/*
  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![29, 11, -10, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, w^2 - 2*w - 5}>;

primesArray := [
[11, 11, -w - 1],
[11, 11, w^2 - 5],
[11, 11, -w^2 + 2*w + 4],
[11, 11, w - 2],
[16, 2, 2],
[19, 19, w^2 - 2*w - 5],
[19, 19, -w^2 + 6],
[25, 5, -2*w^2 + 2*w + 11],
[29, 29, w],
[29, 29, 2*w^2 - w - 10],
[29, 29, -2*w^2 + 3*w + 9],
[29, 29, w - 1],
[31, 31, w^3 - 6*w - 6],
[31, 31, -w^3 + 3*w^2 + 3*w - 11],
[41, 41, w^3 - 4*w^2 - 2*w + 16],
[41, 41, w^3 - 5*w^2 - 2*w + 24],
[59, 59, w^3 - w^2 - 5*w - 2],
[59, 59, 2*w^2 - w - 13],
[61, 61, w^3 - w^2 - 6*w + 3],
[61, 61, -w^3 + 2*w^2 + 5*w - 3],
[71, 71, 2*w^2 - w - 16],
[71, 71, -w^3 + 3*w^2 + 4*w - 10],
[71, 71, -w^3 + 7*w + 4],
[71, 71, w^3 - 5*w^2 - 2*w + 23],
[79, 79, -w^3 + 4*w^2 + 3*w - 16],
[79, 79, w^3 + w^2 - 8*w - 10],
[81, 3, -3],
[131, 131, 4*w^2 - 5*w - 24],
[131, 131, 4*w^2 - 3*w - 25],
[139, 139, w^2 + w - 9],
[139, 139, -w^3 + w^2 + 6*w - 5],
[149, 149, w^3 - w^2 - 4*w + 2],
[149, 149, w^3 - 2*w^2 - 3*w + 2],
[151, 151, -w^3 + 7*w + 3],
[151, 151, -w^3 + 3*w^2 + 4*w - 9],
[179, 179, -w^3 - 2*w^2 + 10*w + 16],
[179, 179, -w^3 + 3*w^2 + 3*w - 6],
[179, 179, w^3 - 6*w - 1],
[179, 179, w^3 - 5*w^2 - 3*w + 23],
[191, 191, w^3 - 6*w^2 - w + 27],
[191, 191, w^3 + 3*w^2 - 10*w - 21],
[199, 199, -w^3 + 2*w^2 + 6*w - 4],
[199, 199, -4*w^2 + 3*w + 22],
[199, 199, 4*w^2 - 5*w - 21],
[199, 199, 3*w^2 - 4*w - 12],
[239, 239, -w^3 + w^2 + 4*w - 6],
[239, 239, 2*w^3 - 5*w^2 - 9*w + 17],
[239, 239, 2*w^3 - w^2 - 13*w - 5],
[239, 239, w^3 - 8*w^2 + 2*w + 41],
[241, 241, -w^3 + 5*w^2 + 4*w - 25],
[241, 241, w^3 - 6*w - 8],
[241, 241, -w^3 + 3*w^2 + 3*w - 13],
[241, 241, w^3 - 5*w^2 + w + 19],
[269, 269, w^2 - 2*w - 10],
[269, 269, w^2 - 11],
[271, 271, w^3 + 2*w^2 - 8*w - 18],
[271, 271, 2*w^3 - 13*w - 10],
[271, 271, 2*w^3 + w^2 - 15*w - 16],
[271, 271, w^3 - 5*w^2 - w + 23],
[289, 17, 2*w^2 - 11],
[289, 17, -2*w^2 + 4*w + 9],
[311, 311, w^3 - 4*w^2 - 3*w + 10],
[311, 311, w^3 - 7*w^2 - w + 34],
[331, 331, w^3 - w^2 - 7*w + 4],
[331, 331, w^3 - 2*w^2 - 6*w + 3],
[349, 349, w^3 + w^2 - 6*w - 12],
