/*
  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![19, 14, -13, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

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

primesArray := [
[4, 2, -2/19*w^3 + 3/19*w^2 + 37/19*w - 3],
[5, 5, 3/19*w^3 + 5/19*w^2 - 27/19*w - 1],
[5, 5, -3/19*w^3 + 14/19*w^2 + 8/19*w - 2],
[9, 3, -2/19*w^3 + 3/19*w^2 + 37/19*w - 4],
[19, 19, -w],
[19, 19, 4/19*w^3 - 6/19*w^2 - 55/19*w + 1],
[19, 19, 4/19*w^3 - 6/19*w^2 - 55/19*w + 2],
[19, 19, w - 1],
[29, 29, -5/19*w^3 - 2/19*w^2 + 64/19*w + 2],
[29, 29, 10/19*w^3 - 34/19*w^2 - 71/19*w + 9],
[29, 29, 4/19*w^3 - 6/19*w^2 - 55/19*w + 4],
[29, 29, -w + 3],
[49, 7, 5/19*w^3 - 17/19*w^2 - 64/19*w + 11],
[49, 7, 6/19*w^3 - 9/19*w^2 - 92/19*w - 1],
[71, 71, 3/19*w^3 + 5/19*w^2 - 46/19*w - 1],
[71, 71, 6/19*w^3 - 9/19*w^2 - 73/19*w],
[71, 71, 6/19*w^3 - 9/19*w^2 - 73/19*w + 4],
[71, 71, -3/19*w^3 + 14/19*w^2 + 27/19*w - 3],
[101, 101, 7/19*w^3 - 1/19*w^2 - 63/19*w - 3],
[101, 101, -9/19*w^3 + 23/19*w^2 + 81/19*w - 5],
[101, 101, 9/19*w^3 - 4/19*w^2 - 100/19*w],
[101, 101, 7/19*w^3 - 20/19*w^2 - 44/19*w + 6],
[121, 11, 6/19*w^3 - 9/19*w^2 - 54/19*w + 1],
[121, 11, -6/19*w^3 + 9/19*w^2 + 54/19*w - 2],
[139, 139, 9/19*w^3 - 4/19*w^2 - 100/19*w - 2],
[139, 139, -7/19*w^3 + 1/19*w^2 + 63/19*w + 1],
[139, 139, 7/19*w^3 - 20/19*w^2 - 44/19*w + 4],
[139, 139, w - 5],
[149, 149, -1/19*w^3 + 11/19*w^2 - 10/19*w - 6],
[149, 149, -3/19*w^3 - 5/19*w^2 + 46/19*w],
[149, 149, -3/19*w^3 + 14/19*w^2 + 27/19*w - 2],
[149, 149, 1/19*w^3 + 8/19*w^2 - 9/19*w - 6],
[169, 13, 14/19*w^3 - 40/19*w^2 - 126/19*w + 13],
[169, 13, 10/19*w^3 - 34/19*w^2 - 52/19*w + 7],
[191, 191, -10/19*w^3 + 15/19*w^2 + 109/19*w - 6],
[191, 191, 6/19*w^3 - 9/19*w^2 - 35/19*w - 2],
[191, 191, -6/19*w^3 + 9/19*w^2 + 35/19*w - 4],
[191, 191, 10/19*w^3 - 15/19*w^2 - 109/19*w],
[211, 211, 1/19*w^3 + 8/19*w^2 - 28/19*w - 8],
[211, 211, 1/19*w^3 - 11/19*w^2 - 9/19*w - 1],
[211, 211, -1/19*w^3 - 8/19*w^2 + 28/19*w - 2],
[211, 211, -1/19*w^3 + 11/19*w^2 + 9/19*w - 9],
[239, 239, 4/19*w^3 - 25/19*w^2 - 17/19*w + 12],
[239, 239, -1/19*w^3 - 8/19*w^2 - 29/19*w + 5],
[239, 239, 10/19*w^3 - 15/19*w^2 - 128/19*w + 1],
[239, 239, -4/19*w^3 - 13/19*w^2 + 55/19*w + 10],
[241, 241, -2/19*w^3 + 3/19*w^2 + 56/19*w - 4],
