/*
  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![44, 2, -15, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[4, 2, w + 2],
[4, 2, -1/2*w^3 + 3/2*w^2 + 9/2*w - 10],
[9, 3, 1/2*w^3 - 5/2*w^2 - 5/2*w + 17],
[9, 3, 3/2*w^3 - 11/2*w^2 - 19/2*w + 28],
[11, 11, 1/2*w^3 - 3/2*w^2 - 9/2*w + 11],
[11, 11, -w - 3],
[25, 5, w^3 - 3*w^2 - 7*w + 15],
[29, 29, w + 1],
[29, 29, 1/2*w^3 - 3/2*w^2 - 9/2*w + 9],
[31, 31, -1/2*w^3 + 5/2*w^2 + 5/2*w - 16],
[31, 31, 1/2*w^3 - 3/2*w^2 - 9/2*w + 5],
[31, 31, -w + 3],
[31, 31, 3/2*w^3 - 11/2*w^2 - 19/2*w + 29],
[59, 59, 2*w^2 - w - 13],
[59, 59, 9/2*w^3 - 31/2*w^2 - 61/2*w + 85],
[61, 61, 2*w^3 - 6*w^2 - 15*w + 31],
[61, 61, -3/2*w^3 + 11/2*w^2 + 21/2*w - 34],
[71, 71, 3/2*w^3 - 11/2*w^2 - 19/2*w + 32],
[71, 71, -1/2*w^3 + 5/2*w^2 + 5/2*w - 13],
[79, 79, -3*w^3 + 10*w^2 + 19*w - 51],
[79, 79, 15/2*w^3 - 51/2*w^2 - 103/2*w + 139],
[101, 101, -2*w^3 + 7*w^2 + 14*w - 37],
[101, 101, -1/2*w^3 + 5/2*w^2 + 7/2*w - 16],
[109, 109, w^3 - 3*w^2 - 6*w + 19],
[109, 109, -w^3 + 4*w^2 + 5*w - 19],
[121, 11, -3/2*w^3 + 9/2*w^2 + 21/2*w - 23],
[139, 139, -w^3 + 4*w^2 + 7*w - 25],
[139, 139, -5/2*w^3 + 17/2*w^2 + 33/2*w - 41],
[149, 149, 1/2*w^3 + 1/2*w^2 - 11/2*w - 8],
[149, 149, 5/2*w^3 - 19/2*w^2 - 35/2*w + 50],
[149, 149, -9/2*w^3 + 31/2*w^2 + 59/2*w - 82],
[149, 149, -5/2*w^3 + 19/2*w^2 + 35/2*w - 56],
[151, 151, -1/2*w^3 + 3/2*w^2 + 3/2*w - 12],
[151, 151, 3/2*w^3 - 9/2*w^2 - 25/2*w + 28],
[179, 179, -1/2*w^3 + 3/2*w^2 + 9/2*w - 3],
[179, 179, w - 5],
[229, 229, -15/2*w^3 + 51/2*w^2 + 105/2*w - 142],
[229, 229, 1/2*w^3 - 7/2*w^2 - 7/2*w + 19],
[229, 229, -17/2*w^3 + 57/2*w^2 + 119/2*w - 156],
[229, 229, -9/2*w^3 + 33/2*w^2 + 55/2*w - 82],
[239, 239, -9*w^3 + 31*w^2 + 59*w - 161],
[239, 239, w^3 + w^2 - 11*w - 19],
[241, 241, 5/2*w^3 - 15/2*w^2 - 37/2*w + 37],
[241, 241, -2*w^3 + 6*w^2 + 13*w - 29],
[251, 251, -7/2*w^3 + 25/2*w^2 + 45/2*w - 67],
[251, 251, 1/2*w^3 - 5/2*w^2 - 13/2*w + 7],
[251, 251, -1/2*w^3 + 1/2*w^2 + 13/2*w + 4],
[251, 251, -1/2*w^3 + 7/2*w^2 + 3/2*w - 23],
[271, 271, -w^3 + 4*w^2 + 5*w - 25],
[271, 271, -5/2*w^3 + 17/2*w^2 + 33/2*w - 47],
[281, 281, 5/2*w^3 - 15/2*w^2 - 39/2*w + 41],
[281, 281, 3/2*w^3 - 7/2*w^2 - 27/2*w + 14],
[289, 17, -7/2*w^3 + 25/2*w^2 + 45/2*w - 62],
[289, 17, 5/2*w^3 - 15/2*w^2 - 39/2*w + 37],
[311, 311, -19/2*w^3 + 65/2*w^2 + 129/2*w - 175],
[311, 311, -7/2*w^3 + 25/2*w^2 + 47/2*w - 71],
[311, 311, 7/2*w^3 - 25/2*w^2 - 49/2*w + 67],
[311, 311, -3/2*w^3 + 11/2*w^2 + 15/2*w - 31],
[331, 331, 19/2*w^3 - 65/2*w^2 - 127/2*w + 173],
[331, 331, -w^3 - w^2 + 10*w + 15],
[349, 349, -5/2*w^3 + 15/2*w^2 + 33/2*w - 35],
[349, 349, 3/2*w^3 - 11/2*w^2 - 21/2*w + 36],
[361, 19, 2*w^3 - 6*w^2 - 14*w + 29],
[361, 19, 2*w^3 - 6*w^2 - 14*w + 31],
[379, 379, 1/2*w^3 - 5/2*w^2 - 1/2*w + 20],
[379, 379, -1/2*w^3 + 5/2*w^2 + 1/2*w - 9],
[401, 401, -7/2*w^3 + 25/2*w^2 + 45/2*w - 69],
[401, 401, w^3 - 3*w^2 - 5*w + 19],
[409, 409, 3/2*w^3 - 7/2*w^2 - 23/2*w + 13],
[409, 409, -11/2*w^3 + 37/2*w^2 + 77/2*w - 100],
[409, 409, 7/2*w^3 - 23/2*w^2 - 47/2*w + 58],
[409, 409, -w^3 + 3*w^2 + 9*w - 9],
[419, 419, 3/2*w^3 - 13/2*w^2 - 17/2*w + 36],
[419, 419, -5/2*w^3 + 19/2*w^2 + 31/2*w - 54],
[421, 421, -w^3 + 4*w^2 + 5*w - 27],
[421, 421, 7/2*w^3 - 23/2*w^2 - 47/2*w + 63],
[439, 439, 2*w^3 - 7*w^2 - 14*w + 35],
[439, 439, -1/2*w^3 + 5/2*w^2 + 7/2*w - 18],
[461, 461, -5/2*w^3 + 17/2*w^2 + 33/2*w - 48],
[461, 461, -13/2*w^3 + 45/2*w^2 + 85/2*w - 119],
[461, 461, -1/2*w^3 + 3/2*w^2 + 1/2*w - 15],
[461, 461, 3/2*w^3 - 11/2*w^2 - 17/2*w + 32],
[479, 479, -3*w^3 + 10*w^2 + 19*w - 53],
[479, 479, 5/2*w^3 - 17/2*w^2 - 33/2*w + 51],
[491, 491, 3*w^2 - 2*w - 23],
[491, 491, 13/2*w^3 - 45/2*w^2 - 87/2*w + 120],
[499, 499, -3*w^3 + 11*w^2 + 21*w - 63],
[499, 499, -5/2*w^3 + 19/2*w^2 + 31/2*w - 50],
[499, 499, 11/2*w^3 - 37/2*w^2 - 75/2*w + 101],
[499, 499, 9*w^3 - 31*w^2 - 58*w + 159],
[509, 509, 3/2*w^3 - 9/2*w^2 - 23/2*w + 29],
[509, 509, w^3 - 3*w^2 - 6*w + 21],
[521, 521, 2*w^3 - 8*w^2 - 12*w + 43],
[521, 521, -3/2*w^3 + 11/2*w^2 + 19/2*w - 36],
[521, 521, 2*w^3 - 8*w^2 - 12*w + 47],
[521, 521, -7/2*w^3 + 23/2*w^2 + 45/2*w - 58],
[541, 541, -1/2*w^3 + 5/2*w^2 + 13/2*w - 6],