[349, 349, -w^3 + 4*w^2 + w - 16],
[361, 19, 4*w^2 - 4*w - 21],
[379, 379, w^3 - 3*w^2 - 3*w + 15],
[379, 379, 2*w^3 - 2*w^2 - 12*w + 1],
[379, 379, -2*w^3 + 4*w^2 + 10*w - 11],
[379, 379, w^3 - 6*w - 10],
[401, 401, -2*w^3 + 7*w^2 + 8*w - 30],
[401, 401, -w^3 + 2*w^2 + 6*w - 2],
[419, 419, 5*w^2 - 4*w - 26],
[419, 419, 5*w^2 - 6*w - 25],
[421, 421, 3*w^2 - w - 20],
[421, 421, -2*w^3 + 2*w^2 + 10*w + 3],
[431, 431, w^2 - 3*w - 5],
[431, 431, w^2 + w - 7],
[449, 449, 2*w^3 - 13*w - 7],
[449, 449, -2*w^3 + 6*w^2 + 7*w - 18],
[461, 461, w^3 - w^2 - 8*w + 3],
[461, 461, -w^3 + 2*w^2 + 7*w - 5],
[491, 491, w^3 - 3*w^2 - 6*w + 15],
[491, 491, w^3 - 9*w - 7],
[499, 499, 4*w^2 - 6*w - 23],
[499, 499, -4*w^2 + 2*w + 25],
[509, 509, 2*w^3 - 3*w^2 - 11*w + 6],
[521, 521, -w^3 + 3*w^2 + 2*w - 12],
[521, 521, -w^3 + 4*w^2 + 2*w - 19],
[529, 23, w^3 - 7*w^2 - w + 35],
[529, 23, -w^3 + 3*w^2 + 4*w - 5],
[541, 541, 2*w^3 - w^2 - 12*w - 5],
[541, 541, w^3 - 2*w^2 - 7*w + 9],
[569, 569, w^3 - 8*w - 2],
[569, 569, -w^3 + 3*w^2 + 5*w - 9],
[599, 599, -w^3 - 4*w^2 + 10*w + 25],
[599, 599, -w^3 + 7*w^2 - 36],
[601, 601, 5*w^2 - 6*w - 26],
[601, 601, -2*w^3 + 8*w^2 + 6*w - 33],
[601, 601, -2*w^3 - 2*w^2 + 16*w + 21],
[601, 601, 5*w^2 - 4*w - 27],
[619, 619, w^3 + 4*w^2 - 11*w - 26],
[619, 619, w^3 - 3*w^2 - 5*w + 18],
[641, 641, -w^3 + 3*w^2 + 6*w - 14],
[641, 641, -w^3 + 9*w + 6],
[659, 659, -w^3 - 4*w^2 + 10*w + 28],
[659, 659, -w^3 + 6*w^2 - w - 27],
[659, 659, -w^3 - 3*w^2 + 8*w + 23],
[659, 659, -w^3 + 7*w^2 - w - 33],
[691, 691, w^3 - w^2 - 8*w + 2],
[691, 691, 2*w^3 - 2*w^2 - 11*w + 1],
[701, 701, -w^3 - w^2 + 7*w + 15],
[701, 701, -w^3 + 4*w^2 + 2*w - 20],
[709, 709, -2*w^3 - w^2 + 15*w + 15],
[709, 709, 2*w^3 - 7*w^2 - 7*w + 27],
[719, 719, -w^3 - 4*w^2 + 13*w + 26],
[719, 719, w^3 - 7*w^2 - 2*w + 34],
[739, 739, 2*w^3 + 2*w^2 - 15*w - 24],
[739, 739, -2*w^3 + 8*w^2 + 5*w - 35],
[751, 751, -w^3 - 3*w^2 + 9*w + 25],
[751, 751, -w^3 + 6*w^2 - 30],
[761, 761, 3*w^3 - 2*w^2 - 16*w - 8],
[761, 761, -3*w^3 + 7*w^2 + 11*w - 23],
[769, 769, -2*w^3 + 4*w^2 + 11*w - 10],
[769, 769, w^3 + 2*w^2 - 9*w - 12],
