/*
  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, 14, -13, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {11, 11, 1/2*w^3 - 2*w^2 - 5/2*w + 11}>;

primesArray := [
[4, 2, w^2 - 2*w - 6],
[4, 2, -w^2 + 7],
[5, 5, -1/2*w^3 + 2*w^2 + 7/2*w - 14],
[5, 5, 1/2*w^3 + 1/2*w^2 - 6*w - 9],
[11, 11, 1/2*w^3 - 2*w^2 - 5/2*w + 11],
[11, 11, 1/2*w^3 + 1/2*w^2 - 5*w - 7],
[31, 31, 1/2*w^2 + 1/2*w - 4],
[31, 31, -1/2*w^2 + 3/2*w + 3],
[41, 41, 1/2*w^2 + 1/2*w - 6],
[41, 41, 2*w^3 - 15/2*w^2 - 25/2*w + 50],
[41, 41, 5/2*w^2 - 1/2*w - 17],
[41, 41, 1/2*w^2 - 3/2*w - 5],
[61, 61, -1/2*w^3 + w^2 + 7/2*w - 1],
[61, 61, -1/2*w^3 + 1/2*w^2 + 4*w - 3],
[71, 71, 1/2*w^3 + w^2 - 11/2*w - 13],
[71, 71, -1/2*w^3 + 5/2*w^2 + 2*w - 17],
[81, 3, -3],
[89, 89, -w^3 + 3/2*w^2 + 13/2*w - 9],
[89, 89, w^3 - 7/2*w^2 - 11/2*w + 20],
[89, 89, -w^3 - 1/2*w^2 + 19/2*w + 12],
[89, 89, 1/2*w^3 + 2*w^2 - 17/2*w - 19],
[109, 109, 1/2*w^3 + 3/2*w^2 - 7*w - 15],
[109, 109, -1/2*w^3 + 1/2*w^2 + 4*w - 5],
[121, 11, -3/2*w^2 + 3/2*w + 10],
[131, 131, 3/2*w^3 - 13/2*w^2 - 9*w + 43],
[131, 131, -3/2*w^3 - 2*w^2 + 35/2*w + 29],
[139, 139, -3/2*w^3 + 11/2*w^2 + 10*w - 39],
[139, 139, 1/2*w^3 + w^2 - 11/2*w - 9],
[139, 139, -1/2*w^3 + 5/2*w^2 + 2*w - 13],
[139, 139, -3/2*w^3 - w^2 + 33/2*w + 25],
[151, 151, -1/2*w^3 + 3/2*w^2 + 3*w - 5],
[151, 151, -1/2*w^3 + 9/2*w + 1],
[179, 179, 5/2*w^2 - 3/2*w - 20],
[179, 179, -w^3 + 5/2*w^2 + 11/2*w - 5],
[179, 179, -w^3 + 1/2*w^2 + 15/2*w - 2],
[179, 179, w^3 + 1/2*w^2 - 19/2*w - 10],
[181, 181, 1/2*w^2 - 3/2*w - 8],
[181, 181, 3/2*w^2 - 1/2*w - 13],
[181, 181, -3/2*w^2 + 5/2*w + 12],
[181, 181, 1/2*w^2 + 1/2*w - 9],
[191, 191, w^3 + 1/2*w^2 - 17/2*w - 9],
[191, 191, 1/2*w^3 - w^2 - 5/2*w + 8],
[199, 199, -1/2*w^3 + 7/2*w^2 + 3*w - 23],
[199, 199, -1/2*w^3 - 2*w^2 + 17/2*w + 17],
[211, 211, w^3 - 5/2*w^2 - 9/2*w + 3],
[211, 211, -w^3 + 1/2*w^2 + 13/2*w - 3],
[229, 229, 3/2*w^3 - 2*w^2 - 21/2*w - 6],
[229, 229, -w^3 + 3*w^2 + 8*w - 23],
[239, 239, -w^3 + 3*w^2 + 4*w - 9],
[239, 239, w^3 - 9*w - 7],
[239, 239, 1/2*w^3 - 3/2*w^2 - 2*w + 13],
[239, 239, -w^3 + 7*w + 3],
[241, 241, 1/2*w^3 + 2*w^2 - 15/2*w - 16],
[241, 241, -1/2*w^3 + 7/2*w^2 + 2*w - 21],
[251, 251, w^3 - 5/2*w^2 - 9/2*w + 10],
[251, 251, w^3 - 9*w - 9],
[269, 269, w^3 - 1/2*w^2 - 17/2*w - 10],
