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

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

primesArray := [
[3, 3, -w^3 + 2*w^2 + 5*w - 3],
[5, 5, w],
[7, 7, -w^3 + 2*w^2 + 4*w - 3],
[13, 13, -w^2 + w + 4],
[13, 13, -w^3 + 2*w^2 + 5*w - 2],
[16, 2, 2],
[17, 17, -w + 2],
[19, 19, -w^3 + 2*w^2 + 3*w - 2],
[27, 3, w^3 - 3*w^2 - 4*w + 7],
[31, 31, w^3 - 3*w^2 - 3*w + 7],
[31, 31, -w^3 + w^2 + 6*w + 1],
[41, 41, w^2 - w - 1],
[43, 43, 2*w^3 - 5*w^2 - 6*w + 4],
[47, 47, -w^3 + 2*w^2 + 5*w - 1],
[53, 53, -w - 3],
[53, 53, -w^3 + 3*w^2 + 3*w - 6],
[59, 59, 2*w^2 - 3*w - 6],
[59, 59, w^3 - w^2 - 7*w - 3],
[79, 79, 2*w^3 - 4*w^2 - 8*w + 3],
[79, 79, w^2 - 2*w - 1],
[101, 101, 2*w^3 - 4*w^2 - 9*w + 4],
[103, 103, -w^3 + 3*w^2 + 4*w - 2],
[103, 103, w^2 - 3*w - 2],
[107, 107, -w^3 + w^2 + 5*w - 1],
[109, 109, -w^3 + 2*w^2 + 4*w - 6],
[121, 11, -w^3 + 2*w^2 + 6*w - 3],
[121, 11, w^2 - 3*w - 3],
[125, 5, -w^3 + w^2 + 7*w + 1],
[127, 127, 2*w^3 - 5*w^2 - 5*w + 7],
[127, 127, 2*w^3 - 5*w^2 - 7*w + 6],
[167, 167, -2*w^3 + 5*w^2 + 7*w - 11],
[169, 13, -w^2 + w + 8],
[181, 181, -2*w^3 + 5*w^2 + 8*w - 7],
[193, 193, -w^3 + 3*w^2 + 3*w - 1],
[197, 197, 2*w^3 - 4*w^2 - 7*w + 8],
[197, 197, 3*w^3 - 7*w^2 - 10*w + 9],
[199, 199, w^3 - 2*w^2 - 3*w - 3],
[211, 211, -w^3 + w^2 + 7*w - 2],
[211, 211, w^3 - w^2 - 6*w + 1],
[227, 227, 2*w^3 - 3*w^2 - 9*w + 3],
[229, 229, 3*w^3 - 6*w^2 - 13*w + 8],
[239, 239, 2*w^3 - 4*w^2 - 9*w + 3],
[239, 239, 2*w^3 - 4*w^2 - 6*w + 1],
[241, 241, 3*w^3 - 7*w^2 - 9*w + 6],
[241, 241, -2*w^3 + 4*w^2 + 6*w - 3],
[251, 251, -4*w^3 + 10*w^2 + 15*w - 16],
[251, 251, -w^3 + 3*w^2 - 3],
[271, 271, -w^3 + 4*w^2 + 4*w - 11],
[271, 271, -3*w^3 + 9*w^2 + 6*w - 13],
[277, 277, -w^3 + 3*w^2 + 2*w - 9],
[281, 281, -w^3 + 4*w^2 + w - 8],
[293, 293, 3*w^3 - 7*w^2 - 11*w + 13],
[293, 293, w^2 - 6],
[293, 293, -4*w^3 + 11*w^2 + 9*w - 14],
[293, 293, 3*w^3 - 5*w^2 - 15*w + 6],
[311, 311, -2*w^3 + 4*w^2 + 7*w - 4],
[313, 313, -w^3 + 4*w^2 + 3*w - 13],
[313, 313, -w^3 + w^2 + 6*w - 2],
[317, 317, w^3 - 4*w^2 - w + 9],
[337, 337, 3*w^3 - 7*w^2 - 8*w + 7],
[343, 7, -2*w^3 + 3*w^2 + 9*w + 1],
[359, 359, -w^3 + 2*w^2 + 4*w - 7],
[359, 359, 2*w^3 - 3*w^2 - 10*w + 4],
[367, 367, 2*w^3 - 4*w^2 - 5*w + 2],
[367, 367, 2*w^3 - 3*w^2 - 11*w - 2],
[367, 367, 2*w - 7],
[367, 367, w^3 - 7*w - 4],
[373, 373, -w^3 + 4*w^2 + w - 14],
[373, 373, -2*w^3 + 4*w^2 + 10*w - 11],
[379, 379, w^3 - 2*w^2 - 2*w - 1],
[383, 383, w^3 - 8*w - 4],
[397, 397, 4*w^3 - 10*w^2 - 12*w + 7],
[397, 397, -2*w^3 + 3*w^2 + 9*w - 2],
[401, 401, -w^3 + 6*w + 8],
[409, 409, 2*w^3 - 3*w^2 - 11*w + 4],
[419, 419, 3*w^3 - 6*w^2 - 15*w + 7],
[419, 419, -w^2 + 4*w + 8],
[431, 431, 5*w^3 - 15*w^2 - 10*w + 18],
[431, 431, 3*w^3 - 4*w^2 - 16*w - 2],
[439, 439, w^3 - w^2 - 4*w - 4],
[443, 443, -4*w^3 + 9*w^2 + 15*w - 16],
[461, 461, 2*w^3 - 2*w^2 - 13*w - 1],
[467, 467, 2*w^3 - 6*w^2 - 2*w + 7],
[467, 467, 3*w^3 - 3*w^2 - 19*w - 6],
[479, 479, w^2 - 8],
[479, 479, -3*w^3 + 7*w^2 + 14*w - 11],
[491, 491, 3*w^3 - 5*w^2 - 16*w + 2],
[499, 499, -w^2 + w - 2],
[509, 509, -w^3 + 5*w^2 - w - 11],
[509, 509, w^3 - 5*w - 1],
[521, 521, -w^3 + 4*w^2 + 3*w - 7],
[521, 521, -w^3 + 2*w^2 + 7*w - 4],
[541, 541, -2*w^3 + 4*w^2 + 11*w - 6],
[557, 557, -w^3 + w^2 + 8*w + 3],
[557, 557, w^3 - 2*w^2 - 2*w - 2],
[563, 563, 2*w^3 - 2*w^2 - 10*w - 3],
[563, 563, 3*w^3 - 5*w^2 - 15*w + 1],