[809, 809, -3*w^3 + 3*w^2 + 16*w + 4],
[809, 809, w^3 + 5*w^2 - 12*w - 30],
[811, 811, -w^3 - 4*w^2 + 10*w + 27],
[811, 811, -2*w^3 + 5*w^2 + 9*w - 14],
[811, 811, -2*w^3 + w^2 + 13*w + 2],
[811, 811, -w^3 + 7*w^2 - w - 32],
[821, 821, 2*w^3 - 8*w^2 - 7*w + 35],
[821, 821, -2*w^3 - 2*w^2 + 17*w + 22],
[829, 829, -3*w^3 + 20*w + 13],
[829, 829, -w^3 + 7*w^2 - w - 36],
[829, 829, w^3 + 4*w^2 - 10*w - 31],
[829, 829, 2*w^3 - 3*w^2 - 11*w + 14],
[859, 859, -w^3 + 7*w^2 - 3*w - 31],
[859, 859, 7*w^2 - 9*w - 39],
[859, 859, 7*w^2 - 5*w - 41],
[859, 859, 3*w^3 - 4*w^2 - 17*w + 2],
[929, 929, w^3 + 5*w^2 - 14*w - 31],
[929, 929, -2*w^3 + 4*w^2 + 10*w - 5],
[929, 929, w^3 - 9*w^2 + 4*w + 44],
[929, 929, -w^3 + 8*w^2 + w - 39],
[941, 941, -w^3 + 7*w^2 - 34],
[941, 941, w^3 + 4*w^2 - 11*w - 28],
[961, 31, -5*w^2 + 5*w + 28],
[971, 971, -3*w^3 + w^2 + 19*w + 14],
[971, 971, -3*w^3 + 8*w^2 + 12*w - 31],
[991, 991, -w^3 - 5*w^2 + 11*w + 35],
[991, 991, w^3 - 8*w^2 + 2*w + 40]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 5*x^3 - 14*x^2 - 68*x + 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/4*e^2 + 1/4*e - 11/2, -3/16*e^3 - 9/16*e^2 + 11/4*e + 17/4, e, 1/8*e^3 + 3/8*e^2 - 3/2*e - 7/2, -1/16*e^3 + 1/16*e^2 + 1/2*e - 15/4, 1, -1/4*e^3 - 1/2*e^2 + 13/4*e + 7/2, 1/16*e^3 - 1/16*e^2 - 3/2*e + 3/4, 1/4*e^2 + 5/4*e - 5/2, 1/8*e^3 + 1/8*e^2 - 11/4*e - 1, -e - 3, 1/16*e^3 - 5/16*e^2 - 7/4*e - 3/4, -1/8*e^3 + 1/8*e^2 + 4*e + 1/2, 2, -1/16*e^3 + 9/16*e^2 - 43/4, 1/4*e^3 - 1/4*e^2 - 5*e + 4, 9/16*e^3 + 27/16*e^2 - 33/4*e - 51/4, -3/16*e^3 - 17/16*e^2 + 5/4*e + 29/4, 7/16*e^3 + 5/16*e^2 - 31/4*e + 15/4, 1/16*e^3 - 9/16*e^2 - e + 27/4, 1/16*e^3 - 5/16*e^2 - 15/4*e + 33/4, 3/8*e^3 + 5/8*e^2 - 6*e - 3/2, 9/16*e^3 + 19/16*e^2 - 31/4*e - 23/4, -3/16*e^3 - 1/16*e^2 + 21/4*e - 3/4, 7/16*e^3 + 5/16*e^2 - 35/4*e - 21/4, -1/8*e^3 + 5/8*e^2 + 7/2*e - 29/2, -1/8*e^3 - 3/8*e^2 + 5/2*e - 1/2, -3/4*e^3 - 9/4*e^2 + 11*e + 17, 1/2*e^2 + 5/2*e - 9, -7/8*e^3 - 5/8*e^2 + 33/2*e + 13/2, -3/2*e^2 - 7/2*e + 15, -1/2*e^3 - 7/4*e^2 + 19/4*e + 33/2, 7/16*e^3 + 9/16*e^2 - 15/2*e - 47/4, 1/2*e^3 + 3/2*e^2 - 8*e - 8, -9/16*e^3 - 19/16*e^2 + 27/4*e - 