/*
  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![4, 6, -7, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {16, 2, 2}>;

primesArray := [
[3, 3, -w + 1],
[4, 2, w],
[4, 2, -1/2*w^3 + 1/2*w^2 + 7/2*w - 3],
[7, 7, -1/2*w^3 + 1/2*w^2 + 5/2*w - 2],
[13, 13, 1/2*w^3 + 1/2*w^2 - 7/2*w - 1],
[17, 17, 1/2*w^3 - 1/2*w^2 - 5/2*w],
[27, 3, -w^3 + 7*w + 1],
[31, 31, w + 3],
[31, 31, -w^2 + 5],
[37, 37, w^2 - 3],
[41, 41, 1/2*w^3 + 1/2*w^2 - 3/2*w - 2],
[47, 47, w^3 - 5*w - 3],
[53, 53, 1/2*w^3 - 1/2*w^2 - 7/2*w],
[61, 61, w^3 - w^2 - 5*w + 3],
[83, 83, w^3 + w^2 - 6*w - 7],
[83, 83, -w^3 + 5*w + 1],
[83, 83, 2*w - 1],
[83, 83, -3/2*w^3 - 1/2*w^2 + 19/2*w + 2],
[89, 89, 1/2*w^3 - 1/2*w^2 - 7/2*w - 2],
[89, 89, w^3 - 5*w + 5],
[97, 97, -1/2*w^3 + 1/2*w^2 + 7/2*w - 6],
[97, 97, -1/2*w^3 + 1/2*w^2 + 9/2*w - 1],
[103, 103, 3/2*w^3 + 3/2*w^2 - 19/2*w - 9],
[109, 109, 1/2*w^3 - 1/2*w^2 - 9/2*w + 4],
[127, 127, 1/2*w^3 + 1/2*w^2 - 3/2*w - 3],
[127, 127, 1/2*w^3 - 1/2*w^2 - 5/2*w + 6],
[139, 139, 1/2*w^3 + 3/2*w^2 - 9/2*w - 3],
[139, 139, 3/2*w^3 - 1/2*w^2 - 21/2*w + 1],
[149, 149, -1/2*w^3 + 1/2*w^2 + 9/2*w - 2],
[151, 151, -1/2*w^3 - 1/2*w^2 + 9/2*w - 3],
[157, 157, 3/2*w^3 - 1/2*w^2 - 21/2*w + 4],
[163, 163, 1/2*w^3 + 3/2*w^2 - 5/2*w - 3],
[167, 167, -1/2*w^3 + 3/2*w^2 + 9/2*w - 9],
[179, 179, w^3 - w^2 - 5*w + 1],
[179, 179, -1/2*w^3 + 1/2*w^2 + 3/2*w - 4],
[181, 181, 5/2*w^3 + 3/2*w^2 - 31/2*w - 10],
[191, 191, -1/2*w^3 + 5/2*w^2 + 7/2*w - 14],
[191, 191, 2*w^3 - 2*w^2 - 12*w + 11],
[193, 193, w^2 + 1],
[193, 193, 1/2*w^3 + 3/2*w^2 - 9/2*w - 6],
[197, 197, 1/2*w^3 + 1/2*w^2 - 11/2*w + 1],
[197, 197, -3/2*w^3 + 3/2*w^2 + 17/2*w - 8],
[199, 199, 3/2*w^3 - 3/2*w^2 - 19/2*w + 7],
[227, 227, 1/2*w^3 + 1/2*w^2 - 7/2*w - 6],
[227, 227, -2*w^3 - w^2 + 14*w + 9],
[229, 229, w^3 - w^2 - 4*w + 3],
[229, 229, 3/2*w^3 - 1/2*w^2 - 17/2*w + 1],
[233, 233, -w^3 - w^2 + 7*w + 9],
[233, 233, 1/2*w^3 + 3/2*w^2 - 5/2*w - 9],
[251, 251, -5/2*w^3 - 1/2*w^2 + 31/2*w + 2],
[257, 257, -3/2*w^3 - 1/2*w^2 + 19/2*w + 1],
[257, 257, 1/2*w^3 + 1/2*w^2 - 11/2*w - 1],
[269, 269, 2*w^3 - 2*w^2 - 13*w + 15],
[269, 269, 1/2*w^3 + 3/2*w^2 - 9/2*w - 4],
[271, 271, 1/2*w^3 + 3/2*w^2 - 5/2*w - 6],
[277, 277, -1/2*w^3 + 5/2*w^2 + 5/2*w - 16],
[281, 281, -2*w^3 + 12*w + 1],
[281, 281, -3/2*w^3 - 1/2*w^2 + 15/2*w - 3],
[283, 283, 5/2*w^3 + 3/2*w^2 - 29/2*w - 8],
[283, 283, 3/2*w^3 + 1/2*w^2 - 15/2*w],
[293, 293, -w^3 - w^2 + 6*w + 1],
[307, 307, -3/2*w^3 + 1/2*w^2 + 13/2*w + 4],
[307, 307, -3/2*w^3 - 1/2*w^2 + 13/2*w - 2],
[311, 311, 2*w^2 + w - 11],
[311, 311, 1/2*w^3 + 3/2*w^2 - 5/2*w - 5],
[313, 313, 3/2*w^3 - 1/2*w^2 - 17/2*w + 2],
[317, 317, w^3 + 2*w^2 - 7*w - 3],
[331, 331, 2*w^2 + 2*w - 9],
[343, 7, -w^3 + 2*w^2 + 5*w - 11],
[349, 349, 5/2*w^3 + 1/2*w^2 - 35/2*w - 6],
[353, 353, 1/2*w^3 + 3/2*w^2 - 7/2*w - 4],
[359, 359, w - 5],
[361, 19, -5/2*w^3 + 1/2*w^2 + 33/2*w],
[361, 19, w^3 - w^2 - 8*w + 3],
[367, 367, -3/2*w^3 + 3/2*w^2 + 15/2*w - 8],
[383, 383, -1/2*w^3 + 1/2*w^2 + 1/2*w - 3],
[397, 397, -3/2*w^3 + 5/2*w^2 + 17/2*w - 16],
[397, 397, 1/2*w^3 - 1/2*w^2 - 5/2*w - 3],
[401, 401, -w^3 + 3*w - 3],
[401, 401, 3/2*w^3 + 3/2*w^2 - 21/2*w - 4],
[409, 409, 3/2*w^3 - 1/2*w^2 - 19/2*w - 3],
[419, 419, -5/2*w^3 - 1/2*w^2 + 29/2*w + 1],
[421, 421, -1/2*w^3 + 3/2*w^2 + 9/2*w - 11],
[421, 421, -3*w + 1],
[431, 431, 1/2*w^3 - 1/2*w^2 + 1/2*w + 2],
[431, 431, w^3 - 7*w - 5],
[433, 433, 1/2*w^3 + 1/2*w^2 - 3/2*w - 6],
[433, 433, w^3 + w^2 - 6*w + 3],
[467, 467, 1/2*w^3 + 3/2*w^2 - 11/2*w - 6],
[487, 487, -2*w^3 + w^2 + 12*w - 9],
[503, 503, 3/2*w^3 - 3/2*w^2 - 19/2*w + 6],
[503, 503, w^2 + 2*w - 7],
[523, 523, w^3 - w^2 - 6*w + 1],
[523, 523, -w^3 + 3*w^2 + 7*w - 19],
[541, 541, -3/2*w^3 + 3/2*w^2 + 15/2*w - 7],
[541, 541, w^3 - w^2 - 8*w + 7],
[547, 547, -1/2*w^3 + 3/2*w^2 + 3/2*w - 8],
[557, 557, 2*w^2 - w - 9],
[563, 563, -3*w^3 - w^2 + 19*w + 5],
[563, 563, 7/2*w^3 + 5/2*w^2 - 45/2*w - 16],
[563, 563, -w^3 + 3*w + 3],
[563, 563, -w^2 + 2*w - 3],
[569, 569, w^3 - w^2 - 6*w - 1],
[571, 571, -5/2*w^3 + 3/2*w^2 + 31/2*w - 11],
[577, 577, 2*w^2 - 2*w - 5],
[587, 587, 3/2*w^3 + 1/2*w^2 - 23/2*w - 3],
[587, 587, -2*w^3 + 14*w + 1],
[599, 599, -3/2*w^3 - 1/2*w^2 + 13/2*w + 4],
[607, 607, -3/2*w^3 - 3/2*w^2 + 21/2*w + 13],
[613, 613, -1/2*w^3 + 3/2*w^2 + 11/2*w - 6],
[613, 613, w^3 + w^2 - 8*w - 7],
[617, 617, 2*w^2 - w - 5],
[625, 5, -5],
[653, 653, -1/2*w^3 + 3/2*w^2 + 9/2*w - 3],
[653, 653, -2*w^3 - w^2 + 14*w + 3],
