/*
  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![8, 1, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {5, 5, -w^3 + 2*w^2 + 3*w - 1}>;

primesArray := [
[2, 2, -w^2 + 2],
[2, 2, -w^3 + 2*w^2 + 4*w - 5],
[5, 5, -w^3 + 2*w^2 + 3*w - 1],
[13, 13, w^3 - 2*w^2 - 3*w + 5],
[17, 17, -w^2 - w + 3],
[19, 19, -w^3 + 3*w^2 + 2*w - 7],
[23, 23, w^3 - 2*w^2 - 3*w + 3],
[31, 31, -w^2 + w + 1],
[47, 47, -w^3 + w^2 + 4*w - 3],
[49, 7, 2*w^3 - 5*w^2 - 7*w + 11],
[53, 53, -3*w^3 + 9*w^2 + 10*w - 31],
[53, 53, w^3 - w^2 - 4*w + 1],
[61, 61, 3*w^3 - 6*w^2 - 13*w + 13],
[61, 61, 2*w^2 - 7],
[71, 71, w^2 - 3*w - 5],
[73, 73, 2*w - 3],
[73, 73, -2*w^3 + 6*w^2 + 6*w - 19],
[79, 79, 2*w^2 - 5],
[81, 3, -3],
[101, 101, 2*w^2 - 4*w - 9],
[107, 107, 2*w - 1],
[109, 109, 2*w^3 - 3*w^2 - 11*w + 7],
[109, 109, 2*w^2 - 2*w - 7],
[125, 5, -2*w^3 + 8*w^2 - 17],
[127, 127, 2*w^2 - 4*w - 7],
[127, 127, -w^2 + 3*w + 1],
[131, 131, w^3 - w^2 - 4*w - 3],
[137, 137, 4*w^3 - 10*w^2 - 14*w + 25],
[139, 139, -w^3 + 2*w^2 + w - 5],
[139, 139, 2*w^2 - 2*w - 5],
[149, 149, -w^3 + 4*w^2 - w - 7],
[149, 149, -2*w^3 + 5*w^2 + 9*w - 19],
[151, 151, -w^3 + 4*w^2 + w - 11],
[151, 151, w^2 + w + 1],
[157, 157, -2*w + 5],
[163, 163, 4*w^3 - 8*w^2 - 18*w + 19],
[163, 163, w^3 - 4*w^2 - w + 9],
[167, 167, -w^3 + 3*w^2 + 2*w - 9],
[167, 167, w^3 - 2*w^2 - 5*w + 1],
[173, 173, w^2 - 3*w - 7],
[173, 173, w^3 - 3*w^2 - 4*w + 5],
[181, 181, -w^3 + 4*w^2 + w - 13],
[191, 191, -w^3 + 2*w^2 + 3*w + 1],
[191, 191, -2*w^3 + 3*w^2 + 9*w - 3],
[193, 193, -w^3 + 4*w^2 + 3*w - 13],
[197, 197, 5*w^3 - 10*w^2 - 21*w + 21],
[211, 211, -w^3 + 3*w^2 + 4*w - 13],
[227, 227, w^3 - 4*w^2 + w + 5],
[229, 229, -w^3 + 5*w^2 - 2*w - 11],
[229, 229, -w^3 + 3*w^2 + 4*w - 3],
[239, 239, -2*w^3 + 4*w^2 + 8*w - 13],
[241, 241, 4*w^3 - 10*w^2 - 16*w + 31],
[241, 241, -5*w^3 + 12*w^2 + 21*w - 31],
[251, 251, -w^3 + 3*w^2 + 2*w - 11],
[263, 263, -2*w^3 + 2*w^2 + 10*w + 1],
[263, 263, -2*w - 5],
[269, 269, w^2 + w - 5],
[269, 269, 4*w^3 - 7*w^2 - 19*w + 15],
[271, 271, 2*w^3 - 5*w^2 - 5*w + 11],
[271, 271, -w^3 + 7*w + 3],
[277, 277, -2*w^3 + 4*w^2 + 6*w - 5],
[277, 277, 4*w^2 - 13],
[281, 281, w^3 - 5*w^2 + 11],
[307, 307, w^3 - 3*w - 3],
[313, 313, 5*w^3 - 10*w^2 - 23*w + 27],
[313, 313, -2*w^3 + 3*w^2 + 9*w - 7],
[317, 317, -3*w^3 + 6*w^2 + 11*w - 15],
[347, 347, -w^3 - w^2 + 1],
[347, 347, -2*w^3 + 6*w^2 + 6*w - 17],
[353, 353, -2*w^3 + 4*w^2 + 8*w - 5],
[353, 353, w^3 - 5*w - 1],
[359, 359, -2*w^2 + 1],
[367, 367, -w^3 + 4*w^2 + 3*w - 17],
[373, 373, 2*w^2 - 2*w - 13],
[397, 397, 5*w^3 - 13*w^2 - 18*w + 39],
[397, 397, 2*w^3 - 5*w^2 - 7*w + 9],
[401, 401, -4*w^3 + 12*w^2 + 8*w - 25],
