/*
  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![2, 4, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w^2 + 3}>;

primesArray := [
[2, 2, w],
[4, 2, -w^3 + 4*w + 1],
[5, 5, w + 1],
[13, 13, -w^2 + 3],
[17, 17, -w^3 + w^2 + 3*w - 1],
[19, 19, -w^3 + 3*w - 1],
[23, 23, -w + 3],
[31, 31, -w^2 - 2*w + 1],
[41, 41, w^3 + w^2 - 5*w - 3],
[43, 43, 2*w - 1],
[53, 53, -w - 3],
[53, 53, w^3 - w^2 - 4*w + 1],
[61, 61, w^3 - 3*w - 5],
[67, 67, w^3 + w^2 - 5*w - 1],
[81, 3, -3],
[83, 83, -w^3 + 5*w - 3],
[89, 89, w^2 + 1],
[97, 97, w^3 - w^2 - 5*w + 1],
[97, 97, 3*w^3 - 5*w^2 - 14*w + 21],
[97, 97, w^3 - 3*w - 3],
[97, 97, w^3 - 7*w - 3],
[103, 103, w^3 - w^2 - 3*w + 5],
[107, 107, -2*w^3 + 2*w^2 + 8*w - 7],
[109, 109, 3*w^3 - 3*w^2 - 13*w + 11],
[113, 113, 2*w^3 - 2*w^2 - 10*w + 7],
[113, 113, w^3 + w^2 - 5*w - 7],
[121, 11, 2*w^2 - w - 9],
[121, 11, 2*w^2 - w - 7],
[125, 5, -w^3 + 2*w^2 + 3*w - 7],
[131, 131, 4*w^3 - 5*w^2 - 18*w + 19],
[137, 137, w^3 + w^2 - 6*w - 5],
[137, 137, 3*w^3 - 13*w - 3],
[139, 139, 3*w^3 - 5*w^2 - 13*w + 19],
[149, 149, 2*w^3 - 9*w + 1],
[163, 163, -2*w^2 + 11],
[163, 163, -w^3 + w^2 + 6*w - 3],
[163, 163, w^3 - 2*w^2 - 5*w + 5],
[163, 163, 3*w^3 - 3*w^2 - 14*w + 15],
[173, 173, 2*w^3 - 10*w + 1],
[173, 173, 2*w^3 - 3*w^2 - 8*w + 13],
[179, 179, 2*w^3 + w^2 - 8*w - 1],
[181, 181, 2*w^3 - 4*w^2 - 9*w + 15],
[193, 193, w^2 - 7],
[199, 199, 2*w^2 - 5],
[211, 211, -2*w^3 + 8*w - 3],
[211, 211, -2*w^3 + w^2 + 8*w - 3],
[223, 223, 3*w - 1],
[223, 223, -2*w^3 - w^2 + 8*w + 5],
[229, 229, -w^3 + 3*w^2 + 4*w - 11],
[233, 233, -w^3 + 3*w^2 - 5],
[233, 233, w^3 + 2*w^2 - 5*w - 3],
[239, 239, 2*w^3 - 7*w - 1],
[241, 241, w^3 - 2*w^2 - 3*w + 11],
[251, 251, 2*w^2 - w - 5],
[251, 251, 2*w^2 + 2*w - 7],
[271, 271, -2*w^3 + 2*w^2 + 7*w - 9],
[283, 283, 6*w^3 - 6*w^2 - 28*w + 23],
[293, 293, 4*w^3 - 4*w^2 - 17*w + 17],
[293, 293, 2*w^3 - 2*w^2 - 7*w + 5],
[307, 307, 6*w^3 - 7*w^2 - 28*w + 29],
[311, 311, w^3 - 2*w^2 - 3*w + 1],
[311, 311, 2*w^2 - 2*w - 7],
[311, 311, 2*w^3 - 4*w^2 - 8*w + 17],
[311, 311, -w^3 + 3*w^2 + 4*w - 9],
[347, 347, -w^3 - w^2 + 6*w - 1],
[347, 347, -3*w^3 + 5*w^2 + 15*w - 21],
[349, 349, 2*w^3 + 2*w^2 - 11*w - 7],
[349, 349, 3*w^3 - w^2 - 15*w + 3],
[353, 353, w^3 + w^2 - 3*w - 5],
[353, 353, -4*w^3 + 16*w + 5],
[361, 19, 6*w^3 - 8*w^2 - 27*w + 31],
[367, 367, 3*w^3 - 3*w^2 - 13*w + 9],
[367, 367, -2*w^2 - w + 9],
[373, 373, -2*w^3 + w^2 + 10*w - 7],
[379, 379, w^3 + 2*w^2 - 7*w - 7],
[383, 383, -w^3 + 2*w^2 + 3*w - 9],
[389, 389, -2*w^3 + 3*w^2 + 10*w - 9],
[389, 389, w^2 - 2*w - 5],
[397, 397, 2*w^3 - w^2 - 10*w - 1],
[397, 397, w - 5],
[409, 409, w^3 - w^2 - 6*w - 3],
[419, 419, w^3 - 7*w + 1],
[419, 419, -3*w^3 + 2*w^2 + 13*w - 11],
[421, 421, w^3 + w^2 - 4*w + 3],
[421, 421, 3*w^3 + w^2 - 14*w - 3],
[431, 431, w^2 + 2*w - 7],
[449, 449, w^3 + w^2 - 3*w - 9],
[457, 457, -2*w^3 + w^2 + 8*w - 9],
[457, 457, 2*w^3 - 2*w^2 - 7*w + 11],
[463, 463, -w^3 - w^2 + 8*w - 3],
[463, 463, 3*w^3 - 3*w^2 - 15*w + 13],
[487, 487, 2*w^3 - 2*w^2 - 7*w + 1],
[487, 487, w^2 - 2*w - 7],
[499, 499, 3*w^3 - 3*w^2 - 12*w + 11],
[503, 503, -3*w^3 + w^2 + 13*w + 5],
[509, 509, -w^3 - w^2 + 7*w + 5],
[521, 521, 2*w^3 + 2*w^2 - 9*w - 5],
[523, 523, -2*w^2 - 1],
[541, 541, 2*w^3 - 2*w^2 - 8*w + 3],
[541, 541, w^3 - 2*w^2 - 7*w + 5],
[547, 547, 3*w^3 - 4*w^2 - 13*w + 13],
[563, 563, w^3 + 3*w^2 - 3*w - 11],
[571, 571, -w^3 + w - 3],
[571, 571, 2*w^3 + w^2 - 10*w - 1],
[587, 587, 3*w^3 - 15*w + 1],
[587, 587, 3*w^3 - w^2 - 14*w - 1],
[593, 593, 3*w^3 - 2*w^2 - 15*w + 7],
[593, 593, -2*w^3 + w^2 + 6*w - 1],
[599, 599, -w^3 + w^2 + 4*w + 3],
[599, 599, 2*w^3 + 2*w^2 - 11*w - 3],
[607, 607, -w - 5],
[607, 607, -w^3 + 4*w^2 + 5*w - 19],
[613, 613, 2*w^3 + w^2 - 8*w + 1],
[617, 617, 3*w^3 - w^2 - 12*w + 5],
[617, 617, -2*w^3 + 12*w - 3],
[619, 619, w^2 - 4*w - 3],