[269, 269, w^3 - 5/2*w^2 - 13/2*w + 18],
[281, 281, -1/2*w^3 + 3*w^2 + 3/2*w - 19],
[281, 281, 1/2*w^3 + 3/2*w^2 - 6*w - 15],
[289, 17, w^3 - 3/2*w^2 - 15/2*w - 1],
[289, 17, -w^3 + 3/2*w^2 + 15/2*w - 9],
[331, 331, 3/2*w^2 - 1/2*w - 14],
[331, 331, -3/2*w^2 + 5/2*w + 13],
[349, 349, -w^3 + 3/2*w^2 + 11/2*w - 1],
[349, 349, w^3 - 5/2*w^2 - 13/2*w + 14],
[359, 359, -w^3 + 5/2*w^2 + 11/2*w - 14],
[359, 359, w^3 - 1/2*w^2 - 15/2*w - 7],
[361, 19, 2*w^2 - 2*w - 15],
[361, 19, -2*w^2 + 2*w + 13],
[379, 379, 5/2*w^2 - 3/2*w - 16],
[379, 379, 5/2*w^2 - 7/2*w - 15],
[389, 389, 7/2*w^2 - 1/2*w - 26],
[389, 389, 1/2*w^3 - 3/2*w^2 - 5*w + 5],
[389, 389, 1/2*w^3 + 2*w^2 - 11/2*w - 18],
[389, 389, 3/2*w^3 - 1/2*w^2 - 13*w - 15],
[409, 409, -w^3 + 7/2*w^2 + 9/2*w - 19],
[409, 409, w^3 + 1/2*w^2 - 17/2*w - 12],
[439, 439, -1/2*w^3 + 5/2*w^2 + w - 15],
[439, 439, 1/2*w^3 + w^2 - 9/2*w - 12],
[461, 461, 1/2*w^3 + 2*w^2 - 15/2*w - 18],
[461, 461, -1/2*w^3 + 7/2*w^2 + 2*w - 23],
[479, 479, -1/2*w^3 + 2*w^2 + 7/2*w - 10],
[479, 479, 1/2*w^3 - 2*w^2 - 5/2*w + 7],
[491, 491, -2*w^3 + 1/2*w^2 + 29/2*w - 1],
[491, 491, 2*w^3 - 11/2*w^2 - 19/2*w + 12],
[499, 499, -w^3 + 5/2*w^2 + 13/2*w - 13],
[499, 499, w^3 - 1/2*w^2 - 17/2*w - 5],
[509, 509, -w^3 + 3/2*w^2 + 15/2*w - 2],
[509, 509, w^3 - 3/2*w^2 - 15/2*w + 6],
[521, 521, -1/2*w^3 + 2*w^2 + 9/2*w - 15],
[521, 521, 1/2*w^3 + 1/2*w^2 - 7*w - 9],
[529, 23, -w^3 + 2*w^2 + 5*w - 9],
[529, 23, -3/2*w^2 + 7/2*w + 12],
[541, 541, 3/2*w^3 + 1/2*w^2 - 14*w - 17],
[541, 541, -3/2*w^3 + 5*w^2 + 17/2*w - 29],
[569, 569, -2*w^3 + 1/2*w^2 + 35/2*w + 18],
[569, 569, -w^3 - 5/2*w^2 + 27/2*w + 27],
[571, 571, -w^3 + w^2 + 8*w + 7],
[571, 571, -1/2*w^3 + 2*w^2 + 3/2*w - 4],
[571, 571, 1/2*w^3 + 1/2*w^2 - 4*w - 1],
[571, 571, -4*w^3 - 7/2*w^2 + 91/2*w + 68],
[599, 599, -1/2*w^3 - 5/2*w^2 + 10*w + 21],
[599, 599, 2*w^3 + 5/2*w^2 - 47/2*w - 37],
[599, 599, 2*w^3 - 17/2*w^2 - 25/2*w + 56],
[599, 599, -1/2*w^3 + 4*w^2 + 7/2*w - 28],
[601, 601, -w^3 - 3/2*w^2 + 23/2*w + 18],
[601, 601, 1/2*w^3 + 7/2*w^2 - 5*w - 27],
[601, 601, 1/2*w^3 - 5*w^2 + 7/2*w + 28],
[601, 601, 2*w^3 - 3*w^2 - 13*w + 21],
[619, 619, -w^3 + 5/2*w^2 + 9/2*w - 8],
[619, 619, w^3 - 1/2*w^2 - 13/2*w - 2],
[631, 631, -5/2*w^3 + 1/2*w^2 + 18*w - 1],
[631, 631, -5/2*w^3 + 7*w^2 + 23/2*w - 15],
