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

NN := ideal<ZF | {15, 15, -w^2 + 2*w + 5}>;

primesArray := [
[3, 3, w - 1],
[4, 2, -w^3 + 2*w^2 + 4*w - 3],
[5, 5, w],
[7, 7, w - 2],
[11, 11, w^2 - 2*w - 4],
[23, 23, -w^2 + 2*w + 1],
[23, 23, w^3 - 2*w^2 - 3*w + 3],
[27, 3, -w^3 + w^2 + 6*w + 2],
[31, 31, -w^3 + 3*w^2 + w - 1],
[31, 31, -w^3 + 2*w^2 + 4*w - 4],
[37, 37, w^2 - 2*w - 6],
[37, 37, w^3 - 2*w^2 - 3*w + 2],
[43, 43, w^2 - 3*w - 2],
[61, 61, -w^3 + 2*w^2 + 2*w - 2],
[73, 73, w^3 - 3*w^2 - 2*w + 3],
[83, 83, -w - 3],
[89, 89, -w^3 + 3*w^2 + 2*w - 2],
[89, 89, 2*w - 1],
[101, 101, w^3 - 4*w^2 + w + 7],
[101, 101, 2*w^2 - 3*w - 3],
[103, 103, w^3 - 7*w - 4],
[107, 107, 3*w^3 - 2*w^2 - 17*w - 12],
[109, 109, -w^3 + 2*w^2 + 5*w - 2],
[113, 113, w^3 - 3*w^2 - 4*w + 4],
[125, 5, w^3 - 2*w^2 - 5*w + 4],
[139, 139, -w^3 + 2*w^2 + w - 4],
[139, 139, -2*w^2 + 3*w + 7],
[149, 149, -w^3 + 3*w^2 + w - 7],
[149, 149, -2*w^3 + 6*w^2 + 5*w - 16],
[157, 157, -2*w^3 + 6*w^2 + 3*w - 11],
[157, 157, 2*w^3 - 3*w^2 - 10*w + 1],
[163, 163, -w^3 + 4*w^2 + w - 9],
[163, 163, 2*w^3 - 5*w^2 - 6*w + 7],
[167, 167, -w^3 + 4*w^2 - 2*w - 6],
[169, 13, -w^3 + 4*w^2 + w - 8],
[169, 13, 2*w^2 - 3*w - 6],
[173, 173, w^3 - w^2 - 4*w - 4],
[181, 181, 2*w^3 - 4*w^2 - 7*w + 4],
[181, 181, w^2 - 2*w + 2],
[191, 191, 2*w^2 - 4*w - 3],
[197, 197, w^3 - 5*w^2 + 17],
[197, 197, -w^3 + w^2 + 4*w - 2],
[211, 211, 2*w^3 - 4*w^2 - 8*w + 3],
[211, 211, -w^3 + 6*w + 8],
[223, 223, w^3 - 4*w^2 - 2*w + 6],
[227, 227, 2*w^3 - 3*w^2 - 10*w - 2],
[229, 229, -w^3 + 3*w^2 + 3*w - 1],
[233, 233, -w^3 + w^2 + 5*w - 1],
[233, 233, -2*w^3 + 7*w^2 + 4*w - 17],
[257, 257, 2*w^3 - 5*w^2 - 5*w + 4],
[263, 263, w^2 - 6],
[263, 263, 3*w^3 - 9*w^2 - 8*w + 24],
[281, 281, 3*w^3 - 4*w^2 - 14*w - 2],
[281, 281, 2*w^3 - 3*w^2 - 8*w + 1],
[289, 17, 2*w^3 - 4*w^2 - 9*w + 9],
[289, 17, -2*w^3 + 5*w^2 + 4*w - 8],
[293, 293, -w - 4],
[307, 307, 3*w^3 - 3*w^2 - 15*w - 7],
[311, 311, w^2 - 8],
[311, 311, -w^3 + 5*w^2 - 2*w - 7],
[337, 337, w^2 - 4*w - 2],
[337, 337, -w^3 + 3*w^2 - 6],
[343, 7, -2*w^3 + 7*w^2 + 3*w - 16],
[347, 347, w^3 - 2*w^2 - 3*w - 3],
[349, 349, -2*w^3 + 5*w^2 + 2*w - 4],
[353, 353, -2*w^3 + 7*w^2 + 4*w - 14],
[353, 353, -w^3 + 4*w^2 - 14],
[359, 359, -w^3 + 3*w^2 - 2*w - 2],
[367, 367, w^2 - 5*w - 4],
[367, 367, w^2 - w - 8],
[379, 379, 3*w^3 - 2*w^2 - 19*w - 11],
