/*
  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, 0, -9, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, -w}>;

primesArray := [
[4, 2, -w^3 + w^2 + 5*w - 6],
[9, 3, -w^2 - w + 4],
[9, 3, w^2 - w - 4],
[11, 11, w + 1],
[11, 11, w - 1],
[19, 19, -w],
[19, 19, -w^2 - w + 6],
[19, 19, -w^2 + w + 6],
[25, 5, 2*w^2 - 9],
[29, 29, -w^3 + 4*w + 2],
[29, 29, -w^3 + 4*w - 2],
[41, 41, 2*w^2 - w - 7],
[41, 41, w^3 - w^2 - 6*w + 4],
[61, 61, -w^3 + 3*w^2 + 6*w - 14],
[61, 61, w^3 + 2*w^2 - 5*w - 8],
[61, 61, -w^3 + 2*w^2 + 5*w - 8],
[61, 61, w^3 + 3*w^2 - 6*w - 14],
[89, 89, -w^3 + w^2 + 6*w - 9],
[89, 89, 2*w^3 - w^2 - 10*w + 10],
[109, 109, -w^3 + 5*w^2 + 7*w - 23],
[109, 109, -w^3 + 6*w - 4],
[121, 11, -3*w^2 + 13],
[131, 131, -w^3 - 4*w^2 + 4*w + 16],
[131, 131, w^3 - 4*w^2 - 4*w + 16],
[139, 139, -w^3 + 2*w^2 + 5*w - 7],
[139, 139, w^3 + 2*w^2 - 5*w - 7],
[149, 149, 2*w^3 - w^2 - 10*w + 7],
[149, 149, 2*w^3 + 2*w^2 - 10*w - 11],
[149, 149, -2*w^3 + 2*w^2 + 10*w - 11],
[149, 149, 2*w^3 + w^2 - 10*w - 7],
[151, 151, 3*w^2 - 2*w - 11],
[151, 151, -w^3 + w^2 + 4*w - 7],
[151, 151, w^3 + w^2 - 4*w - 7],
[151, 151, 3*w^2 + 2*w - 11],
[181, 181, 4*w^2 - w - 21],
[181, 181, 4*w^2 + w - 21],
[191, 191, 3*w^2 - 2*w - 15],
[191, 191, 3*w^2 + 2*w - 15],
[199, 199, 2*w^2 - w - 4],
[199, 199, 2*w^2 + w - 4],
[211, 211, w^3 + 3*w^2 - 6*w - 13],
[211, 211, w^3 - w^2 - 5*w + 9],
[211, 211, -w^3 - w^2 + 5*w + 9],
[211, 211, -w^3 + 3*w^2 + 6*w - 13],
[239, 239, -w^3 + 2*w^2 + 5*w - 14],
[239, 239, w^3 + 2*w^2 - 5*w - 14],
[241, 241, -2*w^3 + 8*w + 1],
[241, 241, 2*w^3 - 8*w + 1],
[251, 251, w^3 - 3*w^2 - 4*w + 15],
[251, 251, -w^3 - 3*w^2 + 4*w + 15],
[269, 269, 2*w^3 - w^2 - 10*w + 5],
[269, 269, 2*w^3 + w^2 - 10*w - 5],
[271, 271, w^3 - 7*w + 2],
[271, 271, -w^3 + 7*w + 2],
[281, 281, 2*w^3 + w^2 - 9*w - 6],
[281, 281, -2*w^3 + w^2 + 9*w - 6],
[289, 17, w^2 + 2*w - 11],
[289, 17, w^2 - 2*w - 11],
[311, 311, 2*w^3 - 9*w + 2],
[311, 311, 2*w^3 - 9*w - 2],
[331, 331, 2*w^3 + 2*w^2 - 10*w - 7],
[331, 331, w^3 - 5*w^2 - 4*w + 25],
[331, 331, -w^3 - 5*w^2 + 4*w + 25],
[331, 331, -2*w^3 + 2*w^2 + 10*w - 7],
