/*
  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![7, 2, -7, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {31,31,w^3 - 4*w^2 - 2*w + 8}>;

primesArray := [
[4, 2, w^3 - 3*w^2 - 3*w + 4],
[7, 7, w],
[7, 7, w^3 - 3*w^2 - 4*w + 5],
[9, 3, w^3 - 4*w^2 - w + 9],
[17, 17, 2*w^3 - 6*w^2 - 7*w + 8],
[17, 17, -w^3 + 3*w^2 + 4*w - 3],
[17, 17, -w + 2],
[23, 23, 2*w^3 - 7*w^2 - 4*w + 12],
[23, 23, -w^2 + 2*w + 3],
[31, 31, -2*w^3 + 7*w^2 + 5*w - 12],
[31, 31, -w^3 + 4*w^2 + 2*w - 8],
[41, 41, 3*w^3 - 10*w^2 - 7*w + 16],
[41, 41, 2*w^3 - 7*w^2 - 5*w + 10],
[49, 7, 2*w^3 - 6*w^2 - 6*w + 9],
[71, 71, w^2 - 2*w - 2],
[71, 71, 2*w^3 - 7*w^2 - 4*w + 13],
[73, 73, 3*w^3 - 11*w^2 - 5*w + 19],
[73, 73, -4*w^3 + 13*w^2 + 10*w - 17],
[79, 79, 3*w^3 - 9*w^2 - 10*w + 13],
[79, 79, -2*w^3 + 6*w^2 + 5*w - 8],
[97, 97, 2*w^3 - 6*w^2 - 8*w + 9],
[97, 97, 2*w - 1],
[113, 113, 2*w^3 - 6*w^2 - 7*w + 6],
[113, 113, -w^3 + 3*w^2 + 2*w - 1],
[137, 137, w^3 - 2*w^2 - 5*w + 2],
[137, 137, 2*w^3 - 7*w^2 - 3*w + 12],
[137, 137, -3*w^3 + 10*w^2 + 7*w - 17],
[137, 137, -w^3 + 2*w^2 + 6*w - 2],
[167, 167, -2*w^3 + 7*w^2 + 6*w - 10],
[167, 167, 3*w^3 - 9*w^2 - 10*w + 10],
[169, 13, -2*w^3 + 7*w^2 + 2*w - 9],
[169, 13, -2*w^3 + 5*w^2 + 10*w - 4],
[193, 193, -2*w^3 + 8*w^2 + 3*w - 13],
[193, 193, -3*w^3 + 11*w^2 + 6*w - 22],
[199, 199, 5*w^3 - 16*w^2 - 14*w + 26],
[199, 199, w^3 - 2*w^2 - 7*w + 3],
[223, 223, -3*w^3 + 8*w^2 + 12*w - 8],
[223, 223, 3*w^3 - 11*w^2 - 6*w + 17],
[223, 223, 2*w^3 - 8*w^2 - 3*w + 18],
[223, 223, -4*w^3 + 13*w^2 + 9*w - 18],
[233, 233, -3*w^3 + 10*w^2 + 6*w - 16],
[233, 233, -2*w^3 + 5*w^2 + 9*w - 6],
[241, 241, -3*w^3 + 9*w^2 + 8*w - 10],
[241, 241, 4*w^3 - 12*w^2 - 13*w + 15],
[257, 257, 2*w^3 - 8*w^2 - w + 16],
[257, 257, 4*w^3 - 13*w^2 - 11*w + 18],
[257, 257, -w^3 + 5*w^2 - 2*w - 9],
[257, 257, -w^3 + 2*w^2 + 4*w + 2],
[263, 263, 4*w^3 - 14*w^2 - 9*w + 27],
[263, 263, 5*w^3 - 15*w^2 - 17*w + 25],
[263, 263, -2*w^3 + 7*w^2 + 6*w - 16],
[263, 263, -2*w^3 + 7*w^2 + 6*w - 9],
[271, 271, -w - 4],
[271, 271, -w^3 + 3*w^2 + 4*w - 9],
[271, 271, 5*w^3 - 16*w^2 - 14*w + 24],
[271, 271, -4*w^3 + 14*w^2 + 9*w - 24],
[281, 281, -2*w^3 + 6*w^2 + 5*w - 4],
[281, 281, -3*w^3 + 9*w^2 + 10*w - 9],
[281, 281, 3*w^3 - 10*w^2 - 8*w + 12],
[281, 281, w^2 - w - 8],
[311, 311, w^2 - 5],
[311, 311, -4*w^3 + 13*w^2 + 12*w - 20],
[313, 313, 6*w^3 - 19*w^2 - 18*w + 31],
[313, 313, -2*w^3 + 5*w^2 + 6*w - 6],
[337, 337, 3*w^3 - 11*w^2 - 3*w + 15],
[337, 337, w^3 - w^2 - 9*w - 5],
[353, 353, -w^3 + 2*w^2 + 5*w - 4],
[353, 353, w^3 - w^2 - 8*w - 3],
[353, 353, 3*w^3 - 10*w^2 - 7*w + 19],
[353, 353, -4*w^3 + 14*w^2 + 7*w - 22],
[361, 19, -3*w^3 + 11*w^2 + 6*w - 18],
[361, 19, 2*w^3 - 8*w^2 - 3*w + 17],
[367, 367, -4*w^3 + 12*w^2 + 13*w - 18],
[367, 367, 3*w^3 - 9*w^2 - 8*w + 13],
[383, 383, 5*w^3 - 17*w^2 - 12*w + 30],
[383, 383, 2*w^2 - 3*w - 5],
[383, 383, 3*w^3 - 9*w^2 - 11*w + 11],
[383, 383, -w^3 + 3*w^2 + w - 1],
[401, 401, 3*w^3 - 11*w^2 - 8*w + 23],
[401, 401, 4*w^3 - 14*w^2 - 11*w + 22],
[431, 431, 2*w^3 - 5*w^2 - 10*w + 8],
[431, 431, 6*w^3 - 20*w^2 - 15*w + 29],
[433, 433, -4*w^3 + 12*w^2 + 13*w - 17],
[433, 433, -3*w^3 + 9*w^2 + 8*w - 12],
[433, 433, 3*w^3 - 11*w^2 - 6*w + 20],
[433, 433, -2*w^3 + 8*w^2 + 3*w - 15],
[439, 439, -3*w^3 + 11*w^2 + 6*w - 19],
[439, 439, -2*w^3 + 8*w^2 + 3*w - 16],
[449, 449, w^3 - 4*w^2 + w + 11],
[449, 449, -w^3 + 2*w^2 + 7*w - 6],
[457, 457, w^2 - 4*w - 3],
[457, 457, 4*w^3 - 15*w^2 - 6*w + 27],