[631, 631, -3*w^2 - 4*w + 7],
[631, 631, -6*w^3 + 4*w^2 + 27*w - 15],
[643, 643, -w^3 + 2*w^2 + 7*w - 7],
[643, 643, 3*w^2 - 11],
[647, 647, 4*w^3 - 4*w^2 - 17*w + 15],
[653, 653, -2*w^3 - 2*w^2 + 10*w + 7],
[653, 653, w^3 - 3*w^2 - 6*w + 7],
[659, 659, -3*w^3 + 2*w^2 + 15*w - 11],
[673, 673, w^3 + w^2 - 3*w - 7],
[677, 677, w^3 - 3*w^2 - 3*w + 13],
[677, 677, 2*w^3 - 2*w^2 - 11*w + 9],
[709, 709, w^3 + w^2 - 2*w - 5],
[751, 751, w^3 - 5*w^2 - 4*w + 21],
[757, 757, -2*w^3 + 2*w^2 + 12*w - 1],
[757, 757, -w^3 + w^2 + 2*w - 7],
[761, 761, 4*w^3 - w^2 - 20*w + 3],
[787, 787, 4*w + 3],
[797, 797, 5*w^3 - 4*w^2 - 23*w + 17],
[809, 809, -2*w^3 + 9*w - 5],
[821, 821, 3*w^2 - 2*w - 9],
[823, 823, -w^2 - 3],
[863, 863, -w^3 + w^2 + 5*w - 9],
[881, 881, -4*w^3 + 5*w^2 + 16*w - 21],
[881, 881, -5*w^3 + 4*w^2 + 23*w - 13],
[887, 887, 2*w^3 - 2*w^2 - 11*w + 7],
[907, 907, -3*w^3 + 2*w^2 + 13*w - 3],
[907, 907, 2*w^2 + 2*w - 9],
[911, 911, -3*w^3 + w^2 + 15*w + 3],
[919, 919, 3*w^2 - 7],
[919, 919, 2*w^3 - 7*w + 7],
[937, 937, 4*w^3 - 4*w^2 - 18*w + 13],
[937, 937, 3*w^3 - w^2 - 13*w + 7],
[977, 977, -4*w^3 + 2*w^2 + 19*w - 9],
[983, 983, -w^3 + 4*w^2 + 5*w - 13],
[983, 983, -2*w^3 + 2*w^2 + 9*w - 3],
[991, 991, -w^3 + 4*w^2 + 3*w - 19],
[991, 991, 2*w - 7],
[1019, 1019, -5*w^3 + 5*w^2 + 23*w - 23],
[1031, 1031, -2*w^3 + 4*w + 3],
[1039, 1039, -3*w^3 + w^2 + 12*w - 3],
[1051, 1051, -4*w^3 - 2*w^2 + 23*w + 9],
[1051, 1051, 5*w^3 - 5*w^2 - 21*w + 21],
[1051, 1051, 4*w^3 - 6*w^2 - 20*w + 23],
[1051, 1051, 4*w^2 + 2*w - 11],
[1061, 1061, -w^3 + 3*w^2 + 8*w - 7],
[1061, 1061, 5*w^3 - 3*w^2 - 24*w + 13],
[1063, 1063, -w^3 - w^2 + 8*w + 5],
[1069, 1069, 2*w^2 - 4*w - 5],
[1087, 1087, 2*w^3 - 2*w^2 - 10*w + 3],
[1091, 1091, 3*w^3 - 17*w + 3],
[1091, 1091, 3*w^3 + w^2 - 12*w - 7],
[1093, 1093, 6*w^3 - 8*w^2 - 26*w + 29],
[1093, 1093, 6*w^3 - 6*w^2 - 26*w + 25],
[1093, 1093, -w^3 + 3*w^2 + 6*w - 15],
[1093, 1093, 2*w^3 - 4*w^2 - 6*w + 13],
[1117, 1117, 2*w^3 + w^2 - 12*w - 7],
[1129, 1129, -w^3 + 3*w^2 + 7*w - 13],
[1129, 1129, 2*w^2 - 3*w + 3],
[1163, 1163, 2*w^3 + 3*w^2 - 10*w - 13],
[1163, 1163, -2*w^3 + 2*w^2 + 6*w - 9],
[1171, 1171, 3*w^2 - 2*w - 7],
[1171, 1171, -3*w^3 + 11*w + 1],
[1181, 1181, -5*w^3 + 6*w^2 + 23*w - 29],
[1187, 1187, 4*w^3 - 2*w^2 - 20*w + 5],
[1187, 1187, 2*w^3 - 3*w^2 - 10*w + 17],
[1201, 1201, 5*w^3 - 3*w^2 - 22*w + 9],
[1213, 1213, 2*w^3 + 2*w^2 - 13*w - 7],
[1213, 1213, 3*w^3 - 11*w + 1],
[1217, 1217, -2*w^3 + 4*w^2 + 11*w - 11],
[1223, 1223, 2*w^3 - w^2 - 12*w + 7],
[1223, 1223, w^3 + 2*w^2 - 5*w - 13],
[1229, 1229, 2*w^3 - 6*w - 5],
[1229, 1229, 3*w^3 + 2*w^2 - 11*w - 7],
[1237, 1237, -w^3 + w^2 + w - 5],
[1237, 1237, -2*w^3 + 10*w - 5],
[1249, 1249, -3*w^3 + 3*w^2 + 13*w - 17],
[1249, 1249, 2*w^3 - 12*w + 1],
[1249, 1249, 4*w^3 - 5*w^2 - 18*w + 17],
[1249, 1249, 4*w^3 - 3*w^2 - 18*w + 15],
[1277, 1277, -3*w^3 + 2*w^2 + 11*w - 3],
[1279, 1279, w^3 + w^2 - 3*w - 11],
[1279, 1279, w^2 - 6*w - 1],
[1279, 1279, -3*w^3 + 4*w^2 + 11*w - 9],
[1279, 1279, 2*w^3 - 11*w + 5],
[1283, 1283, 2*w^3 + 2*w^2 - 9*w - 1],
[1283, 1283, -2*w^3 + 13*w - 5],
[1291, 1291, 3*w^3 - 5*w^2 - 16*w + 19],
[1291, 1291, -2*w^3 + 13*w + 3],
[1301, 1301, -w^3 + 3*w^2 + 2*w - 11],
[1301, 1301, w^3 + 3*w^2 - 7*w - 5],
[1319, 1319, 2*w^3 - 2*w^2 - 8*w + 13],
[1321, 1321, w^3 + 2*w^2 - 7*w - 9],
[1321, 1321, -6*w^3 + 8*w^2 + 28*w - 37],
[1361, 1361, -w^3 + w + 7],
[1361, 1361, -4*w^3 + 2*w^2 + 17*w - 5],
[1367, 1367, -w^3 + w^2 + 8*w - 1],
[1381, 1381, 3*w^3 - w^2 - 11*w + 5],
[1427, 1427, w^3 + 3*w^2 - 5*w - 5],
[1427, 1427, -w^3 + 4*w^2 + 3*w - 13],
[1427, 1427, 6*w^3 - 7*w^2 - 26*w + 33],
[1427, 1427, -4*w^3 + w^2 + 18*w + 7],
[1429, 1429, 3*w^3 - 3*w^2 - 17*w + 17],
[1433, 1433, -w^3 + 5*w - 7],
[1451, 1451, -3*w^3 + w^2 + 15*w - 7],
[1451, 1451, -w^3 + 5*w^2 + 5*w - 19],
[1459, 1459, -5*w + 9],