[349, 349, -w^3 + 5*w^2 + 6*w - 23],
[349, 349, -w^3 - 4*w^2 + 4*w + 20],
[349, 349, -w^3 + 4*w^2 + 4*w - 20],
[349, 349, w^3 + 5*w^2 - 6*w - 23],
[359, 359, -w^3 + w^2 + 6*w - 1],
[359, 359, w^3 + w^2 - 6*w - 1],
[379, 379, -4*w^3 + 4*w^2 + 21*w - 28],
[379, 379, -4*w^2 + 3*w + 17],
[379, 379, -4*w^2 - 3*w + 17],
[379, 379, 2*w^3 + 5*w^2 - 9*w - 24],
[401, 401, -3*w^3 + 5*w^2 + 15*w - 27],
[401, 401, w^3 - 6*w^2 - 8*w + 30],
[409, 409, 2*w^3 + 4*w^2 - 9*w - 15],
[409, 409, -2*w^3 + 4*w^2 + 9*w - 15],
[419, 419, -w - 5],
[419, 419, w - 5],
[421, 421, w^3 - 7*w + 1],
[421, 421, -w^3 + 7*w + 1],
[431, 431, -w^3 + w^2 + 7*w - 5],
[431, 431, -w^3 + 5*w^2 + 6*w - 24],
[431, 431, 4*w^3 - 10*w^2 - 23*w + 52],
[431, 431, w^3 + w^2 - 7*w - 5],
[449, 449, w^3 + 3*w^2 - 6*w - 12],
[449, 449, w^3 - 3*w^2 - 6*w + 12],
[461, 461, w^3 + 3*w^2 - 4*w - 16],
[461, 461, w^3 - 3*w^2 - 4*w + 20],
[461, 461, -w^3 - 3*w^2 + 4*w + 20],
[461, 461, -w^3 + 3*w^2 + 4*w - 16],
[479, 479, 2*w^3 - 12*w + 3],
[479, 479, -2*w^3 + 12*w + 3],
[491, 491, -2*w^3 + 3*w^2 + 8*w - 14],
[491, 491, 2*w^3 + 3*w^2 - 8*w - 14],
[499, 499, 3*w^2 + 2*w - 8],
[499, 499, -3*w^2 + 2*w + 8],
[509, 509, 2*w^3 - 6*w^2 - 13*w + 33],
[509, 509, 3*w^3 - 4*w^2 - 15*w + 24],
[521, 521, 5*w^2 - w - 24],
[521, 521, -5*w^2 - w + 24],
[569, 569, -2*w^3 + 11*w - 7],
[569, 569, w^3 - 6*w^2 - 7*w + 26],
[571, 571, 2*w^3 + w^2 - 10*w + 1],
[571, 571, 3*w^3 - 8*w^2 - 17*w + 39],
[601, 601, -w^3 + 2*w^2 + 6*w - 6],
[601, 601, w^3 + 2*w^2 - 6*w - 6],
[619, 619, 2*w^3 - 3*w^2 - 9*w + 8],
[619, 619, 2*w^3 + 3*w^2 - 9*w - 8],
[631, 631, 2*w^2 - 3*w - 11],
[631, 631, 2*w^2 + 3*w - 11],
[641, 641, 3*w^3 + w^2 - 14*w - 9],
[641, 641, 3*w^3 - 2*w^2 - 15*w + 6],
[661, 661, 6*w^3 - 9*w^2 - 32*w + 51],
[661, 661, 3*w^3 - 2*w^2 - 15*w + 13],
[691, 691, -3*w^3 - 2*w^2 + 15*w + 8],
[691, 691, 3*w^3 - 2*w^2 - 15*w + 8],
[701, 701, 2*w^3 - 11*w + 4],
[701, 701, 3*w^3 + 4*w^2 - 13*w - 15],
[701, 701, -3*w^3 + 4*w^2 + 13*w - 15],
[701, 701, -2*w^3 + 11*w + 4],
[709, 709, w^3 - 2*w^2 - 5*w + 15],
[709, 709, -2*w^3 + 3*w^2 + 8*w - 6],
[709, 709, 2*w^3 + 3*w^2 - 8*w - 6],
[709, 709, -w^3 - 2*w^2 + 5*w + 15],