[541, 541, 9/2*w^3 - 33/2*w^2 - 55/2*w + 80],
[541, 541, -15/2*w^3 + 51/2*w^2 + 101/2*w - 134],
[541, 541, w^3 - w^2 - 7*w + 1],
[571, 571, 13/2*w^3 - 47/2*w^2 - 79/2*w + 116],
[571, 571, -5/2*w^3 + 11/2*w^2 + 43/2*w - 21],
[599, 599, -3*w^3 + 9*w^2 + 19*w - 47],
[599, 599, -15/2*w^3 + 51/2*w^2 + 103/2*w - 137],
[599, 599, -w^3 + 2*w^2 + 11*w + 1],
[599, 599, 5/2*w^3 - 17/2*w^2 - 35/2*w + 38],
[601, 601, -1/2*w^3 + 11/2*w^2 - 5/2*w - 47],
[601, 601, -3/2*w^3 + 13/2*w^2 + 17/2*w - 30],
[619, 619, -w^3 + 5*w^2 + 5*w - 27],
[619, 619, -1/2*w^3 + 1/2*w^2 + 11/2*w - 2],
[619, 619, -2*w^3 + 7*w^2 + 12*w - 39],
[619, 619, -3*w^3 + 11*w^2 + 19*w - 63],
[631, 631, -3/2*w^3 + 13/2*w^2 + 17/2*w - 37],
[631, 631, 5/2*w^3 - 19/2*w^2 - 31/2*w + 53],
[641, 641, 5*w^3 - 16*w^2 - 35*w + 79],
[641, 641, -2*w^3 + 8*w^2 + 12*w - 53],
[641, 641, 3/2*w^3 - 11/2*w^2 - 23/2*w + 34],
[641, 641, 3/2*w^3 - 11/2*w^2 - 23/2*w + 27],
[659, 659, -5/2*w^3 + 19/2*w^2 + 31/2*w - 51],
[659, 659, -3/2*w^3 + 13/2*w^2 + 17/2*w - 39],
[661, 661, -5/2*w^3 + 17/2*w^2 + 33/2*w - 50],
[661, 661, 5*w^3 - 17*w^2 - 30*w + 83],
[661, 661, 5/2*w^3 - 17/2*w^2 - 31/2*w + 46],
[661, 661, 3/2*w^3 - 5/2*w^2 - 21/2*w + 9],
[691, 691, -3/2*w^3 + 13/2*w^2 + 19/2*w - 35],
[691, 691, 3*w^3 - 11*w^2 - 20*w + 63],
[709, 709, 19/2*w^3 - 65/2*w^2 - 127/2*w + 171],
[709, 709, 4*w^3 - 14*w^2 - 28*w + 75],
[709, 709, 1/2*w^3 - 5/2*w^2 - 15/2*w + 4],
[709, 709, -w^3 + 5*w^2 + 7*w - 31],
[739, 739, 11/2*w^3 - 37/2*w^2 - 73/2*w + 95],
[739, 739, -3/2*w^3 + 5/2*w^2 + 25/2*w - 5],
[739, 739, 27/2*w^3 - 91/2*w^2 - 187/2*w + 247],
[739, 739, 4*w^3 - 15*w^2 - 24*w + 73],
[751, 751, -5/2*w^3 + 19/2*w^2 + 35/2*w - 49],
[751, 751, -2*w^2 + 2*w + 21],
[751, 751, -9*w^3 + 31*w^2 + 61*w - 169],
[751, 751, 17/2*w^3 - 57/2*w^2 - 117/2*w + 152],
[761, 761, 5/2*w^3 - 15/2*w^2 - 33/2*w + 39],
[761, 761, -3*w^3 + 9*w^2 + 22*w - 47],
[761, 761, 9/2*w^3 - 29/2*w^2 - 61/2*w + 80],
[761, 761, 25/2*w^3 - 85/2*w^2 - 169/2*w + 227],
[769, 769, 2*w^3 - 8*w^2 - 12*w + 45],
[811, 811, 5/2*w^3 - 19/2*w^2 - 27/2*w + 45],
[811, 811, 1/2*w^3 - 7/2*w^2 + 1/2*w + 29],
[821, 821, 25/2*w^3 - 85/2*w^2 - 173/2*w + 233],
[821, 821, 19/2*w^3 - 63/2*w^2 - 135/2*w + 173],
[829, 829, -7/2*w^3 + 27/2*w^2 + 43/2*w - 79],
[829, 829, 10*w^3 - 34*w^2 - 67*w + 179],
[829, 829, -3/2*w^3 + 1/2*w^2 + 27/2*w + 9],
[829, 829, 5/2*w^3 - 13/2*w^2 - 41/2*w + 31],
[839, 839, -3/2*w^3 + 11/2*w^2 + 25/2*w - 26],
[839, 839, 2*w^3 - 7*w^2 - 16*w + 43],
[841, 29, 5/2*w^3 - 15/2*w^2 - 35/2*w + 39],
[881, 881, -8*w^3 + 27*w^2 + 56*w - 151],
[881, 881, -w^3 + 3*w^2 + 4*w - 21],
[911, 911, 5/2*w^3 - 19/2*w^2 - 33/2*w + 51],
[911, 911, 2*w^3 - 8*w^2 - 13*w + 47],
[919, 919, 3/2*w^3 - 9/2*w^2 - 27/2*w + 29],
[919, 919, -3*w - 5],
[919, 919, 1/2*w^3 - 7/2*w^2 - 5/2*w + 17],
[919, 919, -4*w^3 + 14*w^2 + 27*w - 81],
[929, 929, 7/2*w^3 - 23/2*w^2 - 49/2*w + 58],
[929, 929, 13*w^3 - 44*w^2 - 89*w + 237],
[929, 929, 9/2*w^3 - 29/2*w^2 - 59/2*w + 78],
[929, 929, -w^3 + 2*w^2 + 7*w - 5],
[941, 941, 5/2*w^3 - 19/2*w^2 - 31/2*w + 59],
[941, 941, -3/2*w^3 + 13/2*w^2 + 17/2*w - 31],
[941, 941, w^3 - 3*w^2 - 10*w + 7],
[941, 941, -8*w^3 + 27*w^2 + 56*w - 147],
[971, 971, -9/2*w^3 + 29/2*w^2 + 73/2*w - 92],
[971, 971, -1/2*w^3 + 5/2*w^2 + 17/2*w - 1],
[991, 991, w^3 - 4*w^2 - 3*w + 15],
[991, 991, -1/2*w^3 + 1/2*w^2 + 15/2*w + 6],
[1019, 1019, 3*w^3 - 9*w^2 - 19*w + 45],
[1019, 1019, 3*w^3 - 11*w^2 - 20*w + 53],
[1019, 1019, -1/2*w^3 + 1/2*w^2 + 17/2*w + 9],
[1019, 1019, -1/2*w^3 + 5/2*w^2 - 3/2*w - 4],
[1021, 1021, -3/2*w^3 + 13/2*w^2 + 19/2*w - 37],
[1021, 1021, 3*w^3 - 11*w^2 - 20*w + 61],
[1049, 1049, w^3 - 4*w^2 - 9*w + 15],
[1049, 1049, 5/2*w^3 - 17/2*w^2 - 39/2*w + 54],
[1051, 1051, 3/2*w^3 - 11/2*w^2 - 15/2*w + 21],
[1051, 1051, -1/2*w^3 + 7/2*w^2 + 7/2*w - 23],
[1051, 1051, 9/2*w^3 - 31/2*w^2 - 63/2*w + 83],
[1051, 1051, -1/2*w^3 + 1/2*w^2 + 13/2*w + 8],
[1061, 1061, 2*w^3 - 5*w^2 - 14*w + 19],
[1061, 1061, 9/2*w^3 - 29/2*w^2 - 63/2*w + 72],
[1091, 1091, 3/2*w^3 - 5/2*w^2 - 21/2*w + 10],
[1091, 1091, 3/2*w^3 - 3/2*w^2 - 19/2*w + 7],
[1091, 1091, -13/2*w^3 + 43/2*w^2 + 91/2*w - 116],
[1091, 1091, 19/2*w^3 - 63/2*w^2 - 135/2*w + 174],
[1109, 1109, -5/2*w^3 + 17/2*w^2 + 33/2*w - 53],