[1471, 1471, -4*w^3 + 8*w^2 + 19*w - 29],
[1483, 1483, w^3 + 3*w^2 - 6*w - 5],
[1483, 1483, -3*w^3 - w^2 + 12*w - 1],
[1487, 1487, -2*w^3 + 2*w^2 + 11*w - 15],
[1489, 1489, 2*w^3 - 5*w^2 - 10*w + 23],
[1493, 1493, 3*w^3 - 5*w^2 - 12*w + 25],
[1493, 1493, 3*w^3 - 6*w^2 - 15*w + 25],
[1499, 1499, 2*w^2 + 2*w - 13],
[1499, 1499, w^3 + 3*w^2 - 5*w - 11],
[1523, 1523, -w^3 + 3*w^2 + 5*w - 5],
[1523, 1523, 4*w^3 - w^2 - 14*w + 9],
[1523, 1523, -3*w^3 + w^2 + 14*w - 7],
[1523, 1523, -2*w^3 + 3*w^2 + 6*w - 11],
[1543, 1543, -w^3 + 3*w^2 + w - 9],
[1543, 1543, -4*w^3 + 7*w^2 + 20*w - 29],
[1559, 1559, 2*w^2 - 13],
[1559, 1559, -4*w^2 - w + 17],
[1571, 1571, 5*w - 3],
[1579, 1579, 2*w^2 + 2*w - 11],
[1579, 1579, w^3 + w^2 - 7*w - 11],
[1583, 1583, 2*w^3 - 3*w^2 - 10*w + 7],
[1597, 1597, -3*w^3 + w^2 + 7*w + 1],
[1601, 1601, -5*w^3 + 3*w^2 + 23*w - 13],
[1601, 1601, 5*w^3 - w^2 - 20*w + 9],
[1607, 1607, 2*w^3 - w^2 - 10*w - 5],
[1609, 1609, 3*w^3 - 8*w^2 - 13*w + 37],
[1609, 1609, 3*w^3 - 2*w^2 - 11*w + 7],
[1613, 1613, 2*w^3 - 4*w^2 - 12*w + 13],
[1621, 1621, 5*w - 1],
[1621, 1621, -2*w^3 + 5*w^2 + 8*w - 17],
[1627, 1627, 3*w^3 + w^2 - 14*w - 1],
[1627, 1627, -2*w^3 + 2*w^2 + 9*w + 1],
[1637, 1637, 7*w^3 - 12*w^2 - 31*w + 49],
[1657, 1657, -2*w^3 + 2*w^2 + 9*w - 1],
[1657, 1657, -5*w^3 + w^2 + 21*w - 7],
[1663, 1663, 9*w^3 - 11*w^2 - 39*w + 45],
[1667, 1667, -4*w^3 + 3*w^2 + 16*w - 7],
[1669, 1669, -w^3 - w^2 - w - 3],
[1693, 1693, -3*w^3 + w^2 + 11*w - 1],
[1693, 1693, w^3 + 2*w^2 - 11*w - 5],
[1709, 1709, 3*w^3 + 3*w^2 - 16*w - 5],
[1721, 1721, w^3 - 4*w^2 - 3*w + 11],
[1747, 1747, 2*w^3 - 2*w^2 - 11*w + 1],
[1747, 1747, w^3 + 3*w^2 - 4*w - 7],
[1753, 1753, -6*w^3 + 6*w^2 + 28*w - 21],
[1787, 1787, w^3 + 3*w^2 - 7*w - 13],
[1787, 1787, -w^3 + w^2 + 8*w - 5],
[1789, 1789, 2*w^3 + 3*w^2 - 12*w - 7],
[1789, 1789, 4*w^2 - 15],
[1801, 1801, w^3 + 3*w^2 - 7*w - 7],
[1811, 1811, -3*w^3 + 2*w^2 + 11*w - 5],
[1823, 1823, -w^3 + 3*w^2 + 6*w - 5],
[1831, 1831, 5*w^3 - 4*w^2 - 21*w + 17],
[1867, 1867, 3*w^3 - w^2 - 10*w + 5],
[1871, 1871, 2*w^3 + 2*w^2 - 6*w - 7],
[1873, 1873, -3*w^3 + 3*w^2 + 10*w - 7],
[1879, 1879, 2*w^3 - 4*w^2 - 7*w + 17],
[1889, 1889, 3*w^3 - 17*w - 3],
[1907, 1907, -4*w^3 - 2*w^2 + 19*w + 5],
[1907, 1907, 6*w^3 - 8*w^2 - 29*w + 33],
[1913, 1913, -w^3 + w^2 + 8*w - 3],
[1913, 1913, w^3 + 2*w^2 - 9*w + 3],
[1933, 1933, w^2 + 4*w - 9],
[1933, 1933, 6*w^3 - 6*w^2 - 26*w + 23],
[1951, 1951, w^3 + 2*w^2 - 3*w - 11],
[1951, 1951, -2*w^3 + 2*w^2 + 10*w - 1],
[1973, 1973, -5*w^3 + 7*w^2 + 23*w - 25],
[1979, 1979, -3*w^3 + 5*w^2 + 11*w - 19],
[1987, 1987, 2*w^3 - 2*w^2 - 6*w + 11],
[1993, 1993, -w^3 + 3*w - 7],
[1993, 1993, w^3 + w^2 - 2*w - 9],
[1997, 1997, 5*w^3 - w^2 - 19*w - 5],
[1999, 1999, -3*w^3 + 3*w^2 + 15*w - 19],
[2011, 2011, 5*w^3 - 6*w^2 - 23*w + 21],
[2039, 2039, -2*w^3 + 4*w^2 + 8*w - 9],
[2039, 2039, -3*w^3 + 4*w^2 + 11*w - 17],
[2053, 2053, -2*w^3 + 3*w^2 + 8*w - 5],
[2069, 2069, 5*w^3 - 6*w^2 - 21*w + 19],
[2069, 2069, w^3 - 2*w^2 - 5*w - 3],
[2069, 2069, 5*w^3 - 10*w^2 - 23*w + 43],
[2069, 2069, 3*w^3 - 2*w^2 - 13*w + 1],
[2081, 2081, 7*w^3 - 10*w^2 - 31*w + 37],
[2081, 2081, -3*w^3 + 2*w^2 + 11*w - 15],
[2099, 2099, -4*w^3 + 5*w^2 + 14*w - 13],
[2111, 2111, w^3 + 3*w^2 - 6*w - 9],
[2129, 2129, 4*w^3 - 4*w^2 - 20*w + 15],
[2129, 2129, 4*w^2 - 3*w - 15],
[2141, 2141, -w^3 + w^2 + w - 7],
[2153, 2153, 7*w^3 - 11*w^2 - 30*w + 45],
[2153, 2153, -4*w^3 + 8*w^2 + 17*w - 37],
[2153, 2153, 4*w^3 + w^2 - 14*w - 3],
[2153, 2153, -3*w^3 + 4*w^2 + 13*w - 11],
[2161, 2161, 3*w^3 - 7*w^2 - 11*w + 27],
[2179, 2179, w^3 + 3*w^2 - 5*w - 9],
[2179, 2179, -2*w^3 + 14*w - 7],
[2197, 13, w^3 + 3*w^2 - 6*w - 7],
[2203, 2203, 6*w^3 - 9*w^2 - 28*w + 33],
[2203, 2203, 3*w^3 + 3*w^2 - 13*w - 7],
[2207, 2207, -4*w^3 + 16*w - 5],
[2221, 2221, 5*w^3 - 3*w^2 - 21*w + 13],