[719, 719, -3*w^3 - 3*w^2 + 14*w + 16],
[719, 719, 3*w^3 - 3*w^2 - 14*w + 16],
[739, 739, -w^3 + 4*w - 6],
[739, 739, w^3 - 4*w - 6],
[751, 751, 2*w^3 + w^2 - 11*w - 2],
[751, 751, 3*w^2 + 2*w - 17],
[751, 751, -3*w^2 + 2*w + 17],
[751, 751, -2*w^3 + w^2 + 11*w - 2],
[809, 809, w^3 - 6*w^2 - 3*w + 25],
[809, 809, w^3 + 3*w^2 - 3*w - 15],
[809, 809, w^3 - 3*w^2 - 3*w + 15],
[809, 809, -w^3 - 6*w^2 + 3*w + 25],
[829, 829, -3*w^3 - 4*w^2 + 13*w + 17],
[829, 829, 3*w^3 - 4*w^2 - 13*w + 17],
[841, 29, 5*w^2 - 21],
[859, 859, -4*w^2 - 2*w + 21],
[859, 859, -4*w^2 + 2*w + 21],
[881, 881, -w^3 - 5*w^2 + 6*w + 20],
[881, 881, -6*w^3 + 8*w^2 + 33*w - 49],
[881, 881, -3*w^3 + 6*w^2 + 17*w - 30],
[881, 881, w^3 - 5*w^2 - 6*w + 20],
[911, 911, -3*w^3 + 2*w^2 + 13*w - 13],
[911, 911, 3*w^3 - 15*w - 7],
[911, 911, 2*w^3 + 2*w^2 - 13*w - 8],
[911, 911, -4*w^3 + 10*w^2 + 24*w - 55],
[919, 919, w^3 - 5*w - 6],
[919, 919, -w^3 + 5*w - 6],
[941, 941, -2*w^3 + 7*w^2 + 10*w - 35],
[941, 941, 2*w^3 + 7*w^2 - 10*w - 35],
[961, 31, -2*w^2 + 15],
[961, 31, 5*w^2 - 23],
[991, 991, -w - 6],
[991, 991, 2*w^2 + 3*w - 7],
[991, 991, 2*w^2 - 3*w - 7],
[991, 991, w - 6]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 8*x^3 + 16*x^2 + 4*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/2*e^3 + 7/2*e^2 + 11/2*e - 1/2, e, 1/2*e^3 + 7/2*e^2 + 9/2*e - 7/2, 1/2*e^3 + 9/2*e^2 + 19/2*e - 3/2, -1/2*e^3 - 9/2*e^2 - 19/2*e - 5/2, 1, e^3 + 9*e^2 + 20*e + 3, -1/2*e^3 - 11/2*e^2 - 29/2*e - 5/2, -1/2*e^3 - 7/2*e^2 - 11/2*e + 1/2, -1/2*e^3 - 7/2*e^2 - 5/2*e + 13/2, e^3 + 7*e^2 + 8*e - 4, -2*e^3 - 15*e^2 - 28*e - 5, -2*e^3 - 13*e^2 - 16*e + 3, -1/2*e^3 - 3/2*e^2 + 3/2*e - 7/2, e^3 + 5*e^2 - e - 7, e^3 + 9*e^2 + 23*e + 9, -3*e^3 - 23*e^2 - 40*e - 2, -2*e^3 - 15*e^2 - 26*e - 4, -e^3 - 6*e^2 - 7*e - 3, -3/2*e^3 - 23/2*e^2 - 49/2*e - 13/2, -5/2*e^3 - 33/2*e^2 - 39/2*e + 17/2, 4*e^3 + 28*e^2 + 44*e + 8, 1/2*e^3 + 15/2*e^2 + 53/2*e + 29/2, -e^3 - 11*e^2 - 32*e - 7, 2*e^3 + 12*e^2 + 10*e - 1, 2*e^3 + 16*e^2 + 34*e + 15, 3*e^3 + 22*e^2 + 34*e - 7, -e^3 - 9*e^2 - 22*e - 14, -1/2*e^3 - 3/2*e^2 + 11/2*e - 3/2, 1/2*e^3 + 5/2*e^2 + 9/2*e + 5/2, -2*e^3 - 