/*
  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![2, 2, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {12, 6, -w^3 + 2*w^2 + 4*w - 5}>;

primesArray := [
[2, 2, -w],
[3, 3, w + 1],
[4, 2, -w^2 - w + 1],
[13, 13, -w^2 + 3],
[17, 17, -w + 3],
[27, 3, w^3 - 2*w^2 - 3*w + 5],
[31, 31, -w^2 - 2*w + 1],
[41, 41, -w^2 + 5],
[43, 43, w^3 - w^2 - 5*w - 1],
[43, 43, w^3 - w^2 - 3*w + 1],
[59, 59, -w - 3],
[59, 59, -2*w^3 + 2*w^2 + 9*w - 5],
[59, 59, -w^3 - 3*w^2 - w + 3],
[59, 59, w^3 - 7*w + 1],
[61, 61, -w^3 + w^2 + 2*w + 5],
[61, 61, -w^3 - w^2 + 4*w + 3],
[67, 67, 4*w^3 - 4*w^2 - 18*w + 7],
[73, 73, -2*w^3 + 6*w - 3],
[73, 73, -2*w + 3],
[97, 97, w^3 - 2*w^2 - 5*w + 3],
[101, 101, w^3 + w^2 - 5*w - 7],
[101, 101, -w^3 - w^2 + 4*w + 5],
[103, 103, 2*w^3 - 8*w - 1],
[103, 103, w^2 - 2*w - 7],
[109, 109, -w^3 + 2*w^2 + 3*w - 7],
[113, 113, -w^3 + w^2 + 6*w - 3],
[127, 127, -w^3 + w^2 + 6*w - 1],
[131, 131, -2*w^3 + 3*w^2 + 10*w - 11],
[137, 137, w^3 + w^2 - 6*w - 5],
[139, 139, -3*w^3 + 2*w^2 + 15*w - 1],
[149, 149, w^3 - 3*w^2 - 4*w + 11],
[149, 149, -w^3 + 5*w - 1],
[157, 157, -2*w^3 + 3*w^2 + 6*w - 1],
[163, 163, 2*w^2 - 3],
[163, 163, -2*w^3 + 2*w^2 + 7*w - 5],
[167, 167, -2*w^3 + 2*w^2 + 8*w - 3],
[167, 167, -2*w^2 + 2*w + 9],
[173, 173, 2*w^3 - w^2 - 8*w + 3],
[173, 173, 2*w^3 - 3*w^2 - 6*w + 3],
[181, 181, 2*w^3 - 3*w^2 - 8*w + 5],
[191, 191, -2*w^3 + w^2 + 10*w - 1],
[193, 193, -w^3 + 3*w^2 + 2*w - 7],
[193, 193, w^3 - 3*w - 3],
[193, 193, 2*w^2 - 2*w - 3],
[193, 193, w^3 - 7*w - 1],
[197, 197, w^2 - 2*w - 5],
[211, 211, w^2 - 4*w - 1],
[229, 229, -w^3 + w^2 + 6*w - 7],
[229, 229, -4*w^3 + 6*w^2 + 16*w - 13],
[233, 233, 3*w^2 + 2*w - 5],
[233, 233, 2*w^3 - 2*w^2 - 11*w + 7],
[239, 239, -w^3 + w^2 + 7*w + 1],
[241, 241, w^3 - 3*w^2 - 2*w + 9],
[241, 241, w^3 + w^2 - 9*w - 5],
[251, 251, 2*w^3 - 6*w + 1],
[271, 271, 2*w^3 - w^2 - 8*w - 1],
[271, 271, -2*w^2 + w + 7],
[281, 281, 2*w^3 + 2*w^2 - 5*w - 1],