[1109, 1109, 4*w^3 - 13*w^2 - 26*w + 67],
[1109, 1109, -3/2*w^3 + 11/2*w^2 + 23/2*w - 25],
[1109, 1109, 4*w^3 - 13*w^2 - 28*w + 65],
[1171, 1171, 1/2*w^3 - 1/2*w^2 + 3/2*w + 12],
[1171, 1171, 11/2*w^3 - 35/2*w^2 - 87/2*w + 105],
[1181, 1181, 2*w^3 - 8*w^2 - 13*w + 45],
[1181, 1181, -5/2*w^3 + 19/2*w^2 + 33/2*w - 53],
[1201, 1201, -5/2*w^3 + 15/2*w^2 + 39/2*w - 45],
[1201, 1201, -1/2*w^3 + 3/2*w^2 + 13/2*w - 3],
[1201, 1201, -w^3 + 3*w^2 + 10*w - 21],
[1201, 1201, 3/2*w^3 - 9/2*w^2 - 17/2*w + 29],
[1229, 1229, -5*w^3 + 17*w^2 + 35*w - 91],
[1229, 1229, 3/2*w^3 - 9/2*w^2 - 25/2*w + 17],
[1249, 1249, 5/2*w^3 - 17/2*w^2 - 35/2*w + 40],
[1249, 1249, 3/2*w^3 - 3/2*w^2 - 21/2*w + 4],
[1249, 1249, w^2 - 13],
[1249, 1249, 9*w^3 - 30*w^2 - 63*w + 163],
[1289, 1289, 3/2*w^3 - 1/2*w^2 - 23/2*w - 3],
[1289, 1289, 11*w^3 - 37*w^2 - 76*w + 201],
[1291, 1291, -1/2*w^3 - 7/2*w^2 + 13/2*w + 29],
[1291, 1291, 23/2*w^3 - 79/2*w^2 - 155/2*w + 212],
[1301, 1301, 11*w^3 - 38*w^2 - 73*w + 201],
[1301, 1301, 1/2*w^3 - 5/2*w^2 - 17/2*w + 3],
[1319, 1319, 12*w^3 - 40*w^2 - 85*w + 219],
[1319, 1319, 3/2*w^3 - 1/2*w^2 - 19/2*w - 1],
[1321, 1321, -1/2*w^3 - 1/2*w^2 - 1/2*w - 7],
[1321, 1321, -3*w^3 + 11*w^2 + 15*w - 45],
[1361, 1361, -3*w^3 + 11*w^2 + 21*w - 57],
[1361, 1361, 5/2*w^3 - 15/2*w^2 - 39/2*w + 35],
[1381, 1381, -3*w^3 + 9*w^2 + 23*w - 43],
[1381, 1381, 2*w^3 - 6*w^2 - 12*w + 27],
[1381, 1381, -1/2*w^3 + 3/2*w^2 + 9/2*w - 1],
[1381, 1381, w - 7],
[1399, 1399, 9/2*w^3 - 33/2*w^2 - 57/2*w + 85],
[1399, 1399, -5/2*w^3 + 15/2*w^2 + 41/2*w - 37],
[1409, 1409, 3/2*w^3 - 1/2*w^2 - 21/2*w - 1],
[1409, 1409, -23/2*w^3 + 77/2*w^2 + 161/2*w - 211],
[1409, 1409, 11/2*w^3 - 39/2*w^2 - 71/2*w + 98],
[1409, 1409, 1/2*w^3 - 9/2*w^2 - 1/2*w + 37],
[1429, 1429, 3/2*w^3 - 5/2*w^2 - 33/2*w - 5],
[1429, 1429, 7/2*w^3 - 25/2*w^2 - 37/2*w + 53],
[1439, 1439, -7*w^3 + 23*w^2 + 52*w - 135],
[1439, 1439, -1/2*w^3 - 1/2*w^2 + 1/2*w - 5],
[1451, 1451, 1/2*w^3 + 3/2*w^2 - 13/2*w - 15],
[1451, 1451, -13/2*w^3 + 45/2*w^2 + 85/2*w - 120],
[1459, 1459, -7/2*w^3 + 27/2*w^2 + 43/2*w - 70],
[1459, 1459, 5/2*w^3 - 21/2*w^2 - 29/2*w + 65],
[1471, 1471, 5/2*w^3 - 17/2*w^2 - 37/2*w + 43],
[1471, 1471, -3*w^3 + 10*w^2 + 21*w - 49],
[1481, 1481, -6*w^3 + 21*w^2 + 40*w - 113],
[1481, 1481, -16*w^3 + 54*w^2 + 112*w - 299],
[1481, 1481, 9*w^3 - 30*w^2 - 63*w + 161],
[1481, 1481, 1/2*w^3 - 9/2*w^2 - 3/2*w + 30],
[1489, 1489, -8*w^3 + 26*w^2 + 60*w - 151],
[1489, 1489, -w^3 + 5*w^2 + 6*w - 35],
[1489, 1489, 7/2*w^3 - 25/2*w^2 - 47/2*w + 63],
[1489, 1489, -w^3 + w^2 + 3*w - 13],
[1499, 1499, 3*w^3 - 10*w^2 - 21*w + 45],
[1499, 1499, 9/2*w^3 - 29/2*w^2 - 65/2*w + 75],
[1511, 1511, -11/2*w^3 + 35/2*w^2 + 79/2*w - 91],
[1511, 1511, -5/2*w^3 + 21/2*w^2 + 33/2*w - 54],
[1511, 1511, 7/2*w^3 - 19/2*w^2 - 57/2*w + 42],
[1511, 1511, 5/2*w^3 - 15/2*w^2 - 29/2*w + 41],
[1531, 1531, -4*w^2 + 2*w + 29],
[1531, 1531, 9*w^3 - 31*w^2 - 61*w + 167],
[1549, 1549, -1/2*w^3 + 1/2*w^2 + 13/2*w - 5],
[1549, 1549, -3/2*w^3 + 11/2*w^2 + 15/2*w - 34],
[1559, 1559, 9/2*w^3 - 29/2*w^2 - 63/2*w + 76],
[1559, 1559, 2*w^3 - 5*w^2 - 14*w + 23],
[1571, 1571, 3*w^3 - 9*w^2 - 20*w + 43],
[1571, 1571, 2*w^3 - 7*w^2 - 14*w + 45],
[1601, 1601, -27/2*w^3 + 93/2*w^2 + 179/2*w - 247],
[1601, 1601, 1/2*w^3 - 1/2*w^2 + 5/2*w + 16],
[1609, 1609, -1/2*w^3 + 7/2*w^2 + 1/2*w - 31],
[1609, 1609, 3/2*w^3 - 7/2*w^2 - 23/2*w + 25],
[1621, 1621, 7/2*w^3 - 25/2*w^2 - 45/2*w + 71],
[1621, 1621, -1/2*w^3 + 7/2*w^2 + 3/2*w - 19],
[1681, 41, -3*w^3 + 9*w^2 + 21*w - 43],
[1681, 41, 3*w^3 - 9*w^2 - 21*w + 47],
[1721, 1721, -1/2*w^3 - 3/2*w^2 + 21/2*w + 25],
[1721, 1721, -9/2*w^3 + 33/2*w^2 + 49/2*w - 78],
[1759, 1759, 4*w^3 - 14*w^2 - 25*w + 75],
[1759, 1759, 2*w^3 - 7*w^2 - 12*w + 43],
[1831, 1831, -3/2*w^3 + 7/2*w^2 + 29/2*w - 6],
[1831, 1831, 15/2*w^3 - 51/2*w^2 - 103/2*w + 136],
[1849, 43, -1/2*w^3 + 7/2*w^2 - 3/2*w - 31],
[1849, 43, 15/2*w^3 - 51/2*w^2 - 101/2*w + 138],
[1871, 1871, 3*w^3 - 9*w^2 - 18*w + 43],
[1871, 1871, -7/2*w^3 + 21/2*w^2 + 45/2*w - 51],
[1879, 1879, 21/2*w^3 - 71/2*w^2 - 145/2*w + 195],
[1879, 1879, -w^3 - w^2 + 8*w + 9],
[1889, 1889, 2*w^3 - 4*w^2 - 16*w + 15],
[1889, 1889, -6*w^3 + 20*w^2 + 40*w - 105],