[241, 241, -6/19*w^3 + 9/19*w^2 + 92/19*w],
[241, 241, 6/19*w^3 - 9/19*w^2 - 92/19*w + 5],
[241, 241, -2/19*w^3 + 3/19*w^2 + 56/19*w + 1],
[269, 269, -10/19*w^3 + 34/19*w^2 + 71/19*w - 11],
[269, 269, 8/19*w^3 - 12/19*w^2 - 129/19*w + 11],
[269, 269, -10/19*w^3 + 34/19*w^2 + 71/19*w - 8],
[269, 269, -13/19*w^3 + 48/19*w^2 + 60/19*w - 8],
[289, 17, 30/19*w^3 - 7/19*w^2 - 384/19*w - 14],
[289, 17, 11/19*w^3 - 45/19*w^2 - 80/19*w + 17],
[311, 311, -12/19*w^3 + 37/19*w^2 + 108/19*w - 15],
[311, 311, -24/19*w^3 + 55/19*w^2 + 273/19*w - 22],
[311, 311, -8/19*w^3 - 7/19*w^2 + 110/19*w + 4],
[311, 311, 4/19*w^3 + 13/19*w^2 - 36/19*w - 7],
[331, 331, -1/19*w^3 - 8/19*w^2 + 66/19*w - 2],
[331, 331, -6/19*w^3 + 28/19*w^2 + 35/19*w - 9],
[331, 331, 20/19*w^3 - 68/19*w^2 - 161/19*w + 22],
[331, 331, 16/19*w^3 - 62/19*w^2 - 87/19*w + 15],
[359, 359, 13/19*w^3 - 48/19*w^2 - 79/19*w + 11],
[359, 359, -7/19*w^3 + 20/19*w^2 + 101/19*w - 13],
[359, 359, -15/19*w^3 + 51/19*w^2 + 116/19*w - 17],
[359, 359, 3/19*w^3 + 5/19*w^2 - 84/19*w + 5],
[379, 379, 3/19*w^3 + 5/19*w^2 - 65/19*w - 9],
[379, 379, 25/19*w^3 - 9/19*w^2 - 339/19*w - 11],
[379, 379, -25/19*w^3 + 66/19*w^2 + 282/19*w - 28],
[379, 379, 3/19*w^3 - 14/19*w^2 - 46/19*w + 12],
[389, 389, -13/19*w^3 + 48/19*w^2 + 136/19*w - 24],
[389, 389, 11/19*w^3 - 7/19*w^2 - 118/19*w + 2],
[389, 389, -3/19*w^3 + 14/19*w^2 + 46/19*w - 10],
[389, 389, -13/19*w^3 - 9/19*w^2 + 193/19*w + 15],
[409, 409, 2/19*w^3 - 3/19*w^2 - 56/19*w],
[409, 409, 6/19*w^3 - 9/19*w^2 - 92/19*w + 4],
[409, 409, -6/19*w^3 + 9/19*w^2 + 92/19*w - 1],
[409, 409, -2/19*w^3 + 3/19*w^2 + 56/19*w - 3],
[431, 431, -6/19*w^3 + 9/19*w^2 + 130/19*w - 13],
[431, 431, 12/19*w^3 - 37/19*w^2 - 108/19*w + 11],
[431, 431, 10/19*w^3 - 15/19*w^2 - 166/19*w + 14],
[431, 431, -8/19*w^3 + 31/19*w^2 + 34/19*w - 8],
[461, 461, 5/19*w^3 - 17/19*w^2 - 45/19*w + 1],
[461, 461, -3/19*w^3 + 14/19*w^2 + 8/19*w - 8],
[461, 461, 3/19*w^3 + 5/19*w^2 - 27/19*w - 7],
[461, 461, 5/19*w^3 + 2/19*w^2 - 64/19*w + 2],
[479, 479, 5/19*w^3 - 17/19*w^2 - 26/19*w + 7],
[479, 479, 1/19*w^3 - 11/19*w^2 - 28/19*w + 7],
[479, 479, 7/19*w^3 - 1/19*w^2 - 82/19*w + 1],
[479, 479, -5/19*w^3 + 17/19*w^2 + 64/19*w - 3],
[499, 499, -11/19*w^3 + 7/19*w^2 + 118/19*w + 2],
