/*
  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![20, 10, -11, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {20,10,-w - 2}>;

primesArray := [
[4, 2, -w + 2],
[4, 2, 1/2*w^3 + 1/2*w^2 - 9/2*w - 4],
[5, 5, -1/2*w^3 - 3/2*w^2 + 7/2*w + 10],
[5, 5, -1/2*w^3 - 1/2*w^2 + 5/2*w + 3],
[19, 19, 1/2*w^3 + 1/2*w^2 - 9/2*w - 3],
[19, 19, w - 1],
[31, 31, 1/2*w^3 + 3/2*w^2 - 7/2*w - 6],
[31, 31, -w^3 - 2*w^2 + 7*w + 13],
[49, 7, -2*w^3 - 2*w^2 + 15*w + 9],
[49, 7, 1/2*w^3 + 3/2*w^2 - 9/2*w - 4],
[59, 59, -3/2*w^3 - 3/2*w^2 + 17/2*w + 8],
[59, 59, -1/2*w^3 - 5/2*w^2 + 11/2*w + 9],
[61, 61, -1/2*w^3 - 3/2*w^2 + 9/2*w + 9],
[61, 61, -3/2*w^3 - 5/2*w^2 + 23/2*w + 12],
[71, 71, -3/2*w^3 - 1/2*w^2 + 23/2*w - 1],
[71, 71, 3/2*w^3 + 3/2*w^2 - 19/2*w - 9],
[79, 79, -1/2*w^3 - 3/2*w^2 + 11/2*w + 4],
[79, 79, -5/2*w^3 - 7/2*w^2 + 37/2*w + 18],
[81, 3, -3],
[89, 89, w^3 - 7*w + 1],
[89, 89, -2*w^3 - 3*w^2 + 14*w + 17],
[89, 89, 1/2*w^3 - 1/2*w^2 - 7/2*w + 2],
[89, 89, -5/2*w^3 - 7/2*w^2 + 35/2*w + 18],
[101, 101, -2*w^2 + 11],
[101, 101, 3*w^3 + 5*w^2 - 21*w - 27],
[121, 11, -3/2*w^3 - 3/2*w^2 + 21/2*w + 7],
[121, 11, 3/2*w^3 + 3/2*w^2 - 21/2*w - 8],
[131, 131, -3/2*w^3 - 5/2*w^2 + 21/2*w + 16],
[131, 131, -3/2*w^3 - 5/2*w^2 + 19/2*w + 14],
[139, 139, -2*w^3 - w^2 + 16*w + 1],
[139, 139, 1/2*w^3 + 1/2*w^2 - 3/2*w - 8],
[149, 149, -3/2*w^3 - 7/2*w^2 + 21/2*w + 24],
[149, 149, -w^3 + w^2 + 5*w - 3],
[149, 149, -5*w^3 - 7*w^2 + 37*w + 39],
[149, 149, -3/2*w^3 - 7/2*w^2 + 21/2*w + 14],
[181, 181, -5/2*w^3 - 9/2*w^2 + 37/2*w + 27],
[181, 181, -5/2*w^3 - 7/2*w^2 + 33/2*w + 19],
[181, 181, 3/2*w^3 + 1/2*w^2 - 23/2*w - 2],
[181, 181, w^3 + 3*w^2 - 8*w - 13],
[191, 191, 1/2*w^3 + 5/2*w^2 - 7/2*w - 14],
[191, 191, -5/2*w^3 - 9/2*w^2 + 35/2*w + 24],
[199, 199, -1/2*w^3 - 5/2*w^2 + 7/2*w + 16],
[199, 199, 5/2*w^3 + 9/2*w^2 - 35/2*w - 22],
[211, 211, -2*w^3 - 4*w^2 + 14*w + 19],
[211, 211, -w^3 - 3*w^2 + 7*w + 19],
[229, 229, 3/2*w^3 + 7/2*w^2 - 21/2*w - 17],