[569, 569, 2*w^2 - 3*w - 4],
[569, 569, 3*w^3 - 8*w^2 - 9*w + 7],
[571, 571, -w^3 + 4*w^2 - 11],
[571, 571, 3*w^3 - 8*w^2 - 11*w + 11],
[577, 577, 3*w^3 - 8*w^2 - 5*w + 11],
[577, 577, -4*w^3 + 11*w^2 + 9*w - 9],
[593, 593, 2*w^3 - 7*w^2 - 2*w + 9],
[601, 601, -w^3 + 3*w^2 + 5*w - 9],
[613, 613, 3*w^3 - 5*w^2 - 14*w + 8],
[613, 613, -w^3 + 2*w^2 + 7*w - 3],
[617, 617, -2*w^3 + 6*w^2 + 5*w - 4],
[617, 617, -w^3 + 3*w^2 - w - 6],
[641, 641, 4*w^3 - 9*w^2 - 17*w + 14],
[641, 641, -5*w^3 + 12*w^2 + 18*w - 17],
[643, 643, -3*w^3 + 8*w^2 + 10*w - 13],
[647, 647, 3*w - 4],
[647, 647, -w^3 + 4*w^2 + 2*w - 9],
[653, 653, -w^3 + 2*w^2 + 5*w - 8],
[661, 661, 2*w^3 - 7*w^2 - 5*w + 21],
[661, 661, -w^3 + 2*w^2 + 3*w - 8],
[673, 673, 2*w^3 - 7*w^2 - 5*w + 9],
[673, 673, w^3 - 2*w^2 - 3*w - 4],
[677, 677, -w^3 + 4*w^2 + 3*w - 8],
[683, 683, -3*w^3 + 4*w^2 + 17*w - 1],
[683, 683, -3*w^3 + 7*w^2 + 8*w - 2],
[691, 691, w^2 - 4*w - 2],
[701, 701, 3*w^3 - 6*w^2 - 11*w + 3],
[709, 709, -w^3 + 4*w^2 + 3*w - 17],
[709, 709, w^3 + w^2 - 10*w - 8],
[709, 709, -3*w^3 + 10*w^2 + 2*w - 16],
[709, 709, w^3 - w^2 - 8*w + 1],
[727, 727, -w^3 + 5*w^2 - w - 13],
[727, 727, 2*w^3 - 4*w^2 - 11*w + 2],
[733, 733, -2*w^3 + 3*w^2 + 10*w - 6],
[733, 733, 2*w^3 - 5*w^2 - 5*w + 9],
[739, 739, -3*w^3 + 6*w^2 + 12*w - 8],
[743, 743, 3*w^3 - 4*w^2 - 17*w - 4],
[743, 743, 2*w^3 - 3*w^2 - 8*w + 1],
[743, 743, -3*w^3 + 7*w^2 + 11*w - 7],
[743, 743, -3*w^3 + 6*w^2 + 11*w - 9],
[757, 757, 2*w^3 - 5*w^2 - 6*w + 11],
[757, 757, -w^3 + w^2 + 8*w + 2],
[761, 761, -5*w^3 + 14*w^2 + 13*w - 12],
[769, 769, -3*w^2 + 4*w + 13],
[773, 773, 3*w^3 - 4*w^2 - 18*w + 3],
[797, 797, 4*w^3 - 7*w^2 - 21*w + 8],
[797, 797, -3*w^3 + 7*w^2 + 11*w - 16],
[809, 809, -2*w^2 + 5*w + 11],
[827, 827, w - 6],
[829, 829, -w^3 - w^2 + 11*w + 8],
[841, 29, -3*w^3 + 6*w^2 + 14*w - 6],
[841, 29, -2*w^3 + 5*w^2 + 8*w - 4],
[853, 853, 3*w^3 - 6*w^2 - 9*w + 8],
[857, 857, 2*w^2 - 3*w - 3],
[859, 859, -w^3 + w^2 + 4*w + 7],
[863, 863, 2*w^3 - 3*w^2 - 7*w - 2],
[863, 863, -3*w^3 + 6*w^2 + 10*w - 11],
[881, 881, -3*w^2 + 4*w + 12],
[881, 881, w^2 - 2*w + 3],
[883, 883, 4*w^3 - 13*w^2 - 6*w + 17],
[883, 883, -2*w^3 + 4*w^2 + 12*w - 3],
[887, 887, -3*w^3 + 5*w^2 + 16*w - 8],
[887, 887, 3*w^3 - 5*w^2 - 14*w + 3],
[911, 911, -w^3 + 10*w + 1],
[919, 919, 5*w^3 - 13*w^2 - 14*w + 11],
[929, 929, 2*w^3 - 3*w^2 - 12*w - 4],
[937, 937, -3*w - 7],
[941, 941, -w^3 + 3*w^2 + 3*w - 12],
[947, 947, 2*w^3 - 6*w^2 - 9*w + 8],
[947, 947, -4*w^3 + 7*w^2 + 19*w - 6],
[953, 953, 3*w^2 - w - 9],
[961, 31, 4*w^3 - 12*w^2 - 5*w + 17],
[971, 971, 3*w^3 - 6*w^2 - 16*w + 12],
[983, 983, w^3 + w^2 - 7*w - 12],
[991, 991, -2*w^3 + 7*w^2 + 3*w - 12],
[991, 991, -2*w^3 + 4*w^2 + 12*w - 9],
[997, 997, 3*w^3 - 5*w^2 - 14*w + 9],
[997, 997, -2*w^3 + 3*w^2 + 11*w - 7],
[1013, 1013, -2*w^3 + 6*w^2 + 3*w - 11],
[1019, 1019, -w^3 + 5*w^2 + 2*w - 8],
[1021, 1021, -w^3 + 5*w^2 - w - 17],
[1021, 1021, 3*w^3 - 6*w^2 - 8*w + 3],
[1033, 1033, -2*w^3 + 5*w^2 + 6*w - 13],
[1033, 1033, 3*w^3 - 6*w^2 - 11*w + 4],
[1051, 1051, 3*w^3 - 7*w^2 - 10*w + 13],
[1061, 1061, 2*w^2 - 5*w - 13],
[1063, 1063, -w^3 + w^2 + 5*w + 8],
[1069, 1069, -5*w^3 + 12*w^2 + 19*w - 18],
[1069, 1069, w^3 - 5*w^2 - w + 9],
[1097, 1097, -3*w^3 + 9*w^2 + 8*w - 12],
[1097, 1097, -2*w^3 + 6*w^2 + 7*w - 12],
[1103, 1103, -3*w^3 + 4*w^2 + 15*w - 3],
[1103, 1103, -w^3 + 3*w^2 + 5*w - 11],
