/*
  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![5, -1, -6, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {25, 25, -w^3 + 5*w + 3}>;

primesArray := [
[5, 5, w],
[5, 5, -w^2 + w + 2],
[7, 7, -w^2 + 2],
[13, 13, -w^2 + 3],
[13, 13, w^2 - w - 4],
[16, 2, 2],
[19, 19, -w^2 + w + 1],
[23, 23, -w^3 + 4*w + 2],
[25, 5, -w^3 + w^2 + 3*w - 1],
[29, 29, w^3 - w^2 - 4*w + 1],
[29, 29, -w + 3],
[31, 31, -w^3 + w^2 + 5*w - 2],
[37, 37, w^3 - 4*w - 1],
[37, 37, w^3 - 3*w + 1],
[53, 53, -w^3 + 2*w^2 + 4*w - 6],
[59, 59, w^2 + w - 4],
[61, 61, -2*w^2 + w + 8],
[73, 73, -w^3 + 5*w - 1],
[79, 79, 2*w^2 + w - 6],
[79, 79, 2*w^3 - w^2 - 9*w + 3],
[81, 3, -3],
[83, 83, -w^3 + w^2 + 3*w - 4],
[97, 97, -w^3 + 6*w + 2],
[97, 97, -2*w^3 + w^2 + 10*w - 1],
[97, 97, 3*w^3 - 4*w^2 - 15*w + 13],
[97, 97, w^3 - w^2 - 6*w + 2],
[103, 103, -2*w^3 + 3*w^2 + 11*w - 9],
[109, 109, w^2 + 3*w - 3],
[137, 137, -w^2 + 7],
[149, 149, w^3 - 2*w^2 - 5*w + 4],
[149, 149, w^3 + 2*w^2 - 6*w - 7],
[151, 151, w^3 + w^2 - 5*w - 3],
[163, 163, w^3 - 3*w - 4],
[163, 163, -w^3 + 6*w - 3],
[167, 167, w^3 - 2*w^2 - 6*w + 4],
[169, 13, w^2 + 2*w - 4],
[191, 191, w^3 - w^2 - 5*w - 2],
[191, 191, 2*w^3 - 9*w - 2],
[197, 197, w^3 - 4*w + 4],
[211, 211, -w^3 + 2*w^2 + 6*w - 8],
[223, 223, -w^3 + 3*w^2 + 4*w - 6],
[229, 229, w^3 - w^2 - 5*w - 1],
[251, 251, 2*w^3 - 2*w^2 - 7*w + 3],
[257, 257, 3*w^2 - 14],
[263, 263, w^2 - 3*w - 1],
[263, 263, w^2 - 2*w - 7],
[269, 269, -w^3 + w^2 + 2*w - 4],
[269, 269, 2*w^3 - 11*w - 1],
[271, 271, -w^3 + w^2 + 4*w - 7],
[277, 277, w^3 - w^2 - 3*w + 8],
[311, 311, w^3 + 2*w^2 - 7*w - 11],
[311, 311, -3*w^2 - 2*w + 8],
[317, 317, -2*w^3 + 2*w^2 + 7*w - 4],
[331, 331, w^2 - w - 8],
[331, 331, 3*w^3 - 2*w^2 - 12*w + 9],
[337, 337, -2*w^3 + 2*w^2 + 11*w - 8],
[343, 7, w^3 - 3*w^2 - 4*w + 11],
[347, 347, -2*w^3 + 2*w^2 + 9*w - 4],
[349, 349, -w^3 + 2*w^2 + 2*w - 6],
[353, 353, 2*w^2 - w - 12],
[353, 353, 2*w^3 - w^2 - 9*w - 1],
