/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![3, 6, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {21, 21, 2/3*w^3 - 1/3*w^2 - 5*w - 1}>;

primesArray := [
[3, 3, w],
[7, 7, 1/3*w^3 - 2/3*w^2 - 2*w + 2],
[9, 3, w + 1],
[13, 13, 2/3*w^3 - 1/3*w^2 - 5*w],
[16, 2, 2],
[17, 17, 1/3*w^3 - 2/3*w^2 - 3*w + 5],
[19, 19, -1/3*w^3 + 2/3*w^2 + 2*w - 5],
[23, 23, -1/3*w^3 + 2/3*w^2 + 3*w - 2],
[25, 5, 1/3*w^3 + 1/3*w^2 - 3*w],
[25, 5, -2/3*w^3 + 1/3*w^2 + 5*w - 3],
[29, 29, -2/3*w^3 + 1/3*w^2 + 4*w - 3],
[29, 29, -1/3*w^3 + 2/3*w^2 + 2*w],
[37, 37, 1/3*w^3 + 1/3*w^2 - 2*w - 3],
[41, 41, -2/3*w^3 + 1/3*w^2 + 5*w - 4],
[43, 43, 2/3*w^3 - 1/3*w^2 - 6*w],
[47, 47, 2/3*w^3 - 1/3*w^2 - 4*w],
[59, 59, 1/3*w^3 + 1/3*w^2 - 2*w - 4],
[59, 59, 4/3*w^3 - 5/3*w^2 - 10*w + 9],
[67, 67, -1/3*w^3 + 2/3*w^2 + 3*w],
[71, 71, 2/3*w^3 - 1/3*w^2 - 4*w + 1],
[73, 73, -1/3*w^3 + 2/3*w^2 + w - 3],
[83, 83, -w^3 + w^2 + 8*w - 4],
[83, 83, -2/3*w^3 + 1/3*w^2 + 6*w - 1],
[97, 97, 5/3*w^3 - 4/3*w^2 - 13*w + 6],
[101, 101, -1/3*w^3 + 2/3*w^2 + 4*w - 2],
[127, 127, w^2 - w - 4],
[131, 131, 7/3*w^3 - 8/3*w^2 - 18*w + 15],
[131, 131, 4/3*w^3 - 8/3*w^2 - 10*w + 17],
[137, 137, 1/3*w^3 + 1/3*w^2 - 4*w - 3],
[139, 139, w^2 - 5],
[151, 151, 1/3*w^3 - 2/3*w^2 - 4*w + 6],
[163, 163, -2/3*w^3 + 4/3*w^2 + 6*w - 9],
[163, 163, -2/3*w^3 + 1/3*w^2 + 3*w - 4],
[179, 179, w^2 - 2*w - 4],
[179, 179, w^2 - w - 7],
[191, 191, 1/3*w^3 + 1/3*w^2 - w - 3],
[191, 191, -2/3*w^3 + 1/3*w^2 + 4*w - 6],
[193, 193, -w^3 + 9*w + 1],
[193, 193, 4/3*w^3 - 2/3*w^2 - 9*w],
[197, 197, 2/3*w^3 - 1/3*w^2 - 5*w - 3],
[197, 197, w^3 - 8*w + 1],
[197, 197, w^3 - 7*w + 2],
[197, 197, -4/3*w^3 + 2/3*w^2 + 11*w - 3],
[223, 223, 4/3*w^3 - 5/3*w^2 - 10*w + 8],
[233, 233, -4/3*w^3 + 2/3*w^2 + 9*w - 5],
[241, 241, 1/3*w^3 + 1/3*w^2 - 4*w - 4],
[257, 257, -w^3 - w^2 + 6*w + 5],
[269, 269, -1/3*w^3 - 1/3*w^2 + 5*w],
[269, 269, -2/3*w^3 + 4/3*w^2 + 4*w - 3],
[271, 271, -2/3*w^3 + 4/3*w^2 + 3*w - 6],
[271, 271, w^3 - 8*w - 5],
[277, 277, 2*w^3 - 2*w^2 - 16*w + 13],
[281, 281, 2/3*w^3 + 2/3*w^2 - 5*w],
[281, 281, -w^3 + w^2 + 7*w - 2],
[331, 331, 2/3*w^3 - 4/3*w^2 - 5*w + 4],
[331, 331, 8/3*w^3 - 7/3*w^2 - 20*w + 15],
[343, 7, -4/3*w^3 + 2/3*w^2 + 11*w - 6],
[347, 347, 2/3*w^3 + 2/3*w^2 - 4*w - 5],
[349, 349, -1/3*w^3 + 2/3*w^2 + 3*w - 8],
[349, 349, -1/3*w^3 + 2/3*w^2 - 3],
[353, 353, -5/3*w^3 + 1/3*w^2 + 12*w],
[353, 353, 1/3*w^3 - 5/3*w^2 - 2*w + 14],
[367, 367, -1/3*w^3 + 5/3*w^2 + 4*w - 9],
[367, 367, 1/3*w^3 + 1/3*w^2 + w - 3],
[379, 379, 4/3*w^3 + 1/3*w^2 - 12*w],
[379, 379, -5/3*w^3 + 7/3*w^2 + 13*w - 12],
[383, 383, -5/3*w^3 + 10/3*w^2 + 12*w - 21],
[397, 397, 1/3*w^3 + 4/3*w^2 - 5*w - 6],
[397, 397, 4/3*w^3 + 1/3*w^2 - 8*w],
[397, 397, -1/3*w^3 + 2/3*w^2 + w - 6],
[397, 397, w^3 + w^2 - 8*w - 7],
[401, 401, 5/3*w^3 - 1/3*w^2 - 14*w + 1],
[419, 419, 1/3*w^3 + 4/3*w^2 - 4*w - 3],
[419, 419, 1/3*w^3 - 5/3*w^2 - 3*w + 12],
[433, 433, -w^3 + w^2 + 9*w - 4],
[433, 433, 1/3*w^3 + 4/3*w^2 - 4*w - 9],
[443, 443, 4/3*w^3 - 2/3*w^2 - 8*w - 1],
[443, 443, -4/3*w^3 + 8/3*w^2 + 9*w - 17],
[449, 449, -w^3 + w^2 + 8*w - 2],
[449, 449, 5/3*w^3 - 7/3*w^2 - 13*w + 18],
[457, 457, -w^3 + 3*w^2 + 7*w - 20],
[457, 457, 2/3*w^3 + 2/3*w^2 - 5*w - 9],
[461, 461, 4/3*w^3 - 2/3*w^2 - 9*w + 2],
[461, 461, -2/3*w^3 + 7/3*w^2 + 4*w - 16],
[463, 463, 4/3*w^3 + 1/3*w^2 - 11*w - 1],
[467, 467, w^3 - 2*w^2 - 9*w + 8],
[479, 479, 4/3*w^3 - 8/3*w^2 - 10*w + 15],
[479, 479, 5/3*w^3 - 1/3*w^2 - 14*w],
[491, 491, -1/3*w^3 + 2/3*w^2 + 2*w - 8],
[509, 509, -11/3*w^3 + 13/3*w^2 + 27*w - 27],
[523, 523, -w - 5],
[523, 523, 4/3*w^3 - 2/3*w^2 - 10*w + 9],
[541, 541, 8/3*w^3 - 7/3*w^2 - 21*w + 13],
[547, 547, -2/3*w^3 + 1/3*w^2 + 5*w - 7],
[557, 557, -2/3*w^3 + 1/3*w^2 + 7*w - 3],
[563, 563, w^2 + w - 8],
[563, 563, 1/3*w^3 + 1/3*w^2 - w - 6],
[569, 569, 1/3*w^3 + 4/3*w^2 - 4*w - 6],
[569, 569, -1/3*w^3 + 5/3*w^2 + 2*w - 6],
[571, 571, 5/3*w^3 - 4/3*w^2 - 11*w + 1],