[659, 659, -7/2*w^3 - 1/2*w^2 + 43/2*w + 6],
[661, 661, 5/2*w^3 + 1/2*w^2 - 25/2*w - 5],
[661, 661, w^3 + w^2 - 5*w - 7],
[677, 677, -11/2*w^3 - 3/2*w^2 + 73/2*w + 13],
[683, 683, -7/2*w^3 + 5/2*w^2 + 45/2*w - 16],
[683, 683, -w^3 + 3*w^2 + 5*w - 11],
[691, 691, 2*w^2 - 15],
[691, 691, w^3 + w^2 - 4*w - 5],
[701, 701, -7/2*w^3 - 1/2*w^2 + 47/2*w + 5],
[701, 701, -w^3 + w^2 + 3*w - 5],
[719, 719, 5/2*w^3 + 1/2*w^2 - 27/2*w - 5],
[727, 727, -1/2*w^3 - 1/2*w^2 + 3/2*w - 4],
[733, 733, 3/2*w^3 - 1/2*w^2 - 15/2*w + 1],
[733, 733, -w^3 - 2*w^2 + 5*w + 5],
[733, 733, 1/2*w^3 - 1/2*w^2 + 3/2*w + 2],
[733, 733, 1/2*w^3 + 5/2*w^2 - 5/2*w - 13],
[739, 739, 2*w^2 - 7],
[743, 743, 3/2*w^3 - 1/2*w^2 - 9/2*w + 4],
[743, 743, 2*w^3 - 12*w + 3],
[751, 751, 1/2*w^3 + 1/2*w^2 - 11/2*w - 5],
[751, 751, -1/2*w^3 + 1/2*w^2 + 11/2*w - 2],
[761, 761, -3/2*w^3 + 3/2*w^2 + 23/2*w - 5],
[769, 769, w^3 - 2*w^2 - 5*w + 13],
[787, 787, -3/2*w^3 + 1/2*w^2 + 15/2*w - 3],
[787, 787, 2*w^3 - 10*w - 3],
[797, 797, 3/2*w^3 - 1/2*w^2 - 21/2*w + 7],
[797, 797, -2*w^3 + 2*w^2 + 14*w - 13],
[809, 809, -4*w^3 - 2*w^2 + 25*w + 15],
[821, 821, 2*w^3 + 2*w^2 - 13*w - 11],
[839, 839, -1/2*w^3 + 3/2*w^2 + 3/2*w - 9],
[839, 839, -7/2*w^3 - 5/2*w^2 + 47/2*w + 18],
[841, 29, 3/2*w^3 - 1/2*w^2 - 17/2*w + 10],
[841, 29, 2*w^3 - 15*w + 5],
[853, 853, 2*w^3 - 9*w - 3],
[853, 853, -1/2*w^3 - 3/2*w^2 - 3/2*w + 7],
[857, 857, 3/2*w^3 + 3/2*w^2 - 9/2*w - 4],
[857, 857, w^3 - 3*w^2 - 5*w + 9],
[857, 857, -w^3 + 3*w^2 + 6*w - 13],
[857, 857, 1/2*w^3 + 5/2*w^2 - 7/2*w - 14],
[859, 859, -5/2*w^3 - 1/2*w^2 + 35/2*w + 3],
[863, 863, 2*w^3 - 2*w^2 - 13*w + 9],
[881, 881, 2*w^2 + w - 13],
[881, 881, 5/2*w^3 - 1/2*w^2 - 33/2*w - 1],
[907, 907, -w^3 + 3*w^2 + 6*w - 15],
[919, 919, 3/2*w^3 + 5/2*w^2 - 19/2*w - 4],
[929, 929, 3/2*w^3 + 1/2*w^2 - 19/2*w + 1],
[941, 941, 3/2*w^3 - 1/2*w^2 - 13/2*w],
[947, 947, -1/2*w^3 + 5/2*w^2 + 13/2*w - 11],
[947, 947, -7/2*w^3 - 1/2*w^2 + 43/2*w + 4],
[947, 947, 1/2*w^3 + 1/2*w^2 - 13/2*w - 1],
[947, 947, -4*w - 3],
[953, 953, 3/2*w^3 - 7/2*w^2 - 21/2*w + 22],
[953, 953, 1/2*w^3 + 5/2*w^2 - 3/2*w - 10],
[961, 31, 3/2*w^3 + 3/2*w^2 - 17/2*w - 10],
[967, 967, w^3 - 2*w^2 - 7*w + 7],
[967, 967, 1/2*w^3 - 1/2*w^2 - 5/2*w - 4],
[977, 977, -2*w^3 + 11*w + 1],
[977, 977, 3/2*w^3 + 1/2*w^2 - 25/2*w + 4],
[991, 991, -1/2*w^3 + 1/2*w^2 + 5/2*w - 8],
[997, 997, 1/2*w^3 + 3/2*w^2 - 1/2*w - 7],
[1013, 1013, 2*w^3 - w^2 - 14*w + 5],
[1013, 1013, 3/2*w^3 + 3/2*w^2 - 23/2*w - 3],
[1019, 1019, -1/2*w^3 + 3/2*w^2 + 3/2*w - 12],
[1019, 1019, -3/2*w^3 + 7/2*w^2 + 17/2*w - 22],
[1049, 1049, -5/2*w^3 - 5/2*w^2 + 33/2*w + 15],
[1049, 1049, -3*w^3 - w^2 + 18*w + 3],
[1051, 1051, 3*w^3 - 3*w^2 - 18*w + 17],
[1061, 1061, 3/2*w^3 + 1/2*w^2 - 15/2*w - 7],
[1061, 1061, -3*w - 5],
[1087, 1087, w^2 - 2*w - 7],
[1087, 1087, -w^3 - w^2 + 7*w - 1],
[1091, 1091, 3/2*w^3 - 3/2*w^2 - 23/2*w + 3],
[1093, 1093, 2*w^2 + w + 1],
[1103, 1103, -2*w^3 + w^2 + 12*w - 1],
[1103, 1103, 11/2*w^3 + 3/2*w^2 - 69/2*w - 11],
[1103, 1103, -3/2*w^3 - 3/2*w^2 + 23/2*w + 9],
[1103, 1103, 3/2*w^3 - 3/2*w^2 - 15/2*w + 1],
[1109, 1109, -9/2*w^3 - 1/2*w^2 + 57/2*w + 3],
[1151, 1151, 5/2*w^3 + 5/2*w^2 - 31/2*w - 14],
[1151, 1151, -1/2*w^3 - 1/2*w^2 + 7/2*w - 5],
[1153, 1153, -1/2*w^3 + 5/2*w^2 + 9/2*w - 15],
[1163, 1163, -1/2*w^3 + 3/2*w^2 + 5/2*w - 13],
[1163, 1163, -5/2*w^3 - 1/2*w^2 + 33/2*w + 1],
[1181, 1181, -5/2*w^3 + 1/2*w^2 + 33/2*w - 5],
[1181, 1181, -w^3 + w^2 + 4*w - 9],
[1181, 1181, -5/2*w^3 + 1/2*w^2 + 35/2*w],
[1181, 1181, -3/2*w^3 - 3/2*w^2 + 13/2*w + 6],
[1187, 1187, 7/2*w^3 - 5/2*w^2 - 43/2*w + 15],
[1193, 1193, -3/2*w^3 + 1/2*w^2 + 13/2*w - 2],
[1201, 1201, -2*w^3 - 2*w^2 + 13*w + 5],
[1201, 1201, 3/2*w^3 + 1/2*w^2 - 25/2*w + 2],
[1213, 1213, 3*w^3 - w^2 - 18*w + 3],
[1213, 1213, 3/2*w^3 + 5/2*w^2 - 21/2*w - 6],
[1217, 1217, -1/2*w^3 - 1/2*w^2 + 11/2*w - 7],
[1223, 1223, -1/2*w^3 - 1/2*w^2 - 5/2*w],
[1237, 1237, -w^3 - 2*w^2 + 7*w + 9],
[1259, 1259, -1/2*w^3 + 3/2*w^2 + 11/2*w - 12],
[1279, 1279, 1/2*w^3 + 5/2*w^2 - 3/2*w - 13],
[1279, 1279, 3/2*w^3 - 1/2*w^2 - 11/2*w + 1],
[1297, 1297, -4*w^3 + 25*w - 1],
[1301, 1301, 5/2*w^3 - 1/2*w^2 - 29/2*w + 4],
[1303, 1303, -3/2*w^3 - 1/2*w^2 + 25/2*w + 4],
[1303, 1303, w^3 + w^2 - 7*w + 3],
[1319, 1319, 9/2*w^3 - 9/2*w^2 - 57/2*w + 29],
[1321, 1321, 1/2*w^3 + 5/2*w^2 - 3/2*w - 8],
[1321, 1321, w^2 - 2*w - 9],
[1381, 1381, 1/2*w^3 + 1/2*w^2 - 1/2*w - 7],
[1399, 1399, -1/2*w^3 + 1/2*w^2 + 1/2*w - 7],
[1399, 1399, -w^2 - 3],