[367, 367, -2*w^3 + w^2 + 10*w - 6],
[367, 367, 2*w^3 - 11*w - 3],
[379, 379, -3*w - 1],
[379, 379, w^3 - 2*w - 3],
[383, 383, -w^3 + 2*w^2 + 4*w - 11],
[383, 383, -w^3 - w^2 + 7*w + 4],
[389, 389, 2*w^3 - w^2 - 11*w + 4],
[401, 401, w^3 + 2*w^2 - 7*w - 6],
[401, 401, 2*w^3 - 11*w - 6],
[419, 419, -2*w^3 + w^2 + 8*w - 1],
[421, 421, -w^3 + 3*w^2 + 3*w - 12],
[421, 421, -w^3 - 2*w^2 + 2*w + 6],
[433, 433, w^3 - 2*w^2 - 6*w + 11],
[433, 433, w^2 - 8],
[439, 439, -w^3 + 3*w^2 + 3*w - 7],
[443, 443, w^3 - w^2 - 2*w - 4],
[443, 443, 2*w^3 + w^2 - 7*w - 2],
[443, 443, 2*w^3 - w^2 - 8*w + 2],
[443, 443, -w^2 - 2],
[467, 467, 3*w^3 - 5*w^2 - 15*w + 19],
[467, 467, 2*w^3 - 3*w^2 - 8*w + 13],
[467, 467, -w^3 + 2*w^2 + 3*w - 9],
[467, 467, -w^3 + 3*w^2 + 3*w - 8],
[479, 479, -w^3 - 3*w^2 + 5*w + 13],
[479, 479, -w^3 + w^2 + w - 4],
[487, 487, 3*w^3 - w^2 - 13*w - 2],
[491, 491, w^2 + w - 9],
[499, 499, -3*w^3 + 3*w^2 + 13*w - 7],
[503, 503, 2*w^2 + w - 9],
[509, 509, -2*w^2 - 3*w + 6],
[509, 509, -w^3 + w^2 - 4],
[521, 521, -3*w^2 + w + 12],
[523, 523, 2*w^3 - 8*w - 3],
[529, 23, 2*w^3 - w^2 - 6*w - 1],
[541, 541, -w^3 + w^2 + 7*w - 4],
[547, 547, 2*w^3 - 3*w^2 - 9*w + 7],
[557, 557, 3*w^2 - 2*w - 9],
[557, 557, w^3 - 6*w - 7],
[563, 563, -2*w^3 + w^2 + 6*w - 6],
[577, 577, -w^3 + 4*w^2 + 2*w - 13],
[587, 587, w^2 - 2*w + 3],
[587, 587, w^2 + 2*w - 6],
[593, 593, -2*w^3 + 2*w^2 + 11*w - 3],
[599, 599, w^3 + 3*w^2 - 6*w - 12],
[601, 601, -2*w^3 + w^2 + 9*w + 2],
[607, 607, -w^3 + 3*w^2 + 4*w - 9],
[617, 617, -2*w^3 + w^2 + 7*w - 3],
[619, 619, -2*w^3 + 4*w^2 + 9*w - 11],
[631, 631, 3*w^2 - w - 7],
[631, 631, -w^3 - w^2 + 6*w + 1],
[641, 641, -w^3 + 2*w^2 + 5*w - 1],
[641, 641, -w^3 + 2*w^2 + 2*w - 7],
[643, 643, -3*w^3 + w^2 + 14*w - 4],
[647, 647, -3*w^3 + 5*w^2 + 14*w - 18],
[647, 647, -w^3 + 7*w + 2],
[647, 647, -2*w^3 + 7*w - 1],
[647, 647, 2*w^3 - w^2 - 10*w - 1],
[653, 653, -2*w^3 - w^2 + 11*w + 4],
[659, 659, -w^3 - w^2 + 5*w - 1],
