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

NN := ideal<ZF | {10, 10, w + 2}>;

primesArray := [
[2, 2, w],
[2, 2, -w - 1],
[5, 5, -w^3 + 2*w^2 + 5*w - 1],
[7, 7, -w^3 + 2*w^2 + 5*w - 3],
[29, 29, -w^2 + w + 3],
[29, 29, 2*w^3 - 5*w^2 - 7*w + 9],
[31, 31, -w^3 + 2*w^2 + 5*w + 1],
[41, 41, -w^3 + 4*w^2 - w - 3],
[43, 43, -w^3 + 3*w^2 + 2*w - 5],
[47, 47, w^2 - 3*w - 3],
[53, 53, w^3 - 2*w^2 - 3*w + 1],
[59, 59, w^2 - w - 5],
[61, 61, 5*w^3 - 12*w^2 - 19*w + 17],
[67, 67, 2*w^3 - 5*w^2 - 7*w + 5],
[67, 67, w^3 - 4*w^2 + w + 5],
[67, 67, 3*w^3 - 7*w^2 - 12*w + 11],
[67, 67, -2*w^3 + 4*w^2 + 8*w - 5],
[71, 71, -3*w^3 + 8*w^2 + 9*w - 9],
[71, 71, -w^3 + 3*w^2 + 2*w - 7],
[79, 79, 3*w^3 - 7*w^2 - 14*w + 15],
[81, 3, -3],
[83, 83, -2*w^3 + 6*w^2 + 6*w - 7],
[83, 83, w^3 - w^2 - 6*w - 1],
[89, 89, -4*w^3 + 10*w^2 + 14*w - 13],
[101, 101, -w^3 + w^2 + 8*w + 3],
[107, 107, -w^3 + 3*w^2 + 1],
[125, 5, w^3 - 4*w^2 + 3*w + 1],
[127, 127, -2*w^2 + 6*w + 1],
[127, 127, 2*w - 3],
[131, 131, w^2 - 5*w + 5],
[131, 131, -2*w^3 + 3*w^2 + 9*w - 1],
[137, 137, 4*w^3 - 8*w^2 - 18*w + 5],
[139, 139, -3*w^3 + 8*w^2 + 11*w - 15],
[149, 149, 2*w^3 - 4*w^2 - 10*w + 3],
[157, 157, -2*w^3 + 6*w^2 + 6*w - 11],
[163, 163, -w^3 + 3*w^2 - 3],
[173, 173, -2*w^3 + 4*w^2 + 8*w - 3],
[173, 173, w^3 - 2*w^2 - 5*w - 3],
[179, 179, -w^3 + 4*w^2 + 3*w - 9],
[181, 181, 2*w^3 - 2*w^2 - 14*w - 5],
[193, 193, -2*w + 7],
[193, 193, -2*w^3 + 5*w^2 + 5*w - 5],
[223, 223, 3*w^3 - 9*w^2 - 4*w + 3],
[223, 223, w^3 - 3*w^2 - 4*w + 9],
[233, 233, -7*w^3 + 17*w^2 + 26*w - 25],
[239, 239, w^3 - 3*w^2 - 4*w + 1],
[241, 241, w^2 - 3*w + 3],
[263, 263, 4*w^3 - 10*w^2 - 16*w + 13],
[263, 263, -2*w + 5],
[269, 269, 3*w^3 - 6*w^2 - 13*w + 3],
[269, 269, -4*w^3 + 11*w^2 + 9*w - 7],
[271, 271, 2*w^3 - 5*w^2 - 9*w + 11],
[271, 271, w^3 - w^2 - 8*w + 3],
[277, 277, 2*w^3 - 3*w^2 - 9*w - 1],
[277, 277, w^2 + 3*w - 1],
[277, 277, w^3 - 4*w^2 - w + 9],
[277, 277, w^3 - 4*w^2 + 3*w - 3],
[281, 281, -2*w^2 + 2*w + 3],
[281, 281, w^3 - w^2 - 8*w - 1],
[283, 283, -w^3 + 9*w + 3],
[313, 313, 5*w^3 - 13*w^2 - 20*w + 19],
[313, 313, -4*w - 1],