[2221, 2221, -3*w^3 + 3*w^2 + 16*w - 13],
[2237, 2237, w^3 - w^2 - 3*w - 5],
[2239, 2239, 3*w^3 - w^2 - 16*w + 9],
[2239, 2239, -4*w^3 + 8*w^2 + 18*w - 31],
[2243, 2243, -2*w^3 + 2*w^2 + 12*w - 5],
[2251, 2251, 3*w^3 + 3*w^2 - 14*w - 7],
[2251, 2251, 4*w^3 + 2*w^2 - 15*w - 1],
[2267, 2267, w^3 - 5*w^2 + 15],
[2269, 2269, 3*w^3 - 3*w^2 - 11*w + 3],
[2273, 2273, -w^3 + 5*w^2 + 2*w - 17],
[2281, 2281, -3*w^3 - w^2 + 11*w + 7],
[2287, 2287, -3*w^3 + 4*w^2 + 17*w - 19],
[2287, 2287, 4*w^2 + w - 11],
[2287, 2287, w^3 - 5*w - 7],
[2287, 2287, -5*w^3 + 21*w - 3],
[2293, 2293, w^3 + w^2 - 8*w - 7],
[2293, 2293, -2*w^3 - 2*w^2 + 7*w + 9],
[2293, 2293, -4*w^3 + 6*w^2 + 16*w - 25],
[2293, 2293, -5*w^3 + w^2 + 20*w - 7],
[2309, 2309, 2*w^3 + 2*w^2 - 12*w - 1],
[2311, 2311, -2*w^3 - w^2 + 10*w - 3],
[2333, 2333, -5*w^3 + 7*w^2 + 25*w - 29],
[2341, 2341, 6*w^3 - 7*w^2 - 24*w + 27],
[2341, 2341, -w^3 + 5*w^2 + 6*w - 11],
[2347, 2347, -5*w^3 + 7*w^2 + 23*w - 33],
[2347, 2347, 4*w^3 - 2*w^2 - 16*w + 3],
[2347, 2347, -6*w^3 + 8*w^2 + 26*w - 37],
[2347, 2347, -w^3 - w^2 + 4*w - 5],
[2351, 2351, -3*w^2 - 2*w + 13],
[2357, 2357, 6*w^3 - 4*w^2 - 29*w + 15],
[2357, 2357, 3*w^3 - 4*w^2 - 13*w + 21],
[2371, 2371, 3*w^3 - 13*w + 9],
[2371, 2371, -7*w^3 + 29*w + 9],
[2377, 2377, w^3 - 3*w^2 - 7*w + 15],
[2377, 2377, 4*w^3 - 6*w^2 - 21*w + 25],
[2399, 2399, -3*w^3 + w^2 + 10*w - 1],
[2401, 7, -7],
[2411, 2411, 4*w^3 + w^2 - 16*w + 1],
[2417, 2417, 3*w^3 + 2*w^2 - 15*w - 3],
[2423, 2423, 5*w^3 + w^2 - 20*w - 3],
[2437, 2437, -2*w^3 + 3*w^2 + 12*w - 11],
[2437, 2437, 7*w^3 - 13*w^2 - 32*w + 55],
[2437, 2437, -3*w^3 + w^2 + 14*w - 9],
[2437, 2437, 3*w^3 - 3*w^2 - 9*w + 5],
[2459, 2459, 5*w^2 - 2*w - 23],
[2459, 2459, 5*w^3 - 4*w^2 - 23*w + 19],
[2467, 2467, 4*w^3 + 4*w^2 - 15*w - 11],
[2473, 2473, -w - 7],
[2503, 2503, -4*w^3 + w^2 + 16*w - 11],
[2521, 2521, 3*w^3 - 7*w^2 - 16*w + 27],
[2531, 2531, -2*w^2 + w - 3],
[2539, 2539, w^3 - 3*w^2 - 2*w + 13],
[2543, 2543, -w^3 + 3*w^2 + 5*w - 3],
[2549, 2549, 3*w^3 - 9*w - 5],
[2549, 2549, -3*w^3 + w^2 + 10*w - 3],
[2551, 2551, 2*w^3 - 4*w^2 - 9*w + 21],
[2557, 2557, 3*w^3 - 4*w^2 - 15*w + 11],
[2579, 2579, 6*w^3 - 8*w^2 - 27*w + 29],
[2591, 2591, 2*w^3 - 6*w^2 - 8*w + 23],
[2609, 2609, 6*w^3 - 5*w^2 - 26*w + 17],
[2617, 2617, 7*w^3 - 7*w^2 - 32*w + 31],
[2633, 2633, -5*w^3 + 3*w^2 + 22*w - 15],
[2659, 2659, -6*w^3 - w^2 + 26*w + 13],
[2659, 2659, w^3 + 3*w^2 - 7*w - 17],
[2663, 2663, 2*w^3 - 2*w^2 - 12*w - 5],
[2683, 2683, 4*w^3 - 3*w^2 - 22*w + 21],
[2683, 2683, -w^3 + 3*w^2 + 5*w - 1],
[2687, 2687, -w^3 - w^2 + 3*w - 5],
[2687, 2687, -2*w^3 - w^2 + 12*w - 3],
[2687, 2687, -3*w^3 + 5*w^2 + 15*w - 15],
[2687, 2687, 3*w^3 + 3*w^2 - 16*w - 9],
[2693, 2693, 2*w^3 + 3*w^2 - 10*w - 7],
[2711, 2711, 6*w^3 - 8*w^2 - 28*w + 29],
[2719, 2719, w^3 - 5*w^2 + w + 13],
[2729, 2729, 4*w^3 - 3*w^2 - 16*w + 9],
[2729, 2729, 2*w^2 + 5*w - 5],
[2741, 2741, 3*w^3 + w^2 - 12*w + 3],
[2741, 2741, w^3 - 9*w + 3],
[2749, 2749, w^3 - w^2 - 4*w - 5],
[2749, 2749, -2*w^3 + 13*w - 3],
[2789, 2789, -3*w^3 + 4*w^2 + 11*w - 19],
[2789, 2789, 3*w^3 - w^2 - 17*w + 11],
[2791, 2791, 2*w^3 + 2*w^2 - 9*w - 15],
[2791, 2791, w^3 - 9*w + 1],
[2791, 2791, 7*w^3 - 11*w^2 - 33*w + 47],
[2791, 2791, 3*w^3 - 17*w - 1],
[2801, 2801, 2*w^2 - 4*w - 7],
[2801, 2801, 3*w^3 - w^2 - 7*w + 1],
[2803, 2803, -w^3 + 2*w^2 - w - 5],
[2809, 53, -2*w^3 + 4*w^2 + 7*w - 7],
[2819, 2819, 4*w^2 - 2*w - 13],
[2833, 2833, 2*w^3 + w^2 - 6*w - 7],
[2843, 2843, 6*w^3 - 8*w^2 - 25*w + 35],
[2843, 2843, 4*w^3 - 2*w^2 - 19*w + 11],
[2861, 2861, 8*w^3 - 11*w^2 - 36*w + 49],
[2879, 2879, w^3 - w^2 - 5*w - 5],
[2879, 2879, w^3 + w^2 - w - 13],
[2887, 2887, 7*w^3 + 2*w^2 - 35*w - 9],
[2887, 2887, -4*w^3 + 5*w^2 + 20*w - 15],
[2939, 2939, 4*w^3 - 2*w^2 - 18*w + 1],
[2939, 2939, -w^3 + w^2 + 9*w - 7],
[2953, 2953, -5*w^3 + 9*w^2 + 22*w - 33],