[499, 499, 9/19*w^3 - 23/19*w^2 - 62/19*w + 4],
[499, 499, -9/19*w^3 + 4/19*w^2 + 81/19*w],
[499, 499, -11/19*w^3 + 26/19*w^2 + 99/19*w - 8],
[509, 509, -6/19*w^3 - 10/19*w^2 + 73/19*w + 9],
[509, 509, -1/19*w^3 + 11/19*w^2 + 47/19*w - 7],
[509, 509, -6/19*w^3 - 10/19*w^2 + 73/19*w + 2],
[509, 509, -9/19*w^3 + 4/19*w^2 + 138/19*w - 2],
[529, 23, -4/19*w^3 + 6/19*w^2 + 74/19*w - 1],
[529, 23, 4/19*w^3 - 6/19*w^2 - 74/19*w + 3],
[571, 571, 1/19*w^3 + 8/19*w^2 - 66/19*w + 6],
[571, 571, -9/19*w^3 + 23/19*w^2 + 119/19*w - 15],
[571, 571, -11/19*w^3 + 26/19*w^2 + 156/19*w - 18],
[571, 571, 3/19*w^3 + 5/19*w^2 - 103/19*w + 9],
[599, 599, 6/19*w^3 - 9/19*w^2 - 16/19*w - 2],
[599, 599, -14/19*w^3 + 21/19*w^2 + 164/19*w - 7],
[599, 599, -2/19*w^3 + 22/19*w^2 + 18/19*w - 13],
[599, 599, -6/19*w^3 + 9/19*w^2 + 16/19*w - 3],
[601, 601, 23/19*w^3 - 63/19*w^2 - 264/19*w + 31],
[601, 601, -36/19*w^3 - 3/19*w^2 + 476/19*w + 23],
[601, 601, 36/19*w^3 - 111/19*w^2 - 362/19*w + 46],
[601, 601, -16/19*w^3 + 43/19*w^2 + 125/19*w - 8],
[619, 619, -5/19*w^3 + 17/19*w^2 + 45/19*w - 11],
[619, 619, 5/19*w^3 - 17/19*w^2 - 26/19*w - 2],
[619, 619, -5/19*w^3 - 2/19*w^2 + 45/19*w - 4],
[619, 619, -7/19*w^3 + 20/19*w^2 + 63/19*w - 12],
[691, 691, 7/19*w^3 - 1/19*w^2 - 120/19*w],
[691, 691, 3/19*w^3 - 14/19*w^2 - 65/19*w + 6],
[691, 691, -3/19*w^3 - 5/19*w^2 + 84/19*w + 2],
[691, 691, 7/19*w^3 - 20/19*w^2 - 101/19*w + 6],
[701, 701, -7/19*w^3 + 20/19*w^2 + 63/19*w - 2],
[701, 701, 5/19*w^3 - 17/19*w^2 - 26/19*w + 8],
[701, 701, -7/19*w^3 + 20/19*w^2 + 82/19*w - 4],
[701, 701, -7/19*w^3 + 1/19*w^2 + 82/19*w - 2],
[719, 719, -8/19*w^3 + 31/19*w^2 + 53/19*w - 12],
[719, 719, 8/19*w^3 - 31/19*w^2 - 53/19*w + 6],
[719, 719, -8/19*w^3 - 7/19*w^2 + 91/19*w + 2],
[719, 719, 8/19*w^3 + 7/19*w^2 - 91/19*w - 8],
[739, 739, 9/19*w^3 - 23/19*w^2 - 43/19*w + 3],
[739, 739, -16/19*w^3 + 24/19*w^2 + 163/19*w - 9],
[739, 739, 16/19*w^3 - 24/19*w^2 - 163/19*w],
[739, 739, -9/19*w^3 + 4/19*w^2 + 62/19*w],
[769, 769, 24/19*w^3 - 74/19*w^2 - 254/19*w + 35],
[769, 769, -3/19*w^3 + 33/19*w^2 - 30/19*w - 11],
[769, 769, 3/19*w^3 + 24/19*w^2 - 27/19*w - 11],
[769, 769, 24/19*w^3 + 2/19*w^2 - 330/19*w - 19],
[811, 811, 5/19*w^3 + 2/19*w^2 - 102/19*w],
[811, 811, -5/19*w^3 + 17/19*w^2 + 83/19*w - 7],
