/*
  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 | {13, 13, 1/2*w^3 + 1/2*w^2 - 7/2*w - 1}>;

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^10 - 18*x^8 + 118*x^6 - 333*x^4 + 345*x^2 - 16;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 1/4*e^9 - 7/2*e^7 + 33/2*e^5 - 117/4*e^3 + 57/4*e, -3/4*e^9 + 19/2*e^7 - 79/2*e^5 + 235/4*e^3 - 71/4*e, -2*e^8 + 25*e^6 - 100*e^4 + 130*e^2 - 8, 1, 1/2*e^9 - 6*e^7 + 23*e^5 - 59/2*e^3 + 5/2*e, -2*e^9 + 25*e^7 - 99*e^5 + 122*e^3 + 7*e, -e^6 + 9*e^4 - 19*e^2, -e^9 + 11*e^7 - 36*e^5 + 33*e^3 + 4*e, 5/2*e^9 - 30*e^7 + 112*e^5 - 251/2*e^3 - 33/2*e, -3/2*e^9 + 18*e^7 - 67*e^5 + 149/2*e^3 + 17/2*e, 2*e^9 - 24*e^7 + 93*e^5 - 129*e^3 + 40*e, 1/2*e^9 - 6*e^7 + 21*e^5 - 25/2*e^3 - 57/2*e, e^6 - 8*e^4 + 13*e^2 - 2, 2*e^9 - 26*e^7 + 111*e^5 - 169*e^3 + 53*e, 2*e^8 - 24*e^6 + 91*e^4 - 112*e^2 + 4, 2*e^9 - 26*e^7 + 114*e^5 - 191*e^3 + 85*e, 9*e^8 - 109*e^6 + 416*e^4 - 506*e^2 + 20, -5*e^8 + 60*e^6 - 225*e^4 + 263*e^2 - 6, e^8 - 15*e^6 + 72*e^4 - 107*e^2 - 6, -10*e^8 + 122*e^6 - 475*e^4 + 609*e^2 - 46, -7/2*e^9 + 44*e^7 - 180*e^5 + 519/2*e^3 - 147/2*e, 4*e^9 - 47*e^7 + 169*e^5 - 172*e^3 - 42*e, 3/2*e^9 - 19*e^7 + 78*e^5 - 213/2*e^3 + 19/2*e, 3*e^9 - 38*e^7 + 155*e^5 - 214*e^3 + 44*e, 6*e^8 - 72*e^6 + 274*e^4 - 336*e^2, 3*e^7 - 31*e^5 + 93*e^3 - 77*e, 3*e^9 - 35*e^7 + 122*e^5 - 102*e^3 - 77*e, 5/2*e^9 - 29*e^7 + 103*e^5 - 213/2*e^3 - 33/2*e, -10*e^8 + 122*e^6 - 470*e^4 + 574*e^2 - 8, 11/2*e^9 - 66*e^7 + 247*e^5 - 565/2*e^3 - 29/2*e, 5*e^9 - 61*e^7 + 232*e^5 - 256*e^3 - 71*e, e^9 - 14*e^7 + 62*e^5 - 79*e^3 - 26*e, -3*e^9 + 36*e^7 - 131*e^5 + 120*e^3 + 85*e, e^8 - 14*e^6 + 61*e^4 - 81*e^2 + 4, -1/2*e^9 + 7*e^7 - 38*e^5 + 205/2*e^3 - 241/2*e, -3*e^9 + 36*e^7 - 132*e^5 + 131*e^3 + 52*e, -e^8 + 16*e^6 - 80*e^4 + 127*e^2 - 16, -4*e^8 + 48*e^6 - 183*e^4 + 227*e^2 - 14, 5/2*e^9 - 30*e^7 + 113*e^5 - 263/2*e^3 - 19/2*e, -5/2*e^9 + 32*e^7 - 136*e^5 + 437/2*e^3 - 217/2*e, -13*e^8 + 158*e^6 - 605*e^4 + 734*e^2 - 26, 3*e^9 - 39*e^7 + 169*e^5 - 277*e^3 + 134*e, 4*e^8 - 46*e^6 + 163*e^4 - 178*e^2 - 12, 10*e^8 - 119*e^6 + 445*e^4 - 536*e^2 + 36, -3*e^8 + 39*e^6 - 161*e^4 + 208*e^2 + 6, -3*e^8 + 41*e^6 - 184*e^4 + 278*e^2 - 42, -2*e^8 + 25*e^6 - 98*e^4 + 121*e^2 - 6, 1/2*e^9 - 7*e^7 + 28*e^5 - 23/2*e^3 - 151/2*e, 9*e^8 - 108*e^6 + 406*e^4 - 481*e^2 + 12, -20*e^8 + 242*e^6 - 924*e^4 + 1131*e^2 - 62, 1/2*e^9 - 4*e^7 + 2*e^5 + 77/2*e^3 - 147/2*e, 22*e^8 - 271*e^6 + 1059*e^4 - 1321*e^2 + 30, -3/2*e^9 + 18*e^7 - 63*e^5 + 67/2*e^3 + 203/2*e, 4*e^9 - 49*e^7 + 197*e^5 - 294*e^3 + 120*e, 17*e^8 - 205*e^6 + 782*e^4 - 969*e^2 + 70, e^8 - 11*e^6 + 36*e^4 - 39*e^2 + 10, 4*e^8 - 53*e^6 + 228*e^4 - 321*e^2 + 10, -6*e^9 + 73*e^7 - 284*e^5 + 368*e^3 - 37*e, 2*e^8 - 27*e^6 + 115*e^4 - 156*e^2 + 28, -7*e^8 + 83*e^6 - 308*e^4 + 371*e^2 - 42, -6*e^9 + 72*e^7 - 269*e^5 + 300*e^3 + 45*e, 8*e^8 - 100*e^6 + 400*e^4 - 521*e^2 + 20, -4*e^9 + 51*e^7 - 209*e^5 + 283*e^3 - 30*e, -7*e^9 + 85*e^7 - 321*e^5 + 355*e^3 + 70*e, -10*e^8 + 126*e^6 - 508*e^4 + 668*e^2 - 54, 21*e^8 - 260*e^6 + 1025*e^4 - 1302*e^2 + 46, 2*e^9 - 27*e^7 + 127*e^5 - 246*e^3 + 163*e, -15*e^8 + 186*e^6 - 733*e^4 + 928*e^2 - 40, 5/2*e^9 - 30*e^7 + 117*e^5 - 335/2*e^3 + 115/2*e, 1/2*e^9 - 5*e^7 + 15*e^5 - 21/2*e^3 - 51/2*e, 5*e^8 - 62*e^6 + 246*e^4 - 328*e^2 + 40, 5/2*e^9 - 29*e^7 + 98*e^5 - 127/2*e^3 - 199/2*e, -7/2*e^9 + 40*e^7 - 137*e^5 + 237/2*e^3 + 145/2*e, -13*e^8 + 168*e^6 - 699*e^4 + 956*e^2 - 80, -3*e^8 + 31*e^6 - 89*e^4 + 57*e^2 + 16, 16*e^8 - 191*e^6 + 718*e^4 - 862*e^2 + 14, -20*e^8 + 243*e^6 - 932*e^4 + 1139*e^2 - 50, -12*e^8 + 147*e^6 - 572*e^4 + 710*e^2 + 2, 19*e^8 - 237*e^6 + 943*e^4 - 1218*e^2 + 82, 5/2*e^9 - 34*e^7 + 157*e^5 - 553/2*e^3 + 257/2*e, -4*e^8 + 57*e^6 - 262*e^4 + 390*e^2 - 28, -9/2*e^9 + 54*e^7 - 199*e^5 + 417/2*e^3 + 95/2*e, 1/2*e^9 - 3*e^7 - 6*e^5 + 73/2*e^3 - 9/2*e, -2*e^8 + 21*e^6 - 65*e^4 + 62*e^2 - 16, -15*e^8 + 183*e^6 - 711*e^4 + 898*e^2 - 48, 1/2*e^9 - 8*e^7 + 45*e^5 - 209/2*e^3 + 149/2*e, 21*e^8 - 254*e^6 + 971*e^4 - 1195*e^2 + 66, 10*e^8 - 128*e^6 + 525*e^4 - 699*e^2 + 36, -12*e^8 + 146*e^6 - 560*e^4 + 677*e^2 - 8, -5*e^5 + 37*e^3 - 54*e, -5*e^9 + 58*e^7 - 204*e^5 + 198*e^3 + 50*e, -11*e^9 + 131*e^7 - 481*e^5 + 500*e^3 + 167*e, -3*e^9 + 36*e^7 - 137*e^5 + 174*e^3 - 45*e, 8*e^8 - 94*e^6 + 345*e^4 - 405*e^2 + 14, -29/2*e^9 + 179*e^7 - 713*e^5 + 1947/2*e^3 - 365/2*e, -3*e^8 + 33*e^6 - 112*e^4 + 129*e^2 - 44, -3/2*e^9 + 25*e^7 - 137*e^5 + 537/2*e^3 - 245/2*e, -9*e^8 + 112*e^6 - 447*e^4 + 577*e^2 - 12, 13*e^9 - 162*e^7 + 646*e^5 - 841*e^3 + 49*e, 2*e^9 - 26*e^7 + 116*e^5 - 222*e^3 + 181*e, -7*e^8 + 85*e^6 - 325*e^4 + 394*e^2 - 44, -7/2*e^9 + 43*e^7 - 168*e^5 + 407/2*e^3 + 65/2*e, 3*e^8 - 37*e^6 + 142*e^4 - 167*e^2 - 4, 15/2*e^9 - 92*e^7 + 348*e^5 - 727/2*e^3 - 275/2*e, 3*e^9 - 31*e^7 + 82*e^5 + 7*e^3 - 141*e, 8*e^9 - 94*e^7 + 338*e^5 - 346*e^3 - 77*e, -7*e^9 + 89*e^7 - 377*e^5 + 601*e^3 - 242*e, 4*e^8 - 48*e^6 + 183*e^4 - 234*e^2 + 32, -13/2*e^9 + 78*e^7 - 291*e^5 + 627/2*e^3 + 199/2*e, 17*e^8 - 204*e^6 + 768*e^4 - 906*e^2 + 6, 19/2*e^9 - 118*e^7 + 474*e^5 - 1295/2*e^3 + 193/2*e, 23*e^8 - 285*e^6 + 1125*e^4 - 1427*e^2 + 34, 23/2*e^9 - 145*e^7 + 594*e^5 - 1701/2*e^3 + 459/2*e, 20*e^8 - 251*e^6 + 1004*e^4 - 1280*e^2 + 14, 20*e^8 - 244*e^6 + 942*e^4 - 1170*e^2 + 84, -33*e^8 + 400*e^6 - 1537*e^4 + 1912*e^2 - 106, -13/2*e^9 + 74*e^7 - 251*e^5 + 433/2*e^3 + 223/2*e, 7/2*e^9 - 42*e^7 + 151*e^5 - 245/2*e^3 - 273/2*e, -33*e^8 + 403*e^6 - 1562*e^4 + 1952*e^2 - 84, -10*e^9 + 120*e^7 - 451*e^5 + 520*e^3 + 59*e, -16*e^8 + 198*e^6 - 778*e^4 + 985*e^2 - 60, -6*e^9 + 71*e^7 - 258*e^5 + 260*e^3 + 97*e, -21/2*e^9 + 128*e^7 - 501*e^5 + 1337/2*e^3 - 253/2*e, -23*e^8 + 278*e^6 - 1060*e^4 + 1301*e^2 - 98, 22*e^8 - 273*e^6 + 1075*e^4 - 1353*e^2 + 48, 15*e^8 - 181*e^6 + 686*e^4 - 828*e^2 + 56, 11*e^8 - 130*e^6 + 483*e^4 - 577*e^2 + 46, 11/2*e^9 - 67*e^7 + 246*e^5 - 431/2*e^3 - 365/2*e, -8*e^8 + 96*e^6 - 367*e^4 + 462*e^2 - 34, -7/2*e^9 + 45*e^7 - 175*e^5 + 317/2*e^3 + 323/2*e, 9*e^9 - 110*e^7 + 428*e^5 - 552*e^3 + 77*e, -10*e^8 + 122*e^6 - 471*e^4 + 571*e^2 + 24, 10*e^8 - 131*e^6 + 552*e^4 - 756*e^2 + 40, -18*e^8 + 222*e^6 - 869*e^4 + 1091*e^2 - 48, 2*e^9 - 18*e^7 + 30*e^5 + 59*e^3 - 104*e, 13/2*e^9 - 78*e^7 + 296*e^5 - 719/2*e^3 - 23/2*e, -13*e^8 + 166*e^6 - 680*e^4 + 899*e^2 - 30, -20*e^8 + 243*e^6 - 936*e^4 + 1160*e^2 - 60, 23*e^8 - 284*e^6 + 1116*e^4 - 1428*e^2 + 116, -32*e^8 + 382*e^6 - 1435*e^4 + 1721*e^2 - 50, -25/2*e^9 + 150*e^7 - 561*e^5 + 1245/2*e^3 + 259/2*e, 9/2*e^9 - 57*e^7 + 232*e^5 - 617/2*e^3 + 49/2*e, -15/2*e^9 + 91*e^7 - 346*e^5 + 811/2*e^3 + 31/2*e, -e^6 + 12*e^4 - 39*e^2 + 24, e^8 - 10*e^6 + 35*e^4 - 68*e^2 + 40, -12*e^8 + 142*e^6 - 529*e^4 + 636*e^2 - 22, 15*e^8 - 191*e^6 + 775*e^4 - 1004*e^2 + 10, 14*e^8 - 180*e^6 + 740*e^4 - 977*e^2 + 22, -13/2*e^9 + 90*e^7 - 417*e^5 + 1423/2*e^3 - 577/2*e, 3/2*e^9 - 13*e^7 + 13*e^5 + 203/2*e^3 - 351/2*e, -11/2*e^9 + 69*e^7 - 281*e^5 + 791/2*e^3 - 183/2*e, -19/2*e^9 + 119*e^7 - 483*e^5 + 1353/2*e^3 - 303/2*e, 1/2*e^9 - 9*e^7 + 61*e^5 - 377/2*e^3 + 489/2*e, 2*e^9 - 27*e^7 + 130*e^5 - 273*e^3 + 207*e, -9*e^9 + 111*e^7 - 427*e^5 + 480*e^3 + 116*e, 9/2*e^9 - 60*e^7 + 260*e^5 - 753/2*e^3 + 69/2*e, 5/2*e^9 - 34*e^7 + 153*e^5 - 499/2*e^3 + 221/2*e, 7*e^9 - 88*e^7 + 356*e^5 - 475*e^3 + 39*e, 29*e^8 - 347*e^6 + 1304*e^4 - 1555*e^2 + 40, 17*e^8 - 217*e^6 + 890*e^4 - 1185*e^2 + 50, 7*e^8 - 77*e^6 + 256*e^4 - 253*e^2 - 2, -12*e^9 + 151*e^7 - 620*e^5 + 900*e^3 - 247*e, -27*e^8 + 336*e^6 - 1333*e^4 + 1709*e^2 - 92, 5*e^9 - 61*e^7 + 230*e^5 - 234*e^3 - 127*e, -37*e^8 + 453*e^6 - 1768*e^4 + 2253*e^2 - 124, -17/2*e^9 + 112*e^7 - 491*e^5 + 1575/2*e^3 - 567/2*e, 7/2*e^9 - 47*e^7 + 212*e^5 - 693/2*e^3 + 201/2*e, -33/2*e^9 + 206*e^7 - 820*e^5 + 2099/2*e^3 - 51/2*e, -6*e^7 + 57*e^5 - 137*e^3 + 46*e, -11*e^8 + 132*e^6 - 503*e^4 + 620*e^2 - 24, -e^8 + 11*e^6 - 34*e^4 + 27*e^2 - 30, -18*e^8 + 219*e^6 - 843*e^4 + 1041*e^2 - 62, 15*e^8 - 190*e^6 + 769*e^4 - 996*e^2 + 32, -21/2*e^9 + 128*e^7 - 494*e^5 + 1205/2*e^3 + 55/2*e, -5/2*e^9 + 34*e^7 - 163*e^5 + 653/2*e^3 - 409/2*e, 17*e^8 - 209*e^6 + 812*e^4 - 996*e^2 - 10, 13*e^8 - 160*e^6 + 639*e^4 - 868*e^2 + 92, 20*e^8 - 250*e^6 + 994*e^4 - 1266*e^2 + 60, -17/2*e^9 + 97*e^7 - 331*e^5 + 575/2*e^3 + 297/2*e, 41*e^8 - 507*e^6 + 1995*e^4 - 2529*e^2 + 106, -12*e^8 + 144*e^6 - 546*e^4 + 673*e^2 - 52, 25/2*e^9 - 158*e^7 + 652*e^5 - 1897/2*e^3 + 487/2*e, 7*e^8 - 84*e^6 + 308*e^4 - 333*e^2 - 10, 17*e^8 - 216*e^6 + 877*e^4 - 1158*e^2 + 96, -3*e^8 + 30*e^6 - 80*e^4 + 38*e^2 + 16, -4*e^9 + 56*e^7 - 269*e^5 + 505*e^3 - 295*e, 20*e^8 - 250*e^6 + 994*e^4 - 1264*e^2 + 38, 12*e^9 - 142*e^7 + 522*e^5 - 573*e^3 - 96*e, 11*e^8 - 134*e^6 + 518*e^4 - 641*e^2 + 16, 14*e^8 - 171*e^6 + 672*e^4 - 884*e^2 + 80, -4*e^9 + 46*e^7 - 154*e^5 + 95*e^3 + 176*e, -23*e^8 + 281*e^6 - 1096*e^4 + 1409*e^2 - 106, 4*e^9 - 55*e^7 + 260*e^5 - 477*e^3 + 236*e, 5*e^8 - 59*e^6 + 216*e^4 - 243*e^2 - 16, -9/2*e^9 + 61*e^7 - 271*e^5 + 823/2*e^3 - 155/2*e, -e^8 + 12*e^6 - 43*e^4 + 36*e^2 - 4, -2*e^8 + 33*e^6 - 173*e^4 + 297*e^2 - 68, -41*e^8 + 505*e^6 - 1977*e^4 + 2487*e^2 - 66, -11*e^8 + 138*e^6 - 558*e^4 + 741*e^2 - 50, 1/2*e^9 - 5*e^7 + 22*e^5 - 155/2*e^3 + 227/2*e, -7/2*e^9 + 42*e^7 - 163*e^5 + 437/2*e^3 - 61/2*e, 13*e^8 - 162*e^6 + 636*e^4 - 777*e^2 - 28, -29*e^8 + 352*e^6 - 1347*e^4 + 1640*e^2 - 54, 17*e^8 - 203*e^6 + 768*e^4 - 951*e^2 + 66, 25/2*e^9 - 161*e^7 + 675*e^5 - 1939/2*e^3 + 389/2*e, 3*e^8 - 30*e^6 + 79*e^4 - 29*e^2 - 50, 45/2*e^9 - 274*e^7 + 1059*e^5 - 2663/2*e^3 + 211/2*e, -e^8 + 12*e^6 - 44*e^4 + 45*e^2 + 18, -4*e^9 + 44*e^7 - 141*e^5 + 