/*
  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, 0, -6, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {10, 10, -w^3 + 3*w^2 + 3*w}>;

primesArray := [
[2, 2, w],
[5, 5, -w^3 + 3*w^2 + 2*w - 1],
[5, 5, w - 1],
[11, 11, w^3 - 2*w^2 - 5*w - 1],
[13, 13, -w^3 + 3*w^2 + 3*w - 1],
[17, 17, 2*w^3 - 5*w^2 - 9*w + 5],
[23, 23, -w^3 + 3*w^2 + 4*w - 5],
[25, 5, -w^2 + 3*w + 1],
[29, 29, -2*w^3 + 6*w^2 + 5*w - 1],
[31, 31, -w^2 + 2*w + 1],
[43, 43, -w^3 + 3*w^2 + 2*w - 3],
[43, 43, -w^3 + 2*w^2 + 6*w - 3],
[59, 59, 2*w^3 - 6*w^2 - 7*w + 7],
[73, 73, 3*w^3 - 6*w^2 - 16*w - 5],
[79, 79, -5*w^3 + 14*w^2 + 20*w - 17],
[81, 3, -3],
[83, 83, -w - 3],
[83, 83, w^2 - 3*w - 7],
[89, 89, w^2 - 2*w - 7],
[101, 101, -2*w^3 + 5*w^2 + 8*w - 3],
[101, 101, 5*w^3 - 14*w^2 - 19*w + 17],
[103, 103, -w^3 + w^2 + 8*w + 3],
[103, 103, 2*w^3 - 5*w^2 - 7*w - 1],
[109, 109, 3*w^3 - 7*w^2 - 14*w + 1],
[109, 109, -w^2 + 4*w + 1],
[127, 127, -3*w^3 + 8*w^2 + 11*w - 7],
[127, 127, -w^3 + 2*w^2 + 7*w + 1],
[137, 137, -2*w^3 + 5*w^2 + 9*w - 1],
[139, 139, -w^3 + 3*w^2 + 2*w - 5],
[149, 149, w^3 - 3*w^2 - w - 1],
[149, 149, -2*w^3 + 4*w^2 + 10*w + 1],
[163, 163, w^3 - 4*w^2 - w + 11],
[163, 163, 2*w^3 - 4*w^2 - 10*w + 1],
[173, 173, -4*w^3 + 11*w^2 + 15*w - 13],
[179, 179, 2*w^2 - 6*w - 3],
[179, 179, -w^3 + 2*w^2 + 7*w - 1],
[181, 181, w^3 - 10*w - 9],
[181, 181, -3*w^3 + 8*w^2 + 14*w - 7],
[199, 199, w^3 - 8*w - 9],
[211, 211, 2*w - 3],
[223, 223, -w^3 + 4*w^2 + w - 5],
[227, 227, w^3 - 2*w^2 - 5*w - 5],
[227, 227, -3*w^3 + 7*w^2 + 14*w - 5],
[227, 227, -4*w^3 + 10*w^2 + 19*w - 7],
[227, 227, w^2 - 4*w - 5],
[239, 239, -2*w^3 + 5*w^2 + 7*w - 3],
[251, 251, -w^3 + 4*w^2 + 2*w - 11],
[251, 251, 2*w^3 - 6*w^2 - 6*w + 1],
[257, 257, -3*w^3 + 6*w^2 + 16*w + 3],
[263, 263, -w^3 + 3*w^2 + 3*w - 7],
[263, 263, w^3 - 4*w^2 + 7],
[269, 269, -2*w^3 + 6*w^2 + 6*w - 11],
[269, 269, -w^3 + 4*w^2 - w - 5],
[271, 271, -3*w^3 + 9*w^2 + 11*w - 11],
[271, 271, w^2 - 5*w - 3],
[277, 277, w^3 - 4*w^2 - 2*w + 7],
[277, 277, -2*w^3 + 7*w^2 + 8*w - 9],
[281, 281, 2*w^2 - 4*w - 5],
[281, 281, 2*w^3 - 4*w^2 - 11*w + 1],
[283, 283, -3*w^3 + 8*w^2 + 14*w - 11],
[293, 293, w^3 - 2*w^2 - 6*w + 5],
[307, 307, -2*w^3 + 5*w^2 + 7*w - 1],
[313, 313, 2*w^3 - 4*w^2 - 9*w + 3],
[313, 313, -2*w^3 + 5*w^2 + 6*w - 3],
[317, 317, -2*w^3 + 5*w^2 + 9*w + 1],
[347, 347, 4*w^3 - 9*w^2 - 21*w + 3],
[347, 347, 4*w^3 - 10*w^2 - 18*w + 5],
[349, 349, -2*w^3 + 6*w^2 + 7*w - 3],
[353, 353, -w^3 + 12*w + 5],
[353, 353, w^2 - w - 7],
[359, 359, -2*w^3 + 5*w^2 + 6*w - 7],
[359, 359, -w^3 + 4*w^2 - 11],
[373, 373, -w^3 + 2*w^2 + 6*w + 5],