[317, 317, -3*w^3 + 11*w^2 - 4*w - 3],
[337, 337, 2*w^2 - 4*w - 5],
[337, 337, -2*w^3 + 5*w^2 + 5*w - 3],
[343, 7, 11*w^3 - 28*w^2 - 41*w + 45],
[349, 349, w^3 - 4*w^2 - w + 13],
[349, 349, -4*w^3 + 15*w^2 - 3*w - 7],
[359, 359, w^3 - 3*w^2 - 6*w - 1],
[359, 359, -4*w^3 + 9*w^2 + 17*w - 13],
[367, 367, -6*w^3 + 15*w^2 + 23*w - 21],
[379, 379, -2*w^3 + 5*w^2 + 7*w - 3],
[389, 389, 3*w^3 - 7*w^2 - 10*w + 9],
[397, 397, -w^3 + 3*w^2 + 2*w - 9],
[397, 397, -w^3 + 4*w^2 + w - 5],
[409, 409, w^3 - w^2 - 6*w - 7],
[433, 433, -5*w^3 + 11*w^2 + 20*w - 13],
[439, 439, 4*w^3 - 10*w^2 - 14*w + 11],
[439, 439, -2*w^3 + 7*w^2 - w - 5],
[443, 443, -2*w^3 + 6*w^2 + 4*w - 3],
[443, 443, -9*w^3 + 23*w^2 + 34*w - 37],
[461, 461, 2*w^3 - 6*w^2 - 6*w + 15],
[461, 461, 3*w^3 - 8*w^2 - 13*w + 11],
[463, 463, 4*w^3 - 8*w^2 - 18*w + 7],
[467, 467, 2*w^3 - 6*w^2 - 8*w + 11],
[467, 467, 3*w^3 - 8*w^2 - 9*w + 15],
[479, 479, 2*w^2 - 6*w + 1],
[487, 487, 2*w^2 - 2*w - 9],
[487, 487, w^3 - 7*w - 9],
[491, 491, 3*w^3 - 7*w^2 - 10*w + 1],
[491, 491, -w^3 + w^2 + 6*w - 3],
[499, 499, 2*w^3 - 7*w^2 - w + 9],
[499, 499, -w^3 + 5*w^2 - 17],
[503, 503, -w^3 + 3*w^2 + 6*w - 7],
[521, 521, 2*w^2 - 8*w + 3],
[521, 521, 2*w^2 - 2*w - 5],
[523, 523, 5*w^3 - 10*w^2 - 23*w + 5],
[547, 547, 2*w^3 - 9*w^2 + 5*w + 1],
[557, 557, -w^3 + 2*w^2 + 7*w - 1],
[569, 569, w^3 - 5*w^2 + 6*w + 1],
[571, 571, -6*w^3 + 16*w^2 + 22*w - 27],
[571, 571, -2*w^3 + 5*w^2 + 7*w + 1],
[577, 577, -2*w - 5],
[587, 587, 2*w^3 - 4*w^2 - 6*w - 3],
[593, 593, -8*w^3 + 21*w^2 + 29*w - 33],
[593, 593, -w^3 + 2*w^2 + 5*w - 7],
[601, 601, w^3 - 4*w^2 - 3*w - 1],
[607, 607, -w^3 + 4*w^2 - w - 7],
[613, 613, -w^3 + 5*w^2 - 2*w - 11],
[617, 617, -w^3 + 2*w^2 + 7*w - 3],
[617, 617, -3*w^3 + 8*w^2 + 7*w - 5],
[631, 631, -w^3 + 5*w^2 - 4*w - 7],
[631, 631, -4*w^3 + 11*w^2 + 15*w - 19],
[641, 641, 5*w^3 - 16*w^2 - 3*w + 7],
[641, 641, -6*w^3 + 19*w^2 + 3*w - 5],
[647, 647, -4*w^3 + 9*w^2 + 17*w - 15],
[661, 661, -2*w^3 + 2*w^2 + 12*w + 11],
[673, 673, -2*w^2 - 4*w + 1],
[677, 677, -w^3 + 3*w^2 + 6*w + 3],
[677, 677, -4*w^3 + 7*w^2 + 21*w - 1],
[683, 683, 3*w^3 - 9*w^2 - 6*w + 7],