[457, 457, w^2 - 4*w - 2],
[457, 457, -2*w^3 + 6*w^2 + 8*w - 15],
[479, 479, -4*w^3 + 14*w^2 + 9*w - 29],
[479, 479, -w^3 + 5*w^2 - 6],
[487, 487, -w^3 + 6*w^2 - 3*w - 16],
[487, 487, -5*w^3 + 18*w^2 + 9*w - 29],
[503, 503, -2*w^3 + 7*w^2 + 8*w - 15],
[503, 503, 4*w^3 - 13*w^2 - 14*w + 20],
[521, 521, 3*w^3 - 10*w^2 - 9*w + 10],
[521, 521, -2*w^3 + 7*w^2 + 3*w - 16],
[529, 23, w^3 - 3*w^2 - 3*w - 1],
[599, 599, -4*w^3 + 12*w^2 + 14*w - 15],
[599, 599, 2*w^3 - 6*w^2 - 4*w + 5],
[599, 599, 4*w^3 - 13*w^2 - 12*w + 19],
[599, 599, w^2 - 6],
[601, 601, -4*w^3 + 12*w^2 + 11*w - 19],
[601, 601, 5*w^3 - 15*w^2 - 16*w + 24],
[607, 607, -w^3 + 4*w^2 + w - 13],
[607, 607, -w^3 + 4*w^2 + w - 2],
[617, 617, -4*w^3 + 13*w^2 + 11*w - 16],
[617, 617, 4*w^3 - 13*w^2 - 13*w + 20],
[625, 5, -5],
[631, 631, -3*w^3 + 8*w^2 + 11*w - 9],
[631, 631, 5*w^3 - 16*w^2 - 13*w + 24],
[631, 631, w^3 - 3*w^2 - 4*w - 1],
[631, 631, w - 6],
[641, 641, -3*w^3 + 12*w^2 + w - 17],
[641, 641, w^3 - 6*w^2 + 5*w + 18],
[673, 673, 4*w^3 - 15*w^2 - 6*w + 26],
[673, 673, -2*w^3 + 9*w^2 - 19],
[719, 719, 2*w^2 - 4*w - 5],
[719, 719, 4*w^3 - 14*w^2 - 8*w + 25],
[727, 727, 3*w^3 - 8*w^2 - 11*w + 8],
[727, 727, 6*w^3 - 18*w^2 - 19*w + 20],
[727, 727, -5*w^3 + 14*w^2 + 19*w - 17],
[727, 727, -5*w^3 + 16*w^2 + 13*w - 23],
[743, 743, 4*w^3 - 14*w^2 - 7*w + 23],
[743, 743, w^3 - w^2 - 8*w - 2],
[751, 751, 3*w^3 - 12*w^2 - 3*w + 23],
[751, 751, 3*w^3 - 12*w^2 - 3*w + 22],
[761, 761, 3*w^3 - 10*w^2 - 6*w + 22],
[761, 761, 3*w^3 - 10*w^2 - 6*w + 18],
[761, 761, -5*w^3 + 16*w^2 + 17*w - 25],
[761, 761, -2*w^3 + 5*w^2 + 9*w - 8],
[809, 809, -3*w^3 + 8*w^2 + 13*w - 3],
[809, 809, -2*w^3 + 6*w^2 + 3*w - 12],
[823, 823, -w^3 + 4*w^2 + 2*w - 2],
[823, 823, 2*w^3 - 7*w^2 - 5*w + 18],
[839, 839, 4*w^3 - 13*w^2 - 14*w + 27],
[839, 839, -7*w^3 + 23*w^2 + 19*w - 41],
[857, 857, -3*w^3 + 10*w^2 + 9*w - 12],
[857, 857, -w^3 + 4*w^2 + 3*w - 13],
[863, 863, 3*w^3 - 11*w^2 - 2*w + 20],
[863, 863, -2*w^3 + 4*w^2 + 13*w - 5],
[863, 863, 6*w^3 - 19*w^2 - 18*w + 27],
[863, 863, -2*w^3 + 5*w^2 + 6*w - 2],
[881, 881, -3*w^3 + 10*w^2 + 5*w - 17],
[881, 881, -3*w^3 + 8*w^2 + 13*w - 12],
[887, 887, -3*w^3 + 9*w^2 + 11*w - 9],
[887, 887, w^3 - 3*w^2 - w - 1],
[911, 911, 4*w^3 - 14*w^2 - 10*w + 17],
[911, 911, 5*w^3 - 17*w^2 - 12*w + 22],
[919, 919, -4*w^3 + 15*w^2 + 7*w - 30],
[919, 919, -2*w^2 + 7*w + 2],
[919, 919, w^3 - 5*w^2 + 4*w + 13],
[919, 919, 3*w^3 - 12*w^2 - 4*w + 20],
[929, 929, -4*w^3 + 13*w^2 + 9*w - 22],
[929, 929, 3*w^3 - 8*w^2 - 12*w + 12],
[961, 31, 4*w^3 - 12*w^2 - 12*w + 17],
[967, 967, -2*w^3 + 6*w^2 + 7*w - 2],
[967, 967, -3*w^3 + 8*w^2 + 15*w - 6],
[967, 967, w^3 - 2*w^2 - 5*w - 6],
[967, 967, 7*w^3 - 22*w^2 - 21*w + 36],
[983, 983, 5*w^3 - 17*w^2 - 13*w + 25],
[983, 983, w^3 - 5*w^2 - w + 15],
[991, 991, -6*w^3 + 21*w^2 + 13*w - 34],
[991, 991, 7*w^3 - 22*w^2 - 21*w + 33],
[1009, 1009, w^2 - 2*w + 3],
[1009, 1009, -2*w^3 + 7*w^2 + 4*w - 18],
[1031, 1031, -4*w^3 + 15*w^2 + 4*w - 26],
[1031, 1031, 3*w^2 - 8*w - 9],
[1033, 1033, 2*w^3 - 4*w^2 - 13*w + 3],
[1033, 1033, -5*w^3 + 15*w^2 + 19*w - 23],
[1033, 1033, w^3 - w^2 - 8*w - 10],
[1033, 1033, 4*w^3 - 14*w^2 - 7*w + 15],
[1049, 1049, -7*w^3 + 24*w^2 + 16*w - 34],
[1049, 1049, -5*w^3 + 16*w^2 + 18*w - 24],
[1063, 1063, w^3 - 4*w^2 + 2*w + 8],
[1063, 1063, w^3 - 2*w^2 - 4*w - 8],
[1063, 1063, 7*w^3 - 24*w^2 - 17*w + 39],
[1063, 1063, w^3 - 2*w^2 - 7*w - 6],
[1103, 1103, w^3 - 5*w^2 + 3*w + 17],