[593, 593, -2/3*w^3 + 1/3*w^2 + 4*w - 7],
[593, 593, -2*w^3 + w^2 + 15*w - 1],
[599, 599, -1/3*w^3 + 2/3*w^2 + 5*w - 2],
[601, 601, w^3 - 10*w - 2],
[607, 607, 1/3*w^3 + 4/3*w^2 - 4*w - 4],
[607, 607, 7/3*w^3 - 8/3*w^2 - 18*w + 14],
[617, 617, -5/3*w^3 + 10/3*w^2 + 13*w - 24],
[617, 617, 1/3*w^3 + 4/3*w^2 - 2*w - 6],
[617, 617, w^2 + w - 10],
[617, 617, 1/3*w^3 + 1/3*w^2 - 6*w - 1],
[631, 631, 7/3*w^3 - 2/3*w^2 - 18*w + 3],
[631, 631, -1/3*w^3 + 2/3*w^2 + 5*w - 3],
[641, 641, 5/3*w^3 - 1/3*w^2 - 11*w - 2],
[643, 643, -5/3*w^3 + 7/3*w^2 + 14*w - 13],
[643, 643, -2/3*w^3 + 4/3*w^2 + 6*w - 3],
[653, 653, w^3 - w^2 - 8*w + 1],
[653, 653, 1/3*w^3 + 4/3*w^2 - 3*w - 13],
[659, 659, -1/3*w^3 + 5/3*w^2 + 2*w - 8],
[661, 661, 5/3*w^3 - 4/3*w^2 - 11*w + 9],
[683, 683, 4/3*w^3 - 2/3*w^2 - 7*w + 8],
[709, 709, -2*w^3 + 17*w + 2],
[709, 709, 1/3*w^3 + 1/3*w^2 - 4],
[733, 733, 1/3*w^3 - 2/3*w^2 - 2*w - 3],
[733, 733, 5/3*w^3 - 4/3*w^2 - 14*w + 9],
[733, 733, 2/3*w^3 - 7/3*w^2 - 2*w + 10],
[733, 733, 4/3*w^3 - 8/3*w^2 - 9*w + 18],
[743, 743, w^3 - w^2 - 8*w - 1],
[751, 751, -4/3*w^3 - 1/3*w^2 + 10*w],
[757, 757, 5/3*w^3 - 7/3*w^2 - 11*w + 15],
[757, 757, 2/3*w^3 + 5/3*w^2 - 3*w - 5],
[761, 761, 7/3*w^3 - 5/3*w^2 - 18*w + 11],
[761, 761, 1/3*w^3 + 4/3*w^2 - 4*w - 13],
[773, 773, 1/3*w^3 + 4/3*w^2 - 3*w - 7],
[773, 773, 1/3*w^3 + 1/3*w^2 - 3*w - 7],
[809, 809, w^3 - 8*w + 4],
[811, 811, -2/3*w^3 + 4/3*w^2 + 5*w],
[823, 823, 7/3*w^3 - 5/3*w^2 - 17*w + 9],
[827, 827, -5/3*w^3 + 1/3*w^2 + 12*w - 4],
[827, 827, 1/3*w^3 + 4/3*w^2 - 4*w - 10],
[829, 829, -1/3*w^3 - 1/3*w^2 - w - 3],
[841, 29, -1/3*w^3 - 4/3*w^2 + 2*w + 12],
[857, 857, 4*w^3 - 4*w^2 - 31*w + 26],
[859, 859, 11/3*w^3 - 10/3*w^2 - 27*w + 21],
[863, 863, 7/3*w^3 - 5/3*w^2 - 17*w + 8],
[877, 877, 1/3*w^3 + 4/3*w^2 - 3*w - 6],
[877, 877, -4/3*w^3 + 5/3*w^2 + 11*w - 6],
[881, 881, 5/3*w^3 - 4/3*w^2 - 11*w + 4],
[881, 881, 4/3*w^3 - 11/3*w^2 - 10*w + 24],
[883, 883, 10/3*w^3 - 17/3*w^2 - 25*w + 36],
[883, 883, -4/3*w^3 + 2/3*w^2 + 11*w - 8],
[887, 887, -1/3*w^3 + 2/3*w^2 + 3*w - 9],
[887, 887, 3*w^3 - 2*w^2 - 23*w + 8],
[911, 911, 4/3*w^3 + 1/3*w^2 - 7*w],
[919, 919, -4/3*w^3 + 5/3*w^2 + 7*w - 6],
[929, 929, 2*w^3 + w^2 - 14*w - 8],
[937, 937, -7/3*w^3 + 5/3*w^2 + 15*w - 14],
[941, 941, -1/3*w^3 - 1/3*w^2 + 6*w],
[941, 941, -7/3*w^3 - 1/3*w^2 + 18*w + 3],
[971, 971, -4/3*w^3 + 5/3*w^2 + 10*w - 6],
[977, 977, -7/3*w^3 + 11/3*w^2 + 18*w - 27],
[977, 977, 7/3*w^3 - 2/3*w^2 - 17*w],
[997, 997, 7/3*w^3 - 2/3*w^2 - 17*w + 2],
[1009, 1009, 4/3*w^3 + 1/3*w^2 - 13*w + 3],
[1009, 1009, 2/3*w^3 - 7/3*w^2 - 6*w + 15],
[1013, 1013, 4/3*w^3 + 4/3*w^2 - 12*w - 7],
[1013, 1013, 4/3*w^3 + 1/3*w^2 - 9*w + 2],
[1019, 1019, -1/3*w^3 + 2/3*w^2 + 6*w],
[1021, 1021, w^2 - 11],
[1021, 1021, 10/3*w^3 - 8/3*w^2 - 25*w + 17],
[1021, 1021, -7/3*w^3 + 2/3*w^2 + 19*w + 4],
[1021, 1021, -5/3*w^3 + 7/3*w^2 + 10*w - 13],
[1031, 1031, -w^3 + 2*w^2 + 9*w - 14],
[1031, 1031, -2/3*w^3 + 4/3*w^2 + 4*w - 13],
[1031, 1031, -w^3 + 9*w - 5],
[1031, 1031, w^2 + 2*w - 7],
[1033, 1033, 4/3*w^3 - 11/3*w^2 - 9*w + 26],
[1033, 1033, -4/3*w^3 + 2/3*w^2 + 7*w - 9],
[1061, 1061, -1/3*w^3 + 2/3*w^2 - 5],
[1061, 1061, -5/3*w^3 + 4/3*w^2 + 14*w - 7],
[1069, 1069, 4/3*w^3 + 4/3*w^2 - 11*w - 3],
[1069, 1069, -7/3*w^3 + 14/3*w^2 + 17*w - 30],
[1087, 1087, 2*w^3 - w^2 - 14*w + 2],
[1091, 1091, -w^3 + 10*w - 7],
[1091, 1091, -4/3*w^3 + 5/3*w^2 + 8*w - 2],
[1093, 1093, 2/3*w^3 + 2/3*w^2 - 7*w + 3],
[1103, 1103, 1/3*w^3 - 2/3*w^2 - w - 4],
[1109, 1109, -1/3*w^3 + 2/3*w^2 + 5*w - 9],
[1109, 1109, w^3 + w^2 - 7*w - 11],
[1117, 1117, -w^3 + 10*w - 1],
[1129, 1129, -4/3*w^3 + 8/3*w^2 + 10*w - 11],
[1163, 1163, -4/3*w^3 + 2/3*w^2 + 12*w - 5],
[1163, 1163, -1/3*w^3 + 5/3*w^2 + 2*w - 15],
[1181, 1181, -5/3*w^3 + 4/3*w^2 + 11*w - 7],
[1187, 1187, 10/3*w^3 - 8/3*w^2 - 26*w + 17],
[1193, 1193, -2/3*w^3 + 7/3*w^2 + 5*w - 13],
[1193, 1193, 5/3*w^3 - 1/3*w^2 - 11*w],