[641, 641, 5/2*w^2 - 1/2*w - 20],
[641, 641, -5/2*w^2 + 9/2*w + 18],
[659, 659, w^3 - 7/2*w^2 - 15/2*w + 28],
[659, 659, w^3 + 2*w^2 - 11*w - 21],
[661, 661, -w^3 + 15/2*w^2 + 11/2*w - 52],
[661, 661, -3/2*w^3 + 1/2*w^2 + 13*w + 5],
[661, 661, 3/2*w^3 - 4*w^2 - 19/2*w + 17],
[661, 661, -w^3 - 9/2*w^2 + 35/2*w + 40],
[691, 691, 2*w^3 - 9/2*w^2 - 21/2*w + 5],
[691, 691, -5/2*w^3 - w^2 + 55/2*w + 37],
[701, 701, w^3 - 2*w^2 - 7*w + 3],
[701, 701, w^3 - w^2 - 8*w + 5],
[709, 709, -11/2*w^2 + 3/2*w + 40],
[709, 709, -3/2*w^3 - 3/2*w^2 + 16*w + 23],
[739, 739, 1/2*w^3 - 9/2*w^2 + 3*w + 25],
[739, 739, 1/2*w^3 + 3*w^2 - 9/2*w - 24],
[751, 751, -7/2*w^3 + 12*w^2 + 47/2*w - 83],
[751, 751, -3/2*w^3 + 23/2*w + 3],
[751, 751, -3/2*w^3 + 9/2*w^2 + 7*w - 13],
[751, 751, 7/2*w^3 + 3/2*w^2 - 37*w - 51],
[761, 761, -3/2*w^3 - 1/2*w^2 + 14*w + 19],
[761, 761, -w^3 + 3/2*w^2 + 15/2*w - 4],
[761, 761, -3/2*w^3 + 5*w^2 + 17/2*w - 31],
[769, 769, -3*w^3 - 5/2*w^2 + 69/2*w + 51],
[769, 769, -1/2*w^3 - 4*w^2 + 9/2*w + 27],
[769, 769, -w^3 - 5/2*w^2 + 29/2*w + 25],
[769, 769, 3*w^3 - 23/2*w^2 - 41/2*w + 80],
[809, 809, 1/2*w^3 + 2*w^2 - 13/2*w - 17],
[809, 809, 1/2*w^3 - 7/2*w^2 - w + 21],
[821, 821, 1/2*w^3 - 1/2*w^2 - 6*w + 3],
[821, 821, -1/2*w^3 + w^2 + 11/2*w - 3],
[829, 829, -3/2*w^3 + 3*w^2 + 23/2*w - 10],
[829, 829, -w^3 + 2*w^2 + 5*w + 1],
[829, 829, 7/2*w^2 - 3/2*w - 28],
[829, 829, 3/2*w^3 - 3/2*w^2 - 13*w + 3],
[839, 839, -w^3 - 7/2*w^2 + 17/2*w + 27],
[839, 839, 2*w^3 + 1/2*w^2 - 43/2*w - 27],
[839, 839, 2*w^3 - 13/2*w^2 - 29/2*w + 46],
[839, 839, -w^3 - 3/2*w^2 + 19/2*w + 14],
[841, 29, 5/2*w^2 - 5/2*w - 19],
[841, 29, 5/2*w^2 - 5/2*w - 16],
[859, 859, 1/2*w^3 + w^2 - 5/2*w - 12],
[859, 859, w^3 + 3/2*w^2 - 19/2*w - 19],
[859, 859, -w^3 + 9/2*w^2 + 7/2*w - 26],
[859, 859, 1/2*w^3 - 5/2*w^2 + w + 13],
[881, 881, -1/2*w^3 + 1/2*w^2 + 6*w + 1],
[881, 881, 1/2*w^3 - w^2 - 11/2*w + 7],
[911, 911, -1/2*w^3 + 5/2*w^2 + 3*w - 13],
[911, 911, 1/2*w^3 + w^2 - 13/2*w - 8],
[941, 941, -2*w^3 + 7/2*w^2 + 27/2*w - 21],
[941, 941, 3/2*w^3 - 7*w^2 - 21/2*w + 51],
[941, 941, 3/2*w^3 + 5/2*w^2 - 20*w - 35],
[941, 941, -2*w^3 + 13/2*w^2 + 29/2*w - 48],
[961, 31, -w^2 + w + 13],
[1009, 1009, -5/2*w^2 - 1/2*w + 18],
[1009, 1009, 3/2*w^3 - 3/2*w^2 - 12*w - 5],