[683, 683, -3*w^2 + 11*w - 5],
[701, 701, -w^3 - w^2 + 8*w + 5],
[709, 709, -2*w^3 + 4*w^2 + 12*w - 9],
[709, 709, 6*w^3 - 13*w^2 - 23*w + 17],
[709, 709, -3*w^2 + 7*w + 7],
[709, 709, 5*w^3 - 9*w^2 - 24*w + 1],
[719, 719, 4*w^3 - 10*w^2 - 12*w + 5],
[719, 719, w^2 + w - 5],
[733, 733, 2*w^3 - 4*w^2 - 6*w + 3],
[739, 739, 2*w^3 - 5*w^2 - 7*w + 1],
[739, 739, -3*w^3 + 8*w^2 + 9*w - 7],
[743, 743, 4*w^3 - 12*w^2 - 8*w + 15],
[751, 751, w^3 - 5*w^2 + 2*w + 13],
[751, 751, 2*w^3 - 9*w^2 + 7*w + 7],
[757, 757, 4*w^3 - 9*w^2 - 15*w + 11],
[757, 757, -2*w^3 + 4*w^2 + 6*w - 9],
[769, 769, 3*w^3 - 6*w^2 - 15*w + 5],
[769, 769, 3*w^2 - 5*w - 13],
[773, 773, -w^2 + w - 3],
[773, 773, -2*w^3 + 4*w^2 + 4*w - 3],
[809, 809, 2*w^3 - 7*w^2 + w - 1],
[809, 809, 2*w^3 - 3*w^2 - 7*w + 1],
[821, 821, 2*w^3 - 4*w^2 - 6*w + 1],
[821, 821, -2*w^3 + 2*w^2 + 6*w + 1],
[823, 823, 4*w^3 - 11*w^2 - 11*w + 11],
[827, 827, -3*w^3 + 9*w^2 + 2*w + 1],
[841, 29, 5*w^3 - 9*w^2 - 26*w + 5],
[853, 853, w^2 - 5*w + 7],
[853, 853, 3*w^2 - 3*w - 11],
[857, 857, 3*w^3 - 9*w^2 - 8*w + 15],
[857, 857, -w^3 + 2*w^2 + 7*w + 5],
[859, 859, -w^3 + 6*w^2 - 3*w - 21],
[863, 863, w^3 - w^2 - 4*w - 5],
[877, 877, 3*w^3 - 7*w^2 - 10*w + 5],
[881, 881, -w^3 + 11*w + 1],
[907, 907, 2*w^3 - 3*w^2 - 9*w + 3],
[911, 911, 3*w^3 - 7*w^2 - 10*w - 1],
[937, 937, 6*w^3 - 13*w^2 - 25*w + 13],
[967, 967, 3*w^3 - 7*w^2 - 12*w + 5],
[971, 971, w^2 - 5*w - 5],
[977, 977, w^3 + w^2 - 6*w - 5],
[983, 983, -4*w^3 + 7*w^2 + 19*w - 3],
[991, 991, -w^3 + w^2 + 2*w + 3],
[997, 997, w^3 - 6*w^2 + 9*w + 1],
[1019, 1019, -2*w^3 + 6*w^2 + 4*w - 11],
[1031, 1031, w^3 - 6*w^2 + 7*w + 3],
[1049, 1049, -4*w^3 + 10*w^2 + 12*w - 7],
[1051, 1051, 4*w^3 - 13*w^2 + w + 3],
[1051, 1051, w^3 - 2*w^2 - w - 3],
[1063, 1063, 2*w^3 - 7*w^2 - 3*w + 17],
[1069, 1069, -w^3 + 11*w + 3],
[1069, 1069, -w^3 - w^2 + 10*w + 7],
[1087, 1087, 2*w^3 - w^2 - 13*w - 5],
[1091, 1091, w^3 - 4*w^2 - 3*w + 17],
[1091, 1091, 3*w^3 - 6*w^2 - 13*w + 11],
[1093, 1093, -5*w^3 + 10*w^2 + 21*w - 9],
[1097, 1097, -w^3 + 2*w^2 + 3*w - 7],
[1109, 1109, 2*w^3 - 4*w^2 - 10*w - 5],