[1193, 1193, 2/3*w^3 - 1/3*w^2 - 6*w - 5],
[1193, 1193, 4/3*w^3 + 1/3*w^2 - 10*w - 9],
[1201, 1201, 2/3*w^3 + 5/3*w^2 - 6*w - 11],
[1229, 1229, -2/3*w^3 + 7/3*w^2 + 7*w - 13],
[1231, 1231, 5/3*w^3 - 4/3*w^2 - 11*w - 2],
[1249, 1249, -1/3*w^3 - 1/3*w^2 + 4*w - 6],
[1249, 1249, -5/3*w^3 + 4/3*w^2 + 11*w - 6],
[1259, 1259, 5/3*w^3 - 7/3*w^2 - 13*w + 19],
[1277, 1277, 4/3*w^3 + 1/3*w^2 - 13*w],
[1291, 1291, 5/3*w^3 - 1/3*w^2 - 9*w - 2],
[1297, 1297, -1/3*w^3 + 5/3*w^2 + w - 12],
[1327, 1327, w^3 + 2*w^2 - 7*w - 13],
[1327, 1327, 2/3*w^3 - 1/3*w^2 - 5*w - 5],
[1361, 1361, -1/3*w^3 + 5/3*w^2 + 4*w - 11],
[1367, 1367, 2/3*w^3 - 1/3*w^2 - 8*w],
[1367, 1367, -3*w - 5],
[1399, 1399, -2/3*w^3 + 4/3*w^2 + 3*w - 9],
[1399, 1399, -5/3*w^3 + 4/3*w^2 + 13*w - 3],
[1433, 1433, -5/3*w^3 + 4/3*w^2 + 13*w - 15],
[1433, 1433, -7/3*w^3 + 8/3*w^2 + 19*w - 15],
[1439, 1439, -w^3 + 2*w^2 + 6*w - 14],
[1439, 1439, -1/3*w^3 + 2/3*w^2 - w + 7],
[1447, 1447, 1/3*w^3 + 1/3*w^2 - 5*w - 6],
[1447, 1447, -4/3*w^3 + 2/3*w^2 + 12*w - 3],
[1451, 1451, 4/3*w^3 - 8/3*w^2 - 10*w + 21],
[1451, 1451, w^2 - 3*w - 5],
[1451, 1451, -2/3*w^3 + 4/3*w^2 + 5*w - 13],
[1451, 1451, 10/3*w^3 - 14/3*w^2 - 25*w + 33],
[1453, 1453, 2/3*w^3 - 7/3*w^2 - 5*w + 12],
[1453, 1453, 5/3*w^3 - 1/3*w^2 - 15*w - 2],
[1453, 1453, -4/3*w^3 - 1/3*w^2 + 9*w - 3],
[1453, 1453, 4/3*w^3 + 1/3*w^2 - 7*w - 4],
[1471, 1471, -1/3*w^3 + 2/3*w^2 - 6],
[1489, 1489, -5/3*w^3 + 1/3*w^2 + 13*w - 4],
[1489, 1489, -2/3*w^3 + 1/3*w^2 + 2*w - 6],
[1493, 1493, 7/3*w^3 - 11/3*w^2 - 17*w + 27],
[1493, 1493, -2*w^3 + 2*w^2 + 15*w - 8],
[1511, 1511, 17/3*w^3 - 19/3*w^2 - 43*w + 37],
[1523, 1523, -w^3 + 3*w^2 + 9*w - 11],
[1523, 1523, 1/3*w^3 + 1/3*w^2 - 5*w - 7],
[1523, 1523, -w^3 + 3*w^2 + 8*w - 22],
[1523, 1523, 2/3*w^3 + 5/3*w^2 - 4*w - 5],
[1543, 1543, 1/3*w^3 + 1/3*w^2 - 6*w - 4],
[1543, 1543, -4/3*w^3 + 5/3*w^2 + 10*w - 5],
[1553, 1553, -4/3*w^3 + 5/3*w^2 + 11*w - 5],
[1559, 1559, 2*w^3 - 15*w + 1],
[1559, 1559, -4/3*w^3 + 2/3*w^2 + 5*w - 3],
[1567, 1567, 1/3*w^3 + 1/3*w^2 - 6],
[1571, 1571, 1/3*w^3 + 1/3*w^2 + 3*w - 1],
[1571, 1571, -4/3*w^3 + 5/3*w^2 + 8*w - 14],
[1579, 1579, -10/3*w^3 + 11/3*w^2 + 24*w - 20],
[1583, 1583, 1/3*w^3 - 2/3*w^2 - 2*w - 4],
[1597, 1597, 11/3*w^3 - 19/3*w^2 - 28*w + 42],
[1601, 1601, 10/3*w^3 - 14/3*w^2 - 25*w + 26],
[1613, 1613, 8/3*w^3 - 16/3*w^2 - 19*w + 36],
[1627, 1627, -2/3*w^3 + 1/3*w^2 + 3*w - 9],
[1627, 1627, -2/3*w^3 + 1/3*w^2 + 6*w - 9],
[1637, 1637, 2*w^3 + w^2 - 16*w - 4],
[1657, 1657, -2/3*w^3 + 7/3*w^2 + 6*w - 16],
[1657, 1657, 2/3*w^3 + 2/3*w^2 - 3*w - 6],
[1663, 1663, -3*w + 10],
[1663, 1663, 1/3*w^3 + 4/3*w^2 - w - 9],
[1667, 1667, -2/3*w^3 + 1/3*w^2 + 2*w - 7],
[1667, 1667, 8/3*w^3 - 4/3*w^2 - 22*w + 3],
[1669, 1669, 3*w^3 - 2*w^2 - 22*w + 11],
[1693, 1693, 1/3*w^3 - 2/3*w^2 - 3*w - 4],
[1697, 1697, -4/3*w^3 + 2/3*w^2 + 7*w - 3],
[1709, 1709, 3*w^3 - 3*w^2 - 23*w + 23],
[1741, 1741, -3*w^3 + 2*w^2 + 24*w - 10],
[1747, 1747, -2/3*w^3 + 7/3*w^2 + 4*w - 10],
[1747, 1747, 1/3*w^3 - 8/3*w^2 - 4*w + 9],
[1753, 1753, 4/3*w^3 - 2/3*w^2 - 7*w + 2],
[1753, 1753, 8/3*w^3 - 16/3*w^2 - 19*w + 37],
[1759, 1759, 2*w^2 - w - 5],
[1759, 1759, -5/3*w^3 + 10/3*w^2 + 14*w - 21],
[1777, 1777, -1/3*w^3 + 2/3*w^2 + 6*w - 2],
[1783, 1783, 7/3*w^3 - 8/3*w^2 - 17*w + 12],
[1783, 1783, -5/3*w^3 - 2/3*w^2 + 10*w - 3],
[1787, 1787, -1/3*w^3 - 1/3*w^2 - 2*w - 3],
[1787, 1787, 4/3*w^3 + 1/3*w^2 - 12*w + 2],
[1811, 1811, 5/3*w^3 - 1/3*w^2 - 10*w - 6],
[1811, 1811, -w^3 + w^2 + 5*w - 10],
[1847, 1847, w^2 - 2*w - 10],
[1861, 1861, 2*w^2 - w - 11],
[1867, 1867, w^3 + 2*w^2 - 7*w - 4],
[1871, 1871, -1/3*w^3 + 5/3*w^2 - 9],
[1873, 1873, 2/3*w^3 + 5/3*w^2 - 7*w - 6],
[1877, 1877, -2/3*w^3 + 1/3*w^2 + 8*w - 3],
[1877, 1877, -1/3*w^3 + 2/3*w^2 + 6*w - 3],
[1879, 1879, w^3 + 2*w^2 - 11*w - 8],
[1879, 1879, 8/3*w^3 - 7/3*w^2 - 20*w + 18],
[1901, 1901, -2/3*w^3 + 1/3*w^2 + 2*w + 9],
[1901, 1901, 10/3*w^3 - 11/3*w^2 - 24*w + 21],
