/*
  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![3, 1, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {27, 3, -w^3 + w^2 + 5*w - 1}>;

primesArray := [
[3, 3, w],
[5, 5, -w^3 + w^2 + 4*w + 1],
[7, 7, -w^2 + 2],
[7, 7, -w^2 + w + 2],
[16, 2, 2],
[17, 17, -w^3 + w^2 + 4*w - 2],
[23, 23, -w^3 + 2*w^2 + 3*w - 2],
[27, 3, -w^3 + w^2 + 5*w - 1],
[37, 37, -w^3 + 4*w + 1],
[37, 37, -w^3 + w^2 + 2*w + 1],
[49, 7, -w^3 + 3*w^2 + 2*w - 4],
[61, 61, -w^3 + 2*w^2 + 4*w - 2],
[61, 61, -w^3 + 3*w^2 + 2*w - 7],
[67, 67, w^2 - 3*w - 2],
[71, 71, -w^2 + 5],
[71, 71, 2*w^3 - w^2 - 9*w - 2],
[71, 71, w^2 - 2*w - 5],
[79, 79, w^2 - 3*w - 1],
[79, 79, w^3 - 6*w - 1],
[89, 89, -w^3 + 3*w^2 + 3*w - 7],
[97, 97, -w^3 + 2*w^2 + 2*w - 5],
[101, 101, -w^3 + 3*w^2 + 2*w - 5],
[107, 107, 2*w^3 - 2*w^2 - 9*w - 1],
[109, 109, -w^3 + 5*w + 1],
[125, 5, 2*w^3 - w^2 - 8*w - 2],
[131, 131, -2*w^3 + 3*w^2 + 7*w - 2],
[131, 131, 2*w^2 - 3*w - 7],
[131, 131, 2*w^3 - 3*w^2 - 6*w + 2],
[137, 137, w^3 - 2*w^2 - 3*w + 7],
[139, 139, w^3 - 4*w + 2],
[151, 151, w^2 + w - 4],
[157, 157, 2*w^2 - 2*w - 5],
[157, 157, w^3 - 2*w^2 - 5*w + 7],
[173, 173, 2*w^3 - 3*w^2 - 6*w + 4],
[179, 179, 2*w^3 - 3*w^2 - 9*w + 4],
[181, 181, w^3 + w^2 - 7*w - 4],
[197, 197, w^3 - 3*w^2 - w + 7],
[223, 223, 3*w^3 - 4*w^2 - 11*w + 4],
[223, 223, -w^3 + w^2 + 4*w - 5],
[227, 227, 2*w^3 - 2*w^2 - 9*w - 2],
[229, 229, w^3 - w^2 - 6*w + 2],
[229, 229, 4*w^3 - 6*w^2 - 14*w + 7],
[233, 233, 2*w^3 - 3*w^2 - 5*w + 1],
[239, 239, -w - 4],
[239, 239, -2*w^3 + 4*w^2 + 5*w - 8],
[241, 241, -2*w^3 + w^2 + 10*w + 1],
[251, 251, -w^3 + 2*w^2 + 2*w - 8],
[251, 251, -2*w^3 + 5*w^2 + 5*w - 8],
[257, 257, 2*w^3 - 2*w^2 - 7*w + 1],
[263, 263, w^3 - 4*w^2 - 4*w + 7],
[269, 269, w^2 + 2*w - 4],
[269, 269, 2*w^2 - 7],
[271, 271, 2*w^3 - 3*w^2 - 7*w + 1],
[271, 271, w^2 + 3*w - 2],
[271, 271, 3*w^3 - 5*w^2 - 11*w + 13],
[271, 271, -2*w^3 + 3*w^2 + 9*w - 2],
[277, 277, w^2 - w - 7],
[277, 277, 3*w^3 - 4*w^2 - 13*w + 5],
[277, 277, -w^3 + w^2 + 3*w - 5],
[277, 277, w^3 - 3*w - 4],
[281, 281, w^3 - 2*w^2 - 2*w - 2],
[283, 283, w^2 + 3*w - 1],
[283, 283, w^3 - w^2 - 2*w - 4],
[293, 293, 3*w^3 - 4*w^2 - 12*w + 8],
[293, 293, w^2 + w - 5],
[307, 307, 2*w^3 - 2*w^2 - 6*w - 1],
[307, 307, -w^2 - 2],
[311, 311, -w^3 + 5*w - 1],
[313, 313, 2*w^3 - 2*w^2 - 8*w - 5],
[313, 313, -2*w^3 + 3*w^2 + 5*w - 2],
[317, 317, 2*w^3 - 2*w^2 - 9*w + 4],
[317, 317, w^3 - 2*w^2 - 3*w - 2],
[331, 331, -w^3 + 2*w^2 + w - 5],
[337, 337, -2*w^3 + 3*w^2 + 5*w - 7],
[337, 337, w^3 - 4*w^2 - w + 5],
[349, 349, -2*w^3 + w^2 + 11*w + 1],
[367, 367, -2*w^3 + 2*w^2 + 8*w - 5],
[379, 379, -w^3 + 2*w^2 - 4],