[673, 673, -w^3 + w^2 + 7*w - 2],
[673, 673, -w^3 + w^2 + 2*w - 8],
[683, 683, 2*w^3 - 11*w - 8],
[683, 683, 2*w^3 - 7*w - 3],
[701, 701, 2*w^3 - 4*w^2 - 10*w + 17],
[727, 727, -w^3 + w^2 + 4*w - 8],
[727, 727, 3*w^2 - 11],
[739, 739, 2*w^3 + 3*w^2 - 8*w - 13],
[743, 743, 2*w^2 + w - 12],
[743, 743, 3*w^2 - 2*w - 14],
[751, 751, w^3 + w^2 - 3*w - 7],
[757, 757, -2*w^3 + 10*w - 1],
[761, 761, -2*w^3 + w^2 + 7*w - 1],
[761, 761, 3*w^2 + w - 7],
[773, 773, -2*w^3 + 2*w^2 + 8*w - 1],
[797, 797, -2*w^3 + w^2 + 9*w + 3],
[797, 797, 3*w^3 - 14*w - 8],
[809, 809, 5*w^3 - 5*w^2 - 24*w + 16],
[821, 821, -4*w^3 + 3*w^2 + 18*w - 9],
[821, 821, w^3 - w - 3],
[823, 823, 3*w^2 + w - 8],
[829, 829, -2*w^3 + 3*w^2 + 10*w - 7],
[829, 829, 3*w^2 - 2*w - 16],
[841, 29, 2*w^2 + w - 11],
[853, 853, 3*w^3 - 2*w^2 - 14*w + 3],
[853, 853, 3*w^3 - 5*w^2 - 14*w + 17],
[857, 857, 3*w^2 - w - 8],
[857, 857, 3*w^3 - w^2 - 13*w + 3],
[859, 859, w^3 - 3*w^2 - 6*w + 12],
[877, 877, w^3 + 2*w^2 - 5*w - 4],
[877, 877, w^2 - w - 9],
[881, 881, w^3 + 2*w^2 - 4*w - 12],
[881, 881, w^2 - 4*w - 4],
[883, 883, -3*w^3 + 2*w^2 + 11*w - 8],
[887, 887, -2*w^3 + w^2 + 6*w - 1],
[911, 911, 3*w^2 - w - 17],
[911, 911, 2*w^3 - 2*w^2 - 12*w + 3],
[919, 919, 5*w^3 - 6*w^2 - 22*w + 24],
[919, 919, -2*w^3 + w^2 + 6*w - 7],
[929, 929, 2*w^3 - 7*w - 1],
[937, 937, -w^3 - w^2 + 8*w - 2],
[947, 947, 3*w^3 + w^2 - 15*w - 6],
[947, 947, -2*w^3 + 9*w + 9],
[953, 953, w^3 - 2*w^2 - 6*w + 1],
[977, 977, w^3 - w^2 - 3*w + 9],
[977, 977, w^3 + w^2 - 3*w - 8],
[983, 983, w^3 - 5*w - 7],
[991, 991, -w^3 + 2*w^2 + 2*w - 8],
[991, 991, w^2 + 4*w - 1],
[997, 997, 2*w^3 + w^2 - 12*w - 8]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 7*x^2 + 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e^2 - 3, 0, e, -e^3 + 5*e + 1, e^2 + 2*e - 5, e^3 + e^2 - 4*e - 3, -2*e^2 + e + 6, -2*e^2 + 2*e + 8, e^2 - e + 1, e^2 - 2*e - 2, e^3 + e^2 - 5*e - 3, -e^3 + 2*e^2 + 5*e - 4, e^2 - 4*e - 7, -e^3 + e^2 + 5*e - 1, -e^3 + e^2 + 3*e - 7, -e^3 + 2*e^2 + 5*e - 6, -e^3 + 2*e^2 + 4*e - 6, 2*e^3 - e^2 - 14*e + 1, -2*e^3 + 2*e^2 + 10*e + 2, -e^3 + e^2 + 3*e + 4, 2*e^3 + 3*e^2 - 8*e - 13, 3*e - 6, -3*e^3 - 3*e^2 + 11*e + 11, e^3 - e^2 - 3*e + 9, -3*e^3 - e^2 + 15*e + 1, 2*e^3 - 2*e^2 - 10*e + 6, 2*e^3 - 4*e + 2, e^2 - 1, -5*e^3 - e^2 + 21*e + 1, -6*e^2 - 2*e + 22, -e^3 + 11*e + 11, -4*e^3 + 22*e + 2, -2*e^3 - 4*e^2 + 8*e + 12, e^3 - 2*e^2 - 7*e - 8, -e^3 + 2*e^2 + 7*e - 14, -2*e^3 - 4*e^2 + 8*e + 21, 3*e^3 + e^2 - 19*e - 2, -3*e^3 - 6*e^2 + 17*e + 20, 2*e^2 + 8*e - 15, e^3 - 2*e^2 + e + 12, 2*e^3 - 4*e^2 - 14*e + 18, 2*e^3 + 2*e^2 - 10*e - 3, -e^3 - 3*e^2 + 7*e - 6, -2*e^3 - e^2 + 12*e + 7, e^3 - 2*e^2 - 9*e + 14, 4*e^3 + 4*e^2 - 23*e - 14, 4*e^3 + e^2 - 14*e - 3, -e^3 - 9*e^2 + 3*e + 35, 6*e - 6, 2*e^3 - 2*e^2 - 12*e + 11, -3*e^3 + e^2 + 9*e - 4, -2*e^3 + 2*e^2 + 3*e - 10, -4*e^3 + 3*e^2 + 14*e - 10, e^3 + 6*e^2 - 15*e - 22, -4*e^3 - 2*e^2 + 18*e - 4, -e^3 + e^2 + 3*e - 3, 2*e^3 + 4*e^2 - 8*e - 16, 3*e^3 + 4*e^2 - 14*e + 4, 3*e^3 + 6*e^2 - 20*e - 22, -2*e^3 - 2*e^2 + 10*e - 2, -4*e^3 - 4*e^2 + 18*e + 18, -3*e^3 - 4*e^2 + 13*e + 8, -6*e^3 + 4*e^2 + 21*e - 18, -2*e^3 + 3*e^2 + 12*e - 18, -5*e^3 - 6*e^2 + 27*e + 22, e^3 - 4*e^2 + 4*e + 22, 6*e^3 + 6*e^2 - 28*e - 22, 6*e^2 - 38, e^3 - 5*e^2 - 16*e + 23, -5*e^3 + 5*e^2 + 25*e - 11, 5*e^3 - e^2 - 33*e + 6, 2*e^3 - 12*e^2 - 10*e + 38, 3*e^3 - e^2 - 7*e + 15, -3*e^3 - 5*e^2 + 5*e + 29, e^3 - 4*e^2 - e + 11, -2*e^3 - 10*e^2 + 8*e + 36, -4*e^3 - e^2 + 12*e + 18, -4*e^3 - 2*e^2 + 18*e + 12, 8*e^3 + 2*e^2 - 38*e, 7*e^3 - 29*e + 4, 2*e^3 + 2*e^2 - 13*e + 8, -e^3 - 9*e^2 - e + 36, 2*e^3 + 2*e^2 - 10*e + 18, -3*e^3 + 6*e^2 + 12*e - 16, e^3 - 2*e^2 - 11*e - 14, -e^3 - e^2 - e + 20, 2*e^3 - 8*e - 2, 5*e^3 - 2*e^2 - 23*e + 14, 5*e^3 - e^2 - 17*e + 12, -10*e^3 + 6*e^2 + 48*e - 10, -5*e^2 - e + 11, -3*e^3 - e^2 + 7*e + 11, -e^3 + 7*e^2 + 14*e - 19, 5*e^3 - 7*e^2 - 17*e + 26, -2*e^3 - 11*e^2 + 10*e + 