[379, 379, -w^2 - w - 2],
[379, 379, 3*w^3 - 21*w - 19],
[379, 379, -2*w^3 + 3*w^2 + 10*w + 3],
[383, 383, 3*w^3 - 6*w^2 - 11*w + 7],
[383, 383, w^3 - 5*w^2 + w + 7],
[389, 389, w^2 - 4*w - 1],
[397, 397, -w^3 + w^2 + 6*w - 2],
[397, 397, 2*w^3 - 2*w^2 - 11*w - 8],
[401, 401, 3*w^2 - 4*w - 12],
[409, 409, -2*w^3 + 7*w^2 + 4*w - 16],
[409, 409, 3*w - 2],
[419, 419, 2*w^2 - w - 8],
[433, 433, -w^3 + 4*w^2 + w - 14],
[439, 439, w^3 - w^2 - 7*w - 1],
[439, 439, 3*w^3 - 7*w^2 - 9*w + 9],
[443, 443, -2*w^3 + 5*w^2 + 3*w - 4],
[449, 449, -2*w^3 + 5*w^2 + 4*w - 6],
[457, 457, -2*w^3 + 5*w^2 + 8*w - 12],
[461, 461, 2*w^3 - 3*w^2 - 6*w - 1],
[463, 463, 3*w - 1],
[463, 463, 2*w^3 + w^2 - 16*w - 18],
[479, 479, -w^3 + 5*w^2 + w - 9],
[487, 487, -3*w^3 + 8*w^2 + 7*w - 13],
[503, 503, 3*w^2 - 6*w - 11],
[503, 503, -w^3 + 5*w^2 + w - 13],
[509, 509, -w^3 + 2*w^2 + 6*w - 6],
[521, 521, 2*w^3 - 4*w^2 - 9*w + 6],
[523, 523, -w^3 + 5*w^2 - 8],
[523, 523, w^2 - 4*w - 7],
[529, 23, -w^3 + 4*w^2 - w - 9],
[541, 541, -w^3 + w^2 + 5*w - 3],
[541, 541, -w^3 + 6*w^2 - 3*w - 19],
[547, 547, w^3 - 2*w^2 - w - 2],
[547, 547, -w^3 + 5*w^2 - 14],
[557, 557, -2*w^3 + 4*w^2 + 6*w - 3],
[557, 557, w^3 - 3*w^2 - 5*w + 11],
[563, 563, w^3 - 5*w^2 + 2*w + 9],
[563, 563, -w^3 + 4*w^2 + 2*w - 4],
[569, 569, -2*w^3 + w^2 + 12*w + 12],
[571, 571, -w^2 + 3*w - 4],
[577, 577, w^2 - 2*w + 3],
[587, 587, 3*w^2 - 5*w - 8],
[587, 587, w^3 - 5*w^2 + w + 13],
[613, 613, w^3 + w^2 - 7*w - 11],
[617, 617, w^3 - 9*w - 2],
[643, 643, 2*w^2 - 5*w - 8],
[661, 661, -w^3 + 2*w^2 + 6*w - 2],
[661, 661, -2*w^3 + 6*w^2 + w - 9],
[673, 673, -2*w^3 + 5*w^2 + 8*w - 9],
[677, 677, w^3 - 5*w - 7],
[691, 691, 3*w^3 - 7*w^2 - 9*w + 11],
[701, 701, 3*w^2 - 4*w - 7],
[709, 709, -2*w^2 + 3*w + 12],
[709, 709, -w^3 + 3*w^2 - 9],
[719, 719, -2*w^2 + 6*w - 3],
[727, 727, w^3 - 3*w^2 - 2*w - 1],
[751, 751, -2*w^3 + 3*w^2 + 8*w - 4],
[757, 757, -2*w^3 + 5*w^2 + 7*w - 6],
[757, 757, w^3 - w^2 - 2*w - 3],
[757, 757, -w^3 - w^2 + 10*w + 11],
[757, 757, 6*w^3 - 14*w^2 - 20*w + 23],
[761, 761, -4*w^3 + 10*w^2 + 13*w - 17],
[769, 769, w^3 - 2*w^2 - w - 3],
[769, 769, 5*w^3 - 8*w^2 - 22*w + 2],
[773, 773, -w^3 + 5*w^2 - 3*w - 11],
[787, 787, -2*w^3 + 8*w^2 - w - 13],
[787, 787, 2*w^3 - 2*w^2 - 9*w - 8],
[787, 787, 3*w^3 - 5*w^2 - 12*w + 3],
[787, 787, 4*w^3 - 5*w^2 - 21*w - 4],