[1109, 1109, 5*w^3 - 11*w^2 - 16*w + 6],
[1129, 1129, -w^2 + 3*w - 4],
[1129, 1129, 2*w^3 - 2*w^2 - 10*w - 9],
[1151, 1151, 4*w^3 - 10*w^2 - 14*w + 13],
[1153, 1153, -2*w^3 + 5*w^2 + 11*w - 12],
[1153, 1153, -2*w^3 + 8*w^2 - 13],
[1153, 1153, -w^3 + 3*w^2 + 5*w - 12],
[1153, 1153, 2*w^3 - w^2 - 11*w - 4],
[1163, 1163, -w^3 + w^2 + 9*w - 4],
[1163, 1163, 2*w^3 - 4*w^2 - 10*w - 1],
[1171, 1171, 3*w^3 - 4*w^2 - 18*w - 1],
[1193, 1193, w^3 - 4*w^2 + 2*w + 6],
[1193, 1193, w^3 - 2*w^2 - 5*w - 4],
[1201, 1201, 2*w^3 - 4*w^2 - 9*w - 3],
[1201, 1201, 4*w^3 - 10*w^2 - 13*w + 11],
[1213, 1213, -4*w^3 + 7*w^2 + 19*w - 11],
[1217, 1217, -2*w^3 + 8*w^2 + 2*w - 13],
[1229, 1229, -4*w^3 + 13*w^2 + 6*w - 19],
[1229, 1229, -4*w^3 + 8*w^2 + 16*w - 13],
[1229, 1229, -3*w^3 + 9*w^2 + 6*w - 16],
[1229, 1229, 2*w^3 - 3*w^2 - 6*w - 1],
[1231, 1231, -3*w^3 + 6*w^2 + 16*w - 8],
[1231, 1231, -3*w^2 + w + 12],
[1237, 1237, -2*w^3 + 5*w^2 + 4*w - 9],
[1237, 1237, -w^2 - 3],
[1237, 1237, -w^3 + 6*w - 3],
[1237, 1237, -4*w - 1],
[1249, 1249, -2*w^3 + 4*w^2 + 5*w - 8],
[1283, 1283, 3*w - 8],
[1289, 1289, 2*w^3 - w^2 - 14*w - 6],
[1289, 1289, 4*w^3 - 9*w^2 - 13*w + 8],
[1291, 1291, w^3 - w^2 - 3*w - 4],
[1297, 1297, 4*w^3 - 9*w^2 - 15*w + 9],
[1303, 1303, -w^3 + 4*w^2 - 13],
[1321, 1321, 3*w^3 - 6*w^2 - 13*w + 3],
[1321, 1321, -3*w^3 + 6*w^2 + 11*w - 7],
[1367, 1367, -2*w^3 + 6*w^2 + 4*w - 13],
[1369, 37, 2*w^2 - w - 11],
[1369, 37, w^3 - 2*w^2 - w - 2],
[1373, 1373, -w^3 + 3*w^2 + 8*w - 6],
[1381, 1381, -w^3 + 3*w^2 - 9],
[1381, 1381, 3*w^3 - 8*w^2 - 8*w + 14],
[1427, 1427, 3*w^3 - 7*w^2 - 9*w + 12],
[1429, 1429, 4*w^3 - 9*w^2 - 11*w + 8],
[1433, 1433, -w^3 + 8*w + 12],
[1447, 1447, -4*w^3 + 10*w^2 + 15*w - 14],
[1447, 1447, 6*w^3 - 14*w^2 - 21*w + 19],
[1451, 1451, w^3 - w^2 - 9*w - 4],
[1459, 1459, 3*w^3 - 7*w^2 - 13*w + 9],
[1481, 1481, -3*w^2 + 3*w + 19],
[1481, 1481, -w^3 + 5*w^2 - 3*w - 14],
[1487, 1487, -2*w^3 + 6*w^2 + w - 6],
[1489, 1489, -3*w^3 + 3*w^2 + 19*w + 3],
[1499, 1499, 3*w^2 - 4*w - 7],
[1511, 1511, 4*w^3 - 6*w^2 - 20*w + 3],
[1523, 1523, 8*w^3 - 23*w^2 - 17*w + 27],
[1531, 1531, -w^3 + 2*w^2 + 3*w - 9],
[1531, 1531, 3*w^3 - 9*w^2 - 8*w + 19],
[1549, 1549, -5*w^3 + 11*w^2 + 17*w - 18],
[1553, 1553, 2*w^2 - 9],
[1567, 1567, 2*w^3 - 7*w^2 - 4*w + 22],
[1579, 1579, -2*w^3 + 4*w^2 + 8*w + 3],
[1579, 1579, -3*w^3 + 5*w^2 + 13*w - 1],
[1607, 1607, w^3 - w^2 - 6*w - 8],
[1609, 1609, -4*w^3 + 7*w^2 + 19*w - 2],
[1619, 1619, -5*w^3 + 11*w^2 + 13*w - 9],
[1627, 1627, 4*w^3 - 9*w^2 - 16*w + 11],
[1657, 1657, 2*w^3 - 8*w^2 - w + 12],
[1657, 1657, -4*w^3 + 11*w^2 + 15*w - 17],
[1667, 1667, -w^3 + 5*w^2 - 6],
[1667, 1667, -3*w^3 + 6*w^2 + 10*w - 6],
[1697, 1697, w^3 + w^2 - 9*w - 7],
[1697, 1697, 3*w^3 - 6*w^2 - 13*w + 2],
[1697, 1697, w^3 - 6*w + 1],
[1697, 1697, -5*w^3 + 11*w^2 + 22*w - 17],
[1709, 1709, w^3 - w^2 - 8*w - 9],
[1709, 1709, -w^3 + 3*w^2 - 13],
[1723, 1723, 2*w^3 - 3*w^2 - 7*w - 3],
[1733, 1733, -w^3 + 4*w^2 + 3*w - 19],
[1741, 1741, -w^3 + 5*w^2 - 14],
[1747, 1747, -4*w^3 + 6*w^2 + 22*w + 1],
[1747, 1747, 2*w^3 - 5*w^2 - 9*w + 2],
[1753, 1753, -2*w^3 + 5*w^2 + 3*w - 1],
[1759, 1759, -w^3 + 2*w^2 + 4*w - 9],
[1783, 1783, -3*w^3 + 8*w^2 + 5*w - 12],
[1783, 1783, -2*w^3 + 6*w^2 + 3*w - 12],
[1783, 1783, 2*w^3 - 7*w^2 - 4*w + 17],
[1783, 1783, 2*w^3 - 6*w^2 - w + 7],
[1787, 1787, -5*w^3 + 8*w^2 + 25*w - 3],
[1823, 1823, w^3 - 8*w - 1],