[293, 293, w^3 + w^2 - 5*w - 1],
[293, 293, -3*w + 7],
[307, 307, -w^3 - 2*w^2 + 3*w + 5],
[311, 311, -w^3 + 3*w^2 + 4*w - 7],
[313, 313, 2*w^2 - 2*w - 5],
[317, 317, -2*w^3 + 4*w^2 + 7*w - 13],
[317, 317, -2*w^3 + 3*w^2 + 8*w - 11],
[337, 337, 3*w^3 + 2*w^2 - 7*w + 1],
[347, 347, 2*w^3 - 2*w^2 - 11*w + 1],
[349, 349, -5*w^3 + 6*w^2 + 23*w - 13],
[359, 359, 2*w^3 - 2*w^2 - 12*w + 9],
[359, 359, -2*w^3 + 8*w + 5],
[367, 367, -4*w^3 + 4*w^2 + 19*w - 5],
[367, 367, 2*w^3 + w^2 - 10*w - 7],
[383, 383, 2*w^3 - 2*w^2 - 7*w + 1],
[389, 389, 2*w^2 - w - 5],
[397, 397, 3*w^3 - 4*w^2 - 11*w + 9],
[397, 397, 3*w^3 - 3*w^2 - 12*w + 7],
[397, 397, 2*w^3 - 4*w^2 - 8*w + 15],
[397, 397, -w^3 + 3*w^2 + w - 7],
[409, 409, -w^3 - w^2 + 9*w - 1],
[419, 419, 2*w^2 - 3*w - 7],
[419, 419, 4*w^3 - 3*w^2 - 18*w + 3],
[431, 431, -3*w^3 + w^2 + 17*w - 1],
[431, 431, 3*w^3 - w^2 - 16*w - 1],
[433, 433, -2*w^3 + 5*w^2 + 8*w - 19],
[443, 443, 3*w - 5],
[449, 449, -2*w^3 + 2*w^2 + 9*w - 9],
[449, 449, -5*w^3 + 8*w^2 + 19*w - 17],
[457, 457, -3*w^3 + 3*w^2 + 13*w - 9],
[457, 457, -3*w^3 + 2*w^2 + 13*w + 1],
[457, 457, -2*w^3 + w^2 + 8*w + 7],
[457, 457, 3*w^3 - 3*w^2 - 15*w + 7],
[461, 461, -2*w^3 + 2*w^2 + 9*w + 1],
[479, 479, -w^3 + w^2 + 7*w - 9],
[487, 487, -w^3 - w^2 + w - 3],
[499, 499, -2*w^3 + 7*w - 3],
[499, 499, -5*w^3 + 7*w^2 + 20*w - 15],
[541, 541, -3*w^3 + 4*w^2 + 13*w - 7],
[547, 547, -w^3 - w^2 + 3*w + 5],
[557, 557, 4*w - 1],
[563, 563, 2*w^3 - w^2 - 6*w - 1],
[571, 571, w^3 - w^2 - w - 3],
[587, 587, -2*w^3 + 2*w^2 + 10*w + 1],
[593, 593, 2*w^3 - 2*w^2 - 6*w + 3],
[613, 613, -5*w^3 + 5*w^2 + 23*w - 7],
[613, 613, -w^3 + w^2 + 5*w - 7],
[617, 617, -w - 5],
[619, 619, -3*w^3 + 4*w^2 + 11*w - 7],
[625, 5, -5],
[643, 643, -3*w^3 + 3*w^2 + 14*w - 9],
[643, 643, -3*w^2 - 4*w + 1],
[653, 653, -w^3 + w^2 + 9*w + 3],
[653, 653, -4*w^2 + 6*w + 11],
[659, 659, -w^2 - 4*w + 1],
[661, 661, -3*w^3 + 3*w^2 + 15*w - 5],