[1907, 1907, w^3 + w^2 - 10*w - 8],
[1907, 1907, 5*w^3 - 6*w^2 - 38*w + 35],
[1933, 1933, 1/3*w^3 + 1/3*w^2 - 7],
[1933, 1933, 3*w^3 - w^2 - 22*w + 4],
[1949, 1949, 5/3*w^3 + 2/3*w^2 - 15*w - 9],
[1951, 1951, w^3 + w^2 - 12*w - 1],
[1979, 1979, -7/3*w^3 + 5/3*w^2 + 17*w - 15],
[1993, 1993, -5/3*w^3 + 1/3*w^2 + 12*w - 6],
[1999, 1999, -4/3*w^3 - 1/3*w^2 + 13*w + 7],
[1999, 1999, 2/3*w^3 + 5/3*w^2 - 6*w - 5]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 + x^8 - 42*x^7 - 31*x^6 + 539*x^5 + 196*x^4 - 2176*x^3 - 92*x^2 + 376*x - 56;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 1, e, -749849/1554452*e^8 - 917617/1554452*e^7 + 15654677/777226*e^6 + 30365471/1554452*e^5 - 397989303/1554452*e^4 - 59848882/388613*e^3 + 395344644/388613*e^2 + 110673737/388613*e - 44701380/388613, 470185/1554452*e^8 + 585853/1554452*e^7 - 9805199/777226*e^6 - 19331117/1554452*e^5 + 248898863/1554452*e^4 + 37857019/388613*e^3 - 494165597/777226*e^2 - 68882781/388613*e + 29505633/388613, -110952/388613*e^8 - 139270/388613*e^7 + 9260029/777226*e^6 + 9239783/777226*e^5 - 58821481/388613*e^4 - 73415997/777226*e^3 + 467757443/777226*e^2 + 69279518/388613*e - 26670730/388613, -319358/388613*e^8 - 391270/388613*e^7 + 13322658/388613*e^6 + 12916627/388613*e^5 - 169147391/388613*e^4 - 101260463/388613*e^3 + 671405224/388613*e^2 + 185046398/388613*e - 76587710/388613, -194293/1554452*e^8 - 244001/1554452*e^7 + 2022005/388613*e^6 + 7972569/1554452*e^5 - 102414923/1554452*e^4 - 30288693/777226*e^3 + 203332461/777226*e^2 + 24222600/388613*e - 12697098/388613, 1300909/1554452*e^8 + 1618093/1554452*e^7 - 13554504/388613*e^6 - 53433265/1554452*e^5 + 687465023/1554452*e^4 + 209438021/777226*e^3 - 1362976747/777226*e^2 - 190065044/388613*e + 78581854/388613, -42323/388613*e^8 - 93823/777226*e^7 + 3570197/777226*e^6 + 1548140/388613*e^5 - 45991147/777226*e^4 - 24259011/777226*e^3 + 92310839/388613*e^2 + 22494190/388613*e - 9045968/388613, -131453/1554452*e^8 - 179637/1554452*e^7 + 2726041/777226*e^6 + 6004287/1554452*e^5 - 68557415/1554452*e^4 - 12175894/388613*e^3 + 67037564/388613*e^2 + 23893651/388613*e - 5831580/388613, -441413/777226*e^8 - 275149/388613*e^7 + 18390839/777226*e^6 + 18191505/777226*e^5 - 116525972/388613*e^4 - 143162139/777226*e^3 + 461654450/388613*e^2 + 131665765/388613*e - 52967172/388613, 441413/777226*e^8 + 275149/388613*e^7 - 18390839/777226*e^6 - 18191505/777226*e^5 + 116525972/388613*e^4 + 143162139/777226*e^3 - 461654450/388613*e^2 - 131665765/388613*e + 53744398/388613, 2353027/1554452*e^8 + 2906335/1554452*e^7 - 24535022/388613*e^6 - 96008835/1554452*e^5 + 1245551205/1554452*e^4 + 376859171/777226*e^3 - 2470907325/777226*e^2 - 344224999/388613*e + 142926672/388613, 832247/777226*e^8 + 512435/388613*e^7 - 34723215/777226*e^6 - 33799197/777226*e^5 + 220465560/388613*e^4 + 263985725/777226*e^3 - 875465443/388613*e^2 - 236781834/388613*e + 101102146/388613, -628015/1554452*e^8 - 794263/1554452*e^7 + 6538205/388613*e^6 + 26186955/1554452*e^5 - 331069217/1554452*e^4 - 102363867/777226*e^3 + 654648879/777226*e^2 + 92379695/388613*e - 36617664/388613, 2415867/1554452*e^8 + 2970699/1554452*e^7 - 50388013/777226*e^6 - 97977117/1554452*e^5 + 1279408713/1554452*e^4 + 191398038/388613*e^3 - 1269693716/388613*e^2 - 344942561/388613*e + 145128834/388613, 500389/388613*e^8 + 1221557/777226*e^7 - 41758133/777226*e^6 - 20150667/388613*e^5 + 530327039/777226*e^4 + 315078575/777226*e^3 - 1052705070/388613*e^2 - 284062670/388613*e + 120233042/388613, -26377/777226*e^8 - 28773/777226*e^7 + 545170/388613*e^6 + 851551/777226*e^5 - 13612939/777226*e^4 - 2298041/388613*e^3 + 26408154/388613*e^2 - 793474/388613*e - 3338128/388613, -275493/388613*e^8 - 342507/388613*e^7 + 11484949/388613*e^6 + 22596939/777226*e^5 - 145643645/388613*e^4 - 88318143/388613*e^3 + 1153727805/777226*e^2 + 158393251/388613*e - 63344670/388613, 153438/388613*e^8 + 183479/388613*e^7 - 6416768/388613*e^6 - 12047183/777226*e^5 + 81739092/388613*e^4 + 46416467/388613*e^3 - 650697679/777226*e^2 - 80108119/388613*e + 37106940/388613, -79351/777226*e^8 - 111281/777226*e^7 + 1639011/388613*e^6 + 3723425/777226*e^5 - 41019687/777226*e^4 - 15302245/388613*e^3 + 79965991/388613*e^2 + 32649493/388613*e - 6212936/388613, 2578341/1554452*e^8 + 3183173/1554452*e^7 - 26877162/388613*e^6 - 105099773/1554452*e^5 + 1364054587/1554452*e^4 + 411958947/777226*e^3 - 2705787195/777226*e^2 - 374722829/388613*e + 155946790/388613, -254957/1554452*e^8 - 315833/1554452*e^7 + 2674274/388613*e^6 + 10505359/1554452*e^5 - 137027111/1554452*e^4 - 41955085/777226*e^3 + 137863376/388613*e^2 + 39895987/388613*e - 18069840/388613, -1439869/777226*e^8 - 1769823/777226*e^7 + 30026936/388613*e^6 + 29179715/388613*e^5 - 762141223/777226*e^4 - 227863836/388613*e^3 + 3022802559/777226*e^2 + 408667855/388613*e - 170660128/388613, -127681/1554452*e^8 - 189725/1554452*e^7 + 1318876/388613*e^6 + 6328481/1554452*e^5 - 65950915/1554452*e^4 - 25793407/777226*e^3 + 129161799/777226*e^2 + 27151489/388613*e - 9776046/388613, 1131433/777226*e^8 + 701252/388613*e^7 - 47135357/777226*e^6 - 23092732/388613*e^5 + 298601932/388613*e^4 + 359494283/777226*e^3 - 2364732883/777226*e^2 - 319763372/388613*e + 131115636/388613, -2575221/1554452*e^8 - 3182451/1554452*e^7 + 53669345/777226*e^6 + 104846207/1554452*e^5 - 1361136237/1554452*e^4 - 204279246/388613*e^3 + 1349207599/388613*e^2 + 365926465/388613*e - 155063690/388613, -31167/1554452*e^8 - 42337/1554452*e^7 + 674389/777226*e^6 + 1427673/1554452*e^5 - 18080899/1554452*e^4 - 2983948/388613*e^3 + 19307070/388613*e^2 + 5330695/388613*e - 3597804/388613, 29552/388613*e^8 + 71471/777226*e^7 - 2524097/777226*e^6 - 2406007/777226*e^5 + 33153013/777226*e^4 + 19932329/777226*e^3 - 137416585/777226*e^2 - 20995958/388613*e + 10639610/388613, 142553/388613*e^8 + 167259/388613*e^7 - 5982007/388613*e^6 - 5516358/388613*e^5 + 76623275/388613*e^4 + 43200120/388613*e^3 - 307436848/388613*e^2 - 79089245/388613*e + 38583910/388613, 53300/388613*e^8 + 77103/388613*e^7 - 2190992/388613*e^6 - 2582994/388613*e^5 + 27250823/388613*e^4 + 21166334/388613*e^3 - 106211458/388613*e^2 - 42777530/388613*e + 11639004/388613, -424103/777226*e^8 - 493979/777226*e^7 + 8884626/388613*e^6 + 16388615/777226*e^5 - 227016965/777226*e^4 - 65057283/388613*e^3 + 453281699/388613*e^2 + 124337746/388613*e - 52345404/388613, 2071693/1554452*e^8 + 2561729/1554452*e^7 - 21588163/388613*e^6 - 84651925/1554452*e^5 + 1094911155/1554452*e^4 + 332606045/777226*e^3 - 2170326871/777226*e^2 - 305114362/388613*e + 130182802/388613, -4030653/1554452*e^8 - 4948153/1554452*e^7 + 84136485/777226*e^6 + 163266507/1554452*e^5 - 2138774995/1554452*e^4 - 319361739/388613*e^3 + 2125405298/388613*e^2 + 576157951/388613*e - 245065330/388613, -611341/388613*e^8 - 1500097/777226*e^7 + 25509081/388613*e^6 + 49541117/777226*e^5 - 647970001/777226*e^4 - 194247286/388613*e^3 + 2573167583/777226*e^2 + 354508027/388613*e - 144572094/388613, 663679/777226*e^8 + 410226/388613*e^7 - 27718915/777226*e^6 - 27255939/777226*e^5 + 176220493/388613*e^4 + 217688569/777226*e^3 - 700222578/388613*e^2 - 210874338/388613*e + 79374498/388613, -96692/388613*e^8 - 111059/388613*e^7 + 4052508/388613*e^6 + 3670218/388613*e^5 - 51725844/388613*e^4 - 28731879/388613*e^3 + 205103297/388613*e^2 + 51513382/388613*e - 16823374/388613, 1628105/777226*e^8 + 2029611/777226*e^7 - 33919798/388613*e^6 - 33476259/388613*e^5 + 859806101/777226*e^4 + 261768975/388613*e^3 - 3407010527/777226*e^2 - 470947207/388613*e + 193895984/388613, 10345/1554452*e^8 + 60935/1554452*e^7 - 60233/388613*e^6 - 2107479/1554452*e^5 - 486091/1554452*e^4 + 9038513/777226*e^3 + 4403043/388613*e^2 - 8701689/388613*e - 3705724/388613, -251003/777226*e^8 - 159701/388613*e^7 + 10387543/777226*e^6 + 5203359/388613*e^5 - 65263083/388613*e^4 - 78527275/777226*e^3 + 