/*
  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![1, 5, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {13, 13, -w^2 + 3}>;

primesArray := [
[5, 5, w^3 - 4*w],
[7, 7, w + 1],
[7, 7, -w^3 + 4*w - 1],
[13, 13, -w^2 + 3],
[16, 2, 2],
[17, 17, w^3 + w^2 - 4*w - 2],
[17, 17, -w^3 + 5*w - 2],
[19, 19, -w^2 - w + 4],
[19, 19, -w^2 - w + 1],
[29, 29, 2*w^3 - w^2 - 9*w + 5],
[41, 41, -w^3 + w^2 + 5*w - 3],
[43, 43, w^3 - w^2 - 4*w + 2],
[43, 43, w^3 - 6*w],
[47, 47, -w^3 - w^2 + 5*w],
[49, 7, w^2 + 2*w - 2],
[59, 59, 2*w^3 - 8*w + 3],
[67, 67, w^2 - w - 4],
[79, 79, w^3 - w^2 - 4*w + 1],
[81, 3, -3],
[97, 97, w^3 + w^2 - 5*w + 1],
[107, 107, -2*w^3 + 3*w^2 + 10*w - 12],
[109, 109, 2*w^3 - w^2 - 9*w + 3],
[125, 5, -3*w^3 - w^2 + 12*w + 1],
[127, 127, -2*w^3 + w^2 + 10*w - 8],
[137, 137, -w^3 + w^2 + 3*w - 5],
[139, 139, w^3 - w^2 - 5*w + 1],
[149, 149, 2*w^2 - w - 6],
[149, 149, 2*w^3 + w^2 - 9*w],
[163, 163, w^3 - 5*w - 3],
[163, 163, 2*w^3 - 7*w],
[173, 173, -2*w^3 + 11*w - 3],
[173, 173, -w^3 + w^2 + 4*w - 8],
[179, 179, 2*w^3 - w^2 - 8*w + 5],
[181, 181, 2*w^3 - w^2 - 8*w],
[181, 181, 2*w^3 - 9*w - 3],
[191, 191, 2*w^2 - 5],
[191, 191, -2*w^2 - w + 10],
[193, 193, w^3 + w^2 - 3*w - 4],
[193, 193, -w^3 + 2*w^2 + 3*w - 3],
[197, 197, -2*w^3 + 9*w - 4],
[197, 197, w^2 - 7],
[199, 199, 3*w^3 - 13*w + 1],
[199, 199, w^3 + 2*w^2 - 5*w - 7],
[227, 227, -3*w^3 + w^2 + 13*w - 4],
[227, 227, -w^3 + 2*w^2 + 7*w - 6],
[241, 241, -2*w^3 + 7*w - 1],
[257, 257, -2*w^3 + w^2 + 7*w - 5],
[257, 257, 2*w^3 - w^2 - 7*w + 2],
[263, 263, -w^3 + 7*w - 3],
[269, 269, -2*w^3 + 2*w^2 + 11*w - 8],
[271, 271, w^2 + 3*w - 2],
[271, 271, w^2 + 2*w - 7],
[277, 277, -2*w^3 + 11*w - 2],
[281, 281, 2*w^3 - 11*w + 1],
[283, 283, 2*w^3 - 9*w + 5],
[283, 283, -2*w^3 + w^2 + 11*w - 7],
[289, 17, -w^3 + 2*w^2 + 4*w - 4],
[293, 293, -2*w^3 - w^2 + 7*w + 3],
[307, 307, -3*w^3 + 2*w^2 + 13*w - 9],
[311, 311, -w^3 + w^2 + 3*w - 7],
[313, 313, -w^3 + 2*w^2 + 3*w - 2],
[313, 313, w^2 - w - 8],
[317, 317, -2*w^3 - w^2 + 9*w - 1],