[797, 797, 2*w^2 - w - 9],
[797, 797, -3*w^3 + 7*w^2 + 10*w - 16],
[821, 821, 4*w^3 - 3*w^2 - 24*w - 17],
[821, 821, 2*w^3 - w^2 - 14*w - 6],
[823, 823, 3*w^3 - 5*w^2 - 12*w + 1],
[827, 827, -w^3 + 5*w^2 - 9],
[827, 827, -2*w^3 + 6*w^2 + 2*w - 11],
[827, 827, -2*w^3 + 7*w^2 + 6*w - 13],
[827, 827, w^3 - w^2 - 8*w - 3],
[853, 853, -w^3 + 2*w^2 + 3*w - 8],
[857, 857, -w^3 - 2*w^2 + 11*w + 17],
[857, 857, 3*w^3 - w^2 - 18*w - 16],
[859, 859, -w^3 + w^2 + 7*w + 7],
[859, 859, 5*w^3 - 5*w^2 - 26*w - 12],
[859, 859, 2*w^3 - 4*w^2 - 4*w - 1],
[859, 859, -3*w^2 + 4*w + 3],
[863, 863, w^3 - 3*w^2 - 6*w + 3],
[877, 877, w^3 - 2*w^2 - 3*w - 4],
[877, 877, -w^3 + 5*w^2 - 2*w - 18],
[883, 883, 3*w^3 - 7*w^2 - 7*w + 13],
[883, 883, -4*w^3 + 9*w^2 + 14*w - 12],
[919, 919, -w^3 + 4*w^2 + 3*w - 2],
[919, 919, -w^3 + 3*w^2 + 2*w - 11],
[929, 929, 2*w^3 - 4*w^2 - 9*w + 1],
[937, 937, 3*w^3 - 9*w^2 - 4*w + 9],
[953, 953, -3*w^3 + 9*w^2 + 6*w - 14],
[961, 31, -3*w^3 + 6*w^2 + 11*w - 3],
[967, 967, -w^3 + 5*w^2 - w - 11],
[967, 967, 3*w^3 - 6*w^2 - 9*w + 1],
[977, 977, 2*w^3 - 6*w^2 - 3*w + 18],
[977, 977, 7*w^3 - 16*w^2 - 24*w + 26],
[991, 991, 3*w^2 - 3*w - 8],
[997, 997, 3*w^3 - 5*w^2 - 13*w - 1],
[1009, 1009, 3*w^3 - 12*w^2 - 3*w + 34],
[1013, 1013, -2*w^3 + 8*w^2 + w - 12],
[1013, 1013, -2*w^3 + 3*w^2 + 6*w - 2],
[1049, 1049, 3*w^3 - 5*w^2 - 14*w + 6],
[1069, 1069, w^3 + w^2 - 6*w - 3],
[1069, 1069, 2*w^3 - 8*w^2 - 2*w + 21],
[1087, 1087, 4*w^3 - 5*w^2 - 20*w - 8],
[1091, 1091, -2*w^3 + 8*w^2 - w - 16],
[1103, 1103, -3*w^3 + 6*w^2 + 13*w - 9],
[1117, 1117, 4*w^2 - 5*w - 16],
[1117, 1117, 6*w^3 - 15*w^2 - 18*w + 26],
[1123, 1123, -2*w^3 + 7*w^2 + 2*w - 6],
[1123, 1123, -3*w^3 + 7*w^2 + 5*w - 11],
[1129, 1129, -2*w^2 + 3*w - 2],
[1129, 1129, 5*w^3 - 9*w^2 - 21*w + 9],
[1129, 1129, 3*w^3 - 4*w^2 - 16*w - 2],
[1129, 1129, w^3 + 2*w^2 - 10*w - 12],
[1151, 1151, 2*w^3 + 2*w^2 - 17*w - 21],
[1153, 1153, 6*w^3 - 11*w^2 - 24*w + 7],
[1163, 1163, 3*w^3 - 7*w^2 - 8*w + 7],
[1163, 1163, -3*w^3 + 10*w^2 + 7*w - 28],
[1171, 1171, -2*w^3 + 4*w^2 + 9*w - 12],
[1187, 1187, -3*w^3 + 10*w^2 + 4*w - 22],
[1187, 1187, 5*w^3 - 9*w^2 - 22*w + 9],
[1193, 1193, 3*w^3 - 6*w^2 - 10*w + 8],
[1193, 1193, 3*w^2 - 3*w - 14],
[1193, 1193, 4*w - 3],
[1193, 1193, 3*w^3 - 10*w^2 - 7*w + 27],
[1201, 1201, 4*w^3 - 9*w^2 - 15*w + 18],
[1213, 1213, -w^3 + 4*w^2 - 4*w - 4],
