 - 17*e^3 - 315/2*e^2 - 106*e + 183/2, 9/2*e^5 + 31/2*e^4 - 29*e^3 - 169/2*e^2 + 24*e + 55/2, -7/2*e^5 - 21/2*e^4 + 29*e^3 + 105/2*e^2 - 55*e + 11/2, 17/2*e^5 + 77/2*e^4 - 27*e^3 - 479/2*e^2 - 135*e + 181/2, -17/2*e^5 - 55/2*e^4 + 57*e^3 + 301/2*e^2 - 33*e - 109/2, -5/2*e^5 - 35/2*e^4 - 11*e^3 + 245/2*e^2 + 148*e - 147/2, -7/2*e^5 - 49/2*e^4 - 15*e^3 + 321/2*e^2 + 179*e - 189/2, 4*e^5 + 20*e^4 - 5*e^3 - 125*e^2 - 121*e + 42, -12*e^5 - 42*e^4 + 72*e^3 + 242*e^2 + 33*e - 96, e^5 - 2*e^4 - 20*e^3 + 22*e^2 + 81*e - 15, -7/2*e^5 + 5/2*e^4 + 59*e^3 - 107/2*e^2 - 187*e + 141/2, -6*e^5 - 18*e^4 + 37*e^3 + 82*e^2 + 15*e + 27, 4*e^5 + 12*e^4 - 33*e^3 - 74*e^2 + 31*e + 57, 7/2*e^5 + 41/2*e^4 + 5*e^3 - 247/2*e^2 - 142*e + 129/2, -7*e^5 - 26*e^4 + 39*e^3 + 154*e^2 + 31*e - 54, 5/2*e^5 + 29/2*e^4 - 197/2*e^2 - 85*e + 57/2, 3/2*e^5 + 3/2*e^4 - 17*e^3 + 11/2*e^2 + 38*e - 111/2, -8*e^5 - 16*e^4 + 80*e^3 + 65*e^2 - 136*e + 9, e^5 - 4*e^4 - 27*e^3 + 40*e^2 + 109*e - 39, 3*e^5 + 9*e^4 - 16*e^3 - 36*e^2 - 26*e - 18, -8*e^5 - 25*e^4 + 54*e^3 + 132*e^2 - 20*e - 63, 8*e^5 + 30*e^4 - 42*e^3 - 170*e^2 - 26*e + 70, -7/2*e^5 - 11/2*e^4 + 41*e^3 + 43/2*e^2 - 103*e - 43/2, 8*e^5 + 31*e^4 - 37*e^3 - 173*e^2 - 60*e + 75, 5*e^5 + 12*e^4 - 49*e^3 - 68*e^2 + 79*e + 39];
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;