[1901, 1901, 2*w^2 - 7*w - 29],
[1901, 1901, -19/2*w^3 + 65/2*w^2 + 133/2*w - 182],
[1901, 1901, 1/2*w^3 - 11/2*w^2 - 7/2*w + 30],
[1901, 1901, -1/2*w^3 + 5/2*w^2 + 1/2*w - 24],
[1931, 1931, -1/2*w^3 + 9/2*w^2 + 3/2*w - 34],
[1931, 1931, -5/2*w^3 + 19/2*w^2 + 35/2*w - 59],
[1931, 1931, -5/2*w^3 + 19/2*w^2 + 35/2*w - 47],
[1931, 1931, 6*w^3 - 21*w^2 - 40*w + 109],
[1949, 1949, -3/2*w^3 + 15/2*w^2 + 19/2*w - 46],
[1949, 1949, -11/2*w^3 + 39/2*w^2 + 75/2*w - 105],
[1979, 1979, -7/2*w^3 + 21/2*w^2 + 55/2*w - 51],
[1979, 1979, 3/2*w^3 - 5/2*w^2 - 29/2*w + 7],
[1979, 1979, 9/2*w^3 - 31/2*w^2 - 55/2*w + 81],
[1979, 1979, -2*w^3 + 6*w^2 + 11*w - 27]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^10 - 2*x^9 - 46*x^8 + 98*x^7 + 704*x^6 - 1518*x^5 - 4006*x^4 + 7949*x^3 + 7326*x^2 - 12004*x + 3400;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, 1, e, e, 16447/990128*e^9 + 9619/247532*e^8 - 371797/495064*e^7 - 707687/495064*e^6 + 2920447/247532*e^5 + 8065355/495064*e^4 - 35631315/495064*e^3 - 51634817/990128*e^2 + 29786763/247532*e - 9019997/247532, 16447/990128*e^9 + 9619/247532*e^8 - 371797/495064*e^7 - 707687/495064*e^6 + 2920447/247532*e^5 + 8065355/495064*e^4 - 35631315/495064*e^3 - 51634817/990128*e^2 + 29786763/247532*e - 9019997/247532, 4953/990128*e^9 + 5365/247532*e^8 - 125331/495064*e^7 - 360281/495064*e^6 + 1267065/247532*e^5 + 3207757/495064*e^4 - 22655853/495064*e^3 - 515271/990128*e^2 + 31730861/247532*e - 15364923/247532, -37877/990128*e^9 - 8893/247532*e^8 + 796519/495064*e^7 + 486053/495064*e^6 - 5488909/247532*e^5 - 4126657/495064*e^4 + 52090713/495064*e^3 + 30885579/990128*e^2 - 28069165/247532*e + 9932815/247532, -37877/990128*e^9 - 8893/247532*e^8 + 796519/495064*e^7 + 486053/495064*e^6 - 5488909/247532*e^5 - 4126657/495064*e^4 + 52090713/495064*e^3 + 30885579/990128*e^2 - 28069165/247532*e + 9932815/247532, 4719/495064*e^9 + 4137/123766*e^8 - 96021/247532*e^7 - 319871/247532*e^6 + 652693/123766*e^5 + 4034643/247532*e^4 - 6080639/247532*e^3 - 33371753/495064*e^2 + 989855/123766*e + 2499159/123766, -24647/990128*e^9 - 6107/247532*e^8 + 548405/495064*e^7 + 371999/495064*e^6 - 4122479/247532*e^5 - 3427539/495064*e^4 + 46111875/495064*e^3 + 18150649/990128*e^2 - 35767971/247532*e + 13797133/247532, -24647/990128*e^9 - 6107/247532*e^8 + 548405/495064*e^7 + 371999/495064*e^6 - 4122479/247532*e^5 - 3427539/495064*e^4 + 46111875/495064*e^3 + 18150649/990128*e^2 - 35767971/247532*e + 13797133/247532, 4719/495064*e^9 + 4137/123766*e^8 - 96021/247532*e^7 - 319871/247532*e^6 + 652693/123766*e^5 + 4034643/247532*e^4 - 6080639/247532*e^3 - 33371753/495064*e^2 + 989855/123766*e + 2499159/123766, -55515/990128*e^9 - 13655/247532*e^8 + 1165169/495064*e^7 + 790107/495064*e^6 - 7996327/247532*e^5 - 7558127/495064*e^4 + 75227615/495064*e^3 + 65135045/990128*e^2 - 39247999/247532*e + 8721505/247532, -55515/990128*e^9 - 13655/247532*e^8 + 1165169/495064*e^7 + 790107/495064*e^6 - 7996327/247532*e^5 - 7558127/495064*e^4 + 75227615/495064*e^3 + 65135045/990128*e^2 - 39247999/247532*e + 8721505/247532, -44473/495064*e^9 + 196/61883*e^8 + 949931/247532*e^7 - 240879/247532*e^6 - 6537437/123766*e^5 + 4619559/247532*e^4 + 60283301/247532*e^3 - 27735753/495064*e^2 - 16327387/61883*e + 15118569/123766, -44473/495064*e^9 + 196/61883*e^8 + 949931/247532*e^7 - 240879/247532*e^6 - 6537437/123766*e^5 + 4619559/247532*e^4 + 60283301/247532*e^3 - 27735753/495064*e^2 - 16327387/61883*e + 15118569/123766, -10579/123766*e^9 - 15263/123766*e^8 + 227061/61883*e^7 + 260796/61883*e^6 - 3241663/61883*e^5 - 2900696/61883*e^4 + 16469850/61883*e^3 + 22007789/123766*e^2 - 38069917/123766*e + 5019937/61883, -10579/123766*e^9 - 15263/123766*e^8 + 227061/61883*e^7 + 260796/61883*e^6 - 3241663/61883*e^5 - 2900696/61883*e^4 + 16469850/61883*e^3 + 22007789/123766*e^2 - 38069917/123766*e + 5019937/61883, 34657/495064*e^9 + 3631/123766*e^8 - 729583/247532*e^7 - 120393/247532*e^6 + 4963159/123766*e^5 + 261857/247532*e^4 - 45185913/247532*e^3 - 8978879/495064*e^2 + 22254395/123766*e - 7168215/123766, 34657/495064*e^9 + 3631/123766*e^8 - 729583/247532*e^7 - 120393/247532*e^6 + 4963159/123766*e^5 + 261857/247532*e^4 - 45185913/247532*e^3 - 8978879/495064*e^2 + 