/*
  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![1, 3, -4, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {23, 23, -w^2 + 3}>;

primesArray := [
[3, 3, w + 1],
[7, 7, w^3 - 2*w^2 - 3*w + 1],
[11, 11, -w - 2],
[11, 11, -w^3 + 2*w^2 + 3*w - 2],
[13, 13, -w^2 + 2*w + 2],
[16, 2, 2],
[19, 19, -w^3 + 2*w^2 + 4*w - 1],
[23, 23, -w^2 + w + 3],
[23, 23, -w^2 + 3],
[27, 3, w^3 - 3*w^2 - w + 4],
[29, 29, -w^3 + 3*w^2 + 3*w - 5],
[29, 29, w^3 - 2*w^2 - 4*w],
[47, 47, -w^3 + 2*w^2 + 3*w - 5],
[47, 47, w^2 - 3*w - 2],
[61, 61, -w^3 + 2*w^2 + 5*w - 3],
[67, 67, -w^3 + w^2 + 5*w - 1],
[67, 67, -2*w^3 + 3*w^2 + 11*w - 7],
[83, 83, -2*w + 3],
[89, 89, 3*w^3 - 7*w^2 - 9*w + 9],
[97, 97, w^3 - w^2 - 4*w - 3],
[97, 97, -w^3 + 5*w + 3],
[103, 103, -w^3 + 3*w^2 + w - 5],
[103, 103, -w^3 + 2*w^2 + 3*w + 2],
[121, 11, 2*w^3 - 5*w^2 - 3*w + 2],
[127, 127, -2*w^3 + 4*w^2 + 6*w - 5],
[131, 131, w^2 - 3*w - 3],
[137, 137, -2*w^3 + 3*w^2 + 9*w - 3],
[139, 139, 2*w - 5],
[149, 149, -2*w^3 + 4*w^2 + 9*w - 4],
[151, 151, 3*w^3 - 5*w^2 - 12*w + 3],
[151, 151, w^3 - w^2 - 6*w + 3],
[163, 163, -2*w^3 + 5*w^2 + 5*w - 9],
[167, 167, -w^3 + 2*w^2 + 6*w - 7],
[179, 179, -w^3 + 4*w^2 - w - 5],
[179, 179, -3*w^3 + 8*w^2 + 7*w - 12],
[181, 181, 2*w^3 - 5*w^2 - 4*w + 2],
[199, 199, w^3 - 5*w - 5],
[211, 211, 2*w^2 - 3*w - 7],
[223, 223, -2*w^3 + 3*w^2 + 8*w - 4],
[223, 223, -2*w^3 + 3*w^2 + 10*w - 5],
[223, 223, w^3 - 2*w^2 - 6*w + 8],
[223, 223, 2*w^3 - 2*w^2 - 9*w],
[227, 227, 2*w^3 - 3*w^2 - 7*w + 2],
[229, 229, -w^3 + 3*w^2 + 4*w - 2],
[229, 229, -2*w^3 + 2*w^2 + 8*w - 1],
[233, 233, -2*w^3 + 4*w^2 + 7*w - 1],
[251, 251, w^2 + w - 4],
[257, 257, -2*w^3 + 4*w^2 + 9*w - 2],
[263, 263, -w^3 + 2*w^2 + 6*w - 2],
[263, 263, w^2 - 3*w - 6],
[263, 263, 2*w^3 - 4*w^2 - 7*w],
[263, 263, -w^3 + 8*w],
[271, 271, 2*w^2 - w - 5],
[271, 271, -3*w^3 + 7*w^2 + 8*w - 6],
[277, 277, -w^3 + 2*w^2 + w - 4],
[283, 283, -2*w^3 + 3*w^2 + 9*w - 4],
[307, 307, -w^3 + 3*w^2 + w - 8],
[311, 311, 2*w^2 - 6*w - 3],
[313, 313, -w^3 + 2*w^2 + 2*w + 4],
[313, 313, -4*w^3 + 9*w^2 + 12*w - 12],
[313, 313, w^2 - 2*w - 7],
[313, 313, 4*w^3 - 8*w^2 - 14*w + 11],
[317, 317, 2*w^3 - w^2 - 12*w - 5],
[317, 317, w^3 - 6*w],
[331, 331, -w^3 + 2*w^2 + 5*w - 8],
[343, 7, -2*w^3 + 3*w^2 + 6*w - 3],
[347, 347, -3*w^3 + 5*w^2 + 14*w - 8],
[353, 353, -2*w^3 + 5*w^2 + 6*w - 5],
[359, 359, 2*w^3 - 4*w^2 - 5*w + 2],
[367, 367, 2*w^2 - 7*w + 1],
[367, 367, -w^3 + 3*w^2 + 4*w - 8],
[367, 367, 3*w^2 - 6*w - 8],
[367, 367, -3*w^3 + 6*w^2 + 11*w - 5],
[379, 379, 2*w^3 - 4*w^2 - 5*w + 3],
[379, 379, w^3 - 9*w + 2],
[397, 397, -w^3 + 4*w^2 - 2*w - 5],
[409, 409, 2*w^2 - 3*w - 4],
[419, 419, w^3 - w^2 - 8*w + 4],
[419, 419, 2*w^3 - 3*w^2 - 10*w + 6],
[439, 439, -2*w^3 + 5*w^2 + 5*w - 3],
[449, 449, 2*w^2 - 2*w - 5],
[457, 457, 2*w^3 - 2*w^2 - 9*w - 6],
[461, 461, 2*w^3 - 3*w^2 - 6*w + 1],
[461, 461, -w^3 + 4*w^2 + w - 5],
[463, 463, -w^3 + 4*w^2 - 12],
[463, 463, -2*w^3 + 5*w^2 + 4*w - 8],
[479, 479, w^3 - 5*w - 6],
[487, 487, -3*w^3 + 8*w^2 + 8*w - 13],
[487, 487, -4*w^3 + 6*w^2 + 17*w - 4],
[491, 491, -w^3 + 4*w^2 - 6],
[499, 499, -4*w^3 + 10*w^2 + 10*w - 15],
[499, 499, -2*w^3 + 4*w^2 + 10*w - 3],
[503, 503, w^2 + w - 5],
[529, 23, 3*w^3 - 4*w^2 - 12*w],
[541, 541, -2*w^3 + 7*w^2 - 10],
[547, 547, w^3 - 3*w^2 + 2*w - 4],
[547, 547, -2*w^3 + 2*w^2 + 11*w - 1],
[557, 557, -w^3 + 10*w - 2],
[557, 557, 3*w^3 - 4*w^2 - 16*w + 5],
[563, 563, -2*w^3 + 5*w^2 + 9*w - 11],
[563, 563, -3*w^3 + 6*w^2 + 13*w - 7],
[571, 571, -w^3 + 2*w^2 + 3*w - 7],
[571, 571, -3*w^3 + 5*w^2 + 13*w + 1],
[577, 577, -2*w^3 + 5*w^2 + 7*w - 5],
[577, 577, -2*w^3 + 6*w^2 + 6*w - 7],
[587, 587, 3*w^3 - 6*w^2 - 13*w + 10],
[587, 587, -w^3 + 4*w^2 + 2*w - 7],
[599, 599, -3*w^3 + 6*w^2 + 8*w - 8],
[617, 617, -4*w^3 + 7*w^2 + 15*w - 3],
[617, 617, -w^3 + 4*w^2 + w - 6],
[625, 5, -5],
[631, 631, w^2 - 4*w - 4],
[641, 641, -2*w^3 + 3*w^2 + 11*w - 1],
[641, 641, -3*w^3 + 5*w^2 + 12*w - 7],
[643, 643, -w^3 + 4*w^2 - w - 8],