[383, 383, -3*w^3 + 6*w^2 + 9*w - 14],
[383, 383, w - 5],
[389, 389, 4*w^3 - 6*w^2 - 14*w + 11],
[397, 397, -2*w^3 + w^2 + 11*w - 1],
[397, 397, -3*w^3 + 2*w^2 + 12*w + 1],
[401, 401, 3*w - 7],
[409, 409, w^2 - 4*w - 1],
[421, 421, -2*w^3 + 4*w^2 + 7*w - 5],
[421, 421, -2*w^3 + 4*w^2 + 4*w - 7],
[443, 443, 2*w^3 - 2*w^2 - 10*w + 1],
[449, 449, -w^3 - w^2 + 2*w - 2],
[461, 461, 3*w^3 - 3*w^2 - 12*w + 4],
[463, 463, -2*w^3 + 2*w^2 + 6*w - 1],
[463, 463, -3*w^2 + 5*w + 4],
[467, 467, w^2 + w - 10],
[467, 467, 3*w^3 - 3*w^2 - 14*w + 1],
[479, 479, -2*w^3 + w^2 + 6*w + 1],
[491, 491, w^3 + 2*w^2 - 7*w - 8],
[499, 499, 4*w^3 - 5*w^2 - 16*w + 8],
[499, 499, w^3 - 3*w - 5],
[503, 503, -w^3 + 3*w^2 + w - 10],
[503, 503, w^3 - 3*w^2 - 4*w + 1],
[523, 523, -w^3 + 4*w - 4],
[541, 541, w^3 - 5*w^2 + 2*w + 10],
[547, 547, -3*w^3 + 6*w^2 + 11*w - 10],
[547, 547, -3*w^3 + 6*w^2 + 7*w - 10],
[557, 557, -w^3 + 3*w^2 + 4*w - 10],
[557, 557, 3*w^2 - 6*w - 8],
[557, 557, -w^3 + 4*w^2 - 11],
[557, 557, -2*w^3 + 4*w^2 + 5*w - 11],
[569, 569, w^3 + w^2 - 6*w - 11],
[571, 571, 3*w^3 - 4*w^2 - 10*w + 2],
[571, 571, w^3 - 3*w^2 - 5*w + 2],
[577, 577, 2*w^3 - w^2 - 7*w + 4],
[587, 587, 2*w^3 - 11*w - 4],
[607, 607, 2*w^3 - 3*w^2 - 9*w + 1],
[613, 613, -3*w^3 + 3*w^2 + 14*w - 5],
[617, 617, 2*w^3 - 3*w^2 - 8*w + 10],
[617, 617, 3*w^3 - 5*w^2 - 8*w + 10],
[643, 643, -w^3 + w^2 + 7*w + 1],
[643, 643, w^3 + 2*w^2 - 6*w - 8],
[661, 661, -3*w^3 + 5*w^2 + 8*w - 7],
[673, 673, -3*w^3 + 3*w^2 + 13*w - 4],
[683, 683, -w^3 + 2*w^2 - 5],
[683, 683, w^3 - w^2 - 4*w - 5],
[691, 691, w^3 + w^2 - 6*w - 2],
[691, 691, -w^3 + 3*w^2 - 7],
[709, 709, w^2 - w - 8],
[727, 727, -2*w^3 + 4*w^2 + 7*w - 4],
[733, 733, 3*w^3 - 5*w^2 - 7*w + 2],
[739, 739, -w^3 + 4*w^2 + w - 8],
[743, 743, 2*w^3 - 3*w^2 - 10*w + 7],
[751, 751, 3*w^2 - 3*w - 7],
[761, 761, w^2 + 3*w - 5],
[773, 773, 3*w^2 - w - 8],
[787, 787, w^3 - 5*w - 8],
[797, 797, -w^3 - w^2 + 7*w + 2],
[809, 809, 3*w^3 - 4*w^2 - 11*w - 4],
[811, 811, 3*w^2 - 4*w - 13],
[827, 827, -4*w^3 + 7*w^2 + 12*w - 7],
[827, 827, 2*w^2 + w - 7],
[839, 839, -2*w^2 + 5*w + 7],
[857, 857, -3*w^3 + 4*w^2 + 10*w - 4],
[857, 857, 2*w^3 - 3*w^2 - 8*w - 1],
[859, 859, -5*w^3 + 9*w^2 + 18*w - 16],
[863, 863, -2*w^3 + 4*w^2 + 6*w - 1],
[863, 863, -2*w^3 + w^2 + 8*w - 2],
[877, 877, w^3 - 3*w - 7],
[883, 883, -w^3 + w^2 + 7*w - 2],
[907, 907, 2*w^3 - w^2 - 12*w - 5],
[907, 907, 3*w^3 - 14*w - 7],
[911, 911, 5*w^3 - 5*w^2 - 21*w + 4],
[911, 911, w^3 - w^2 - w - 4],
[929, 929, 3*w^2 - 2*w - 7],
[941, 941, 4*w^3 - 5*w^2 - 14*w + 1],
[941, 941, 2*w^3 - 5*w^2 - 8*w + 7],
[953, 953, -w^3 - 3*w^2 + 9*w + 10],
[953, 953, -3*w^3 + 2*w^2 + 15*w - 1],