[1409, 1409, -5/2*w^3 - 3/2*w^2 + 25/2*w],
[1423, 1423, 5/2*w^3 - 3/2*w^2 - 31/2*w + 5],
[1423, 1423, -w^3 + 9*w - 3],
[1439, 1439, w^3 + 2*w^2 - 7*w - 7],
[1439, 1439, -5/2*w^3 - 5/2*w^2 + 33/2*w + 7],
[1447, 1447, 3/2*w^3 - 5/2*w^2 - 17/2*w + 9],
[1451, 1451, 1/2*w^3 + 5/2*w^2 - 7/2*w - 5],
[1459, 1459, -1/2*w^3 + 1/2*w^2 + 13/2*w],
[1459, 1459, 1/2*w^3 + 3/2*w^2 - 17/2*w - 3],
[1481, 1481, 3/2*w^3 - 3/2*w^2 - 21/2*w + 4],
[1487, 1487, 7/2*w^3 - 1/2*w^2 - 43/2*w + 1],
[1499, 1499, -3/2*w^3 - 5/2*w^2 + 11/2*w + 8],
[1523, 1523, -5/2*w^3 + 1/2*w^2 + 29/2*w - 1],
[1523, 1523, -3/2*w^3 + 1/2*w^2 + 11/2*w + 5],
[1543, 1543, -2*w^3 + 10*w - 3],
[1567, 1567, -1/2*w^3 - 5/2*w^2 + 9/2*w + 5],
[1567, 1567, 2*w^3 - 15*w - 3],
[1579, 1579, 1/2*w^3 + 3/2*w^2 - 3/2*w - 10],
[1579, 1579, -3/2*w^3 + 3/2*w^2 + 13/2*w - 10],
[1597, 1597, -1/2*w^3 - 3/2*w^2 + 11/2*w + 8],
[1601, 1601, 2*w^3 - w^2 - 10*w + 1],
[1601, 1601, 3/2*w^3 + 1/2*w^2 - 21/2*w + 2],
[1613, 1613, -1/2*w^3 - 1/2*w^2 + 5/2*w - 5],
[1619, 1619, 5/2*w^3 - 1/2*w^2 - 29/2*w + 3],
[1619, 1619, 3*w^3 - 3*w^2 - 17*w + 19],
[1621, 1621, w^3 + w^2 - w - 5],
[1621, 1621, 3/2*w^3 + 5/2*w^2 - 23/2*w - 7],
[1637, 1637, -7/2*w^3 + 1/2*w^2 + 45/2*w - 4],
[1657, 1657, -7/2*w^3 - 5/2*w^2 + 45/2*w + 14],
[1669, 1669, -w^3 - w^2 + 2*w + 5],
[1693, 1693, 11/2*w^3 + 1/2*w^2 - 71/2*w - 4],
[1699, 1699, 7/2*w^3 - 9/2*w^2 - 47/2*w + 27],
[1699, 1699, -1/2*w^3 + 5/2*w^2 + 11/2*w - 14],
[1733, 1733, w^3 - 9*w + 1],
[1741, 1741, 5/2*w^3 - 1/2*w^2 - 31/2*w - 4],
[1747, 1747, 7/2*w^3 - 5/2*w^2 - 45/2*w + 12],
[1777, 1777, -3/2*w^3 - 1/2*w^2 + 23/2*w - 3],
[1777, 1777, -1/2*w^3 - 1/2*w^2 - 3/2*w - 3],
[1787, 1787, -2*w^3 - 2*w^2 + 15*w + 5],
[1801, 1801, -1/2*w^3 - 5/2*w^2 + 11/2*w + 6],
[1801, 1801, 1/2*w^3 - 1/2*w^2 - 9/2*w - 5],
[1823, 1823, w^3 + 3*w^2 - 7*w - 17],
[1823, 1823, 5/2*w^3 + 5/2*w^2 - 33/2*w - 9],
[1847, 1847, -w^3 - 3*w^2 + 7*w + 5],
[1861, 1861, -1/2*w^3 + 1/2*w^2 + 9/2*w - 10],
[1861, 1861, -3/2*w^3 + 3/2*w^2 + 17/2*w - 2],
[1871, 1871, -w^3 + w^2 + 10*w - 11],
[1873, 1873, -5*w^3 - w^2 + 31*w + 7],
[1879, 1879, -2*w^3 - w^2 + 8*w + 5],
[1889, 1889, w^3 + w^2 - 7*w - 11],
[1889, 1889, -3/2*w^3 - 1/2*w^2 + 9/2*w + 4],
[1901, 1901, 4*w^3 - 4*w^2 - 26*w + 27],
[1901, 1901, 3/2*w^3 + 3/2*w^2 - 21/2*w - 2],
[1901, 1901, -1/2*w^3 + 1/2*w^2 + 13/2*w - 1],
[1901, 1901, w^2 + 4*w - 3],
[1907, 1907, 1/2*w^3 - 5/2*w^2 - 5/2*w + 19],
[1913, 1913, -2*w^3 + 2*w^2 + 12*w - 7],
[1931, 1931, -w^3 - 3*w^2 + 6*w + 5],
[1973, 1973, 4*w^3 - 4*w^2 - 25*w + 21],
[1973, 1973, 1/2*w^3 + 3/2*w^2 - 3/2*w - 12],
[1979, 1979, -w^3 + 2*w^2 + 9*w - 9],
[1979, 1979, -3/2*w^3 + 3/2*w^2 + 23/2*w - 8],
[1987, 1987, 1/2*w^3 + 5/2*w^2 - 5/2*w - 7],
[1993, 1993, -3*w^3 + 17*w - 3],
[1993, 1993, -1/2*w^3 - 5/2*w^2 + 9/2*w + 11],
[1999, 1999, -2*w^3 + 12*w - 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 - 2*x^8 - 17*x^7 + 30*x^6 + 86*x^5 - 104*x^4 - 163*x^3 + 66*x^2 + 44*x - 16;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1, 1, e^8 - 3/2*e^7 - 18*e^6 + 43/2*e^5 + 100*e^4 - 61*e^3 - 202*e^2 - 33/2*e + 42, -3/4*e^8 + e^7 + 51/4*e^6 - 14*e^5 - 129/2*e^4 + 36*e^3 + 469/4*e^2 + 12*e - 22, e^8 - 3/2*e^7 - 17*e^6 + 43/2*e^5 + 86*e^4 - 61*e^3 - 158*e^2 - 7/2*e + 32, 1/2*e^8 - e^7 - 17/2*e^6 + 14*e^5 + 43*e^4 - 39*e^3 - 165/2*e^2 - 3*e + 16, 7/4*e^8 - 3*e^7 - 123/4*e^6 + 43*e^5 + 331/2*e^4 - 124*e^3 - 1337/4*e^2 - 17*e + 72, 1/2*e^8 - e^7 - 17/2*e^6 + 14*e^5 + 43*e^4 - 39*e^3 - 165/2*e^2 - 2*e + 16, 2*e^8 - 3*e^7 - 35*e^6 + 43*e^5 + 186*e^4 - 122*e^3 - 359*e^2 - 20*e + 70, -e^8 + 2*e^7 + 18*e^6 - 29*e^5 - 100*e^4 + 90*e^3 + 207*e^2 - 7*e - 42, -e^8 + 3/2*e^7 + 17*e^6 - 43/2*e^5 - 85*e^4 + 62*e^3 + 149*e^2 - 3/2*e - 26, 5/2*e^8 - 4*e^7 - 87/2*e^6 + 57*e^5 + 229*e^4 - 161*e^3 - 883/2*e^2 - 26*e + 86, 3/4*e^8 - e^7 - 55/4*e^6 + 14*e^5 + 157/2*e^4 - 36*e^3 - 649/4*e^2 - 25*e + 38, 1/2*e^7 + e^6 - 15/2*e^5 - 14*e^4 + 27*e^3 + 48*e^2 + 11/2*e - 2, -5/2*e^8 + 4*e^7 + 87/2*e^6 - 57*e^5 - 230*e^4 + 160*e^3 + 905/2*e^2 + 27*e - 96, 11/4*e^8 - 4*e^7 - 195/4*e^6 + 57*e^5 + 531/2*e^4 - 156*e^3 - 2125/4*e^2 - 59*e + 116, -1/2*e^8 + e^7 + 19/2*e^6 - 14*e^5 - 57*e^4 + 39*e^3 + 255/2*e^2 + 14*e - 28, -5/2*e^8 + 4*e^7 + 87/2*e^6 - 58*e^5 - 229*e^4 + 172*e^3 + 879/2*e^2 + 5*e - 82, -5/2*e^8 + 4*e^7 + 87/2*e^6 - 57*e^5 - 229*e^4 + 160*e^3 + 883/2*e^2 + 29*e - 82, 3/2*e^8 - 3*e^7 - 53/2*e^6 + 44*e^5 + 144*e^4 - 139*e^3 - 593/2*e^2 + 19*e + 62, e^6 - 14*e^4 + 44*e^2 + 11*e - 