102*e^3 + 90*e, -7/2*e^9 + 37*e^7 - 97*e^5 - 105/2*e^3 + 591/2*e, 23*e^9 - 283*e^7 + 1108*e^5 - 1412*e^3 + 111*e, 3*e^9 - 37*e^7 + 144*e^5 - 176*e^3 + 12*e, -6*e^8 + 63*e^6 - 193*e^4 + 175*e^2 - 16, -46*e^8 + 560*e^6 - 2163*e^4 + 2707*e^2 - 174, 33*e^8 - 413*e^6 + 1650*e^4 - 2138*e^2 + 118, 16*e^9 - 197*e^7 + 760*e^5 - 868*e^3 - 186*e, -16*e^8 + 199*e^6 - 786*e^4 + 988*e^2 - 40, 9*e^8 - 100*e^6 + 332*e^4 - 317*e^2 - 8, -1/2*e^9 + 8*e^7 - 26*e^5 - 135/2*e^3 + 529/2*e, 11*e^8 - 136*e^6 + 544*e^4 - 730*e^2 + 74, 7/2*e^9 - 45*e^7 + 187*e^5 - 517/2*e^3 + 71/2*e, 4*e^6 - 29*e^4 + 46*e^2 - 40, 50*e^8 - 620*e^6 + 2450*e^4 - 3129*e^2 + 136, -2*e^8 + 18*e^6 - 42*e^4 + 37*e^2 - 46, -e^9 + 10*e^7 - 11*e^5 - 120*e^3 + 256*e, 3*e^9 - 39*e^7 + 167*e^5 - 244*e^3 + 24*e, -4*e^9 + 53*e^7 - 239*e^5 + 413*e^3 - 172*e, 46*e^8 - 558*e^6 + 2139*e^4 - 2629*e^2 + 128, -5*e^9 + 58*e^7 - 212*e^5 + 270*e^3 - 90*e, -e^9 + 9*e^7 - 32*e^5 + 106*e^3 - 189*e, -8*e^9 + 103*e^7 - 441*e^5 + 693*e^3 - 243*e, -9*e^9 + 114*e^7 - 477*e^5 + 728*e^3 - 227*e, -1/2*e^9 + 7*e^7 - 11*e^5 - 263/2*e^3 + 705/2*e, -31*e^8 + 378*e^6 - 1463*e^4 + 1838*e^2 - 112, 6*e^9 - 83*e^7 + 392*e^5 - 710*e^3 + 343*e, -36*e^8 + 432*e^6 - 1633*e^4 + 1982*e^2 - 108, e^9 - 15*e^7 + 73*e^5 - 119*e^3 + 21*e, 4*e^8 - 45*e^6 + 153*e^4 - 150*e^2 - 24, -17*e^9 + 206*e^7 - 783*e^5 + 933*e^3 - 20*e, -14*e^9 + 178*e^7 - 728*e^5 + 981*e^3 - 100*e, 9*e^9 - 106*e^7 + 398*e^5 - 524*e^3 + 147*e, 26*e^8 - 316*e^6 + 1219*e^4 - 1505*e^2 + 12, 45*e^8 - 553*e^6 + 2166*e^4 - 2760*e^2 + 126, 20*e^8 - 238*e^6 + 897*e^4 - 1110*e^2 + 98, 14*e^8 - 176*e^6 + 705*e^4 - 899*e^2 - 30, -13*e^8 + 163*e^6 - 646*e^4 + 826*e^2 - 114, 22*e^8 - 275*e^6 + 1096*e^4 - 1411*e^2 + 36, -8*e^8 + 94*e^6 - 346*e^4 + 392*e^2 + 36, 29/2*e^9 - 171*e^7 + 616*e^5 - 1221/2*e^3 - 433/2*e, 19/2*e^9 - 111*e^7 + 394*e^5 - 759/2*e^3 - 273/2*e, -27*e^8 + 340*e^6 - 1372*e^4 + 1799*e^2 - 74, 11/2*e^9 - 