[953, 953, -3*w^3 + 5*w^2 + 9*w - 2],
[953, 953, 3*w^3 - 3*w^2 - 11*w + 1],
[961, 31, -3*w^2 + 5*w + 10],
[961, 31, -2*w^3 + 3*w^2 + 4*w - 7],
[967, 967, -w^3 - w^2 + 8*w + 1],
[977, 977, -2*w^3 + 3*w^2 + 10*w - 4],
[983, 983, 4*w^3 - 6*w^2 - 12*w + 11],
[983, 983, -w^3 + 2*w^2 + 7*w - 5],
[997, 997, 3*w^3 - 8*w^2 - 5*w + 13],
[1009, 1009, -2*w^3 + 4*w^2 + 6*w - 13],
[1009, 1009, 4*w^3 - 5*w^2 - 15*w + 1],
[1019, 1019, w^3 - 4*w^2 - w + 16],
[1031, 1031, -w^3 - w^2 + 6*w + 1],
[1033, 1033, -w^3 + 3*w^2 - 8],
[1049, 1049, -2*w^3 + 6*w^2 + 3*w - 11],
[1049, 1049, -2*w^3 + 4*w^2 + 3*w - 7],
[1051, 1051, -2*w^3 + 3*w^2 + 10*w - 5],
[1063, 1063, -3*w^3 + 3*w^2 + 11*w - 8],
[1063, 1063, -w^2 + 2*w - 4],
[1069, 1069, -3*w^3 + 4*w^2 + 9*w - 1],
[1087, 1087, w^2 - 5*w - 2],
[1087, 1087, 3*w^3 - 3*w^2 - 9*w + 5],
[1093, 1093, -2*w^3 + w^2 + 7*w + 8],
[1093, 1093, w^3 - 8*w - 2],
[1109, 1109, -5*w^3 + 8*w^2 + 17*w - 14],
[1109, 1109, -4*w^3 + 7*w^2 + 15*w - 11],
[1123, 1123, 4*w^3 - 6*w^2 - 14*w + 5],
[1129, 1129, 3*w^3 - 2*w^2 - 8*w + 4],
[1151, 1151, 4*w^3 - 5*w^2 - 12*w + 10],
[1151, 1151, -w^3 - 3*w^2 + 9*w + 11],
[1171, 1171, -w^3 - 3*w^2 + 8*w + 14],
[1171, 1171, -3*w^3 + 3*w^2 + 7*w - 8],
[1181, 1181, -4*w^2 + 5*w + 5],
[1181, 1181, -3*w^3 + 6*w^2 + 9*w - 7],
[1193, 1193, -2*w^3 + 6*w^2 + 3*w - 10],
[1193, 1193, -4*w^3 + 6*w^2 + 17*w - 11],
[1201, 1201, 3*w^3 - 4*w^2 - 6*w + 5],
[1201, 1201, -3*w^3 + 3*w^2 + 15*w - 5],
[1201, 1201, -2*w^3 + 5*w^2 + 4*w - 16],
[1201, 1201, 3*w^3 - 3*w^2 - 8*w - 4],
[1213, 1213, -3*w^3 + 4*w^2 + 14*w - 4],
[1217, 1217, -w^3 + 4*w^2 - 13],
[1229, 1229, -4*w^3 + 4*w^2 + 18*w - 5],
[1231, 1231, -w^3 + 2*w^2 + 7*w - 4],
[1277, 1277, 3*w^3 - 5*w^2 - 9*w - 1],
[1279, 1279, 4*w^3 - 5*w^2 - 14*w - 2],
[1291, 1291, 3*w^3 - 6*w^2 - 8*w + 5],
[1291, 1291, w^3 - 5*w^2 + 14],
[1301, 1301, -4*w^3 + 9*w^2 + 13*w - 20],
[1321, 1321, 3*w^3 - 4*w^2 - 9*w + 5],
[1367, 1367, -3*w^3 + 4*w^2 + 11*w + 1],
[1367, 1367, -w^3 + 2*w^2 - 8],
[1369, 37, -3*w^3 + 5*w^2 + 11*w - 5],
[1373, 1373, -2*w^3 - w^2 + 6*w - 1],
[1409, 1409, -2*w^3 + 6*w^2 + 4*w - 13],
[1439, 1439, -2*w^3 + 5*w^2 + 5*w - 17],
[1447, 1447, 6*w^3 - 9*w^2 - 23*w + 19],
[1447, 1447, 2*w^3 - 3*w^2 - 8*w + 11],
[1453, 1453, -w^3 + w^2 - 3*w - 2],
[1453, 1453, 2*w^3 - w^2 - 10*w + 2],
[1459, 1459, 3*w^3 - 2*w^2 - 5*w + 5],
[1471, 1471, -w^3 - w^2 + 6*w - 1],
[1481, 1481, w^3 - 6*w^2 + 20],
[1483, 1483, 2*w^3 - 4*w^2 - 5*w + 14],
[1483, 1483, -2*w^3 + 5*w^2 + 6*w - 17],
[1483, 1483, w^3 - 2*w^2 - 4*w + 10],
[1483, 1483, 3*w^3 - 4*w^2 - 9*w + 4],
[1487, 1487, -w^3 + 5*w^2 - 7],
[1487, 1487, -3*w^3 + 5*w^2 + 13*w - 5],
[1489, 1489, -w^2 - w - 4],