[401, 401, -3*w^3 + 7*w^2 + 8*w - 13],
[421, 421, -4*w^3 + 11*w^2 + 13*w - 31],
[433, 433, 2*w^3 - 12*w - 15],
[433, 433, w^3 - w^2 - 6*w + 5],
[439, 439, 5*w^3 - 13*w^2 - 12*w + 19],
[439, 439, -2*w^3 + 3*w^2 + 7*w + 1],
[449, 449, -w^3 + 4*w^2 + 3*w - 11],
[461, 461, -4*w^3 + 11*w^2 + 7*w - 19],
[461, 461, -w^3 + w^2 + 2*w + 1],
[463, 463, -w^3 + 2*w^2 + 7*w - 1],
[463, 463, -4*w^3 + 11*w^2 + 7*w - 13],
[467, 467, 2*w^2 - 2*w - 3],
[467, 467, 2*w^3 - 4*w^2 - 6*w + 11],
[487, 487, -3*w^3 + 11*w^2 + 2*w - 23],
[487, 487, w^3 - w + 5],
[491, 491, w^3 - 2*w^2 - 3*w + 9],
[499, 499, w^3 - 3*w^2 - 4*w + 1],
[503, 503, -6*w^3 + 11*w^2 + 27*w - 21],
[503, 503, -3*w^3 + 8*w^2 + 5*w - 9],
[521, 521, 2*w^2 - 4*w - 11],
[541, 541, -3*w^3 + 7*w^2 + 8*w - 11],
[557, 557, 5*w^3 - 11*w^2 - 18*w + 21],
[557, 557, 4*w^3 - 7*w^2 - 17*w + 9],
[557, 557, 4*w^3 - 8*w^2 - 16*w + 15],
[557, 557, -3*w^3 + 6*w^2 + 11*w - 9],
[563, 563, -2*w^3 + 6*w^2 + 8*w - 13],
[569, 569, -2*w^3 + 3*w^2 + 7*w - 11],
[587, 587, 3*w^3 - 8*w^2 - 9*w + 15],
[601, 601, -3*w^2 + 5*w + 9],
[607, 607, -w^3 + w^2 - 5],
[619, 619, -2*w^3 + 4*w^2 + 6*w - 13],
[631, 631, -2*w^3 + 7*w^2 + 5*w - 23],
[643, 643, 2*w^3 - 2*w^2 - 12*w - 5],
[659, 659, -w^3 + 6*w^2 + 3*w - 11],
[659, 659, -3*w^2 + w + 9],
[661, 661, -2*w^3 + 4*w^2 + 8*w - 3],
[661, 661, -w^3 + 4*w^2 + 5*w - 7],
[673, 673, w^2 - w - 9],
[673, 673, 2*w^3 - 7*w^2 - 3*w + 17],
[677, 677, 9*w^3 - 20*w^2 - 37*w + 51],
[701, 701, -7*w^3 + 18*w^2 + 27*w - 57],
[709, 709, 4*w^3 - 6*w^2 - 14*w + 17],
[709, 709, w^3 + w^2 - 6*w - 17],
[719, 719, -2*w^3 + 6*w^2 + 4*w - 15],
[719, 719, -2*w^3 + 3*w^2 + 7*w + 3],
[727, 727, -w^3 + 3*w^2 + 6*w - 3],
[743, 743, -2*w^3 + 2*w^2 + 6*w - 1],
[757, 757, 6*w^3 - 16*w^2 - 22*w + 51],
[757, 757, w^3 + w^2 - 10*w - 11],
[761, 761, w^2 + w - 7],
[761, 761, 2*w^2 - 17],
[761, 761, 2*w^3 - 6*w^2 - 2*w + 13],
[761, 761, 2*w^2 - 4*w - 3],
[769, 769, -w^3 - w^2 + 4*w + 7],
[769, 769, w^3 + w^2 - 4*w - 1],
[787, 787, 3*w^3 - 6*w^2 - 11*w + 3],
[809, 809, -w^3 + 3*w^2 + 2*w - 1],
[809, 809, 2*w^3 - w^2 - 9*w - 3],
[809, 809, 2*w^3 - 7*w^2 - 3*w + 15],
[809, 809, 2*w^3 - 2*w^2 - 8*w - 1],
[811, 811, -8*w^3 + 19*w^2 + 33*w - 57],
[811, 811, -2*w^3 + 2*w^2 + 4*w - 11],
[823, 823, 2*w^3 - w^2 - 7*w + 1],
[823, 823, -3*w^3 + 8*w^2 + 7*w - 9],
[829, 829, -w^3 + 5*w - 1],
[853, 853, -w^3 + 5*w^2 - 2*w - 15],
[853, 853, -2*w^3 + 4*w^2 + 4*w - 9],
[853, 853, 2*w^3 - w^2 - 11*w - 3],
[853, 853, 2*w^3 - 5*w^2 - 7*w + 5],
[857, 857, -2*w^3 + 5*w^2 + 7*w - 7],
[859, 859, w^3 - 2*w^2 - w + 9],
[859, 859, w^3 - 2*w^2 - w - 1],
[877, 877, 2*w^3 - 6*w^2 - 8*w + 25],