69*e^7 + 281*e^5 - 763/2*e^3 + 25/2*e, 19/2*e^9 - 115*e^7 + 428*e^5 - 881/2*e^3 - 297/2*e, -22*e^8 + 261*e^6 - 968*e^4 + 1132*e^2 - 2, -11*e^9 + 136*e^7 - 531*e^5 + 639*e^3 + 65*e, 9*e^9 - 101*e^7 + 328*e^5 - 224*e^3 - 223*e, -31/2*e^9 + 183*e^7 - 669*e^5 + 1477/2*e^3 + 119/2*e, -27/2*e^9 + 163*e^7 - 627*e^5 + 1593/2*e^3 - 133/2*e, 3*e^7 - 40*e^5 + 173*e^3 - 255*e, 21*e^8 - 255*e^6 + 980*e^4 - 1199*e^2 + 2, -13*e^8 + 159*e^6 - 612*e^4 + 729*e^2 + 34, -8*e^8 + 93*e^6 - 333*e^4 + 366*e^2 + 12, -19/2*e^9 + 119*e^7 - 477*e^5 + 1259/2*e^3 - 143/2*e, 46*e^8 - 565*e^6 + 2210*e^4 - 2810*e^2 + 170, 2*e^9 - 27*e^7 + 124*e^5 - 213*e^3 + 84*e, -7*e^9 + 86*e^7 - 338*e^5 + 434*e^3 - 16*e, 5*e^9 - 57*e^7 + 204*e^5 - 253*e^3 + 82*e, -26*e^8 + 314*e^6 - 1206*e^4 + 1528*e^2 - 138, -17/2*e^9 + 94*e^7 - 308*e^5 + 525/2*e^3 + 215/2*e, -3*e^9 + 43*e^7 - 200*e^5 + 315*e^3 - 76*e, 11*e^8 - 141*e^6 + 583*e^4 - 784*e^2 + 2, 4*e^9 - 47*e^7 + 170*e^5 - 183*e^3 + 6*e, -23*e^8 + 289*e^6 - 1164*e^4 + 1517*e^2 - 30, 3/2*e^9 - 17*e^7 + 40*e^5 + 231/2*e^3 - 747/2*e, -31*e^8 + 383*e^6 - 1497*e^4 + 1863*e^2 - 66, 49*e^8 - 606*e^6 + 2386*e^4 - 3037*e^2 + 126, 7/2*e^9 - 33*e^7 + 64*e^5 + 171/2*e^3 - 389/2*e, 21/2*e^9 - 132*e^7 + 536*e^5 - 1455/2*e^3 + 171/2*e, -22*e^8 + 272*e^6 - 1078*e^4 + 1405*e^2 - 92, -23/2*e^9 + 143*e^7 - 571*e^5 + 1503/2*e^3 - 95/2*e, 11*e^9 - 138*e^7 + 565*e^5 - 816*e^3 + 199*e, 9/2*e^9 - 62*e^7 + 290*e^5 - 1043/2*e^3 + 527/2*e, 27/2*e^9 - 170*e^7 + 686*e^5 - 1821/2*e^3 + 159/2*e, -25*e^9 + 314*e^7 - 1257*e^5 + 1607*e^3 - 9*e, -8*e^9 + 104*e^7 - 445*e^5 + 665*e^3 - 133*e, 5*e^9 - 54*e^7 + 152*e^5 + 25*e^3 - 375*e, -34*e^8 + 409*e^6 - 1555*e^4 + 1915*e^2 - 166, 19/2*e^9 - 120*e^7 + 502*e^5 - 1555/2*e^3 + 609/2*e, 30*e^8 - 365*e^6 + 1411*e^4 - 1756*e^2 + 48];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, 1/2*w^3 + 1/2*w^2 - 7/2*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;