[1021, 1021, w^3 + w^2 - 10*w - 19],
[1021, 1021, w^3 - 5/2*w^2 - 9/2*w + 12],
[1031, 1031, -1/2*w^3 + 3/2*w^2 + 5*w - 9],
[1031, 1031, -w^3 + 3/2*w^2 + 13/2*w - 2],
[1049, 1049, -3/2*w^3 - 1/2*w^2 + 16*w + 23],
[1049, 1049, -2*w^3 + 9/2*w^2 + 23/2*w - 6],
[1049, 1049, -2*w^3 + 3/2*w^2 + 29/2*w - 8],
[1049, 1049, 3/2*w^3 - 5*w^2 - 21/2*w + 37],
[1051, 1051, -2*w^3 + 11/2*w^2 + 17/2*w - 10],
[1051, 1051, -3*w^3 - 1/2*w^2 + 59/2*w + 36],
[1051, 1051, -3*w^3 + 19/2*w^2 + 39/2*w - 62],
[1051, 1051, 2*w^3 - 1/2*w^2 - 27/2*w + 2],
[1061, 1061, 11/2*w^2 - 3/2*w - 38],
[1061, 1061, 11/2*w^2 - 19/2*w - 34],
[1069, 1069, 1/2*w^3 - 4*w^2 - 5/2*w + 25],
[1069, 1069, 2*w^3 - 20*w - 25],
[1069, 1069, -2*w^3 + 6*w^2 + 14*w - 43],
[1069, 1069, 1/2*w^3 + 5/2*w^2 - 9*w - 19],
[1109, 1109, -5/2*w^3 + 13/2*w^2 + 12*w - 13],
[1109, 1109, -5/2*w^3 + w^2 + 35/2*w - 3],
[1151, 1151, 13/2*w^2 - 3/2*w - 46],
[1151, 1151, 1/2*w^3 + 3/2*w^2 - 5*w - 17],
[1151, 1151, 1/2*w^3 - 3*w^2 - 1/2*w + 20],
[1151, 1151, 13/2*w^2 - 23/2*w - 41],
[1171, 1171, w^3 - 3/2*w^2 - 13/2*w + 12],
[1171, 1171, 1/2*w^3 - 1/2*w^2 - 6*w + 1],
[1171, 1171, -1/2*w^3 + w^2 + 11/2*w - 5],
[1171, 1171, -1/2*w^3 + 5*w^2 - 7/2*w - 26],
[1201, 1201, 1/2*w^3 + 1/2*w^2 - 6*w - 3],
[1201, 1201, -1/2*w^3 + 2*w^2 + 7/2*w - 8],
[1229, 1229, 1/2*w^3 + 5/2*w^2 - 8*w - 21],
[1229, 1229, -1/2*w^3 + 4*w^2 + 3/2*w - 26],
[1249, 1249, -w^3 + 7/2*w^2 + 9/2*w - 12],
[1249, 1249, w^3 + 1/2*w^2 - 17/2*w - 5],
[1279, 1279, 1/2*w^3 + 3/2*w^2 - 3*w - 15],
[1279, 1279, -1/2*w^3 + 3*w^2 - 3/2*w - 16],
[1289, 1289, 5/2*w^2 - 9/2*w - 19],
[1289, 1289, w^3 - 9/2*w^2 - 15/2*w + 32],
[1289, 1289, -w^3 - 3/2*w^2 + 27/2*w + 21],
[1289, 1289, 5/2*w^2 - 1/2*w - 21],
[1291, 1291, -1/2*w^3 + 5*w^2 - 9/2*w - 23],
[1291, 1291, 1/2*w^3 + 7/2*w^2 - 4*w - 23],
[1301, 1301, -3/2*w^3 + 2*w^2 + 21/2*w - 10],
[1301, 1301, -3/2*w^3 + 5/2*w^2 + 10*w - 1],
[1319, 1319, 4*w^3 - 29/2*w^2 - 49/2*w + 93],
[1319, 1319, -4*w^3 - 5/2*w^2 + 83/2*w + 58],
[1321, 1321, -17/2*w^2 + 31/2*w + 52],
[1321, 1321, 17/2*w^2 - 3/2*w - 59],
[1361, 1361, -3*w^3 - 7/2*w^2 + 71/2*w + 58],
[1361, 1361, -3/2*w^3 - 3*w^2 + 39/2*w + 34],
[1369, 37, 3/2*w^3 - 1/2*w^2 - 13*w - 17],
[1369, 37, 3/2*w^3 + 1/2*w^2 - 11*w - 3],
[1381, 1381, w^3 - 5*w^2 - 2*w + 27],
[1381, 1381, w^3 + 2*w^2 - 9*w - 21],