[811, 811, 5/19*w^3 + 2/19*w^2 - 102/19*w - 2],
[811, 811, -5/19*w^3 + 17/19*w^2 + 83/19*w - 5],
[821, 821, -6/19*w^3 - 10/19*w^2 + 111/19*w + 13],
[821, 821, 1/19*w^3 + 8/19*w^2 + 10/19*w - 12],
[821, 821, -1/19*w^3 + 11/19*w^2 - 29/19*w - 11],
[821, 821, -6/19*w^3 + 28/19*w^2 + 73/19*w - 18],
[839, 839, -2*w^3 + 26*w + 23],
[839, 839, 18/19*w^3 - 65/19*w^2 - 143/19*w + 24],
[839, 839, -18/19*w^3 - 11/19*w^2 + 219/19*w + 14],
[839, 839, 2*w^3 - 6*w^2 - 20*w + 47],
[859, 859, 10/19*w^3 - 34/19*w^2 - 109/19*w + 19],
[859, 859, 3/19*w^3 + 24/19*w^2 - 46/19*w - 14],
[859, 859, -17/19*w^3 + 35/19*w^2 + 210/19*w - 14],
[859, 859, -10/19*w^3 - 4/19*w^2 + 147/19*w + 12],
[911, 911, -1/19*w^3 + 11/19*w^2 + 28/19*w - 9],
[911, 911, -5/19*w^3 + 17/19*w^2 + 64/19*w - 1],
[911, 911, 5/19*w^3 + 2/19*w^2 - 83/19*w + 3],
[911, 911, 1/19*w^3 + 8/19*w^2 - 47/19*w - 7],
[941, 941, 11/19*w^3 - 7/19*w^2 - 99/19*w - 5],
[941, 941, -1/19*w^3 + 11/19*w^2 + 28/19*w - 10],
[941, 941, -13/19*w^3 + 10/19*w^2 + 136/19*w - 4],
[941, 941, -6/19*w^3 + 28/19*w^2 - 3/19*w - 6],
[961, 31, 10/19*w^3 - 15/19*w^2 - 90/19*w + 2],
[961, 31, -10/19*w^3 + 15/19*w^2 + 90/19*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 4*x^3 - 28*x^2 + 16*x + 16;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [0, 1/24*e^3 + 1/3*e^2 - 1/3*e + 1/3, 1/24*e^3 + 1/3*e^2 - 1/3*e - 11/3, -2, e, -1/12*e^3 - 2/3*e^2 - 1/3*e + 4/3, 1/3*e^3 + 5/3*e^2 - 23/3*e - 4/3, -1/4*e^3 - e^2 + 7*e - 4, -7/24*e^3 - 11/6*e^2 + 10/3*e + 17/3, 1/8*e^3 + 1/2*e^2 - 2*e + 5, 3/8*e^3 + 3/2*e^2 - 10*e + 3, -13/24*e^3 - 17/6*e^2 + 34/3*e - 1/3, 1/4*e^3 + e^2 - 8*e - 2, -1/4*e^3 - e^2 + 8*e - 6, -7/12*e^3 - 8/3*e^2 + 44/3*e - 2/3, 7/12*e^3 + 8/3*e^2 - 44/3*e + 14/3, 1/12*e^3 + 2/3*e^2 + 4/3*e - 10/3, -1/12*e^3 - 2/3*e^2 - 4/3*e - 2/3, -11/24*e^3 - 8/3*e^2 + 23/3*e - 5/3, 5/24*e^3 + 2/3*e^2 - 17/3*e + 35/3, 5/24*e^3 + 2/3*e^2 - 17/3*e - 1/3, -11/24*e^3 - 8/3*e^2 + 23/3*e + 31/3, -1/3*e^3 - 5/3*e^2 + 20/3*e - 32/3, 1/3*e^3 + 5/3*e^2 - 20/3*e - 28/3, 7/12*e^3 + 8/3*e^2 - 47/3*e - 4/3, -1/2*e^3 - 2*e^2 + 15*e - 8, -1/4*e^3 - e^2 + 9*e - 4, 1/6*e^3 + 1/3*e^2 - 25/3*e + 4/3, 11/24*e^3 + 13/6*e^2 - 32/3*e + 17/3, -5/24*e^3 - 7/6*e^2 + 8/3*e - 11/3, -11/24*e^3 - 13/6*e^2 + 32/3*e + 7/3, 5/24*e^3 + 7/6*e^2 - 8/3*e - 13/3, -1/2*e^3 - 2*e^2 + 16*e + 8, 1/2*e^3 + 2*e^2 - 16*e + 16, -1/3*e^3 - 7/6*e^2 + 29/3*e - 14/3, 7/12*e^3 + 19/6*e^2 - 35/3*e - 4/3, 1/3*e^3 + 13/6*e^2 - 11/3*e - 22/3, -1/12*e^3 - 1/6*e^2 + 5/3*e - 20/3, -7/12*e^3 - 11/3*e^2 + 29/3*e + 16/3, -e^2 - 5*e + 8, 1/4*e^3 + 2*e^2 - 3*e - 16, 1/3*e^3 + 8/3*e^2 - 5/3*e - 52/3, -3/4*e^3 - 9/2*e^2 + 15*e + 4, -3/2*e^2 - 9*e + 10, -3/4*e^3 - 9/2*e^2 + 15*e + 20, -3/2*e^2 - 9*e + 26, 5/6*e^3 + 11/3*e^2 - 56/3*e + 26/3, 7/6*e^3 + 19/3*e^2 - 64/3*e - 14/3, -7/6*e^3 - 19/3*e^2 + 64/3*e + 14/3, -5/6*e^3 - 11/3*e^2 + 56/3*e - 26/3, 25/24*e^3 + 41/6*e^2 - 46/3*e - 41/3, 3/8*e^3 + 7/2*e^2 - 2*e - 31, 1/8*e^3 + 5/2*e^2 + 6*e - 17, 19/24*e^3 + 35/6*e^2 - 22/3*e - 95/3, e^3 + 4*e^2 - 32*e + 16, -e^3 - 4*e^2 + 32*e, -5/3*e^3 - 28/3*e^2 + 100/3*e + 8/3, 2/3*e^3 + 4/3*e^2 - 76/3*e + 40/3, -1/3*e^3 - 8/3*e^2 + 20/3*e + 88/3, -2/3*e^3 - 16/3*e^2 + 4/3*e + 104/3, e^2 + 7*e - 8, 5/12*e^3 + 7/3*e^2 - 31/3*e - 8/3, 1/3*e^3 + 2/3*e^2 - 41/3*e + 44/3, -3/4*e^3 - 4*e^2 + 17*e + 8, -1/12*e^3 - 7/6*e^2 - 1/3*e - 8/3, -3/4*e^3 - 9/2*e^2 + 13*e + 24, -1/2*e^3 - 7/2*e^2 + 5*e - 2, 1/6*e^3 - 1/6*e^2 - 25/3*e + 82/3, 1/6*e^3 + 7/3*e^2 + 11/3*e - 80/3, 3/4*e^3 + 5*e^2 - 11*e - 24, -1/2*e^3 - 4*e^2 + 3*e + 12, -5/12*e^3 - 10/3*e^2 + 13/3*e + 32/3, -29/24*e^3 - 17/3*e^2 + 89/3*e - 53/3, 1/8*e^3 + e^2 + 3*e + 13, 9/8*e^3 + 5*e^2 - 29*e - 7, -5/24*e^3 - 5/3*e^2 - 7/3*e + 55/3, -7/4*e^3 - 10*e^2 + 30*e + 14, -11/12*e^3 - 10/3*e^2 + 70/3*e - 58/3, 11/12*e^3 + 10/3*e^2 - 70/3*e + 58/3, 7/4*e^3 + 10*e^2 - 30*e - 14, 19/12*e^3 + 29/3*e^2 - 74/3*e - 82/3, -5/12*e^3 - 1/3*e^2 + 46/3*e - 58/3, -5/12*e^3 - 1/3*e^2 + 46/3*e - 94/3, 19/12*e^3 + 29/3*e^2 - 74/3*e - 46/3, -7/8*e^3 - 7/2*e^2 + 26*e - 23, 25/24*e^3 + 29/6*e^2 - 82/3*e - 35/3, -5/24*e^3 - 1/6*e^2 + 38/3*e + 19/3, 3/8*e^3 + 3/2*e^2 - 14*e + 15, -e^3 - 9/2*e^2 + 27*e - 10, 11/12*e^3 + 23/6*e^2 - 79/3*e + 4/3, -1/3*e^3 - 7/6*e^2 + 41/3*e + 10/3, 1/4*e^3 + 1/2*e^2 - 13*e + 12, -1/6*e^3 - 1/3*e^2 + 7/3*e - 76/3, -3/2*e^3 - 8*e^2 + 31*e - 8, 3/4*e^3 + 5*e^2 - 7*e - 28, 11/12*e^3 + 10/3*e^2 - 79/3*e + 4/3, -31/24*e^3 - 16/3*e^2 + 103/3*e - 1/3, 17/24*e^3 + 14/3*e^2 - 17/3*e - 73/3, 41/24*e^3 + 26/3*e^2 - 113/3*e + 35/3, -7/24*e^3 - 4/3*e^2 + 7/3*e - 61/3, -1/3*e^3 - 8/3*e^2 + 