[647, 647, 3*w^3 - 4*w^2 - 15*w + 3],
[647, 647, -3*w^3 + 5*w^2 + 10*w - 6],
[661, 661, 3*w^2 - 5],
[661, 661, 2*w^3 - 2*w^2 - 10*w + 1],
[677, 677, 5*w^3 - 13*w^2 - 9*w + 13],
[683, 683, w^3 - w^2 - 8*w + 1],
[683, 683, w^3 - w^2 - 8*w - 7],
[701, 701, -3*w^3 + 10*w^2 + w - 13],
[701, 701, -w^3 + w^2 - 4],
[701, 701, w^3 - w^2 - 8*w + 2],
[701, 701, -4*w^3 + 6*w^2 + 20*w - 7],
[709, 709, 4*w^3 - 7*w^2 - 18*w + 3],
[719, 719, w^3 - 3*w^2 - 2*w - 2],
[727, 727, 3*w^3 - 6*w^2 - 13*w + 9],
[733, 733, 4*w^3 - 5*w^2 - 17*w - 1],
[739, 739, -3*w^3 + 5*w^2 + 8*w + 1],
[739, 739, w - 6],
[751, 751, -3*w^3 + 6*w^2 + 9*w - 7],
[761, 761, -w - 5],
[761, 761, -3*w^3 + 6*w^2 + 14*w - 17],
[773, 773, -3*w + 7],
[787, 787, w^3 - 3*w - 4],
[797, 797, -4*w^3 + 6*w^2 + 16*w + 1],
[797, 797, -w^3 + 2*w^2 + 7*w - 4],
[809, 809, -3*w^3 + 7*w^2 + 10*w - 8],
[809, 809, 3*w^2 - 3*w - 13],
[811, 811, 3*w^3 - 6*w^2 - 7*w],
[821, 821, 3*w^3 - 5*w^2 - 10*w],
[839, 839, w^3 - 12*w + 11],
[839, 839, -3*w^3 + 5*w^2 + 15*w - 12],
[841, 29, 2*w^3 - 2*w^2 - 12*w + 3],
[853, 853, 2*w^2 - 5*w - 6],
[857, 857, w^3 - 2*w^2 - 3*w - 4],
[857, 857, -2*w^3 + 2*w^2 + 9*w + 9],
[859, 859, -2*w^3 + 4*w^2 + 10*w - 7],
[863, 863, 2*w^3 - 7*w^2 + 7],
[881, 881, -2*w^3 + 2*w^2 + 9*w - 2],
[883, 883, -3*w^3 + 10*w^2 - 8],
[887, 887, 2*w^2 - 7],
[887, 887, 2*w^3 - 2*w^2 - 13*w + 2],
[907, 907, -w^2 + 2*w - 4],
[907, 907, 4*w^3 - 11*w^2 - 7*w + 10],
[907, 907, -4*w^3 + 8*w^2 + 17*w - 9],
[907, 907, -3*w^3 + 6*w^2 + 9*w - 5],
[911, 911, w^3 + w^2 - 7*w - 12],
[919, 919, 2*w^3 - 5*w^2 - 7*w + 4],
[929, 929, w^3 - w^2 - 6*w - 6],
[947, 947, 5*w^3 - 8*w^2 - 20*w + 6],
[967, 967, -3*w^3 + 5*w^2 + 13*w - 8],
[977, 977, -4*w^3 + 7*w^2 + 15*w - 6],
[997, 997, -3*w^3 + 7*w^2 + 6*w - 6],
[1009, 1009, w^2 - 4*w - 6],
[1019, 1019, -w^2 + 4*w - 6],
[1019, 1019, -2*w^3 + w^2 + 14*w + 6],
[1031, 1031, 4*w^3 - 11*w^2 - 8*w + 17],
[1031, 1031, -w^3 + 2*w^2 - 7],
[1033, 1033, w^3 - 5*w - 8],
[1039, 1039, 2*w^3 - 5*w^2 - 8*w + 4],
[1039, 1039, -w^3 + 5*w^2 - 3*w - 7],
[1049, 1049, -3*w^3 + 6*w^2 + 12*w - 4],
[1051, 1051, 2*w^3 - 4*w^2 - 8*w - 3],
[1051, 1051, w^3 + w^2 - 8*w - 7],
[1061, 1061, -w^3 + 5*w^2 - 14],
[1061, 1061, -4*w^3 + 9*w^2 + 11*w - 6],
[1063, 1063, -2*w^3 + 5*w^2 + 4*w - 11],
[1069, 1069, -3*w^3 + 7*w^2 + 6*w - 5],
[1091, 1091, -2*w^3 + 2*w^2 + 13*w - 1],
[1093, 1093, 5*w^3 - 9*w^2 - 18*w + 3],
[1097, 1097, 5*w^3 - 11*w^2 - 15*w + 12],
[1109, 1109, -3*w^3 + 5*w^2 + 12*w - 10],
[1117, 1117, -3*w^3 + 4*w^2 + 14*w - 3],
[1129, 1129, -3*w^3 + 3*w^2 + 18*w + 1],
[1129, 1129, 3*w^3 - 7*w^2 - 8*w + 4],
[1151, 1151, w^2 + w - 7],
[1163, 1163, -w^3 + 3*w^2 + 5*w - 9],
[1163, 1163, -w^3 - w^2 + 14*w - 8],
[1181, 1181, -3*w^3 + 7*w^2 + 9*w - 6],
[1181, 1181, 5*w^3 - 10*w^2 - 17*w + 14],
[1187, 1187, w^3 - 4*w^2 + w - 2],
[1187, 1187, -2*w^3 + 14*w + 5],
[1193, 1193, -5*w^3 + 13*w^2 + 11*w - 18],
[1201, 1201, -2*w^3 + 5*w^2 + 9*w - 12],
[1201, 1201, -3*w^3 + 5*w^2 + 9*w - 6],
[1217, 1217, 2*w^3 - w^2 - 10*w - 8],
[1223, 1223, 3*w^2 - 2*w - 4],
[1229, 1229, -4*w^3 + 9*w^2 + 10*w - 11],
[1237, 1237, -w^3 + 2*w^2 + 2*w - 8],
[1259, 1259, -4*w^3 + 10*w^2 + 12*w - 13],
[1259, 1259, w^2 - w - 9],
[1259, 1259, w^3 - 9*w - 9],
[1259, 1259, w^3 - 3*w^2 + w - 5],
[1277, 1277, -4*w^3 + 8*w^2 + 13*w - 7],
[1279, 1279, -3*w^3 + 5*w^2 + 13*w - 11],
[1289, 1289, 2*w^2 - 8*w + 1],
[1291, 1291, -3*w + 10],
[1291, 1291, -2*w^3 + 4*w^2 + 11*w - 9],
[1297, 1297, -2*w^3 + 3*w^2 + 6*w + 5],
[1297, 1297, 3*w^3 - 5*w^2 - 9*w],
[1319, 1319, -2*w^3 + 3*w^2 + 9*w + 5],
[1367, 1367, -3*w^3 + 7*w^2 + 10*w - 7],
[1373, 1373, -4*w^3 + 9*w^2 + 16*w - 23],
[1373, 1373, 3*w^3 - 5*w^2 - 10*w + 3],
[1423, 1423, 4*w^3 - 8*w^2 - 13*w + 9],
[1423, 1423, 3*w^3 - 6*w^2 - 11*w + 2],
[1427, 1427, -w^3 - w^2 + 8*w],
[1429, 1429, -3*w^3 + 6*w^2 + 8*w - 3],
[1429, 1429, -3*w^3 + 2*w^2 + 13*w + 6],
[1439, 1439, w^3 - 2*w^2 - 5],
[1447, 1447, -2*w^3 + 6*w^2 + 3*w - 16],
[1459, 1459, -4*w^3 + 9*w^2 + 12*w - 8],