[877, 877, -4*w^3 + 8*w^2 + 16*w - 19],
[881, 881, -4*w^2 + 6*w + 13],
[881, 881, w^3 + 2*w^2 - 7*w - 9],
[887, 887, -4*w^2 + 4*w + 11],
[887, 887, -w^3 - w^2 + 6*w + 11],
[887, 887, w^3 - 6*w^2 - w + 29],
[887, 887, 4*w^3 - 9*w^2 - 19*w + 31],
[907, 907, -3*w^3 + 9*w^2 + 12*w - 25],
[907, 907, -w^3 + 2*w^2 + 7*w - 7],
[929, 929, -7*w^3 + 13*w^2 + 32*w - 29],
[941, 941, -2*w^3 + 10*w^2 - 2*w - 23],
[941, 941, 4*w^3 - 9*w^2 - 13*w + 13],
[967, 967, -w^3 + 2*w^2 - w - 1],
[977, 977, w^3 - 5*w^2 + 15],
[983, 983, w^3 + 2*w^2 - 9*w - 11],
[983, 983, 2*w^3 - 5*w^2 - 5*w + 13],
[997, 997, -4*w^3 + 10*w^2 + 12*w - 23],
[1009, 1009, 2*w^2 + 2*w + 1],
[1019, 1019, 2*w^3 - 2*w^2 - 6*w + 3],
[1031, 1031, w^3 - 9*w - 7],
[1033, 1033, 2*w^3 - 2*w^2 - 8*w + 1],
[1063, 1063, -w^3 - w - 7],
[1063, 1063, 4*w^3 - 8*w^2 - 12*w + 5],
[1091, 1091, 2*w^3 - 2*w^2 - 4*w + 1],
[1093, 1093, 3*w^3 - 4*w^2 - 13*w - 1],
[1093, 1093, w^3 - w^2 - 4*w - 7],
[1097, 1097, w^3 + 3*w^2 - 2*w - 9],
[1109, 1109, w^2 + w - 9],
[1109, 1109, -2*w^3 + 5*w^2 + 3*w - 11],
[1123, 1123, -5*w^3 + 16*w^2 + 7*w - 27],
[1123, 1123, 8*w^3 - 17*w^2 - 35*w + 47],
[1163, 1163, 3*w^3 - 3*w^2 - 14*w - 5],
[1171, 1171, 2*w^3 - w^2 - 7*w - 1],
[1171, 1171, -3*w^3 + 6*w^2 + 9*w - 7],
[1181, 1181, 3*w^3 - 7*w^2 - 10*w + 21],
[1187, 1187, w^3 - 2*w^2 - 7*w + 3],
[1201, 1201, -3*w^3 + 5*w^2 + 10*w - 3],
[1201, 1201, -3*w^2 + 3*w + 19],
[1213, 1213, w^3 - 5*w^2 + 4*w + 7],
[1217, 1217, w^3 - 4*w^2 - 5*w + 5],
[1223, 1223, -w^3 - w^2 + 2*w + 5],
[1223, 1223, w^3 - 5*w^2 + 4*w + 5],
[1231, 1231, 7*w^3 - 15*w^2 - 28*w + 33],
[1237, 1237, w^3 - 5*w^2 + 2*w + 9],
[1249, 1249, -2*w^3 + 2*w^2 + 10*w - 3],
[1259, 1259, -4*w^3 + 8*w^2 + 16*w - 13],
[1283, 1283, w^3 - 2*w^2 - 3*w - 3],
[1289, 1289, -w^3 + w^2 + 4*w - 7],
[1291, 1291, 2*w^3 - 5*w^2 - 9*w + 9],
[1297, 1297, -10*w^3 + 24*w^2 + 38*w - 63],
[1301, 1301, 3*w^3 - 8*w^2 - 11*w + 17],
[1301, 1301, 2*w^2 - 4*w - 1],
[1307, 1307, w^3 - w^2 + 7],
[1327, 1327, -6*w^3 + 12*w^2 + 24*w - 31],
[1327, 1327, -w^3 + 2*w^2 - w - 5],
[1327, 1327, 8*w^3 - 16*w^2 - 34*w + 35],
[1327, 1327, 3*w^3 - 8*w^2 - 3*w + 15],
[1367, 1367, -w^3 + 2*w^2 + 7*w - 5],
[1369, 37, 7*w^3 - 17*w^2 - 18*w + 19],
[1369, 37, w^3 - 7*w^2 + 6*w + 19],
[1381, 1381, 6*w^2 - 6*w - 35],
[1427, 1427, -2*w^3 + 3*w^2 + 11*w + 1],
[1427, 1427, -2*w^3 + 7*w^2 + 7*w - 13],
[1427, 1427, 3*w^3 - 8*w^2 - 9*w + 9],
[1427, 1427, -12*w^3 + 26*w^2 + 50*w - 65],
[1433, 1433, w^3 - 7*w^2 - 4*w + 11],
[1439, 1439, -5*w^3 + 12*w^2 + 21*w - 39],
[1453, 1453, -w^3 - w^2 - 2*w - 7],
[1453, 1453, 2*w^3 - 4*w^2 - 6*w - 1],
[1459, 1459, 3*w^2 - 7*w - 15],
[1483, 1483, 4*w^3 - 15*w^2 - 9*w + 55],