[229, 229, -3/2*w^3 - 7/2*w^2 + 21/2*w + 21],
[229, 229, -1/2*w^3 - 3/2*w^2 + 13/2*w + 1],
[229, 229, 5/2*w^3 + 7/2*w^2 - 41/2*w - 24],
[241, 241, -1/2*w^3 - 5/2*w^2 + 7/2*w + 12],
[241, 241, 3/2*w^3 + 3/2*w^2 - 23/2*w - 3],
[241, 241, w^3 + w^2 - 6*w - 1],
[241, 241, -5/2*w^3 - 9/2*w^2 + 35/2*w + 26],
[251, 251, 2*w^3 + 2*w^2 - 13*w - 9],
[251, 251, -5/2*w^3 - 5/2*w^2 + 37/2*w + 11],
[269, 269, -2*w^3 - 3*w^2 + 12*w + 17],
[269, 269, 5/2*w^3 + 11/2*w^2 - 37/2*w - 21],
[271, 271, 3/2*w^3 + 5/2*w^2 - 23/2*w - 9],
[271, 271, 1/2*w^3 + 3/2*w^2 - 9/2*w - 12],
[281, 281, 5/2*w^3 + 9/2*w^2 - 39/2*w - 28],
[281, 281, 3/2*w^3 + 7/2*w^2 - 25/2*w - 14],
[311, 311, 1/2*w^3 + 7/2*w^2 - 5/2*w - 24],
[311, 311, 3*w^3 + 5*w^2 - 22*w - 27],
[311, 311, 3/2*w^3 + 5/2*w^2 - 17/2*w - 12],
[311, 311, 1/2*w^3 + 5/2*w^2 - 9/2*w - 13],
[349, 349, 3*w^3 + 4*w^2 - 21*w - 21],
[349, 349, -3/2*w^3 - 1/2*w^2 + 21/2*w + 2],
[359, 359, -w^3 - 3*w^2 + 7*w + 17],
[359, 359, -2*w^3 - 4*w^2 + 14*w + 21],
[361, 19, -2*w^3 - 2*w^2 + 14*w + 11],
[389, 389, -2*w^3 - 2*w^2 + 13*w + 11],
[389, 389, 5/2*w^3 + 5/2*w^2 - 37/2*w - 13],
[401, 401, 3/2*w^3 + 7/2*w^2 - 21/2*w - 19],
[401, 401, 5/2*w^3 + 9/2*w^2 - 37/2*w - 23],
[401, 401, -w^3 - 3*w^2 + 8*w + 17],
[431, 431, 5/2*w^3 + 5/2*w^2 - 29/2*w - 13],
[431, 431, -15/2*w^3 - 23/2*w^2 + 107/2*w + 61],
[439, 439, -3/2*w^3 + 1/2*w^2 + 21/2*w - 6],
[439, 439, -9/2*w^3 - 13/2*w^2 + 63/2*w + 32],
[449, 449, -w^2 - 2*w + 1],
[449, 449, -1/2*w^3 - 3/2*w^2 + 3/2*w + 14],
[461, 461, -2*w^3 - 2*w^2 + 12*w + 9],
[461, 461, -3*w^3 - 3*w^2 + 23*w + 13],
[479, 479, 7/2*w^3 + 9/2*w^2 - 47/2*w - 26],
[479, 479, 3/2*w^3 + 5/2*w^2 - 27/2*w - 16],
[479, 479, -7/2*w^3 - 9/2*w^2 + 51/2*w + 24],
[479, 479, 3/2*w^3 + 1/2*w^2 - 19/2*w - 3],
[491, 491, 2*w^3 + 4*w^2 - 15*w - 23],
[491, 491, -7/2*w^3 - 7/2*w^2 + 53/2*w + 17],
[491, 491, -1/2*w^3 - 5/2*w^2 + 11/2*w + 6],
[491, 491, -3/2*w^3 - 7/2*w^2 + 23/2*w + 17],
[499, 499, 3*w^2 - 19],
[499, 499, 3/2*w^3 + 5/2*w^2 - 19/2*w - 16],