8/3*e + 82/3, 1/3*e^3 + 8/3*e^2 - 8/3*e + 2/3, -7/12*e^3 - 11/3*e^2 + 11/3*e + 16/3, 1/2*e^3 + 3*e^2 - 3*e, -7/4*e^3 - 8*e^2 + 43*e - 8, 11/6*e^3 + 26/3*e^2 - 131/3*e + 20/3, -1/4*e^3 + 1/2*e^2 + 13*e - 4, 4/3*e^3 + 49/6*e^2 - 65/3*e + 2/3, 13/12*e^3 + 43/6*e^2 - 41/3*e - 124/3, 3/2*e^2 + 5*e - 42, 3/4*e^3 + 5*e^2 - 8*e - 26, 7/12*e^3 + 5/3*e^2 - 56/3*e + 14/3, -1/12*e^3 + 1/3*e^2 + 8/3*e - 74/3, -5/4*e^3 - 7*e^2 + 24*e - 2, 2/3*e^3 + 4/3*e^2 - 73/3*e + 4/3, e^3 + 7*e^2 - 9*e - 48, -7/4*e^3 - 10*e^2 + 33*e - 8, 1/12*e^3 + 5/3*e^2 + 1/3*e - 136/3, -7/12*e^3 - 2/3*e^2 + 65/3*e - 80/3, 2*e^3 + 12*e^2 - 33*e - 20, -7/4*e^3 - 11*e^2 + 25*e + 32, 1/3*e^3 - 1/3*e^2 - 41/3*e + 104/3, -3/8*e^3 - 2*e^2 + 13*e + 21, -5/24*e^3 - 8/3*e^2 - 25/3*e + 91/3, -41/24*e^3 - 26/3*e^2 + 119/3*e - 29/3, 9/8*e^3 + 4*e^2 - 35*e + 5, -7/12*e^3 - 8/3*e^2 + 20/3*e - 74/3, 5/4*e^3 + 8*e^2 - 12*e - 30, 11/4*e^3 + 14*e^2 - 60*e + 10, -25/12*e^3 - 26/3*e^2 + 164/3*e - 26/3, 7/6*e^3 + 13/3*e^2 - 91/3*e + 52/3, 11/6*e^3 + 32/3*e^2 - 89/3*e - 64/3, -25/12*e^3 - 35/3*e^2 + 113/3*e + 28/3, -11/12*e^3 - 10/3*e^2 + 67/3*e - 76/3, -e^3 - 7*e^2 + 12*e + 32, -1/3*e^3 - 11/3*e^2 - 4/3*e + 100/3, 1/3*e^3 + 11/3*e^2 + 4/3*e - 64/3, e^3 + 7*e^2 - 12*e - 20, 3/2*e^3 + 10*e^2 - 19*e - 52, 1/12*e^3 + 8/3*e^2 + 19/3*e - 160/3, 1/6*e^3 - 5/3*e^2 - 43/3*e + 40/3, -7/4*e^3 - 11*e^2 + 27*e + 8, -13/8*e^3 - 13/2*e^2 + 50*e - 11, -23/24*e^3 - 19/6*e^2 + 110/3*e - 53/3, 43/24*e^3 + 47/6*e^2 - 154/3*e + 37/3, 9/8*e^3 + 9/2*e^2 - 38*e + 3, -2*e^3 - 19/2*e^2 + 45*e + 2, 4/3*e^3 + 43/6*e^2 - 65/3*e - 34/3, 25/12*e^3 + 61/6*e^2 - 137/3*e + 44/3, -5/4*e^3 - 13/2*e^2 + 21*e - 12, -7/3*e^3 - 35/3*e^2 + 149/3*e - 56/3, -4/3*e^3 - 20/3*e^2 + 71/3*e - 68/3, 25/12*e^3 + 29/3*e^2 - 143/3*e - 16/3, 19/12*e^3 + 26/3*e^2 - 77/3*e - 64/3, -5/4*e^3 - 5*e^2 + 38*e - 6, -7/12*e^3 - 5/3*e^2 + 74/3*e - 50/3, 17/12*e^3 + 19/3*e^2 - 118/3*e + 34/3, 3/4*e^3 + 3*e^2 - 26*e - 2, 1/8*e^3 + 2*e^2 + 7*e - 17, -13/24*e^3 - 4/3*e^2 + 61/3*e - 43/3, 11/24*e^3 + 8/3*e^2 - 35/3*e - 31/3, 9/8*e^3 + 6*e^2 - 25*e - 5, 3*e^3 + 15*e^2 - 60*e + 2, -3*e^3 - 15*e^2 + 60*e - 10];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {4, 2, -2/19*w^3 + 3/19*w^2 + 37/19*w - 3}>] := -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;