[331, 331, w^3 + 2*w^2 - 5*w - 3],
[337, 337, w^3 + 2*w^2 - 6*w - 4],
[337, 337, w^3 - 2*w^2 - 6*w + 5],
[347, 347, 2*w^3 + 2*w^2 - 9*w - 3],
[347, 347, -2*w^3 + w^2 + 7*w - 4],
[349, 349, w^2 - 2*w - 4],
[353, 353, -w^3 + 2*w^2 + 2*w - 6],
[359, 359, -w^3 + w^2 + 7*w - 6],
[359, 359, -w^2 - w - 2],
[361, 19, 2*w^3 - w^2 - 11*w],
[367, 367, -2*w^3 + 2*w^2 + 8*w - 5],
[373, 373, w^3 + w^2 - 2*w - 4],
[379, 379, 2*w^3 - w^2 - 11*w + 4],
[389, 389, 3*w^2 + w - 11],
[397, 397, 3*w^3 - 13*w + 5],
[401, 401, w - 5],
[409, 409, 3*w^3 - 12*w + 4],
[409, 409, 3*w^3 + 2*w^2 - 14*w - 4],
[421, 421, 2*w^3 - w^2 - 10*w + 1],
[421, 421, w^3 + 2*w^2 - 5*w - 4],
[439, 439, -2*w^3 + 10*w - 5],
[443, 443, -2*w^3 + 12*w - 5],
[443, 443, -4*w^3 + w^2 + 18*w - 9],
[443, 443, -2*w^3 + w^2 + 7*w - 9],
[443, 443, 3*w^3 + 2*w^2 - 14*w - 5],
[449, 449, w^3 + 2*w^2 - 2*w - 6],
[449, 449, -2*w^3 + w^2 + 11*w - 5],
[461, 461, 2*w^3 + 2*w^2 - 9*w - 4],
[467, 467, -w^3 + w^2 + 7*w - 5],
[487, 487, 2*w^2 + 4*w - 7],
[491, 491, -w^3 + w^2 + 7*w - 4],
[503, 503, 2*w^3 + w^2 - 9*w + 2],
[521, 521, w^2 - 3*w - 3],
[523, 523, -w^3 + 2*w^2 + 5*w - 4],
[523, 523, -w^3 + 8*w - 5],
[563, 563, w^3 + 3*w^2 - w - 8],
[563, 563, w^3 - 2*w^2 - 8*w + 4],
[563, 563, -w^3 + w^2 + 2*w - 5],
[563, 563, -3*w^3 - w^2 + 10*w - 2],
[577, 577, 3*w^3 - 16*w],
[587, 587, w^3 + 2*w^2 - 6],
[587, 587, -w^3 + 4*w - 6],
[601, 601, -w - 5],
[607, 607, w^3 - 2*w^2 - 4*w + 1],
[613, 613, 4*w^3 - 17*w],
[617, 617, -3*w^3 + w^2 + 14*w - 2],
[617, 617, -4*w^3 + 17*w - 4],
[619, 619, -3*w^3 + 12*w - 2],
[643, 643, 3*w^3 - w^2 - 12*w + 3],
[643, 643, -2*w^3 - w^2 + 11*w - 2],
[647, 647, 2*w^3 + w^2 - 9*w + 3],
[653, 653, w^3 + w^2 - 7*w - 4],
[661, 661, -w^3 + 4*w^2 + 6*w - 15],
[661, 661, 3*w^3 - 2*w^2 - 12*w + 10],
[677, 677, -3*w^2 - 2*w + 12],
[691, 691, -2*w^3 + 3*w^2 + 11*w - 10],
[691, 691, -3*w^3 + 16*w - 3],
[701, 701, 3*w^3 - 13*w + 6],
[719, 719, -3*w^2 + w + 10],
[727, 727, w^3 - 3*w^2 - 4*w + 6],
[743, 743, -w^3 + 2*w^2 + 5*w - 3],