[1459, 1459, -w^3 + w^2 + 9*w - 4],
[1471, 1471, -2*w^3 + 5*w^2 + 8*w - 3],
[1481, 1481, 2*w^3 - 6*w^2 - 4*w + 3],
[1493, 1493, w^3 - 3*w^2 - 4],
[1493, 1493, -2*w^3 + 5*w^2 + 10*w - 7],
[1511, 1511, -5*w^3 + 7*w^2 + 22*w - 3],
[1531, 1531, -3*w^3 + 6*w^2 + 14*w - 6],
[1549, 1549, w^2 - 3*w - 9],
[1549, 1549, 5*w^3 - 10*w^2 - 16*w + 6],
[1553, 1553, 3*w^3 - 3*w^2 - 16*w],
[1553, 1553, -4*w^3 + 11*w^2 + 5*w - 5],
[1559, 1559, -5*w^3 + 12*w^2 + 12*w - 13],
[1567, 1567, -w - 6],
[1571, 1571, -2*w^3 + w^2 + 16*w + 5],
[1571, 1571, -3*w^3 + 6*w^2 + 15*w - 1],
[1579, 1579, -4*w^3 + 8*w^2 + 16*w - 7],
[1597, 1597, -w^3 + 2*w^2 + 8*w - 5],
[1597, 1597, -w^3 + 3*w^2 + 3*w - 12],
[1601, 1601, 3*w^3 - 6*w^2 - 14*w + 11],
[1607, 1607, -3*w^3 + 4*w^2 + 13*w - 4],
[1619, 1619, 4*w^2 - 3*w - 15],
[1621, 1621, -2*w^3 + 4*w^2 + 11*w - 3],
[1621, 1621, -w^2 - w - 4],
[1637, 1637, -2*w^3 + 3*w^2 + 15*w - 18],
[1637, 1637, 7*w^3 - 18*w^2 - 14*w + 19],
[1657, 1657, -2*w^3 + w^2 + 19*w - 12],
[1657, 1657, -3*w^3 + 4*w^2 + 10*w - 2],
[1657, 1657, 3*w^3 - 3*w^2 - 20*w + 8],
[1657, 1657, -3*w^3 + 6*w^2 + 7*w - 6],
[1667, 1667, -w^3 + 3*w^2 + 6*w - 9],
[1667, 1667, -w^3 + 3*w^2 - w - 6],
[1669, 1669, 3*w^2 - 7*w - 8],
[1669, 1669, 4*w^2 - 14*w + 5],
[1681, 41, -3*w^3 + 5*w^2 + 15*w - 7],
[1681, 41, -w^3 + 2*w^2 + 8*w - 2],
[1693, 1693, w^3 - w^2 - w - 4],
[1697, 1697, 3*w^3 - 6*w^2 - 11*w],
[1697, 1697, 5*w^3 - 7*w^2 - 21*w + 1],
[1699, 1699, 2*w^2 - 5*w - 8],
[1709, 1709, -2*w^3 + 7*w^2 + 6*w - 13],
[1721, 1721, -2*w^3 + w^2 + 13*w],
[1723, 1723, 2*w^2 - 5*w - 11],
[1723, 1723, -2*w^3 + 4*w^2 + 6*w - 11],
[1733, 1733, -4*w^3 + 7*w^2 + 14*w - 2],
[1733, 1733, -2*w^3 + 2*w^2 + 9*w - 5],
[1747, 1747, w^3 - 2*w^2 - 3*w - 5],
[1747, 1747, -4*w^3 + 9*w^2 + 10*w - 14],
[1759, 1759, 2*w^3 - 3*w^2 - 12*w - 6],
[1759, 1759, -2*w^3 + 4*w^2 + 3*w - 7],
[1777, 1777, -3*w^3 + 4*w^2 + 14*w - 4],
[1783, 1783, -2*w^3 + 6*w^2 + w - 8],
[1787, 1787, -6*w^3 + 10*w^2 + 23*w - 10],
[1787, 1787, 2*w^3 - w^2 - 18*w + 11],
[1789, 1789, -w^3 + 2*w^2 + 4*w - 9],
[1823, 1823, -5*w^3 + 9*w^2 + 22*w - 8],
[1831, 1831, -6*w^3 + 17*w^2 + 9*w - 18],
[1831, 1831, -4*w^3 + 11*w^2 + 9*w - 16],
[1849, 43, -w^3 + 5*w^2 - 2*w - 9],
[1849, 43, 3*w^2 - 2*w - 6],
[1861, 1861, -w^2 + 2*w - 5],
[1867, 1867, w^3 - 2*w^2 - 2*w - 6],
[1867, 1867, 5*w^3 - 7*w^2 - 24*w + 4],
[1867, 1867, -2*w^3 + 5*w^2 + 2*w - 7],
[1867, 1867, 6*w^3 - 13*w^2 - 18*w + 12],
[1871, 1871, 2*w^2 - w - 11],
[1873, 1873, -w^3 + 5*w^2 - 2*w - 6],
[1873, 1873, -w^3 + 2*w^2 + 8*w - 3],
[1877, 1877, -4*w^3 + 4*w^2 + 21*w + 2],
[1879, 1879, 4*w - 7],
[1879, 1879, 5*w^3 - 9*w^2 - 22*w + 5],
[1879, 1879, 3*w^2 - 3*w - 5],
[1879, 1879, -3*w^3 + 7*w^2 + 12*w - 20],
[1901, 1901, 3*w^2 - 4*w - 5],
[1907, 1907, -2*w^3 + 8*w^2 - 2*w - 11],
[1931, 1931, -w^3 + 4*w - 5],
[1933, 1933, -w^3 + 4*w^2 - w - 11],
[1933, 1933, w^3 - 4*w - 7],
[1933, 1933, 6*w^3 - 16*w^2 - 9*w + 12],
[1933, 1933, -4*w^2 + 13*w - 6],
[1949, 1949, 2*w^3 - 2*w^2 - 12*w + 5],
[1949, 1949, -w^3 + 2*w^2 - w - 5],
[1951, 1951, -3*w^3 + 6*w^2 + 15*w - 4],
[1951, 1951, 4*w^3 - 8*w^2 - 10*w + 13],
[1973, 1973, -w^3 - w^2 + 13*w - 1],
[1979, 1979, 3*w^3 - 6*w^2 - 12*w - 1],
[1979, 1979, 4*w^3 - 6*w^2 - 20*w + 9],
[1993, 1993, 2*w^2 - 5*w - 9],
[2027, 2027, -w^3 - w^2 + 13*w - 6],
[2027, 2027, -7*w^3 + 17*w^2 + 18*w - 19],
[2027, 2027, -3*w^3 + 8*w^2 + 11*w - 17],
[2027, 2027, 5*w^3 - 9*w^2 - 18*w + 4],
[2039, 2039, -3*w^3 + 4*w^2 + 15*w - 5],
[2081, 2081, -w^3 + 5*w^2 - 2*w - 7],
[2081, 2081, -6*w^3 + 16*w^2 + 12*w - 21],
[2083, 2083, -4*w^3 + 6*w^2 + 15*w - 8],
[2099, 2099, -4*w^3 + 9*w^2 + 10*w - 8],
[2113, 2113, 3*w^3 - 5*w^2 - 9*w + 3],
[2113, 2113, -w^3 + w^2 + 11*w - 11],
[2131, 2131, -w^3 + 5*w^2 - w - 6],
[2131, 2131, -w^3 + 3*w^2 - 2*w - 5],
[2137, 2137, 2*w^3 - 5*w^2 - 3],
[2143, 2143, -3*w^3 + 9*w^2 + 8*w - 18],
[2143, 2143, -3*w^3 + 5*w^2 + 8*w - 5],
[2153, 2153, 6*w^3 - 15*w^2 - 12*w + 10],
[2153, 2153, -7*w^3 + 16*w^2 + 21*w - 19],