[1117, 1117, 5*w^3 - 12*w^2 - 21*w + 17],
[1151, 1151, -w^3 + 9*w + 1],
[1151, 1151, 16*w^3 - 40*w^2 - 58*w + 63],
[1171, 1171, w^3 - 4*w^2 - 5*w + 1],
[1187, 1187, 2*w^3 - 6*w^2 - 1],
[1223, 1223, 4*w - 5],
[1229, 1229, 9*w^3 - 17*w^2 - 44*w + 9],
[1231, 1231, 4*w^3 - 12*w^2 + 2*w + 3],
[1231, 1231, 2*w^2 - 7],
[1237, 1237, -4*w^3 + 9*w^2 + 15*w - 7],
[1237, 1237, 3*w^3 - 6*w^2 - 11*w - 1],
[1249, 1249, 3*w^2 - 5*w - 5],
[1249, 1249, 4*w^3 - 8*w^2 - 16*w - 1],
[1249, 1249, -3*w^3 + 5*w^2 + 12*w - 3],
[1249, 1249, w^2 - 3*w - 9],
[1259, 1259, 3*w^3 - 9*w^2 - 10*w + 23],
[1279, 1279, -8*w^3 + 19*w^2 + 31*w - 29],
[1279, 1279, 2*w^2 - 6*w - 7],
[1289, 1289, 2*w^2 - 8*w + 7],
[1289, 1289, w^3 - 2*w^2 - 5*w - 5],
[1291, 1291, w^3 - 7*w^2 + 10*w + 5],
[1291, 1291, -2*w^3 + 6*w^2 + 2*w - 7],
[1297, 1297, -4*w^3 + 6*w^2 + 20*w + 5],
[1297, 1297, 4*w^3 - 10*w^2 - 10*w + 1],
[1303, 1303, 3*w^3 - 6*w^2 - 13*w + 9],
[1307, 1307, -w^3 + 3*w^2 - 7],
[1307, 1307, 4*w^3 - 7*w^2 - 21*w - 3],
[1319, 1319, w^3 + w^2 - 6*w - 9],
[1319, 1319, -7*w^3 + 18*w^2 + 23*w - 21],
[1327, 1327, 3*w^3 - 5*w^2 - 16*w + 3],
[1327, 1327, -w^3 + 3*w^2 - 9],
[1373, 1373, -4*w^3 + 9*w^2 + 15*w - 9],
[1381, 1381, 3*w^3 - 7*w^2 - 14*w + 17],
[1399, 1399, -2*w^3 + 3*w^2 + 11*w - 3],
[1409, 1409, -16*w^3 + 41*w^2 + 61*w - 63],
[1427, 1427, -5*w^3 + 11*w^2 + 24*w - 7],
[1429, 1429, 5*w^3 - 11*w^2 - 20*w + 11],
[1429, 1429, -2*w^3 + 8*w^2 - 6*w - 1],
[1447, 1447, 5*w^3 - 16*w^2 - w + 3],
[1451, 1451, -13*w^3 + 33*w^2 + 48*w - 49],
[1451, 1451, -6*w^3 + 15*w^2 + 21*w - 17],
[1459, 1459, -7*w^3 + 18*w^2 + 21*w - 15],
[1481, 1481, -w^3 + 5*w^2 - 2*w - 9],
[1483, 1483, -4*w^3 + 11*w^2 + 13*w - 13],
[1483, 1483, w^2 + 3*w - 5],
[1483, 1483, w^3 - 4*w^2 + w + 9],
[1483, 1483, 4*w^3 - 11*w^2 - 9*w + 5],
[1487, 1487, -3*w^2 + 5*w + 1],
[1489, 1489, w^3 - 3*w^2 + 4*w + 1],
[1489, 1489, -7*w^3 + 19*w^2 + 22*w - 27],
[1493, 1493, 5*w^3 - 10*w^2 - 21*w + 1],
[1493, 1493, -2*w^3 + 4*w^2 + 10*w - 11],
[1511, 1511, -4*w^3 + 9*w^2 + 17*w - 7],
[1523, 1523, -2*w^3 + 8*w^2 + 2*w - 3],
[1531, 1531, w^3 - 3*w^2 + 2*w - 3],
[1531, 1531, -5*w^3 + 11*w^2 + 22*w - 9],