[1399, 1399, w^3 + 5/2*w^2 - 19/2*w - 25],
[1399, 1399, w^3 - 11/2*w^2 - 3/2*w + 31],
[1439, 1439, -2*w^3 + 13/2*w^2 + 23/2*w - 40],
[1439, 1439, 4*w^3 + 3*w^2 - 43*w - 65],
[1451, 1451, 1/2*w^3 + 9/2*w^2 - 5*w - 35],
[1451, 1451, -1/2*w^3 + 6*w^2 - 11/2*w - 35],
[1459, 1459, -3/2*w^3 + 9/2*w^2 + 9*w - 25],
[1459, 1459, -3/2*w^3 + 1/2*w^2 + 10*w + 7],
[1471, 1471, 3/2*w^3 - 11/2*w^2 - 7*w + 31],
[1471, 1471, -3/2*w^3 - w^2 + 27/2*w + 20],
[1489, 1489, -3*w^3 - 3*w^2 + 34*w + 51],
[1489, 1489, 1/2*w^3 + 5/2*w^2 - 6*w - 25],
[1489, 1489, -1/2*w^3 + 4*w^2 - 1/2*w - 28],
[1489, 1489, 3*w^3 - 12*w^2 - 19*w + 79],
[1499, 1499, 1/2*w^3 - 7/2*w^2 + 2*w + 19],
[1499, 1499, 1/2*w^3 + 5/2*w^2 - 6*w - 21],
[1499, 1499, -1/2*w^3 + 4*w^2 - 1/2*w - 24],
[1499, 1499, -w^3 + 5/2*w^2 + 11/2*w - 17],
[1511, 1511, -2*w^3 + 15/2*w^2 + 17/2*w - 37],
[1511, 1511, -2*w^3 - 3/2*w^2 + 35/2*w + 23],
[1531, 1531, 1/2*w^3 - 3*w^2 - 9/2*w + 16],
[1531, 1531, -2*w^3 + 9/2*w^2 + 23/2*w - 8],
[1531, 1531, -2*w^3 + 3/2*w^2 + 29/2*w - 6],
[1531, 1531, w^3 + 5/2*w^2 - 21/2*w - 25],
[1549, 1549, -1/2*w^3 + 1/2*w^2 + 3*w - 7],
[1549, 1549, 1/2*w^3 - w^2 - 5/2*w - 4],
[1579, 1579, w^3 + 3/2*w^2 - 19/2*w - 13],
[1579, 1579, -w^3 + 9/2*w^2 + 7/2*w - 20],
[1609, 1609, -11/2*w^2 + 1/2*w + 37],
[1609, 1609, 11/2*w^2 - 21/2*w - 32],
[1619, 1619, 1/2*w^3 - 3/2*w^2 - w + 15],
[1619, 1619, w^3 - 3/2*w^2 - 21/2*w + 1],
[1621, 1621, -2*w^3 + 15/2*w^2 + 23/2*w - 49],
[1621, 1621, 1/2*w^3 + 3/2*w^2 - 9*w - 19],
[1669, 1669, 1/2*w^3 + 5/2*w^2 - 7*w - 23],
[1669, 1669, 1/2*w^3 - 4*w^2 - 1/2*w + 27],
[1699, 1699, -w^3 + 1/2*w^2 + 17/2*w - 2],
[1699, 1699, -w^3 + 5/2*w^2 + 13/2*w - 6],
[1709, 1709, 1/2*w^3 + 3/2*w^2 - 2*w - 15],
[1709, 1709, -1/2*w^3 + 3*w^2 - 5/2*w - 15],
[1721, 1721, 5/2*w^3 - w^2 - 45/2*w - 20],
[1721, 1721, -5/2*w^3 + 13/2*w^2 + 17*w - 41],
[1741, 1741, -w^3 + 5/2*w^2 + 13/2*w - 3],
[1741, 1741, -5/2*w^3 - 3*w^2 + 61/2*w + 48],
[1741, 1741, -5*w^3 + 35/2*w^2 + 65/2*w - 116],
[1741, 1741, -w^3 + 5*w^2 + 6*w - 31],
[1759, 1759, 4*w^3 - 16*w^2 - 26*w + 111],
[1759, 1759, 3*w^3 + 3/2*w^2 - 61/2*w - 43],
[1789, 1789, w^3 + 1/2*w^2 - 17/2*w - 14],
[1789, 1789, -w^3 + 7/2*w^2 + 9/2*w - 21],
[1801, 1801, w^3 + 5/2*w^2 - 23/2*w - 24],
[1801, 1801, w^3 + 11/2*w^2 - 15/2*w - 38],