[2963, 2963, w^3 - 5*w^2 - 5*w + 17],
[3001, 3001, 4*w^2 + w - 19],
[3037, 3037, w^3 + w^2 - 7],
[3037, 3037, w^3 + 4*w^2 - 9*w - 9],
[3041, 3041, -5*w^3 + 7*w^2 + 22*w - 33],
[3041, 3041, -4*w^3 - w^2 + 16*w + 9],
[3041, 3041, -2*w^3 - 4*w^2 + 14*w + 13],
[3041, 3041, -2*w^3 + 3*w^2 + 6*w - 13],
[3049, 3049, -w^3 + 3*w^2 + 7*w - 3],
[3067, 3067, w^3 + w^2 - 4*w - 11],
[3067, 3067, -3*w^3 + 6*w^2 + 13*w - 29],
[3079, 3079, 2*w^3 + 5*w^2 - 6*w - 17],
[3079, 3079, 2*w^3 + 3*w^2 - 10*w - 9],
[3079, 3079, w^3 + 3*w^2 - 7*w - 15],
[3079, 3079, -5*w^3 + w^2 + 21*w + 9],
[3089, 3089, -4*w^3 + w^2 + 16*w - 3],
[3089, 3089, 5*w^3 - 2*w^2 - 25*w + 5],
[3109, 3109, -w^2 - 5],
[3109, 3109, w^2 - 2*w - 11],
[3119, 3119, 5*w^3 - 9*w^2 - 24*w + 31],
[3119, 3119, 2*w^3 - 4*w - 5],
[3137, 3137, -w^3 + 3*w^2 + 2*w - 15],
[3167, 3167, -2*w^3 - 2*w^2 + 15*w + 7],
[3167, 3167, 4*w^3 - 4*w^2 - 21*w + 19],
[3169, 3169, -5*w^2 - 4*w + 9],
[3181, 3181, -3*w^3 + 2*w^2 + 9*w - 3],
[3187, 3187, -4*w^3 + 2*w^2 + 16*w - 7],
[3187, 3187, 2*w^3 + 2*w^2 - 10*w + 1],
[3191, 3191, -5*w^3 + 4*w^2 + 23*w - 23],
[3191, 3191, 5*w^3 + w^2 - 18*w - 5],
[3203, 3203, -2*w^3 + 6*w^2 + 10*w - 27],
[3229, 3229, 4*w^3 + w^2 - 18*w - 1],
[3229, 3229, 7*w^3 - 5*w^2 - 31*w + 17],
[3257, 3257, 5*w^3 - 11*w^2 - 23*w + 47],
[3271, 3271, 6*w^3 - 11*w^2 - 30*w + 43],
[3271, 3271, 6*w - 1],
[3271, 3271, -5*w^3 - w^2 + 21*w + 11],
[3271, 3271, 2*w^3 - 13*w + 1],
[3299, 3299, -w^3 + 3*w^2 + 8*w - 15],
[3299, 3299, -2*w^2 + 6*w + 5],
[3301, 3301, -8*w^3 - w^2 + 34*w + 13],
[3307, 3307, -w^3 + 2*w^2 + 9*w - 3],
[3307, 3307, -8*w^3 + 6*w^2 + 35*w - 25],
[3313, 3313, 6*w^3 - 6*w^2 - 25*w + 31],
[3323, 3323, -6*w^3 + w^2 + 26*w - 3],
[3323, 3323, -2*w^2 - 3*w + 15],
[3347, 3347, 2*w^3 - 6*w^2 - 5*w + 17],
[3373, 3373, 4*w^3 - 4*w^2 - 18*w + 11],
[3373, 3373, 4*w^3 - 6*w^2 - 20*w + 19],
[3389, 3389, 4*w^2 - 11],
[3389, 3389, 3*w^3 - 7*w^2 - 15*w + 21],
[3391, 3391, 4*w^3 - 15*w + 1],
[3391, 3391, 4*w^3 - 7*w^2 - 16*w + 29],
[3433, 3433, -5*w^2 + 2*w + 19],
[3433, 3433, -w^3 + 4*w^2 + 7*w - 17],
[3449, 3449, 2*w^2 - 3*w - 11],
[3449, 3449, 3*w^3 - w^2 - 15*w - 7],
[3463, 3463, 4*w^2 - w - 13],
[3463, 3463, -5*w^3 - 3*w^2 + 23*w + 7],
[3467, 3467, 3*w^3 + 2*w^2 - 13*w - 1],
[3469, 3469, 4*w^2 - 3*w - 9],
[3491, 3491, w^3 - 5*w^2 - 7*w + 11],
[3499, 3499, 5*w^3 - 2*w^2 - 23*w + 1],
[3499, 3499, -8*w^3 + 9*w^2 + 34*w - 37],
[3511, 3511, w^3 - 5*w^2 - 8*w + 11],
[3511, 3511, -2*w^3 + 6*w^2 + 4*w - 13],
[3517, 3517, w^3 + 4*w^2 - 3*w - 9],
[3527, 3527, w^2 - 4*w - 7],
[3529, 3529, -8*w^3 + 10*w^2 + 38*w - 39],
[3529, 3529, 2*w^3 - 5*w - 7],
[3539, 3539, 6*w^3 - 4*w^2 - 28*w + 11],
[3539, 3539, 4*w^2 - w - 9],
[3541, 3541, 6*w^3 - 23*w - 9],
[3547, 3547, -3*w^3 - 2*w^2 + 11*w + 9],
[3557, 3557, -3*w^3 + 2*w^2 + 15*w + 3],
[3571, 3571, 3*w^3 - w^2 - 15*w + 11],
[3581, 3581, 6*w^3 - 2*w^2 - 28*w + 1],
[3593, 3593, 5*w^3 - w^2 - 24*w + 5],
[3593, 3593, -2*w^3 + 3*w^2 + 4*w - 9],
[3607, 3607, 3*w^3 - 4*w^2 - 17*w + 11],
[3607, 3607, -w^3 - 5*w^2 + 6*w + 21],
[3617, 3617, 2*w^3 + 4*w^2 - 7*w - 7],
[3617, 3617, 2*w^2 + 4*w - 11],
[3623, 3623, -5*w^3 + 4*w^2 + 25*w - 17],
[3631, 3631, 8*w^3 - 14*w^2 - 36*w + 57],
[3631, 3631, -2*w^3 + 4*w^2 + 13*w - 7],
[3637, 3637, 6*w^3 - 3*w^2 - 26*w + 7],
[3637, 3637, w^3 + 2*w^2 - w - 9],
[3659, 3659, 4*w^3 - 4*w^2 - 14*w + 13],
[3677, 3677, -w^3 + w^2 + 3*w - 11],
[3691, 3691, -4*w^3 + 5*w^2 + 14*w - 19],
[3697, 3697, -4*w^3 + 18*w - 3],
[3719, 3719, w^3 + 5*w^2 - 2*w - 9],
[3733, 3733, 3*w^2 + 2*w - 15],
[3733, 3733, -5*w^3 + 5*w^2 + 22*w - 15],
[3767, 3767, w^3 + w^2 - w - 9],
[3767, 3767, -3*w^3 + 3*w^2 + 13*w - 5],
[3793, 3793, 2*w^2 + 3*w - 13],
[3793, 3793, 5*w^3 - 3*w^2 - 25*w + 9],
[3797, 3797, -9*w^3 - w^2 + 39*w + 15],
[3797, 3797, 4*w^3 - 6*w^2 - 21*w + 21],
[3803, 3803, -6*w^3 + 4*w^2 + 26*w - 15],