514878677/777226*e^2 + 61938720/388613*e - 32914882/388613, 495653/777226*e^8 + 592743/777226*e^7 - 10358867/388613*e^6 - 19550535/777226*e^5 + 263548567/777226*e^4 + 76455265/388613*e^3 - 521955974/388613*e^2 - 139263044/388613*e + 51446894/388613, -1696387/777226*e^8 - 2096729/777226*e^7 + 35370296/388613*e^6 + 69307775/777226*e^5 - 897819719/777226*e^4 - 272362856/388613*e^3 + 1782315608/388613*e^2 + 497424252/388613*e - 205715498/388613, -3517987/1554452*e^8 - 4371651/1554452*e^7 + 73296675/777226*e^6 + 144112705/1554452*e^5 - 1858360153/1554452*e^4 - 281138295/388613*e^3 + 1841981175/388613*e^2 + 502170723/388613*e - 209780878/388613, -242625/1554452*e^8 - 343869/1554452*e^7 + 2473698/388613*e^6 + 11210031/1554452*e^5 - 121730571/1554452*e^4 - 42078657/777226*e^3 + 117182793/388613*e^2 + 30359739/388613*e - 11958684/388613, -511311/388613*e^8 - 1269459/777226*e^7 + 42566419/777226*e^6 + 20893328/388613*e^5 - 538747103/777226*e^4 - 324355497/777226*e^3 + 1066040618/388613*e^2 + 284693681/388613*e - 123172350/388613, -156485/388613*e^8 - 179451/388613*e^7 + 6552607/388613*e^6 + 5928199/388613*e^5 - 83661222/388613*e^4 - 46231369/388613*e^3 + 333997635/388613*e^2 + 81430249/388613*e - 40307580/388613, -2817365/1554452*e^8 - 3425817/1554452*e^7 + 29407879/388613*e^6 + 113189987/1554452*e^5 - 1495419239/1554452*e^4 - 444113699/777226*e^3 + 1486364379/388613*e^2 + 404250333/388613*e - 172683040/388613, -497434/388613*e^8 - 628778/388613*e^7 + 20714360/388613*e^6 + 20773680/388613*e^5 - 262380822/388613*e^4 - 163156254/388613*e^3 + 1040044902/388613*e^2 + 296564424/388613*e - 119547956/388613, -2278357/777226*e^8 - 2806849/777226*e^7 + 47510824/388613*e^6 + 92593349/777226*e^5 - 1206101253/777226*e^4 - 362060734/388613*e^3 + 2393695214/388613*e^2 + 654377520/388613*e - 271772812/388613, -1173927/1554452*e^8 - 1532409/1554452*e^7 + 24371095/777226*e^6 + 50507401/1554452*e^5 - 614758127/1554452*e^4 - 98638692/388613*e^3 + 606774206/388613*e^2 + 175599535/388613*e - 71761314/388613, 1001686/388613*e^8 + 1232728/388613*e^7 - 41836492/388613*e^6 - 81623141/777226*e^5 + 532098064/388613*e^4 + 322333983/388613*e^3 - 4232678655/777226*e^2 - 598571093/388613*e + 243666350/388613, -27139/777226*e^8 - 15222/388613*e^7 + 1100461/777226*e^6 + 858177/777226*e^5 - 6695762/388613*e^4 - 2931443/777226*e^3 + 25291783/388613*e^2 - 9912905/388613*e + 445468/388613, 1112673/777226*e^8 + 1358305/777226*e^7 - 23216146/388613*e^6 - 44840177/777226*e^5 + 589800145/777226*e^4 + 175532882/388613*e^3 - 1170872763/388613*e^2 - 317850736/388613*e + 133150626/388613, -195743/777226*e^8 - 231881/777226*e^7 + 4084749/388613*e^6 + 3757483/388613*e^5 - 103783663/777226*e^4 - 27979166/388613*e^3 + 411352325/777226*e^2 + 45361495/388613*e - 19628534/388613, -87803/777226*e^8 - 51138/388613*e^7 + 3709537/777226*e^6 + 3390967/777226*e^5 - 24001856/388613*e^4 - 26264227/777226*e^3 + 97686074/388613*e^2 + 20656643/388613*e - 14963372/388613, -198065/388613*e^8 - 233913/388613*e^7 + 8264598/388613*e^6 + 7648375/388613*e^5 - 105021423/388613*e^4 - 58471374/388613*e^3 + 418046032/388613*e^2 + 100958020/388613*e - 53663952/388613, -190989/777226*e^8 - 107668/388613*e^7 + 7975797/777226*e^6 + 7052375/777226*e^5 - 50608854/388613*e^4 - 53793903/777226*e^3 + 199388917/388613*e^2 + 44620111/388613*e - 18780764/388613, -680296/388613*e^8 - 837002/388613*e^7 + 28357307/388613*e^6 + 27553430/388613*e^5 - 359654983/388613*e^4 - 214372318/388613*e^3 + 1426470232/388613*e^2 + 383791372/388613*e - 165754566/388613, -367357/777226*e^8 - 421559/777226*e^7 + 7702496/388613*e^6 + 13840935/777226*e^5 - 197003717/777226*e^4 - 53494698/388613*e^3 + 394033307/388613*e^2 + 97021842/388613*e - 50364494/388613, 3690177/1554452*e^8 + 4513633/1554452*e^7 - 77043705/777226*e^6 - 149534879/1554452*e^5 + 1958695507/1554452*e^4 + 295943761/388613*e^3 - 1945328444/388613*e^2 - 556388319/388613*e + 225132532/388613, 