[661, 661, w^3 + w^2 - 7*w - 11],
[673, 673, 2*w^3 - 3*w^2 - 12*w + 1],
[673, 673, -2*w^3 + w^2 + 10*w - 3],
[683, 683, -w^3 - 2*w^2 + 5*w + 11],
[691, 691, 4*w^3 - 3*w^2 - 18*w + 5],
[709, 709, w^3 - 3*w - 7],
[719, 719, 2*w^3 - 8*w + 1],
[727, 727, -3*w^3 + 6*w^2 + 13*w - 19],
[733, 733, w^3 - w^2 + 3],
[739, 739, -w^3 + w^2 + 3*w - 7],
[751, 751, -w^2 - 3],
[773, 773, 2*w^3 - 2*w^2 - 11*w + 3],
[787, 787, w^3 - w^2 - 3*w - 5],
[797, 797, -2*w^3 + 2*w^2 + 5*w - 1],
[797, 797, -7*w^3 + 8*w^2 + 31*w - 15],
[809, 809, 4*w^3 - 6*w^2 - 17*w + 13],
[809, 809, 3*w^2 - 4*w - 5],
[811, 811, 2*w^2 + 4*w + 3],
[811, 811, -3*w^3 + 3*w^2 + 12*w - 5],
[821, 821, 2*w^3 - 10*w - 1],
[821, 821, 6*w^3 - 10*w^2 - 24*w + 29],
[823, 823, -5*w^3 + 3*w^2 + 24*w - 1],
[823, 823, w^3 + w^2 + w + 3],
[827, 827, -4*w^3 + 3*w^2 + 16*w - 7],
[829, 829, 2*w^3 - 4*w^2 - 6*w + 11],
[841, 29, w^3 - 3*w^2 - 5*w + 3],
[841, 29, -3*w^3 + 3*w^2 + 13*w - 1],
[857, 857, -4*w^3 + 6*w^2 + 13*w - 7],
[857, 857, -3*w^3 + w^2 + 15*w + 1],
[859, 859, -4*w^2 + 3*w + 17],
[863, 863, -4*w^3 + 8*w^2 + 14*w - 21],
[877, 877, -w^3 - w^2 + 6*w - 1],
[881, 881, -5*w - 1],
[883, 883, -4*w^3 + 4*w^2 + 21*w - 13],
[883, 883, 2*w^3 - 11*w - 1],
[887, 887, -3*w^3 - w^2 + 8*w - 5],
[907, 907, w^3 - w^2 - w + 5],
[911, 911, 2*w^2 - 5*w - 5],
[919, 919, -w^3 + 3*w^2 + 6*w - 3],
[919, 919, 5*w^2 + 4*w - 9],
[929, 929, w^3 + 3*w^2 - 8*w - 17],
[937, 937, -4*w^3 + 4*w^2 + 17*w - 7],
[953, 953, -2*w^3 + 2*w^2 + 7*w + 5],
[953, 953, 3*w^3 - 4*w^2 - 11*w + 3],
[967, 967, 4*w^3 - 2*w^2 - 17*w + 3],
[971, 971, w^3 - 3*w^2 - 8*w + 1],
[983, 983, -2*w^3 - 2*w^2 + 12*w + 13],
[991, 991, -2*w^3 - 2*w^2 + 8*w + 7],
[991, 991, w^3 + w^2 - 10*w + 1]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

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