22254395/123766*e - 7168215/123766, 14161/247532*e^9 + 5/123766*e^8 - 294911/123766*e^7 + 86947/123766*e^6 + 1969110/61883*e^5 - 1764593/123766*e^4 - 17244223/123766*e^3 + 10581205/247532*e^2 + 16845369/123766*e - 3247078/61883, 14161/247532*e^9 + 5/123766*e^8 - 294911/123766*e^7 + 86947/123766*e^6 + 1969110/61883*e^5 - 1764593/123766*e^4 - 17244223/123766*e^3 + 10581205/247532*e^2 + 16845369/123766*e - 3247078/61883, 54943/495064*e^9 + 24405/123766*e^8 - 1176033/247532*e^7 - 1681455/247532*e^6 + 8461033/123766*e^5 + 18545563/247532*e^4 - 88562387/247532*e^3 - 132139169/495064*e^2 + 55607647/123766*e - 15302745/123766, 54943/495064*e^9 + 24405/123766*e^8 - 1176033/247532*e^7 - 1681455/247532*e^6 + 8461033/123766*e^5 + 18545563/247532*e^4 - 88562387/247532*e^3 - 132139169/495064*e^2 + 55607647/123766*e - 15302745/123766, 56629/990128*e^9 + 7865/247532*e^8 - 1215847/495064*e^7 - 297013/495064*e^6 + 8685965/247532*e^5 + 300001/495064*e^4 - 89890777/495064*e^3 + 15776437/990128*e^2 + 64565753/247532*e - 29427471/247532, -29389/247532*e^9 - 17027/123766*e^8 + 621919/123766*e^7 + 553035/123766*e^6 - 4297124/61883*e^5 - 6031117/123766*e^4 + 40422181/123766*e^3 + 51855687/247532*e^2 - 37848027/123766*e + 3043080/61883, -29389/247532*e^9 - 17027/123766*e^8 + 621919/123766*e^7 + 553035/123766*e^6 - 4297124/61883*e^5 - 6031117/123766*e^4 + 40422181/123766*e^3 + 51855687/247532*e^2 - 37848027/123766*e + 3043080/61883, 41993/495064*e^9 + 9693/61883*e^8 - 912215/247532*e^7 - 1330141/247532*e^6 + 6752879/123766*e^5 + 14404961/247532*e^4 - 74875501/247532*e^3 - 95630415/495064*e^2 + 27147778/61883*e - 16560845/123766, 1913/13028*e^9 + 296/3257*e^8 - 40855/6514*e^7 - 15183/6514*e^6 + 283409/3257*e^5 + 127945/6514*e^4 - 2662685/6514*e^3 - 1322491/13028*e^2 + 1348395/3257*e - 425725/3257, 41993/495064*e^9 + 9693/61883*e^8 - 912215/247532*e^7 - 1330141/247532*e^6 + 6752879/123766*e^5 + 14404961/247532*e^4 - 74875501/247532*e^3 - 95630415/495064*e^2 + 27147778/61883*e - 16560845/123766, 1913/13028*e^9 + 296/3257*e^8 - 40855/6514*e^7 - 15183/6514*e^6 + 283409/3257*e^5 + 127945/6514*e^4 - 2662685/6514*e^3 - 1322491/13028*e^2 + 1348395/3257*e - 425725/3257, -10715/495064*e^9 + 363/123766*e^8 + 212361/247532*e^7 - 110817/247532*e^6 - 1284231/123766*e^5 + 1969349/247532*e^4 + 8229699/247532*e^3 - 9879555/495064*e^2 + 1106331/123766*e - 781251/123766, -10715/495064*e^9 + 363/123766*e^8 + 212361/247532*e^7 - 110817/247532*e^6 - 1284231/123766*e^5 + 1969349/247532*e^4 + 8229699/247532*e^3 - 9879555/495064*e^2 + 1106331/123766*e - 781251/123766, -13507/495064*e^9 - 1062/61883*e^8 + 289157/247532*e^7 + 104767/247532*e^6 - 2052973/123766*e^5 - 759071/247532*e^4 + 20770935/247532*e^3 + 4964165/495064*e^2 - 6552303/61883*e + 5629435/123766, -13507/495064*e^9 - 1062/61883*e^8 + 289157/247532*e^7 + 104767/247532*e^6 - 2052973/123766*e^5 - 759071/247532*e^4 + 20770935/247532*e^3 + 4964165/495064*e^2 - 6552303/61883*e + 5629435/123766, -26383/247532*e^9 - 4639/61883*e^8 + 550053/123766*e^7 + 262413/123766*e^6 - 3649878/61883*e^5 - 2734507/123766*e^4 + 30575397/123766*e^3 + 32737881/247532*e^2 - 8230447/61883*e + 64395/61883, 145/61883*e^9 - 9251/123766*e^8 - 1416/61883*e^7 + 192691/61883*e^6 - 133229/61883*e^5 - 2406298/61883*e^4 + 2343142/61883*e^3 + 7864709/61883*e^2 - 11835063/123766*e + 872295/61883, -26383/247532*e^9 - 4639/61883*e^8 + 550053/123766*e^7 + 262413/123766*e^6 - 3649878/61883*e^5 - 2734507/123766*e^4 + 30575397/123766*e^3 + 32737881/247532*e^2 - 8230447/61883*e + 64395/61883, 145/61883*e^9 - 9251/123766*e^8 - 1416/61883*e^7 + 192691/61883*e^6 - 133229/61883*e^5 - 2406298/61883*e^4 + 2343142/61883*e^3 + 7864709/61883*e^2 - 11835063/123766*e + 872295/61883, 75751/495064*e^9 + 19357/123766*e^8 - 1649785/247532*e^7 - 1200079/247532*e^6 + 12006085/123766*e^5 + 11754751/247532*e^4 - 126929103/247532*e^3 - 77774217/495064*e^2 + 88056661/123766*e - 32460665/123766, 75751/495064*e^9 + 19357/123766*e^8 - 1649785/247532*e^7 - 1200079/247532*e^6 + 12006085/123766*e^5 + 11754751/247532*e^4 - 126929103/247532*e^3 - 77774217/495064*e^2 + 88056661/123766*e - 32460665/123766, -26851/495064*e^9 - 7095/123766*e^8 + 608673/247532*e^7 + 466999/247532*e^6 - 