[1831, 1831, 13/2*w^2 - 1/2*w - 46],
[1831, 1831, -3/2*w^3 + 8*w^2 + 19/2*w - 53],
[1831, 1831, 9/2*w^3 - 33/2*w^2 - 28*w + 109],
[1831, 1831, -13/2*w^2 + 25/2*w + 40],
[1861, 1861, 5/2*w^2 - 9/2*w - 8],
[1861, 1861, 5/2*w^2 - 1/2*w - 10],
[1871, 1871, -1/2*w^3 - 1/2*w^2 + 7*w + 15],
[1871, 1871, -1/2*w^3 - 4*w^2 + 7/2*w + 26],
[1879, 1879, 1/2*w^3 - 9/2*w^2 - w + 25],
[1879, 1879, -w^3 - 5/2*w^2 + 23/2*w + 27],
[1879, 1879, w^3 - 11/2*w^2 - 7/2*w + 35],
[1879, 1879, 1/2*w^3 + 3*w^2 - 17/2*w - 20],
[1889, 1889, w^3 + 5/2*w^2 - 25/2*w - 27],
[1889, 1889, -w^3 + 11/2*w^2 + 9/2*w - 36],
[1901, 1901, -3*w^3 - w^2 + 32*w + 41],
[1901, 1901, 3*w^3 - 10*w^2 - 21*w + 69],
[1931, 1931, 2*w^3 - 15/2*w^2 - 13/2*w + 26],
[1931, 1931, -w^3 + 11/2*w^2 + 5/2*w - 31],
[1951, 1951, -w^3 + 1/2*w^2 + 13/2*w + 7],
[1951, 1951, 2*w^2 - 19],
[1979, 1979, -1/2*w^3 + 4*w^2 + 1/2*w - 25],
[1979, 1979, 7/2*w^3 + w^2 - 73/2*w - 47],
[1979, 1979, 7/2*w^3 - 23/2*w^2 - 24*w + 79],
[1979, 1979, 1/2*w^3 + 5/2*w^2 - 7*w - 21],
[1999, 1999, -w^3 + 15/2*w^2 + 11/2*w - 54],
[1999, 1999, -w^3 + 15/2*w^2 - 9/2*w - 37]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 + x^2 - 9*x + 3;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1, e, 1/2*e^2 + e - 3/2, -1/2*e^2 - e + 7/2, -1, -e - 1, -e^2 + e + 10, e^2 + 4*e - 7, 3/2*e^2 + 4*e - 15/2, e^2 + 3*e - 6, -3*e - 5, -1/2*e^2 + 17/2, -2*e^2 - 5*e + 11, -2*e^2 - 3*e + 7, -e^2 - 2*e - 5, e^2 + 3*e - 2, -e^2 + 11, -3/2*e^2 - e + 33/2, e - 1, e^2 + e + 4, 1/2*e^2 + 2*e - 13/2, 1/2*e^2 + e + 9/2, -3/2*e^2 - 2*e - 5/2, 1/2*e^2 + 3*e - 11/2, -2*e^2 - 7*e + 11, -e^2 + 11, -e^2 - 4*e + 17, 2*e^2 + 2*e - 12, -e + 11, -3*e^2 - 6*e + 11, 4*e^2 + 8*e - 16, e^2 + e + 4, -3*e^2 - 6*e + 15, -4*e^2 - 10*e + 20, 2*e + 10, e^2 + 6*e + 5, -1/2*e^2 + 6*e + 17/2, -3/2*e^2 - 9*e + 17/2, 9/2*e^2 + 8*e - 57/2, 1/2*e^2 + 3*e + 9/2, 5*e^2 + 7*e - 18, -4*e - 6, 2*e^2 + 4*e - 10, 2*e^2 + e - 13, 4, -2*e^2 + 24, 7/2*e^2 + 8*e - 27/2, -5/2*e^2 - 6*e + 53/2, -3*e^2 - 7*e + 24, 5*e^2 + 8*e - 23, 3*e^2 + 5*e - 6, -2*e^2 - 6*e + 18, 7/2*e^2 + 5*e - 49/2, 1/2*e^2 + 7/2, e^2 + 3*e - 2, -e^2 + 4*e + 1, 4*e^2 + 6*e - 24, -2*e^2 - 4*e + 20, 2*e^2 + 2*e - 18, -3*e^2 - 6*e + 27, -9/2*e^2 - 10*e + 57/2, 1/2*e^2 + 5*e - 19/2, -e^2 + 2*e + 31, 6*e - 8, 5/2*e^2 + 3*e + 5/2, -3/2*e^2 - 9*e + 