[1489, 1489, w^2 + 3*w + 5],
[1493, 1493, -3*w^3 + 2*w^2 + 12*w - 2],
[1499, 1499, -w^3 + 5*w^2 - w - 11],
[1511, 1511, 2*w^2 + 2*w - 7],
[1511, 1511, -w^3 + 5*w^2 - w - 8],
[1549, 1549, 2*w^3 - 4*w^2 - 7*w + 1],
[1559, 1559, w^3 - 2*w^2 - 2*w - 4],
[1567, 1567, 3*w^2 - w - 13],
[1567, 1567, -w^3 + w^2 + 3*w - 7],
[1583, 1583, -2*w^3 + 6*w^2 + 6*w - 11],
[1583, 1583, w^3 - 5*w^2 - 8*w + 8],
[1597, 1597, -4*w^3 + 5*w^2 + 10*w + 2],
[1597, 1597, 4*w^3 - 6*w^2 - 11*w + 1],
[1601, 1601, 5*w - 2],
[1607, 1607, -3*w^3 + 6*w^2 + 6*w - 10],
[1609, 1609, -2*w^3 + 6*w^2 + 5*w - 14],
[1613, 1613, 3*w^3 - 3*w^2 - 10*w + 2],
[1613, 1613, -4*w^3 + 9*w^2 + 12*w - 17],
[1621, 1621, -w^2 - 4],
[1621, 1621, 4*w^3 - 5*w^2 - 18*w + 5],
[1627, 1627, -3*w^3 + 4*w^2 + 8*w - 5],
[1627, 1627, -2*w^3 + 5*w - 5],
[1663, 1663, 4*w^3 - 2*w^2 - 19*w - 4],
[1667, 1667, 3*w^3 - 7*w^2 - 6*w + 4],
[1669, 1669, -5*w^3 + 9*w^2 + 14*w - 14],
[1669, 1669, -3*w^3 + 6*w^2 + 13*w - 8],
[1693, 1693, w^3 + w^2 - 4*w - 10],
[1697, 1697, 3*w^3 - 2*w^2 - 14*w - 10],
[1699, 1699, -2*w^3 + 4*w^2 + 3*w - 8],
[1709, 1709, -4*w^3 + 5*w^2 + 14*w - 7],
[1709, 1709, 4*w^3 - 6*w^2 - 13*w + 8],
[1721, 1721, w^3 - 2*w^2 - 4*w - 4],
[1753, 1753, w^3 - 3*w^2 - 2*w - 2],
[1753, 1753, -2*w^3 + 6*w^2 + 4*w - 11],
[1759, 1759, w^3 + 2*w^2 - 8*w - 4],
[1777, 1777, 2*w^3 + w^2 - 15*w - 10],
[1787, 1787, w^3 - w^2 - 8*w - 1],
[1789, 1789, 2*w^3 - 4*w^2 - 4*w + 11],
[1801, 1801, -w^3 + 5*w^2 - w - 16],
[1811, 1811, 2*w^3 - 3*w^2 - 5*w - 5],
[1823, 1823, 3*w^2 - 11],
[1823, 1823, w - 7],
[1831, 1831, w^3 - 2*w^2 - 3*w - 4],
[1847, 1847, -w^3 - w^2 + 9*w - 2],
[1849, 43, 6*w^3 - 9*w^2 - 21*w + 11],
[1849, 43, -2*w^3 + w^2 + 9*w - 5],
[1861, 1861, 2*w^3 - 3*w^2 - 11*w + 2],
[1871, 1871, 5*w - 1],
[1871, 1871, 3*w^3 - 6*w^2 - 14*w + 13],
[1873, 1873, 2*w^3 - 11*w - 2],
[1877, 1877, w^2 - 5*w + 8],
[1877, 1877, 3*w^3 - 2*w^2 - 6*w + 7],
[1889, 1889, 2*w^3 - w^2 - 9*w + 4],
[1901, 1901, -w^3 - w^2 + 7*w - 1],
[1901, 1901, -3*w^3 + 3*w^2 + 12*w - 8],
[1901, 1901, -2*w^3 + 6*w^2 + 5*w - 13],
[1901, 1901, -4*w^3 + 3*w^2 + 15*w - 1],
[1933, 1933, -3*w^3 + 2*w^2 + 10*w + 2],
[1949, 1949, 3*w^3 - 3*w^2 - 15*w - 5],
[1949, 1949, -w^3 + 5*w^2 + 2*w - 8],
[1973, 1973, -w^3 + w^2 + 5*w - 8],
[1979, 1979, -2*w^3 + 5*w^2 + 3*w - 11],
[1987, 1987, -w^3 + w^2 - 3*w - 1],
[1993, 1993, -2*w^3 + 6*w^2 + 2*w - 13],
[1997, 1997, -4*w^3 + 8*w^2 + 10*w - 7],
[1997, 1997, w^3 - w - 5],
[1999, 1999, -w^3 + 2*w^2 - w - 11],