[499, 499, -9/2*w^3 - 15/2*w^2 + 63/2*w + 38],
[499, 499, -1/2*w^3 + 1/2*w^2 + 9/2*w - 1],
[509, 509, 2*w^2 - w - 7],
[509, 509, 7/2*w^3 + 11/2*w^2 - 51/2*w - 33],
[521, 521, 4*w^3 + 5*w^2 - 32*w - 31],
[521, 521, -1/2*w^3 + 1/2*w^2 - 1/2*w + 4],
[541, 541, 1/2*w^3 - 7/2*w^2 - 3/2*w + 18],
[541, 541, -15/2*w^3 - 23/2*w^2 + 109/2*w + 62],
[541, 541, -w^3 - 3*w^2 + 7*w + 11],
[541, 541, 2*w^3 + 4*w^2 - 14*w - 27],
[569, 569, 1/2*w^3 + 5/2*w^2 - 7/2*w - 21],
[569, 569, -5/2*w^3 - 5/2*w^2 + 31/2*w + 14],
[571, 571, 2*w^2 + 2*w - 11],
[571, 571, 2*w^3 + 4*w^2 - 12*w - 23],
[601, 601, w^3 + 3*w^2 - 8*w - 9],
[601, 601, 3*w^3 + 2*w^2 - 23*w - 7],
[601, 601, -3/2*w^3 - 7/2*w^2 + 23/2*w + 19],
[601, 601, -2*w^3 - 4*w^2 + 15*w + 21],
[631, 631, -5/2*w^3 - 3/2*w^2 + 33/2*w + 3],
[631, 631, -13/2*w^3 - 19/2*w^2 + 101/2*w + 58],
[641, 641, 13/2*w^3 + 19/2*w^2 - 97/2*w - 52],
[641, 641, -3/2*w^3 - 9/2*w^2 + 23/2*w + 16],
[659, 659, 1/2*w^3 - 1/2*w^2 - 1/2*w + 2],
[659, 659, 3/2*w^3 + 7/2*w^2 - 15/2*w - 11],
[661, 661, -3/2*w^3 - 3/2*w^2 + 25/2*w + 3],
[661, 661, -1/2*w^3 - 1/2*w^2 + 3/2*w - 1],
[691, 691, -2*w^3 - 7*w^2 + 14*w + 49],
[691, 691, -1/2*w^3 - 7/2*w^2 + 13/2*w + 14],
[691, 691, -13/2*w^3 - 19/2*w^2 + 101/2*w + 59],
[691, 691, -7/2*w^3 - 13/2*w^2 + 49/2*w + 36],
[701, 701, 3/2*w^3 + 11/2*w^2 - 21/2*w - 38],
[701, 701, -2*w^3 - 2*w^2 + 10*w + 11],
[709, 709, 1/2*w^3 + 5/2*w^2 - 9/2*w - 17],
[709, 709, -3*w^3 - 3*w^2 + 22*w + 13],
[709, 709, -3*w^3 - 5*w^2 + 22*w + 23],
[709, 709, -5/2*w^3 - 5/2*w^2 + 33/2*w + 11],
[751, 751, -3/2*w^3 + 3/2*w^2 + 15/2*w - 4],
[751, 751, 15/2*w^3 + 21/2*w^2 - 111/2*w - 59],
[751, 751, 1/2*w^3 - 1/2*w^2 - 9/2*w - 2],
[751, 751, -3/2*w^3 - 5/2*w^2 + 19/2*w + 19],
[769, 769, -9/2*w^3 - 15/2*w^2 + 65/2*w + 38],
[769, 769, -1/2*w^3 - 7/2*w^2 + 9/2*w + 21],
[809, 809, -7/2*w^3 - 13/2*w^2 + 51/2*w + 42],
[809, 809, -5/2*w^3 - 3/2*w^2 + 27/2*w + 14],
[809, 809, -3*w^3 - 5*w^2 + 21*w + 29],
[809, 809, 2*w^2 - 9],