4692973/123766*e^5 - 4546183/247532*e^4 + 54938439/247532*e^3 + 19996765/495064*e^2 - 46320685/123766*e + 20321809/123766, -26851/495064*e^9 - 7095/123766*e^8 + 608673/247532*e^7 + 466999/247532*e^6 - 4692973/123766*e^5 - 4546183/247532*e^4 + 54938439/247532*e^3 + 19996765/495064*e^2 - 46320685/123766*e + 20321809/123766, 20301/495064*e^9 + 12919/123766*e^8 - 410247/247532*e^7 - 916261/247532*e^6 + 2688349/123766*e^5 + 10677801/247532*e^4 - 22462589/247532*e^3 - 87700755/495064*e^2 + 1735641/123766*e + 7066969/123766, -80209/990128*e^9 - 36405/247532*e^8 + 1732155/495064*e^7 + 2541169/495064*e^6 - 12635189/247532*e^5 - 28326653/495064*e^4 + 135514621/495064*e^3 + 198918463/990128*e^2 - 89869933/247532*e + 26439939/247532, -80209/990128*e^9 - 36405/247532*e^8 + 1732155/495064*e^7 + 2541169/495064*e^6 - 12635189/247532*e^5 - 28326653/495064*e^4 + 135514621/495064*e^3 + 198918463/990128*e^2 - 89869933/247532*e + 26439939/247532, 20301/495064*e^9 + 12919/123766*e^8 - 410247/247532*e^7 - 916261/247532*e^6 + 2688349/123766*e^5 + 10677801/247532*e^4 - 22462589/247532*e^3 - 87700755/495064*e^2 + 1735641/123766*e + 7066969/123766, -9293/123766*e^9 - 3390/61883*e^8 + 191334/61883*e^7 + 86316/61883*e^6 - 2522679/61883*e^5 - 732581/61883*e^4 + 10733763/61883*e^3 + 7898341/123766*e^2 - 7583428/61883*e + 1671276/61883, -9293/123766*e^9 - 3390/61883*e^8 + 191334/61883*e^7 + 86316/61883*e^6 - 2522679/61883*e^5 - 732581/61883*e^4 + 10733763/61883*e^3 + 7898341/123766*e^2 - 7583428/61883*e + 1671276/61883, -1319/123766*e^9 - 621/61883*e^8 + 30340/61883*e^7 + 18331/61883*e^6 - 498756/61883*e^5 - 101395/61883*e^4 + 3414938/61883*e^3 - 1647231/123766*e^2 - 7628845/61883*e + 3926912/61883, -1319/123766*e^9 - 621/61883*e^8 + 30340/61883*e^7 + 18331/61883*e^6 - 498756/61883*e^5 - 101395/61883*e^4 + 3414938/61883*e^3 - 1647231/123766*e^2 - 7628845/61883*e + 3926912/61883, 23281/123766*e^9 + 18497/123766*e^8 - 499484/61883*e^7 - 274148/61883*e^6 + 7022851/61883*e^5 + 2693567/61883*e^4 - 34215042/61883*e^3 - 22609173/123766*e^2 + 75762419/123766*e - 12792485/61883, 23281/123766*e^9 + 18497/123766*e^8 - 499484/61883*e^7 - 274148/61883*e^6 + 7022851/61883*e^5 + 2693567/61883*e^4 - 34215042/61883*e^3 - 22609173/123766*e^2 + 75762419/123766*e - 12792485/61883, 85559/990128*e^9 + 40151/247532*e^8 - 1856437/495064*e^7 - 2931927/495064*e^6 + 13434535/247532*e^5 + 34610379/495064*e^4 - 138823707/495064*e^3 - 266908089/990128*e^2 + 78268351/247532*e - 14961397/247532, -13227/495064*e^9 - 4357/123766*e^8 + 304861/247532*e^7 + 342027/247532*e^6 - 2394697/123766*e^5 - 4200555/247532*e^4 + 28726515/247532*e^3 + 29728117/495064*e^2 - 23971993/123766*e + 6502929/123766, 85559/990128*e^9 + 40151/247532*e^8 - 1856437/495064*e^7 - 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47113010/61883*e + 5614045/61883, -863/6514*e^9 - 2109/6514*e^8 + 18320/3257*e^7 + 38546/3257*e^6 - 263588/3257*e^5 - 451719/3257*e^4 + 1395017/3257*e^3 + 3362609/6514*e^2 - 3320083/6514*e + 146059/3257, 2021/123766*e^9 + 7989/61883*e^8 - 56387/61883*e^7 - 325210/61883*e^6 + 1214604/61883*e^5 + 3870920/61883*e^4 - 10936374/61883*e^3 - 20207641/123766*e^2 + 25485845/61883*e - 12098604/61883, -863/6514*e^9 - 2109/6514*e^8 + 18320/3257*e^7 + 38546/3257*e^6 - 263588/3257*e^5 - 451719/3257*e^4 + 1395017/3257*e^3 + 3362609/6514*e^2 - 3320083/6514*e + 146059/3257, 2021/123766*e^9 + 7989/61883*e^8 - 56387/61883*e^7 - 325210/61883*e^6 + 1214604/61883*e^5 + 3870920/61883*e^4 - 10936374/61883*e^3 - 20207641/123766*e^2 + 25485845/61883*e - 12098604/61883, 71281/495064*e^9 + 8593/123766*e^8 - 1546871/247532*e^7 - 275817/247532*e^6 + 11158239/123766*e^5 - 712683/247532*e^4 - 116757325/247532*e^3 + 24408617/495064*e^2 + 87551627/123766*e - 40592791/123766, 71281/495064*e^9 + 8593/123766*e^8 - 1546871/247532*e^7 - 275817/247532*e^6 + 11158239/123766*e^5 - 712683/247532*e^4 - 116757325/247532*e^3 + 24408617/495064*e^2 + 87551627/123766*e - 40592791/123766, -89287/990128*e^9 + 14333/247532*e^8 + 1802941/495064*e^7 - 1641489/495064*e^6 - 11091295/247532*e^5 + 23148637/495064*e^4 + 73849707/495064*e^3 - 121959319/990128*e^2 + 1332837/247532*e + 11019085/247532, -89287/990128*e^9 + 14333/247532*e^8 + 1802941/495064*e^7 - 1641489/495064*e^6 - 11091295/247532*e^5 + 23148637/495064*e^4 + 73849707/495064*e^3 - 121959319/990128*e^2 + 1332837/247532*e + 11019085/247532, 1271/123766*e^9 - 10990/61883*e^8 - 10047/61883*e^7 + 480262/61883*e^6 - 322935/61883*e^5 - 6315560/61883*e^4 + 6333667/61883*e^3 + 45112007/123766*e^2 - 15940901/61883*e - 768020/61883, 1271/123766*e^9 - 10990/61883*e^8 - 10047/61883*e^7 + 480262/61883*e^6 - 322935/61883*e^5 - 6315560/61883*e^4 + 6333667/61883*e^3 + 45112007/123766*e^2 - 15940901/61883*e - 768020/61883, -67201/495064*e^9 - 41131/123766*e^8 + 1492807/247532*e^7 + 2884353/247532*e^6 - 11664389/123766*e^5 - 31152649/247532*e^4 + 144029673/247532*e^3 + 184665063/495064*e^2 - 125905003/123766*e + 41616119/123766, -67201/495064*e^9 - 41131/123766*e^8 + 1492807/247532*e^7 + 2884353/247532*e^6 - 11664389/123766*e^5 - 31152649/247532*e^4 + 144029673/247532*e^3 + 184665063/495064*e^2 - 125905003/123766*e + 41616119/123766, 11661/495064*e^9 + 9970/61883*e^8 - 222955/247532*e^7 - 1518701/247532*e^6 + 1510567/123766*e^5 + 18283705/247532*e^4 - 15537141/247532*e^3 - 134073483/495064*e^2 + 1120461/61883*e + 7324279/123766, 11661/495064*e^9 + 9970/61883*e^8 - 222955/247532*e^7 - 1518701/247532*e^6 + 1510567/123766*e^5 + 18283705/247532*e^4 - 15537141/247532*e^3 - 134073483/495064*e^2 + 1120461/61883*e + 7324279/123766, -62663/495064*e^9 - 24721/123766*e^8 + 1273477/247532*e^7 + 1681775/247532*e^6 - 8215579/123766*e^5 - 19665987/247532*e^4 + 63811195/247532*e^3 + 183824441/495064*e^2 + 1926601/123766*e - 16259735/123766, -62663/495064*e^9 - 24721/123766*e^8 + 1273477/247532*e^7 + 1681775/247532*e^6 - 8215579/123766*e^5 - 19665987/247532*e^4 + 63811195/247532*e^3 + 183824441/495064*e^2 + 1926601/123766*e - 16259735/123766, 142811/990128*e^9 + 10559/247532*e^8 - 3086849/495064*e^7 + 52757/495064*e^6 + 21997559/247532*e^5 - 10105689/495064*e^4 - 222987679/495064*e^3 + 119608571/990128*e^2 + 157023175/247532*e - 79640121/247532, 142811/990128*e^9 + 10559/247532*e^8 - 3086849/495064*e^7 + 52757/495064*e^6 + 21997559/247532*e^5 - 10105689/495064*e^4 - 222987679/495064*e^3 + 119608571/990128*e^2 + 157023175/247532*e - 79640121/247532, 47565/990128*e^9 + 92527/247532*e^8 - 1045263/495064*e^7 - 6881245/495064*e^6 + 9309281/247532*e^5 + 78175777/495064*e^4 - 149363305/495064*e^3 - 469041251/990128*e^2 + 165773111/247532*e - 31369851/247532, -282211/990128*e^9 - 74621/247532*e^8 + 6089185/495064*e^7 + 4636891/495064*e^6 - 43669763/247532*e^5 - 46203911/495064*e^4 + 448188279/495064*e^3 + 328022029/990128*e^2 - 288283785/247532*e + 114562949/247532, -118919/990128*e^9 - 21637/247532*e^8 + 2531461/495064*e^7 + 1114655/495064*e^6 - 17806747/247532*e^5 - 8681083/495064*e^4 + 178201515/495064*e^3 + 60670569/990128*e^2 - 117340677/247532*e + 44699553/247532, -118919/990128*e^9 - 21637/247532*e^8 + 2531461/495064*e^7 + 1114655/495064*e^6 - 17806747/247532*e^5 - 8681083/495064*e^4 + 178201515/495064*e^3 + 60670569/990128*e^2 - 117340677/247532*e + 44699553/247532, -123731/990128*e^9 + 22927/247532*e^8 + 2601049/495064*e^7 - 2167189/495064*e^6 - 16782067/247532*e^5 + 26767393/495064*e^4 + 122687367/495064*e^3 - 97588147/990128*e^2 - 5991913/247532*e - 2195955/247532, -123731/990128*e^9 + 22927/247532*e^8 + 2601049/495064*e^7 - 2167189/495064*e^6 - 16782067/247532*e^5 + 26767393/495064*e^4 + 122687367/495064*e^3 - 97588147/990128*e^2 - 5991913/247532*e - 2195955/247532, 19200/61883*e^9 + 6350/61883*e^8 - 825533/61883*e^7 - 38465/61883*e^6 + 11665578/61883*e^5 - 1487340/61883*e^4 - 58001413/61883*e^3 + 7173112/61883*e^2 + 77338567/61883*e - 34342594/61883, 19200/61883*e^9 + 6350/61883*e^8 - 825533/61883*e^7 - 38465/61883*e^6 + 11665578/61883*e^5 - 1487340/61883*e^4 - 58001413/61883*e^3 + 7173112/61883*e^2 + 77338567/61883*e - 34342594/61883, -140257/247532*e^9 - 27833/61883*e^8 + 3046211/123766*e^7 + 1572127/123766*e^6 - 21916103/61883*e^5 - 13684561/123766*e^4 + 225157601/123766*e^3 + 91560039/247532*e^2 - 148900284/61883*e + 56106535/61883, -140257/247532*e^9 - 27833/61883*e^8 + 3046211/123766*e^7 + 1572127/123766*e^6 - 21916103/61883*e^5 - 13684561/123766*e^4 + 225157601/123766*e^3 + 91560039/247532*e^2 - 148900284/61883*e + 56106535/61883, 