[1483, 1483, w^3 - 3*w - 5],
[1493, 1493, -2*w^3 + 8*w^2 + 2*w - 23],
[1499, 1499, -w^3 + 9*w + 1],
[1499, 1499, w^3 - 4*w^2 + 3*w + 3],
[1499, 1499, 5*w^3 - 12*w^2 - 13*w + 17],
[1499, 1499, w^3 + w^2 - 8*w - 15],
[1511, 1511, -7*w^3 + 17*w^2 + 26*w - 47],
[1531, 1531, w^3 - 4*w^2 - 7*w + 13],
[1543, 1543, 4*w^3 - 7*w^2 - 21*w + 21],
[1543, 1543, 5*w^2 + 3*w - 7],
[1579, 1579, 4*w^3 - 9*w^2 - 13*w + 17],
[1583, 1583, -2*w^3 + 6*w^2 + 6*w - 23],
[1597, 1597, -4*w^3 + 11*w^2 + 13*w - 25],
[1609, 1609, 4*w^3 - 11*w^2 - 11*w + 27],
[1609, 1609, -4*w^3 + 13*w^2 + 5*w - 21],
[1637, 1637, -6*w^3 + 16*w^2 + 20*w - 45],
[1667, 1667, 5*w^2 - 3*w - 13],
[1667, 1667, -5*w^2 - w + 11],
[1697, 1697, -w^3 + 3*w^2 + 8*w + 1],
[1699, 1699, -w^3 + 4*w^2 - w - 13],
[1699, 1699, 5*w^3 - 14*w^2 - 9*w + 25],
[1721, 1721, 2*w^3 - 7*w^2 - w + 17],
[1723, 1723, -3*w^3 + 6*w^2 + 9*w - 11],
[1733, 1733, w^3 - w^2 - 8*w + 3],
[1747, 1747, w^3 - 2*w^2 - 7*w - 5],
[1753, 1753, -w^3 + 3*w^2 - 11],
[1759, 1759, -w^3 + 6*w^2 - w - 15],
[1759, 1759, -4*w^3 + 8*w^2 + 14*w - 19],
[1783, 1783, 3*w^3 - 6*w^2 - 7*w + 13],
[1783, 1783, -5*w^2 - 3*w + 13],
[1787, 1787, -2*w^3 + 7*w^2 + 5*w - 21],
[1787, 1787, w^2 - 3*w - 11],
[1787, 1787, -2*w - 7],
[1787, 1787, -w^3 + 3*w^2 + 4*w - 15],
[1789, 1789, w^3 - w^2 - 8*w - 1],
[1789, 1789, -4*w^2 + 8*w + 13],
[1801, 1801, -3*w^3 + 9*w^2 + 6*w - 11],
[1823, 1823, -2*w^3 + 5*w^2 + 5*w - 15],
[1831, 1831, 3*w^3 - 5*w^2 - 12*w + 9],
[1831, 1831, 3*w^3 - 3*w^2 - 18*w + 1],
[1831, 1831, w^3 - 5*w^2 - 2*w + 5],
[1831, 1831, w^3 - 5*w^2 - 2*w + 13],
[1847, 1847, 2*w^3 - 2*w^2 - 8*w - 9],
[1847, 1847, -3*w^3 + 5*w^2 + 14*w - 17],
[1849, 43, 3*w^3 - 3*w^2 - 12*w + 5],
[1849, 43, 3*w^3 - 2*w^2 - 19*w - 5],
[1861, 1861, 4*w^3 - 13*w^2 - 5*w + 27],
[1867, 1867, w^3 + 2*w^2 - 5*w - 5],
[1871, 1871, -w^3 + 5*w^2 + 2*w - 15],
[1871, 1871, 2*w^3 - w^2 - 15*w + 1],
[1873, 1873, w^3 - 7*w - 11],
[1873, 1873, -4*w^3 + 9*w^2 + 11*w - 17],
[1877, 1877, -4*w - 1],
[1877, 1877, 4*w + 9],
[1889, 1889, -3*w^3 + 7*w^2 + 6*w - 7],
[1901, 1901, 2*w^3 - 8*w^2 - 2*w + 17],
[1901, 1901, -3*w^2 + 7*w + 7],
[1913, 1913, 4*w^3 - 14*w^2 - 2*w + 27],
[1931, 1931, w^2 + 3*w - 7],
[1931, 1931, 3*w^3 - 3*w^2 - 12*w - 5],
[1933, 1933, -2*w^3 + 7*w^2 + 7*w - 17],
[1951, 1951, -2*w^3 + 6*w^2 + 4*w - 17],
[1951, 1951, -5*w^3 + 13*w^2 + 20*w - 43],
[1973, 1973, -2*w^3 + 9*w^2 - w - 19],
[1979, 1979, -4*w^3 + 10*w^2 + 18*w - 37],
[1979, 1979, -2*w^3 + 6*w^2 + 8*w - 11],
[1987, 1987, -w^3 + 3*w^2 + 6*w - 13],
[1997, 1997, -4*w^3 + 11*w^2 + 13*w - 29],
[1997, 1997, 4*w - 1],
[1997, 1997, 9*w^3 - 23*w^2 - 22*w + 35],
[1997, 1997, -6*w^3 + 18*w^2 + 8*w - 31],