[2161, 2161, 3*w^3 - 6*w^2 - 7*w + 3],
[2179, 2179, 3*w^3 - 6*w^2 - 7*w + 13],
[2179, 2179, -w^3 + 4*w^2 - 3*w - 6],
[2197, 13, 5*w^3 - 12*w^2 - 10*w + 6],
[2203, 2203, 3*w^2 - 4*w - 6],
[2207, 2207, 2*w^3 - 7*w^2 + w + 9],
[2209, 47, -w^3 + 4*w^2 - 2*w - 8],
[2213, 2213, -3*w^3 + 2*w^2 + 18*w + 2],
[2221, 2221, -2*w^3 + 2*w^2 + 10*w - 5],
[2237, 2237, 3*w^2 - 4*w - 8],
[2237, 2237, -3*w^3 + 6*w^2 + 14*w - 8],
[2243, 2243, -w^3 + 3*w^2 + 5*w - 12],
[2273, 2273, -3*w^3 + 4*w^2 + 8*w - 4],
[2287, 2287, 4*w^3 - 8*w^2 - 18*w + 5],
[2287, 2287, -3*w^3 + 6*w^2 + 14*w - 18],
[2309, 2309, -2*w^3 + 5*w^2 + 7],
[2311, 2311, -w^3 + 5*w^2 - w - 11],
[2311, 2311, 3*w^3 - 5*w^2 - 12*w - 5],
[2333, 2333, -3*w^3 + 4*w^2 + 13*w - 5],
[2333, 2333, w^2 - 5*w - 8],
[2347, 2347, -w^2 + 7*w - 8],
[2347, 2347, 4*w^3 - 7*w^2 - 16*w + 12],
[2351, 2351, 5*w^3 - 13*w^2 - 8*w + 6],
[2351, 2351, -4*w^3 + 7*w^2 + 18*w - 1],
[2357, 2357, -4*w^3 + 8*w^2 + 12*w - 5],
[2357, 2357, -w^3 + 4*w^2 - w - 16],
[2371, 2371, -5*w^3 + 7*w^2 + 20*w - 3],
[2371, 2371, -6*w^3 + 16*w^2 + 11*w - 19],
[2377, 2377, 4*w^3 - 6*w^2 - 15*w + 3],
[2377, 2377, -4*w^3 + 7*w^2 + 16*w - 14],
[2381, 2381, 5*w^3 - 9*w^2 - 18*w + 10],
[2381, 2381, w^3 - 3*w - 6],
[2383, 2383, w^3 - 3*w^2 - 8*w + 3],
[2383, 2383, -2*w^3 + 4*w^2 + 11*w - 6],
[2383, 2383, -w^3 + 3*w^2 + 8*w - 6],
[2383, 2383, w^3 + 2*w^2 - 16*w + 5],
[2423, 2423, 2*w^2 - 9],
[2423, 2423, 4*w^3 - 7*w^2 - 14*w + 4],
[2441, 2441, -2*w^3 + 5*w^2 + 9*w - 14],
[2441, 2441, -3*w^3 + 4*w^2 + 16*w - 7],
[2447, 2447, -4*w^3 + 9*w^2 + 17*w - 9],
[2447, 2447, -w^3 - w^2 + 12*w - 4],
[2459, 2459, w^2 - w - 10],
[2473, 2473, -2*w^3 + 7*w^2 + 2*w - 9],
[2477, 2477, 4*w^3 - 7*w^2 - 14*w + 5],
[2503, 2503, -2*w^3 + 6*w^2 + 7*w - 5],
[2521, 2521, w^3 + 2*w^2 - 17*w + 8],
[2521, 2521, -w^3 + 7*w^2 - 11*w + 1],
[2531, 2531, -5*w^3 + 10*w^2 + 18*w - 8],
[2539, 2539, 6*w^3 - 11*w^2 - 29*w + 25],
[2551, 2551, 2*w^3 + w^2 - 15*w - 3],
[2557, 2557, -3*w^3 + 3*w^2 + 16*w - 1],
[2557, 2557, 4*w^3 - 6*w^2 - 20*w + 3],
[2579, 2579, w^3 + w^2 - 6*w - 11],
[2591, 2591, 4*w^3 - 6*w^2 - 15*w + 2],
[2591, 2591, 4*w^3 - 5*w^2 - 22*w + 6],
[2609, 2609, 6*w^3 - 15*w^2 - 14*w + 15],
[2609, 2609, w^3 + w^2 - 5*w - 9],
[2617, 2617, -2*w^3 + 2*w^2 + 11*w - 6],
[2617, 2617, -2*w^3 + 3*w^2 + 9*w + 6],
[2617, 2617, 3*w^3 - 6*w^2 - 6*w + 4],
[2617, 2617, 3*w^3 - 6*w^2 - 13*w + 1],
[2621, 2621, 2*w^3 + w^2 - 11*w - 11],
[2659, 2659, -2*w^3 + 3*w^2 + 13*w - 15],
[2663, 2663, w^3 + 2*w^2 - 11*w - 7],
[2671, 2671, 4*w^3 - 5*w^2 - 15*w + 1],
[2683, 2683, -2*w^3 + 7*w^2 + 3*w - 14],
[2687, 2687, -2*w^3 + w^2 + 13*w - 1],
[2699, 2699, -2*w^3 + 7*w^2 + 2*w - 11],
[2699, 2699, -4*w^3 + 5*w^2 + 17*w - 3],
[2707, 2707, -4*w^3 + 10*w^2 + 10*w - 9],
[2711, 2711, -4*w^3 + 7*w^2 + 13*w - 8],
[2713, 2713, 4*w^3 - 7*w^2 - 17*w + 13],
[2729, 2729, -2*w^3 + 3*w^2 + 13*w - 8],
[2731, 2731, -2*w^3 + 4*w^2 + 10*w - 15],
[2741, 2741, 3*w^3 - 3*w^2 - 15*w + 1],
[2749, 2749, -2*w^3 + 4*w^2 + 3*w - 10],
[2749, 2749, 5*w^2 - 14*w],
[2753, 2753, -3*w^3 + 8*w^2 + 7*w - 21],
[2767, 2767, -2*w^3 + 7*w^2 + 4*w - 10],
[2777, 2777, -w^3 + 5*w^2 - 4*w - 9],
[2789, 2789, -w^3 + w^2 + 10*w - 5],
[2797, 2797, -2*w^3 + 5*w^2 + 3*w + 4],
[2797, 2797, -7*w^3 + 18*w^2 + 18*w - 27],
[2797, 2797, -3*w^3 + 9*w^2 + 6*w - 13],
[2797, 2797, w^3 - w^2 - 2*w - 8],
[2801, 2801, -5*w^3 + 14*w^2 + 11*w - 24],
[2803, 2803, -2*w^3 + w^2 + 14*w - 2],
[2803, 2803, w^3 + w^2 - 9*w + 2],
[2837, 2837, -4*w^3 + 7*w^2 + 19*w - 5],
[2837, 2837, -2*w - 7],
[2843, 2843, 5*w^3 - 14*w^2 - 6*w + 8],
[2843, 2843, -3*w^3 + 6*w^2 + 14*w],
[2843, 2843, -w^3 + 5*w^2 + w - 10],
[2843, 2843, -w^3 + 5*w^2 - 11],
[2851, 2851, -w^3 + w^2 + 3*w - 7],
[2851, 2851, -w^3 + 3*w^2 - w - 10],
[2857, 2857, 2*w^3 - 3*w^2 - 10*w - 6],
[2857, 2857, -5*w^3 + 9*w^2 + 20*w - 4],
[2861, 2861, -w^2 - 2*w - 5],
[2879, 2879, 4*w^2 - 11],
[2903, 2903, -3*w^3 + 4*w^2 + 15*w - 6],
[2903, 2903, 2*w^2 - 6*w - 7],
[2909, 2909, w^3 + w^2 - 11*w + 3],