10, -7/4*e^8 + 2*e^7 + 123/4*e^6 - 28*e^5 - 329/2*e^4 + 69*e^3 + 1265/4*e^2 + 43*e - 72, -e^8 + 2*e^7 + 17*e^6 - 29*e^5 - 87*e^4 + 90*e^3 + 172*e^2 - 20*e - 34, 2*e^8 - 3*e^7 - 35*e^6 + 43*e^5 + 186*e^4 - 122*e^3 - 359*e^2 - 22*e + 72, -3/2*e^8 + 5/2*e^7 + 53/2*e^6 - 73/2*e^5 - 144*e^4 + 110*e^3 + 579/2*e^2 + 13/2*e - 50, -3/2*e^8 + 3*e^7 + 51/2*e^6 - 43*e^5 - 130*e^4 + 128*e^3 + 509/2*e^2 - 11*e - 48, -e^8 + 2*e^7 + 18*e^6 - 29*e^5 - 101*e^4 + 86*e^3 + 214*e^2 + 21*e - 40, -7/4*e^8 + 3*e^7 + 127/4*e^6 - 43*e^5 - 359/2*e^4 + 126*e^3 + 1517/4*e^2 + 14*e - 86, -3*e^8 + 5*e^7 + 53*e^6 - 71*e^5 - 287*e^4 + 199*e^3 + 577*e^2 + 46*e - 112, -5/2*e^8 + 4*e^7 + 91/2*e^6 - 57*e^5 - 258*e^4 + 158*e^3 + 1083/2*e^2 + 71*e - 118, 3*e^8 - 5*e^7 - 53*e^6 + 72*e^5 + 286*e^4 - 210*e^3 - 566*e^2 - 25*e + 104, -1/2*e^8 + 1/2*e^7 + 19/2*e^6 - 15/2*e^5 - 58*e^4 + 19*e^3 + 255/2*e^2 + 67/2*e - 22, -5*e^8 + 8*e^7 + 89*e^6 - 114*e^5 - 487*e^4 + 322*e^3 + 983*e^2 + 68*e - 204, -e^8 + 3/2*e^7 + 16*e^6 - 45/2*e^5 - 72*e^4 + 74*e^3 + 112*e^2 - 89/2*e - 26, 3/4*e^8 - e^7 - 55/4*e^6 + 15*e^5 + 157/2*e^4 - 47*e^3 - 645/4*e^2 - 6*e + 34, -7/4*e^8 + 2*e^7 + 123/4*e^6 - 29*e^5 - 333/2*e^4 + 80*e^3 + 1325/4*e^2 + 24*e - 84, 3*e^8 - 4*e^7 - 54*e^6 + 57*e^5 + 301*e^4 - 154*e^3 - 611*e^2 - 71*e + 140, 4*e^8 - 13/2*e^7 - 70*e^6 + 187/2*e^5 + 372*e^4 - 273*e^3 - 725*e^2 - 33/2*e + 148, -5/2*e^8 + 4*e^7 + 89/2*e^6 - 58*e^5 - 244*e^4 + 173*e^3 + 985/2*e^2 + 7*e - 98, 4*e^8 - 7*e^7 - 70*e^6 + 101*e^5 + 373*e^4 - 301*e^3 - 738*e^2 - 2*e + 142, e^5 - e^4 - 14*e^3 + 11*e^2 + 38*e - 2, 15/4*e^8 - 6*e^7 - 267/4*e^6 + 86*e^5 + 733/2*e^4 - 245*e^3 - 2985/4*e^2 - 61*e + 160, e^4 + 2*e^3 - 6*e^2 - 12*e - 12, 3/2*e^8 - 5/2*e^7 - 51/2*e^6 + 69/2*e^5 + 130*e^4 - 88*e^3 - 505/2*e^2 - 73/2*e + 62, -e^8 + 2*e^7 + 18*e^6 - 28*e^5 - 101*e^4 + 75*e^3 + 218*e^2 + 44*e - 42, -23/4*e^8 + 9*e^7 + 403/4*e^6 - 129*e^5 - 1073/2*e^4 + 371*e^3 + 4177/4*e^2 + 36*e - 206, e^7 + e^6 - 15*e^5 - 15*e^4 + 54*e^3 + 63*e^2 - 6*e - 14, -5*e^8 + 15/2*e^7 + 89*e^6 - 215/2*e^5 - 486*e^4 + 307*e^3 + 967*e^2 + 113/2*e - 200, 3*e^8 - 5*e^7 - 55*e^6 + 72*e^5 + 316*e^4 - 208*e^3 - 674*e^2 - 61*e + 144, -13/2*e^8 + 11*e^7 + 229/2*e^6 - 158*e^5 - 616*e^4 + 460*e^3 + 2449/2*e^2 + 52*e - 246, -15/2*e^8 + 13*e^7 + 265/2*e^6 - 186*e^5 - 717*e^4 + 537*e^3 + 2887/2*e^2 + 77*e - 298, -2*e^8 + 4*e^7 + 35*e^6 - 57*e^5 - 188*e^4 + 165*e^3 + 388*e^2 + 14*e - 74, -3*e^8 + 5*e^7 + 53*e^6 - 71*e^5 - 288*e^4 + 198*e^3 + 585*e^2 + 51*e - 110, -5/2*e^8 + 9/2*e^7 + 89/2*e^6 - 131/2*e^5 - 243*e^4 + 200*e^3 + 973/2*e^2 - 1/2*e - 86, 2*e^8 - 4*e^7 - 35*e^6 + 57*e^5 + 187*e^4 - 164*e^3 - 380*e^2 - 29*e + 74, 1/4*e^8 - e^7 - 17/4*e^6 + 15*e^5 + 45/2*e^4 - 54*e^3 - 219/4*e^2 + 29*e + 18, -5/2*e^8 + 4*e^7 + 87/2*e^6 - 57*e^5 - 228*e^4 + 162*e^3 + 867/2*e^2 + 12*e - 78, 2*e^8 - 4*e^7 - 38*e^6 + 58*e^5 + 229*e^4 - 177*e^3 - 514*e^2 - 28*e + 108, -5/4*e^8 + 3*e^7 + 89/4*e^6 - 42*e^5 - 247/2*e^4 + 116*e^3 + 1107/4*e^2 + 37*e - 68, 3*e^8 - 4*e^7 - 55*e^6 + 57*e^5 + 314*e^4 - 155*e^3 - 648*e^2 - 77*e + 154, 7/4*e^8 - 3*e^7 - 123/4*e^6 + 43*e^5 + 329/2*e^4 - 125*e^3 - 1289/4*e^2 - 7*e + 52, -4*e^8 + 6*e^7 + 71*e^6 - 86*e^5 - 386*e^4 + 246*e^3 + 765*e^2 + 32*e - 156, 21/4*e^8 - 8*e^7 - 373/4*e^6 + 115*e^5 + 1019/2*e^4 - 328*e^3 - 4091/4*e^2 - 77*e + 216, -5*e^8 + 9*e^7 + 88*e^6 - 129*e^5 - 474*e^4 + 376*e^3 + 958*e^2 + 40*e - 200, 7/2*e^8 - 5*e^7 - 123/2*e^6 + 72*e^5 + 328*e^4 - 206*e^3 - 1261/2*e^2 - 33*e + 134, -4*e^8 + 6*e^7 + 71*e^6 - 86*e^5 - 385*e^4 + 245*e^3 + 755*e^2 + 50*e - 154, 3/4*e^8 - 2*e^7 - 59/4*e^6 + 29*e^5 + 189/2*e^4 - 90*e^3 - 941/4*e^2 - 14*e + 56, -21/4*e^8 + 8*e^7 + 369/4*e^6 - 114*e^5 - 991/2*e^4 + 315*e^3 + 3931/4*e^2 + 95*e - 208, -1/2*e^8 + 3/2*e^7 + 19/2*e^6 - 41/2*e^5 - 58*e^4 + 55*e^3 + 283/2*e^2 + 49/2*e - 28, -29/4*e^8 + 11*e^7 + 521/4*e^6 - 157*e^5 - 1449/2*e^4 + 438*e^3 + 5931/4*e^2 + 148*e - 314, 3/4*e^8 - e^7 - 47/4*e^6 + 15*e^5 + 101/2*e^4 - 49*e^3 - 285/4*e^2 + 36*e + 12, e^8 - e^7 - 15*e^6 + 15*e^5 + 56*e^4 - 47*e^3 - 41*e^2 + 38*e + 2, -1/2*e^8 - 1/2*e^7 + 17/2*e^6 + 15/2*e^5 - 42*e^4 - 32*e^3 + 113/2*e^2 + 37/2*e - 8, 6*e^8 - 9*e^7 - 108*e^6 + 128*e^5 + 602*e^4 - 351*e^3 - 1233*e^2 - 146*e + 264, 3/2*e^8 - 2*e^7 - 51/2*e^6 + 27*e^5 + 129*e^4 - 63*e^3 - 483/2*e^2 - 27*e + 68, -11/2*e^8 + 8*e^7 + 195/2*e^6 - 115*e^5 - 530*e^4 + 326*e^3 + 2103/2*e^2 + 76*e - 218, 2*e^8 - 3*e^7 - 35*e^6 + 43*e^5 + 186*e^4 - 122*e^3 - 361*e^2 - 20*e + 78, -1/2*e^8 + e^7 + 