[1549, 1549, 6*w^3 - 17*w^2 - 19*w + 31],
[1553, 1553, 13*w^3 - 34*w^2 - 47*w + 55],
[1559, 1559, -2*w^2 + 2*w + 13],
[1559, 1559, -w^3 + 11*w - 3],
[1571, 1571, -2*w^3 + 5*w^2 + 3*w + 3],
[1583, 1583, 3*w^3 - 10*w^2 + 3*w + 1],
[1597, 1597, -3*w^3 + 12*w^2 - 3*w - 7],
[1597, 1597, 3*w^3 - 8*w^2 - 3*w + 3],
[1601, 1601, -5*w^3 + 9*w^2 + 26*w - 1],
[1601, 1601, -4*w^3 + 12*w^2 + 2*w - 5],
[1607, 1607, -4*w^3 + 12*w^2 + 6*w - 5],
[1613, 1613, 6*w^3 - 17*w^2 - 19*w + 35],
[1637, 1637, 3*w^3 - 7*w^2 - 8*w + 7],
[1637, 1637, -2*w^3 + 4*w^2 + 12*w - 1],
[1657, 1657, -3*w^3 + 6*w^2 + 11*w - 5],
[1657, 1657, w^3 - 2*w^2 - 7*w + 11],
[1657, 1657, -2*w^3 + 7*w^2 + 3*w - 9],
[1657, 1657, 3*w^3 - 8*w^2 - 5*w + 5],
[1667, 1667, w^3 - w^2 - 4*w - 7],
[1667, 1667, -w^3 + 2*w^2 + 9*w - 9],
[1667, 1667, -w^3 + 6*w^2 - 5*w - 1],
[1667, 1667, 6*w^3 - 14*w^2 - 24*w + 15],
[1669, 1669, 2*w^3 - 16*w - 9],
[1693, 1693, -4*w^3 + 13*w^2 + 7*w - 21],
[1693, 1693, 3*w^3 - 6*w^2 - 9*w + 7],
[1723, 1723, 3*w^3 - 8*w^2 - 7*w + 11],
[1723, 1723, 2*w^3 - 6*w^2 - 8*w + 13],
[1741, 1741, -3*w^2 + 3*w + 5],
[1741, 1741, -w^3 + 5*w^2 - 2*w + 1],
[1747, 1747, -w^3 + 2*w^2 - w - 9],
[1759, 1759, -w^3 + 6*w^2 - 7*w + 1],
[1759, 1759, 5*w^3 - 14*w^2 - 13*w + 19],
[1777, 1777, -9*w^3 + 22*w^2 + 33*w - 31],
[1783, 1783, 4*w^3 - 14*w^2 + 11],
[1783, 1783, 4*w^3 - 12*w^2 - 10*w + 27],
[1787, 1787, 2*w^3 - 9*w^2 + 5*w + 11],
[1787, 1787, -3*w^3 + 4*w^2 + 15*w + 9],
[1789, 1789, w^2 - 5*w - 11],
[1801, 1801, 4*w^3 - 12*w^2 - 10*w + 19],
[1811, 1811, w^3 + 2*w^2 - 7*w - 3],
[1861, 1861, -5*w^3 + 11*w^2 + 24*w - 19],
[1861, 1861, w^3 - 6*w^2 + 3*w + 5],
[1871, 1871, 4*w^3 - 10*w^2 - 16*w + 11],
[1873, 1873, 3*w^2 - 5*w - 11],
[1873, 1873, -4*w^3 + 10*w^2 + 12*w - 9],
[1879, 1879, -4*w^3 + 11*w^2 + 11*w - 17],
[1889, 1889, -w^2 - 5*w + 1],
[1901, 1901, w^3 - 5*w^2 - 2*w + 7],
[1907, 1907, -3*w^3 + 11*w^2 + 2*w - 5],
[1907, 1907, -w^3 + 7*w - 1],
[1907, 1907, -9*w^3 + 19*w^2 + 40*w - 19],
[1907, 1907, -2*w^3 + 3*w^2 + 13*w - 9],
[1913, 1913, w^3 - 2*w^2 - 9*w - 1],
[1913, 1913, 4*w^3 - 13*w^2 - 5*w + 13],
[1931, 1931, -w^3 + 7*w^2 - 6*w - 21],