[1999, 1999, w^2 - 10]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^5 + 3*x^4 - 5*x^3 - 11*x^2 + 8*x + 7;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -e^4 - 5*e^3 - 3*e^2 + 11*e + 8, -e^4 - 4*e^3 + 8*e + 2, e^4 + 4*e^3 - 8*e - 3, e^4 + 3*e^3 - 4*e^2 - 9*e + 3, 2*e^4 + 8*e^3 - 16*e - 9, 2*e^4 + 8*e^3 + 2*e^2 - 14*e - 12, 1, -2*e^4 - 6*e^3 + 6*e^2 + 12*e - 6, -e^4 - 4*e^3 - e^2 + 6*e + 6, e^4 + 5*e^3 + e^2 - 16*e - 2, 2*e^2 + 3*e - 12, -e^4 - 3*e^3 + 2*e^2 + 7*e + 7, -e^4 - e^3 + 8*e^2 - e - 12, -3*e^3 - 10*e^2 + 5*e + 12, 3*e^4 + 13*e^3 + 2*e^2 - 29*e - 9, -2*e^4 - 11*e^3 - 7*e^2 + 26*e + 12, -e^4 - e^3 + 8*e^2 - 8, -2*e^4 - 7*e^3 + 6*e^2 + 21*e - 4, 2*e^4 + 9*e^3 + e^2 - 21*e - 6, -e^4 - e^3 + 10*e^2 - 2*e - 19, -e^4 - e^3 + 10*e^2 + 5*e - 10, e^3 + 6*e^2 + 2*e - 8, 5*e^4 + 24*e^3 + 11*e^2 - 52*e - 36, -4*e^4 - 19*e^3 - 5*e^2 + 50*e + 17, -2*e^4 - 9*e^3 - 3*e^2 + 18*e + 6, e^3 + 3*e^2 - 2*e + 1, e^2 - 2*e - 7, -e^4 - 6*e^3 - 4*e^2 + 14*e + 15, e^4 + 4*e^3 - e^2 - 8*e - 10, 5*e^4 + 19*e^3 - 4*e^2 - 41*e - 13, -6*e^4 - 25*e^3 - 3*e^2 + 48*e + 25, -4*e^4 - 18*e^3 - 3*e^2 + 42*e + 19, -2*e^4 - 7*e^3 + 5*e^2 + 18*e + 8, -4*e^4 - 20*e^3 - 13*e^2 + 45*e + 39, 4*e^4 + 17*e^3 + 4*e^2 - 39*e - 23, -4*e^4 - 21*e^3 - 12*e^2 + 50*e + 23, -4*e^4 - 16*e^3 + 2*e^2 + 36*e + 5, -6*e^4 - 23*e^3 + e^2 + 47*e + 16, -3*e^2 - 12*e + 1, 5*e^4 + 20*e^3 - 3*e^2 - 42*e - 14, 3*e^4 + 15*e^3 + 10*e^2 - 34*e - 35, 12*e^4 + 50*e^3 + 7*e^2 - 105*e - 52, -11*e^4 - 47*e^3 - 9*e^2 + 94*e + 57, -e^4 + 3*e^3 + 20*e^2 - 11*e - 38, 4*e^4 + 14*e^3 - 8*e^2 - 26*e + 3, e^4 + 6*e^3 - 27*e - 5, -8*e^4 - 32*e^3 + 4*e^2 + 70*e + 7, -8*e^4 - 34*e^3 - 10*e^2 + 68*e + 39, 4*e^4 + 12*e^3 - 9*e^2 - 16*e + 10, 6*e^4 + 31*e^3 + 22*e^2 - 73*e - 61, 7*e^4 + 31*e^3 + 13*e^2 - 62*e - 56, 3*e^4 + 8*e^3 - 15*e^2 - 23*e + 24, 7*e^4 + 34*e^3 + 13*e^2 - 77*e - 36, 12*e^4 + 49*e^3 - 108*e - 29, -9*e^4 - 41*e^3 - 14*e^2 + 86*e + 49, 4*e^4 + 21*e^3 + 18*e^2 - 46*e - 59, -7*e^4 - 28*e^3 - 2*e^2 + 55*e + 37, -e^4 - 7*e^3 - 10*e^2 + 7*e + 23, -3*e^4 - 6*e^3 + 16*e^2 + 12*e - 10, 5*e^4 + 22*e^3 + 2*e^2 - 54*e - 13, -2*e^4 - 6*e^3 + 10*e^2 + 15*e - 15, 5*e^4 + 24*e^3 + 11*e^2 - 51*e - 33, -12*e^4 - 53*e^3 - 12*e^2 + 120*e + 45, 2*e^3 + 9*e^2 - e - 25, 6*e^4 + 20*e^3 - 16*e^2 - 45*e + 9, 5*e^4 + 16*e^3 - 10*e^2 - 32*e + 6, -3*e^4 - 14*e^3 - 5*e^2 + 26*e + 14, 6*e^4 + 33*e^3 + 25*e^2 - 75*e - 60, 2*e^4 + 7*e^3 - 9*e^2 - 23*e + 12, 2*e^4 + 11*e^3 + 12*e^2 - 26*e - 29, 3*e^4 + 13*e^3 - e^2 - 40*e - 5, -6*e^4 - 19*e^3 + 11*e^2 + 37*e + 14, 3*e^4 + 15*e^3 + 9*e^2 - 28*e - 37, 2*e^4 + 18*e^3 + 29*e^2 - 40*e - 62, -6*e^4 - 27*e^3 - 10*e^2 + 64*e + 40, -6*e^4 - 32*e^3 - 22*e^2 + 75*e + 63, 6*e^4 + 25*e^3 - 61*e - 9, -4*e^4 - 18*e^3 - 12*e^2 + 26*e + 32, 5*e^4 + 16*e^3 - 16*e^2 - 37*e + 1, -8*e^4 - 42*e^3 - 28*e^2 + 91*e + 75, -13*e^4 - 56*e^3 - 