[1999, 1999, 6*w^3 - 19*w^2 - 9*w + 37]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + 2*x^4 - 3*x^3 - 5*x^2 + x + 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e^4 + 2*e^3 - 3*e^2 - 4*e + 1, e, -1, 3, -e^4 + 6*e^2 - e - 5, 4*e^4 + 6*e^3 - 12*e^2 - 11*e + 5, e^4 + 2*e^3 - 2*e^2 - 6*e - 1, 4*e^4 + 3*e^3 - 18*e^2 - 5*e + 11, 2*e^4 + 3*e^3 - 9*e^2 - 7*e + 7, -4*e^4 - 6*e^3 + 9*e^2 + 11*e + 4, e^4 - 2*e^3 - 5*e^2 + 8*e + 2, 5*e^4 + 9*e^3 - 13*e^2 - 17*e, -3*e^3 - 3*e^2 + 9*e, -3*e^4 - 3*e^3 + 15*e^2 + 6*e - 6, -e^4 - e^3 + e^2 + 2*e + 5, e^4 - 6*e^2 + e + 11, -3*e + 3, -6*e^4 - 11*e^3 + 16*e^2 + 20*e - 2, -2*e^4 + 12*e^2 + e - 10, 6*e^4 + 15*e^3 - 12*e^2 - 33*e, 8*e^4 + 14*e^3 - 27*e^2 - 31*e + 13, -4*e^4 - 8*e^3 + 15*e^2 + 25*e - 12, -e^4 - 4*e^3 - e^2 + 6*e + 1, 6*e^2 + 3*e - 6, -e^4 - e^3 + 5*e^2 + 3*e - 6, 6*e^4 + 9*e^3 - 19*e^2 - 23*e + 3, 8*e^4 + 12*e^3 - 24*e^2 - 25*e - 2, 5*e^4 + 9*e^3 - 16*e^2 - 22*e + 3, -2*e^4 + 3*e^3 + 15*e^2 - 14*e - 10, e^4 - 9*e^2 + 4*e + 20, -e^4 - e^3 + 2*e^2 + 7*e + 7, 3*e^4 - 10*e^2 + 4*e - 3, e^4 + 8*e^3 + 4*e^2 - 23*e - 11, 3*e^3 + 5*e^2 - 16*e - 8, -3*e^4 + 3*e^3 + 21*e^2 - 6*e - 15, -9*e^4 - 17*e^3 + 28*e^2 + 35*e - 16, -4*e^4 - 6*e^3 + 15*e^2 + 14*e - 5, 2*e^4 - 3*e^3 - 12*e^2 + 17*e + 19, -6*e^4 - 9*e^3 + 21*e^2 + 24*e - 9, -3*e^3 + 12*e + 6, 6*e^3 + 6*e^2 - 21*e, -7*e^4 - 12*e^3 + 24*e^2 + 23*e - 2, -14*e^4 - 21*e^3 + 48*e^2 + 42*e - 16, 4*e^4 - 3*e^3 - 20*e^2 + 12*e + 8, -4*e^4 - 3*e^3 + 11*e^2 - 4*e - 4, 6*e^4 + 5*e^3 - 28*e^2 - 4*e + 24, 3*e^4 + 3*e^3 - 17*e^2 - 8*e + 14, -10*e^4 - 21*e^3 + 33*e^2 + 50*e - 8, -4*e^4 - 9*e^3 + 18*e^2 + 32*e - 14, e^4 + 12*e^3 + 3*e^2 - 35*e - 4, -2*e^4 - e^3 + 4*e^2 - 15*e + 5, 11*e^4 + 24*e^3 - 27*e^2 - 46*e + 7, -e^4 - 9*e^3 - 3*e^2 + 23*e + 7, -2*e^4 - 6*e^3 + 4*e + 14, -3*e^4 - 3*e^3 + 11*e^2 + 7*e - 6, 8*e^4 + 8*e^3 - 35*e^2 - 10*e + 11, -3*e^3 - 15*e^2 + 3*e + 21, e^4 + 10*e + 2, -4*e^4 + 21*e^2 - 7*e - 2, -8*e^4 - 18*e^3 + 24*e^2 + 52*e - 4, e^4 - 3*e^3 - 8*e^2 + 21*e + 5, 4*e^4 + 2*e^3 - 16*e^2 + 8*e + 9, 3*e^4 + 6*e^3 - 8*e^2 - 7*e, -3*e^4 - 3*e^3 + 3*e^2 - 3*e + 12, 2*e^4 + 9*e^3 + 3*e^2 - 25*e + 1, -15*e^4 - 30*e^3 + 39*e^2 + 57*e - 15, -12*e^4 - 14*e^3 + 46*e^2 + 33*e - 27, -e^4 - 2*e^3 + 6*e^2 + 5*e - 8, 6*e^4 + 9*e^3 - 17*e^2 - 18*e - 8, -9*e^4 - 15*e^3 + 21*e^2 + 27*e + 3, 5*e^4 + 6*e^3 - 24*e^2 - e + 31, -10*e^4 - 15*e^3 + 38*e^2 + 20*e - 24, 20*e^4 + 33*e^3 - 57*e^2 - 58*e + 13, -e^4 - 8*e^3 - 5*e^2 + 35*e + 13, 5*e^4 + 21*e^3 - 3*e^2 - 58*e - 5, 5*e^4 + 6*e^3 - 27*e^2 - 22*e + 31, -8*e^4 - 17*e^3 + 17*e^2 + 28*e + 4, 10*e^4 + 26*e^3 - 14*e^2 - 50*e - 6, e^4 - 7*e^3 - 7*e^2 + 23*e - 3, 3*e^4 + 3*e^3 - 12*e^2 - 15*e - 3, -7*e^4 - 9*e^3 + 18*e^2 + 11*e - 5, -12*e^4 - 18*e^3 + 36*e^2 + 36*e, 15*e^4 + 21*e^3 - 42*e^2 - 39*e + 3, 4*e^3 - 3*e^2 - 18*e + 18, 10*e^4 + 3*e^3 - 45*e^2 - 2*e + 20, 3*e^4 + 15*e^3 + 3*e^2 - 36*e - 18, -11*e^4 - 19*e^3 + 44*e^2 + 46*e - 22, 3*e^4 + 14*e^3 - 7*e^2 - 36*e + 15, 17*e^4 + 21*e^3 - 63*e^2 - 43*e + 28, -13*e^4 - 12*e^3 + 48*e^2 + 11*e - 20, -8*e^4 - 17*e^3 + 16*e^2 + 33*e + 23, 9*e^4 + 19*e^3 - 33*e^2 - 55*e + 26, 5*e^4 + 3*e^3 - 12*e^2 + 12*e - 8, 5*e^4 + 20*e^3 - 9*e^2 - 48*e + 11, -e^4 - 3*e^3 - 3*e^2 - 16*e + 10, -9*e^4 - 18*e^3 + 33*e^2 + 45*e - 9, 3*e^4 + 6*e^3 - 9*e^2 - 6*e + 9, 8*e^4 + 18*e^3 - 24*e^2 - 34*e + 25, -e^4 + 8*e^3 + 8*e^2 - 40*e - 17, -e^4 + 5*e^3 + 15*e^2 - 18, 7*e^4 + 11*e^3 - 25*e^2 - 20*e + 17, -20*e^4 - 30*e^3 + 70*e^2 + 73*e - 12, -8*e^4 - 9*e^3 + 27*e^2 + 7*e + 2, -4*e^4 - 11*e^3 + 12*e^2 + 34*e + 5, -10*e^4 - 12*e^3 + 27*e^2 + 17*e - 11, 17*e^4 + 24*e^3 - 54*e^2 - 25*e + 28, 12*e^4 + 27*e^3 - 42*e^2 - 69*e + 21, e^4 - 3*e^2 + 10*e + 23, -7*e^4 - 18*e^3 + 17*e^2 + 36*e + 1, -e^4 - 15*e^3 + 6*e^2 + 56*e - 11, 7*e^4 + 16*e^3 - 30*e^2 - 54*e + 33, -2*e^4 + e^3 + 19*e^2 + 16*e - 25, 8*e^4 - 51*e^2 + 5*e + 43, -11*e^4 - 15*e^3 + 45*e^2 + 25*e - 49, -6*e^4 - 3*e^3 + 20*e^2 - 9*e + 13, 7*e^4 + 22*e^3 - 11*e^2 - 51*e + 17, -4*e^4 - 12*e^3 + 12*e^2 + 53*e - 11, e^4 + 4*e^3 + 4*e^2 - e - 37, 6*e^4 + 16*e^3 - 4*e^2 - 36*e - 5, -6*e^4 + 8*e^3 + 50*e^2 - 17*e - 42, -26*e^4 - 42*e^3 + 78*e^2 + 91*e - 19, -13*e^4 - 24*e^3 + 42*e^2 + 44*e - 23, -14*e^4 - 18*e^3 + 45*e^2 + 25*e - 4, -3*e^4 + e^3 + 17*e^2 + 3*e - 10, -21*e^4 - 30*e^3 + 67*e^2 + 49*e - 28, 25*e^4 + 47*e^3 - 63*e^2 - 91*e + 9, -9*e^4 - 21*e^3 + 12*e^2 + 42*e - 3, 11*e^4 + 33*e^3 - 18*e^2 - 76*e - 2, 10*e^4 + 15*e^3 - 48*e^2 - 41*e + 20, 2*e^4 - 3*e^3 - 12*e^2 + 17*e + 34, e^4 + 28*e + 2, -3*e^4 - 6*e^3 - 12*e^2 - 3*e + 30, 8*e^4 + 27*e^3 - 24*e^2 - 91*e + 10, 13*e^4 + 15*e^3 - 44*e^2 - 16*e + 42, 26*e^4 + 39*e^3 - 89*e^2 - 77*e + 25, -9*e^4 - 2*e^3 + 49*e^2 - 11*e - 48, 2*e^4 - 2*e^3 - 27*e^2 - 13*e + 25, -22*e^4 - 36*e^3 + 63*e^2 + 86*e - 8, -17*e^4 - 24*e^3 + 42*e^2 + 25*e + 5, -14*e^4 - 28*e^3 + 35*e^2 + 58*e - 25, 13*e^4 + 12*e^3 - 60*e^2 - 8*e + 50, 16*e^4 + 21*e^3 - 51*e^2 - 44*e + 17, e^4 + 3*e^3 + 21*e^2 + 19*e - 52, -6*e^4 - 3*e^3 + 33*e^2 - 3*e - 12, 18*e^4 + 39*e^3 - 54*e^2 - 102*e + 24, e^4 + 6*e^3 + 9*e^2 - 20*e - 10, -21*e^4 - 36*e^3 + 51*e^2 + 60*e + 6, -3*e^4 - 3*e^3 + 27*e^2 + 12*e - 45, -8*e^4 - 9*e^3 + 27*e^2 + 28*e - 7, 2*e^4 + 8*e^3 - 13*e^2 - 21*e + 6, -14*e^4 - 16*e^3 + 41*e^2 + 20*e + 3, e^4 + 6*e^3 + 3*e^2 - 8*e + 26, -19*e^4 - 27*e^3 + 60*e^2 + 59*e - 11, 14*e^4 + 30*e^3 - 27*e^2 - 67*e - 20, 20*e^4 + 30*e^3 - 66*e^2 - 67*e + 13, -15*e^4 - 24*e^3 + 60*e^2 + 60*e - 39, 10*e^4 + 15*e^3 - 39*e^2 - 23*e + 20, -3*e^4 + 2*e^3 + 22*e^2 - 9*e - 31, 5*e^4 + 21*e^3 - 5*e^2 - 60*e - 6, 8*e^4 + 33*e^3 + 6*e^2 - 73*e - 14, 7*e^4 - 42*e^2 + 22*e + 41, -21*e^4 - 36*e^3 + 75*e^2 + 87*e - 27, 19*e^4 + 49*e^3 - 44*e^2 - 105*e + 25, 14*e^4 + 31*e^3 - 36*e^2 - 68*e + 17, -5*e^4 - 18*e^3 + 18*e^2 + 61*e - 4, -15*e^4 - 33*e^3 + 30*e^2 + 66*e + 24, -16*e^4 - 27*e^3 + 42*e^2 + 65*e - 2, -5*e^4 - 29*e^3 - 17*e^2 + 75*e + 18, -21*e^4 - 39*e^3 + 57*e^2 + 64*e - 3, -19*e^4 - 36*e^3 + 56*e^2 + 73*e - 6, -29*e^4 - 43*e^3 + 107*e^2 + 79*e - 46, -3*e^4 + 18*e^2 - 3*e - 45, 14*e^4 + 15*e^3 - 66*e^2 - 40*e + 40, -33*e^4 - 51*e^3 + 105*e^2 + 90*e - 36, -13*e^3 - 7*e^2 + 45*e - 6, 11*e^4 - 50*e^2 + 34*e + 46, -8*e^4 + 3*e^3 + 33*e^2 - 20*e + 2, -2*e^4 + 3*e^3 + 21*e^2 - 5*e - 13, 9*e^4 + 12*e^3 - 33*e^2 - 30*e + 39, -9*e^4 - 21*e^3 + 24*e^2 + 51*e + 15, 10*e^4 + 24*e^3 - 33*e^2 - 77*e + 14, -19*e^4 - 22*e^3 + 56*e^2 + 35*e + 19, -16*e^4 - 18*e^3 + 56*e^2 + 28*e - 34, -13*e^4 - 20*e^3 + 46*e^2 + 25*e + 4, -22*e^4 - 42*e^3 + 72*e^2 + 113*e - 17, -2*e^4 + 23*e^2 - 12*e - 53, -10*e^4 - 9*e^3 + 59*e^2 + 19*e - 61, -2*e^4 - 2*e^3 + 4*e^2 - 6*e - 16, 17*e^4 + 29*e^3 - 64*e^2 - 63*e + 26, -6*e^4 + 3*e^3 + 36*e^2 - 27*e - 33, -20*e^4 - 45*e^3 + 45*e^2 + 91*e - 19, 20*e^4 + 18*e^3 - 81*e^2 - 46*e + 40, e^4 - 24*e^3 - 27*e^2 + 67*e + 11, 9*e^3 + 15*e^2 - 3*e - 18, -e^4 - 6*e^3 - 6*e^2 + 5*e + 40, 19*e^4 + 24*e^3 - 51*e^2 - 44*e - 22, 9*e^4 + 9*e^3 - 47*e^2 - 18*e + 22, -7*e^4 - 12*e^3 + 25*e^2 + 29*e - 22, 12*e^4 + 24*e^3 - 27*e^2 - 72*e - 27, 4*e^3 - 11*e^2 - 19*e + 29, -14*e^4 - 12*e^3 + 45*e^2 - 8*e - 16, -28*e^4 - 36*e^3 + 81*e^2 + 53*e - 8, -21*e^4 - 39*e^3 + 45*e^2 + 63*e + 3, -32*e^4 - 49*e^3 + 114*e^2 + 116*e - 39, 7*e^4 + 8*e^3 - 36*e^2 + 2*e + 54, 22*e^4 + 36*e^3 - 76*e^2 - 83*e + 30, 12*e^4 + 3*e^3 - 62*e^2 - 6*e + 50, -16*e^4 - 50*e^3 + 26*e^2 + 120*e + 10, -20*e^4 - 14*e^3 + 91*e^2 + 39*e - 44, 9*e^4 + 2*e^3 - 52*e^2 + 4*e + 52, 37*e^4 + 63*e^3 - 118*e^2 - 147*e + 24, -2*e^4 + 6*e^3 + 27*e^2 - 20*e - 19, -27*e^4 - 57*e^3 + 66*e^2 + 135*e - 15, 9*e^4 + 36*e^3 - 15*e^2 - 114*e - 6, 14*e^4 + 27*e^3 - 42*e^2 - 43*e + 10, 19*e^4 + 12*e^3 - 84*e^2 - 5*e + 38, -11*e^4 - 9*e^3 + 34*e^2 - 2*e + 6, 24*e^4 + 35*e^3 - 61*e^2 - 71*e - 18, -9*e^4 - 2*e^3 + 43*e^2 - 22*e - 27, -e^4 - 3*e^3 + 12*e^2 + 14*e - 41, -28*e^4 - 48*e^3 + 90*e^2 + 125*e - 32, 8*e^4 + 15*e^3 - 18*e^2 - 49*e - 38, 9*e^4 + 9*e^3 - 37*e^2 - 10*e + 35, -25*e^4 - 47*e^3 + 77*e^2 + 132*e - 17, -22*e^4 - 13*e^3 + 98*e^2 + 11*e - 56, -20*e^4 - 