/*
  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 | {9, 9, w^3 - 3*w^2 - 2*w + 7}>;

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^6 - 20*x^4 + 104*x^2 - 144;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, e, -1/24*e^5 + 5/6*e^3 - 10/3*e, -1/4*e^4 + 4*e^2 - 10, -1/4*e^4 + 4*e^2 - 10, -1/6*e^5 + 17/6*e^3 - 28/3*e, -1/12*e^5 + 7/6*e^3 - 2/3*e, -1/12*e^5 + 5/3*e^3 - 20/3*e, 1/2*e^4 - 7*e^2 + 16, -1/4*e^5 + 9/2*e^3 - 16*e, -1/4*e^4 + 2*e^2 + 8, 7/24*e^5 - 29/6*e^3 + 43/3*e, 1/12*e^5 - 2/3*e^3 - 10/3*e, -3/8*e^5 + 6*e^3 - 16*e, 1/24*e^5 - 5/6*e^3 + 19/3*e, -3/4*e^4 + 12*e^2 - 34, 11/24*e^5 - 23/3*e^3 + 65/3*e, -5/24*e^5 + 8/3*e^3 - 11/3*e, -3/4*e^4 + 12*e^2 - 36, -7/24*e^5 + 13/3*e^3 - 34/3*e, -1/3*e^5 + 37/6*e^3 - 74/3*e, -1/4*e^4 + e^2 + 10, 3/8*e^5 - 13/2*e^3 + 20*e, 3/8*e^5 - 6*e^3 + 17*e, -5/24*e^5 + 11/3*e^3 - 47/3*e, e^2 - 8, -e^2 + 12, 1/2*e^4 - 9*e^2 + 32, 7/4*e^4 - 28*e^2 + 76, e^2, 11/24*e^5 - 23/3*e^3 + 89/3*e, -5/4*e^4 + 21*e^2 - 58, -e^5 + 16*e^3 - 44*e, 3/2*e^4 - 22*e^2 + 48, -1/4*e^4 + 2*e^2 + 4, 1/4*e^4 - 6*e^2 + 16, -1/4*e^4 + 7*e^2 - 24, 17/24*e^5 - 73/6*e^3 + 107/3*e, -1/2*e^4 + 5*e^2 + 2, 3/4*e^5 - 11*e^3 + 26*e, 3/4*e^4 - 14*e^2 + 44, 5/8*e^5 - 19/2*e^3 + 21*e, 3/4*e^5 - 12*e^3 + 32*e, 3/4*e^4 - 11*e^2 + 38, -5/6*e^5 + 79/6*e^3 - 122/3*e, 2/3*e^5 - 65/6*e^3 + 82/3*e, 17/24*e^5 - 32/3*e^3 + 68/3*e, 11/24*e^5 - 20/3*e^3 + 38/3*e, 3/4*e^4 - 14*e^2 + 44, 1/24*e^5 - 11/6*e^3 + 37/3*e, -5/4*e^4 + 21*e^2 - 58, -7/4*e^4 + 25*e^2 - 54, -17/24*e^5 + 61/6*e^3 - 59/3*e, -3/8*e^5 + 11/2*e^3 - 12*e, 5/24*e^5 - 8/3*e^3 - 1/3*e, 3/8*e^5 - 9/2*e^3, 1/2*e^4 - 7*e^2 + 18, 3/4*e^5 - 12*e^3 + 30*e, 3/2*e^4 - 23*e^2 + 68, -5/4*e^4 + 19*e^2 - 70, -7/24*e^5 + 10/3*e^3 + 20/3*e, -1/2*e^4 + 7*e^2 - 4, 1/4*e^4 - 4*e^2 + 26, 3/2*e^4 - 27*e^2 + 84, -3/8*e^5 + 11/2*e^3 - 7*e, 1/4*e^4 - 8*e^2 + 40, -e^4 + 18*e^2 - 66, -2/3*e^5 + 34/3*e^3 - 94/3*e, -1/2*e^3 + 14*e, -1/4*e^4 + 4*e^2 - 18, 1/3*e^5 - 37/6*e^3 + 44/3*e, 1/2*e^4 - 8*e^2 + 20, e^4 - 17*e^2 + 44, 1/2*e^3, 5/12*e^5 - 41/6*e^3 + 76/3*e, -1/4*e^4 + 6*e^2 - 6, -5/8*e^5 + 9*e^3 - 19*e, -5/24*e^5 + 11/3*e^3 - 50/3*e, -3/4*e^4 + 16*e^2 - 56, -3/4*e^4 + 14*e^2 - 28, 1/4*e^4 - 7*e^2 + 20, 5/8*e^5 - 12*e^3 + 46*e, 9/4*e^4 - 32*e^2 + 78, 3/2*e^4 - 28*e^2 + 76, -2/3*e^5 + 28/3*e^3 - 52/3*e, 1/4*e^5 - 9/2*e^3 + 6*e, -1/3*e^5 + 19/6*e^3 + 22/3*e, -5/8*e^5 + 21/2*e^3 - 37*e, -17/24*e^5 + 79/6*e^3 - 140/3*e, 5/24*e^5 - 19/6*e^3 + 2/3*e, 7/12*e^5 - 55/6*e^3 + 74/3*e, 1/2*e^4 - 8*e^2 + 28, 5/4*e^4 - 18*e^2 + 62, -5/12*e^5 + 19/3*e^3 - 28/3*e, -3/4*e^4 + 8*e^2 - 6, -1/2*e^4 + 11*e^2 - 48, -35/24*e^5 + 151/6*e^3 - 224/3*e, 23/24*e^5 - 97/6*e^3 + 143/3*e, -5/4*e^4 + 22*e^2 - 54, 1/2*e^4 - 7*e^2 + 40, -9/4*e^4 + 37*e^2 - 112, -23/24*e^5 + 85/6*e^3 - 95/3*e, -5/2*e^4 + 38*e^2 - 96, -1/4*e^4 + 3*e^2 + 10, 5/4*e^4 - 21*e^2 + 62, 11/24*e^5 - 20/3*e^3 + 26/3*e, -3/4*e^4 + 10*e^2 - 4, -2*e^4 + 31*e^2 - 60, -5/12*e^5 + 16/3*e^3 - 34/3*e, e^4 - 17*e^2 + 52, -7/4*e^4 + 31*e^2 - 106, 9/4*e^4 - 34*e^2 + 96, 2*e^4 - 31*e^2 + 88, -3/4*e^4 + 14*e^2 - 58, -3/2*e^4 + 16*e^2 + 4, -5/24*e^5 + 17/3*e^3 - 116/3*e, 3/2*e^4 - 18*e^2 + 12, 5/24*e^5 - 11/3*e^3 + 35/3*e, 3/2*e^4 - 29*e^2 + 80, 13/24*e^5 - 28/3*e^3 + 88/3*e, 5/4*e^4 - 20*e^2 + 80, -3/2*e^3 + 8*e, -e^4 + 23*e^2 - 92, e^4 - 14*e^2 + 16, -1/2*e^4 + 7*e^2 + 26, 1/12*e^5 + 7/3*e^3 - 94/3*e, 7/4*e^4 - 23*e^2 + 26, -1/12*e^5 - 1/3*e^3 + 58/3*e, -9/8*e^5 + 20*e^3 - 73*e, e^4 - 17*e^2 + 36, 3/4*e^4 - 9*e^2 - 20, -1/8*e^5 + 10*e, -3/8*e^5 + 8*e^3 - 29*e, 15/4*e^4 - 56*e^2 + 148, 11/8*e^5 - 41/2*e^3 + 47*e, -7/4*e^4 + 26*e^2 - 60, -5/2*e^4 + 39*e^2 - 80, -1/2*e^4 + 5*e^2 + 40, -2*e^3 + 18*e, 5/6*e^5 - 41/3*e^3 + 110/3*e, -3/4*e^4 + 8*e^2 + 20, -1/24*e^5 + 17/6*e^3 - 58/3*e, -13/4*e^4 + 51*e^2 - 124, 7/24*e^5 - 35/6*e^3 + 79/3*e, -3/4*e^5 + 23/2*e^3 - 32*e, -11/4*e^4 + 43*e^2 - 106, -1/4*e^4 - 4*e^2 + 54, -1/12*e^5 + 13/6*e^3 - 26/3*e, 13/12*e^5 - 91/6*e^3 + 92/3*e, 7/4*e^4 - 24*e^2 + 56, e^4 - 12*e^2 + 14, 3/4*e^4 - 14*e^2 + 60, -5/2*e^4 + 45*e^2 - 136, -1/2*e^5 + 19/2*e^3 - 32*e, 2*e^4 - 27*e^2 + 60, -17/12*e^5 + 70/3*e^3 - 226/3*e, 4/3*e^5 - 127/6*e^3 + 152/3*e, 1/4*e^4 + 4*e^2 - 56, 11/4*e^4 - 42*e^2 + 108, -1/3*e^5 + 31/6*e^3 - 50/3*e, -1/2*e^5 + 11*e^3 - 58*e, 1/3*e^5 - 43/6*e^3 + 80/3*e, 2*e^4 - 32*e^2 + 64, 17/8*e^5 - 35*e^3 + 100*e, -5/8*e^5 + 12*e^3 - 43*e, 7/8*e^5 - 27/2*e^3 + 42*e, -1/24*e^5 + 4/3*e^3 - 46/3*e, 1/2*e^4 - 2*e^2 - 32, 5/6*e^5 - 44/3*e^3 + 152/3*e, -7/4*e^4 + 24*e^2 - 48, 19/24*e^5 - 34/3*e^3 + 64/3*e, 3/4*e^4 - 17*e^2 + 74, -53/24*e^5 + 211/6*e^3 - 296/3*e, -9/4*e^4 + 38*e^2 - 96, 4*e^2 - 66, -3/8*e^5 + 9/2*e^3 - 2*e, -7/24*e^5 + 22/3*e^3 - 118/3*e, 49/24*e^5 - 94/3*e^3 + 247/3*e, 7/2*e^4 - 61*e^2 + 178, -13/12*e^5 + 91/6*e^3 - 98/3*e, 9/2*e^4 - 71*e^2 + 180, -17/12*e^5 + 73/3*e^3 - 202/3*e, 2*e^4 - 36*e^2 + 98, 35/24*e^5 - 80/3*e^3 + 269/3*e, 2*e^5 - 31*e^3 + 82*e, 7/12*e^5 - 67/6*e^3 + 158/3*e, -11/4*e^4 + 38*e^2 - 96, -23/24*e^5 + 85/6*e^3 - 74/3*e, -1/8*e^5 + 5/2*e^3 - 30*e, -3/4*e^4 + 14*e^2 - 8, 9/4*e^4 - 32*e^2 + 78, 5/24*e^5 - 19/6*e^3 + 41/3*e, 11/12*e^5 - 101/6*e^3 + 172/3*e, 4*e^4 - 61*e^2 + 164, 13/6*e^5 - 227/6*e^3 + 346/3*e, 1/2*e^4 - 4*e^2 + 8, -13/12*e^5 + 115/6*e^3 - 212/3*e, 15/8*e^5 - 30*e^3 + 77*e, 7/4*e^4 - 30*e^2 + 72, 7/24*e^5 - 4/3*e^3 - 65/3*e, 2*e^4 - 28*e^2 + 26, -5/8*e^5 + 11*e^3 - 34*e, 5/24*e^5 - 31/6*e^3 + 125/3*e, 1/24*e^5 - 4/3*e^3 + 37/3*e, -5/4*e^4 + 18*e^2 - 24, 11/4*e^4 - 36*e^2 + 60, -7/6*e^5 + 64/3*e^3 - 190/3*e, -7/2*e^4 + 53*e^2 - 120, -19/24*e^5 + 34/3*e^3 - 79/3*e, -1/4*e^4 - 3*e^2 + 50, 19/12*e^5 - 157/6*e^3 + 248/3*e, 1/12*e^5 - 19/6*e^3 + 98/3*e, -29/24*e^5 + 59/3*e^3 - 206/3*e, 5/4*e^5 - 23*e^3 + 78*e, 17/12*e^5 - 155/6*e^3 + 274/3*e, -e^5 + 35/2*e^3 - 50*e, 5/4*e^4 - 13*e^2 - 10, 19/24*e^5 - 40/3*e^3 + 85/3*e, -1/2*e^4 + 18*e^2 - 92, 3/8*e^5 - 10*e^3 + 54*e, -1/2*e^4 - 2*e^2 + 40, 7/2*e^4 - 54*e^2 + 132, 3/4*e^4 - 3*e^2 - 30, 5/12*e^5 - 19/3*e^3 + 28/3*e, -31/24*e^5 + 70/3*e^3 - 229/3*e, 7/4*e^4 - 26*e^2 + 24, 3*e^4 - 51*e^2 + 156, 1/4*e^5 - 13/2*e^3 + 38*e, 31/24*e^5 - 119/6*e^3 + 145/3*e, -1/6*e^5 + 11/6*e^3 + 8/3*e, -3/2*e^5 + 51/2*e^3 - 74*e, 1/4*e^4 + 2*e^2 - 34, 9/4*e^4 - 37*e^2 + 108, 1/4*e^5 - 6*e^3 + 40*e, -5/2*e^4 + 33*e^2 - 64, -17/4*e^4 + 65*e^2 - 158, -1/2*e^4 + 11*e^2 - 32, -1/2*e^4 + 12*e^2 - 52, -9/4*e^4 + 30*e^2 - 36, 23/12*e^5 - 185/6*e^3 + 280/3*e, 3/4*e^5 - 15*e^3 + 52*e, 2*e^4 - 26*e^2 + 60, e^5 - 16*e^3 + 30*e, -3/4*e^4 + 9*e^2 - 2, 2*e^4 - 28*e^2 + 90, 5/12*e^5 - 19/3*e^3 + 58/3*e, 11/12*e^5 - 43/3*e^3 + 154/3*e, -1/12*e^5 + 25/6*e^3 - 140/3*e, 1/4*e^4 - 2*e^2 + 6, 17/12*e^5 - 143/6*e^3 + 220/3*e, 13/4*e^4 - 53*e^2 + 122, -5/8*e^5 + 29/2*e^3 - 80*e, -5/2*e^4 + 44*e^2 - 92, -47/24*e^5 + 181/6*e^3 - 260/3*e, 11/4*e^4 - 41*e^2 + 108, -3/4*e^4 + 15*e^2 - 72, -13/12*e^5 + 53/3*e^3 - 122/3*e, 9/2*e^4 - 71*e^2 + 198, 5/8*e^5 - 23/2*e^3 + 55*e, -13/24*e^5 + 47/6*e^3 - 52/3*e, -37/24*e^5 + 137/6*e^3 - 172/3*e, 3/8*e^5 - 7*e^3 + 27*e, -11/24*e^5 + 32/3*e^3 - 206/3*e, 3/2*e^4 - 23*e^2 + 54, 13/8*e^5 - 27*e^3 + 94*e, -43/24*e^5 + 155/6*e^3 - 184/3*e, 5/12*e^5 - 17/6*e^3 - 38/3*e, 8*e^2 - 68, -15/8*e^5 + 63/2*e^3 - 113*e, -31/24*e^5 + 61/3*e^3 - 124/3*e, -15/4*e^4 + 53*e^2 - 132, 3/4*e^4 - 8*e^2 - 30, -25/24*e^5 + 107/6*e^3 - 235/3*e, 3/8*e^5 - 7/2*e^3 - 27*e, -5/4*e^4 + 13*e^2 + 14, -3*e^4 + 42*e^2 - 90, -7/2*e^4 + 49*e^2 - 130, -3/2*e^3 + 24*e, 2*e^4 - 39*e^2 + 128, -2*e^4 + 36*e^2 - 78, -17/24*e^5 + 55/6*e^3 + 4/3*e, -7/24*e^5 + 10/3*e^3 + 56/3*e, 23/24*e^5 - 97/6*e^3 + 164/3*e, 5/24*e^5 - 49/6*e^3 + 191/3*e, -1/2*e^5 + 6*e^3 - 14*e, -1/4*e^5 + 5*e^3 - 16*e, -55/24*e^5 + 233/6*e^3 - 361/3*e, 1/2*e^4 - 9*e^2 - 4, 5/4*e^4 - 14*e^2 + 32, 4*e^4 - 70*e^2 + 204, 23/8*e^5 - 93/2*e^3 + 140*e, 43/24*e^5 - 91/3*e^3 + 316/3*e, 35/12*e^5 - 139/3*e^3 + 400/3*e, 3/2*e^5 - 27*e^3 + 108*e];
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;

// 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;