/*
  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 | {3, 3, w}>;

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^3 - x^2 - 5*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, -e^2 + e + 5, e + 2, 3*e^2 - 5*e - 9, 2*e^2 - 2*e - 8, -e^2 + 1, 2*e^2 - 4*e - 4, -e^2 - 2*e + 5, 2*e^2 - 4*e - 5, -e^2 + 4*e + 3, -2*e^2 + 6*e + 5, -3*e^2 + 4*e + 9, e^2 - 1, -e^2 + 10, e^2 + 2*e - 5, -e^2 + 2*e + 11, e^2 - 2*e + 2, e^2 - 4*e + 5, e^2 - 2*e - 15, e^2 - 6*e - 3, e^2 - 2*e + 7, -e^2 - 1, -e^2 - 4*e + 11, -2*e^2 + 2*e + 10, -7*e^2 + 6*e + 21, -3*e^2 + 6*e + 19, -e^2 + 8*e + 3, 3*e^2 - 3*e - 7, 6*e^2 - 2*e - 21, 3*e^2 - 2*e - 3, 4*e^2 - 14*e - 12, -5*e^2 + 14*e + 15, -5*e^2 + 9*e + 21, 4*e^2 - 3*e - 18, 2*e^2 - 6*e - 1, -4*e^2 + 8*e + 12, 6*e + 4, -8*e^2 + 8*e + 24, 6*e + 8, 2*e^2 - 2*e - 26, -4*e^2 + 8*e + 20, -8*e^2 + 12*e + 16, 3*e^2 - 2*e + 5, e^2 - 12*e + 5, -5*e^2 + 2*e + 17, 7*e^2 - 14*e - 31, 2*e^2 + e - 10, -7*e^2 + 5*e + 17, 2*e^2 - 9*e + 2, -e^2 - e + 19, 5*e^2 - 12*e - 16, 3*e^2 + 2*e - 27, -9*e^2 + 16*e + 26, 6*e^2 - 14*e - 4, -6*e^2 + 10*e + 4, -5*e^2 + 10*e + 37, -2*e^2 + 10*e - 2, -12*e^2 + 22*e + 32, 5*e^2 + 2*e - 26, 4*e^2 - 9*e - 28, -10*e + 9, 6*e^2 - 12*e - 2, -4*e^2 + 2*e + 13, -2*e^2 + 2*e - 2, -e^2 + 10*e + 5, 4*e^2 - 2*e + 6, -e^2 + 4*e + 25, 4*e^2 + 4*e - 22, 3*e^2 - 2*e - 29, -9*e^2 + 8*e + 27, -13*e^2 + 20*e + 41, -e^2 - 9*e + 11, e^2 + e + 15, 8*e^2 - 18*e - 22, 7*e^2 - 14*e - 13, e^2 + 2*e - 15, -6*e^2 + 18*e + 24, 7*e^2 - 6*e - 7, -7*e^2 + 18*e + 34, -14*e^2 + 26*e + 46, -3*e^2 - 7*e + 9, 3*e^2 - 12*e - 7, -e^2 + 6*e - 13, 9*e^2 - 12*e - 40, -8*e^2 + 18*e + 38, -2*e^2 + 10*e, 3*e^2 - 16*e - 2, -2*e^2 - 8*e + 13, 8*e^2 - 6*e - 22, 3*e^2 - 16*e + 3, e^2 - 18*e + 7, 7*e^2 - 19, 3*e^2 + 8*e - 17, 8*e^2 - 6*e - 8, 7*e^2 - 16*e - 43, -2*e^2 + 8*e - 18, 11*e^2 - 7*e - 47, -5*e^2 + 4*e + 11, -6*e^2 + 4*e, e^2 - 7*e - 5, 4*e^2 - 14*e - 24, -7*e^2 - 4*e + 27, e^2 + 2*e + 4, 8*e^2 - 6*e - 30, e^2 + 6*e - 25, 5*e^2 - 6*e + 5, 5*e^2 - 6*e - 39, -12*e^2 + 22*e + 41, e^2 + 2*e - 30, -7*e^2 + 8*e + 37, 2*e^2 - 7*e - 18, 5*e^2 - 22*e - 16, -2*e^2 - e - 8, -e^2 + 2*e + 29, -7*e^2 + 43, -4*e^2 + 20*e + 2, -13*e^2 + 29*e + 39, -e^2 - 8*e + 33, 5*e^2 - 21*e - 23, 8*e + 10, -e^2 + 6*e + 27, 10*e^2 - 10*e, -2*e^2 + 10*e - 15, 4*e^2 + 4*e - 28, 20*e^2 - 34*e - 60, -12*e^2 + 6*e + 34, 2*e^2 + 6*e - 16, -13*e^2 + 17*e + 31, -9*e^2 + 6*e + 21, -7*e^2 + 20*e + 27, -2*e^2 + 14*e + 10, 15*e^2 - 20*e - 31, -6*e^2 + 19*e + 18, -8*e^2 + 12*e + 46, -6*e^2 + 12*e + 12, -7*e^2 + 15, 14*e^2 - 33*e - 54, 12*e^2 - 30*e - 51, -9*e^2 + 8*e + 13, -9*e^2 + 22*e + 40, 3*e^2 + 4*e - 49, e^2 + 10*e - 21, -14*e^2 + 15*e + 48, 5*e^2 - 16*e - 29, -10*e + 20, 13*e^2 - 4*e - 45, -2*e^2 - 8*e - 11, 4*e - 29, 9*e^2 - 4*e - 42, -8*e^2 + 10*e - 8, -e^2 + 11, -2*e^2 - 6*e + 28, 6*e^2 - 20*e - 