[761, 761, -w^3 + 3*w^2 + 3*w - 12],
[761, 761, 3*w^3 - 2*w^2 - 14*w + 7],
[769, 769, -3*w^3 + 3*w^2 + 16*w - 13],
[769, 769, w^3 + 2*w^2 - 7*w - 6],
[773, 773, w^3 - 8*w + 4],
[773, 773, 4*w^2 + w - 16],
[787, 787, w^2 + 4*w - 2],
[797, 797, -2*w^3 + 3*w^2 + 10*w - 10],
[797, 797, -w^3 - w^2 + 4*w - 4],
[797, 797, w^3 + 2*w^2 - 3*w - 9],
[797, 797, -3*w^3 + 3*w^2 + 11*w - 10],
[821, 821, w^2 + w - 9],
[827, 827, w^3 - 8*w + 1],
[827, 827, -2*w^3 + w^2 + 9*w - 11],
[827, 827, 3*w^3 - 14*w + 5],
[827, 827, 2*w^3 + w^2 - 7*w - 5],
[829, 829, -2*w^3 - 2*w^2 + 9*w - 1],
[839, 839, -3*w^3 + 15*w - 5],
[853, 853, -4*w^3 + 15*w - 8],
[853, 853, -w^3 + 2*w^2 + 6*w - 3],
[859, 859, -w^3 + w^2 + w - 4],
[859, 859, w^3 + w^2 - w - 5],
[881, 881, -w^3 + w^2 + 2*w - 6],
[881, 881, w^3 + w^2 - 2*w - 6],
[883, 883, w^3 + 3*w^2 - 3*w - 6],
[887, 887, w^3 + 3*w^2 - 3*w - 8],
[919, 919, 2*w^3 - w^2 - 13*w],
[929, 929, -w^3 - 3*w^2 + 2*w + 10],
[929, 929, w^3 + w^2 - 5*w - 8],
[941, 941, w^3 + 2*w^2 - 3*w - 10],
[953, 953, -w^3 + 5*w - 7],
[953, 953, -2*w^3 + 3*w^2 + 9*w - 9],
[953, 953, -2*w^3 - w^2 + 10*w - 3],
[953, 953, -3*w^3 + 11*w - 1],
[961, 31, -w^3 + 3*w^2 + 2*w - 10],
[961, 31, -3*w^2 + 2*w + 7],
[967, 967, w^3 + 3*w^2 - 3*w - 7],
[967, 967, 4*w^3 + w^2 - 20*w + 1],
[983, 983, 2*w^3 - w^2 - 12*w + 8],
[983, 983, 2*w^3 + w^2 - 6*w - 4],
[991, 991, 2*w^3 - 12*w + 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 7*x^3 + 13*x^2 - x - 13;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -2*e^3 - 9*e^2 - 5*e + 8, e^2 + 4*e + 1, 1, e^3 + 5*e^2 + e - 11, e^3 + 3*e^2 - 2*e - 6, e + 3, -3*e^3 - 15*e^2 - 11*e + 14, e^3 + 4*e^2 + e - 5, 3*e^3 + 13*e^2 + 4*e - 22, -e^3 - 4*e^2 - 5*e - 4, -3*e^3 - 13*e^2 - 5*e + 12, -e^3 - 6*e^2 - 8*e + 3, e^3 + 8*e^2 + 10*e - 13, 4*e^3 + 16*e^2 + 3*e - 22, 2*e^3 + 9*e^2 + e - 16, 7*e^3 + 30*e^2 + 10*e - 38, 5*e^3 + 23*e^2 + 11*e - 28, -5*e^3 - 25*e^2 - 14*e + 35, -e^3 - 3*e^2 + 5*e + 6, e^3 + 8*e^2 + 14*e - 5, -e^3 - 5*e^2 - 2*e + 17, 5*e^3 + 28*e^2 + 21*e - 40, -4*e^3 - 