[1823, 1823, -3*w^3 + 5*w^2 + 14*w + 6],
[1831, 1831, -3*w^3 + 8*w^2 + 14*w - 18],
[1847, 1847, w^3 - 3*w^2 - w + 14],
[1847, 1847, -2*w^3 + 2*w^2 + 11*w + 11],
[1861, 1861, 2*w^2 - w - 12],
[1871, 1871, 4*w^3 - 8*w^2 - 19*w + 9],
[1871, 1871, -w^3 + 5*w^2 - w - 7],
[1873, 1873, -3*w^3 + 5*w^2 + 13*w - 4],
[1877, 1877, -4*w^3 + 8*w^2 + 16*w - 7],
[1877, 1877, 3*w^3 - 5*w^2 - 15*w - 2],
[1879, 1879, -w^3 + 4*w^2 - w - 12],
[1907, 1907, w^3 - w^2 - 10*w - 6],
[1931, 1931, -w^3 + 5*w^2 - w - 19],
[1931, 1931, -6*w^3 + 17*w^2 + 12*w - 22],
[1951, 1951, 4*w^3 - 10*w^2 - 14*w + 21],
[1951, 1951, -2*w^3 + 4*w^2 + 13*w - 13],
[1993, 1993, 2*w^3 - 2*w^2 - 11*w + 1],
[1993, 1993, -w^3 + 6*w + 14]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 3*x^5 - 12*x^4 + 15*x^3 + 38*x^2 + 14*x + 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, -1, e, -e^5 + 3*e^4 + 12*e^3 - 16*e^2 - 36*e - 9, e^2 - 2*e - 5, 4/3*e^5 - 14/3*e^4 - 41/3*e^3 + 79/3*e^2 + 38*e + 11/3, -5/3*e^5 + 16/3*e^4 + 58/3*e^3 - 89/3*e^2 - 61*e - 34/3, 7/3*e^5 - 23/3*e^4 - 77/3*e^3 + 124/3*e^2 + 76*e + 41/3, -2*e^5 + 7*e^4 + 20*e^3 - 39*e^2 - 52*e - 4, e^5 - 3*e^4 - 13*e^3 + 18*e^2 + 44*e + 6, -4/3*e^5 + 14/3*e^4 + 41/3*e^3 - 79/3*e^2 - 40*e - 17/3, 2*e^5 - 6*e^4 - 24*e^3 + 31*e^2 + 73*e + 19, -8/3*e^5 + 25/3*e^4 + 94/3*e^3 - 137/3*e^2 - 96*e - 52/3, e^5 - 3*e^4 - 12*e^3 + 16*e^2 + 39*e + 1, -1/3*e^5 + 2/3*e^4 + 17/3*e^3 - 13/3*e^2 - 22*e - 11/3, e^5 - 3*e^4 - 12*e^3 + 15*e^2 + 38*e + 12, 8/3*e^5 - 28/3*e^4 - 82/3*e^3 + 161/3*e^2 + 71*e + 7/3, -1/3*e^5 + 5/3*e^4 + 5/3*e^3 - 34/3*e^2 + 7/3, 5/3*e^5 - 16/3*e^4 - 55/3*e^3 + 83/3*e^2 + 52*e + 31/3, 11/3*e^5 - 37/3*e^4 - 115/3*e^3 + 197/3*e^2 + 106*e + 61/3, -3*e^5 + 11*e^4 + 29*e^3 - 64*e^2 - 76*e - 3, -10/3*e^5 + 32/3*e^4 + 113/3*e^3 - 169/3*e^2 - 115*e - 80/3, 13/3*e^5 - 41/3*e^4 - 152/3*e^3 + 226/3*e^2 + 159*e + 101/3, -1/3*e^5 + 2/3*e^4 + 14/3*e^3 + 2/3*e^2 - 19*e - 50/3, -5*e^5 + 17*e^4 + 53*e^3 - 94*e^2 - 156*e - 25, -4/3*e^5 + 14/3*e^4 + 41/3*e^3 - 85/3*e^2 - 35*e + 19/3, -2/3*e^5 + 7/3*e^4 + 22/3*e^3 - 47/3*e^2 - 22*e + 14/3, -e^4 + 4*e^3 + 7*e^2 - 20*e - 9, -e^3 + 2*e^2 + 7*e - 4, 3*e^5 - 10*e^4 - 33*e^3 + 56*e^2 + 102*e + 7, -3*e^5 + 11*e^4 + 29*e^3 - 63*e^2 - 78*e - 4, -4/3*e^5 + 14/3*e^4 + 41/3*e^3 - 79/3*e^2 - 41*e - 47/3, 16/3*e^5 - 53/3*e^4 - 173/3*e^3 + 283/3*e^2 + 169*e + 98/3, 10/3*e^5 - 35/3*e^4 - 98/3*e^3 + 181/3*e^2 + 92*e + 77/3, -14/3*e^5 + 49/3*e^4 + 142/3*e^3 - 278/3*e^2 - 126*e - 28/3, -e^5 + 3*e^4 + 12*e^3 - 15*e^2 - 33*e - 26, 5/3*e^5 - 13/3*e^4 - 67/3*e^3 + 56/3*e^2 + 78*e + 58/3, 10/3*e^5 - 32/3*e^4 - 113/3*e^3 + 163/3*e^2 + 121*e + 104/3, e^5 - 3*e^4 - 12*e^3 + 17*e^2 + 29*e + 2, -6*e^5 + 20*e^4 + 64*e^3 - 107*e^2 - 184*e - 28, 3*e^5 - 10*e^4 - 31*e^3 + 50*e^2 + 86*e + 20, -2/3*e^5 + 7/3*e^4 + 19/3*e^3 - 35/3*e^2 - 8*e - 52/3, -17/3*e^5 + 55/3*e^4 + 187/3*e^3 - 290/3*e^2 - 182*e - 103/3, e^3 - 15*e - 6, 4*e^5 - 14*e^4 - 42*e^3 + 83*e^2 + 116*e + 5, -13/3*e^5 + 44/3*e^4 + 137/3*e^3 - 235/3*e^2 - 132*e - 119/3, -8*e^5 + 26*e^4 + 89*e^3 - 143*e^2 - 258*e - 47, 11/3*e^5 - 37/3*e^4 - 118/3*e^3 + 212/3*e^2 + 103*e - 2/3, 7/3*e^5 - 23/3*e^4 - 74/3*e^3 + 121/3*e^2 + 67*e + 32/3, -8/3*e^5 + 28/3*e^4 + 79/3*e^3 - 155/3*e^2 - 67*e - 22/3, -2*e^5 + 8*e^4 + 16*e^3 - 47*e^2 - 29*e + 7, -8/3*e^5 + 25/3*e^4 + 91/3*e^3 - 137/3*e^2 - 81*e - 52/3, -1/3*e^5 + 5/3*e^4 + 2/3*e^3 - 25/3*e^2 + 3*e + 10/3, -22/3*e^5 + 68/3*e^4 + 