[1103, 1103, -w^3 + 5*w^2 - 3*w - 3],
[1103, 1103, -w^3 + 5*w^2 - 2*w - 6],
[1103, 1103, -2*w^3 + 8*w^2 + w - 19],
[1129, 1129, -6*w^3 + 18*w^2 + 20*w - 27],
[1129, 1129, -4*w^3 + 12*w^2 + 10*w - 17],
[1151, 1151, w^3 - w^2 - 7*w - 1],
[1151, 1151, 5*w^3 - 17*w^2 - 11*w + 29],
[1153, 1153, -2*w^3 + 8*w^2 + 3*w - 23],
[1153, 1153, -6*w^3 + 18*w^2 + 20*w - 25],
[1193, 1193, -2*w^3 + 5*w^2 + 9*w - 10],
[1193, 1193, 3*w^3 - 10*w^2 - 6*w + 20],
[1201, 1201, 6*w^3 - 19*w^2 - 16*w + 24],
[1201, 1201, 4*w^3 - 11*w^2 - 14*w + 9],
[1217, 1217, 7*w^3 - 22*w^2 - 21*w + 30],
[1217, 1217, 7*w^3 - 23*w^2 - 18*w + 39],
[1223, 1223, -2*w^3 + 9*w^2 - 15],
[1223, 1223, -3*w^3 + 9*w^2 + 13*w - 9],
[1223, 1223, -5*w^3 + 17*w^2 + 12*w - 34],
[1223, 1223, 4*w^3 - 15*w^2 - 6*w + 30],
[1231, 1231, -3*w^3 + 7*w^2 + 12*w - 3],
[1231, 1231, -8*w^3 + 26*w^2 + 21*w - 38],
[1289, 1289, -4*w^3 + 14*w^2 + 7*w - 24],
[1289, 1289, -w^3 + w^2 + 8*w + 1],
[1297, 1297, -4*w^3 + 12*w^2 + 11*w - 15],
[1297, 1297, -5*w^3 + 15*w^2 + 16*w - 20],
[1303, 1303, -3*w^3 + 10*w^2 + 7*w - 23],
[1303, 1303, -w^3 + 2*w^2 + 5*w - 8],
[1319, 1319, -8*w^3 + 28*w^2 + 15*w - 43],
[1319, 1319, w^3 + w^2 - 12*w - 12],
[1321, 1321, w^2 - 2*w - 10],
[1321, 1321, 2*w^3 - 7*w^2 - 4*w + 5],
[1367, 1367, -5*w^3 + 17*w^2 + 14*w - 26],
[1367, 1367, 2*w^3 - 8*w^2 - 5*w + 19],
[1399, 1399, -2*w^3 + 7*w^2 + 3*w - 4],
[1399, 1399, w^3 - 2*w^2 - 6*w - 6],
[1409, 1409, w^3 - 2*w^2 - 3*w - 3],
[1409, 1409, 3*w^3 - 10*w^2 - 5*w + 18],
[1409, 1409, -5*w^3 + 16*w^2 + 15*w - 22],
[1409, 1409, -3*w^3 + 8*w^2 + 13*w - 13],
[1423, 1423, 7*w^3 - 24*w^2 - 18*w + 44],
[1423, 1423, -8*w^3 + 26*w^2 + 21*w - 40],
[1433, 1433, -7*w^3 + 24*w^2 + 17*w - 45],
[1433, 1433, -3*w^3 + 12*w^2 + 3*w - 29],
[1439, 1439, -2*w^3 + 6*w^2 + 4*w - 3],
[1439, 1439, -4*w^3 + 12*w^2 + 14*w - 13],
[1471, 1471, -w - 6],
[1471, 1471, -w^3 + 3*w^2 + 4*w - 11],
[1471, 1471, w^3 - 2*w^2 - 8*w + 4],
[1471, 1471, w^2 - 5*w - 4],
[1489, 1489, -2*w^3 + 9*w^2 + 2*w - 17],
[1489, 1489, -6*w^3 + 21*w^2 + 14*w - 38],
[1543, 1543, -7*w^3 + 24*w^2 + 19*w - 45],
[1543, 1543, 2*w^3 - 3*w^2 - 15*w - 4],
[1543, 1543, -5*w^3 + 18*w^2 + 6*w - 26],
[1543, 1543, -3*w^3 + 12*w^2 + 7*w - 20],
[1553, 1553, -4*w^3 + 14*w^2 + 11*w - 20],
[1553, 1553, -3*w^3 + 11*w^2 + 8*w - 25],
[1559, 1559, 2*w^2 - 4*w - 3],
[1559, 1559, 4*w^3 - 14*w^2 - 8*w + 27],
[1567, 1567, -4*w^3 + 15*w^2 + 3*w - 20],
[1567, 1567, w^3 - 12*w - 10],
[1601, 1601, -3*w^3 + 10*w^2 + 10*w - 12],
[1601, 1601, 2*w^3 - 7*w^2 - 7*w + 18],
[1607, 1607, -7*w^3 + 23*w^2 + 20*w - 36],
[1607, 1607, -6*w^3 + 19*w^2 + 18*w - 26],
[1607, 1607, 2*w^2 - w - 9],
[1607, 1607, -2*w^3 + 5*w^2 + 6*w - 1],
[1609, 1609, -4*w^3 + 15*w^2 + 8*w - 24],
[1609, 1609, 4*w^3 - 15*w^2 - 8*w + 31],
[1663, 1663, -6*w^3 + 19*w^2 + 13*w - 26],
[1663, 1663, -7*w^3 + 20*w^2 + 26*w - 26],
[1681, 41, -5*w^3 + 15*w^2 + 15*w - 17],
[1697, 1697, -w^3 + 9*w + 6],
[1697, 1697, -7*w^3 + 24*w^2 + 15*w - 39],
[1721, 1721, -5*w^3 + 18*w^2 + 13*w - 38],
[1721, 1721, -w^3 + 6*w^2 - 5*w - 20],
[1721, 1721, -w^3 + w^2 + 6*w + 11],
[1721, 1721, 6*w^3 - 20*w^2 - 15*w + 24],
[1753, 1753, -3*w^3 + 9*w^2 + 13*w - 17],
[1753, 1753, w^3 - 3*w^2 - 7*w + 3],
[1753, 1753, -6*w^3 + 22*w^2 + 12*w - 45],
[1753, 1753, -6*w^3 + 21*w^2 + 16*w - 40],
[1759, 1759, -2*w^3 + 6*w^2 + w - 8],
[1759, 1759, 7*w^3 - 21*w^2 - 26*w + 33],
[1759, 1759, -3*w^3 + 10*w^2 + 10*w - 26],
[1759, 1759, -5*w^3 + 17*w^2 + 12*w - 37],