13/2, e^2 + 2*e - 5, 2*e - 10, 3/2*e^2 + 9*e - 33/2, -7/2*e^2 - 5*e + 9/2, -2*e^2 + 4*e + 8, -2*e^2 + 3*e + 25, 7/2*e^2 + 8*e - 51/2, 4*e^2 + 13*e - 21, e^2 - e - 2, 3/2*e^2 + 5*e - 65/2, -2*e^2 - 9*e - 1, 5*e - 1, 3*e^2 - 2*e - 19, -3*e^2 - 2*e + 41, -2*e^2 - 5*e + 3, -4*e^2 - e + 27, -4*e^2 - 10*e + 38, -3*e^2 + 41, 6*e^2 + 12*e - 30, 4*e + 16, e^2 - 5, -7*e^2 - 7*e + 44, -5/2*e^2 - 7*e + 67/2, 3/2*e^2 + e - 21/2, -6*e^2 - 12*e + 24, 5*e^2 + 10*e - 37, -3/2*e^2 + e + 65/2, 11/2*e^2 + 12*e - 59/2, -6*e^2 - 13*e + 31, 2*e^2 - 7*e - 25, -1/2*e^2 - 3*e + 19/2, -9/2*e^2 - 8*e + 77/2, 3*e^2 + 4*e - 43, e^2 + 2*e - 17, 6*e^2 + 8*e - 10, 5*e^2 + 9*e - 16, -4*e^2 - e + 19, -4*e^2 - 10*e + 24, -6*e^2 - 9*e + 33, 6*e^2 + 8*e - 36, 1/2*e^2 + 3*e - 7/2, -9/2*e^2 - 11*e + 59/2, -3/2*e^2 - 9*e + 21/2, -5/2*e^2 - 6*e + 65/2, 8*e + 12, 5*e^2 + 12*e - 49, e^2 + 8*e + 23, -e^2 - 10*e - 17, 2*e^2 + e + 5, 2*e^2 - 5*e - 25, e^2 + 5*e - 24, -8*e - 6, -3*e^2 - 9*e + 22, 3/2*e^2 - 9*e - 29/2, 11/2*e^2 + 14*e - 51/2, 4*e^2 + 3*e - 15, -3*e^2 + 10*e + 39, -6*e^2 - 9*e + 29, -13/2*e^2 - 21*e + 99/2, -5/2*e^2 - 16*e + 41/2, 2*e^2 + 12*e, -10*e^2 - 16*e + 44, 5*e^2 - 2*e - 49, -5*e^2 - 6*e + 35, 7*e^2 + 7*e - 44, -13*e - 21, -7*e^2 - 18*e + 53, 4*e^2 + 4*e - 8, 2*e^2 - 4*e - 12, e^2 + 14*e - 17, 42, -1/2*e^2 - 8*e - 3/2, 7/2*e^2 + 2*e - 19/2, 7/2*e^2 - e - 73/2, -1/2*e^2 - 10*e + 17/2, 3/2*e^2 - 8*e + 5/2, -17/2*e^2 - 14*e + 109/2, -3/2*e^2 - 14*e + 43/2, -7/2*e^2 + 7*e + 53/2, -1/2*e^2 + 4*e + 25/2, -5*e^2 - 6*e + 45, 4*e^2 - 4*e - 46, 11/2*e^2 + 16*e - 47/2, -2*e^2 + 6*e + 42, -4*e^2 - 10*e + 56, -4*e^2, 16*e + 20, 6*e^2 + 6*e - 34, e^2 + 2*e - 57, -5*e^2 - 13*e + 24, -3*e^2 - 8*e + 7, -6*e^2 - 22*e + 38, -3*e^2 - 12*e + 5, -11/2*e^2 - 7*e + 81/2, 11/2*e^2 - 6*e - 95/2, -7*e^2 - 12*e + 25, -10*e + 2, 11/2*e^2 + 18*e - 27/2, 1/2*e^2 - 3*e - 55/2, 5/2*e^2 + 5*e + 29/2, 5/2*e^2 + 9*e + 33/2, 5/2*e^2 - 7*e - 67/2, 7*e^2 + 18*e - 59, 6*e^2 + 8*e - 8, -9/2*e^2 - 6*e + 109/2, 15/2*e^2 + 13*e - 45/2, 7*e^2 + 18*e - 27, 2*e^2 + 15*e - 19, 17/2*e^2 + 16*e - 81/2, -8*e^2 - 10*e + 36, -6*e^2 - 26*e + 42, -19/2*e^2 - 14*e + 87/2, -e - 13, e^2 - 12*e - 29, 3*e^2 + 4*e - 15, 2*e^2 + 10*e - 28, -19/2*e^2 - 20*e + 127/2, 13/2*e^2 + 6*e - 45/2, -19/2*e^2 - 14*e + 103/2, 8*e^2 + 25*e - 57, 4*e^2 - 11*e - 41, -17/2*e^2 - 13*e + 67/2, -4*e^2 - 5*e - 27, -4*e^2 - 21*e + 23, 4*e^2 + 4*e - 24, -10*e^2 - 6*e + 64, 2*e^2 - 4*e + 6, -9*e^2 - 8*e + 61, 4*e^2 + 4*e - 32, -11*e^2 - 12*e + 71, 2*e^2 + 8*e - 18, -3*e^2 - 8*e + 23, -11*e^2 - 18*e + 67, 7*e^2 - 65, e^2 + 3*e + 14, 4*e^2 + 7*e - 15, 11/2*e^2 + 14*e - 55/2, -9/2*e^2 - 9*e + 63/2, 10*e^2 + 11*e - 59, 18, 15/2*e^2 + 15*e - 33/2, -9/2*e^2 + 129/2, 15/2*e^2 + 9*e - 117/2, -13/2*e^2 - e + 131/2, 10*e^2 + 18*e - 58, -3*e + 9, -4*e^2 - 13*e + 13, 4*e^2 + 3*e - 63, 6*e^2 - 54, -2*e^2 - 9*e + 5, -5*e^2 - 6*e + 1, 3*e^2 - 10*e - 23, -2*e^2 - 5*e + 57, -6*e^2 - 23*e + 25, 3*e^2 + 17*e - 18, -2*e^2 - 9*e + 57, -8*e + 14, -7*e^2 - 10*e + 19, -8*e^2 - 10*e + 38, 3*e - 29, 8*e^2 + 11*e - 41, 5*e^2 + 12*e - 49, -e^2 - 15*e + 6, -2*e^2 - 10*e + 18, -6*e^2 - 20*e + 62, -3*e^2 - 12*e + 7, 4*e^2 + 18*e - 20, 6*e^2 + 12*e - 46, -17/2*e^2 - 26*e + 93/2, 8*e - 14, 9*e^2 + 26*e - 53, -17/2*e^2 - 16*e + 53/2, -2*e^2 + e + 15, -9*e^2 + 65, -4*e + 16, -8*e^2 - 24*e + 32, 2*e^2 + 10*e - 22, -4*e^2 - 11*e + 29, -5*e^2 - 18*e - 1, 4*e^2 - 50, -11*e^2 - 18*e + 37, 9*e^2 + 17*e - 76, -6*e^2 + e + 25, 2*e^2 - e - 3, -7*e^2 - 12*e + 7, -10*e^2 - 29*e + 65, -4*e^2 + 12*e + 34, -e^2 - 8*e + 11, -5*e^2 + 7, -11*e - 27, -3*e^2 + 3*e + 38, -8*e^2 - 3*e + 43, 25/2*e^2 + 10*e - 137/2, -3/2*e^2 + 2*e + 67/2, -4*e^2 - 18*e + 10, e^2 + 2*e - 11, 17/2*e^2 + 3*e - 95/2, -17/2*e^2 - 4*e + 45/2, 5/2*e^2 + 17*e + 9/2, 17/2*e^2 + 5*e - 99/2, 8*e^2 + 8*e - 14, 4*e^2 + 9*e - 37, -2*e^2 + 3*e + 23, -8*e^2 - 32*e + 58, -11*e^2 - 20*e + 55, 4*e^2 - 14*e - 68, -9/2*e^2 - 7*e + 55/2, -3/2*e^2 + 12*e + 15/2, -16*e^2 - 19*e + 91, -6*e^2 - 9*e + 1, -9*e^2 - 18*e + 7, 2*e^2 - 6*e - 34, 7*e^2 + 7*e - 80, 9*e^2 + 4*e - 61, 2*e^2 + e - 9, 12*e^2 + 17*e - 69, -6*e^2 - 30*e + 48, -e^2 + 5*e - 30, 5*e^2 + 16*e - 3, -6*e^2 - 2*e - 8, 7*e^2 + 12*e - 25, 12*e^2 + 25*e - 67, 12*e^2 + 31*e - 79, -5*e^2 - 13*e + 8, 4*e^2 - 10*e - 52, e^2 + 10*e - 53, -6*e - 36, -8*e - 44, 11*e^2 + 20*e - 41, -12*e^2 - 9*e + 67, -9*e^2 - 16*e + 75, 10*e^2 + 12*e - 22, 2*e^2 - 20*e - 46, 6*e^2 + 19*e - 55, 3*e^2 + 7*e - 24, 2*e^2 + 12];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {11, 11, 1/2*w^3 - 2*w^2 - 5/2*w + 11}>] := 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;