[2927, 2927, -3*w^3 + 2*w^2 + 11*w - 2],
[2927, 2927, -w^3 + 3*w^2 - w - 9],
[2939, 2939, w^3 - w^2 - 5*w - 8],
[2953, 2953, 2*w^2 - 10*w + 13],
[2953, 2953, 4*w^3 - 9*w^2 - 11*w + 4],
[2957, 2957, 2*w^3 - w^2 - 16*w + 6],
[2999, 2999, 6*w^3 - 12*w^2 - 20*w + 17],
[3001, 3001, 4*w^3 - 7*w^2 - 15*w - 5],
[3001, 3001, 3*w^2 - 8*w - 7],
[3011, 3011, 3*w^3 - 2*w^2 - 11*w - 5],
[3023, 3023, -5*w^3 + 8*w^2 + 26*w - 16],
[3037, 3037, -w^3 + 4*w^2 - 2*w - 9],
[3037, 3037, -5*w^3 + 9*w^2 + 18*w - 9],
[3041, 3041, 5*w^3 - 10*w^2 - 15*w + 4],
[3061, 3061, -5*w^3 + 7*w^2 + 17*w + 3],
[3061, 3061, -3*w^3 + 6*w^2 + 15*w - 5],
[3061, 3061, 5*w^3 - 10*w^2 - 16*w + 13],
[3061, 3061, 5*w^3 - 9*w^2 - 25*w + 21],
[3067, 3067, -5*w^3 + 12*w^2 + 13*w - 11],
[3067, 3067, 7*w^3 - 12*w^2 - 27*w + 12],
[3079, 3079, 3*w^2 - 2*w - 13],
[3089, 3089, 3*w^3 - 3*w^2 - 19*w + 6],
[3089, 3089, -2*w^3 + 8*w^2 - 3*w - 11],
[3089, 3089, 2*w^3 - 5*w^2 - w - 7],
[3089, 3089, 5*w^3 - 8*w^2 - 17*w + 10],
[3109, 3109, -5*w^3 + 11*w^2 + 21*w - 14],
[3119, 3119, 2*w^2 + 3*w - 7],
[3121, 3121, -w^3 + 5*w^2 - 10],
[3121, 3121, -3*w^3 + 10*w^2 + 7*w - 23],
[3137, 3137, -6*w^3 + 17*w^2 + 6*w - 9],
[3167, 3167, w^3 - w^2 - 4*w - 9],
[3187, 3187, -7*w^3 + 18*w^2 + 12*w - 12],
[3187, 3187, -5*w^3 + 10*w^2 + 23*w - 27],
[3187, 3187, w^3 + 2*w^2 - 14*w + 1],
[3187, 3187, -4*w^3 + 8*w^2 + 11*w - 3],
[3203, 3203, 4*w^3 - 7*w^2 - 20*w + 14],
[3203, 3203, -5*w^3 + 6*w^2 + 28*w - 6],
[3203, 3203, 6*w^3 - 9*w^2 - 24*w + 8],
[3203, 3203, -w^3 + 5*w^2 - 9*w + 10],
[3221, 3221, -3*w^3 + 9*w^2 + 7*w - 16],
[3221, 3221, -w^3 + 2*w^2 + 4*w - 10],
[3229, 3229, -4*w^3 + 10*w^2 + 15*w - 21],
[3229, 3229, 6*w^3 - 11*w^2 - 21*w + 4],
[3251, 3251, w^3 + 2*w^2 - 10*w - 10],
[3271, 3271, 3*w^3 - 8*w^2 - 14*w + 11],
[3299, 3299, 4*w^3 - 7*w^2 - 17*w - 5],
[3301, 3301, w^2 - 2*w - 10],
[3307, 3307, -4*w^3 + 9*w^2 + 13*w - 8],
[3313, 3313, -4*w^3 + 9*w^2 + 8*w - 3],
[3313, 3313, w^3 - 10*w - 7],
[3323, 3323, 5*w^3 - 7*w^2 - 20*w + 2],
[3331, 3331, -4*w^3 + 6*w^2 + 19*w - 8],
[3343, 3343, -3*w^3 + 4*w^2 + 14*w - 6],
[3343, 3343, 3*w^3 - 4*w^2 - 18*w + 8],
[3347, 3347, 2*w^3 - 7*w^2 - 9*w + 14],
[3361, 3361, -2*w^3 + 7*w^2 + 3*w - 11],
[3373, 3373, -2*w^3 + 7*w^2 + 4*w - 13],
[3391, 3391, -3*w^3 + 8*w^2 + 7*w - 5],
[3407, 3407, -3*w^3 + 5*w^2 + 16*w - 9],
[3449, 3449, -w^3 + w^2 + 10*w - 2],
[3457, 3457, 4*w^2 - 8*w - 7],
[3457, 3457, w^3 - 2*w^2 - 2*w - 7],
[3467, 3467, -5*w^3 + 7*w^2 + 20*w + 1],
[3469, 3469, -w^3 + 2*w^2 + 5*w - 11],
[3469, 3469, -2*w^3 + 7*w^2 + 4*w - 12],
[3481, 59, w^3 - 3*w - 7],
[3481, 59, w^2 + 2*w - 9],
[3491, 3491, -4*w^3 + 8*w^2 + 20*w - 23],
[3499, 3499, -5*w^3 + 11*w^2 + 14*w - 21],
[3527, 3527, w^3 + 2*w^2 - 10*w - 6],
[3529, 3529, -4*w^3 + 10*w^2 + 11*w - 10],
[3533, 3533, -4*w^3 + 7*w^2 + 19*w - 6],
[3539, 3539, 5*w^3 - 11*w^2 - 21*w + 26],
[3547, 3547, -2*w^3 + 5*w^2 + w - 7],
[3559, 3559, -w^3 - w^2 + 3*w + 7],
[3571, 3571, -3*w^3 + 3*w^2 + 14*w - 2],
[3581, 3581, w^3 - w^2 - 5],
[3581, 3581, -5*w^3 + 8*w^2 + 17*w + 2],
[3593, 3593, -4*w^3 + 8*w^2 + 17*w - 5],
[3607, 3607, -w^3 + 3*w^2 + 6*w - 11],
[3617, 3617, -6*w^3 + 10*w^2 + 30*w - 21],
[3631, 3631, 5*w^2 - 10*w - 13],
[3637, 3637, 2*w^3 + w^2 - 19*w - 8],
[3643, 3643, -4*w^3 + 7*w^2 + 21*w - 1],
[3643, 3643, 3*w^2 - 3*w - 19],
[3659, 3659, 2*w^3 + w^2 - 18*w - 10],
[3671, 3671, 4*w^3 - 8*w^2 - 14*w - 1],
[3677, 3677, -3*w^3 + 7*w^2 + 6*w - 12],
[3677, 3677, -3*w^3 + 4*w^2 + 13*w - 7],
[3691, 3691, 2*w^3 - 17*w + 1],
[3691, 3691, -5*w^3 + 10*w^2 + 16*w - 12],
[3697, 3697, -5*w^3 + 16*w^2 + 2*w - 17],
[3701, 3701, 2*w^3 - 18*w + 3],
[3709, 3709, w^3 - 4*w - 10],
[3727, 3727, 6*w^3 - 11*w^2 - 26*w + 11],
[3767, 3767, 7*w^3 - 18*w^2 - 15*w + 18],
[3767, 3767, -w^3 - 2*w^2 + 7*w + 9],
[3779, 3779, 7*w^3 - 14*w^2 - 28*w + 15],
[3779, 3779, -3*w^3 + 5*w^2 + 16*w - 3],
[3797, 3797, -7*w^3 + 19*w^2 + 14*w - 23],
[3821, 3821, -3*w^3 + 10*w^2 + 2*w - 13],
[3821, 3821, -6*w^3 + 10*w^2 + 24*w + 1],