260/3*e^3 - 367/3*e^2 - 272*e - 167/3, -4/3*e^5 + 14/3*e^4 + 41/3*e^3 - 67/3*e^2 - 47*e - 74/3, e^5 - 5*e^4 - 5*e^3 + 34*e^2 + 6*e - 27, -16/3*e^5 + 56/3*e^4 + 161/3*e^3 - 313/3*e^2 - 139*e - 23/3, -13/3*e^5 + 50/3*e^4 + 113/3*e^3 - 283/3*e^2 - 90*e + 25/3, -4/3*e^5 + 11/3*e^4 + 56/3*e^3 - 61/3*e^2 - 72*e - 35/3, -2/3*e^5 + 7/3*e^4 + 19/3*e^3 - 41/3*e^2 - 13*e + 8/3, -5*e^5 + 17*e^4 + 54*e^3 - 97*e^2 - 161*e - 22, 7/3*e^5 - 26/3*e^4 - 68/3*e^3 + 163/3*e^2 + 53*e - 61/3, 19/3*e^5 - 62/3*e^4 - 209/3*e^3 + 328/3*e^2 + 214*e + 134/3, -2/3*e^5 + 10/3*e^4 + 4/3*e^3 - 47/3*e^2 + 16*e - 1/3, 2/3*e^5 - 7/3*e^4 - 19/3*e^3 + 44/3*e^2 + 9*e - 77/3, 4*e^5 - 14*e^4 - 42*e^3 + 85*e^2 + 109*e - 11, 11/3*e^5 - 34/3*e^4 - 130/3*e^3 + 191/3*e^2 + 129*e + 49/3, -11*e^5 + 36*e^4 + 121*e^3 - 194*e^2 - 361*e - 64, -7/3*e^5 + 26/3*e^4 + 62/3*e^3 - 130/3*e^2 - 55*e - 50/3, 6*e^5 - 19*e^4 - 68*e^3 + 101*e^2 + 204*e + 51, -2*e^5 + 6*e^4 + 23*e^3 - 27*e^2 - 73*e - 16, -7/3*e^5 + 26/3*e^4 + 62/3*e^3 - 133/3*e^2 - 45*e - 47/3, -2*e^5 + 6*e^4 + 25*e^3 - 32*e^2 - 87*e - 32, -2*e^4 + 7*e^3 + 19*e^2 - 39*e - 21, e^5 - 4*e^4 - 8*e^3 + 22*e^2 + 13*e - 8, -3*e^5 + 8*e^4 + 42*e^3 - 47*e^2 - 142*e - 23, 8/3*e^5 - 34/3*e^4 - 61/3*e^3 + 206/3*e^2 + 51*e - 80/3, 4*e^5 - 13*e^4 - 43*e^3 + 64*e^2 + 126*e + 45, 26/3*e^5 - 85/3*e^4 - 286/3*e^3 + 446/3*e^2 + 288*e + 190/3, -1/3*e^5 + 2/3*e^4 + 17/3*e^3 - 16/3*e^2 - 20*e - 38/3, 4*e^5 - 15*e^4 - 37*e^3 + 87*e^2 + 89*e - 7, 2*e^5 - 4*e^4 - 32*e^3 + 16*e^2 + 114*e + 45, 20/3*e^5 - 58/3*e^4 - 247/3*e^3 + 296/3*e^2 + 261*e + 196/3, 38/3*e^5 - 127/3*e^4 - 406/3*e^3 + 695/3*e^2 + 382*e + 184/3, 19/3*e^5 - 71/3*e^4 - 176/3*e^3 + 409/3*e^2 + 153*e - 13/3, 11/3*e^5 - 34/3*e^4 - 127/3*e^3 + 167/3*e^2 + 128*e + 115/3, 3*e^5 - 9*e^4 - 36*e^3 + 43*e^2 + 123*e + 44, -11/3*e^5 + 37/3*e^4 + 118/3*e^3 - 197/3*e^2 - 120*e - 58/3, -6*e^5 + 18*e^4 + 72*e^3 - 95*e^2 - 214*e - 61, -1/3*e^5 - 1/3*e^4 + 29/3*e^3 + 11/3*e^2 - 48*e - 38/3, 14/3*e^5 - 52/3*e^4 - 130/3*e^3 + 305/3*e^2 + 99*e - 59/3, -e^4 + 3*e^3 + 10*e^2 - 12*e - 17, 3*e^5 - 10*e^4 - 33*e^3 + 54*e^2 + 109*e + 34, 6*e^5 - 19*e^4 - 71*e^3 + 112*e^2 + 221*e + 18, 10/3*e^5 - 35/3*e^4 - 107/3*e^3 + 217/3*e^2 + 99*e + 20/3, -e^5 + 3*e^4 + 14*e^3 - 21*e^2 - 51*e - 4, 2/3*e^5 - 7/3*e^4 - 19/3*e^3 + 32/3*e^2 + 27*e + 7/3, 25/3*e^5 - 80/3*e^4 - 284/3*e^3 + 424/3*e^2 + 301*e + 215/3, -10/3*e^5 + 35/3*e^4 + 101/3*e^3 - 196/3*e^2 - 91*e + 19/3, 28/3*e^5 - 89/3*e^4 - 323/3*e^3 + 481/3*e^2 + 341*e + 206/3, 13/3*e^5 - 41/3*e^4 - 152/3*e^3 + 226/3*e^2 + 155*e + 89/3, 17/3*e^5 - 55/3*e^4 - 190/3*e^3 + 287/3*e^2 + 201*e + 154/3, -1/3*e^5 + 5/3*e^4 + 8/3*e^3 - 52/3*e^2 - 5*e + 91/3, 2/3*e^5 - 13/3*e^4 + 5/3*e^3 + 80/3*e^2 - 19*e - 98/3, 2/3*e^5 - 7/3*e^4 - 16/3*e^3 + 23/3*e^2 + 18*e + 52/3, 6*e^5 - 20*e^4 - 65*e^3 + 109*e^2 + 203*e + 27, -13/3*e^5 + 41/3*e^4 + 149/3*e^3 - 217/3*e^2 - 154*e - 98/3, 2*e^5 - 8*e^4 - 17*e^3 + 46*e^2 + 41*e - 4, -3*e^5 + 10*e^4 + 34*e^3 - 61*e^2 - 105*e - 1, -7/3*e^5 + 23/3*e^4 + 77/3*e^3 - 133/3*e^2 - 77*e + 34/3, 16/3*e^5 - 50/3*e^4 - 185/3*e^3 + 271/3*e^2 + 188*e + 71/3, 17/3*e^5 - 58/3*e^4 - 175/3*e^3 + 305/3*e^2 + 169*e + 139/3, 20/3*e^5 - 61/3*e^4 - 244/3*e^3 + 344/3*e^2 + 260*e + 145/3, -22/3*e^5 + 68/3*e^4 + 263/3*e^3 - 379/3*e^2 - 277*e - 155/3, 19/3*e^5 - 71/3*e^4 - 170/3*e^3 + 397/3*e^2 + 133*e - 22/3, -37/3*e^5 + 122/3*e^4 + 401/3*e^3 - 661/3*e^2 - 386*e - 179/3, -4*e^5 + 12*e^4 + 47*e^3 - 59*e^2 - 147*e - 17, -3*e^5 + 11*e^4 + 29*e^3 - 63*e^2 - 78*e - 15, 13/3*e^5 - 47/3*e^4 - 125/3*e^3 + 259/3*e^2 + 118*e + 38/3, -8/3*e^5 + 25/3*e^4 + 100/3*e^3 - 152/3*e^2 - 108*e - 28/3, -32/3*e^5 + 100/3*e^4 + 376/3*e^3 - 542/3*e^2 - 396*e - 235/3, 2*e^4 - 7*e^3 - 16*e^2 + 31*e + 29, 11/3*e^5 - 40/3*e^4 - 100/3*e^3 + 203/3*e^2 + 83*e + 55/3, 17/3*e^5 - 52/3*e^4 - 205/3*e^3 + 278/3*e^2 + 214*e + 190/3, -38/3*e^5 + 121/3*e^4 + 439/3*e^3 - 665/3*e^2 - 458*e - 280/3, 8/3*e^5 - 25/3*e^4 - 94/3*e^3 + 140/3*e^2 + 88*e + 1/3, -1/3*e^5 - 4/3*e^4 + 35/3*e^3 + 50/3*e^2 - 47*e - 122/3, -16/3*e^5 + 47/3*e^4 + 197/3*e^3 - 244/3*e^2 - 211*e - 125/3, 13/3*e^5 - 50/3*e^4 - 119/3*e^3 + 307/3*e^2 + 95*e - 37/3, 3*e^5 - 13*e^4 - 21*e^3 + 79*e^2 + 35*e - 48, 20/3*e^5 - 79/3*e^4 - 172/3*e^3 + 464/3*e^2 + 144*e - 65/3, 25/3*e^5 - 92/3*e^4 - 239/3*e^3 + 532/3*e^2 + 208*e - 4/3, -2*e^4 + 9*e^3 + 15*e^2 - 55*e - 17, -e^5 + 3*e^4 + 11*e^3 - 11*e^2 - 36*e - 31, -11/3*e^5 + 40/3*e^4 + 112/3*e^3 - 242/3*e^2 - 104*e + 44/3, -10*e^5 + 33*e^4 + 110*e^3 - 181*e^2 - 316*e - 68, 22/3*e^5 - 71/3*e^4 - 245/3*e^3 + 379/3*e^2 + 251*e + 104/3, -4/3*e^5 + 11/3*e^4 + 47/3*e^3 - 49/3*e^2 - 41*e - 62/3, 8*e^5 - 29*e^4 - 76*e^3 + 158*e^2 + 201*e + 15, 1/3*e^5 - 8/3*e^4 + 10/3*e^3 + 49/3*e^2 - 23*e - 19/3, -5*e^5 + 17*e^4 + 54*e^3 - 100*e^2 - 161*e + 3, -e^5 + 6*e^4 - e^3 - 34*e^2 + 23*e + 8, 11/3*e^5 - 46/3*e^4 - 82/3*e^3 + 281/3*e^2 + 44*e - 77/3, 8/3*e^5 - 25/3*e^4 - 97/3*e^3 + 164/3*e^2 + 95*e - 83/3, 14/3*e^5 - 52/3*e^4 - 130/3*e^3 + 308/3*e^2 + 94*e - 32/3, 32/3*e^5 - 112/3*e^4 - 334/3*e^3 + 647/3*e^2 + 318*e + 103/3, -7/3*e^5 + 20/3*e^4 + 83/3*e^3 - 85/3*e^2 - 86*e - 101/3, -7*e^5 + 22*e^4 + 82*e^3 - 120*e^2 - 250*e - 58, 4/3*e^5 - 14/3*e^4 - 29/3*e^3 + 40/3*e^2 + 17*e + 80/3, -2/3*e^5 + 1/3*e^4 + 49/3*e^3 - 14/3*e^2 - 65*e - 79/3, 2/3*e^5 + 2/3*e^4 - 61/3*e^3 - 7/3*e^2 + 81*e + 55/3, -4*e^5 + 13*e^4 + 47*e^3 - 73*e^2 - 155*e - 44, -11/3*e^5 + 43/3*e^4 + 91/3*e^3 - 230/3*e^2 - 72*e + 2/3, 20/3*e^5 - 61/3*e^4 - 241/3*e^3 + 329/3*e^2 + 244*e + 160/3, -10/3*e^5 + 32/3*e^4 + 107/3*e^3 - 160/3*e^2 - 87*e - 122/3, 3*e^5 - 11*e^4 - 30*e^3 + 67*e^2 + 80*e + 24, 16/3*e^5 - 47/3*e^4 - 206/3*e^3 + 262/3*e^2 + 232*e + 203/3, -6*e^5 + 21*e^4 + 59*e^3 - 105*e^2 - 165*e - 61, 31/3*e^5 - 98/3*e^4 - 359/3*e^3 + 529/3*e^2 + 371*e + 221/3, 31/3*e^5 - 95/3*e^4 - 371/3*e^3 + 523/3*e^2 + 370*e + 185/3, 2/3*e^5 + 2/3*e^4 - 58/3*e^3 - 19/3*e^2 + 83*e + 91/3, 8*e^5 - 25*e^4 - 95*e^3 + 137*e^2 + 303*e + 64, 2*e^5 - 6*e^4 - 24*e^3 + 31*e^2 + 82*e + 3, -16/3*e^5 + 65/3*e^4 + 122/3*e^3 - 373/3*e^2 - 68*e + 127/3, 8*e^5 - 27*e^4 - 85*e^3 + 146*e^2 + 247*e + 16, -1/3*e^5 - 7/3*e^4 + 56/3*e^3 + 47/3*e^2 - 89*e - 158/3, -14/3*e^5 + 55/3*e^4 + 115/3*e^3 - 317/3*e^2 - 79*e + 122/3, -23/3*e^5 + 70/3*e^4 + 274/3*e^3 - 371/3*e^2 - 277*e - 181/3, -8/3*e^5 + 28/3*e^4 + 76/3*e^3 - 143/3*e^2 - 62*e + 11/3, 5*e^5 - 17*e^4 - 55*e^3 + 107*e^2 + 157*e - 8, 7/3*e^5 - 26/3*e^4 - 68/3*e^3 + 154/3*e^2 + 71*e - 40/3, 8*e^5 - 27*e^4 - 87*e^3 + 156*e^2 + 259*e + 19, 10*e^5 - 35*e^4 - 101*e^3 + 190*e^2 + 283*e + 51, 16/3*e^5 - 56/3*e^4 - 152/3*e^3 + 283/3*e^2 + 122*e + 131/3, 4/3*e^5 - 11/3*e^4 - 56/3*e^3 + 88/3*e^2 + 54*e - 61/3, -4*e^5 + 14*e^4 + 41*e^3 - 83*e^2 - 106*e - 11, 23/3*e^5 - 82/3*e^4 - 235/3*e^3 + 479/3*e^2 + 216*e + 31/3, 29/3*e^5 - 100/3*e^4 - 298/3*e^3 + 554/3*e^2 + 269*e + 