11*e^2 - 5*e + 10, 9/2*e^3 + 61/2*e^2 + 87/2*e - 13/2, 9/2*e^3 + 65/2*e^2 + 111/2*e + 3/2, -5/2*e^3 - 41/2*e^2 - 89/2*e - 21/2, 3*e^3 + 20*e^2 + 20*e - 17, -1/2*e^3 - 5/2*e^2 + 15/2*e + 31/2, -7/2*e^3 - 47/2*e^2 - 59/2*e + 5/2, -2*e^3 - 15*e^2 - 31*e - 16, 3/2*e^3 + 21/2*e^2 + 47/2*e + 37/2, 5*e^3 + 35*e^2 + 48*e - 6, 2*e^2 + 6*e + 2, -5*e^3 - 36*e^2 - 55*e - 4, -2*e^3 - 13*e^2 - 22*e - 17, -3*e^3 - 23*e^2 - 39*e + 7, -e + 11, -1/2*e^3 - 7/2*e^2 - 9/2*e + 29/2, -e^3 - 10*e^2 - 22*e - 10, 5/2*e^3 + 41/2*e^2 + 77/2*e - 21/2, -5/2*e^3 - 43/2*e^2 - 85/2*e - 13/2, 2*e^3 + 18*e^2 + 37*e - 6, 3/2*e^3 + 23/2*e^2 + 29/2*e - 43/2, -5/2*e^3 - 37/2*e^2 - 51/2*e - 3/2, -8*e^3 - 60*e^2 - 100*e - 12, -2*e^3 - 10*e^2 - 10*e - 22, -9/2*e^3 - 67/2*e^2 - 113/2*e - 13/2, -2*e^3 - 12*e^2 - 15*e - 8, 7/2*e^3 + 61/2*e^2 + 137/2*e + 9/2, 1/2*e^3 - 5/2*e^2 - 49/2*e - 45/2, -e^3 - 8*e^2 - 16*e - 15, -1/2*e^3 - 5/2*e^2 - 1/2*e - 21/2, 6*e^3 + 49*e^2 + 104*e + 16, -2*e^3 - 11*e^2 - 11*e - 1, -11/2*e^3 - 83/2*e^2 - 143/2*e - 1/2, 4*e^3 + 21*e^2 + 6*e - 26, -3/2*e^3 - 33/2*e^2 - 85/2*e - 31/2, 6*e^3 + 46*e^2 + 84*e + 20, 3*e^3 + 17*e^2 + 15*e + 9, 7/2*e^3 + 61/2*e^2 + 129/2*e - 5/2, 11/2*e^3 + 75/2*e^2 + 111/2*e - 1/2, 6*e^3 + 43*e^2 + 71*e + 4, 4*e^3 + 31*e^2 + 61*e - 1, -4*e^3 - 32*e^2 - 66*e - 19, -3*e^3 - 17*e^2 - 11*e + 6, 7/2*e^3 + 43/2*e^2 + 43/2*e - 43/2, -15/2*e^3 - 101/2*e^2 - 145/2*e - 13/2, -17/2*e^3 - 123/2*e^2 - 207/2*e - 31/2, e^3 + e^2 - 10*e + 19, 17/2*e^3 + 131/2*e^2 + 229/2*e + 29/2, 3*e^3 + 22*e^2 + 35*e + 2, e^3 + 6*e^2 + 9*e + 8, 7*e^3 + 58*e^2 + 124*e + 15, 7/2*e^3 + 31/2*e^2 - 17/2*e - 65/2, -3*e^3 - 21*e^2 - 39*e - 31, -11/2*e^3 - 71/2*e^2 - 71/2*e + 49/2, -2*e^3 - 17*e^2 - 47*e - 24, -6*e^3 - 42*e^2 - 60*e - 10, 3*e^3 + 21*e^2 + 30*e + 5, 3/2*e^3 + 21/2*e^2 + 39/2*e + 31/2, -5*e^3 - 40*e^2 - 76*e - 21, 6*e^3 + 44*e^2 + 72*e - 8, 3*e^3 + 19*e^2 + 27*e - 3, -1/2*e^3 + 3/2*e^2 + 31/2*e - 25/2, 1/2*e^3 - 7/2*e^2 - 67/2*e - 35/2, 2*e^3 + 21*e^2 + 61*e + 28, -2*e^2 - 4*e + 8, 4*e^3 + 30*e^2 + 48*e - 4, 3/2*e^3 + 35/2*e^2 + 81/2*e - 45/2, -15/2*e^3 - 119/2*e^2 - 213/2*e - 31/2, -6*e^3 - 53*e^2 - 119*e - 24, 1/2*e^3 + 