[1213, 1213, w^3 - 9*w + 1],
[1217, 1217, w^3 - w^2 - w - 3],
[1217, 1217, -3*w^3 + 20*w + 18],
[1223, 1223, 6*w^3 - 11*w^2 - 24*w + 6],
[1223, 1223, 3*w^2 - 6*w - 13],
[1229, 1229, 3*w^3 - 6*w^2 - 13*w + 8],
[1231, 1231, -w^3 + 4*w^2 - w - 12],
[1259, 1259, -2*w^3 + 6*w^2 + 5*w - 8],
[1259, 1259, 3*w^2 - 6*w - 4],
[1277, 1277, -4*w^3 + 11*w^2 + 8*w - 19],
[1279, 1279, -3*w^3 + 5*w^2 + 11*w + 1],
[1283, 1283, -4*w^3 + 9*w^2 + 13*w - 14],
[1291, 1291, -2*w^3 + 6*w^2 + 6*w - 9],
[1291, 1291, -2*w^3 - 2*w^2 + 18*w + 21],
[1297, 1297, 3*w^3 - 8*w^2 - 5*w + 8],
[1319, 1319, -w^3 + 6*w^2 - 3*w - 9],
[1321, 1321, -4*w^3 + 9*w^2 + 15*w - 12],
[1327, 1327, w^2 + 2*w - 7],
[1331, 11, -2*w^3 + 5*w^2 + 2*w - 9],
[1367, 1367, -w^3 + 7*w^2 - 6*w - 11],
[1367, 1367, -w^3 + w^2 + 8*w - 3],
[1369, 37, -2*w^3 + 6*w^2 + 7*w - 9],
[1373, 1373, -2*w^3 + 6*w^2 + 2*w - 1],
[1409, 1409, 5*w^3 - 16*w^2 - 11*w + 38],
[1423, 1423, -2*w^3 + 3*w^2 + 11*w - 2],
[1433, 1433, -2*w^3 + w^2 + 9*w + 2],
[1439, 1439, w^3 - 3*w^2 - 6*w + 4],
[1439, 1439, -3*w^3 + 6*w^2 + 12*w - 4],
[1447, 1447, 4*w^3 - 13*w^2 - 7*w + 24],
[1453, 1453, -w^3 + 5*w^2 - 2*w - 16],
[1459, 1459, 3*w^3 - 7*w^2 - 4*w + 4],
[1471, 1471, -w^3 + 4*w^2 - 3*w - 8],
[1487, 1487, -2*w^2 + w + 14],
[1489, 1489, -3*w^3 + 7*w^2 + 5*w - 13],
[1489, 1489, -4*w^2 + 6*w + 21],
[1493, 1493, -2*w^3 + 7*w^2 + 2*w - 3],
[1499, 1499, 6*w^3 - 14*w^2 - 20*w + 21],
[1531, 1531, 3*w^3 - 7*w^2 - 8*w + 8],
[1531, 1531, 2*w^2 + w - 8],
[1531, 1531, -3*w^3 + 7*w^2 + 8*w - 11],
[1531, 1531, 2*w^3 - 3*w^2 - 6*w - 4],
[1553, 1553, -w^3 + 6*w^2 - 4*w - 12],
[1567, 1567, 4*w^3 - 5*w^2 - 20*w - 2],
[1571, 1571, 4*w^3 - 8*w^2 - 13*w + 13],
[1571, 1571, 5*w^3 - 5*w^2 - 25*w - 9],
[1579, 1579, -2*w^3 + 6*w^2 - 9],
[1579, 1579, -3*w^3 + 8*w^2 + 5*w - 9],
[1583, 1583, 4*w^2 - 4*w - 17],
[1597, 1597, 3*w^3 - 6*w^2 - 9*w + 2],
[1601, 1601, 2*w^2 - w - 11],
[1609, 1609, 4*w^3 - 5*w^2 - 18*w - 7],
[1613, 1613, -w^3 + 6*w^2 - 3*w - 16],
[1613, 1613, -2*w^3 + 8*w^2 - 13],
[1619, 1619, w^3 - 2*w^2 - 7*w + 1],
[1621, 1621, -w^3 + w^2 + 2*w + 12],
[1621, 1621, 6*w^3 - 11*w^2 - 25*w + 8],
[1637, 1637, 3*w^3 - 5*w^2 - 14*w - 1],
[1681, 41, -2*w^3 + 5*w^2 + 9*w - 4],
[1681, 41, w^3 - 4*w^2 + 5*w + 6],
[1693, 1693, 3*w^3 - 3*w^2 - 14*w - 2],
[1697, 1697, 2*w^3 + w^2 - 16*w - 21],
[1697, 1697, w^3 + w^2 - 8*w - 4],