[811, 811, 3*w^3 + 5*w^2 - 20*w - 27],
[811, 811, 1/2*w^3 - 7/2*w^2 - 9/2*w + 31],
[811, 811, w^3 - w^2 - 2*w + 1],
[811, 811, 3/2*w^3 + 5/2*w^2 - 19/2*w - 18],
[821, 821, 9/2*w^3 + 13/2*w^2 - 67/2*w - 34],
[821, 821, 3/2*w^3 + 3/2*w^2 - 25/2*w - 1],
[829, 829, 1/2*w^3 + 5/2*w^2 - 1/2*w - 21],
[829, 829, 4*w^3 + 2*w^2 - 32*w - 3],
[839, 839, -1/2*w^3 - 1/2*w^2 + 3/2*w - 2],
[839, 839, -3/2*w^3 - 3/2*w^2 + 25/2*w + 2],
[841, 29, -5/2*w^3 - 5/2*w^2 + 35/2*w + 11],
[841, 29, 5/2*w^3 + 5/2*w^2 - 35/2*w - 14],
[881, 881, -9/2*w^3 - 13/2*w^2 + 63/2*w + 36],
[881, 881, 1/2*w^3 + 5/2*w^2 - 3/2*w - 21],
[911, 911, -1/2*w^3 + 3/2*w^2 + 7/2*w - 7],
[911, 911, 7/2*w^3 + 13/2*w^2 - 49/2*w - 38],
[911, 911, 7*w^3 + 11*w^2 - 51*w - 61],
[911, 911, -7/2*w^3 - 11/2*w^2 + 49/2*w + 31],
[919, 919, -1/2*w^3 + 7/2*w^2 + 3/2*w - 16],
[919, 919, 15/2*w^3 + 23/2*w^2 - 109/2*w - 64],
[929, 929, -3/2*w^3 - 7/2*w^2 + 19/2*w + 23],
[929, 929, -5/2*w^3 - 7/2*w^2 + 33/2*w + 21],
[941, 941, -1/2*w^3 - 7/2*w^2 + 11/2*w + 18],
[941, 941, -5*w^3 - 8*w^2 + 37*w + 43],
[961, 31, 5/2*w^3 + 5/2*w^2 - 35/2*w - 13],
[971, 971, 2*w^3 + w^2 - 14*w - 3],
[971, 971, -5/2*w^3 - 9/2*w^2 + 35/2*w + 31],
[971, 971, 7/2*w^3 + 9/2*w^2 - 49/2*w - 22],
[971, 971, 3/2*w^3 + 1/2*w^2 - 27/2*w + 6],
[991, 991, 5/2*w^3 + 7/2*w^2 - 39/2*w - 16],
[991, 991, 5/2*w^3 + 7/2*w^2 - 37/2*w - 14],
[1009, 1009, 1/2*w^3 + 7/2*w^2 - 9/2*w - 18],
[1009, 1009, 9/2*w^3 + 15/2*w^2 - 65/2*w - 41],
[1019, 1019, 2*w^3 + 5*w^2 - 14*w - 21],
[1019, 1019, 5/2*w^3 + 11/2*w^2 - 35/2*w - 36],
[1021, 1021, 1/2*w^3 + 3/2*w^2 - 7/2*w - 2],
[1021, 1021, w^3 + 2*w^2 - 7*w - 17],
[1039, 1039, -3/2*w^3 - 5/2*w^2 + 27/2*w + 8],
[1039, 1039, -7/2*w^3 - 13/2*w^2 + 51/2*w + 31],
[1039, 1039, -3*w^3 - 4*w^2 + 25*w + 19],
[1039, 1039, -4*w^3 - 6*w^2 + 29*w + 29],
[1051, 1051, 3/2*w^3 + 7/2*w^2 - 27/2*w - 13],
[1051, 1051, -7/2*w^3 - 9/2*w^2 + 53/2*w + 22],
[1061, 1061, 1/2*w^3 + 3/2*w^2 - 1/2*w - 17],
[1061, 1061, -3*w^3 - w^2 + 24*w - 3],