[3803, 3803, 2*w - 9],
[3823, 3823, -5*w^3 + 10*w^2 + 23*w - 39],
[3833, 3833, w^3 + w^2 - 5*w - 11],
[3847, 3847, 3*w^3 - 5*w^2 - 11*w + 21],
[3851, 3851, 4*w^2 - 2*w - 9],
[3851, 3851, -2*w^3 + 6*w^2 + 9*w - 21],
[3853, 3853, -w^3 + w^2 + 7*w - 13],
[3863, 3863, 4*w^3 - 6*w^2 - 18*w + 19],
[3881, 3881, -4*w^3 + 14*w + 7],
[3889, 3889, -3*w^3 + w^2 + 12*w - 13],
[3889, 3889, -4*w^3 + 2*w^2 + 21*w - 9],
[3889, 3889, 5*w^3 - 6*w^2 - 25*w + 25],
[3889, 3889, 3*w^3 - 7*w^2 - 12*w + 21],
[3907, 3907, -3*w^3 + 9*w^2 + 14*w - 39],
[3917, 3917, 3*w^3 + 3*w^2 - 13*w - 17],
[3917, 3917, 9*w^3 - 11*w^2 - 40*w + 41],
[3917, 3917, -w^3 + w^2 + 4*w - 11],
[3917, 3917, 3*w^3 - 3*w^2 - 11*w - 1],
[3923, 3923, -2*w^3 + 4*w^2 + 12*w - 21],
[3923, 3923, w^3 + 3*w^2 - 11*w - 7],
[3931, 3931, -3*w^3 + 19*w - 7],
[3947, 3947, -4*w^3 + 4*w^2 + 22*w - 9],
[3947, 3947, -5*w^3 - w^2 + 17*w + 7],
[3947, 3947, -w^3 - 4*w^2 + 3*w + 17],
[3947, 3947, -2*w^3 - 4*w^2 + 12*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^14 - 5*x^13 - 12*x^12 + 89*x^11 + 15*x^10 - 575*x^9 + 271*x^8 + 1694*x^7 - 1131*x^6 - 2383*x^5 + 1514*x^4 + 1486*x^3 - 644*x^2 - 284*x + 24;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -343/6332*e^13 + 969/6332*e^12 + 2761/3166*e^11 - 16431/6332*e^10 - 29731/6332*e^9 + 97967/6332*e^8 + 60121/6332*e^7 - 62584/1583*e^6 - 25459/6332*e^5 + 269523/6332*e^4 - 12782/1583*e^3 - 50643/3166*e^2 + 12150/1583*e + 2873/1583, 2699/3166*e^13 - 3284/1583*e^12 - 24901/1583*e^11 + 56955/1583*e^10 + 171278/1583*e^9 - 698739/3166*e^8 - 1122535/3166*e^7 + 923264/1583*e^6 + 927025/1583*e^5 - 1030268/1583*e^4 - 704964/1583*e^3 + 764929/3166*e^2 + 164340/1583*e - 10342/1583, -1, 599/1583*e^13 - 1443/1583*e^12 - 11051/1583*e^11 + 25233/1583*e^10 + 75209/1583*e^9 - 156769/1583*e^8 - 237097/1583*e^7 + 423021/1583*e^6 + 353307/1583*e^5 - 490159/1583*e^4 - 211870/1583*e^3 + 198988/1583*e^2 + 23528/1583*e - 8282/1583, 2286/1583*e^13 - 5692/1583*e^12 - 84077/3166*e^11 + 197911/3166*e^10 + 287503/1583*e^9 - 1218707/3166*e^8 - 1865391/3166*e^7 + 3241431/3166*e^6 + 3026933/3166*e^5 - 1827790/1583*e^4 - 2231309/3166*e^3 + 1376305/3166*e^2 + 241207/1583*e - 17960/1583, 417/6332*e^13 - 735/6332*e^12 - 2216/1583*e^11 + 12767/6332*e^10 + 73325/6332*e^9 - 75711/6332*e^8 - 299853/6332*e^7 + 84537/3166*e^6 + 626095/6332*e^5 - 67505/6332*e^4 - 299057/3166*e^3 - 31241/1583*e^2 + 41146/1583*e + 7902/1583, 769/1583*e^13 - 3565/3166*e^12 - 13894/1583*e^11 + 60029/3166*e^10 + 185441/3166*e^9 - 175164/1583*e^8 - 293081/1583*e^7 + 422148/1583*e^6 + 949109/3166*e^5 - 776585/3166*e^4 - 376005/1583*e^3 + 163597/3166*e^2 + 102294/1583*e + 7132/1583, -3126/1583*e^13 + 7742/1583*e^12 + 57064/1583*e^11 - 134268/1583*e^10 - 385577/1583*e^9 + 823767/1583*e^8 + 1228458/1583*e^7 - 2177296/1583*e^6 - 1952456/1583*e^5 + 2423331/1583*e^4 + 1422489/1583*e^3 - 879228/1583*e^2 - 308496/1583*e + 17222/1583, 6691/3166*e^13 - 16235/3166*e^12 - 61950/1583*e^11 + 283123/3166*e^10 + 856327/3166*e^9 - 1752909/3166*e^8 - 2824019/3166*e^7 + 2354531/1583*e^6 + 4703927/3166*e^5 - 5415003/3166*e^4 - 1810525/1583*e^3 + 1070279/1583*e^2 + 424360/1583*e - 47892/1583, 3041/1583*e^13 - 7968/1583*e^12 - 54851/1583*e^11 + 138605/1583*e^10 + 364553/1583*e^9 - 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22066224/1583*e^4 - 13958673/1583*e^3 + 15596349/3166*e^2 + 3225480/1583*e - 111804/1583, 46509/6332*e^13 - 106211/6332*e^12 - 219197/1583*e^11 + 1852551/6332*e^10 + 6205625/6332*e^9 - 11455663/6332*e^8 - 21025445/6332*e^7 + 15291361/3166*e^6 + 35734775/6332*e^5 - 34361549/6332*e^4 - 13753239/3166*e^3 + 3114848/1583*e^2 + 1588005/1583*e - 64098/1583, -37123/6332*e^13 + 92765/6332*e^12 + 169121/1583*e^11 - 1607673/6332*e^10 - 4561195/6332*e^9 + 9869697/6332*e^8 + 14481183/6332*e^7 - 13111897/3166*e^6 - 22770177/6332*e^5 + 29712639/6332*e^4 + 