[1783, 1783, -5*w^3 + 18*w^2 + 11*w - 29],
[1783, 1783, 4*w^3 - 15*w^2 - 7*w + 26],
[1783, 1783, 3*w^3 - 12*w^2 - 5*w + 26],
[1783, 1783, -3*w^3 + 12*w^2 + 4*w - 24],
[1801, 1801, 8*w^3 - 24*w^2 - 27*w + 33],
[1801, 1801, 4*w^3 - 12*w^2 - 16*w + 19],
[1801, 1801, 4*w - 1],
[1801, 1801, 7*w^3 - 21*w^2 - 23*w + 33],
[1831, 1831, w^3 - 2*w^2 - 8*w + 2],
[1831, 1831, w^2 - 5*w - 2],
[1847, 1847, 6*w^3 - 20*w^2 - 15*w + 25],
[1847, 1847, w^3 - w^2 - 6*w - 10],
[1849, 43, -6*w^3 + 21*w^2 + 14*w - 37],
[1849, 43, -2*w^3 + 9*w^2 + 2*w - 18],
[1871, 1871, -4*w^3 + 13*w^2 + 8*w - 22],
[1871, 1871, 4*w^3 - 13*w^2 - 12*w + 15],
[1871, 1871, 4*w^3 - 11*w^2 - 16*w + 17],
[1871, 1871, w^2 - 10],
[1873, 1873, -w^3 + 5*w^2 - 20],
[1873, 1873, -4*w^3 + 14*w^2 + 9*w - 15],
[1913, 1913, -2*w^3 + 6*w^2 + 3*w - 4],
[1913, 1913, 5*w^3 - 15*w^2 - 18*w + 19],
[1913, 1913, 6*w^3 - 20*w^2 - 11*w + 30],
[1913, 1913, 9*w^3 - 30*w^2 - 23*w + 45],
[1993, 1993, -8*w^3 + 28*w^2 + 17*w - 43],
[1993, 1993, 4*w^3 - 13*w^2 - 16*w + 19],
[1993, 1993, -6*w^3 + 21*w^2 + 14*w - 36],
[1993, 1993, -2*w^3 + 9*w^2 + 2*w - 19],
[1999, 1999, -7*w^3 + 22*w^2 + 20*w - 30],
[1999, 1999, 4*w^3 - 11*w^2 - 13*w + 10]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 38*x^6 + 455*x^4 - 1696*x^2 + 256;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/464*e^6 + 3/232*e^4 + 201/464*e^2 - 72/29, -11/696*e^6 + 149/348*e^4 - 655/232*e^2 + 98/87, 1/1392*e^6 - 1/232*e^4 - 665/1392*e^2 + 188/87, e, -91/5568*e^7 + 1433/2784*e^5 - 26717/5568*e^3 + 1441/116*e, 1/2784*e^7 - 1/464*e^5 - 665/2784*e^3 + 94/87*e, -7/5568*e^7 + 253/2784*e^5 - 2779/1856*e^3 + 2135/348*e, 59/5568*e^7 - 1105/2784*e^5 + 26653/5568*e^3 - 545/29*e, 37/2784*e^7 - 575/1392*e^5 + 9731/2784*e^3 - 165/29*e, -1/1392*e^6 + 1/232*e^4 + 665/1392*e^2 - 536/87, 1, -7/464*e^7 + 353/696*e^5 - 7147/1392*e^3 + 5639/348*e, 25/2784*e^7 - 539/1392*e^5 + 14927/2784*e^3 - 686/29*e, 85/1392*e^6 - 1183/696*e^4 + 5441/464*e^2 - 956/87, 13/696*e^7 - 121/174*e^5 + 5159/696*e^3 - 2347/116*e, 121/5568*e^7 - 585/928*e^5 + 28111/5568*e^3 - 1415/174*e, 43/696*e^6 - 593/348*e^4 + 8525/696*e^2 - 372/29, 3/116*e^6 - 85/174*e^4 + 395/348*e^2 - 163/87, 1/58*e^6 - 38/87*e^4 + 673/174*e^2 - 1201/87, -1/58*e^6 + 3/29*e^4 + 143/58*e^2 + 4/29, -5/174*e^6 + 73/87*e^4 - 1315/174*e^2 + 316/29, -1/116*e^6 + 3/58*e^4 + 201/116*e^2 - 172/29, -43/2784*e^7 + 1057/1392*e^5 - 9647/928*e^3 + 3034/87*e, -7/2784*e^7 + 123/464*e^5 - 13441/2784*e^3 + 7025/348*e, 5/348*e^7 - 131/174*e^5 + 1405/116*e^3 - 4969/87*e, -67/5568*e^7 + 1361/2784*e^5 - 31541/5568*e^3 + 2017/116*e, 39/1856*e^7 - 2207/2784*e^5 + 56755/5568*e^3 - 3839/87*e, -25/696*e^7 + 197/174*e^5 - 2385/232*e^3 + 8017/348*e, 17/2784*e^7 - 17/464*e^5 - 8521/2784*e^3 + 2468/87*e, -149/5568*e^7 + 845/928*e^5 - 51251/5568*e^3 + 7571/348*e, 35/696*e^6 - 337/348*e^4 + 853/696*e^2 + 636/29, -1/696*e^6 + 119/348*e^4 - 2351/696*e^2 - 425/29, 19/232*e^6 - 635/348*e^4 + 4783/696*e^2 + 1364/87, 19/464*e^6 - 635/696*e^4 + 4783/1392*e^2 + 1204/87, 5/58*e^6 - 73/29*e^4 + 967/58*e^2 + 38/29, 19/696*e^6 - 173/348*e^4 + 821/696*e^2 - 132/29, -4/87*e^6 + 37/29*e^4 - 646/87*e^2 - 722/87, -17/348*e^6 + 75/58*e^4 - 2615/348*e^2 - 169/87, -5/696*e^6 + 131/348*e^4 - 1057/232*e^2 + 614/87, 11/174*e^6 - 149/87*e^4 + 655/58*e^2 - 914/87, 11/2784*e^7 - 149/1392*e^5 + 191/928*e^3 + 2077/348*e, 1/4*e^5 - 11/2*e^3 + 95/4*e, -13/696*e^6 + 271/348*e^4 - 1913/232*e^2 + 970/87, -1/3*e^4 + 16/3*e^2 - 5/3, 7/928*e^7 - 253/464*e^5 + 