[1069, 1069, -7/2*w^3 - 9/2*w^2 + 47/2*w + 23],
[1069, 1069, 2*w^3 + 3*w^2 - 14*w - 23],
[1129, 1129, 3*w^3 + 4*w^2 - 23*w - 19],
[1129, 1129, 5/2*w^3 + 7/2*w^2 - 37/2*w - 13],
[1151, 1151, w^3 + w^2 - 4*w - 13],
[1151, 1151, -5*w^3 - 8*w^2 + 35*w + 43],
[1229, 1229, -3*w^3 - 4*w^2 + 19*w + 23],
[1229, 1229, -3/2*w^3 - 9/2*w^2 + 19/2*w + 31],
[1249, 1249, 3*w^3 + 6*w^2 - 21*w - 23],
[1249, 1249, 7/2*w^3 + 1/2*w^2 - 55/2*w + 7],
[1279, 1279, 2*w^3 + 2*w^2 - 17*w - 9],
[1279, 1279, -3*w^3 - 6*w^2 + 21*w + 29],
[1319, 1319, 3/2*w^3 + 9/2*w^2 - 19/2*w - 22],
[1319, 1319, -w^3 + w^2 + 9],
[1319, 1319, 5/2*w^3 + 11/2*w^2 - 33/2*w - 33],
[1319, 1319, 9*w^3 + 14*w^2 - 65*w - 77],
[1321, 1321, 3*w^3 + 3*w^2 - 16*w - 19],
[1321, 1321, -3*w^3 - 8*w^2 + 21*w + 53],
[1361, 1361, -11*w^3 - 17*w^2 + 79*w + 89],
[1361, 1361, -9/2*w^3 - 11/2*w^2 + 59/2*w + 24],
[1409, 1409, -3/2*w^3 - 1/2*w^2 + 15/2*w + 3],
[1409, 1409, -5*w^3 - 6*w^2 + 39*w + 31],
[1409, 1409, 7/2*w^3 + 13/2*w^2 - 51/2*w - 27],
[1409, 1409, 1/2*w^3 + 5/2*w^2 - 9/2*w - 21],
[1429, 1429, -1/2*w^3 - 1/2*w^2 + 13/2*w - 1],
[1429, 1429, 7/2*w^3 + 13/2*w^2 - 49/2*w - 34],
[1429, 1429, 1/2*w^3 - 5/2*w^2 - 5/2*w + 17],
[1429, 1429, 11/2*w^3 + 17/2*w^2 - 79/2*w - 42],
[1439, 1439, -17/2*w^3 - 25/2*w^2 + 125/2*w + 71],
[1439, 1439, 9/2*w^3 + 17/2*w^2 - 65/2*w - 51],
[1451, 1451, 3/2*w^3 - 3/2*w^2 - 15/2*w + 7],
[1451, 1451, -7/2*w^3 - 5/2*w^2 + 47/2*w + 7],
[1459, 1459, 3/2*w^3 + 11/2*w^2 - 25/2*w - 18],
[1459, 1459, -7/2*w^3 - 9/2*w^2 + 55/2*w + 22],
[1459, 1459, 3*w^3 + 5*w^2 - 22*w - 21],
[1459, 1459, w^3 + 7*w^2 - 5*w - 53],
[1471, 1471, -1/2*w^3 - 9/2*w^2 + 7/2*w + 37],
[1471, 1471, -21/2*w^3 - 31/2*w^2 + 151/2*w + 82],
[1481, 1481, -11/2*w^3 - 15/2*w^2 + 81/2*w + 38],
[1481, 1481, 3/2*w^3 - 1/2*w^2 - 17/2*w + 4],
[1489, 1489, -7/2*w^3 - 7/2*w^2 + 45/2*w + 22],
[1489, 1489, 9/2*w^3 + 9/2*w^2 - 67/2*w - 26],
[1499, 1499, 7/2*w^3 + 9/2*w^2 - 45/2*w - 26],
[1499, 1499, 7*w^3 + 11*w^2 - 49*w - 53],
[1511, 1511, -7/2*w^3 - 9/2*w^2 + 51/2*w + 19],