15/2*e^6 - 15*e^5 - 30*e^4 + 51*e^3 + 93/2*e^2 - 42*e - 14, -9/4*e^8 + 4*e^7 + 165/4*e^6 - 57*e^5 - 469/2*e^4 + 163*e^3 + 1947/4*e^2 + 50*e - 98, -5/4*e^8 + 3*e^7 + 89/4*e^6 - 42*e^5 - 249/2*e^4 + 115*e^3 + 1143/4*e^2 + 42*e - 74, 5/4*e^8 - 2*e^7 - 93/4*e^6 + 28*e^5 + 275/2*e^4 - 72*e^3 - 1231/4*e^2 - 59*e + 76, 19/2*e^8 - 14*e^7 - 335/2*e^6 + 200*e^5 + 904*e^4 - 557*e^3 - 3581/2*e^2 - 147*e + 378, -3*e^8 + 11/2*e^7 + 51*e^6 - 159/2*e^5 - 258*e^4 + 239*e^3 + 483*e^2 - 53/2*e - 96, -7*e^8 + 21/2*e^7 + 122*e^6 - 301/2*e^5 - 643*e^4 + 428*e^3 + 1226*e^2 + 125/2*e - 234, 2*e^8 - 7/2*e^7 - 37*e^6 + 101/2*e^5 + 217*e^4 - 145*e^3 - 482*e^2 - 113/2*e + 114, -5/2*e^8 + 4*e^7 + 85/2*e^6 - 57*e^5 - 218*e^4 + 158*e^3 + 849/2*e^2 + 35*e - 98, 1/2*e^8 - 1/2*e^7 - 17/2*e^6 + 13/2*e^5 + 41*e^4 - 14*e^3 - 123/2*e^2 - 7/2*e + 24, -e^8 + e^7 + 17*e^6 - 15*e^5 - 86*e^4 + 41*e^3 + 149*e^2 + 22*e - 20, e^7 + e^6 - 13*e^5 - 17*e^4 + 29*e^3 + 87*e^2 + 53*e - 28, -9*e^8 + 29/2*e^7 + 157*e^6 - 417/2*e^5 - 831*e^4 + 606*e^3 + 1617*e^2 + 87/2*e - 326, -3*e^8 + 4*e^7 + 53*e^6 - 57*e^5 - 287*e^4 + 154*e^3 + 568*e^2 + 62*e - 128, -7/4*e^8 + 2*e^7 + 119/4*e^6 - 28*e^5 - 299/2*e^4 + 70*e^3 + 1049/4*e^2 + 25*e - 60, -2*e^8 + 4*e^7 + 35*e^6 - 56*e^5 - 185*e^4 + 157*e^3 + 364*e^2 + 16*e - 52, -5*e^8 + 17/2*e^7 + 85*e^6 - 241/2*e^5 - 430*e^4 + 339*e^3 + 799*e^2 + 41/2*e - 132, -e^8 + 2*e^7 + 21*e^6 - 28*e^5 - 142*e^4 + 74*e^3 + 341*e^2 + 92*e - 66, 41/4*e^8 - 16*e^7 - 713/4*e^6 + 229*e^5 + 1881/2*e^4 - 651*e^3 - 7307/4*e^2 - 86*e + 360, -e^6 + e^5 + 14*e^4 - 13*e^3 - 48*e^2 + 22*e + 22, -1/2*e^8 + 25/2*e^6 - 100*e^4 - 8*e^3 + 515/2*e^2 + 88*e - 68, -1/2*e^7 + 17/2*e^5 + e^4 - 38*e^3 - 14*e^2 + 71/2*e + 6, 6*e^8 - 10*e^7 - 107*e^6 + 144*e^5 + 587*e^4 - 423*e^3 - 1188*e^2 - 50*e + 252, 7/2*e^8 - 11/2*e^7 - 127/2*e^6 + 155/2*e^5 + 357*e^4 - 213*e^3 - 1477/2*e^2 - 131/2*e + 162, -3/2*e^8 + 5/2*e^7 + 57/2*e^6 - 73/2*e^5 - 170*e^4 + 112*e^3 + 725/2*e^2 + 29/2*e - 72, -3*e^8 + 5*e^7 + 55*e^6 - 71*e^5 - 316*e^4 + 198*e^3 + 676*e^2 + 67*e - 144, -e^8 + 2*e^7 + 16*e^6 - 28*e^5 - 72*e^4 + 78*e^3 + 126*e^2 - 6*e - 38, -9*e^8 + 12*e^7 + 159*e^6 - 171*e^5 - 859*e^4 + 466*e^3 + 1677*e^2 + 150*e - 348, -23/4*e^8 + 9*e^7 + 411/4*e^6 - 127*e^5 - 1133/2*e^4 + 343*e^3 + 4629/4*e^2 + 138*e - 240, 3*e^8 - 6*e^7 - 53*e^6 + 86*e^5 + 287*e^4 - 255*e^3 - 587*e^2 - 11*e + 116, -5*e^8 + 7*e^7 + 91*e^6 - 100*e^5 - 515*e^4 + 274*e^3 + 1060*e^2 + 129*e - 228, -2*e^7 - 3*e^6 + 29*e^5 + 44*e^4 - 99*e^3 - 176*e^2 + 38, 7/2*e^8 - 6*e^7 - 127/2*e^6 + 87*e^5 + 356*e^4 - 262*e^3 - 1451/2*e^2 - 20*e + 142, -3/2*e^8 + 3*e^7 + 51/2*e^6 - 45*e^5 - 129*e^4 + 152*e^3 + 485/2*e^2 - 65*e - 58, 13/2*e^8 - 23/2*e^7 - 227/2*e^6 + 331/2*e^5 + 603*e^4 - 489*e^3 - 2389/2*e^2 - 11/2*e + 240, 10*e^8 - 17*e^7 - 177*e^6 + 244*e^5 + 959*e^4 - 712*e^3 - 1918*e^2 - 78*e + 390, 5*e^8 - 7*e^7 - 89*e^6 + 100*e^5 + 488*e^4 - 279*e^3 - 984*e^2 - 70*e + 230, -5*e^8 + 9*e^7 + 90*e^6 - 131*e^5 - 501*e^4 + 401*e^3 + 1031*e^2 + 11*e - 200, -5*e^8 + 9*e^7 + 87*e^6 - 130*e^5 - 460*e^4 + 389*e^3 + 916*e^2 - 10*e - 194, -4*e^8 + 13/2*e^7 + 71*e^6 - 187/2*e^5 - 386*e^4 + 270*e^3 + 772*e^2 + 121/2*e - 156, -2*e^8 + 3*e^7 + 35*e^6 - 44*e^5 - 185*e^4 + 136*e^3 + 346*e^2 - 24*e - 58, -23/4*e^8 + 8*e^7 + 403/4*e^6 - 114*e^5 - 1077/2*e^4 + 311*e^3 + 4205/4*e^2 + 103*e - 228, e^8 - 2*e^7 - 17*e^6 + 30*e^5 + 86*e^4 - 106*e^3 - 161*e^2 + 72*e + 28, e^8 - 3*e^7 - 18*e^6 + 44*e^5 + 102*e^4 - 144*e^3 - 231*e^2 + 36*e + 20, 31/4*e^8 - 12*e^7 - 547/4*e^6 + 171*e^5 + 1481/2*e^4 - 476*e^3 - 5961/4*e^2 - 126*e + 336, -13/2*e^8 + 21/2*e^7 + 229/2*e^6 - 299/2*e^5 - 618*e^4 + 417*e^3 + 2469/2*e^2 + 229/2*e - 244, 2*e^8 - 5/2*e^7 - 37*e^6 + 69/2*e^5 + 215*e^4 - 80*e^3 - 457*e^2 - 225/2*e + 112, -11/2*e^8 + 19/2*e^7 + 193/2*e^6 - 273/2*e^5 - 518*e^4 + 395*e^3 + 2073/2*e^2 + 133/2*e - 194, 10*e^8 - 18*e^7 - 176*e^6 + 257*e^5 + 949*e^4 - 738*e^3 - 1927*e^2 - 114*e + 400, 4*e^8 - 6*e^7 - 70*e^6 + 85*e^5 + 372*e^4 - 231*e^3 - 717*e^2 - 71*e + 138, 6*e^8 - 23/2*e^7 - 106*e^6 + 329/2*e^5 + 573*e^4 - 481*e^3 - 1159*e^2 - 99/2*e + 212, -9*e^8 + 27/2*e^7 + 159*e^6 - 387/2*e^5 - 858*e^4 + 551*e^3 + 1685*e^2 + 195/2*e - 344, 7/2*e^8 - 6*e^7 - 125/2*e^6 + 85*e^5 + 345*e^4 - 233*e^3 - 1425/2*e^2 - 98*e + 142, 15/4*e^8 - 6*e^7 - 279/4*e^6 + 86*e^5 + 817/2*e^4 - 243*e^3 - 3509/4*e^2 - 111*e + 180, 17/2*e^8 - 13*e^7 - 297/2*e^6 + 184*e^5 + 789*e^4 - 503*e^3 - 3089/2*e^2 - 134*e + 328, -6*e^8 + 10*e^7 + 105*e^6 - 142*e^5 - 