44, -8*e^2 + 12*e + 8, 9*e^2 - 14*e + 5, 6*e^2 + 2*e - 24, -5*e^2 + 2*e + 51, 11*e^2 - 20*e - 42, 18*e^2 - 23*e - 34, 12*e^2 - 20*e - 46, -2*e^2 - 2*e + 42, 9*e^2 - 11*e - 15, 2*e^2 - 6*e - 20, 4*e^2 + 8*e - 9, 7*e^2 - 12*e + 2, e^2 + 4*e - 33, -8*e^2 + 22*e + 42, 7*e^2 - 22*e - 2, -10*e^2 + 13*e + 40, -12*e^2 + 10*e + 54, -5*e^2 + 2*e + 15, 9*e^2 - 20*e - 51, 3*e^2 + 9*e - 7, 5*e^2 + 2*e - 22, 3*e - 36, -19*e^2 + 16*e + 64, 7*e^2 - 31*e - 9, -17*e - 10, -15*e^2 + 6*e + 61, 18*e^2 - 24*e - 53, -2*e^2 - 23, -15*e^2 + 30*e + 64, -17*e^2 + 8*e + 59, -14*e^2 + 14*e + 55, e^2 + 7*e + 19, -4*e^2 + 16*e + 30, -4*e^2 + 4*e - 22, 10*e^2 - 18*e - 22, -10*e^2 + 31*e + 32, 5*e^2 + 8*e - 48, e^2 + 12*e - 39, 10*e^2 - 8*e - 47, -3*e^2 + 2*e - 1, 7*e^2 - 16*e - 24, 4*e^2 + e - 18, 10*e^2 - 24*e - 36, 14*e^2 - 24*e - 35, 8*e^2 - 11*e - 18, 6*e^2 + 11*e - 16, 3*e^2 - 6*e - 7, 14*e^2 - 34*e - 30, -12*e^2 + 6*e + 36, 3*e^2 - 25*e - 9, 9*e^2 - 24*e - 25, 5*e^2 + 4*e - 52, -e^2 + 6*e + 42, -9*e^2 + 24*e + 35, -8*e^2 + 6*e + 5, -9*e^2 + 16*e + 21, 12*e^2 - 16*e - 37, -4*e^2 + 30*e + 7, 25*e^2 - 32*e - 57, -18*e^2 + 36*e + 52, -8*e^2 + 6*e + 20, 9*e^2 - 25*e - 11, 3*e^2 + 2, -17*e^2 + 29*e + 47, 4*e^2 - e - 28, -4*e^2 + 5*e + 56, 4*e^2 - 16*e + 22, -4*e^2 - 6*e + 33, 13*e^2 - 21*e - 51, 7*e^2 - 34*e - 23, 11*e^2 - 7*e - 69, 6*e^2 - 19*e - 10, 8*e^2 + 3*e - 34, -19*e^2 + 39*e + 51, 7*e^2 - 16*e - 23, 12*e^2 - 11*e - 30, -6*e^2 - 10*e + 66, -19*e^2 + 50*e + 63, -2*e^2 + 4*e + 36, -22*e + 11, 12*e^2 - 46*e - 44, 19*e^2 - 36*e - 61, 2*e^2 - 2*e + 12, 23*e^2 - 41*e - 75, 16*e^2 - 22*e - 50, 11*e^2 - 30*e - 37, -14*e^2 + 35*e + 72, 4*e^2 - 34, 14*e^2 - 10*e - 47, -6*e^2 + 14*e - 7, 4*e^2 - 4*e - 24, 12*e^2 + 4*e - 64, -11*e^2 + 4*e + 35, -e^2 + 6*e - 7, 11*e^2 - 2*e - 66, -11*e^2 + 26*e + 21, -3*e^2 - 15*e + 1, -8*e^2 + 10*e - 29, e^2 + 2*e - 59, -2*e^2 + 23*e + 4, -21*e^2 + 34*e + 89, -11*e^2 + 20*e + 11, -3*e^2 - 2*e - 21, 3*e^2 + 16*e - 48, 4*e^2 - 20*e + 11, -6*e + 26, 10*e^2 - 30*e - 38, 9*e^2 - 6*e - 13, 8*e^2 - 14*e - 26, -15*e^2 + 3*e + 69, -7*e^2 - 8*e + 33, -9*e^2 - 10*e + 22, 8*e + 17, -11*e^2 + 2*e + 41, -2*e^2 - 14*e - 4, -19*e^2 + 16*e + 63, -17*e^2 + 14*e + 31, -19*e^2 + 50*e + 55, -13*e^2 + 10*e + 59, 4*e^2 + 2*e + 6, e^2 + 6*e + 30, -14*e^2 + 30*e + 48, -6*e^2 + 8*e - 4, -6*e^2 + 12*e - 24, 23*e^2 - 41*e - 53, -5*e^2 + 6*e + 33, 7*e^2 - e - 9, -16*e^2 + 30*e + 68, -20*e^2 + 28*e + 75, 7*e^2 - 85, e^2 - 25, -15*e^2 + 40*e + 33, -7*e^2 + 20*e + 58, 17*e^2 - 35*e - 63, 12*e^2 - 32*e - 20, -2*e^2 + 16*e - 9, -8*e^2 + 4*e + 36, 9*e^2 + 12*e - 47, 2*e^2 + 3*e + 30, e^2 - 2*e - 17, -26*e^2 + 19*e + 72, -19*e^2 + 30*e + 87, -11*e^2 + 30*e + 17, -14*e^2 + 12*e + 30, 7*e^2 - 20*e + 7];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w}>] := 1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;