29/2*e^2 + 117/2*e + 37/2, -7/2*e^3 - 47/2*e^2 - 65/2*e + 1/2, -7/2*e^3 - 51/2*e^2 - 89/2*e - 15/2, 3*e^3 + 19*e^2 + 23*e - 3, 4*e^3 + 30*e^2 + 54*e + 6, 5*e^3 + 36*e^2 + 55*e + 14, 2*e^3 + 13*e^2 + 22*e + 27, -7/2*e^3 - 43/2*e^2 - 43/2*e + 3/2, -4*e^3 - 31*e^2 - 61*e - 19, -5/2*e^3 - 35/2*e^2 - 51/2*e - 7/2, -3/2*e^3 - 21/2*e^2 - 37/2*e - 21/2, -11/2*e^3 - 65/2*e^2 - 53/2*e + 49/2, -13/2*e^3 - 103/2*e^2 - 211/2*e - 33/2, 11*e^3 + 78*e^2 + 116*e - 4, 11/2*e^3 + 75/2*e^2 + 131/2*e + 53/2, 8*e^3 + 67*e^2 + 135*e + 22, -3/2*e^3 - 43/2*e^2 - 127/2*e + 1/2, -1/2*e^3 - 21/2*e^2 - 99/2*e - 73/2, -3/2*e^3 - 7/2*e^2 + 55/2*e + 53/2, 17/2*e^3 + 121/2*e^2 + 199/2*e + 47/2, -3/2*e^3 - 3/2*e^2 + 67/2*e + 87/2, -7/2*e^3 - 67/2*e^2 - 177/2*e - 29/2, 17/2*e^3 + 117/2*e^2 + 175/2*e + 31/2, 4*e^3 + 35*e^2 + 83*e + 22, 4*e^3 + 28*e^2 + 32*e, -2*e^3 - 14*e^2 - 10*e + 42, 5/2*e^3 + 21/2*e^2 - 23/2*e - 47/2, -3/2*e^3 - 29/2*e^2 - 49/2*e + 57/2, 13/2*e^3 + 99/2*e^2 + 159/2*e + 9/2, -3/2*e^3 - 31/2*e^2 - 79/2*e + 27/2, 2*e^3 + 19*e^2 + 45*e + 29, -17/2*e^3 - 131/2*e^2 - 219/2*e + 15/2, -8*e^3 - 51*e^2 - 55*e + 34, -13/2*e^3 - 101/2*e^2 - 209/2*e - 33/2, 3/2*e^3 + 33/2*e^2 + 65/2*e - 29/2, -12*e^3 - 85*e^2 - 134*e - 26, 8*e^3 + 60*e^2 + 114*e + 17, 9*e^3 + 59*e^2 + 73*e - 22, -10*e^3 - 69*e^2 - 108*e - 32, -2*e^3 - e^2 + 37*e + 18, -23/2*e^3 - 187/2*e^2 - 371/2*e - 39/2, 1, 2*e^3 + 13*e^2 + 20*e - 9, 4*e^3 + 29*e^2 + 46*e - 15, -12*e^3 - 91*e^2 - 155*e + 1, 3*e^3 + 24*e^2 + 56*e + 22, 11/2*e^3 + 71/2*e^2 + 75/2*e - 39/2, -5/2*e^3 - 21/2*e^2 - 9/2*e - 19/2, -4*e^3 - 31*e^2 - 45*e + 30, -3*e^3 - 13*e^2 + 10*e + 14, -11/2*e^3 - 93/2*e^2 - 207/2*e - 65/2, 9/2*e^3 + 69/2*e^2 + 101/2*e - 11/2, 9/2*e^3 + 83/2*e^2 + 205/2*e + 39/2, e^3 - 3*e^2 - 42*e - 36, 17/2*e^3 + 107/2*e^2 + 131/2*e + 17/2, 25/2*e^3 + 187/2*e^2 + 331/2*e + 57/2, 21/2*e^3 + 147/2*e^2 + 231/2*e - 33/2, 4*e^3 + 28*e^2 + 44*e - 35, -21/2*e^3 - 161/2*e^2 - 273/2*e - 1/2, 13/2*e^3 + 85/2*e^2 + 105/2*e - 31/2, 6*e^3 + 45*e^2 + 85*e + 12, 7*e^2 + 21*e - 18];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, -w}>] := -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;