9265/928*e^3 - 2831/58*e, 3/1856*e^7 + 437/2784*e^5 - 18049/5568*e^3 + 817/174*e, -19/2784*e^7 + 521/1392*e^5 - 18917/2784*e^3 + 1164/29*e, 3/928*e^7 - 125/464*e^5 + 4037/928*e^3 - 2091/116*e, -1/232*e^7 + 3/116*e^5 + 201/232*e^3 - 144/29*e, -7/928*e^7 + 253/464*e^5 - 8337/928*e^3 + 2019/58*e, -71/1392*e^7 + 1025/696*e^5 - 5451/464*e^3 + 3725/174*e, 1/24*e^7 - 17/12*e^5 + 335/24*e^3 - 71/2*e, -11/696*e^6 + 149/348*e^4 - 191/232*e^2 - 1990/87, 41/696*e^6 - 157/116*e^4 + 3359/696*e^2 + 1061/87, -14/87*e^6 + 374/87*e^4 - 2377/87*e^2 + 192/29, -91/1392*e^6 + 1201/696*e^4 - 5503/464*e^2 + 1742/87, -5/5568*e^7 - 449/2784*e^5 + 15389/5568*e^3 - 31/58*e, -1/87*e^7 + 41/174*e^5 - 89/87*e^3 + 247/58*e, 23/928*e^7 - 787/1392*e^5 + 4691/2784*e^3 + 5209/348*e, 5/96*e^7 - 95/48*e^5 + 737/32*e^3 - 230/3*e, -109/2784*e^7 + 573/464*e^5 - 30523/2784*e^3 + 2369/87*e, -71/5568*e^7 + 1141/2784*e^5 - 18673/5568*e^3 + 125/29*e, -71/696*e^6 + 303/116*e^4 - 11249/696*e^2 + 1318/87, 5/87*e^6 - 146/87*e^4 + 1141/87*e^2 - 719/29, -4/87*e^6 + 140/87*e^4 - 428/29*e^2 + 1946/87, 1/87*e^6 - 2/29*e^4 - 143/87*e^2 - 385/87, -23/348*e^7 + 359/174*e^5 - 2055/116*e^3 + 2917/87*e, -1/32*e^7 + 41/48*e^5 - 565/96*e^3 + 22/3*e, -5/192*e^7 + 103/96*e^5 - 2371/192*e^3 + 131/4*e, -131/2784*e^7 + 2017/1392*e^5 - 37237/2784*e^3 + 2337/58*e, 41/1392*e^6 - 587/696*e^4 + 2357/464*e^2 - 760/87, -7/232*e^6 + 295/348*e^4 - 4595/696*e^2 - 298/87, 11/232*e^6 - 149/116*e^4 + 2429/232*e^2 - 910/29, -23/696*e^6 + 301/348*e^4 - 3265/696*e^2 - 640/29, 293/5568*e^7 - 4823/2784*e^5 + 30945/1856*e^3 - 13327/348*e, 127/1392*e^7 - 707/232*e^5 + 42217/1392*e^3 - 14453/174*e, -103/2784*e^7 + 683/464*e^5 - 49825/2784*e^3 + 21737/348*e, -19/1392*e^7 + 289/696*e^5 - 1975/464*e^3 + 1097/87*e, -83/1392*e^7 + 1409/696*e^5 - 10215/464*e^3 + 6610/87*e, 91/2784*e^7 - 1549/1392*e^5 + 10607/928*e^3 - 10705/348*e, -57/1856*e^7 + 3761/2784*e^5 - 104365/5568*e^3 + 12607/174*e, 97/1856*e^7 - 6209/2784*e^5 + 160981/5568*e^3 - 36497/348*e, 13/174*e^6 - 155/87*e^4 + 1679/174*e^2 + 118/29, -4/87*e^6 + 82/87*e^4 - 182/87*e^2 - 318/29, 17/232*e^6 - 733/348*e^4 + 8309/696*e^2 + 1834/87, 11/174*e^6 - 91/87*e^4 + 109/174*e^2 - 208/29, -9/116*e^6 + 143/58*e^4 - 1903/116*e^2 - 388/29, 11/696*e^6 + 83/348*e^4 - 6851/696*e^2 + 383/29, -37/1856*e^7 + 2189/2784*e^5 - 44041/5568*e^3 + 815/87*e, -19/5568*e^7 + 521/2784*e^5 - 21701/5568*e^3 + 1135/58*e, -211/1392*e^6 + 2953/696*e^4 - 12311/464*e^2 - 460/87, -23/174*e^6 + 301/87*e^4 - 3613/174*e^2 + 224/29, 5/116*e^6 - 277/174*e^4 + 5105/348*e^2 - 784/87, -3/29*e^6 + 76/29*e^4 - 499/29*e^2 + 430/29, 5/58*e^7 - 277/87*e^5 + 6149/174*e^3 - 9224/87*e, 1/5568*e^7 - 931/2784*e^5 + 14317/1856*e^3 - 3143/87*e, 1/696*e^6 - 1/116*e^4 + 31/696*e^2 - 1190/87, -53/348*e^6 + 623/174*e^4 - 2017/116*e^2 - 1634/87, -7/464*e^7 + 643/696*e^5 - 22691/1392*e^3 + 15275/174*e, -1/87*e^7 - 17/174*e^5 + 270/29*e^3 - 11903/174*e, 353/5568*e^7 - 6163/2784*e^5 + 131815/5568*e^3 - 4261/58*e, 41/2784*e^7 - 1399/1392*e^5 + 48367/2784*e^3 - 8907/116*e, -19/348*e^6 + 289/174*e^4 - 1279/116*e^2 - 1354/87, -67/2784*e^7 + 1129/1392*e^5 - 7111/928*e^3 + 1126/87*e, 65/2784*e^7 - 529/464*e^5 + 48647/2784*e^3 - 7723/87*e, 5/696*e^7 - 40/87*e^5 + 5143/696*e^3 - 3129/116*e, 1/32*e^7 - 49/48*e^5 + 1013/96*e^3 - 223/6*e, -71/696*e^6 + 1141/348*e^4 - 17281/696*e^2 + 188/29, 115/696*e^6 - 1621/348*e^4 + 7839/232*e^2 - 4291/87, -5/29*e^6 + 117/29*e^4 - 619/29*e^2 + 359/29, 19/232*e^6 - 635/348*e^4 + 4783/696*e^2 + 668/87, -403/5568*e^7 + 2027/928*e^5 - 116197/5568*e^3 + 25939/348*e, 13/1392*e^7 + 193/696*e^5 - 16069/1392*e^3 + 2052/29*e, 217/1392*e^6 - 1145/232*e^4 + 54751/1392*e^2 - 4096/87, -113/464*e^6 + 4265/696*e^4 - 48325/1392*e^2 - 454/87, 31/348*e^6 - 325/174*e^4 + 475/116*e^2 + 2722/87, 7/696*e^6 - 137/348*e^4 + 3233/696*e^2 - 969/29, 17/174*e^6 - 75/29*e^4 + 2963/174*e^2 - 2272/87, -143/2784*e^7 + 2749/1392*e^5 - 68233/2784*e^3 + 3098/29*e, -271/2784*e^7 + 1431/464*e^5 - 81481/2784*e^3 + 14215/174*e, 65/1392*e^6 - 297/232*e^4 + 15239/1392*e^2 - 830/87, 95/696*e^6 - 1445/348*e^4 + 6859/232*e^2 - 1574/87, 17/192*e^7 - 307/96*e^5 + 6967/192*e^3 - 134*e, -37/696*e^7 + 60/29*e^5 - 17503/696*e^3 + 29003/348*e, -25/174*e^6 + 307/87*e^4 - 1167/58*e^2 + 1318/87, -71/1392*e^6 + 1141/696*e^4 - 21457/1392*e^2 + 384/29, -21/232*e^6 + 179/116*e^4 + 45/232*e^2 - 1139/29, -1/87*e^6 + 2/29*e^4 + 143/87*e^2 - 1094/87, 143/2784*e^7 - 2749/1392*e^5 + 59881/2784*e^3 - 1677/29*e, -5/116*e^7 + 51/29*e^5 - 2359/116*e^3 + 3413/58*e, 11/87*e^6 - 298/87*e^4 + 655/29*e^2 - 3046/87, -25/696*e^6 + 307/348*e^4 - 877/232*e^2 - 410/87, 127/5568*e^7 - 2237/2784*e^5 + 44537/5568*e^3 - 485/29*e, 67/1392*e^7 - 1535/696*e^5 + 41981/1392*e^3 - 12911/116*e, -179/5568*e^7 + 3553/2784*e^5 - 94405/5568*e^3 + 9769/116*e, 15/928*e^7 - 367/1392*e^5 - 8581/2784*e^3 + 7271/174*e, -7/348*e^7 + 79/174*e^5 - 227/116*e^3 + 362/87*e, -7/696*e^7 + 311/348*e^5 - 12977/696*e^3 + 6143/58*e, 65/696*e^6 - 659/348*e^4 + 517/232*e^2 + 4720/87, 37/348*e^6 - 153/58*e^4 + 4975/348*e^2 - 886/87, 19/696*e^7 - 217/174*e^5 + 11609/696*e^3 - 6357/116*e, 91/5568*e^7 - 737/2784*e^5 - 9475/5568*e^3 + 1295/58*e, 7/464*e^7 - 137/232*e^5 + 3233/464*e^3 - 1689/58*e, -137/1392*e^7 + 2383/696*e^5 - 52735/1392*e^3 + 7779/58*e, -497/5568*e^7 + 8915/2784*e^5 - 66461/1856*e^3 + 11035/87*e, 137/1392*e^7 - 717/232*e^5 + 38351/1392*e^3 - 10693/174*e, -107/1856*e^7 + 1945/928*e^5 - 47629/1856*e^3 + 12277/116*e, 27/928*e^7 - 1171/1392*e^5 + 23159/2784*e^3 - 3464/87*e, -37/5568*e^7 - 353/2784*e^5 + 12223/1856*e^3 - 7625/174*e, -55/5568*e^7 + 2021/2784*e^5 - 70145/5568*e^3 + 1439/29*e, -527/5568*e^7 + 8773/2784*e^5 - 169225/5568*e^3 + 9247/116*e, 37/928*e^7 - 1957/1392*e^5 + 39401/2784*e^3 - 6073/174*e, -7/1856*e^7 + 295/2784*e^5 - 10163/5568*e^3 + 6811/348*e, 343/5568*e^7 - 6365/2784*e^5 + 137537/5568*e^3 - 7747/116*e, -4/29*e^6 + 111/29*e^4 - 820/29*e^2 + 1163/29, -9/116*e^6 + 371/174*e^4 - 4549/348*e^2 + 2258/87, -1/58*e^6 - 20/87*e^4 + 1531/174*e^2 - 3961/87, 23/174*e^6 - 110/29*e^4 + 4541/174*e^2 - 1513/87, -33/928*e^7 + 1457/1392*e^5 - 24181/2784*e^3 + 3341/174*e, 85/1392*e^7 - 549/232*e^5 + 36739/1392*e^3 - 6118/87*e, -25/696*e^6 + 655/348*e^4 - 4357/232*e^2 + 373/87, -23/348*e^6 + 81/58*e^4 - 1409/348*e^2 + 452/87, 2/87*e^6 - 70/87*e^4 + 330/29*e^2 - 5410/87, 79/696*e^6 - 311/116*e^4 + 10801/696*e^2 - 746/87, 5/696*e^6 - 5/116*e^4 - 541/696*e^2 + 662/87, -1/116*e^7 + 93/116*e^5 - 1829/116*e^3 + 9781/116*e, 575/5568*e^7 - 2895/928*e^5 + 143801/5568*e^3 - 8677/174*e, 13/348*e^6 + 19/174*e^4 - 4237/348*e^2 + 1277/29, -23/87*e^6 + 602/87*e^4 - 3787/87*e^2 - 16/29, -229/1392*e^6 + 3239/696*e^4 - 46307/1392*e^2 + 826/29, 181/696*e^6 - 2399/348*e^4 + 32291/696*e^2 - 1694/29, 23/1856*e^7 - 439/2784*e^5 - 10621/5568*e^3 + 5693/348*e, -671/5568*e^7 + 3687/928*e^5 - 216377/5568*e^3 + 37673/348*e, 10/87*e^6 - 350/87*e^4 + 1070/29*e^2 - 5474/87, -1/696*e^6 - 229/348*e^4 + 8089/696*e^2 - 686/29, 53/232*e^6 - 2101/348*e^4 + 24185/696*e^2 + 2770/87, -23/116*e^6 + 787/174*e^4 - 6083/348*e^2 - 3458/87, 13/232*e^7 - 196/87*e^5 + 21509/696*e^3 - 51341/348*e, -53/5568*e^7 + 2015/2784*e^5 - 21969/1856*e^3 + 3367/87*e, -43/232*e^6 + 593/116*e^4 - 8525/232*e^2 + 478/29, 67/232*e^6 - 2575/348*e^4 + 28271/696*e^2 + 2902/87, 1/174*e^6 - 61/87*e^4 + 571/58*e^2 - 1048/87, 25/174*e^6 - 365/87*e^4 + 5183/174*e^2 - 652/29, -301/2784*e^7 + 5195/1392*e^5 - 105611/2784*e^3 + 11565/116*e, 505/5568*e^7 - 10331/2784*e^5 + 249743/5568*e^3 - 8179/58*e, -35/2784*e^7 + 267/464*e^5 - 21269/2784*e^3 + 6209/174*e, -127/5568*e^7 + 2701/2784*e^5 - 19795/1856*e^3 + 2881/174*e, -35/696*e^6 + 35/116*e^4 + 7963/696*e^2 - 2198/87, 13/232*e^6 - 581/348*e^4 + 12577/696*e^2 - 5491/87, -47/2784*e^7 + 395/464*e^5 - 31385/2784*e^3 + 8167/348*e, 171/1856*e^7 - 3065/928*e^5 + 73741/1856*e^3 - 19733/116*e, -1/29*e^6 + 76/87*e^4 - 499/87*e^2 - 2296/87, -1/12*e^6 + 11/6*e^4 - 47/12*e^2 - 14, -35/2784*e^7 + 499/464*e^5 - 49109/2784*e^3 + 4975/87*e, -3/464*e^7 + 259/696*e^5 - 7007/1392*e^3 + 1672/87*e, -21/116*e^6 + 295/58*e^4 - 4479/116*e^2 + 1434/29, -197/1392*e^6 + 2215/696*e^4 - 18403/1392*e^2 - 262/29, -233/2784*e^7 + 4063/1392*e^5 - 84943/2784*e^3 + 9957/116*e, -359/5568*e^7 + 2215/928*e^5 - 153809/5568*e^3 + 9401/87*e, 53/2784*e^7 - 1319/1392*e^5 + 13617/928*e^3 - 12511/174*e, 449/2784*e^7 - 6799/1392*e^5 + 119479/2784*e^3 - 11827/116*e, 347/5568*e^7 - 2435/928*e^5 + 197981/5568*e^3 - 53345/348*e, 59/5568*e^7 - 1105/2784*e^5 + 32221/5568*e^3 - 893/29*e, -11/348*e^6 + 69/58*e^4 - 3821/348*e^2 + 1385/87, 133/1392*e^6 - 863/696*e^4 - 9565/1392*e^2 + 1336/29, -73/1856*e^7 + 1817/2784*e^5 + 13859/5568*e^3 - 15565/348*e, -409/5568*e^7 + 8651/2784*e^5 - 230063/5568*e^3 + 5057/29*e, 13/348*e^6 - 271/174*e^4 + 1681/116*e^2 - 1070/87, 53/464*e^6 - 2101/696*e^4 + 26969/1392*e^2 + 776/87, 7/116*e^6 - 79/58*e^4 + 681/116*e^2 + 740/29, -5/87*e^6 + 88/87*e^4 - 187/29*e^2 + 4390/87, -109/5568*e^7 + 1487/2784*e^5 - 25883/5568*e^3 + 2053/116*e, 43/348*e^7 - 398/87*e^5 + 5819/116*e^3 - 25489/174*e, 13/696*e^6 - 271/348*e^4 + 521/232*e^2 + 3032/87, -283/1392*e^6 + 4097/696*e^4 - 56333/1392*e^2 + 284/29, -455/5568*e^7 + 6701/2784*e^5 - 34011/1856*e^3 + 5317/348*e, -181/5568*e^7 + 1573/928*e^5 - 144115/5568*e^3 + 21179/174*e, -7/174*e^6 - 37/87*e^4 + 3727/174*e^2 - 1518/29, -39/232*e^6 + 1859/348*e^4 - 34483/696*e^2 + 6352/87, -7/696*e^7 + 427/348*e^5 - 5563/232*e^3 + 19415/174*e, -403/5568*e^7 + 7937/2784*e^5 - 186725/5568*e^3 + 12745/116*e, -187/2784*e^7 + 4273/1392*e^5 - 37119/928*e^3 + 11248/87*e, -51/1856*e^7 + 4171/2784*e^5 - 136751/5568*e^3 + 10963/87*e, 19/464*e^6 - 57/232*e^4 - 3355/464*e^2 + 92/29, 79/696*e^6 - 311/116*e^4 + 8713/696*e^2 - 1616/87, -781/5568*e^7 + 12551/2784*e^5 - 249947/5568*e^3 + 4304/29*e, 59/928*e^7 - 1105/464*e^5 + 27581/928*e^3 - 3850/29*e, 2/87*e^7 + 5/174*e^5 - 953/87*e^3 + 4233/58*e, -91/1392*e^7 + 497/232*e^5 - 32053/1392*e^3 + 31531/348*e, 7/174*e^6 - 79/87*e^4 + 517/58*e^2 - 5422/87, 41/232*e^6 - 471/116*e^4 + 5679/232*e^2 - 1897/29, 55/696*e^6 - 745/348*e^4 + 3275/232*e^2 - 6058/87, 31/696*e^6 - 325/348*e^4 + 475/232*e^2 - 1162/87, 31/232*e^6 - 1439/348*e^4 + 21211/696*e^2 - 1978/87, -1/48*e^6 + 11/24*e^4 + 217/48*e^2 - 40, -245/696*e^6 + 3287/348*e^4 - 14673/232*e^2 + 6596/87, -9/232*e^6 - 151/348*e^4 + 13547/696*e^2 - 4874/87, -119/348*e^6 + 1633/174*e^4 - 22249/348*e^2 + 1684/29, -15/464*e^6 + 45/232*e^4 + 1623/464*e^2 - 442/29, 77/5568*e^7 - 3943/2784*e^5 + 165019/5568*e^3 - 4787/29*e, -647/5568*e^7 + 10525/2784*e^5 - 217489/5568*e^3 + 16169/116*e, 151/928*e^7 - 2541/464*e^5 + 49921/928*e^3 - 8007/58*e, -279/1856*e^7 + 4549/928*e^5 - 90081/1856*e^3 + 3910/29*e, -31/1392*e^6 + 1021/696*e^4 - 9523/464*e^2 + 3452/87, -13/696*e^6 + 129/116*e^4 - 10843/696*e^2 + 1289/87, 127/1392*e^7 - 1889/696*e^5 + 32009/1392*e^3 - 2343/58*e, 131/5568*e^7 - 2713/2784*e^5 + 84565/5568*e^3 - 4315/58*e, -57/464*e^7 + 3181/696*e^5 - 73277/1392*e^3 + 32825/174*e, 45/1856*e^7 - 1295/928*e^5 + 44315/1856*e^3 - 13551/116*e, -73/1856*e^7 + 3905/2784*e^5 - 72445/5568*e^3 + 2179/174*e, 21/464*e^7 - 295/232*e^5 + 4595/464*e^3 - 605/29*e, 13/116*e^6 - 523/174*e^4 + 6893/348*e^2 + 1517/87, 43/464*e^6 - 1315/696*e^4 + 9335/1392*e^2 - 688/87, -1/348*e^6 - 55/174*e^4 + 433/348*e^2 + 513/29, -41/348*e^6 + 157/58*e^4 - 4751/348*e^2 - 556/87, 43/174*e^6 - 159/29*e^4 + 3421/174*e^2 + 6208/87, -113/928*e^7 + 5773/1392*e^5 - 120245/2784*e^3 + 48131/348*e, 37/348*e^7 - 302/87*e^5 + 4133/116*e^3 - 22333/174*e, 49/696*e^7 - 959/348*e^5 + 24023/696*e^3 - 4144/29*e, 17/2784*e^7 - 365/464*e^5 + 37415/2784*e^3 - 16315/348*e, -857/5568*e^7 + 14171/2784*e^5 - 90597/1856*e^3 + 22421/174*e, -1/8*e^7 + 43/12*e^5 - 701/24*e^3 + 409/6*e, -1/464*e^6 + 241/696*e^4 - 5429/1392*e^2 + 538/87, -33/116*e^6 + 1225/174*e^4 - 13973/348*e^2 - 266/87, -439/1392*e^6 + 5957/696*e^4 - 86225/1392*e^2 + 2514/29, 10/87*e^6 - 49/29*e^4 - 386/87*e^2 + 1109/87, -23/232*e^6 + 301/116*e^4 - 3033/232*e^2 - 528/29, -17/348*e^6 - 65/174*e^4 + 7709/348*e^2 - 1748/29, -223/696*e^6 + 2641/348*e^4 - 9187/232*e^2 + 223/87, -51/464*e^6 + 2315/696*e^4 - 38383/1392*e^2 + 2324/87, 85/1392*e^6 - 2111/696*e^4 + 50195/1392*e^2 - 1556/29, -443/1392*e^6 + 5969/696*e^4 - 26463/464*e^2 + 1918/87, 227/696*e^6 - 923/116*e^4 + 30701/696*e^2 - 82/87, -1/24*e^6 + 5/4*e^4 - 199/24*e^2 + 53/3, -7/87*e^6 + 72/29*e^4 - 1957/87*e^2 + 346/87, 31/348*e^6 - 325/174*e^4 + 939/116*e^2 + 634/87, -41/696*e^6 + 157/116*e^4 - 6143/696*e^2 - 2366/87, -21/58*e^6 + 295/29*e^4 - 4363/58*e^2 + 2346/29, 13/348*e^6 - 155/174*e^4 + 1679/348*e^2 - 434/29, -35/348*e^6 + 569/174*e^4 - 3223/116*e^2 + 1288/87, -109/1392*e^7 + 1603/696*e^5 - 8473/464*e^3 + 3589/174*e, 11/96*e^7 - 173/48*e^5 + 1087/32*e^3 - 1253/12*e, 49/696*e^6 - 611/348*e^4 + 7319/696*e^2 + 264/29, 9/116*e^6 - 313/174*e^4 + 1997/348*e^2 + 1628/87, -595/5568*e^7 + 8977/2784*e^5 - 52471/1856*e^3 + 24701/348*e, 205/1392*e^7 - 1133/232*e^5 + 69691/1392*e^3 - 14704/87*e, 5/174*e^7 - 349/174*e^5 + 2043/58*e^3 - 29185/174*e, -19/348*e^7 + 665/348*e^5 - 2497/116*e^3 + 24773/348*e, -11/348*e^6 + 149/174*e^4 - 887/116*e^2 + 544/87, 149/696*e^6 - 1607/348*e^4 + 4169/232*e^2 + 1678/87, -131/5568*e^7 + 1321/2784*e^5 - 1045/5568*e^3 - 554/29*e, -115/5568*e^7 - 117/928*e^5 + 93179/5568*e^3 - 45197/348*e, 67/928*e^7 - 4547/1392*e^5 + 123391/2784*e^3 - 30971/174*e, 113/1856*e^7 - 7049/2784*e^5 + 187525/5568*e^3 - 26221/174*e, 1/12*e^6 - 19/6*e^4 + 109/4*e^2 + 64/3, -119/232*e^6 + 4667/348*e^4 - 57235/696*e^2 + 1198/87, -17/87*e^6 + 150/29*e^4 - 2963/87*e^2 + 4196/87, 155/696*e^6 - 2321/348*e^4 + 11887/232*e^2 - 2591/87, -27/116*e^6 + 997/174*e^4 - 9935/348*e^2 - 998/87, -89/696*e^6 + 321/116*e^4 - 7631/696*e^2 - 752/87];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {31,31,w^3 - 4*w^2 - 2*w + 8}>] := -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;