31, -3*e^3 + 16*e + 2, -4*e^3 + 6*e^2 + 16*e - 20, -2*e^3 + 5*e^2 - 2*e - 14, 3*e^3 + e^2 - 19*e + 5, -3*e^3 - 5*e^2 + 17*e + 38, 6*e^3 - 8*e^2 - 22*e + 26, 8*e^3 - 4*e^2 - 38*e + 12, -2*e^3 - 8*e^2 + 16*e + 16, -e^3 + 7*e^2 + 14*e - 33, -3*e^3 + 29*e + 12, 2*e^2 + 6*e + 14, -3*e^3 + 4*e^2 + 22*e, -3*e^3 + 13*e^2 + 9*e - 55, -2*e^3 - 4*e^2 + 6*e - 8, 2*e^3 - 24*e, 8*e^3 + 4*e^2 - 32*e - 26, 2*e^3 + 5*e^2 - 11*e + 9, 3*e^3 - 9*e^2 - 9*e + 51, e^3 + 8*e^2 - 11*e - 20, 10*e^3 - e^2 - 36*e + 6, 6*e^3 - e^2 - 38*e + 2, 2*e^3 - 12*e^2 - 4*e + 48, 18*e^2 + 3*e - 68, e^3 - 11*e^2 - 11*e + 45, 5*e^3 + e^2 - 19*e - 24, -8*e^3 + 4*e^2 + 32*e - 18, 2*e^3 + e^2 - 11*e + 19, 4*e^3 - 16*e + 22, -2*e^3 + 8*e^2 + 3*e - 30, 4*e^3 - 5*e^2 - 16*e + 23, 4*e^3 + 8*e^2 - 18*e - 24, -e^3 + 6*e^2 + 3*e + 2, -e^3 + 8*e^2 + 11*e - 40, 2*e^3 + 2*e - 14, -6*e^3 - 6*e^2 + 30*e + 28, -8*e^2 - 12*e + 28, 2*e^3 + 3*e^2 - 28*e - 14, -4*e^3 - 3*e^2 + 20*e + 19, 7*e^3 - 7*e^2 - 45*e + 5, -3*e^2 - 6*e + 31, -4*e^3 + 13*e^2 + 20*e - 43, 4*e^3 + 10*e^2 - 32*e - 26, -e^3 - 3*e^2 - e + 33, -2*e^3 - 13*e^2 + 51, -5*e^3 + 5*e^2 + 24*e - 25, -4*e^2 - 8*e + 28, -7*e^3 + 10*e^2 + 38*e - 14, 3*e^3 + 9*e^2 - 13*e - 19, -2*e^3 + 5*e^2 + 26*e - 13, -2*e^3 + 9*e^2 + 16*e - 11, 2*e^3 - 8*e^2 + 2*e + 30, -7*e^3 - e^2 + 34*e + 13, 8*e^3 - 6*e^2 - 32*e + 31, 2*e^3 - 6*e^2 + 46, e^3 - 4*e^2 + 5*e - 1, -3*e^3 + e^2 + 25*e + 1, 6*e^3 - 2*e^2 - 36*e - 7, -4*e^2 + 6, e^3 - 12*e^2 + 5*e + 42, e^3 + 2*e^2 - 17*e - 8, -2*e^3 - 2*e^2 + 15*e + 16, 3*e^2 + 18*e - 18, -2*e^3 - 6*e^2 + 18*e + 40, -3*e^3 + 13*e + 12, -3*e^2 - 9*e + 7, 2*e^3 + 14*e^2 - 4*e - 46, -2*e^3 - 9*e^2 + 10*e + 18, 3*e^3 - 2*e^2 - 13*e + 26, -11*e^3 + 5*e^2 + 39*e - 29, -3*e^3 - 11*e^2 + 7*e + 31, -8*e^3 - 7*e^2 + 46*e + 21, -4*e + 12, 4*e^3 + 8*e^2 - 24*e - 46, -6*e^3 + 2*e^2 + 17*e - 16, 6*e^3 - e^2 - 26*e - 18];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

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