/*
  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 | {29, 29, w + 1}>;

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

heckeEigenvaluesArray := [e, e + 2, 4*e + 5, -3, -3*e - 5, -e - 3, e + 2, 1, 6*e + 8, 3, -3, 2*e + 3, -3, 4*e + 5, 6*e + 5, -5*e - 13, -13*e - 19, -e + 6, 5*e + 6, -9*e - 14, 7*e + 16, 5*e + 7, -e + 9, -e - 11, 9*e + 17, 4*e + 10, 6*e + 15, 1, -5*e + 1, 11*e + 13, e + 13, e - 1, 4*e + 12, -6*e + 4, -9*e - 23, e + 15, -6*e - 24, -6*e - 12, -4*e - 4, -16*e - 22, 7*e + 4, 15*e + 30, 4*e - 11, 10*e + 17, -4*e + 1, -14*e - 30, 10*e + 4, -27, -8*e - 5, -14*e - 21, -24*e - 37, -14*e - 25, e + 14, e - 24, 4*e + 18, -15*e - 23, 6*e + 36, 3*e - 1, -10*e + 5, 4*e + 7, 10*e + 15, -18*e - 19, 7*e + 16, -4*e - 25, -5*e - 15, -23*e - 43, 3*e - 3, -17*e - 15, 14*e + 28, -e - 14, 16*e + 32, 19*e + 22, -9*e - 20, 3*e + 20, -6*e + 16, 18*e + 26, -27*e - 35, -7*e - 23, -13*e - 27, -2*e + 11, 10*e + 23, 5*e + 13, -23*e - 47, -9*e - 41, -12*e + 2, 4*e + 34, -10*e - 24, -12*e - 20, -7*e - 23, -9*e + 7, -5*e + 24, 19*e + 48, -e + 7, 10*e + 46, -3*e + 19, -18*e - 20, 20*e + 13, -16*e - 11, 2*e - 7, -4*e + 17, e - 13, -3*e - 25, -27*e - 22, 5*e - 3, -5*e - 1, e - 26, 8*e - 3, -10*e + 9, 5*e - 14, 12*e + 17, -8*e + 3, -21*e - 52, -23*e - 53, -3*e + 33, 10*e + 18, 6*e - 16, -21*e - 19, -7*e + 13, -27*e - 40, -17*e - 26, 18*e + 30, 12*e + 41, -4*e - 18, 26*e + 19, 8*e - 22, -8*e - 36, -2*e - 33, 25*e + 26, 2*e + 13, -11*e - 40, -e + 15, 3*e - 17, -17*e + 1, -9*e - 27, 14*e + 52, 6*e + 8, -17*e - 39, -15*e - 5, 8*e - 26, 22, 17*e + 33, -33*e - 45, -32, 12*e - 6, 20*e + 30, 29*e + 58, -17*e - 60, 27*e + 18, -19*e - 46, -33*e - 52, 15*e + 34, 12*e - 12, 26*e + 24, -1, 6*e + 44, 14*e + 30, 10*e + 13, -20*e - 57, 4*e + 2, -6*e - 40, 15*e + 6, e - 36, -12*e - 23, -2*e - 15, 14*e - 3, -24*e - 57, -10*e + 22, -26*e - 24, -17*e - 21, -25*e - 27, -12*e - 37, -16*e - 29, 7*e, -9*e - 24, -29*e - 34, 5*e - 48, -23*e - 59, -e + 45, 3*e - 18, -31*e - 38, -4*e + 26, 14*e + 12, -4*e - 1, 14*e + 55, -16*e - 51, -22*e - 35, -16, -8*e - 2, -33*e - 36, -52*e - 81, 3*e + 16, 20*e + 69, 24*e + 47, -8*e - 9, 12*e + 23, -26*e - 33, -18*e - 40, -4*e - 20, 6*e - 18, 4*e + 18, -10*e + 3, 4*e + 9, 10*e + 25, 44*e + 75, -22*e - 41, -6*e - 5, 11*e + 37, 13*e - 27, -7*e - 61, 27*e + 39, -28*e - 73, -14*e - 21, 40*e + 71, 2*e - 45, 18*e + 68, 8*e - 6, -21*e - 31, -25*e - 31, -37*e - 55, -3*e - 27, 32*e + 63, 10*e - 27, 29*e + 39, 11*e - 33, -66, 2*e - 4, 5*e + 13, -21*e - 49, -11*e + 12, -27*e - 74, -2*e + 64, 38*e + 50, 5*e - 8, 25*e + 26, 45*e + 73, -3*e + 1, -e + 15, 31*e + 67, 4*e - 43, -28*e - 57, -17*e - 55, 7*e + 35, 14*e + 33, -17*e - 36, 13*e - 6, -31, 14*e + 39, -30*e - 66, -24*e - 38, -6*e - 9, 22*e + 14, -30*e - 58, -19*e - 57, 43*e + 48, 5*e - 19, -e + 4, 8*e + 40, 38*e + 76, 9*e - 25, 37*e + 57, 22*e + 53, 24*e + 29, 19*e + 71, 25*e + 23, -6*e + 4, 20*e + 82, -6*e - 11, 18*e + 39, 3*e - 18, -e + 28, -e - 16, 10*e - 23, -e + 17, -3*e - 27, -2*e + 9, 26*e + 57, 39*e + 59, 29*e + 93, 6*e + 39, -28*e - 33, -36*e - 43, -18*e - 61, -4*e + 6, -4*e - 66, 36*e + 38, -38*e - 82, 7*e + 15, -20*e + 30, -42*e - 68, 5*e - 9, 6*e + 2, -18*e - 1, 40*e + 103, 36*e + 60, -3*e - 18, -e + 38, 24*e + 17, -34*e - 65, 11, -28*e - 61];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {29, 29, 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;