[1511, 1511, -3/2*w^3 - 1/2*w^2 + 19/2*w - 2],
[1511, 1511, -5*w^3 - 3*w^2 + 39*w + 9],
[1511, 1511, 1/2*w^3 - 9/2*w^2 - 5/2*w + 26],
[1559, 1559, -9/2*w^3 - 13/2*w^2 + 65/2*w + 31],
[1559, 1559, -3/2*w^3 - 5/2*w^2 + 27/2*w + 4],
[1559, 1559, -6*w^3 - 8*w^2 + 43*w + 37],
[1559, 1559, w^3 - w^2 - 6*w + 9],
[1571, 1571, 9/2*w^3 + 17/2*w^2 - 65/2*w - 39],
[1571, 1571, -2*w^3 - 6*w^2 + 15*w + 39],
[1601, 1601, -3*w^3 - 5*w^2 + 24*w + 33],
[1601, 1601, -7/2*w^3 - 9/2*w^2 + 51/2*w + 14],
[1609, 1609, 4*w^3 + 5*w^2 - 28*w - 23],
[1609, 1609, -3/2*w^3 - 3/2*w^2 + 13/2*w + 8],
[1609, 1609, -13/2*w^3 - 19/2*w^2 + 91/2*w + 48],
[1609, 1609, -5/2*w^3 - 3/2*w^2 + 35/2*w + 4],
[1619, 1619, 5/2*w^3 + 7/2*w^2 - 29/2*w - 14],
[1619, 1619, -11/2*w^3 - 17/2*w^2 + 75/2*w + 44],
[1621, 1621, 2*w^3 + 2*w^2 - 16*w - 3],
[1621, 1621, w^3 + w^2 - 5*w + 1],
[1669, 1669, -6*w^3 - 6*w^2 + 46*w + 31],
[1669, 1669, -1/2*w^3 - 9/2*w^2 + 15/2*w + 11],
[1681, 41, 3*w^3 + 3*w^2 - 21*w - 13],
[1681, 41, 3*w^3 + 3*w^2 - 21*w - 17],
[1699, 1699, 7/2*w^3 + 7/2*w^2 - 39/2*w - 23],
[1699, 1699, -7/2*w^3 - 17/2*w^2 + 49/2*w + 56],
[1709, 1709, -1/2*w^3 + 1/2*w^2 + 13/2*w - 1],
[1709, 1709, 1/2*w^3 + 3/2*w^2 - 1/2*w - 12],
[1721, 1721, 5/2*w^3 + 9/2*w^2 - 39/2*w - 19],
[1721, 1721, -3*w^3 - 3*w^2 + 23*w + 11],
[1741, 1741, 5/2*w^3 + 9/2*w^2 - 31/2*w - 26],
[1741, 1741, -3*w^3 - 5*w^2 + 21*w + 31],
[1759, 1759, w^3 + 5*w^2 - 6*w - 33],
[1759, 1759, 9/2*w^3 + 17/2*w^2 - 61/2*w - 41],
[1789, 1789, 5/2*w^3 + 11/2*w^2 - 35/2*w - 28],
[1789, 1789, 2*w^3 + 5*w^2 - 14*w - 29],
[1801, 1801, w^3 + 3*w^2 - 4*w - 23],
[1801, 1801, 3/2*w^3 + 9/2*w^2 - 21/2*w - 26],
[1801, 1801, 3*w^3 + 6*w^2 - 21*w - 31],
[1801, 1801, 11/2*w^3 + 17/2*w^2 - 79/2*w - 48],
[1811, 1811, 1/2*w^3 + 3/2*w^2 - 5/2*w - 1],
[1811, 1811, 1/2*w^3 + 3/2*w^2 - 5/2*w - 16],
[1831, 1831, -9/2*w^3 - 13/2*w^2 + 63/2*w + 37],
[1831, 1831, -w^3 - 3*w^2 + 5*w + 23],
[1849, 43, 6*w^3 + 8*w^2 - 44*w - 43],