1/4, -13/16*e^3 - 23/16*e^2 + 53/4*e + 31/4, 7/8*e^3 + 21/8*e^2 - 25/2*e - 49/2, 3/4*e^3 + 2*e^2 - 45/4*e - 11/2, -3/8*e^3 - 5/8*e^2 + 7*e + 27/2, -1/4*e^3 - e^2 + 19/4*e - 7/2, -1/16*e^3 - 3/16*e^2 + 9/4*e + 43/4, e^2 + e - 10, -3/4*e^3 - 7/4*e^2 + 25/2*e + 10, -13/16*e^3 - 31/16*e^2 + 55/4*e + 91/4, -3/8*e^3 - 7/8*e^2 + 15/4*e + 7, -7/16*e^3 - 21/16*e^2 + 27/4*e + 21/4, -3/8*e^3 - 13/8*e^2 + 9*e + 39/2, 1/4*e^3 + 3/4*e^2 - 7*e - 13, 1/2*e^3 - 3/4*e^2 - 29/4*e + 55/2, 1/4*e^3 - 7/4*e^2 - 17/2*e + 21, 7/16*e^3 - 3/16*e^2 - 41/4*e - 13/4, -1/2*e^3 + 3/4*e^2 + 37/4*e - 41/2, e^3 + 9/4*e^2 - 67/4*e - 57/2, -7/16*e^3 - 5/16*e^2 + 39/4*e + 17/4, -9/16*e^3 - 31/16*e^2 + 3*e + 61/4, 1/8*e^3 - 11/8*e^2 - 13/4*e + 27, 5/4*e^3 + 2*e^2 - 79/4*e - 45/2, -1/4*e^3 - e^2 + 11/4*e - 3/2, 1/4*e^3 - 5/4*e^2 - 5*e + 29, -1/8*e^3 + 15/8*e^2 + 31/4*e - 27, -1/16*e^3 + 13/16*e^2 + 1/4*e - 105/4, 3/8*e^3 + 3/8*e^2 - 33/4*e - 6, -1/4*e^3 + 1/4*e^2 + 5*e - 15, -1/8*e^3 - 19/8*e^2 - 1/2*e + 43/2, -1/16*e^3 + 21/16*e^2 + 11/4*e - 25/4, 5/8*e^3 + 1/8*e^2 - 49/4*e - 10, 7/16*e^3 + 29/16*e^2 - 17/4*e - 37/4, 1/8*e^3 + 13/8*e^2 + 3/4*e - 20, 7/4*e^2 - 5/4*e - 73/2, -1/4*e^3 - 2*e^2 - 1/4*e + 21/2, -1/8*e^3 + 7/8*e^2 - 1/4*e - 28, 3/16*e^3 + 17/16*e^2 + 7/4*e - 5/4, 11/16*e^3 + 49/16*e^2 - 31/4*e - 125/4, 3/16*e^3 - 3/16*e^2 - 13/2*e + 101/4, -5/16*e^3 + 17/16*e^2 + 21/4*e - 89/4, -9/8*e^3 + 1/8*e^2 + 22*e - 11/2, 1/8*e^3 + 9/8*e^2 - 7/4*e - 21, 13/16*e^3 + 31/16*e^2 - 51/4*e - 71/4, 9/8*e^3 + 3/8*e^2 - 45/2*e - 15/2, -1/16*e^3 + 5/16*e^2 - 5/4*e - 57/4, 3/8*e^3 + 13/8*e^2 - 2*e - 73/2, -3/4*e^3 - 3/4*e^2 + 33/2*e - 3, -11/16*e^3 + 27/16*e^2 + 21/2*e - 165/4, -17/16*e^3 - 15/16*e^2 + 47/2*e + 41/4, -5/16*e^3 + 9/16*e^2 + 11/4*e - 117/4, 5/8*e^3 + 15/8*e^2 - 5/2*e - 35/2, -1/4*e^3 - 11/4*e^2 + 7*e + 33, 11/8*e^3 + 11/8*e^2 - 105/4*e - 14, 23/16*e^3 + 25/16*e^2 - 49/2*e - 75/4, 1/16*e^3 + 27/16*e^2 + 17/4*e - 107/4, 7/16*e^3 + 45/16*e^2 - 5/4*e - 93/4, 3/8*e^3 + 35/8*e^2 - 5/4*e - 37, -3/8*e^3 + 1/8*e^2 + 47/4*e + 16, -3/16*e^3 - 5/16*e^2 + 6*e - 49/4, -5/4*e^3 - 5/2*e^2 + 61/4*e + 21/2, -19/16*e^3 - 29/16*e^2 + 41/2*e + 103/4, 2*e^2 + e - 44, 19/16*e^3 + 49/16*e^2 - 93/4*e - 101/4, 5/16*e^3 + 23/16*e^2 - 15/4*e - 131/4, -1/2*e^3 + 2*e^2 + 15/2*e - 46, -11/16*e^3 - 45/16*e^2 + 10*e + 87/4, -5/16*e^3 - 7/16*e^2 + 19/4*e + 7/4, -1/4*e^3 - 3/4*e^2 + 11*e + 24, -1/8*e^3 + 5/8*e^2 + 15/2*e - 29/2, 27/16*e^3 + 65/16*e^2 - 91/4*e - 105/4, 15/16*e^3 + 9/16*e^2 - 21*e - 71/4, -11/16*e^3 + 3/16*e^2 + 11*e - 117/4, 5/8*e^3 + 7/8*e^2 - 35/2*e - 35/2, 2*e^3 + 2*e^2 - 39*e - 26, -13/8*e^3 - 23/8*e^2 + 53/2*e + 67/2, -21/16*e^3 - 55/16*e^2 + 87/4*e + 179/4, 7/8*e^3 + 21/8*e^2 - 23/2*e - 53/2, -1/8*e^3 + 21/8*e^2 + 19/2*e - 57/2, 11/8*e^3 + 41/8*e^2 - 35/2*e - 101/2, -3/2*e^3 - 9/2*e^2 + 22*e + 36, 3/16*e^3 - 11/16*e^2 - 2*e + 109/4, 15/16*e^3 + 81/16*e^2 - 27/2*e - 171/4, -3/2*e^3 - 5/2*e^2 + 23*e + 18, 1/4*e^3 - 1/4*e^2 - 12*e - 1, -3/8*e^3 + 7/8*e^2 + 21/2*e - 7/2, 9/8*e^3 + 11/8*e^2 - 31/2*e - 11/2, 7/8*e^3 + 9/8*e^2 - 15*e - 83/2, e^3 + e^2 - 11*e + 4, -13/8*e^3 - 31/8*e^2 + 45/2*e + 83/2, 1/16*e^3 + 7/16*e^2 + 6*e + 47/4, -e^3 - 3*e^2 + 20*e + 31, 3/8*e^3 + 15/8*e^2 - 7/4*e - 12, -5/8*e^3 + 7/8*e^2 + 29/4*e - 48, 17/16*e^3 + 51/16*e^2 - 49/4*e - 135/4, 23/16*e^3 + 29/16*e^2 - 97/4*e - 25/4, e^3 - 3/4*e^2 - 71/4*e + 25/2, -1/4*e^3 - 11/4*e^2 + 9*e + 33, -9/8*e^3 - 13/8*e^2 + 65/4*e - 9, -5/16*e^3 - 27/16*e^2 + 15/2*e + 29/4, -25/16*e^3 + 1/16*e^2 + 26*e - 139/4, -1/2*e^3 - 7/4*e^2 + 51/4*e + 69/2, -21/16*e^3 - 19/16*e^2 + 26*e + 5/4, 31/16*e^3 + 89/16*e^2 - 32*e - 175/4, -5/8*e^3 - 19/8*e^2 + 9*e + 75/2, -9/8*e^3 - 7/8*e^2 + 18*e - 7/2, 3/2*e^3 + 9/4*e^2 - 121/4*e - 49/2, 3/8*e^3 - 7/8*e^2 - 11/2*e + 43/2, -3/16*e^3 - 9/16*e^2 + 31/4*e - 71/4, -3/8*e^3 - 11/8*e^2 + 37/4*e + 25, 5/16*e^3 - 29/16*e^2 + 227/4, -1/2*e^3 + 7/2*e^2 + 19*e - 50, 1/16*e^3 - 13/16*e^2 + 11/4*e + 73/4, -13/16*e^3 + 29/16*e^2 + 41/2*e - 91/4, 5/16*e^3 - 13/16*e^2 - 12*e + 95/4, 3/4*e^3 + 3*e^2 - 9/4*e - 81/2, -13/8*e^3 - 27/8*e^2 + 17*e + 29/2, -3/2*e^3 - 3/4*e^2 + 135/4*e - 1/2, -7/4*e^3 - 1/2*e^2 + 127/4*e - 7/2, -e^3 - 3/4*e^2 + 57/4*e - 11/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, w^2 - 2*w - 5}>] := -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;