/*
  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 | {16, 2, 2}>;

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^4 - 13*x^2 + 24;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, 1, e, -e + 2, -1/2*e^3 + 7/2*e, -1/2*e^3 + 9/2*e + 2, e^2 - e - 6, e^2 + e - 4, -e^3 + 10*e, 1/2*e^3 - e^2 - 7/2*e + 8, -e^2 + 2*e + 6, -e^2 + 2*e + 6, 2*e^2 + e - 10, 1/2*e^3 - e^2 - 7/2*e + 8, -e^2 - 2*e + 10, -1/2*e^3 - e^2 + 7/2*e + 6, 1/2*e^3 - 13/2*e + 6, -1/2*e^3 + 11/2*e + 2, -3*e^2 - 2*e + 20, 3/2*e^3 + e^2 - 27/2*e - 4, e^3 - 2*e^2 - 9*e + 8, e^2 + e - 4, 1/2*e^3 - 2*e^2 - 17/2*e + 12, e^3 + e^2 - 11*e - 4, -1/2*e^3 + 2*e^2 + 9/2*e - 18, 1/2*e^3 + 3*e^2 - 9/2*e - 18, -1/2*e^3 + 1/2*e - 4, 3/2*e^3 + 2*e^2 - 27/2*e - 10, e^3 - 8*e, e^2 - 5*e - 6, -1/2*e^3 + 19/2*e - 6, e^3 + e^2 - 11*e, 1/2*e^3 - 2*e^2 - 9/2*e + 20, e^3 - 3*e^2 - 11*e + 12, -e^3 + e^2 + 8*e - 10, -1/2*e^3 - 2*e^2 + 21/2*e + 14, -1/2*e^3 - e^2 + 15/2*e + 14, e^3 + 2*e^2 - 9*e - 10, e^2 + 4*e - 6, e^2 - 6, e^3 - e^2 - 16*e + 6, -2*e^3 + e^2 + 18*e + 2, -3/2*e^3 + 25/2*e - 4, -e^2 + e + 14, -1/2*e^3 + e^2 + 3/2*e + 2, -e^3 + 3*e^2 + 12*e - 16, e^2 - 3*e + 8, 3*e^2 - e - 16, -2*e^2 - 2*e + 26, e^3 - 11*e, e^3 - 6*e - 6, e^3 - 2*e^2 - 9*e + 18, 1/2*e^3 - 3*e^2 - 5/2*e + 26, -e^3 - 3*e^2 + 10*e + 12, e^2 + 3*e - 6, 2*e^2 + 2, e^3 - e^2 - 10*e - 4, -4*e^2 + 2*e + 24, e^3 + 2*e^2 - 8*e - 18, 1/2*e^3 - 2*e^2 - 9/2*e + 20, 3*e^2 + 3*e - 24, -2*e^3 + e^2 + 21*e - 6, -5*e^2 - e + 24, e^3 - 3*e^2 - 2*e + 30, 2*e^2 + 2*e - 36, 3/2*e^3 + e^2 - 25/2*e - 18, -2*e^3 + e^2 + 18*e - 10, 3*e^2 - 40, -2*e^2 + 2*e - 6, -3/2*e^3 - e^2 + 15/2*e + 12, e^3 - 3*e^2 - 7*e + 20, e^3 - 3*e^2 - 12*e + 20, -2*e^2 + 2, -e^3 + 14*e + 8, -e^2 + 9*e + 14, -e^3 + 2*e^2 + 7*e - 24, e^3 + e^2 - 5*e + 6, 2*e^3 - 23*e - 6, -2*e^3 - e^2 + 12*e + 14, -2*e^2 + 2*e + 14, 1/2*e^3 - 19/2*e - 4, e^3 + e^2 - 12*e + 12, 3/2*e^3 - 3*e^2 - 29/2*e + 12, e^3 - 2*e^2 - 7*e + 26, -2*e^3 - 3*e^2 + 22*e + 14, -2*e - 6, 2*e^2 - 5*e, 1/2*e^3 + 4*e^2 - 19/2*e - 28, -2*e^2 + 4*e + 8, -e^3 + 3*e^2 + 10*e - 10, e^3 + 3*e^2 - 2*e - 40, -2*e^3 + 32*e + 2, e^3 + 6*e^2 - 7*e - 40, 5/2*e^3 + 3*e^2 - 35/2*e - 28, -e^3 + 2*e^2 + 3*e - 24, 4*e^2 + 4*e - 18, 2*e^3 + 4*e^2 - 13*e - 36, 3*e^3 - e^2 - 24*e + 20, -e^3 + 2*e^2 + 19*e - 10, 2*e^3 - 4*e^2 - 15*e + 26, 1/2*e^3 + e^2 - 27/2*e + 8, 3/2*e^3 - e^2 - 45/2*e - 6, -e^3 + e^2 + 6*e - 28, -1/2*e^3 + e^2 + 11/2*e - 10, 1/2*e^3 - 8*e^2 - 13/2*e + 48, 1/2*e^3 - 5*e^2 - 3/2*e + 30, -3/2*e^3 - 4*e^2 + 25/2*e + 24, -1/2*e^3 + 3*e^2 + 13/2*e - 24, -3*e^3 - 2*e^2 + 29*e + 12, 3*e^3 + 3*e^2 - 24*e - 24, 2*e^3 + e^2 - 21*e + 8, -3*e^3 - 4*e^2 + 23*e + 32, -e^3 - 3*e^2 + 9*e + 38, 1/2*e^3 - 2*e^2 - 33/2*e + 18, 3*e^3 - 2*e^2 - 26*e - 6, 3/2*e^3 - e^2 - 33/2*e - 16, e^3 - 4*e^2 - 11*e + 14, e^3 + 3*e^2 - 8*e + 2, -1/2*e^3 + 1/2*e - 10, e^3 - 2*e^2 - 21*e + 26, -3*e^3 - 2*e^2 + 33*e, -e - 18, -2*e^2 + 5*e + 42, e^3 + 3*e^2 - 10*e - 12, 3/2*e^3 - 51/2*e - 4, 2*e^2 + 2*e - 30, -e^3 - 3*e^2 + 9*e + 18, 3*e^3 - 4*e^2 - 28*e + 26, -e^3 - 3*e^2 + 18*e + 20, -6*e^2 + 2*e + 26, e^2 - 2*e - 28, -1/2*e^3 + e^2 + 9/2*e - 12, -e^3 - 2*e^2 + 5*e + 14, -2*e^3 + 8*e, -5/2*e^3 + 47/2*e - 18, 2*e^3 - 4*e^2 - 20*e + 36, -4*e^2 + 6*e + 38, -e^3 + 8*e^2 + 9*e - 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3*e^2 - 5/2*e + 48, 1/2*e^3 + e^2 + 25/2*e - 18, -e^3 - 11*e^2 + 3*e + 84, -3*e^2 + e - 24, -2*e^2 - 10*e + 84, e^3 - 9*e^2 - 10*e + 20, 5*e^3 + 3*e^2 - 52*e + 2, 3/2*e^3 - 3/2*e + 18, 2*e^3 + 4*e^2 - 12*e - 54, -4*e^3 + 13*e^2 + 28*e - 94, 2*e^3 + 2*e^2 - 16*e - 48, 6*e^3 - 12*e^2 - 66*e + 50, -6*e^3 - 9*e^2 + 54*e + 44, -2*e^3 + 8*e^2 + 25*e - 22, 3/2*e^3 + 6*e^2 - 23/2*e - 96, -2*e^3 + 7*e^2 + 22*e - 60, 5/2*e^3 + 12*e^2 - 59/2*e - 66, e^3 - 4*e^2 - 7*e, -7/2*e^3 + 49/2*e + 2, 3*e^3 - e^2 - 12*e + 32, 1/2*e^3 + 13*e^2 - 35/2*e - 76, 2*e^3 + 4*e^2 - 12*e - 34, -4*e^3 + 4*e^2 + 30*e + 2, 6*e^3 + 2*e^2 - 60*e - 16, 3*e^3 - 39*e + 44, 4*e^3 + 4*e^2 - 24*e - 30, 9/2*e^3 + 4*e^2 - 101/2*e - 54, -3*e^3 + 7*e^2 + 39*e - 40, 6*e^3 + 4*e^2 - 54*e + 2, -5*e^2 - 13*e + 96, 4*e^3 - 2*e^2 - 40*e + 42, 7/2*e^3 + 6*e^2 - 51/2*e - 60, -4*e^3 + 7*e^2 + 35*e - 24, -3*e^3 + 6*e^2 + 27*e - 30, -1/2*e^3 + 12*e^2 + 21/2*e - 58, -2*e^3 + e^2 + 22*e - 34, -7/2*e^3 + 5*e^2 + 65/2*e - 22, -4*e^2 - 8*e + 68, -e^3 - 7*e^2 + 10*e, -3*e^3 + 10*e^2 + 29*e - 84, 3*e^3 - 9*e^2 - 8*e + 96, 5*e^3 + 6*e^2 - 52*e + 26, -3*e^3 + 11*e^2 + 19*e - 88, e^3 + 8*e^2 - 36, -e^3 - 5*e^2 + 56, 3*e^3 + 2*e^2 - 27*e + 50, 4*e^3 - 4*e^2 - 36*e + 56, -e^3 - 3*e^2 + 14*e + 68, -5/2*e^3 + 4*e^2 + 17/2*e - 60, 9/2*e^3 + e^2 - 71/2*e - 42, -e^2 + 12*e - 52, -2*e^2 - 2*e + 8, 2*e^3 + e^2 - 15*e - 46, 15/2*e^3 + e^2 - 155/2*e + 2, -9/2*e^3 + 5*e^2 + 99/2*e - 18, 5/2*e^3 - 4*e^2 - 93/2*e + 54, -3/2*e^3 + 8*e^2 - 1/2*e - 60, -e^3 + 3*e + 8, -e^3 - 5*e^2 + 19*e + 50, -e^3 - 10*e^2 + 15*e + 54, -5*e^3 - 8*e^2 + 64*e + 30, 3*e^3 + 9*e^2 - 34*e - 28, -8*e^3 + e^2 + 85*e - 4, 7/2*e^3 - 12*e^2 - 81/2*e + 44, 3/2*e^3 + 3*e^2 + 21/2*e - 22, 5/2*e^3 - 5*e^2 - 69/2*e + 54, 3*e^3 - 2*e^2 - 50*e + 6, 4*e^2 + 4*e - 64, -5*e^3 + 12*e^2 + 51*e - 46, -11/2*e^3 + 13*e^2 + 97/2*e - 60, 2*e^3 - 2*e^2 - 18*e - 40, -13/2*e^3 + 113/2*e + 12, 2*e^3 - 10*e^2 - 20*e + 80, 3/2*e^3 + e^2 - 25/2*e + 14, -6*e^3 + 3*e^2 + 57*e + 20, -3*e^3 + 51*e + 8, 13*e^2 + 8*e - 70, 2*e^3 - 5*e^2 - 33*e + 54, -7/2*e^3 + 5*e^2 + 63/2*e - 28, -3/2*e^3 - 7*e^2 + 35/2*e + 44, 5*e^3 + 3*e^2 - 44*e + 18, -4*e^3 + 12*e + 12, 3*e^3 + 6*e^2 - 15*e - 40, -4*e^2 - 26*e + 74, e^3 + 2*e^2 - 8*e + 18, -4*e^3 - 8*e^2 + 38*e + 38, -e^3 + 3*e^2 - 5*e - 42, -11/2*e^3 + 2*e^2 + 139/2*e + 6, -5/2*e^3 - 6*e^2 + 71/2*e + 48, -2*e^3 + 6*e^2 + 8*e - 40, -2*e^3 - 5*e^2 + 49*e + 32, -e^3 - 2*e^2 + 10*e - 54, -5/2*e^3 + 12*e^2 + 71/2*e - 42, -e^3 - 6*e^2 + 19*e + 36, -8*e^3 + 4*e^2 + 78*e - 22, 4*e^3 + 4*e^2 - 42*e - 34, -7*e^3 + 4*e^2 + 48*e - 40, 4*e^3 - 32*e + 26, 2*e^3 - 38*e - 60, 3*e^3 - 39*e - 18, -4*e^3 + 6*e^2 + 42*e - 4, -2*e^3 - 4*e + 32, -8*e^3 + 7*e^2 + 87*e, -5*e^3 - 3*e^2 + 53*e - 22, 3*e^3 - 8*e^2 - 22*e + 44, -3*e^3 + e^2 + 20*e - 24, 6*e^2 - 22*e - 36, 4*e^3 + 12*e^2 - 47*e - 76, 3/2*e^3 + 4*e^2 - 29/2*e - 76, -5*e^3 - 2*e^2 + 42*e + 18, e^3 - 2*e^2 - 24*e + 6, e^3 - 4*e^2 - 7*e + 18, 1/2*e^3 - 16*e^2 - 45/2*e + 108, 2*e^3 + 10*e^2 - 12*e - 88, -3/2*e^3 + 4*e^2 - 1/2*e - 90, 11*e^3 - 2*e^2 - 93*e - 16, 5*e^3 + e^2 - 48*e + 12, 1/2*e^3 + 3*e^2 + 1/2*e - 72, e^3 - 7*e^2 - 13*e + 86, -3*e^2 - 9*e - 42, -9*e^2 + 96, 4*e^3 + e^2 - 54*e - 16, -4*e^3 + e^2 + 68*e - 34, 5/2*e^3 - 2*e^2 - 69/2*e + 74, -1/2*e^3 - 10*e^2 + 29/2*e + 80, 18*e + 14, 2*e^2 - 16*e + 12, -6*e^3 - e^2 + 84*e - 6, -2*e^3 - 6*e^2 + 32*e + 30, 7*e^3 + 3*e^2 - 61*e - 6, 1/2*e^3 + 6*e^2 + 7/2*e - 96, 2*e^3 - 10*e^2 - 2*e + 66, 3/2*e^3 + 2*e^2 - 75/2*e - 4, -11/2*e^3 - 5*e^2 + 117/2*e + 36, 4*e^3 - 4*e^2 - 26*e + 66, 6*e^3 - 9*e^2 - 53*e + 78, -3/2*e^3 + 3*e^2 + 21/2*e + 24];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, w}>] := -1;
ALEigenvalues[ideal<ZF | {4, 2, -w^3 + 4*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;