/*
  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 | {14, 14, w}>;

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^2 + 2*x - 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, -1, e, 1/2*e - 1, -e - 2, e - 2, 3/2*e + 2, e, -7/2*e - 5, -e - 6, -e - 4, -5/2*e - 8, -2*e + 4, -1/2*e + 8, -e - 4, 1/2*e + 3, 4*e + 8, -3/2*e - 3, 1/2*e - 11, 1/2*e + 5, -3*e - 8, -2*e - 2, 9/2*e + 6, -4*e - 8, -3/2*e, 14, 3/2*e + 2, 5*e + 2, 3/2*e + 1, -8*e - 6, -7/2*e - 16, 9/2*e + 11, -11/2*e - 14, -7/2*e + 1, -1/2*e + 16, -4*e, 17/2*e + 6, 11/2*e + 7, -7/2*e + 6, 7/2*e - 15, 15/2*e + 16, -3*e + 6, -2*e - 12, -11/2*e - 8, -6*e - 8, 2*e - 22, -5*e - 4, -11/2*e - 8, 7/2*e - 9, 3/2*e - 6, 8*e + 18, -6*e - 14, -3*e - 4, -17/2*e - 8, 11/2*e + 19, 1/2*e - 15, -3/2*e - 8, 13/2*e - 7, 1/2*e - 1, -15/2*e - 10, 3/2*e + 14, 9/2*e - 4, 7*e + 12, 13/2*e + 15, 5*e + 2, 5/2*e - 8, 11*e + 18, e + 18, -5/2*e - 1, 3*e - 8, -4*e - 18, -2*e - 30, e - 12, 11*e + 6, 4*e - 8, 8*e + 26, 8*e + 2, 8*e + 18, -23/2*e - 7, -2*e - 20, 21/2*e - 2, 11/2*e - 4, e - 6, -29/2*e - 9, 9/2*e - 15, -9/2*e + 2, -15/2*e - 8, 15/2*e - 2, -5*e - 14, 2*e + 10, -5/2*e - 26, 3*e + 2, -16, 7/2*e - 18, 9*e - 8, -1/2*e + 1, 11*e + 2, 16, 5/2*e - 24, 2*e + 30, 14*e + 20, -10*e - 22, -5*e - 20, 13/2*e + 23, -23/2*e - 12, -5/2*e - 1, 12*e - 6, 2*e - 6, 23/2*e + 7, -13/2*e + 5, -2*e - 2, -e - 36, e + 8, -17/2*e - 36, -5/2*e - 16, 6, -5*e + 6, 6*e - 6, -5*e + 10, 7/2*e + 14, 2*e - 30, 3*e + 4, 9/2*e + 35, -34, 21/2*e + 16, -2*e + 22, -14*e - 12, -3/2*e - 27, -4*e - 14, 13/2*e - 28, 5*e + 14, -14*e - 22, -13/2*e - 22, -19/2*e - 11, -21/2*e + 13, 13/2*e + 19, -15/2*e + 17, 5/2*e + 13, 8, -21/2*e - 8, -25/2*e + 12, 6*e + 36, 5*e + 8, -6*e - 38, 39/2*e + 15, -10*e - 10, -22, 18*e + 14, -7/2*e - 5, -6*e, 7/2*e + 26, -5*e - 18, -3/2*e + 21, -3/2*e + 6, 6*e + 28, 2*e + 32, 19/2*e + 40, 9*e - 12, 7*e - 14, -16*e - 16, 3*e - 46, -15/2*e + 17, -14*e - 32, 10*e + 8, -4*e - 12, -11*e - 26, -7*e + 28, 15*e - 2, 3/2*e + 1, 9*e + 32, -35/2*e - 28, -2, -27/2*e - 23, 2*e + 20, -7*e + 10, -29/2*e + 5, 15*e + 26, -4*e - 40, 16*e + 30, -4*e + 18, 13/2*e + 43, -7*e - 38, -8*e, e, -4*e - 44, 19/2*e + 3, -18*e - 6, -5/2*e - 1, 11*e + 6, 13/2*e - 37, -45/2*e - 14, -11/2*e - 46, 8*e - 4, -25/2*e - 6, 3*e + 2, 18*e + 34, -12*e - 10, 3/2*e - 38, 14*e - 2, -25/2*e - 21, 12*e + 34, 25/2*e + 19, 12*e - 8, 15/2*e + 10, 7/2*e + 7, -2*e - 20, 33/2*e + 29, -6*e + 22, -28*e - 30, 5*e + 42, -4*e + 54, 24*e + 36, 15*e + 20, -9*e, 21/2*e - 30, -7/2*e - 14, 28*e + 28, 35/2*e + 11, -22*e - 8, -3*e, -16*e + 14, -22*e - 44, -11/2*e - 24, -10*e - 10, 31/2*e + 36, 6*e - 14, -39/2*e - 8, -12*e - 42, -11*e - 12, 27*e + 24, -13*e - 56, 18*e + 2, 21*e + 8, -22*e - 20, 3/2*e - 26, 8*e - 6, 3*e - 36, -10*e - 4, -15*e - 20, 41/2*e + 5, 5*e - 2, 68, 3/2*e + 2, -15/2*e - 13, 21*e + 4, -8*e - 26, -4*e + 4, 27/2*e - 18, 19*e + 10, 11/2*e - 54, 2*e - 46, -15/2*e - 65, 11/2*e - 26, -25/2*e - 24, -6*e + 50, e + 36, 61/2*e + 30, 39/2*e + 34, 19/2*e - 20, 11*e + 28, 15*e + 36, -2*e + 32, 37/2*e - 5, 1/2*e - 1, 27/2*e + 29, -17*e - 54, -9*e + 14, -31/2*e + 19, -31/2*e - 45, 29/2*e + 40, -15/2*e + 31, 12*e + 22, 11*e + 32, -15*e - 32, -17/2*e - 1, -21/2*e + 21, -13*e + 34, 7/2*e + 17, -47/2*e - 37, -17/2*e + 7, -6*e - 16, -e + 40, 3*e - 2, -9/2*e + 54, 4*e + 16, 15/2*e - 44, -8*e - 38, -5*e + 48, 14*e - 8, 12*e + 18, -19/2*e + 3, 9*e + 6, 4*e + 14, -2*e - 16, -7*e + 10, -43/2*e - 54, -9*e + 46, -20*e + 14];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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