[1849, 43, 1/2*w^3 - 3/2*w^2 + 1/2*w - 3],
[1861, 1861, 5/2*w^3 + 7/2*w^2 - 33/2*w - 23],
[1861, 1861, 5/2*w^3 + 9/2*w^2 - 33/2*w - 27],
[1861, 1861, -2*w^3 + 13*w - 1],
[1861, 1861, 2*w^3 + 3*w^2 - 12*w - 7],
[1871, 1871, 3/2*w^3 + 5/2*w^2 - 15/2*w - 14],
[1871, 1871, 3/2*w^3 + 1/2*w^2 - 27/2*w - 1],
[1901, 1901, -4*w^3 - 5*w^2 + 28*w + 27],
[1901, 1901, 5/2*w^3 + 3/2*w^2 - 35/2*w - 8],
[1931, 1931, -2*w^3 + w^2 + 10*w + 1],
[1931, 1931, 17/2*w^3 + 23/2*w^2 - 127/2*w - 66],
[1949, 1949, -w^3 - 2*w^2 + 3*w + 17],
[1949, 1949, -4*w^2 + 2*w + 17],
[1949, 1949, -w^3 - 5*w^2 + 7*w + 27],
[1949, 1949, -5*w^3 - 9*w^2 + 35*w + 49],
[1951, 1951, -13/2*w^3 - 17/2*w^2 + 103/2*w + 52],
[1951, 1951, 11/2*w^3 + 13/2*w^2 - 87/2*w - 41],
[1979, 1979, -15/2*w^3 - 19/2*w^2 + 119/2*w + 59],
[1979, 1979, 9/2*w^3 + 17/2*w^2 - 61/2*w - 47],
[1999, 1999, 3*w^3 + 3*w^2 - 20*w - 17],
[1999, 1999, 7/2*w^3 + 7/2*w^2 - 51/2*w - 19]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^2 + x - 4;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, e - 1, 1, -2*e - 2, -4, -2*e - 2, -e - 5, e - 1, -6, 2*e - 2, e + 9, -3*e - 1, -2*e - 4, 4*e, -8, -5*e + 3, 3*e - 1, -2*e - 4, 4*e + 6, e + 11, -6*e - 8, 2, 4*e + 10, 2*e + 4, -e + 1, e - 5, e - 11, -3*e - 3, 3*e - 1, -2*e - 6, e - 9, -4*e - 14, 10, -3*e - 5, -10, -2*e + 4, 7*e + 5, 4*e + 6, -5*e + 3, 4*e - 8, -e + 7, -e - 1, -4, 2*e - 2, -2*e + 8, -2*e + 16, 4*e + 2, -22, -2*e + 12, -3*e + 7, e - 13, -2*e - 8, -e + 7, -e - 25, -6*e - 12, -6*e, e + 17, -4*e - 8, -8*e - 10, -10, -6*e + 10, -16*e - 8, 11*e + 7, -e - 13, -2*e + 20, 2*e + 12, e + 13, 9*e + 13, 6, -9*e + 9, -5*e - 3, 7*e + 5, 3*e - 19, 11*e - 3, 4*e - 4, -e + 11, -2*e - 2, 2*e - 10, 2*e, -4*e - 2, 17*e + 11, 6*e, -13*e - 9, -4*e - 32, 5*e + 17, -3*e + 25, -12, -7*e + 21, -4*e, 8*e - 8, 11*e - 5, -12*e - 20, 8*e - 12, -12*e, -6*e - 12, -14, -6*e + 8, 2*e - 16, 2*e - 4, 15*e + 21, -2*e - 16, 8*e + 10, -3*e - 21, 18*e, 2*e + 6, -4*e - 12, 8*e + 18, 18*e + 8, -10*e - 12, -4*e - 26, 7*e - 5, -4*e + 4, -14, -4*e + 14, -4*e - 8, -3*e + 5, -3*e - 21, -2*e - 20, -9*e - 1, 8*e - 12, 4*e + 24, 6*e + 18, -19*e - 17, 2*e + 4, -7*e + 11, -4*e + 22, -8*e + 2, 18, 15*e + 19, -2*e - 2, 14*e + 14, -24, 2*e + 16, 5*e + 7, -4*e + 6, 16*e - 6, -e + 13, -2*e - 16, 17*e + 17, 4*e, 16*e + 20, -4, -14*e - 20, 7*e + 29, e + 31, -7*e - 5, -3*e + 1, 15*e + 27, -e + 21, 11*e - 23, 7*e - 31, 12*e - 2, 2*e - 26, -6*e + 34, -13*e - 9, -7*e - 3, -17*e + 3, 8, 18*e - 4, -5*e + 29, 8*e - 26, -12*e - 26, -8*e - 6, -12*e - 20, 4*e, 36, 6*e + 10, 11*e + 31, 4*e - 4, 15*e - 23, -2, -6*e - 54, -5*e - 5, 14*e - 8, -21*e - 3, -6*e + 14, 11*e - 17, -9*e - 29, 4*e - 20, -20*e + 4, 22*e + 18, -4*e + 18, 4*e + 10, 12*e + 6, -2, 12*e + 6, 20*e + 10, -11*e + 13, -4*e + 24, 15*e + 29, -17*e - 31, -16*e + 2, -19*e + 19, -6*e - 14, -10*e + 22, -8*e - 32, -3*e + 41, -5*e - 9, 11*e + 7, 24*e - 2, -12*e - 6, 4*e + 50, 21*e + 7, 28*e + 18, -2*e - 4, -21*e + 13, 9*e - 21, 8*e - 22, 21*e - 1, 4*e + 30, -18*e - 24, -12*e - 24, 17*e + 1, 4, -6*e - 42, -3*e - 27, -17*e - 9, -36, -16*e, 28*e - 4, 18*e + 22, 4*e - 54, -28*e - 14, 2*e + 4, -24*e - 14, -7*e - 59, -20*e - 32, -20*e + 8, 2*e + 58, 3*e - 49, 14*e - 34, -10*e - 2, 16*e - 16, 10*e + 10, -10*e + 38, e - 23, -e - 61, -e + 1, 32*e + 14, 10*e - 24, 20*e + 18, -3*e + 63, 19*e + 17, e - 7, -8*e - 12, 4*e - 50, 3*e - 19, 15*e - 3, 31*e + 9, -5*e - 43, -11*e + 39, -8*e - 20, -18*e + 22, -11*e - 33, -e - 67, -24*e - 2, -4*e + 50, 2*e - 40, 3*e + 41, 12*e + 12, 7*e + 15, -10*e - 32, -18*e - 12, 25*e + 11, -6*e + 12, -6*e - 56, e + 19, -18*e - 30, 36*e + 24, 0, -12*e + 36, -11*e - 1, 16*e + 22, -16*e - 38, -7*e + 27, -14*e + 32, 21*e + 31, -11*e - 31, -8*e - 16, 8*e - 38, -4*e - 34, -13*e - 29, 8*e + 12, -4*e - 54, 2*e - 8, -12*e - 26, 18, -4*e - 4, 5*e - 43, 19*e + 35, -6*e + 10, -10*e - 66, -10*e - 58];
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 | {5,5,w^3 + 2*w^2 - 7*w - 9}>] := -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;