[3851, 3851, -4*w^3 + 8*w^2 + 15*w - 4],
[3863, 3863, -w^3 + 5*w^2 - 7*w - 5],
[3863, 3863, -2*w^3 + 8*w^2 - w - 13],
[3881, 3881, -3*w^3 + 7*w^2 + 12*w - 5],
[3881, 3881, 4*w^3 - 5*w^2 - 23*w + 9],
[3889, 3889, 4*w^3 - 8*w^2 - 18*w + 13],
[3889, 3889, w^3 - w^2 - 11*w - 5],
[3907, 3907, -7*w^3 + 20*w^2 + 10*w - 21],
[3911, 3911, 4*w^3 - 7*w^2 - 13*w + 3],
[3917, 3917, 3*w^3 - 2*w^2 - 22*w + 3],
[3917, 3917, 6*w^3 - 11*w^2 - 27*w + 22],
[3923, 3923, -3*w^3 + 8*w^2 + 11*w - 19],
[3923, 3923, -3*w^3 + 9*w^2 + 9*w - 22],
[3931, 3931, -5*w^3 + 8*w^2 + 21*w - 9],
[3943, 3943, -5*w^3 + 10*w^2 + 18*w - 7],
[3989, 3989, 7*w^3 - 13*w^2 - 30*w + 22],
[3989, 3989, -w^2 + 4*w - 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^12 - 42*x^10 + 708*x^8 - 6120*x^6 + 28584*x^4 - 68400*x^2 + 65536;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -29/1024*e^11 + 545/512*e^9 - 3917/256*e^7 + 13353/128*e^5 - 42817/128*e^3 + 25671/64*e, 1/16*e^10 - 19/8*e^8 + 69/2*e^6 - 238*e^4 + 1551/2*e^2 - 951, -1/2*e^3 + 4*e, -53/1024*e^11 + 985/512*e^9 - 7013/256*e^7 + 23777/128*e^5 - 76345/128*e^3 + 46223/64*e, -1/4*e^6 + 11/2*e^4 - 37*e^2 + 77, -23/1024*e^11 + 419/512*e^9 - 2919/256*e^7 + 9659/128*e^5 - 30211/128*e^3 + 17941/64*e, 11/512*e^11 - 199/256*e^9 + 1371/128*e^7 - 4447/64*e^5 + 13383/64*e^3 - 7377/32*e, -1, -5/512*e^11 + 105/256*e^9 - 853/128*e^7 + 3313/64*e^5 - 12137/64*e^3 + 8287/32*e, -9/256*e^11 + 173/128*e^9 - 1273/64*e^7 + 4453/32*e^5 - 14733/32*e^3 + 9227/16*e, -27/1024*e^11 + 503/512*e^9 - 3563/256*e^7 + 11887/128*e^5 - 37015/128*e^3 + 21473/64*e, -1/4*e^8 + 15/2*e^6 - 159/2*e^4 + 349*e^2 - 530, -37/512*e^11 + 681/256*e^9 - 4789/128*e^7 + 15985/64*e^5 - 50377/64*e^3 + 29919/32*e, 3/16*e^10 - 55/8*e^8 + 387/4*e^6 - 651*e^4 + 4177/2*e^2 - 2547, 1/16*e^10 - 19/8*e^8 + 69/2*e^6 - 473/2*e^4 + 1505/2*e^2 - 877, 1/8*e^10 - 19/4*e^8 + 275/4*e^6 - 939/2*e^4 + 1500*e^2 - 1788, -1/2*e^6 + 23/2*e^4 - 80*e^2 + 164, 49/1024*e^11 - 901/512*e^9 + 6369/256*e^7 - 21549/128*e^5 + 69605/128*e^3 - 43011/64*e, 3/16*e^10 - 55/8*e^8 + 387/4*e^6 - 1299/2*e^4 + 4133/2*e^2 - 2475, 85/1024*e^11 - 1593/512*e^9 + 11461/256*e^7 - 39297/128*e^5 + 127385/128*e^3 - 77487/64*e, -9/64*e^11 + 165/32*e^9 - 1157/16*e^7 + 3857/8*e^5 - 12157/8*e^3 + 7215/4*e, 1/8*e^10 - 9/2*e^8 + 247/4*e^6 - 401*e^4 + 1226*e^2 - 1408, 7/256*e^11 - 147/128*e^9 + 1175/64*e^7 - 4427/32*e^5 + 15555/32*e^3 - 10069/16*e, 33/512*e^11 - 597/256*e^9 + 4145/128*e^7 - 13757/64*e^5 + 43477/64*e^3 - 25971/32*e, 55/1024*e^11 - 963/512*e^9 + 6407/256*e^7 - 20187/128*e^5 + 60387/128*e^3 - 34677/64*e, 3/8*e^10 - 55/4*e^8 + 385/2*e^6 - 1279*e^4 + 4014*e^2 - 4752, 3/16*e^10 - 55/8*e^8 + 385/4*e^6 - 639*e^4 + 4005/2*e^2 - 2365, 1/16*e^10 - 17/8*e^8 + 109/4*e^6 - 325/2*e^4 + 877/2*e^2 - 407, 121/1024*e^11 - 2285/512*e^9 + 16553/256*e^7 - 57173/128*e^5 + 187277/128*e^3 - 116059/64*e, -5/16*e^10 + 93/8*e^8 - 661/4*e^6 + 1113*e^4 - 7041/2*e^2 + 4157, 7/512*e^11 - 115/256*e^9 + 727/128*e^7 - 2251/64*e^5 + 7027/64*e^3 - 4581/32*e, 1/16*e^10 - 21/8*e^8 + 169/4*e^6 - 323*e^4 + 2325/2*e^2 - 1561, 23/256*e^11 - 419/128*e^9 + 2903/64*e^7 - 9483/32*e^5 + 28979/32*e^3 - 16485/16*e, 19/512*e^11 - 335/256*e^9 + 2243/128*e^7 - 7047/64*e^5 + 20463/64*e^3 - 10857/32*e, 25/1024*e^11 - 461/512*e^9 + 3209/256*e^7 - 10421/128*e^5 + 31149/128*e^3 - 16699/64*e, 1/4*e^8 - 7*e^6 + 68*e^4 - 270*e^2 + 376, -131/1024*e^11 + 2431/512*e^9 - 17235/256*e^7 + 57911/128*e^5 - 183007/128*e^3 + 107993/64*e, 1/4*e^6 - 5*e^4 + 30*e^2 - 62, 1/4*e^8 - 31/4*e^6 + 85*e^4 - 391*e^2 + 640, 3/8*e^10 - 27/2*e^8 + 371/2*e^6 - 2419/2*e^4 + 3726*e^2 - 4334, 1/4*e^10 - 19/2*e^8 + 277/2*e^6 - 1923/2*e^4 + 3151*e^2 - 3872, -3/128*e^11 + 55/64*e^9 - 379/32*e^7 + 1199/16*e^5 - 3351/16*e^3 + 1537/8*e, -115/512*e^11 + 2127/256*e^9 - 15075/128*e^7 + 50887/64*e^5 - 162735/64*e^3 + 98217/32*e, -1/16*e^10 + 17/8*e^8 - 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367665/128*e^3 + 227543/64*e, -61/128*e^11 + 1113/64*e^9 - 7749/32*e^7 + 25529/16*e^5 - 78865/16*e^3 + 45391/8*e, -21/16*e^10 + 393/8*e^8 - 2823/4*e^6 + 4837*e^4 - 31467/2*e^2 + 19371, -11/32*e^11 + 205/16*e^9 - 1463/8*e^7 + 4963/4*e^5 - 15877/4*e^3 + 9497/2*e, 9/8*e^10 - 85/2*e^8 + 615*e^6 - 8465/2*e^4 + 13768*e^2 - 16914, -135/512*e^11 + 2611/256*e^9 - 19447/128*e^7 + 69291/64*e^5 - 234451/64*e^3 + 149381/32*e, 353/512*e^11 - 6581/256*e^9 + 46929/128*e^7 - 158941/64*e^5 + 507925/64*e^3 - 304915/32*e, e^10 - 75/2*e^8 + 539*e^6 - 3683*e^4 + 11857*e^2 - 14290, -1/8*e^10 + 15/4*e^8 - 169/4*e^6 + 473/2*e^4 - 738*e^2 + 1074, -187/512*e^11 + 3447/256*e^9 - 24395/128*e^7 + 82415/64*e^5 - 264183/64*e^3 + 160129/32*e, -1/4*e^8 + 37/4*e^6 - 121*e^4 + 657*e^2 - 1222, -1/128*e^11 - 19/64*e^9 + 423/32*e^7 - 2459/16*e^5 + 11315/16*e^3 - 9141/8*e, -3/2*e^10 + 221/4*e^8 - 1561/2*e^6 + 5256*e^4 - 16779*e^2 + 20282, -7/16*e^10 + 135/8*e^8 - 1001/4*e^6 + 3549/2*e^4 - 11973/2*e^2 + 7691, -43/128*e^11 + 815/64*e^9 - 5891/32*e^7 + 20111/16*e^5 - 64303/16*e^3 + 38473/8*e, 33/256*e^11 - 645/128*e^9 + 4849/64*e^7 - 17389/32*e^5 + 58821/32*e^3 - 37011/16*e, 15/1024*e^11 - 315/512*e^9 + 2591/256*e^7 - 10451/128*e^5 + 40667/128*e^3 - 28477/64*e, -3/4*e^10 + 57/2*e^8 - 825/2*e^6 + 2818*e^4 - 9006*e^2 + 10740, -153/512*e^11 + 2861/256*e^9 - 20489/128*e^7 + 69973/64*e^5 - 227725/64*e^3 + 141435/32*e, -3/8*e^10 + 29/2*e^8 - 214*e^6 + 1496*e^4 - 4928*e^2 + 6108, -1/4*e^10 + 35/4*e^8 - 117*e^6 + 1479/2*e^4 - 2175*e^2 + 2380, -27/512*e^11 + 439/256*e^9 - 2667/128*e^7 + 7727/64*e^5 - 22807/64*e^3 + 14817/32*e, 7/8*e^10 - 65/2*e^8 + 463*e^6 - 3145*e^4 + 10114*e^2 - 12208, 5/8*e^10 - 97/4*e^8 + 361*e^6 - 2553*e^4 + 8487*e^2 - 10528, -1/16*e^10 + 13/8*e^8 - 51/4*e^6 + 16*e^4 + 313/2*e^2 - 425, -19/16*e^10 + 349/8*e^8 - 1227/2*e^6 + 8203/2*e^4 - 25901/2*e^2 + 15389, -505/1024*e^11 + 9389/512*e^9 - 66921/256*e^7 + 226901/128*e^5 - 725517/128*e^3 + 435163/64*e, -3/8*e^10 + 57/4*e^8 - 419/2*e^6 + 2961/2*e^4 - 4997*e^2 + 6420, 15/8*e^10 - 277/4*e^8 + 3905/4*e^6 - 13055/2*e^4 + 20603*e^2 - 24512, -9/64*e^11 + 161/32*e^9 - 1093/16*e^7 + 3477/8*e^5 - 10201/8*e^3 + 5535/4*e, -39/512*e^11 + 595/256*e^9 - 3255/128*e^7 + 7627/64*e^5 - 14035/64*e^3 + 3397/32*e, 1/2*e^10 - 18*e^8 + 493/2*e^6 - 1597*e^4 + 4881*e^2 - 5608, -5/8*e^10 + 95/4*e^8 - 1385/4*e^6 + 2407*e^4 - 7924*e^2 + 9790, 1/2*e^8 - 14*e^6 + 261/2*e^4 - 454*e^2 + 436, 151/512*e^11 - 2787/256*e^9 + 19751/128*e^7 - 66843/64*e^5 + 215139/64*e^3 - 132245/32*e, 111/128*e^11 - 2043/64*e^9 + 14351/32*e^7 - 47755/16*e^5 + 149523/16*e^3 - 87621/8*e, -53/1024*e^11 + 921/512*e^9 - 6245/256*e^7 + 21089/128*e^5 - 72313/128*e^3 + 50191/64*e, -391/512*e^11 + 7283/256*e^9 - 51991/128*e^7 + 176715/64*e^5 - 568083/64*e^3 + 343749/32*e, -1/4*e^10 + 37/4*e^8 - 132*e^6 + 900*e^4 - 2900*e^2 + 3518, 21/16*e^10 - 391/8*e^8 + 2789/4*e^6 - 4734*e^4 + 30411/2*e^2 - 18447, 7/16*e^10 - 125/8*e^8 + 855/4*e^6 - 1393*e^4 + 8571/2*e^2 - 4923, 7/8*e^10 - 135/4*e^8 + 501*e^6 - 3549*e^4 + 11890*e^2 - 14988, 51/512*e^11 - 943/256*e^9 + 6659/128*e^7 - 22215/64*e^5 + 68719/64*e^3 - 37897/32*e, -457/1024*e^11 + 8573/512*e^9 - 61305/256*e^7 + 207269/128*e^5 - 658141/128*e^3 + 392267/64*e, 243/1024*e^11 - 4527/512*e^9 + 32259/256*e^7 - 109415/128*e^5 + 352271/128*e^3 - 214313/64*e, -13/16*e^10 + 231/8*e^8 - 1559/4*e^6 + 4953/2*e^4 - 14675/2*e^2 + 8049, -7/16*e^10 + 139/8*e^8 - 1063/4*e^6 + 1944*e^4 - 13515/2*e^2 + 8889, 295/1024*e^11 - 5683/512*e^9 + 41975/256*e^7 - 147467/128*e^5 + 489363/128*e^3 - 305669/64*e, -3/16*e^10 + 53/8*e^8 - 355/4*e^6 + 555*e^4 - 3197/2*e^2 + 1717, -121/1024*e^11 + 2349/512*e^9 - 17385/256*e^7 + 61141/128*e^5 - 204237/128*e^3 + 128731/64*e, -1/4*e^10 + 39/4*e^8 - 147*e^6 + 1065*e^4 - 3678*e^2 + 4784, -81/512*e^11 + 1477/256*e^9 - 10273/128*e^7 + 33837/64*e^5 - 104997/64*e^3 + 61155/32*e, -9/8*e^10 + 43*e^8 - 631*e^6 + 8831/2*e^4 - 14641*e^2 + 18306, -3/16*e^10 + 55/8*e^8 - 389/4*e^6 + 666*e^4 - 4475/2*e^2 + 2955, 9/16*e^10 - 153/8*e^8 + 977/4*e^6 - 2903/2*e^4 + 7953/2*e^2 - 3959, 39/1024*e^11 - 883/512*e^9 + 7543/256*e^7 - 30219/128*e^5 + 112659/128*e^3 - 78469/64*e, 115/256*e^11 - 2079/128*e^9 + 14323/64*e^7 - 46727/32*e^5 + 143743/32*e^3 - 83385/16*e, 75/256*e^11 - 1383/128*e^9 + 9707/64*e^7 - 32239/32*e^5 + 101159/32*e^3 - 60225/16*e, 9/16*e^10 - 167/8*e^8 + 297*e^6 - 2013*e^4 + 12931/2*e^2 - 7863, 3/8*e^10 - 53/4*e^8 + 178*e^6 - 1132*e^4 + 3412*e^2 - 3988, 3/128*e^11 - 47/64*e^9 + 275/32*e^7 - 751/16*e^5 + 1831/16*e^3 - 465/8*e, -5/16*e^10 + 93/8*e^8 - 657/4*e^6 + 1087*e^4 - 6607/2*e^2 + 3649, 9/8*e^10 - 83/2*e^8 + 2335/4*e^6 - 7771/2*e^4 + 12149*e^2 - 14194, 3/2*e^10 - 221/4*e^8 + 3117/4*e^6 - 5232*e^4 + 16640*e^2 - 19978, -1/4*e^11 + 73/8*e^9 - 509/4*e^7 + 840*e^5 - 5201/2*e^3 + 3031*e, -827/1024*e^11 + 15447/512*e^9 - 110603/256*e^7 + 376847/128*e^5 - 1212471/128*e^3 + 732545/64*e, 9/8*e^10 - 167/4*e^8 + 1185/2*e^6 - 3989*e^4 + 12635*e^2 - 15018, -11/8*e^10 + 205/4*e^8 - 733*e^6 + 9995/2*e^4 - 16139*e^2 + 19674, -37/512*e^11 + 617/256*e^9 - 3893/128*e^7 + 11665/64*e^5 - 33737/64*e^3 + 18815/32*e, -15/512*e^11 + 411/256*e^9 - 3871/128*e^7 + 16179/64*e^5 - 60603/64*e^3 + 40989/32*e, -17/16*e^10 + 317/8*e^8 - 2257/4*e^6 + 7625/2*e^4 - 24341/2*e^2 + 14655, 3/8*e^10 - 57/4*e^8 + 837/4*e^6 - 2947/2*e^4 + 4937*e^2 - 6278, -135/512*e^11 + 2515/256*e^9 - 18007/128*e^7 + 61803/64*e^5 - 203219/64*e^3 + 127589/32*e, -3/256*e^11 + 95/128*e^9 - 1011/64*e^7 + 4727/32*e^5 - 19727/32*e^3 + 14825/16*e, 15/256*e^11 - 299/128*e^9 + 2239/64*e^7 - 7699/32*e^5 + 23643/32*e^3 - 12733/16*e, 5/8*e^10 - 43/2*e^8 + 280*e^6 - 3431/2*e^4 + 4935*e^2 - 5398, 1/8*e^10 - 4*e^8 + 44*e^6 - 363/2*e^4 + 113*e^2 + 512, -383/512*e^11 + 7019/256*e^9 - 49199/128*e^7 + 164035/64*e^5 - 518603/64*e^3 + 311309/32*e, -191/256*e^11 + 3483/128*e^9 - 24223/64*e^7 + 79763/32*e^5 - 246923/32*e^3 + 142637/16*e, 11/8*e^10 - 51*e^8 + 1447/2*e^6 - 9749/2*e^4 + 15484*e^2 - 18422, 3/16*e^10 - 41/8*e^8 + 179/4*e^6 - 199/2*e^4 - 737/2*e^2 + 1303, 1/4*e^10 - 10*e^8 + 309/2*e^6 - 2275/2*e^4 + 3920*e^2 - 4980, 285/512*e^11 - 5217/256*e^9 + 36525/128*e^7 - 121545/64*e^5 + 382465/64*e^3 - 226727/32*e, 1/32*e^11 - 21/16*e^9 + 161/8*e^7 - 557/4*e^5 + 1729/4*e^3 - 1009/2*e, -909/1024*e^11 + 16977/512*e^9 - 121085/256*e^7 + 409049/128*e^5 - 1298161/128*e^3 + 769111/64*e, 17/16*e^10 - 325/8*e^8 + 2375/4*e^6 - 8239/2*e^4 + 26975/2*e^2 - 16617, -715/1024*e^11 + 13223/512*e^9 - 93467/256*e^7 + 313439/128*e^5 - 989639/128*e^3 + 584241/64*e, -243/1024*e^11 + 4591/512*e^9 - 33027/256*e^7 + 111847/128*e^5 - 352143/128*e^3 + 204969/64*e, 5/64*e^11 - 105/32*e^9 + 857/16*e^7 - 3357/8*e^5 + 12409/8*e^3 - 8515/4*e, -1/4*e^11 + 19/2*e^9 - 551/4*e^7 + 1887/2*e^5 - 3017*e^3 + 3555*e, -255/1024*e^11 + 4715/512*e^9 - 33359/256*e^7 + 112195/128*e^5 - 356427/128*e^3 + 211789/64*e, -99/1024*e^11 + 1887/512*e^9 - 13619/256*e^7 + 46167/128*e^5 - 146175/128*e^3 + 86393/64*e, -5/256*e^11 + 153/128*e^9 - 1557/64*e^7 + 6945/32*e^5 - 27625/32*e^3 + 19775/16*e, 7/8*e^10 - 63/2*e^8 + 867/2*e^6 - 5675/2*e^4 + 8792*e^2 - 10228, 15/16*e^10 - 277/8*e^8 + 1961/4*e^6 - 3308*e^4 + 21165/2*e^2 - 12771, -107/512*e^11 + 2023/256*e^9 - 14683/128*e^7 + 50591/64*e^5 - 163207/64*e^3 + 97457/32*e, 137/256*e^11 - 2525/128*e^9 + 17769/64*e^7 - 59333/32*e^5 + 187085/32*e^3 - 111035/16*e, 423/1024*e^11 - 7795/512*e^9 + 54903/256*e^7 - 183883/128*e^5 + 584275/128*e^3 - 352901/64*e, 1/8*e^10 - 9/2*e^8 + 121/2*e^6 - 729/2*e^4 + 907*e^2 - 642, -17/16*e^10 + 323/8*e^8 - 2357/4*e^6 + 8191/2*e^4 - 26867/2*e^2 + 16515, 1/16*e^10 - 19/8*e^8 + 129/4*e^6 - 369/2*e^4 + 761/2*e^2 - 61, -3/8*e^10 + 55/4*e^8 - 193*e^6 + 2591/2*e^4 - 4172*e^2 + 5178, -13/8*e^10 + 60*e^8 - 847*e^6 + 11371/2*e^4 - 18075*e^2 + 21748, 95/512*e^11 - 1707/256*e^9 + 11695/128*e^7 - 37891/64*e^5 + 114795/64*e^3 - 63917/32*e, 7/4*e^10 - 64*e^8 + 891*e^6 - 5861*e^4 + 18100*e^2 - 20906, 5/8*e^10 - 24*e^8 + 1411/4*e^6 - 2456*e^4 + 8006*e^2 - 9734];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, -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;