144239/388613*e^8 + 314373/777226*e^7 - 6099259/388613*e^6 - 10526547/777226*e^5 + 157820545/777226*e^4 + 42757424/388613*e^3 - 637866361/777226*e^2 - 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333905614/388613, 433465/1554452*e^8 + 517569/1554452*e^7 - 4522104/388613*e^6 - 17004493/1554452*e^5 + 228601983/1554452*e^4 + 65287673/777226*e^3 - 443244939/777226*e^2 - 55693919/388613*e + 17527880/388613, -584568/388613*e^8 - 1508133/777226*e^7 + 48579107/777226*e^6 + 50003689/777226*e^5 - 613097957/777226*e^4 - 397186033/777226*e^3 + 2419293083/777226*e^2 + 373385478/388613*e - 150182044/388613, 1966161/1554452*e^8 + 2189549/1554452*e^7 - 20681340/388613*e^6 - 72102397/1554452*e^5 + 1062767251/1554452*e^4 + 280625347/777226*e^3 - 2129532929/777226*e^2 - 256885203/388613*e + 117580878/388613, -1617539/777226*e^8 - 966501/388613*e^7 + 67687017/777226*e^6 + 63748677/777226*e^5 - 431459556/388613*e^4 - 497838429/777226*e^3 + 1717542258/388613*e^2 + 448269056/388613*e - 181427832/388613, 2057185/1554452*e^8 + 2597233/1554452*e^7 - 42761743/777226*e^6 - 85143879/1554452*e^5 + 1080369295/1554452*e^4 + 163011744/388613*e^3 - 1067800090/388613*e^2 - 265843444/388613*e + 125687774/388613, 4132081/1554452*e^8 + 4882941/1554452*e^7 - 86534865/777226*e^6 - 160541823/1554452*e^5 + 2209797683/1554452*e^4 + 310360043/388613*e^3 - 2205402240/388613*e^2 - 540677978/388613*e + 257880730/388613, 3749759/1554452*e^8 + 4653471/1554452*e^7 - 78249769/777226*e^6 - 154247141/1554452*e^5 + 1988496685/1554452*e^4 + 305254685/388613*e^3 - 1976633826/388613*e^2 - 564304969/388613*e + 242510080/388613, 2435100/388613*e^8 + 3029705/388613*e^7 - 101455284/388613*e^6 - 100047644/388613*e^5 + 1285841157/388613*e^4 + 784060802/388613*e^3 - 5096440171/388613*e^2 - 1418223170/388613*e + 578286218/388613, 351299/388613*e^8 + 739031/777226*e^7 - 29794885/777226*e^6 - 24243721/777226*e^5 + 387069681/777226*e^4 + 186227205/777226*e^3 - 1571809357/777226*e^2 - 158743014/388613*e + 87490810/388613, -1215427/388613*e^8 - 1507137/388613*e^7 + 50698934/388613*e^6 + 49859504/388613*e^5 - 643506861/388613*e^4 - 393437442/388613*e^3 + 2551940219/388613*e^2 + 731954706/388613*e - 288752308/388613, -893766/388613*e^8 - 2246113/777226*e^7 + 74436379/777226*e^6 + 36988130/388613*e^5 - 942529845/777226*e^4 - 576996821/777226*e^3 + 1865289169/388613*e^2 + 517022449/388613*e - 212377124/388613, 3335137/388613*e^8 + 8195329/777226*e^7 - 278332577/777226*e^6 - 135298196/388613*e^5 + 3535227915/777226*e^4 + 2121496595/777226*e^3 - 7020420367/388613*e^2 - 1932484394/388613*e + 806322690/388613, -1997743/1554452*e^8 - 2402623/1554452*e^7 + 41875863/777226*e^6 + 79620929/1554452*e^5 - 1070961749/1554452*e^4 - 157188686/388613*e^3 + 1072324067/388613*e^2 + 289898800/388613*e - 143135698/388613, 5426433/777226*e^8 + 6601403/777226*e^7 - 113237469/388613*e^6 - 108884251/388613*e^5 + 2877192985/777226*e^4 + 851442106/388613*e^3 - 11427019327/777226*e^2 - 1547124509/388613*e + 654181966/388613, -2455474/388613*e^8 - 6139587/777226*e^7 + 204680867/777226*e^6 + 101495704/388613*e^5 - 2595114117/777226*e^4 - 1597978683/777226*e^3 + 5143195601/388613*e^2 + 1468284093/388613*e - 587343246/388613, 6630947/777226*e^8 + 8180997/777226*e^7 - 138344380/388613*e^6 - 270119785/777226*e^5 + 3514147871/777226*e^4 + 1058875546/388613*e^3 - 6978608219/388613*e^2 - 1924208519/388613*e + 809693590/388613, 714111/388613*e^8 + 875966/388613*e^7 - 29804495/388613*e^6 - 28825632/388613*e^5 + 378817717/388613*e^4 + 223201852/388613*e^3 - 1507544938/388613*e^2 - 386577422/388613*e + 180234178/388613, 6485467/1554452*e^8 + 8137367/1554452*e^7 - 135004739/777226*e^6 - 268250929/1554452*e^5 + 3418496165/1554452*e^4 + 523255827/388613*e^3 - 3382597535/388613*e^2 - 935083137/388613*e + 372149162/388613, 1585981/777226*e^8 + 988508/388613*e^7 - 65926583/777226*e^6 - 64596243/777226*e^5 + 416208381/388613*e^4 + 491040215/777226*e^3 - 