52*e^3 + 42*e^2 + 109*e - 7, -17*e^4 - 47*e^3 + 34*e^2 + 112*e + 14, 12*e^4 - 54*e^2 + 12*e + 27, 8*e^4 - 9*e^3 - 30*e^2 + 62*e + 13, 17*e^4 - 78*e^2 + 38*e + 64, 35*e^4 + 60*e^3 - 99*e^2 - 124*e + 22, -11*e^4 + 4*e^3 + 61*e^2 - 45*e - 53, 19*e^4 + 31*e^3 - 51*e^2 - 74*e - 13, -17*e^4 - 33*e^3 + 38*e^2 + 64*e + 3, 11*e^4 + 18*e^3 - 51*e^2 - 31*e + 64, 10*e^4 - 6*e^3 - 60*e^2 + 7*e + 50, 19*e^4 + 21*e^3 - 60*e^2 - 41*e - 10, 37*e^4 + 55*e^3 - 98*e^2 - 86*e + 5, -2*e^4 - 13*e^3 - 17*e^2 + 45*e + 29, -6*e^4 - 18*e^3 - 3*e^2 + 30*e + 54, 17*e^4 + 33*e^3 - 60*e^2 - 73*e + 43, -24*e^4 - 39*e^3 + 72*e^2 + 51*e - 6, 27*e^4 + 66*e^3 - 72*e^2 - 150*e + 24, 25*e^4 + 42*e^3 - 69*e^2 - 77*e - 18, -45*e^4 - 72*e^3 + 128*e^2 + 139*e - 36, -9*e^4 - 16*e^3 + 22*e^2 + 51*e - 4, 19*e^4 + 33*e^3 - 69*e^2 - 110*e + 41, 26*e^4 + 25*e^3 - 108*e^2 - 47*e + 42, -10*e^4 - 17*e^3 + 47*e^2 + 35*e - 55, -e^4 - 6*e^3 + 12*e^2 + 17*e - 17, 2*e^4 - 12*e^3 - 24*e^2 + 32*e + 16, -12*e^4 - 34*e^3 - e^2 + 69*e + 33, -6*e^4 - 4*e^3 + 49*e^2 + 14*e - 63, 19*e^4 + 21*e^3 - 73*e^2 - 35*e + 70, 21*e^4 + 17*e^3 - 67*e^2 - 4*e + 12, -31*e^4 - 46*e^3 + 92*e^2 + 84*e - 8, -7*e^4 + 38*e^2 - 26*e - 25, -25*e^4 - 33*e^3 + 82*e^2 + 42*e - 75, -39*e^4 - 60*e^3 + 111*e^2 + 87*e - 30, -20*e^4 - 23*e^3 + 79*e^2 + 45*e - 53, 10*e^4 - 46*e^2 - 7*e - 4, 2*e^4 - 22*e^3 - 43*e^2 + 51*e + 53, e^4 - 21*e^3 - 21*e^2 + 94*e + 41, -6*e^4 - 24*e^3 - 3*e^2 + 45*e + 66, 3*e^4 + 15*e^3 + 21*e^2 - 33*e - 18, -15*e^4 - 35*e^3 + 43*e^2 + 80*e - 2, -25*e^4 - 28*e^3 + 86*e^2 + 48*e - 6, -17*e^4 - 42*e^3 + 39*e^2 + 115*e - 19, -20*e^4 - 33*e^3 + 45*e^2 + 37*e + 26, -3*e^4 + 7*e^3 + 15*e^2 - 28*e - 44, -e^4 + 10*e^3 + 5*e^2 - 30*e - 17, -3*e^4 + 10*e^3 + 50*e^2 - 6*e - 74, -40*e^4 - 64*e^3 + 128*e^2 + 114*e - 59, -12*e^4 - 32*e^3 + 20*e^2 + 81*e + 23, 13*e^4 + 11*e^3 - 62*e^2 - 12*e + 77, -24*e^4 - 54*e^3 + 63*e^2 + 144*e - 30, 10*e^4 + 33*e^3 + 4*e^2 - 73*e - 42, 8*e^4 - e^3 - 25*e^2 + 31*e - 5, 28*e^4 + 28*e^3 - 118*e^2 - 71*e + 64, 8*e^4 + 30*e^3 - 9*e^2 - 61*e + 4, -11*e^4 - 21*e^3 + 36*e^2 + 58*e + 20, -12*e^4 - 30*e^3 + 20*e^2 + 67*e + 27, -13*e^4 - 12*e^3 + 18*e^2 - 13*e + 49, 31*e^4 + 60*e^3 - 72*e^2 - 101*e - 25, -14*e^4 - 36*e^3 + 54*e^2 + 112*e - 10, 12*e^4 + 21*e^3 - 51*e^2 - 48*e + 33, 33*e^4 + 48*e^3 - 123*e^2 - 99*e + 45, 3*e^4 - 12*e^3 - 33*e^2 + 30*e + 9, 4*e^3 - 10*e^2 - 53*e + 39, 10*e^4 + 9*e^3 - 38*e^2 - 14*e + 51, -13*e^4 - 3*e^3 + 40*e^2 - 31*e - 10, 22*e^4 + 20*e^3 - 101*e^2 - 35*e + 63, 35*e^4 + 66*e^3 - 90*e^2 - 142*e + 4];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {5, 5, -w^3 + 2*w^2 + 3*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;