[1931, 1931, -2*w^3 + 5*w^2 + 3*w - 7],
[1951, 1951, -11*w^3 + 28*w^2 + 41*w - 41],
[1951, 1951, -w^3 + 7*w - 3],
[1979, 1979, 2*w^3 - 16*w - 7],
[1993, 1993, -2*w^2 + 8*w + 1],
[1993, 1993, -6*w - 1],
[1999, 1999, -2*w^3 + 5*w^2 + w - 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [-1, e, 1, e^3 - 5*e, -2*e^2, 4*e^2 - 10, -4*e^3 + 14*e, -6, e^3 - e, 3*e^3 - 9*e, -2*e^2, 2*e^3 - 16*e, -4*e^2 + 10, 2*e^3 - 10*e, -2*e^3 + 6*e, -2*e^3 + 12*e, e^3 - 11*e, 4*e^3 - 18*e, -5*e^3 + 17*e, e^3 + 3*e, -4*e^2 + 6, -5*e^3 + 21*e, 2*e, 2*e^2 - 8, -14, -3*e^3 + 17*e, -2*e^2 + 4, -7*e^3 + 31*e, -5*e^3 + 19*e, 8*e, -5*e^3 + 15*e, -4*e^2 + 2, 4*e^3 - 10*e, -8*e^2 + 26, 4*e^2 - 22, -2*e^3 + 12*e, -14, 4*e^2 - 10, -5*e^3 + 27*e, -2*e^2 - 8, -2*e^2 - 12, 8*e^2 - 26, 5*e^3 - 31*e, -6*e^3 + 20*e, -8*e^2 + 30, 6*e^3 - 26*e, 4*e^2 + 2, 6*e^3 - 20*e, -e^3 + e, 4*e^2 - 14, 4*e^2 - 18, 5*e^3 - 23*e, 5*e^3 - 21*e, 12*e^2 - 34, -14, 2*e^2 + 8, 4*e^2 + 10, -2*e^2, 10*e^2 - 24, -9*e^3 + 51*e, 12*e^2 - 38, -4*e^2 - 14, 4*e^2 - 30, -2*e^2, 4*e^2 - 10, e^3 - 7*e, -4*e^2 + 6, -8*e^2 + 22, -3*e^3 + 27*e, -13*e^3 + 51*e, 11*e^3 - 51*e, 4*e^3 - 12*e, 6*e^2 - 4, -6*e^2 + 4, -8*e^2 + 26, 8*e^2 - 6, -8*e^2 + 22, 2*e^3 - 2*e, -9*e^3 + 53*e, 6*e^3 - 34*e, -15*e^3 + 63*e, 8*e^2 - 10, 18*e^2 - 44, 8*e^3 - 54*e, 8*e^3 - 34*e, 11*e^3 - 35*e, 11*e^3 - 59*e, 9*e^3 - 23*e, 6*e^3 - 22*e, 4*e^3 - 2*e, -9*e^3 + 47*e, -e^3 + 21*e, 17*e^3 - 67*e, -9*e^3 + 41*e, -16*e^2 + 38, 10*e^2 - 12, 11*e^3 - 37*e, -5*e^3 + 21*e, 4*e^2 - 10, -4*e^2 - 2, -9*e^3 + 53*e, -13*e^3 + 41*e, 8*e^2 - 18, -3*e^3 + 5*e, -6*e^2 - 12, -12*e^2 + 18, -12*e^2 + 30, 2*e^3 + 4*e, -8*e^2 + 18, 4*e^2 - 38, 6*e^2 - 36, -10*e^3 + 58*e, 5*e^3 - 37*e, -4*e^2 - 26, 12*e^2 - 30, -16*e^3 + 58*e, -10*e^2 + 28, -8*e^2 + 6, 14, -14, 12*e^3 - 34*e, 8*e^3 - 18*e, -4*e^2 + 50, 12*e^2 - 30, -16*e^2 + 42, -18, 4*e^2 + 30, 2*e^3 - 8*e, -19*e^3 + 71*e, -6, -3*e^3 - 9*e, 7*e^3 - 29*e, 22*e^3 - 86*e, 4*e^3 - 18*e, 6*e^3 - 28*e, 24*e^2 - 58, -8*e^2 + 6, -30, 12*e^2 - 22, -18*e^2 + 48, -4*e^2 + 22, 2*e^2 - 16, 20*e^2 - 54, -6*e^2 + 8, -4*e^2 + 14, 7*e^3 - 27*e, 2*e^3 - 