1641085896/388613*e^2 - 403374250/388613*e + 185694184/388613, 4362939/1554452*e^8 + 5477931/1554452*e^7 - 45381656/388613*e^6 - 180898727/1554452*e^5 + 2295552141/1554452*e^4 + 710005381/777226*e^3 - 4537449875/777226*e^2 - 651142647/388613*e + 270257944/388613, 4920619/777226*e^8 + 2987834/388613*e^7 - 205458537/777226*e^6 - 197165729/777226*e^5 + 1305816074/388613*e^4 + 1541983167/777226*e^3 - 5187021691/388613*e^2 - 1392623581/388613*e + 588568068/388613, -2925379/777226*e^8 - 1785442/388613*e^7 + 122030241/777226*e^6 + 117784841/777226*e^5 - 774693810/388613*e^4 - 919988505/777226*e^3 + 3076531016/388613*e^2 + 823879556/388613*e - 363774232/388613, -843825/388613*e^8 - 1080859/388613*e^7 + 35141171/388613*e^6 + 71623659/777226*e^5 - 445256532/388613*e^4 - 283565896/388613*e^3 + 3533893901/777226*e^2 + 525252819/388613*e - 213618712/388613, -10972283/1554452*e^8 - 13472327/1554452*e^7 + 114532984/388613*e^6 + 445683835/1554452*e^5 - 5823717345/1554452*e^4 - 1755976195/777226*e^3 + 11574486759/777226*e^2 + 1619386749/388613*e - 669729980/388613, 5700949/777226*e^8 + 6970837/777226*e^7 - 118927301/388613*e^6 - 115027754/388613*e^5 + 3020496615/777226*e^4 + 900599092/388613*e^3 - 11993290857/777226*e^2 - 1636809035/388613*e + 680104778/388613, 3144942/388613*e^8 + 7676341/777226*e^7 - 262516313/777226*e^6 - 126712553/388613*e^5 + 3335299219/777226*e^4 + 1985668425/777226*e^3 - 6625398178/388613*e^2 - 1808946789/388613*e + 761967568/388613, -1272057/388613*e^8 - 1502804/388613*e^7 + 53243210/388613*e^6 + 49536967/388613*e^5 - 679006637/388613*e^4 - 386371610/388613*e^3 + 2704057876/388613*e^2 + 698631588/388613*e - 301471072/388613, -1334645/777226*e^8 - 1588035/777226*e^7 + 27907301/388613*e^6 + 52653597/777226*e^5 - 710819887/777226*e^4 - 209183669/388613*e^3 + 1413081499/388613*e^2 + 404776074/388613*e - 155834218/388613, -1621939/777226*e^8 - 2043629/777226*e^7 + 33719222/388613*e^6 + 33652427/388613*e^5 - 852157831/777226*e^4 - 261830761/388613*e^3 + 3364094909/777226*e^2 + 459856507/388613*e - 176903664/388613, 3776743/1554452*e^8 + 4578007/1554452*e^7 - 78806365/777226*e^6 - 151172961/1554452*e^5 + 2002626325/1554452*e^4 + 295977567/388613*e^3 - 1990568533/388613*e^2 - 534665554/388613*e + 242813938/388613, 105475/388613*e^8 + 150209/388613*e^7 - 4399171/388613*e^6 - 5018508/388613*e^5 + 55784771/388613*e^4 + 41588072/388613*e^3 - 221496463/388613*e^2 - 94869471/388613*e + 30643472/388613, -10748/1697*e^8 - 13187/1697*e^7 + 448466/1697*e^6 + 434619/1697*e^5 - 5695946/1697*e^4 - 3387411/1697*e^3 + 22623941/1697*e^2 + 6023256/1697*e - 2590362/1697, 1050159/388613*e^8 + 1288536/388613*e^7 - 43799664/388613*e^6 - 42583015/388613*e^5 + 555786712/388613*e^4 + 333961267/388613*e^3 - 2202715257/388613*e^2 - 602505012/388613*e + 245505794/388613, 11315025/1554452*e^8 + 14086233/1554452*e^7 - 117883667/388613*e^6 - 465546897/1554452*e^5 + 5978068271/1554452*e^4 + 1830268909/777226*e^3 - 11852936087/777226*e^2 - 1680728449/388613*e + 692780320/388613, 9470413/1554452*e^8 + 11687409/1554452*e^7 - 197552473/777226*e^6 - 386004987/1554452*e^5 + 5017262259/1554452*e^4 + 757013297/388613*e^3 - 4981708646/388613*e^2 - 1377967435/388613*e + 569459042/388613, -1026319/388613*e^8 - 1233197/388613*e^7 + 42899588/388613*e^6 + 40675404/388613*e^5 - 546361576/388613*e^4 - 317329229/388613*e^3 + 2178161563/388613*e^2 + 566899556/388613*e - 265307626/388613, -10388281/1554452*e^8 - 12757751/1554452*e^7 + 108326435/388613*e^6 + 421085215/1554452*e^5 - 5499704505/1554452*e^4 - 1649326261/777226*e^3 + 5454349478/388613*e^2 + 1500347526/388613*e - 616745348/388613, 3970933/777226*e^8 + 4884511/777226*e^7 - 82903495/388613*e^6 - 161551791/777226*e^5 + 2107835837/777226*e^4 + 636187028/388613*e^3 - 4189702486/388613*e^2 - 1168473506/388613*e + 481274728/388613];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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