7947041/3166*e^3 - 2784645/1583*e^2 - 706248/1583*e + 16900/1583, 31903/1583*e^13 - 80478/1583*e^12 - 578758/1583*e^11 + 1395298/1583*e^10 + 3878691/1583*e^9 - 8564577/1583*e^8 - 12249346/1583*e^7 + 22717158/1583*e^6 + 19389466/1583*e^5 - 25661821/1583*e^4 - 14270286/1583*e^3 + 9888258/1583*e^2 + 3325092/1583*e - 353734/1583, 56925/6332*e^13 - 144295/6332*e^12 - 260473/1583*e^11 + 2510463/6332*e^10 + 7100077/6332*e^9 - 15488751/6332*e^8 - 23127313/6332*e^7 + 20699223/3166*e^6 + 38581175/6332*e^5 - 47326649/6332*e^4 - 15374947/3166*e^3 + 4703918/1583*e^2 + 2017746/1583*e - 252720/1583, -37801/3166*e^13 + 94985/3166*e^12 + 343455/1583*e^11 - 1643327/3166*e^10 - 4622105/3166*e^9 + 10050239/3166*e^8 + 14728075/3166*e^7 - 13237718/1583*e^6 - 23720679/3166*e^5 + 29489897/3166*e^4 + 8974823/1583*e^3 - 5502686/1583*e^2 - 2146354/1583*e + 215762/1583, -19611/3166*e^13 + 23393/1583*e^12 + 184231/1583*e^11 - 404783/1583*e^10 - 1303348/1583*e^9 + 4933387/3166*e^8 + 8899253/3166*e^7 - 6403914/1583*e^6 - 7743745/1583*e^5 + 6809354/1583*e^4 + 6276278/1583*e^3 - 4332569/3166*e^2 - 1590686/1583*e - 24948/1583, 49851/3166*e^13 - 128501/3166*e^12 - 901465/3166*e^11 + 1117975/1583*e^10 + 6015469/3166*e^9 - 6903140/1583*e^8 - 9454377/1583*e^7 + 36987437/3166*e^6 + 14933206/1583*e^5 - 42449211/3166*e^4 - 22026679/3166*e^3 + 16666305/3166*e^2 + 2477373/1583*e - 259372/1583, -15006/1583*e^13 + 36961/1583*e^12 + 275108/1583*e^11 - 645334/1583*e^10 - 1874614/1583*e^9 + 4010529/1583*e^8 + 6078167/1583*e^7 - 10879218/1583*e^6 - 10018220/1583*e^5 + 12789454/1583*e^4 + 7849191/1583*e^3 - 5299928/1583*e^2 - 2031528/1583*e + 268658/1583, 32367/6332*e^13 - 84701/6332*e^12 - 141778/1583*e^11 + 1456565/6332*e^10 + 3592083/6332*e^9 - 8843109/6332*e^8 - 10372911/6332*e^7 + 11567759/3166*e^6 + 14407573/6332*e^5 - 25796967/6332*e^4 - 4322945/3166*e^3 + 2502435/1583*e^2 + 303574/1583*e - 89304/1583, -152445/6332*e^13 + 380123/6332*e^12 + 696333/1583*e^11 - 6596103/6332*e^10 - 18881169/6332*e^9 + 40519659/6332*e^8 + 60708641/6332*e^7 - 53733091/3166*e^6 - 98661151/6332*e^5 + 120918837/6332*e^4 + 37707401/3166*e^3 - 11387283/1583*e^2 - 4585812/1583*e + 222694/1583, 30489/1583*e^13 - 75708/1583*e^12 - 558966/1583*e^11 + 1314155/1583*e^10 + 3806944/1583*e^9 - 8074488/1583*e^8 - 12299395/1583*e^7 + 21412820/1583*e^6 + 19975379/1583*e^5 - 24098602/1583*e^4 - 15000961/1583*e^3 + 9224119/1583*e^2 + 3540110/1583*e - 414972/1583, -3599/1583*e^13 + 9669/1583*e^12 + 65505/1583*e^11 - 168649/1583*e^10 - 444017/1583*e^9 + 1042048/1583*e^8 + 1445925/1583*e^7 - 2777273/1583*e^6 - 2458329/1583*e^5 + 3117093/1583*e^4 + 2103165/1583*e^3 - 1110144/1583*e^2 - 656656/1583*e - 32798/1583, 7141/3166*e^13 - 21743/3166*e^12 - 60731/1583*e^11 + 373021/3166*e^10 + 736885/3166*e^9 - 2245121/3166*e^8 - 2015335/3166*e^7 + 2861472/1583*e^6 + 2722589/3166*e^5 - 5906081/3166*e^4 - 914122/1583*e^3 + 868081/1583*e^2 + 123268/1583*e + 29590/1583, 72673/3166*e^13 - 182803/3166*e^12 - 663019/1583*e^11 + 3175447/3166*e^10 + 8963863/3166*e^9 - 19529099/3166*e^8 - 28643379/3166*e^7 + 25910474/1583*e^6 + 45908643/3166*e^5 - 58132819/3166*e^4 - 17029963/1583*e^3 + 10796767/1583*e^2 + 3942020/1583*e - 222560/1583, 34009/6332*e^13 - 86183/6332*e^12 - 154691/1583*e^11 + 1491379/6332*e^10 + 4168697/6332*e^9 - 9105427/6332*e^8 - 13285953/6332*e^7 + 11906877/3166*e^6 + 21301459/6332*e^5 - 25926101/6332*e^4 - 7847087/3166*e^3 + 2221396/1583*e^2 + 734843/1583*e + 17112/1583, 13488/1583*e^13 - 34274/1583*e^12 - 245699/1583*e^11 + 592572/1583*e^10 + 1660225/1583*e^9 - 3617948/1583*e^8 - 5327348/1583*e^7 + 9501907/1583*e^6 + 8673844/1583*e^5 - 10558213/1583*e^4 - 6687498/1583*e^3 + 3938460/1583*e^2 + 1679024/1583*e - 35278/1583, 31399/3166*e^13 - 80831/3166*e^12 - 282576/1583*e^11 + 1401813/3166*e^10 + 3738797/3166*e^9 - 8608001/3166*e^8 - 11564537/3166*e^7 + 