[1721, 1721, -3*w^3 + 9*w^2 + 6*w - 13],
[1723, 1723, 2*w^3 - 2*w^2 - 12*w + 1],
[1723, 1723, 5*w^2 - 7*w - 26],
[1741, 1741, -2*w^3 + 2*w^2 + 9*w - 1],
[1753, 1753, -w^3 + 4*w^2 + 5*w - 3],
[1753, 1753, 3*w^3 + 2*w^2 - 24*w - 28],
[1759, 1759, -w^3 + 7*w + 11],
[1777, 1777, 2*w^3 - w^2 - 13*w - 4],
[1777, 1777, -2*w^3 + 6*w^2 + 5*w - 7],
[1787, 1787, -w^3 + 2*w^2 + 3*w - 9],
[1801, 1801, -2*w^3 + 3*w^2 + 8*w + 7],
[1861, 1861, -2*w^3 - w^2 + 14*w + 14],
[1867, 1867, -w^3 + w^2 + 8*w - 1],
[1871, 1871, -w^3 + 3*w^2 + 2*w - 12],
[1873, 1873, 3*w^3 - 8*w^2 - 9*w + 22],
[1877, 1877, -3*w^3 + 7*w^2 + 13*w - 7],
[1877, 1877, 3*w^3 - w^2 - 19*w - 11],
[1877, 1877, -7*w^3 + 17*w^2 + 23*w - 29],
[1877, 1877, 5*w^2 - 7*w - 8],
[1879, 1879, 4*w^3 - 8*w^2 - 16*w + 7],
[1931, 1931, -2*w^3 + 6*w^2 + 9*w - 9],
[1933, 1933, 5*w^3 - 13*w^2 - 14*w + 21],
[1951, 1951, -w^2 + w - 4],
[1973, 1973, w^3 - 6*w - 12],
[1973, 1973, 3*w^3 - 11*w^2 + 19],
[1993, 1993, -5*w^3 + 12*w^2 + 17*w - 19],
[1993, 1993, -6*w^3 + 15*w^2 + 19*w - 26],
[1999, 1999, -3*w^3 + 7*w^2 + 4*w - 9],
[1999, 1999, 3*w^3 - 8*w^2 - 11*w + 17]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - x^3 - 9*x^2 + 9*x - 2;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, 1, 2, 4*e^3 - 2*e^2 - 36*e + 18, -2*e^3 + e^2 + 19*e - 14, -e^3 + 11*e - 2, -e^3 + 7*e - 2, -5*e^3 + 3*e^2 + 48*e - 24, e^3 - 7*e + 2, 4*e^3 - 2*e^2 - 38*e + 18, 2*e^3 - 3*e^2 - 19*e + 18, -2*e^3 + e^2 + 17*e - 2, -e^3 + 11*e, -3*e^3 + 27*e - 10, 2*e^3 - 18*e + 14, -2*e^3 + e^2 + 21*e - 18, 2*e^3 - 20*e, -8*e^3 + 4*e^2 + 74*e - 34, 6*e^3 - 4*e^2 - 60*e + 34, -3*e^3 + 25*e - 6, -3*e^3 + 3*e^2 + 28*e - 14, 6*e^3 - e^2 - 55*e + 20, -3*e^3 + 27*e - 6, -2*e^2 + 2*e + 4, 4*e^3 - 2*e^2 - 34*e + 16, 2*e^3 - 20*e + 8, e^3 - 4*e^2 - 9*e + 20, 6, 3*e^3 - 3*e^2 - 30*e + 26, e^3 - 2*e^2 - 11*e + 18, -5*e^3 + 4*e^2 + 47*e - 20, -14*e^3 + 8*e^2 + 128*e - 60, 2*e^3 - 26*e, -14*e^3 + 8*e^2 + 134*e - 66, 9*e^3 - 3*e^2 - 84*e + 38, 8*e^3 - 4*e^2 - 80*e + 34, -9*e^3 + 2*e^2 + 79*e - 26, 2*e^3 - 4*e^2 - 16*e + 22, -17*e^3 + 8*e^2 + 155*e - 74, 10*e^3 - 4*e^2 - 90*e + 50, -7*e^3 + 4*e^2 + 63*e - 38, -e^3 + 2*e^2 + 9*e - 22, 12*e^3 - 6*e^2 - 114*e + 62, 2*e^3 - 22*e, 13*e^3 - 4*e^2 - 123*e + 50, 2*e^2 - 6, -11*e^3 + 6*e^2 + 99*e - 44, -4*e^3 + 34*e - 6, 18*e^3 - 10*e^2 - 166*e + 84, 22*e^3 - 12*e^2 - 200*e + 98, -18*e^3 + 8*e^2 + 