18*e, -10*e^2 - 8, -22, -10*e^2 + 32, 42, -2*e^2 + 52, -4*e^3 + 12*e, -11*e^3 + 37*e, 8*e^2 + 18, -16*e^2 + 50, 16*e^3 - 72*e, 19*e^3 - 93*e, -8*e^2 + 14, 9*e^3 - 37*e, -2*e^3 + 8*e, 14*e^2 - 60, 19*e^3 - 73*e, 8*e^3 - 26*e, -10*e^2 + 28, 16*e^3 - 68*e, -9*e^3 + 45*e, -12*e^2 + 62, -22*e^3 + 84*e, 16*e^3 - 60*e, -4*e^3 - 10*e, 6*e^2 - 40, -10*e^2 + 24, -15*e^3 + 51*e, -5*e^3 + 47*e, 3*e^3 + 7*e, 4*e^2 + 6, 10, -8*e^2 + 46, 14*e^2 - 44, -11*e^3 + 37*e, 6*e^3 - 12*e, 8*e^3 - 56*e, 13*e^3 - 71*e, 27*e^3 - 107*e, 22, -19*e^3 + 59*e, e^3 + 17*e, -54, -8*e^2 - 10, -22, 4*e^2 + 14, 42, 16*e^2 - 50, -20*e^3 + 72*e, 15*e^3 - 75*e, 13*e^3 - 39*e, -4*e^2 + 22, 20*e^2 - 70, 11*e^3 - 67*e, -3*e^3 + 35*e, -6*e^2 - 16, -8*e^2 + 46, -3*e^3 + 5*e, -21*e^3 + 97*e, 10*e^3 - 46*e, -17*e^3 + 65*e, 13*e^3 - 75*e, -3*e^3 - 3*e, -30*e^3 + 110*e, 4*e^2 - 2, 10*e^2 - 20, -11*e^3 + 35*e, -16*e^2 + 50, e^3 - 7*e, 2*e^2 - 12, -8*e^2 + 10, 4*e^3 + 8*e, 19*e^3 - 89*e, -7*e^3 + 9*e, 20*e^3 - 64*e, 12*e^2 - 10, -14*e^3 + 32*e, -20*e^3 + 76*e, -15*e^3 + 83*e, 16*e^3 - 66*e, 22*e^3 - 108*e, 12*e^2 - 38, 4*e^2 - 42, 16*e^2 - 74, 8*e^2 + 22, -8*e^3 + 40*e, 15*e^3 - 63*e, 5*e^3 - 15*e, -8*e^3 + 2*e, -22, -12*e^2 + 26, 22*e^3 - 96*e, -19*e^3 + 85*e, 24*e^3 - 108*e, -e^3 - 7*e, 8*e^2 + 38, 2, -42, -8*e^2 - 26, 25*e^3 - 115*e, 2*e^2 - 32, -24*e^2 + 66, 6*e^2 + 36, 12*e^2 - 62, 32*e^2 - 86, -22*e^2 + 32, 8*e^2 + 10, 19*e^3 - 101*e, 29*e^3 - 125*e, -8*e^3 + 40*e, 11*e^3 - 57*e, 16*e^2 - 58, 62, -32*e^2 + 86, -5*e^3 + 17*e, 26*e^3 - 90*e, 22*e^2 - 64, -18*e^2 + 44, 12*e^3 - 44*e, -2*e^3 + 40*e, 33*e^3 - 119*e, 2*e^2 - 56, 8*e^3 - 18*e, -e^3 + 7*e, -16*e^3 + 72*e, -12*e^3 + 42*e, 4*e^2 - 54, 20*e^2 - 58, -e^3 + 39*e, -4*e^2 + 58, 16*e^2 - 14, -4*e^3 + 28*e, 4*e^2 + 26, -36*e^2 + 90, -8*e^3 + 32*e, 8*e^2 + 10, 4*e^2 - 10, -15*e^3 + 107*e, -e^3 - 11*e, 12*e^3 - 88*e, -e^3 - 31*e, 16*e^2 - 70, 38, -15*e^3 + 57*e, -25*e^3 + 93*e, 4*e^3 - 32*e, 17*e^3 - 89*e, 22*e^3 - 114*e, 14, -6*e^2 - 12, 21*e^3 - 107*e];
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 | {5, 5, -w^3 + 2*w^2 + 5*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;