23*e^2 - 16*e + 38, 3*e^3 + 11*e^2 + 3*e - 15, -e^3 - 9*e^2 - 11*e + 21, -4*e^3 - 23*e^2 - 22*e + 18, e^3 + 2*e^2 - 2*e + 2, 5*e^3 + 28*e^2 + 25*e - 34, 2*e^3 + 12*e^2 + 13*e - 23, 6*e^3 + 26*e^2 + 10*e - 29, -3*e^3 - 6*e^2 + 13*e + 9, -e^3 + e^2 + 5*e - 24, 7*e^3 + 31*e^2 + 10*e - 41, e^3 + 2*e^2 + 5*e + 13, -2*e^3 - 10*e^2 - 8*e + 3, 7*e^3 + 31*e^2 + 17*e - 38, -9*e^3 - 33*e^2 + 3*e + 51, -e^3 - 2*e^2 - 9, -8*e^3 - 38*e^2 - 20*e + 61, 11*e^3 + 51*e^2 + 32*e - 50, e^2 + 9*e, e^3 + 11*e^2 + 24*e - 5, 6*e^3 + 30*e^2 + 24*e - 43, -7*e^3 - 32*e^2 - 22*e + 16, 2*e^3 + 3*e^2 - 5*e + 3, -14*e^3 - 62*e^2 - 32*e + 67, -3*e^3 - 16*e^2 - 10*e + 30, 5*e^3 + 13*e^2 - 6*e + 7, -14*e^3 - 63*e^2 - 27*e + 87, 2*e^3 + 8*e^2 + 6*e - 3, 7*e^3 + 38*e^2 + 31*e - 49, 5*e^3 + 20*e^2 - e - 33, -3*e^3 - 25*e^2 - 38*e + 19, -11*e^3 - 51*e^2 - 27*e + 51, 3*e^3 + 23*e^2 + 33*e - 20, 3*e^3 + 9*e^2 - 9*e - 20, -3*e^3 - 17*e^2 - 20*e + 5, -15*e^3 - 69*e^2 - 40*e + 75, 2*e^3 + 12*e^2 + 3*e - 42, -9*e^3 - 38*e^2 - 19*e + 32, -6*e^3 - 24*e^2 - 2*e + 45, 3*e^2 + 12*e + 9, -6*e^3 - 33*e^2 - 27*e + 42, -4*e^3 - 12*e^2 + 7*e + 14, 8*e^2 + 18*e - 7, 17*e^3 + 77*e^2 + 30*e - 108, 10*e + 13, -9*e^3 - 50*e^2 - 44*e + 54, -3*e^3 - 15*e^2 - 4*e + 39, -11*e^3 - 51*e^2 - 20*e + 71, e^3 + 3*e^2 - 13*e - 23, 5*e^3 + 28*e^2 + 23*e - 40, -14*e^3 - 64*e^2 - 28*e + 88, -e^3 + 6*e^2 + 21*e - 23, -8*e^3 - 30*e^2 - 2*e + 49, -5*e^3 - 29*e^2 - 15*e + 65, 14*e^3 + 64*e^2 + 28*e - 86, 12*e^3 + 53*e^2 + 21*e - 76, 7*e^3 + 33*e^2 + 22*e - 24, -e^3 - 6*e^2 - 5*e - 6, -3*e^3 - 7*e^2 + 8*e, -3*e^3 - 9*e^2 + 4*e - 8, -4*e^3 - 25*e^2 - 32*e + 25, -e^3 - 3*e^2 + 3*e - 10, -2*e^3 - 14*e^2 - 9*e + 21, 10*e^3 + 42*e^2 + 14*e - 55, -15*e^3 - 62*e^2 - 25*e + 57, -11*e^3 - 52*e^2 - 23*e + 74, 9*e^3 + 39*e^2 + 7*e - 60, 11*e^3 + 44*e^2 + 15*e - 42, -6*e^3 - 27*e^2 - 11*e + 50, -4*e^3 - 27*e^2 - 35*e + 8, 3*e^3 + 17*e^2 + 2*e - 34, 10*e^3 + 39*e^2 + 12*e - 40, -9*e^3 - 51*e^2 - 56*e + 49, 5*e^3 + 31*e^2 + 37*e - 30, 2*e^3 + 15*e^2 + 21*e - 25, 3*e^3 + 17*e^2 + 