heckeEigenvaluesArray := [e, 1, 1, e^3 + e^2 - 6*e - 4, -2*e - 2, e^3 - e^2 - 6*e + 2, e^2 - e, -2*e^3 + 12*e + 8, -e^3 - 2*e^2 + 9*e + 10, e^3 - 2*e^2 - 7*e + 4, e^3 - 7*e + 2, e^3 - 5*e - 4, 2*e^3 + 2*e^2 - 10*e - 8, 3*e^3 - 21*e - 10, e^3 - 7*e + 4, -2*e^3 + 12*e + 2, -2*e - 4, -2*e^3 - 3*e^2 + 13*e + 12, -2*e^3 + 2*e^2 + 12*e - 4, 2*e^2 - 2*e - 18, -2*e^3 - 2*e^2 + 10*e + 14, -2*e^2 + 2*e + 10, 2*e^3 + 2*e^2 - 20*e - 12, -e^3 + e - 2, -e^3 + e^2 + 6*e + 6, -3*e^2 - e + 14, 2*e^2 - 2*e - 14, -3*e^3 - 2*e^2 + 21*e + 12, -3*e^3 - 4*e^2 + 19*e + 16, e^3 - 3*e^2 - 2*e + 16, e^3 - 4*e^2 - 5*e + 18, -e^3 + 5*e + 6, -2*e + 8, 2*e + 2, -e^3 + 2*e^2 + 9*e - 10, -2*e^3 + 3*e^2 + 9*e - 14, 2*e^3 + 4*e^2 - 12*e - 16, -e^3 + e^2 - 2*e - 10, -e^3 + e^2 + 4*e - 2, -e^3 + 2*e^2 + 13*e - 4, -6*e^3 + e^2 + 35*e + 6, -2*e^2 - 4*e + 16, 5*e^2 + 3*e - 16, 2*e^3 - 2*e^2 - 4*e + 14, -2*e^3 + e^2 + 9*e + 4, 3*e^3 - e^2 - 18*e - 2, -5*e^3 - e^2 + 28*e + 12, 3*e^3 + 2*e^2 - 19*e - 20, -4*e^2 + 4*e + 20, -2*e^3 - 3*e^2 + 3*e + 20, -2*e^3 + 12*e + 18, 4*e^3 + e^2 - 25*e - 10, 2*e^3 - 2*e^2 - 4*e + 10, 2*e^3 + 3*e^2 - 13*e - 8, 3*e^3 + 3*e^2 - 20*e - 4, -4*e^3 - 6*e^2 + 24*e + 24, -2*e^3 + 2*e^2 + 16*e, 4*e^2 - 14, 6*e^3 - 40*e - 8, -3*e^3 + 2*e^2 + 19*e - 12, 3*e^3 - 4*e^2 - 17*e + 2, -2*e^2 + 2*e + 12, -2*e^3 + 20*e - 4, -2*e^3 + 6*e^2 + 12*e - 30, -2*e^3 - 2*e^2 + 20*e + 18, -8*e^3 + 44*e + 6, -4*e^3 + 24*e + 20, -5*e^3 - 3*e^2 + 30*e + 6, -4*e^3 - e^2 + 19*e + 4, e^3 - 3*e + 4, -4*e^3 - 4*e^2 + 24*e + 24, 3*e^3 - 4*e^2 - 11*e + 18, 6*e^3 + 2*e^2 - 30*e - 18, 3*e^3 + 2*e^2 - 19*e - 16, -3*e^3 + 4*e^2 + 15*e - 18, -5*e^3 - 3*e^2 + 36*e + 10, e^3 - 11*e - 12, 2*e^3 - 6*e^2 - 6*e + 26, 4*e^3 - 4*e^2 - 26*e, -e^3 + 3*e^2 + 14*e - 12, -3*e^3 - e^2 + 24*e - 4, 3*e^3 + 4*e^2 - 17*e - 24, 2*e^3 - 4*e^2 - 12*e + 8, -6*e^3 - 2*e^2 + 42*e + 20, -e^3 - 2*e^2 + 9*e - 6, -e^3 - 4*e^2 - e + 22, -e^3 - 2*e^2 + 3*e - 8, 2*e^3 + 2*e^2 - 4*e - 18, 2*e^3 + 7*e^2 - 17*e - 24, 2*e^3 - 12*e - 2, 8*e^3 - 48*e - 30, -5*e^3 + 2*e^2 + 33*e + 12, -4*e^2 + 4*e + 10, 2*e^3 + 2*e^2 - 10*e - 