10*e^2 + 128*e + 49, 3*e^4 + 7*e^3 - 15*e^2 - 15*e + 9, -4*e^4 - 25*e^3 - 27*e^2 + 51*e + 53, 8*e^4 + 38*e^3 + 20*e^2 - 80*e - 71, -2*e^4 - 11*e^3 - 4*e^2 + 29*e - 15, -8*e^4 - 38*e^3 - 19*e^2 + 85*e + 42, -12*e^4 - 50*e^3 - e^2 + 118*e + 25, -3*e^4 - 9*e^3 + 5*e^2 + 2*e - 4, e^4 - 7*e^2 + 15*e + 16, -4*e^4 - 13*e^3 + 6*e^2 + 18*e + 2, 4*e^4 + 24*e^3 + 22*e^2 - 57*e - 42, -13*e^4 - 52*e^3 + 2*e^2 + 110*e + 31, 8*e^4 + 31*e^3 + 5*e^2 - 49*e - 34, e^4 + e^3 - 10*e^2 - 8*e + 25, 2*e^4 + 7*e^3 - 11*e^2 - 29*e + 9, 3*e^4 + 8*e^3 - 12*e^2 - 20*e - 2, -6*e^4 - 26*e^3 - 2*e^2 + 76*e + 17, -e^4 + 7*e^3 + 31*e^2 - 12*e - 37, -4*e^4 - 19*e^3 - 13*e^2 + 34*e + 47, 8*e^4 + 21*e^3 - 35*e^2 - 52*e + 24, 2*e^4 + 13*e^3 + 14*e^2 - 26*e - 13, 6*e^4 + 20*e^3 - 8*e^2 - 35*e - 15, -2*e^4 - 12*e^3 - 18*e^2 + 23*e + 39, 5*e^4 + 26*e^3 + 13*e^2 - 66*e - 40, 10*e^4 + 48*e^3 + 23*e^2 - 111*e - 74, 14*e^4 + 49*e^3 - 22*e^2 - 105*e - 9, -18*e^4 - 78*e^3 - 16*e^2 + 163*e + 87, e^4 + 8*e^3 + 4*e^2 - 27*e + 14, 8*e^4 + 32*e^3 - 65*e - 33, -5*e^4 - 24*e^3 - 8*e^2 + 66*e + 26, 2*e^4 + 4*e^3 - 15*e^2 - 22*e - 3, e^4 + 13*e^3 + 23*e^2 - 34*e - 48, -8*e^4 - 33*e^3 + 7*e^2 + 83*e + 11, -8*e^4 - 35*e^3 - 10*e^2 + 81*e + 34, -2*e^4 - 3*e^3 + 16*e^2 + 22*e - 12, 4*e^4 + 23*e^3 + 14*e^2 - 60*e - 22, -12*e^4 - 51*e^3 - 6*e^2 + 125*e + 37, -10*e^4 - 35*e^3 + 16*e^2 + 74*e + 3, -3*e^4 - 11*e^3 - 2*e^2 + 12*e - 4, -2*e^4 - 16*e^3 - 24*e^2 + 34*e + 48, -2*e^4 - 8*e^3 + 2*e^2 + 35*e + 9, 6*e^4 + 15*e^3 - 29*e^2 - 33*e + 41, -11*e^4 - 53*e^3 - 31*e^2 + 104*e + 88, -e^4 - 4*e^3 + 8*e^2 + 22*e - 7, -e^4 - 8*e^3 - 10*e^2 + 5*e - 1, 13*e^4 + 57*e^3 + 12*e^2 - 124*e - 64, -8*e^3 - 23*e^2 + 22*e + 26, 14*e^4 + 51*e^3 - 18*e^2 - 97*e - 3, -15*e^4 - 67*e^3 - 23*e^2 + 145*e + 92, -6*e^4 - 24*e^3 + 6*e^2 + 52*e + 4, 2*e^4 + 9*e^3 - 9*e^2 - 45*e + 25, -e^4 - 11*e^3 - 22*e^2 + 37*e + 59, -13*e^4 - 62*e^3 - 17*e^2 + 157*e + 53, -2*e^4 - 9*e^3 - 3*e^2 + 22*e + 40, -e^4 - 12*e^3 - 15*e^2 + 46*e + 18, -5*e^4 - 14*e^3 + 8*e^2 + 12*e + 5, 15*e^4 + 57*e^3 - 17*e^2 - 126*e - 34, 6*e^4 + 26*e^3 + 16*e^2 - 35*e - 44, -6*e^4 - 37*e^3 - 39*e^2 + 94*e + 95, 3*e^4 + 2*e^3 - 31*e^2 - 3*e + 23, 19*e^4 + 86*e^3 + 30*e^2 - 183*e - 120, -3*e^4 - 12*e^3 - e^2 + 12*e - 9, -10*e^4 - 41*e^3 + 4*e^2 + 87*e + 3, 7*e^4 + 39*e^3 + 33*e^2 - 77*e - 65, -4*e^4 - 18*e^3 - 6*e^2 + 53*e + 22, e^4 + 2*e^3 - 6*e^2 - 20, 3*e^4 + 5*e^3 - 16*e^2 + 6*e + 9, 6*e^4 + 26*e^3 - 2*e^2 - 59*e - 1, -13*e^4 - 63*e^3 - 39*e^2 + 118*e + 127, -2*e^4 - 8*e^3 + 3*e^2 + 31*e + 27, 4*e^4 + 22*e^3 + 10*e^2 - 47*e, 17*e^4 + 61*e^3 - 18*e^2 - 130*e - 37, 20*e^4 + 74*e^3 - 18*e^2 - 154*e - 30, 18*e^4 + 82*e^3 + 23*e^2 - 181*e - 92, -15*e^4 - 62*e^3 - 6*e^2 + 138*e + 58, 10*e^4 + 40*e^3 + 3*e^2 - 82*e - 43, 3*e^4 + 14*e^3 + 15*e^2 - 17*e - 44, 3*e^4 + 16*e^3 + 15*e^2 - 35*e - 54, 7*e^4 + 23*e^3 - 14*e^2 - 36*e + 6, -16*e^4 - 56*e^3 + 26*e^2 + 111*e - 11, -4*e^4 - 15*e^3 - 2*e^2 + 37*e + 35, -10*e^4 - 42*e^3 - 11*e^2 + 85*e + 60, 9*e^4 + 36*e^3 + 4*e^2 - 61*e - 58, -10*e^4 - 31*e^3 + 39*e^2 + 76*e - 30, 10*e^4 + 42*e^3 + 3*e^2 - 88*e - 16, -6*e^2 + 3*e + 34, 15*e^4 + 64*e^3 + 3*e^2 - 145*e - 31, -6*e^4 - 19*e^3 + 21*e^2 + 52*e - 24, -4*e^4 - 21*e^3 - 27*e^2 + 28*e + 66, -7*e^4 - 18*e^3 + 23*e^2 + 18*e - 19, e^4 - 10*e^3 - 40*e^2 + 20*e + 47, -13*e^4 - 54*e^3 + 6*e^2 + 140*e + 22, -6*e^4 - 24*e^3 - 5*e^2 + 55*e + 56, -25*e^4 - 112*e^3 - 35*e^2 + 252*e + 126, -4*e^4 - 29*e^3 - 42*e^2 + 58*e + 105, 9*e^4 + 45*e^3 + 27*e^2 - 100*e - 92, 5*e^4 + 28*e^3 + 29*e^2 - 47*e - 87, 4*e^4 + 20*e^3 + 6*e^2 - 69*e - 33, -8*e^4 - 38*e^3 - 16*e^2 + 78*e + 30, -18*e^4 - 75*e^3 + 5*e^2 + 177*e + 34, -11*e^4 - 42*e^3 + 69*e + 43, 5*e^4 + 26*e^3 + 11*e^2 - 79*e - 18, -11*e^4 - 56*e^3 - 37*e^2 + 133*e + 101, 20*e^4 + 88*e^3 + 19*e^2 - 195*e - 90, -15*e^4 - 63*e^3 + 8*e^2 + 161*e + 31, 23*e^4 + 93*e^3 - 3*e^2 - 199*e - 56, 7*e^4 + 29*e^3 + 14*e^2 - 53*e - 74, -19*e^4 - 68*e^3 + 32*e^2 + 156*e + 10, -19*e^4 - 90*e^3 - 31*e^2 + 203*e + 100, 13*e^4 + 56*e^3 + 15*e^2 - 104*e - 68, -20*e^4 - 89*e^3 - 36*e^2 + 180*e + 148, 8*e^4 + 34*e^3 - 6*e^2 - 91*e + 4, -8*e^4 - 26*e^3 + 14*e^2 + 58*e + 39, -9*e^4 - 36*e^3 + 2*e^2 + 73*e + 44, -9*e^4 - 47*e^3 - 38*e^2 + 85*e + 104, 2*e^4 + 12*e^3 + 14*e^2 - 19*e - 56, 5*e^4 + 16*e^3 - 10*e^2 - 39*e - 6, 28*e^4 + 115*e^3 + 7*e^2 - 246*e - 105, -4*e^4 - 24*e^3 - 30*e^2 + 45*e + 79, -e^4 - 6*e^3 + 2*e^2 + 33*e - 27, 11*e^4 + 56*e^3 + 37*e^2 - 137*e - 110, 6*e^4 + 36*e^3 + 38*e^2 - 69*e - 102, -16*e^4 - 57*e^3 + 22*e^2 + 110*e + 17, 17*e^4 + 78*e^3 + 22*e^2 - 186*e - 66, -31*e^4 - 123*e^3 - e^2 + 261*e + 103, 16*e^4 + 62*e^3 - 16*e^2 - 131*e - 12, -26*e^4 - 103*e^3 + e^2 + 216*e + 92, -4*e^4 - 23*e^3 - 24*e^2 + 37*e + 31, 12*e^4 + 40*e^3 - 23*e^2 - 88*e - 1, -14*e^4 - 55*e^3 + 2*e^2 + 109*e + 52, -3*e^4 - 12*e^3 - e^2 + 41*e + 31, 7*e^4 + 33*e^3 + 6*e^2 - 97*e - 39, -21*e^4 - 87*e^3 - 9*e^2 + 184*e + 105, -2*e^4 - 11*e^3 - 3*e^2 + 44*e - 17, -14*e^4 - 65*e^3 - 22*e^2 + 136*e + 86, -6*e^4 - 17*e^3 + 11*e^2 + 13*e + 10, -16*e^4 - 69*e^3 - 26*e^2 + 131*e + 103, -20*e^4 - 86*e^3 - 2*e^2 + 216*e + 42, 2*e^4 + 12*e^3 + 23*e^2 - 13*e - 89, 6*e^3 + e^2 - 34*e + 30, -3*e^4 - 3*e^3 + 35*e^2 + 10*e - 60, -28*e^4 - 111*e^3 + 12*e^2 + 233*e + 60, 4*e^4 + 18*e^3 - 2*e^2 - 46*e - 24, 24*e^4 + 95*e^3 - 6*e^2 - 214*e - 40, 16*e^4 + 66*e^3 - 