82/3, 7*e^5 - 25*e^4 - 69*e^3 + 142*e^2 + 192*e - 5, -17*e^5 + 58*e^4 + 177*e^3 - 316*e^2 - 506*e - 78, -9*e^5 + 28*e^4 + 105*e^3 - 152*e^2 - 310*e - 59, -19/3*e^5 + 65/3*e^4 + 203/3*e^3 - 373/3*e^2 - 188*e - 56/3, -25/3*e^5 + 86/3*e^4 + 257/3*e^3 - 490/3*e^2 - 218*e - 20/3, -40/3*e^5 + 137/3*e^4 + 419/3*e^3 - 748/3*e^2 - 415*e - 236/3, 11/3*e^5 - 43/3*e^4 - 100/3*e^3 + 278/3*e^2 + 77*e - 131/3, 8*e^5 - 25*e^4 - 97*e^3 + 146*e^2 + 305*e + 35, -4*e^5 + 15*e^4 + 37*e^3 - 89*e^2 - 88*e + 18, -38/3*e^5 + 124/3*e^4 + 418/3*e^3 - 677/3*e^2 - 397*e - 205/3, -e^5 + 2*e^4 + 15*e^3 - 4*e^2 - 50*e - 22, -11/3*e^5 + 43/3*e^4 + 97/3*e^3 - 248/3*e^2 - 84*e - 7/3, -16/3*e^5 + 53/3*e^4 + 170/3*e^3 - 271/3*e^2 - 168*e - 128/3, -26/3*e^5 + 88/3*e^4 + 277/3*e^3 - 479/3*e^2 - 275*e - 151/3, 4/3*e^5 - 17/3*e^4 - 38/3*e^3 + 130/3*e^2 + 30*e - 31/3, -4*e^5 + 12*e^4 + 49*e^3 - 65*e^2 - 149*e - 63, 3*e^5 - 12*e^4 - 25*e^3 + 66*e^2 + 70*e + 6, -5*e^5 + 14*e^4 + 67*e^3 - 80*e^2 - 227*e - 44, -40/3*e^5 + 137/3*e^4 + 422/3*e^3 - 748/3*e^2 - 421*e - 233/3, 14/3*e^5 - 37/3*e^4 - 190/3*e^3 + 191/3*e^2 + 198*e + 202/3, 2*e^5 - 8*e^4 - 15*e^3 + 45*e^2 + 15*e + 9, 32/3*e^5 - 100/3*e^4 - 373/3*e^3 + 539/3*e^2 + 375*e + 223/3, -22/3*e^5 + 71/3*e^4 + 242/3*e^3 - 379/3*e^2 - 228*e - 113/3, -50/3*e^5 + 166/3*e^4 + 547/3*e^3 - 920/3*e^2 - 535*e - 331/3, -17/3*e^5 + 55/3*e^4 + 193/3*e^3 - 311/3*e^2 - 202*e - 46/3, -8*e^5 + 26*e^4 + 90*e^3 - 149*e^2 - 258*e - 33, -e^5 + 5*e^4 + 2*e^3 - 21*e^2 + 13*e - 18, -17/3*e^5 + 67/3*e^4 + 148/3*e^3 - 407/3*e^2 - 113*e + 101/3, 28/3*e^5 - 86/3*e^4 - 332/3*e^3 + 475/3*e^2 + 322*e + 158/3, -40/3*e^5 + 140/3*e^4 + 407/3*e^3 - 775/3*e^2 - 383*e - 149/3, 6*e^5 - 21*e^4 - 61*e^3 + 120*e^2 + 171*e + 1, 7*e^5 - 25*e^4 - 70*e^3 + 141*e^2 + 207*e + 20, -3*e^3 + 10*e^2 + 14*e - 28, e^5 - 6*e^4 + 3*e^3 + 26*e^2 - 42*e - 16, 56/3*e^5 - 190/3*e^4 - 592/3*e^3 + 1049/3*e^2 + 570*e + 283/3, 9*e^5 - 27*e^4 - 111*e^3 + 149*e^2 + 364*e + 52, -2/3*e^5 + 7/3*e^4 + 25/3*e^3 - 56/3*e^2 - 21*e + 95/3, -14/3*e^5 + 40/3*e^4 + 172/3*e^3 - 188/3*e^2 - 180*e - 136/3, -34/3*e^5 + 113/3*e^4 + 365/3*e^3 - 619/3*e^2 - 331*e - 134/3, -2*e^4 + 9*e^3 + 14*e^2 - 57*e - 10, 7*e^5 - 24*e^4 - 75*e^3 + 140*e^2 + 222*e - 12, -5*e^5 + 17*e^4 + 55*e^3 - 100*e^2 - 172*e - 17, -1/3*e^5 + 2/3*e^4 + 17/3*e^3 + 5/3*e^2 - 36*e - 47/3, -35/3*e^5 + 115/3*e^4 + 391/3*e^3 - 656/3*e^2 - 384*e - 139/3, 26/3*e^5 - 85/3*e^4 - 286/3*e^3 + 446/3*e^2 + 289*e + 247/3, 32/3*e^5 - 103/3*e^4 - 364/3*e^3 + 566/3*e^2 + 375*e + 181/3, 20/3*e^5 - 70/3*e^4 - 196/3*e^3 + 392/3*e^2 + 156*e - 20/3, 17/3*e^5 - 52/3*e^4 - 208/3*e^3 + 287/3*e^2 + 239*e + 178/3, -47/3*e^5 + 160/3*e^4 + 484/3*e^3 - 848/3*e^2 - 443*e - 244/3, -1/3*e^5 + 5/3*e^4 + 8/3*e^3 - 31/3*e^2 - 15*e + 16/3, -37/3*e^5 + 125/3*e^4 + 392/3*e^3 - 682/3*e^2 - 367*e - 113/3, -8/3*e^5 + 16/3*e^4 + 127/3*e^3 - 44/3*e^2 - 177*e - 232/3, -47/3*e^5 + 157/3*e^4 + 511/3*e^3 - 884/3*e^2 - 498*e - 223/3, 12*e^5 - 37*e^4 - 142*e^3 + 195*e^2 + 446*e + 98, 10/3*e^5 - 29/3*e^4 - 131/3*e^3 + 163/3*e^2 + 149*e + 38/3, -40/3*e^5 + 128/3*e^4 + 458/3*e^3 - 700/3*e^2 - 477*e - 200/3, e^4 - 4*e^3 - 12*e^2 + 26*e + 28, -8*e^5 + 29*e^4 + 77*e^3 - 162*e^2 - 209*e - 18, -6*e^5 + 14*e^4 + 87*e^3 - 64*e^2 - 299*e - 103, -10/3*e^5 + 35/3*e^4 + 101/3*e^3 - 202/3*e^2 - 94*e + 37/3, -14/3*e^5 + 55/3*e^4 + 124/3*e^3 - 326/3*e^2 - 105*e + 29/3, 4*e^5 - 10*e^4 - 58*e^3 + 