160*e - 70, -20*e^3 + 12*e^2 + 180*e - 92, 9*e^3 - 2*e^2 - 89*e + 24, 2*e^3 - 2*e^2 - 24*e + 12, -6*e^3 + 6*e^2 + 60*e - 46, -4*e^3 + 4*e^2 + 38*e - 18, 15*e^3 - 6*e^2 - 131*e + 54, 22*e^3 - 12*e^2 - 196*e + 96, -21*e^3 + 11*e^2 + 188*e - 90, 11*e^3 - 4*e^2 - 105*e + 44, -16*e^3 + 10*e^2 + 148*e - 86, -10*e^3 + 2*e^2 + 98*e - 26, -17*e^3 + 9*e^2 + 160*e - 76, -6*e^3 + 6*e^2 + 54*e - 40, 3*e^3 - 4*e^2 - 31*e + 24, -3*e^3 - e^2 + 28*e + 2, -3*e^3 + 4*e^2 + 29*e - 14, 5*e^2 + 3*e - 30, -6*e^3 + 6*e^2 + 54*e - 46, 6*e^3 - e^2 - 53*e + 22, 12*e^3 - 6*e^2 - 114*e + 68, -7*e^3 + 3*e^2 + 70*e - 28, 4*e^3 - 3*e^2 - 31*e + 26, -3*e^3 - 2*e^2 + 27*e + 14, -4*e^3 + 4*e^2 + 34*e - 22, -5*e^3 + 6*e^2 + 43*e - 46, -2*e^3 + 26*e + 2, 19*e^3 - 11*e^2 - 174*e + 86, -6*e^3 + 54*e - 18, 20*e^3 - 10*e^2 - 176*e + 78, 16*e^3 - 11*e^2 - 149*e + 84, 8*e^3 - 8*e^2 - 82*e + 66, -20*e^3 + 10*e^2 + 182*e - 86, -2*e^3 - 2*e^2 + 22*e + 12, 4*e^3 - 46*e - 4, -11*e^3 + 4*e^2 + 97*e - 50, -20*e^3 + 14*e^2 + 186*e - 100, -9*e^3 + 83*e - 32, 13*e^3 - 6*e^2 - 117*e + 76, -18*e^3 + 11*e^2 + 165*e - 90, -3*e^3 + 4*e^2 + 23*e - 30, 7*e^3 - 5*e^2 - 66*e + 30, 10*e^3 - 9*e^2 - 89*e + 58, -6*e^3 + 4*e^2 + 52*e - 30, 7*e^3 - 7*e^2 - 68*e + 36, e^3 - 2*e^2 - 9*e, -20*e^3 + 13*e^2 + 187*e - 98, 14*e^3 - 10*e^2 - 126*e + 74, -10*e^3 + 2*e^2 + 84*e - 16, 15*e^3 - 5*e^2 - 142*e + 78, 24*e^3 - 8*e^2 - 222*e + 100, -26*e^3 + 14*e^2 + 240*e - 110, 14*e^3 - 10*e^2 - 124*e + 82, -5*e^3 + 5*e^2 + 48*e - 14, 12*e^3 - 9*e^2 - 111*e + 72, -8*e^3 + 6*e^2 + 66*e - 38, -12*e^3 + 110*e - 40, 8*e^3 - 3*e^2 - 75*e + 50, 16*e^3 - 4*e^2 - 148*e + 46, -10*e^3 + 6*e^2 + 98*e - 66, 26*e^3 - 14*e^2 - 248*e + 130, -12*e^3 + 2*e^2 + 110*e - 36, 7*e^3 - 2*e^2 - 55*e + 20, -18*e^3 + 10*e^2 + 168*e - 62, 28*e^3 - 18*e^2 - 260*e + 134, 29*e^3 - 15*e^2 - 276*e + 136, 17*e^3 - 4*e^2 - 161*e + 66, -24*e^3 + 10*e^2 + 214*e - 94, 4*e^3 - 4*e^2 - 38*e + 62, -2*e^3 + 2*e^2 + 28*e - 18, -2*e^3 + 6*e - 12, 33*e^3 - 17*e^2 - 304*e + 142, 12*e^3 - 2*e^2 - 104*e + 38, -5*e^3 + e^2 + 42*e + 2, 30*e^3 - 15*e^2 - 281*e + 142, 7*e^3 - 3*e^2 - 54*e + 30, 4*e^3 - 3*e^2 - 37*e + 2, -6*e^3 + 6*e^2 + 54*e - 16, -14*e^3 + 10*e^2 + 120*e - 70, 20*e^3 - 10*e^2 - 188*e + 90, -26*e^3 + 15*e^2 + 247*e - 104, -8*e^3 + 6*e^2 + 76*e - 40, -4*e^3 - 2*e^2 + 42*e + 18, -22*e^3 + 7*e^2 + 195*e - 78, -6*e^3 + 54*e - 42, -16*e^3 + 6*e^2 + 150*e - 80, 12*e^3 + 2*e^2 - 110*e + 16, 8*e^3 - 6*e^2 - 72*e + 28, 14*e^3 - 8*e^2 - 122*e + 88, 5*e^3 - 3*e^2 - 46*e + 22, -7*e^3 + 6*e^2 + 59*e - 56, -15*e^3 + 9*e^2 + 150*e - 78, -31*e^3 + 16*e^2 + 271*e - 142, 12*e^3 - e^2 - 101*e + 10, -21*e^3 + 8*e^2 + 195*e - 104, 8*e^3 - 4*e^2 - 82*e + 38, -5*e^3 + 8*e^2 + 39*e - 46, -2*e^2 - 2*e, -12*e^3 + 8*e^2 + 116*e - 46, 13*e^3 - 8*e^2 - 129*e + 84, 11*e^3 - 7*e^2 - 98*e + 58, -4*e^3 - 4*e^2 + 36*e + 8, -4*e^3 + 2*e^2 + 36*e - 16, 21*e^3 - 16*e^2 - 193*e + 122, -22*e^3 + 9*e^2 + 199*e - 94, 9*e^3 - 2*e^2 - 77*e + 6, -5*e^3 - e^2 + 40*e + 26, -19*e^3 + 12*e^2 + 179*e - 96, 13*e^3 - 4*e^2 - 107*e + 38, -2*e^3 - 2*e^2 + 8*e + 28, -15*e^3 + 133*e - 30, -15*e^3 + 6*e^2 + 141*e - 46, 12*e^3 - 7*e^2 - 105*e + 52, -20*e^3 + 10*e^2 + 198*e - 88, -10*e - 10, 12*e^3 - 6*e^2 - 94*e + 58, -13*e^3 + 4*e^2 + 125*e - 58, 8*e^3 + 2*e^2 - 70*e + 4, 13*e^3 - 121*e + 24, 12*e^3 - 8*e^2 - 102*e + 50, 29*e^3 - 13*e^2 - 266*e + 114, -4*e^3 + 5*e^2 + 49*e - 48, -11*e^3 + 5*e^2 + 94*e - 46, 21*e^3 - 9*e^2 - 198*e + 90, 6*e^3 + e^2 - 59*e + 24, -3*e^3 + 8*e^2 + 21*e - 56, -26*e^3 + 19*e^2 + 245*e - 134, 3*e^3 - 3*e^2 - 20*e + 42, 2*e^3 + 2*e^2 - 32*e - 24, 8*e^3 - 6*e^2 - 58*e + 40, 13*e^3 - 10*e^2 - 129*e + 98, -38*e^3 + 20*e^2 + 350*e - 162, -34*e^3 + 15*e^2 + 307*e - 142, -12*e^3 + 10*e^2 + 118*e - 64, -16*e^3 + 6*e^2 + 152*e - 60, 23*e^3 - 11*e^2 - 214*e + 98, 31*e^3 - 14*e^2 - 267*e + 116, -16*e^3 + 5*e^2 + 153*e - 76, 7*e^3 - 59*e + 16, 6*e^3 - 6*e^2 - 54*e + 30, 36*e^3 - 18*e^2 - 320*e + 144, 4*e^3 - 8*e^2 - 28*e + 68, 8*e^3 - 10*e^2 - 74*e + 64, -6*e^3 + 12*e^2 + 62*e - 68, 8*e^3 - 10*e^2 - 82*e + 80, 14*e^3 - 10*e^2 - 136*e + 68, -12*e^3 + 11*e^2 + 115*e - 82, 7*e^3 - 7*e^2 - 70*e + 58, -12*e^3 + 12*e^2 + 112*e - 104, -2*e^3 + 12*e^2 + 10*e - 70, e^3 + 6*e^2 - 17*e - 44, 10*e^3 - 4*e^2 - 102*e + 28, -43*e^3 + 28*e^2 + 403*e - 206, -6*e^2 - 2*e + 34, -36*e^3 + 18*e^2 + 322*e - 170, -13*e^3 + 12*e^2 + 119*e - 50, -29*e^3 + 12*e^2 + 273*e - 146, -22*e^3 + 6*e^2 + 194*e - 80, -2*e^3 + 2*e^2 + 14*e + 18, -23*e^3 + 16*e^2 + 225*e - 138, -27*e^3 + 10*e^2 + 239*e - 86, -5*e^3 + 2*e^2 + 61*e, 15*e^3 - 12*e^2 - 137*e + 66, -4*e^3 + 3*e^2 + 43*e - 66, -20*e^3 + 6*e^2 + 172*e - 56, -12*e^3 + 10*e^2 + 100*e - 58, 20*e^3 - 12*e^2 - 178*e + 108, -5*e^3 + 3*e^2 + 42*e + 4, 10*e^3 - 8*e^2 - 110*e + 78, -32*e^3 + 12*e^2 + 282*e - 108, -2*e^3 + 38*e, -6*e^3 - 2*e^2 + 54*e + 14, -13*e^3 + 12*e^2 + 107*e - 98, -17*e^3 + 12*e^2 + 141*e - 90, 10*e^3 - 6*e^2 - 78*e + 70, -36*e^3 + 16*e^2 + 324*e - 154, -4*e^3 + 9*e^2 + 43*e - 46, e^2 + 5*e - 38, 25*e^3 - 18*e^2 - 225*e + 142, 14*e^3 - 6*e^2 - 128*e + 36, -16*e^3 + 16*e^2 + 166*e - 134, -8*e^3 + 2*e^2 + 86*e - 42, 30*e^3 - 15*e^2 - 283*e + 154, -36*e^3 + 18*e^2 + 316*e - 180, -17*e^3 + 4*e^2 + 167*e - 40, -35*e^3 + 20*e^2 + 303*e - 176, -53*e^3 + 33*e^2 + 488*e - 238, 6*e^3 + 2*e^2 - 54*e - 14, -14*e^3 + 5*e^2 + 135*e - 58, 13*e^3 - 2*e^2 - 129*e + 26, -38*e^3 + 16*e^2 + 346*e - 180, -22*e^3 + 4*e^2 + 208*e - 92, -11*e^3 + 5*e^2 + 104*e - 78, -e^3 + 7*e^2 + 6*e - 14, -13*e^3 + 8*e^2 + 119*e - 50, 19*e^3 - 4*e^2 - 177*e + 50, -22*e^3 + 6*e^2 + 188*e - 68, 11*e^3 + 6*e^2 - 107*e - 4, 18*e^3 - 3*e^2 - 155*e + 18, 2*e^2 - 4*e - 42, -32*e^3 + 18*e^2 + 310*e - 146, 32*e^3 - 12*e^2 - 304*e + 118, 22*e^3 - 6*e^2 - 210*e + 54, -11*e^3 + 6*e^2 + 105*e - 44, 11*e^3 - 6*e^2 - 101*e + 72, 17*e^3 - 12*e^2 - 175*e + 106, -37*e^3 + 20*e^2 + 329*e - 140, 25*e^3 - 14*e^2 - 243*e + 86, -25*e^3 + 16*e^2 + 219*e - 128, -14*e^3 + 12*e^2 + 146*e - 94, -8*e^3 + 74*e - 40, -10*e^3 + 7*e^2 + 91*e - 32, 14*e^3 - 8*e^2 - 122*e + 14, -18*e^3 + 14*e^2 + 160*e - 82, 32*e^3 - 12*e^2 - 312*e + 122, 14*e^3 - 6*e^2 - 120*e + 4, 32*e^3 - 8*e^2 - 284*e + 112, -44*e^3 + 20*e^2 + 392*e - 180, 18*e^3 - 13*e^2 - 167*e + 52, 3*e^3 + e^2 - 26*e + 14, 14*e^3 - 12*e^2 - 148*e + 102, 9*e^3 - 8*e^2 - 81*e + 58, 26*e^3 - 13*e^2 - 229*e + 106, -26*e^3 + 16*e^2 + 234*e - 90, -8*e^3 + 18*e^2 + 80*e - 106, 21*e^3 - 15*e^2 - 188*e + 102, -28*e^3 + 24*e^2 + 264*e - 176, 42*e^3 - 22*e^2 - 396*e + 208, -36*e^3 + 14*e^2 + 330*e - 138, -13*e^3 + 6*e^2 + 125*e - 68, 35*e^3 - 22*e^2 - 327*e + 168, -56*e^3 + 32*e^2 + 536*e - 266, 14*e^3 - 4*e^2 - 140*e + 64, -36*e^3 + 16*e^2 + 352*e - 160, -9*e^3 + 4*e^2 + 95*e - 38, 10*e^3 - 4*e^2 - 76*e + 48, -2*e^3 + 6*e^2 + 32*e - 28, -16*e^3 + 10*e^2 + 126*e - 86, -8*e^3 + 4*e^2 + 72*e - 90, 31*e^3 - 14*e^2 - 279*e + 152, -11*e^3 + 14*e^2 + 103*e - 112, 43*e^3 - 14*e^2 - 395*e + 162, 34*e^3 - 16*e^2 - 314*e + 164];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w - 1}>] := -1;
ALEigenvalues[ideal<ZF | {5, 5, w}>] := -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;