11424385/1583*e^6 + 17766117/3166*e^5 - 25874693/3166*e^4 - 6293463/1583*e^3 + 5076054/1583*e^2 + 1343014/1583*e - 283466/1583, 33474/1583*e^13 - 79318/1583*e^12 - 623738/1583*e^11 + 1380490/1583*e^10 + 4347848/1583*e^9 - 8512995/1583*e^8 - 14487723/1583*e^7 + 22668128/1583*e^6 + 24376576/1583*e^5 - 25511680/1583*e^4 - 18968125/1583*e^3 + 9471653/1583*e^2 + 4577030/1583*e - 222446/1583, -6635/3166*e^13 + 21375/3166*e^12 + 108493/3166*e^11 - 183785/1583*e^10 - 606851/3166*e^9 + 1111480/1583*e^8 + 687822/1583*e^7 - 5725021/3166*e^6 - 573420/1583*e^5 + 6037733/3166*e^4 + 101795/3166*e^3 - 1805257/3166*e^2 + 99587/1583*e - 50546/1583, -14719/3166*e^13 + 35601/3166*e^12 + 272329/3166*e^11 - 303888/1583*e^10 - 1888569/3166*e^9 + 1809112/1583*e^8 + 3157790/1583*e^7 - 8982889/3166*e^6 - 5446787/1583*e^5 + 8639387/3166*e^4 + 9018163/3166*e^3 - 1931035/3166*e^2 - 1244245/1583*e - 43672/1583, -67183/3166*e^13 + 81706/1583*e^12 + 618370/1583*e^11 - 1417841/1583*e^10 - 4233975/1583*e^9 + 17413227/3166*e^8 + 27527103/3166*e^7 - 23057059/1583*e^6 - 22490301/1583*e^5 + 25831959/1583*e^4 + 17025650/1583*e^3 - 19274809/3166*e^2 - 4087152/1583*e + 216774/1583, -71621/3166*e^13 + 89364/1583*e^12 + 658003/1583*e^11 - 1559967/1583*e^10 - 4492447/1583*e^9 + 19347663/3166*e^8 + 29073687/3166*e^7 - 26062480/1583*e^6 - 23569753/1583*e^5 + 30143783/1583*e^4 + 17568473/1583*e^3 - 24192369/3166*e^2 - 4030478/1583*e + 513774/1583, 43899/3166*e^13 - 113523/3166*e^12 - 793099/3166*e^11 + 982013/1583*e^10 + 5291327/3166*e^9 - 6007049/1583*e^8 - 8326252/1583*e^7 + 31705715/3166*e^6 + 13178290/1583*e^5 - 35646193/3166*e^4 - 19499845/3166*e^3 + 13800317/3166*e^2 + 2332239/1583*e - 249594/1583, -9017/3166*e^13 + 20517/3166*e^12 + 168237/3166*e^11 - 176540/1583*e^10 - 1169833/3166*e^9 + 1068039/1583*e^8 + 1918303/1583*e^7 - 5495657/3166*e^6 - 3053379/1583*e^5 + 5786675/3166*e^4 + 4036855/3166*e^3 - 1831549/3166*e^2 - 299693/1583*e + 34014/1583, 14094/1583*e^13 - 34283/1583*e^12 - 260061/1583*e^11 + 593200/1583*e^10 + 1792738/1583*e^9 - 3625136/1583*e^8 - 5927339/1583*e^7 + 9516357/1583*e^6 + 10046218/1583*e^5 - 10471490/1583*e^4 - 8128994/1583*e^3 + 3708616/1583*e^2 + 2173488/1583*e - 76876/1583, -26305/1583*e^13 + 61728/1583*e^12 + 492198/1583*e^11 - 1077395/1583*e^10 - 3449816/1583*e^9 + 6677821/1583*e^8 + 11565364/1583*e^7 - 17955766/1583*e^6 - 19527975/1583*e^5 + 20616650/1583*e^4 + 15110352/1583*e^3 - 8037200/1583*e^2 - 3602340/1583*e + 303496/1583, 1460/1583*e^13 - 5523/1583*e^12 - 22554/1583*e^11 + 103081/1583*e^10 + 104579/1583*e^9 - 708399/1583*e^8 - 78908/1583*e^7 + 2227326/1583*e^6 - 474467/1583*e^5 - 3190663/1583*e^4 + 909465/1583*e^3 + 1690981/1583*e^2 - 391226/1583*e - 120156/1583, -3673/1583*e^13 + 17287/3166*e^12 + 68847/1583*e^11 - 293561/3166*e^10 - 979971/3166*e^9 + 863075/1583*e^8 + 1722066/1583*e^7 - 2069143/1583*e^6 - 6455109/3166*e^5 + 3485727/3166*e^4 + 2969278/1583*e^3 - 100889/3166*e^2 - 857176/1583*e - 110724/1583, 56653/3166*e^13 - 141219/3166*e^12 - 515189/1583*e^11 + 2437593/3166*e^10 + 6943519/3166*e^9 - 14845115/3166*e^8 - 22187351/3166*e^7 + 19392680/1583*e^6 + 35933529/3166*e^5 - 42451263/3166*e^4 - 13746121/1583*e^3 + 7551015/1583*e^2 + 3417796/1583*e - 159380/1583, -64387/3166*e^13 + 82704/1583*e^12 + 582401/1583*e^11 - 1435592/1583*e^10 - 3890222/1583*e^9 + 17656539/3166*e^8 + 24502319/3166*e^7 - 23483938/1583*e^6 - 19396442/1583*e^5 + 26624974/1583*e^4 + 14328748/1583*e^3 - 20495897/3166*e^2 - 3242902/1583*e + 305316/1583, 118/1583*e^13 + 1987/3166*e^12 - 8324/1583*e^11 - 23793/3166*e^10 + 231143/3166*e^9 + 23082/1583*e^8 - 606933/1583*e^7 + 147060/1583*e^6 + 2629047/3166*e^5 - 1062671/3166*e^4 - 1103304/1583*e^3 + 719361/3166*e^2 + 322144/1583*e + 85156/1583];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w^2 + 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;