22, 2*e^3 - 8*e^2 - 18*e + 26, -3*e^3 - 3*e^2 + 12*e, -5*e^3 + 2*e^2 + 25*e + 12, -e^3 + 15*e + 10, -11*e^3 + e^2 + 62*e + 10, 3*e^3 - 5*e^2 - 18*e + 36, -10*e^2 + 2*e + 32, 2*e^2 + 14*e - 18, -4*e^3 + 6*e^2 + 30*e - 14, 2*e^3 + 8*e^2 - 4*e - 46, -3*e^3 + 2*e^2 + 13*e + 14, 8*e^3 + e^2 - 39*e - 10, 3*e^3 - 17*e + 18, -2*e^2 + 8*e + 22, 2*e^3 + 2*e^2 - 20*e + 8, 5*e^3 + 5*e^2 - 42*e - 34, -2*e^3 + 4*e - 6, -e^3 - 2*e^2 + 5*e + 8, 4*e^3 + 2*e^2 - 30*e - 26, -5*e^3 + 39*e + 4, 3*e^3 - 6*e^2 - 11*e + 44, -6*e^3 - 5*e^2 + 35*e + 22, 4*e^3 + 2*e^2 - 38*e - 14, 8*e^3 - 4*e^2 - 44*e + 12, 7*e^3 - 3*e^2 - 52*e - 16, -11*e^3 + e^2 + 56*e - 2, -2*e^3 + 6*e^2 + 12*e + 8, 2*e^3 - 5*e^2 - 7*e + 38, -5*e^3 + 2*e^2 + 37*e + 12, -2*e^3 + 16*e + 18, 4*e^3 - 4*e^2 - 26*e + 4, -2*e^3 - 2*e^2 + 10*e + 10, 5*e^3 + e^2 - 50*e - 12, -2*e^3 - 2*e^2 + 8*e - 12, -5*e^3 + 2*e^2 + 13*e - 8, -5*e^3 + 2*e^2 + 39*e + 4, -8*e^3 + 2*e^2 + 62*e + 16, e^3 + 4*e^2 - 3*e - 22, 2*e^3 + 4*e^2 - 14*e - 16, 5*e^3 - 2*e^2 - 41*e - 4, -5*e^3 - 4*e^2 + 35*e + 4, 8*e^3 + 7*e^2 - 41*e - 34, -4*e^3 - 12*e^2 + 18*e + 50, -3*e^3 - 8*e^2 + 33*e + 50, -3*e^3 - 9*e^2 + 26*e + 30, -6*e^3 + 3*e^2 + 35*e - 12, -6*e^3 + 42*e - 6, 2*e^3 + 8*e^2 - 10*e - 18, 8*e^3 + 6*e^2 - 50*e - 42, 5*e^3 - 47*e - 18, 2*e^3 - 2*e^2 - 6*e - 12, -10*e^3 + 2*e^2 + 72*e + 30, -6*e^2 - 2*e + 6, 7*e^3 + 11*e^2 - 38*e - 44, 6*e^2 + 12*e - 38, 13*e^3 + 4*e^2 - 89*e - 38, 3*e^3 - 21*e + 6, -6*e^3 - 4*e^2 + 36*e + 40, -11*e^3 - 2*e^2 + 65*e + 26, 6*e^3 - 6*e^2 - 24*e + 26, -6*e^3 + 7*e^2 + 29*e - 20, 4*e^3 + 10*e^2 - 24*e - 20, -14*e^3 - 2*e^2 + 78*e + 24, 8*e^3 - 2*e^2 - 36*e + 6, 8*e^3 - 4*e^2 - 48*e - 4, -6*e^3 + 2*e^2 + 34*e, 4*e^3 + 4*e^2 - 40*e - 28, -2*e^3 - 4*e^2 + 30*e + 30, -6*e^3 + 8*e^2 + 34*e - 8];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {3, 3, w + 1}>] := -1;
ALEigenvalues[ideal<ZF | {4, 2, -w^2 - w + 1}>] := -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;