156977/495064*e^9 + 18257/123766*e^8 - 3204671/247532*e^7 - 718001/247532*e^6 + 20499539/123766*e^5 + 5355721/247532*e^4 - 156401921/247532*e^3 - 128047791/495064*e^2 + 18815625/123766*e + 12178477/123766, 156977/495064*e^9 + 18257/123766*e^8 - 3204671/247532*e^7 - 718001/247532*e^6 + 20499539/123766*e^5 + 5355721/247532*e^4 - 156401921/247532*e^3 - 128047791/495064*e^2 + 18815625/123766*e + 12178477/123766, -229557/990128*e^9 - 89203/247532*e^8 + 5180823/495064*e^7 + 6244949/495064*e^6 - 40020557/247532*e^5 - 67279913/495064*e^4 + 470738985/495064*e^3 + 397793451/990128*e^2 - 382801095/247532*e + 142501435/247532, -229557/990128*e^9 - 89203/247532*e^8 + 5180823/495064*e^7 + 6244949/495064*e^6 - 40020557/247532*e^5 - 67279913/495064*e^4 + 470738985/495064*e^3 + 397793451/990128*e^2 - 382801095/247532*e + 142501435/247532, -83029/495064*e^9 - 44857/123766*e^8 + 1856887/247532*e^7 + 3351857/247532*e^6 - 14161861/123766*e^5 - 39547817/247532*e^4 + 161740921/247532*e^3 + 280248203/495064*e^2 - 113879351/123766*e + 31018175/123766, -83029/495064*e^9 - 44857/123766*e^8 + 1856887/247532*e^7 + 3351857/247532*e^6 - 14161861/123766*e^5 - 39547817/247532*e^4 + 161740921/247532*e^3 + 280248203/495064*e^2 - 113879351/123766*e + 31018175/123766, 5992/61883*e^9 + 1385/123766*e^8 - 218984/61883*e^7 + 67799/61883*e^6 + 2234496/61883*e^5 - 1350648/61883*e^4 - 2518645/61883*e^3 - 54569/61883*e^2 - 43252853/123766*e + 9682547/61883, 221799/495064*e^9 + 67531/123766*e^8 - 4647025/247532*e^7 - 4429179/247532*e^6 + 31714187/123766*e^5 + 49205599/247532*e^4 - 291533803/247532*e^3 - 429998233/495064*e^2 + 122789635/123766*e - 13789681/123766, 221799/495064*e^9 + 67531/123766*e^8 - 4647025/247532*e^7 - 4429179/247532*e^6 + 31714187/123766*e^5 + 49205599/247532*e^4 - 291533803/247532*e^3 - 429998233/495064*e^2 + 122789635/123766*e - 13789681/123766, 5992/61883*e^9 + 1385/123766*e^8 - 218984/61883*e^7 + 67799/61883*e^6 + 2234496/61883*e^5 - 1350648/61883*e^4 - 2518645/61883*e^3 - 54569/61883*e^2 - 43252853/123766*e + 9682547/61883, 29433/247532*e^9 + 53455/123766*e^8 - 697247/123766*e^7 - 2036305/123766*e^6 + 5965540/61883*e^5 + 23531859/123766*e^4 - 82914459/123766*e^3 - 138193159/247532*e^2 + 160106327/123766*e - 30945888/61883, -15191/247532*e^9 - 7259/61883*e^8 + 325547/123766*e^7 + 481265/123766*e^6 - 2365214/61883*e^5 - 4980403/123766*e^4 + 25506673/123766*e^3 + 31335917/247532*e^2 - 17928314/61883*e + 7110631/61883, -15191/247532*e^9 - 7259/61883*e^8 + 325547/123766*e^7 + 481265/123766*e^6 - 2365214/61883*e^5 - 4980403/123766*e^4 + 25506673/123766*e^3 + 31335917/247532*e^2 - 17928314/61883*e + 7110631/61883, 29433/247532*e^9 + 53455/123766*e^8 - 697247/123766*e^7 - 2036305/123766*e^6 + 5965540/61883*e^5 + 23531859/123766*e^4 - 82914459/123766*e^3 - 138193159/247532*e^2 + 160106327/123766*e - 30945888/61883, 10008/61883*e^9 + 14913/61883*e^8 - 417391/61883*e^7 - 481465/61883*e^6 + 5754095/61883*e^5 + 5123721/61883*e^4 - 27673833/61883*e^3 - 20371251/61883*e^2 + 28668436/61883*e - 2986810/61883, 10008/61883*e^9 + 14913/61883*e^8 - 417391/61883*e^7 - 481465/61883*e^6 + 5754095/61883*e^5 + 5123721/61883*e^4 - 27673833/61883*e^3 - 20371251/61883*e^2 + 28668436/61883*e - 2986810/61883, -7500/61883*e^9 - 8282/61883*e^8 + 307970/61883*e^7 + 267392/61883*e^6 - 4017324/61883*e^5 - 3043031/61883*e^4 + 16245143/61883*e^3 + 15554048/61883*e^2 - 5614786/61883*e - 6068400/61883, -128279/495064*e^9 - 4437/61883*e^8 + 2713733/247532*e^7 + 8203/247532*e^6 - 18618555/123766*e^5 + 5334393/247532*e^4 + 172873675/247532*e^3 - 20384287/495064*e^2 - 47203557/61883*e + 32385115/123766, -128279/495064*e^9 - 4437/61883*e^8 + 2713733/247532*e^7 + 8203/247532*e^6 - 18618555/123766*e^5 + 5334393/247532*e^4 + 172873675/247532*e^3 - 20384287/495064*e^2 - 47203557/61883*e + 32385115/123766, -7500/61883*e^9 - 8282/61883*e^8 + 307970/61883*e^7 + 267392/61883*e^6 - 4017324/61883*e^5 - 3043031/61883*e^4 + 16245143/61883*e^3 + 15554048/61883*e^2 - 5614786/61883*e - 6068400/61883];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, w + 2}>] := -1;
ALEigenvalues[ideal<ZF | {4, 2, -1/2*w^3 + 3/2*w^2 + 9/2*w - 10}>] := -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;