24*e - 19, 15*e^3 + 67*e^2 + 25*e - 96, -6*e^3 - 31*e^2 - 27*e + 25, -3*e^3 - 8*e^2 + 22*e + 34, -11*e^3 - 47*e^2 - 30*e + 36, 10*e^3 + 42*e^2 + 17*e - 41, 7*e^3 + 23*e^2 - 6*e - 24, 22*e^3 + 107*e^2 + 70*e - 115, -14*e^3 - 71*e^2 - 45*e + 75, 23*e^3 + 103*e^2 + 42*e - 138, 7*e^3 + 20*e^2 - 16*e - 13, -19*e^3 - 94*e^2 - 55*e + 124, -5*e^3 - 32*e^2 - 30*e + 49, -18*e^3 - 86*e^2 - 39*e + 122, 8*e^3 + 30*e^2 - e - 30, 8*e^3 + 33*e^2 + 8*e - 59, -18*e^3 - 73*e^2 - 18*e + 92, 15*e^3 + 75*e^2 + 51*e - 75, -e^3 - 8*e^2 - 24*e - 20, -17*e^3 - 72*e^2 - 32*e + 57, -3*e^3 - 13*e^2 - 22*e + 4, -14*e^3 - 61*e^2 - 21*e + 109, 4*e^3 + 14*e^2 - 10*e - 27, -4*e^3 - 9*e^2 - e - 27, 17*e^3 + 71*e^2 + 9*e - 118, 3*e^3 + 12*e^2 + 5*e - 15, -e^3 - 4*e^2 - 13*e - 30, 14*e^3 + 72*e^2 + 50*e - 79, -2*e^3 - 3*e^2 + 9*e + 3, -23*e^3 - 109*e^2 - 70*e + 125, 14*e^3 + 67*e^2 + 33*e - 117, 5*e^3 + 14*e^2 - 11*e - 12, 17*e^3 + 80*e^2 + 47*e - 77, 7*e^3 + 21*e^2 - 26*e - 51, -8*e^3 - 22*e^2 + 9*e - 7, -2*e^3 - 16*e^2 - 37*e - 8, -8*e^3 - 36*e^2 - 16*e + 88, 7*e^3 + 37*e^2 + 22*e - 84, 14*e^3 + 71*e^2 + 58*e - 71, -9*e^3 - 47*e^2 - 43*e + 66, 11*e^3 + 46*e^2 + 13*e - 40, -10*e^3 - 39*e^2 - 21*e + 16, 9*e^3 + 45*e^2 + 21*e - 71, 12*e^3 + 62*e^2 + 51*e - 69, -15*e^3 - 72*e^2 - 47*e + 57, -4*e^3 - 23*e^2 - 40*e + 11, -7*e^3 - 21*e^2 - e - 19, 4*e^3 + 25*e^2 + 45*e - 19, 11*e^3 + 53*e^2 + 49*e - 36, -3*e^3 - 12*e^2 - 5*e - 1, -4*e^3 - 14*e^2 + 3*e + 34, -22*e^3 - 106*e^2 - 54*e + 148, -9*e^3 - 45*e^2 - 32*e + 26, -4*e^3 - 7*e^2 + 39*e + 20, -6*e^3 - 23*e^2 - 5*e - 9, 15*e^3 + 78*e^2 + 64*e - 69, 7*e^3 + 15*e^2 - 31*e - 31, 10*e^3 + 34*e^2 - 3*e - 36, -2*e^3 - 5*e^2 + 4*e + 12, -7*e^3 - 26*e^2 - 4*e + 35, 10*e^3 + 35*e^2 + 2*e - 36, -6*e^3 - 38*e^2 - 42*e + 34, 23*e^3 + 112*e^2 + 80*e - 112, -2*e^3 - 14*e^2 - 38*e - 14, 17*e^3 + 73*e^2 + 28*e - 100, -16*e^3 - 78*e^2 - 46*e + 81, -11*e^3 - 42*e^2 + 8*e + 71];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {13, 13, -w^2 + 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;