558*e^4 + 396*e^3 + 1091*e^2 + 97*e - 204, 3/2*e^8 - 1/2*e^7 - 51/2*e^6 + 11/2*e^5 + 130*e^4 + 13*e^3 - 459/2*e^2 - 157/2*e + 66, 5/2*e^8 - 5/2*e^7 - 87/2*e^6 + 73/2*e^5 + 228*e^4 - 104*e^3 - 829/2*e^2 - 5/2*e + 98, 11/4*e^8 - 3*e^7 - 199/4*e^6 + 42*e^5 + 559/2*e^4 - 97*e^3 - 2257/4*e^2 - 134*e + 118, -9/2*e^8 + 9*e^7 + 157/2*e^6 - 130*e^5 - 416*e^4 + 396*e^3 + 1665/2*e^2 - 32*e - 166, 7*e^8 - 12*e^7 - 123*e^6 + 173*e^5 + 660*e^4 - 515*e^3 - 1316*e^2 + 4*e + 260, 15/2*e^8 - 23/2*e^7 - 265/2*e^6 + 327/2*e^5 + 718*e^4 - 450*e^3 - 2871/2*e^2 - 301/2*e + 294, e^7 - e^6 - 16*e^5 + 15*e^4 + 68*e^3 - 46*e^2 - 70*e + 22, -6*e^8 + 10*e^7 + 104*e^6 - 143*e^5 - 546*e^4 + 409*e^3 + 1069*e^2 + 49*e - 214, 1/2*e^8 - 2*e^7 - 23/2*e^6 + 30*e^5 + 87*e^4 - 104*e^3 - 479/2*e^2 - e + 38, 23/4*e^8 - 9*e^7 - 403/4*e^6 + 132*e^5 + 1073/2*e^4 - 406*e^3 - 4165/4*e^2 + 31*e + 206, 3*e^8 - 6*e^7 - 51*e^6 + 85*e^5 + 262*e^4 - 241*e^3 - 526*e^2 - 31*e + 100, -9/2*e^8 + 17/2*e^7 + 161/2*e^6 - 245/2*e^5 - 446*e^4 + 368*e^3 + 1871/2*e^2 + 19/2*e - 214, -3*e^8 + 5*e^7 + 53*e^6 - 71*e^5 - 289*e^4 + 195*e^3 + 596*e^2 + 78*e - 142, -5/2*e^8 + 3*e^7 + 89/2*e^6 - 43*e^5 - 243*e^4 + 114*e^3 + 939/2*e^2 + 71*e - 74, -27/4*e^8 + 10*e^7 + 491/4*e^6 - 143*e^5 - 1387/2*e^4 + 401*e^3 + 5725/4*e^2 + 144*e - 318, -e^8 + 4*e^7 + 17*e^6 - 58*e^5 - 86*e^4 + 192*e^3 + 189*e^2 - 84*e - 42, 6*e^8 - 19/2*e^7 - 102*e^6 + 271/2*e^5 + 518*e^4 - 381*e^3 - 970*e^2 - 71/2*e + 168, -11/2*e^8 + 7*e^7 + 193/2*e^6 - 100*e^5 - 512*e^4 + 274*e^3 + 1917/2*e^2 + 71*e - 170, -29/4*e^8 + 11*e^7 + 509/4*e^6 - 158*e^5 - 1359/2*e^4 + 452*e^3 + 5295/4*e^2 + 68*e - 274, -17/2*e^8 + 13*e^7 + 303/2*e^6 - 186*e^5 - 828*e^4 + 529*e^3 + 3307/2*e^2 + 107*e - 362, 1/4*e^8 - 2*e^7 - 21/4*e^6 + 27*e^5 + 77/2*e^4 - 69*e^3 - 555/4*e^2 - 59*e + 52, -27/2*e^8 + 39/2*e^7 + 483/2*e^6 - 557/2*e^5 - 1331*e^4 + 772*e^3 + 5343/2*e^2 + 529/2*e - 570, -4*e^8 + 6*e^7 + 68*e^6 - 88*e^5 - 343*e^4 + 271*e^3 + 618*e^2 - 62*e - 118, -9/4*e^8 + 3*e^7 + 157/4*e^6 - 43*e^5 - 421/2*e^4 + 114*e^3 + 1667/4*e^2 + 67*e - 70, -5/2*e^8 + 9/2*e^7 + 95/2*e^6 - 133/2*e^5 - 288*e^4 + 209*e^3 + 1297/2*e^2 + 47/2*e - 154, 11/2*e^8 - 8*e^7 - 199/2*e^6 + 116*e^5 + 558*e^4 - 337*e^3 - 2277/2*e^2 - 93*e + 244, 15/4*e^8 - 5*e^7 - 267/4*e^6 + 71*e^5 + 729/2*e^4 - 194*e^3 - 2889/4*e^2 - 61*e + 186, 9/2*e^8 - 7*e^7 - 159/2*e^6 + 101*e^5 + 428*e^4 - 293*e^3 - 1657/2*e^2 - 51*e + 130, -13/4*e^8 + 4*e^7 + 233/4*e^6 - 57*e^5 - 643/2*e^4 + 154*e^3 + 2555/4*e^2 + 54*e - 144, 5*e^8 - 7*e^7 - 87*e^6 + 100*e^5 + 461*e^4 - 272*e^3 - 896*e^2 - 96*e + 180, -25/2*e^8 + 39/2*e^7 + 439/2*e^6 - 561/2*e^5 - 1174*e^4 + 807*e^3 + 4587/2*e^2 + 239/2*e - 450, -11/4*e^8 + 6*e^7 + 195/4*e^6 - 87*e^5 - 525/2*e^4 + 275*e^3 + 2085/4*e^2 - 52*e - 80, 1/2*e^8 + e^7 - 23/2*e^6 - 16*e^5 + 83*e^4 + 72*e^3 - 359/2*e^2 - 114*e + 58, -3/2*e^8 + e^7 + 49/2*e^6 - 15*e^5 - 115*e^4 + 43*e^3 + 355/2*e^2 - 29*e - 42, -7/2*e^8 + 5*e^7 + 119/2*e^6 - 71*e^5 - 300*e^4 + 195*e^3 + 1089/2*e^2 + 31*e - 130, 8*e^8 - 13*e^7 - 142*e^6 + 185*e^5 + 773*e^4 - 523*e^3 - 1552*e^2 - 112*e + 328, 17/2*e^8 - 13*e^7 - 305/2*e^6 + 187*e^5 + 843*e^4 - 536*e^3 - 3395/2*e^2 - 127*e + 348, -1/2*e^8 + 19/2*e^6 - e^5 - 53*e^4 + 13*e^3 + 145/2*e^2 - 40*e + 10, 14*e^8 - 24*e^7 - 248*e^6 + 343*e^5 + 1348*e^4 - 983*e^3 - 2725*e^2 - 182*e + 570, -3/2*e^8 + 3*e^7 + 53/2*e^6 - 42*e^5 - 145*e^4 + 117*e^3 + 625/2*e^2 + 25*e - 76, 19/2*e^8 - 35/2*e^7 - 339/2*e^6 + 503/2*e^5 + 933*e^4 - 743*e^3 - 3849/2*e^2 - 135/2*e + 380, -5*e^8 + 8*e^7 + 91*e^6 - 115*e^5 - 515*e^4 + 330*e^3 + 1074*e^2 + 108*e - 250, -23/2*e^8 + 17*e^7 + 401/2*e^6 - 244*e^5 - 1059*e^4 + 691*e^3 + 4057/2*e^2 + 127*e - 398, -1/2*e^8 - 1/2*e^7 + 25/2*e^6 + 13/2*e^5 - 99*e^4 - 25*e^3 + 491/2*e^2 + 155/2*e - 82, -2*e^8 + 4*e^7 + 36*e^6 - 57*e^5 - 202*e^4 + 167*e^3 + 436*e^2 + 21*e - 88, -15/2*e^8 + 11*e^7 + 265/2*e^6 - 158*e^5 - 718*e^4 + 445*e^3 + 2853/2*e^2 + 119*e - 298, -41/4*e^8 + 17*e^7 + 729/4*e^6 - 243*e^5 - 1989/2*e^4 + 698*e^3 + 8003/4*e^2 + 101*e - 390, 9/4*e^8 - 5*e^7 - 161/4*e^6 + 75*e^5 + 447/2*e^4 - 251*e^3 - 1875/4*e^2 + 52*e + 88, 13*e^8 - 19*e^7 - 227*e^6 + 270*e^5 + 1204*e^4 - 741*e^3 - 2330*e^2 - 190*e + 478, 4*e^8 - 9/2*e^7 - 73*e^6 + 127/2*e^5 + 414*e^4 - 160*e^3 - 838*e^2 - 237/2*e + 192, 15/2*e^8 - 11*e^7 - 263/2*e^6 + 156*e^5 + 705*e^4 - 421*e^3 - 2795/2*e^2 - 148*e + 296, 29/4*e^8 - 14*e^7 - 513/4*e^6 + 199*e^5 + 1397/2*e^4 - 569*e^3 - 5799/4*e^2 - 101*e + 304, 2*e^8 - 4*e^7 - 37*e^6 + 55*e^5 + 215*e^4 - 142*e^3 - 477*e^2 - 89*e + 120, 3*e^8 - 13/2*e^7 - 50*e^6 + 187/2*e^5 + 245*e^4 - 285*e^3 - 463*e^2 + 125/2*e + 92, 3*e^8 - 4*e^7 - 50*e^6 + 59*e^5 + 240*e^4 - 181*e^3 - 383*e^2 + 48*e + 64, -9*e^8 + 15*e^7 + 154*e^6 - 215*e^5 - 790*e^4 + 620*e^3 + 1494*e^2 + 32*e - 272, 3/2*e^8 - 7/2*e^7 - 49/2*e^6 + 97/2*e^5 + 115*e^4 - 130*e^3 - 413/2*e^2 - 47/2*e + 22, -17/2*e^8 + 14*e^7 + 295/2*e^6 - 200*e^5 - 777*e^4 + 568*e^3 + 3049/2*e^2 + 81*e - 316, 19/2*e^8 - 33/2*e^7 - 335/2*e^6 + 473/2*e^5 + 901*e^4 - 687*e^3 - 3573/2*e^2 - 157/2*e + 344, 35/4*e^8 - 13*e^7 - 615/4*e^6 + 183*e^5 + 1647/2*e^4 - 486*e^3 - 6469/4*e^2 - 207*e + 336, 3/2*e^8 - 3*e^7 - 57/2*e^6 + 42*e^5 + 173*e^4 - 115*e^3 - 805/2*e^2 - 63*e + 100, 9/4*e^8 - 5*e^7 - 157/4*e^6 + 71*e^5 + 421/2*e^4 - 202*e^3 - 1763/4*e^2 - 53*e + 66, 21/2*e^8 - 16*e^7 - 373/2*e^6 + 228*e^5 + 1021*e^4 - 632*e^3 - 4143/2*e^2 - 206*e + 444, 1/4*e^8 - 25/4*e^6 + 2*e^5 + 99/2*e^4 - 20*e^3 - 491/4*e^2 + 7*e + 28, -7*e^8 + 10*e^7 + 120*e^6 - 144*e^5 - 617*e^4 + 407*e^3 + 1144*e^2 + 66*e - 202, -6*e^8 + 8*e^7 + 106*e^6 - 113*e^5 - 570*e^4 + 299*e^3 + 1099*e^2 + 131*e - 222, -2*e^8 + 2*e^7 + 36*e^6 - 27*e^5 - 202*e^4 + 53*e^3 + 417*e^2 + 97*e - 86, -21/4*e^8 + 8*e^7 + 353/4*e^6 - 115*e^5 - 879/2*e^4 + 332*e^3 + 3207/4*e^2 - 16*e - 190, -8*e^8 + 13*e^7 + 136*e^6 - 186*e^5 - 688*e^4 + 537*e^3 + 1275*e^2 - 17*e - 264, -11*e^8 + 18*e^7 + 189*e^6 - 258*e^5 - 973*e^4 + 749*e^3 + 1837*e^2 - 6*e - 366, 4*e^8 - 5*e^7 - 72*e^6 + 71*e^5 + 400*e^4 - 188*e^3 - 792*e^2 - 99*e + 158, -17/2*e^8 + 12*e^7 + 297/2*e^6 - 170*e^5 - 786*e^4 + 461*e^3 + 3009/2*e^2 + 135*e - 306, 2*e^7 - e^6 - 30*e^5 + 14*e^4 + 116*e^3 - 27*e^2 - 109*e + 34, -27/2*e^8 + 47/2*e^7 + 477/2*e^6 - 669/2*e^5 - 1290*e^4 + 953*e^3 + 5205/2*e^2 + 313/2*e - 540, -9/2*e^8 + 8*e^7 + 165/2*e^6 - 116*e^5 - 472*e^4 + 353*e^3 + 1995/2*e^2 + 20*e - 198, 7/2*e^8 - 5*e^7 - 129/2*e^6 + 71*e^5 + 369*e^4 - 192*e^3 - 1503/2*e^2 - 116*e + 144, -23/4*e^8 + 9*e^7 + 415/4*e^6 - 130*e^5 - 1157/2*e^4 + 378*e^3 + 4713/4*e^2 + 78*e - 262, 2*e^8 - 2*e^7 - 35*e^6 + 29*e^5 + 183*e^4 - 90*e^3 - 324*e^2 + 56*e + 76, -11/4*e^8 + 4*e^7 + 207/4*e^6 - 57*e^5 - 607/2*e^4 + 159*e^3 + 2529/4*e^2 + 81*e - 136, -11/2*e^8 + 8*e^7 + 197/2*e^6 - 113*e^5 - 545*e^4 + 303*e^3 + 2229/2*e^2 + 136*e - 264, 13*e^8 - 22*e^7 - 229*e^6 + 314*e^5 + 1234*e^4 - 894*e^3 - 2477*e^2 - 178*e + 522, -9/2*e^8 + 7*e^7 + 155/2*e^6 - 99*e^5 - 405*e^4 + 268*e^3 + 1593/2*e^2 + 68*e - 194, -3/2*e^8 + 2*e^7 + 49/2*e^6 - 30*e^5 - 112*e^4 + 97*e^3 + 327/2*e^2 - 48*e - 32, 15/2*e^8 - 25/2*e^7 - 253/2*e^6 + 361/2*e^5 + 634*e^4 - 534*e^3 - 2351/2*e^2 + 83/2*e + 234, -2*e^8 + 3/2*e^7 + 37*e^6 - 41/2*e^5 - 212*e^4 + 36*e^3 + 412*e^2 + 223/2*e - 86, -53/4*e^8 + 19*e^7 + 945/4*e^6 - 269*e^5 - 2589/2*e^4 + 728*e^3 + 10347/4*e^2 + 290*e - 554, 10*e^8 - 35/2*e^7 - 175*e^6 + 501/2*e^5 + 931*e^4 - 729*e^3 - 1837*e^2 - 97/2*e + 380, 5/4*e^8 - e^7 - 73/4*e^6 + 11*e^5 + 135/2*e^4 + 9*e^3 - 315/4*e^2 - 94*e + 10, -11/2*e^8 + 10*e^7 + 187/2*e^6 - 144*e^5 - 475*e^4 + 428*e^3 + 1793/2*e^2 - 32*e - 144, -6*e^8 + 8*e^7 + 105*e^6 - 114*e^5 - 561*e^4 + 305*e^3 + 1099*e^2 + 130*e - 256, 27/4*e^8 - 12*e^7 - 483/4*e^6 + 173*e^5 + 1337/2*e^4 - 506*e^3 - 5529/4*e^2 - 111*e + 290, 3*e^8 - 7*e^7 - 48*e^6 + 102*e^5 + 215*e^4 - 330*e^3 - 357*e^2 + 166*e + 72, 29/4*e^8 - 12*e^7 - 521/4*e^6 + 174*e^5 + 1453/2*e^4 - 516*e^3 - 6027/4*e^2 - 64*e + 316, 2*e^8 - 3*e^7 - 33*e^6 + 43*e^5 + 158*e^4 - 124*e^3 - 269*e^2 + 22*e + 24, -13/4*e^8 + 3*e^7 + 229/4*e^6 - 44*e^5 - 613/2*e^4 + 125*e^3 + 2291/4*e^2 + 5*e - 128, -1/2*e^8 + 5/2*e^7 + 23/2*e^6 - 75/2*e^5 - 87*e^4 + 130*e^3 + 495/2*e^2 - 7/2*e - 66, 14*e^8 - 23*e^7 - 247*e^6 + 328*e^5 + 1334*e^4 - 929*e^3 - 2674*e^2 - 172*e + 572, -9*e^8 + 29/2*e^7 + 160*e^6 - 417/2*e^5 - 875*e^4 + 598*e^3 + 1768*e^2 + 267/2*e - 354, -4*e^8 + 7*e^7 + 72*e^6 - 98*e^5 - 401*e^4 + 265*e^3 + 831*e^2 + 104*e - 156, 5*e^8 - 8*e^7 - 87*e^6 + 114*e^5 + 456*e^4 - 322*e^3 - 866*e^2 - 60*e + 186, 9*e^8 - 15*e^7 - 163*e^6 + 213*e^5 + 920*e^4 - 594*e^3 - 1937*e^2 - 207*e + 412, -13/2*e^8 + 19/2*e^7 + 229/2*e^6 - 269/2*e^5 - 618*e^4 + 359*e^3 + 2465/2*e^2 + 337/2*e - 270, -3*e^8 + 4*e^7 + 55*e^6 - 60*e^5 - 314*e^4 + 190*e^3 + 635*e^2 + 11*e - 104, 21/2*e^8 - 17*e^7 - 367/2*e^6 + 243*e^5 + 976*e^4 - 687*e^3 - 3843/2*e^2 - 141*e + 384, -e^7 - 6*e^6 + 14*e^5 + 86*e^4 - 41*e^3 - 297*e^2 - 90*e + 64, -8*e^8 + 13*e^7 + 138*e^6 - 187*e^5 - 711*e^4 + 547*e^3 + 1312*e^2 + 14*e - 216, -3/2*e^8 + 1/2*e^7 + 49/2*e^6 - 9/2*e^5 - 114*e^4 - 18*e^3 + 335/2*e^2 + 97/2*e - 26, -19/2*e^8 + 13*e^7 + 339/2*e^6 - 185*e^5 - 935*e^4 + 500*e^3 + 3795/2*e^2 + 209*e - 432, -25/2*e^8 + 21*e^7 + 441/2*e^6 - 298*e^5 - 1191*e^4 + 831*e^3 + 4793/2*e^2 + 199*e - 504, -23/2*e^8 + 17*e^7 + 405/2*e^6 - 241*e^5 - 1088*e^4 + 655*e^3 + 4279/2*e^2 + 229*e - 462, 7/2*e^8 - 13/2*e^7 - 125/2*e^6 + 185/2*e^5 + 341*e^4 - 269*e^3 - 1359/2*e^2 - 69/2*e + 132, -21/2*e^8 + 16*e^7 + 371/2*e^6 - 229*e^5 - 1003*e^4 + 650*e^3 + 3975/2*e^2 + 121*e - 378, -3/2*e^8 + 5/2*e^7 + 49/2*e^6 - 75/2*e^5 - 118*e^4 + 125*e^3 + 445/2*e^2 - 153/2*e - 72, 2*e^8 - e^7 - 31*e^6 + 13*e^5 + 129*e^4 - 12*e^3 - 153*e^2 - 12*e + 20, 33/2*e^8 - 23*e^7 - 587/2*e^6 + 331*e^5 + 1601*e^4 - 940*e^3 - 6317/2*e^2 - 215*e + 664, -3/4*e^8 + 3*e^7 + 47/4*e^6 - 41*e^5 - 95/2*e^4 + 114*e^3 + 253/4*e^2 + 13*e + 46, 14*e^8 - 22*e^7 - 245*e^6 + 314*e^5 + 1305*e^4 - 883*e^3 - 2557*e^2 - 176*e + 518, -6*e^8 + 10*e^7 + 110*e^6 - 141*e^5 - 631*e^4 + 380*e^3 + 1347*e^2 + 214*e - 290, -6*e^8 + 10*e^7 + 105*e^6 - 143*e^5 - 557*e^4 + 410*e^3 + 1078*e^2 + 52*e - 182, -8*e^8 + 15*e^7 + 138*e^6 - 213*e^5 - 718*e^4 + 606*e^3 + 1404*e^2 + 69*e - 238, -17/2*e^8 + 13*e^7 + 305/2*e^6 - 187*e^5 - 844*e^4 + 537*e^3 + 3423/2*e^2 + 120*e - 378, 11/2*e^8 - 10*e^7 - 199/2*e^6 + 142*e^5 + 563*e^4 - 396*e^3 - 2421/2*e^2 - 142*e + 260, -17/2*e^8 + 16*e^7 + 301/2*e^6 - 230*e^5 - 819*e^4 + 686*e^3 + 3355/2*e^2 + 4*e - 324, 25/2*e^8 - 39/2*e^7 - 441/2*e^6 + 555/2*e^5 + 1188*e^4 - 776*e^3 - 4691/2*e^2 - 373/2*e + 456, 7*e^8 - 10*e^7 - 125*e^6 + 144*e^5 + 684*e^4 - 412*e^3 - 1341*e^2 - 92*e + 238, 6*e^8 - 19/2*e^7 - 104*e^6 + 275/2*e^5 + 548*e^4 - 407*e^3 - 1070*e^2 + 17/2*e + 210, -41/2*e^8 + 34*e^7 + 723/2*e^6 - 487*e^5 - 1949*e^4 + 1405*e^3 + 7769/2*e^2 + 178*e - 766, 8*e^8 - 23/2*e^7 - 147*e^6 + 329/2*e^5 + 844*e^4 - 454*e^3 - 1758*e^2 - 465/2*e + 372, 15/2*e^8 - 12*e^7 - 261/2*e^6 + 171*e^5 + 690*e^4 - 478*e^3 - 2707/2*e^2 - 103*e + 304, -10*e^8 + 16*e^7 + 181*e^6 - 230*e^5 - 1013*e^4 + 667*e^3 + 2067*e^2 + 143*e - 418, -5/2*e^8 + 7/2*e^7 + 89/2*e^6 - 103/2*e^5 - 243*e^4 + 157*e^3 + 951/2*e^2 - 19/2*e - 72, -4*e^8 + 6*e^7 + 70*e^6 - 84*e^5 - 372*e^4 + 222*e^3 + 730*e^2 + 80*e - 192, -2*e^8 + 4*e^7 + 37*e^6 - 56*e^5 - 210*e^4 + 156*e^3 + 425*e^2 + 49*e - 52, -23/4*e^8 + 9*e^7 + 403/4*e^6 - 131*e^5 - 1071/2*e^4 + 394*e^3 + 4097/4*e^2 - 3*e - 192, -3/2*e^8 + 9/2*e^7 + 55/2*e^6 - 131/2*e^5 - 159*e^4 + 208*e^3 + 719/2*e^2 - 37/2*e - 8, -11/2*e^8 + 19/2*e^7 + 199/2*e^6 - 269/2*e^5 - 560*e^4 + 371*e^3 + 2363/2*e^2 + 287/2*e - 272, 3*e^8 - 6*e^7 - 56*e^6 + 85*e^5 + 333*e^4 - 242*e^3 - 767*e^2 - 69*e + 212, -e^8 + 17*e^6 + 2*e^5 - 86*e^4 - 38*e^3 + 141*e^2 + 100*e - 46, 11*e^8 - 17*e^7 - 190*e^6 + 243*e^5 + 986*e^4 - 694*e^3 - 1866*e^2 - 38*e + 400, -7/2*e^8 + 5*e^7 + 117/2*e^6 - 69*e^5 - 287*e^4 + 173*e^3 + 1031/2*e^2 + 53*e - 138, 6*e^8 - 10*e^7 - 108*e^6 + 143*e^5 + 606*e^4 - 398*e^3 - 1273*e^2 - 168*e + 242, 15/2*e^8 - 12*e^7 - 251/2*e^6 + 171*e^5 + 618*e^4 - 482*e^3 - 2227/2*e^2 - 14*e + 190, 11/2*e^8 - 15/2*e^7 - 197/2*e^6 + 217/2*e^5 + 546*e^4 - 303*e^3 - 2197/2*e^2 - 273/2*e + 216, 9*e^8 - 23/2*e^7 - 159*e^6 + 325/2*e^5 + 860*e^4 - 422*e^3 - 1686*e^2 - 453/2*e + 344, e^7 - 5*e^6 - 18*e^5 + 71*e^4 + 96*e^3 - 226*e^2 - 210*e + 54, -14*e^8 + 39/2*e^7 + 249*e^6 - 559/2*e^5 - 1359*e^4 + 779*e^3 + 2681*e^2 + 477/2*e - 534, -1/2*e^8 - 1/2*e^7 + 21/2*e^6 + 7/2*e^5 - 73*e^4 + 11*e^3 + 329/2*e^2 - 53/2*e - 68, -7/4*e^8 + 2*e^7 + 147/4*e^6 - 29*e^5 - 505/2*e^4 + 74*e^3 + 2473/4*e^2 + 132*e - 168, e^8 - 7/2*e^7 - 18*e^6 + 99/2*e^5 + 100*e^4 - 147*e^3 - 218*e^2 - 57/2*e + 8, -14*e^8 + 22*e^7 + 244*e^6 - 316*e^5 - 1289*e^4 + 913*e^3 + 2490*e^2 + 69*e - 478, 13*e^8 - 22*e^7 - 231*e^6 + 316*e^5 + 1256*e^4 - 922*e^3 - 2504*e^2 - 108*e + 492, 59/4*e^8 - 27*e^7 - 1035/4*e^6 + 386*e^5 + 2777/2*e^4 - 1117*e^3 - 11245/4*e^2 - 131*e + 572, 19/2*e^8 - 12*e^7 - 331/2*e^6 + 172*e^5 + 871*e^4 - 474*e^3 - 3259/2*e^2 - 106*e + 324, -31/2*e^8 + 25*e^7 + 543/2*e^6 - 361*e^5 - 1448*e^4 + 1059*e^3 + 5673/2*e^2 + 83*e - 554, 23/2*e^8 - 17*e^7 - 409/2*e^6 + 239*e^5 + 1117*e^4 - 636*e^3 - 4477/2*e^2 - 264*e + 466, 7*e^8 - 9*e^7 - 128*e^6 + 128*e^5 + 727*e^4 - 342*e^3 - 1485*e^2 - 204*e + 336];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, w}>] := -1;
ALEigenvalues[ideal<ZF | {4, 2, -1/2*w^3 + 1/2*w^2 + 7/2*w - 3}>] := -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;