9*e^2 - 168*e - 20, -e^4 + 2*e^3 + 10*e^2 - 10*e + 34, 2*e^4 + 16*e^3 + 22*e^2 - 38*e - 5, 4*e^4 + 28*e^3 + 36*e^2 - 45*e - 56, 5*e^4 + 22*e^3 + 6*e^2 - 40*e - 47, -21*e^4 - 97*e^3 - 34*e^2 + 218*e + 96, 11*e^4 + 43*e^3 + 6*e^2 - 75*e - 55, -14*e^4 - 60*e^3 - 14*e^2 + 109*e + 67, -4*e^4 - 18*e^3 - 11*e^2 + 25*e + 17, -8*e^4 - 50*e^3 - 50*e^2 + 120*e + 119, -19*e^4 - 86*e^3 - 21*e^2 + 215*e + 75, 4*e^4 + 26*e^3 + 22*e^2 - 85*e - 74, 7*e^3 + 17*e^2 - 9*e + 8, 2*e^4 + 5*e^3 + 10*e - 39, -6*e^4 - 20*e^3 + 11*e^2 + 34*e + 17, 18*e^4 + 71*e^3 - 9*e^2 - 140*e - 35, 20*e^4 + 82*e^3 + 2*e^2 - 182*e - 67, 14*e^4 + 65*e^3 + 27*e^2 - 143*e - 117, -22*e^4 - 94*e^3 - 14*e^2 + 202*e + 63, -17*e^4 - 54*e^3 + 42*e^2 + 99*e - 30, 14*e^4 + 79*e^3 + 66*e^2 - 175*e - 148, 7*e^4 + 35*e^3 + 21*e^2 - 80*e - 40, -3*e^3 - 14*e^2 - 17*e + 30, e^4 - 10*e^3 - 45*e^2 + 30*e + 92, -4*e^3 - 17*e^2 + 3*e + 35, -3*e^4 - 19*e^3 - 21*e^2 + 49*e + 16, 13*e^4 + 70*e^3 + 54*e^2 - 152*e - 133, 11*e^4 + 68*e^3 + 67*e^2 - 151*e - 142, 22*e^4 + 83*e^3 - 11*e^2 - 160*e - 47, -29*e^4 - 115*e^3 + 3*e^2 + 237*e + 98, 9*e^4 + 41*e^3 + 7*e^2 - 102*e - 43, 6*e^4 + 17*e^3 - 31*e^2 - 60*e + 13, -13*e^4 - 62*e^3 - 10*e^2 + 169*e + 27, -6*e^4 - 25*e^3 - 10*e^2 + 49*e + 65, 20*e^4 + 86*e^3 + 22*e^2 - 174*e - 100, 3*e^4 + 3*e^3 - 22*e^2 - 7*e - 11, e^4 + 12*e^3 + 18*e^2 - 25*e + 22, 15*e^4 + 64*e^3 + 11*e^2 - 143*e - 76, -3*e^4 - 31*e^3 - 58*e^2 + 63*e + 113, -e^4 - 8*e^3 + 3*e^2 + 42*e - 34, -12*e^4 - 58*e^3 - 24*e^2 + 134*e + 64, e^4 + 14*e^3 + 33*e^2 - 12*e - 33, 10*e^4 + 25*e^3 - 39*e^2 - 45*e + 28, -4*e^4 - 10*e^3 + 32*e^2 + 47*e - 26, 21*e^4 + 91*e^3 + 19*e^2 - 208*e - 104, -2*e^4 - 16*e^3 - 11*e^2 + 60*e + 33, -7*e^3 - 20*e^2 + 5*e + 2, 13*e^4 + 49*e^3 - 6*e^2 - 113*e - 36, e^4 + 11*e^3 + 19*e^2 - 43*e - 35, -13*e^4 - 57*e^3 - 8*e^2 + 143*e + 67, 21*e^4 + 84*e^3 + 7*e^2 - 164*e - 79, 7*e^4 + 33*e^3 + 24*e^2 - 61*e - 107, 14*e^4 + 55*e^3 - 110*e - 72, 2*e^4 + 11*e^3 + 29*e^2 + 10*e - 56, -9*e^4 - 52*e^3 - 48*e^2 + 119*e + 119, 8*e^4 + 58*e^3 + 75*e^2 - 136*e - 155, -6*e^4 - 41*e^3 - 39*e^2 + 114*e + 84, e^4 + e^3 - e^2 + 17*e + 18, 12*e^4 + 62*e^3 + 38*e^2 - 142*e - 95, 19*e^4 + 76*e^3 - 17*e^2 - 187*e - 32, 27*e^4 + 109*e^3 - e^2 - 221*e - 62, -16*e^4 - 77*e^3 - 41*e^2 + 180*e + 116, 21*e^4 + 93*e^3 + 4*e^2 - 243*e - 46, 12*e^4 + 52*e^3 - 135*e - 21, -13*e^4 - 46*e^3 + 14*e^2 + 104*e + 66, -5*e^4 - 29*e^3 - 17*e^2 + 72*e + 9, 6*e^4 + 21*e^3 - 14*e^2 - 54*e - 25, -7*e^4 - 25*e^3 + 28*e^2 + 88*e - 42, -4*e^4 - 26*e^3 - 22*e^2 + 49*e - 7];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {27, 3, -w^3 + 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;