53*e^2 + 200*e + 69, 17*e^5 - 51*e^4 - 206*e^3 + 270*e^2 + 644*e + 131, -18*e^5 + 58*e^4 + 205*e^3 - 324*e^2 - 622*e - 118, -19/3*e^5 + 65/3*e^4 + 206/3*e^3 - 361/3*e^2 - 226*e - 68/3, -19/3*e^5 + 65/3*e^4 + 197/3*e^3 - 358/3*e^2 - 196*e + 25/3, 4*e^5 - 14*e^4 - 38*e^3 + 71*e^2 + 98*e + 11, -19/3*e^5 + 59/3*e^4 + 221/3*e^3 - 304/3*e^2 - 241*e - 191/3, 44/3*e^5 - 136/3*e^4 - 523/3*e^3 + 740/3*e^2 + 558*e + 280/3, 23/3*e^5 - 79/3*e^4 - 229/3*e^3 + 416/3*e^2 + 204*e + 52/3, 12*e^5 - 42*e^4 - 121*e^3 + 234*e^2 + 333*e + 26, 10*e^5 - 31*e^4 - 116*e^3 + 156*e^2 + 364*e + 103, -26/3*e^5 + 91/3*e^4 + 271/3*e^3 - 545/3*e^2 - 246*e - 4/3, 13/3*e^5 - 47/3*e^4 - 131/3*e^3 + 295/3*e^2 + 122*e - 118/3, -3*e^4 + 11*e^3 + 30*e^2 - 64*e - 57, 37/3*e^5 - 116/3*e^4 - 428/3*e^3 + 619/3*e^2 + 427*e + 233/3, -44/3*e^5 + 148/3*e^4 + 472/3*e^3 - 815/3*e^2 - 456*e - 235/3, e^5 - 3*e^4 - 12*e^3 + 11*e^2 + 33*e + 60, -29/3*e^5 + 100/3*e^4 + 301/3*e^3 - 566/3*e^2 - 270*e - 142/3, 10*e^5 - 34*e^4 - 104*e^3 + 182*e^2 + 306*e + 72, -32/3*e^5 + 97/3*e^4 + 385/3*e^3 - 509/3*e^2 - 405*e - 208/3, 16/3*e^5 - 53/3*e^4 - 182/3*e^3 + 307/3*e^2 + 188*e + 146/3, 44/3*e^5 - 145/3*e^4 - 481/3*e^3 + 773/3*e^2 + 479*e + 325/3, -38/3*e^5 + 124/3*e^4 + 421/3*e^3 - 659/3*e^2 - 419*e - 349/3, 7/3*e^5 - 23/3*e^4 - 71/3*e^3 + 112/3*e^2 + 60*e - 43/3, -9*e^5 + 29*e^4 + 102*e^3 - 159*e^2 - 309*e - 86, 2/3*e^5 - 19/3*e^4 + 29/3*e^3 + 122/3*e^2 - 71*e - 125/3, 7/3*e^5 - 23/3*e^4 - 74/3*e^3 + 103/3*e^2 + 81*e + 176/3, -13*e^5 + 42*e^4 + 145*e^3 - 221*e^2 - 440*e - 64, -22/3*e^5 + 80/3*e^4 + 206/3*e^3 - 442/3*e^2 - 170*e + 4/3, -23/3*e^5 + 70/3*e^4 + 274/3*e^3 - 362/3*e^2 - 281*e - 106/3, 11/3*e^5 - 49/3*e^4 - 70/3*e^3 + 287/3*e^2 + 33*e - 86/3, 19/3*e^5 - 59/3*e^4 - 221/3*e^3 + 328/3*e^2 + 210*e + 98/3, 8/3*e^5 - 37/3*e^4 - 37/3*e^3 + 191/3*e^2 + 7*e - 2/3, 20/3*e^5 - 61/3*e^4 - 241/3*e^3 + 329/3*e^2 + 253*e + 82/3, -8/3*e^5 + 25/3*e^4 + 94/3*e^3 - 110/3*e^2 - 118*e - 163/3, -29/3*e^5 + 91/3*e^4 + 328/3*e^3 - 461/3*e^2 - 321*e - 295/3, 38/3*e^5 - 136/3*e^4 - 376/3*e^3 + 758/3*e^2 + 348*e + 97/3, -37/3*e^5 + 125/3*e^4 + 386/3*e^3 - 655/3*e^2 - 381*e - 314/3, 8/3*e^5 - 22/3*e^4 - 103/3*e^3 + 107/3*e^2 + 127*e + 76/3, -35/3*e^5 + 121/3*e^4 + 358/3*e^3 - 668/3*e^2 - 307*e - 160/3, 8*e^5 - 28*e^4 - 79*e^3 + 151*e^2 + 205*e + 36, -26/3*e^5 + 85/3*e^4 + 277/3*e^3 - 431/3*e^2 - 263*e - 238/3, 5/3*e^5 - 28/3*e^4 - 7/3*e^3 + 164/3*e^2 - 25*e - 182/3, -24*e^5 + 80*e^4 + 259*e^3 - 438*e^2 - 745*e - 118, 32/3*e^5 - 112/3*e^4 - 319/3*e^3 + 593/3*e^2 + 299*e + 151/3, 35/3*e^5 - 112/3*e^4 - 397/3*e^3 + 602/3*e^2 + 392*e + 214/3, 5/3*e^5 - 22/3*e^4 - 31/3*e^3 + 110/3*e^2 + 21*e + 142/3, 14*e^5 - 48*e^4 - 146*e^3 + 260*e^2 + 418*e + 66, 2/3*e^5 - 7/3*e^4 - 19/3*e^3 + 29/3*e^2 + 27*e + 112/3, -e^4 + 5*e^3 + 5*e^2 - 29*e + 5, -2*e^5 + 8*e^4 + 17*e^3 - 51*e^2 - 17*e + 3, 3*e^5 - 8*e^4 - 38*e^3 + 31*e^2 + 132*e + 25, -19/3*e^5 + 53/3*e^4 + 251/3*e^3 - 298/3*e^2 - 266*e - 161/3, 23*e^5 - 77*e^4 - 247*e^3 + 418*e^2 + 736*e + 137, -2*e^5 + 5*e^4 + 30*e^3 - 39*e^2 - 92*e + 29, -14/3*e^5 + 40/3*e^4 + 184/3*e^3 - 245/3*e^2 - 203*e - 13/3, 6*e^5 - 23*e^4 - 51*e^3 + 120*e^2 + 113*e + 35, 32/3*e^5 - 103/3*e^4 - 358/3*e^3 + 563/3*e^2 + 331*e + 160/3, -41/3*e^5 + 151/3*e^4 + 397/3*e^3 - 875/3*e^2 - 362*e - 19/3];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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