[379, 379, -3*w^3 + 8*w^2 + 14*w - 5],
[397, 397, 6*w^3 - 14*w^2 - 30*w + 7],
[401, 401, 3*w^3 - 5*w^2 - 20*w - 7],
[409, 409, 3*w^3 - 7*w^2 - 12*w - 3],
[409, 409, 2*w^2 - 4*w - 7],
[421, 421, w^3 - 11*w - 11],
[421, 421, -w^3 + 3*w^2 + w - 7],
[431, 431, 2*w^3 - 3*w^2 - 14*w - 3],
[431, 431, -4*w^3 + 11*w^2 + 15*w - 15],
[433, 433, 2*w^3 - 5*w^2 - 6*w + 1],
[439, 439, -4*w^3 + 11*w^2 + 13*w - 9],
[439, 439, -2*w^3 + 6*w^2 + 8*w - 11],
[443, 443, 2*w - 5],
[443, 443, -6*w^3 + 17*w^2 + 24*w - 21],
[449, 449, w^3 - 2*w^2 - 8*w + 3],
[449, 449, 4*w^3 - 12*w^2 - 15*w + 17],
[457, 457, -2*w^3 + 4*w^2 + 9*w + 5],
[461, 461, -3*w^3 + 6*w^2 + 15*w - 1],
[461, 461, -3*w^3 + 7*w^2 + 17*w + 1],
[463, 463, -4*w^3 + 10*w^2 + 17*w - 7],
[463, 463, -3*w^3 + 10*w^2 + 9*w - 17],
[467, 467, 4*w^3 - 8*w^2 - 21*w - 7],
[467, 467, w^3 - 6*w^2 + 3*w + 19],
[479, 479, -w^3 + w^2 + 8*w + 1],
[487, 487, -4*w^3 + 11*w^2 + 14*w - 15],
[487, 487, 3*w^3 - 6*w^2 - 15*w - 5],
[491, 491, w^2 - 5],
[503, 503, 2*w^3 - 5*w^2 - 11*w + 3],
[523, 523, -3*w^3 + 7*w^2 + 13*w + 1],
[547, 547, 2*w^3 - 3*w^2 - 12*w - 3],
[547, 547, 2*w^3 - 3*w^2 - 12*w - 9],
[557, 557, -3*w^3 + 9*w^2 + 9*w - 11],
[557, 557, -6*w^3 + 15*w^2 + 27*w - 13],
[563, 563, -3*w^3 + 7*w^2 + 14*w - 7],
[571, 571, -5*w^3 + 11*w^2 + 26*w - 3],
[593, 593, -6*w^3 + 15*w^2 + 26*w - 7],
[599, 599, 7*w^3 - 18*w^2 - 29*w + 17],
[599, 599, -2*w^3 + 6*w^2 + 8*w - 3],
[601, 601, -7*w^3 + 20*w^2 + 28*w - 23],
[601, 601, 3*w^3 - 7*w^2 - 12*w + 5],
[607, 607, 5*w^3 - 12*w^2 - 24*w + 9],
[607, 607, w^3 - 5*w^2 + 15],
[613, 613, -w^3 + 5*w^2 - 2*w - 9],
[617, 617, w^3 + w^2 - 13*w - 15],
[617, 617, 2*w^3 - 4*w^2 - 8*w - 1],
[619, 619, 5*w^3 - 10*w^2 - 27*w - 7],
[619, 619, 2*w^3 - 4*w^2 - 10*w + 3],
[643, 643, w^3 - 5*w^2 + 5*w + 5],
[643, 643, 4*w^3 - 9*w^2 - 20*w - 1],
[643, 643, 3*w^3 - 7*w^2 - 14*w - 3],
[643, 643, -3*w^3 + 7*w^2 + 13*w - 3],
[647, 647, -3*w^3 + 8*w^2 + 12*w - 3],
[653, 653, -w^3 + 9*w + 13],
[653, 653, -w^3 + 4*w^2 + 3*w - 9],
[653, 653, -2*w^3 + 7*w^2 + 4*w - 7],
[653, 653, w^2 - 4*w - 9],
[673, 673, -w^3 + 5*w^2 + w - 13],
[683, 683, w^3 - 3*w^2 - 3*w - 3],
[701, 701, 5*w^3 - 13*w^2 - 23*w + 9],
[709, 709, 2*w^3 - w^2 - 19*w - 15],
[719, 719, -5*w^3 + 15*w^2 + 20*w - 19],
[727, 727, -w - 5],
[751, 751, -4*w^3 + 10*w^2 + 15*w - 9],
[751, 751, 6*w^3 - 17*w^2 - 22*w + 21],
[757, 757, 3*w^3 - 6*w^2 - 18*w - 1],
[757, 757, -w^3 + 4*w^2 + 2*w + 1],
[769, 769, w^2 - 2*w + 3],
[773, 773, 5*w^3 - 12*w^2 - 23*w + 9],
[787, 787, 2*w^3 - 4*w^2 - 5*w + 1],
[797, 797, -3*w^3 + 8*w^2 + 10*w - 3],
[809, 809, 2*w^2 - 5*w - 1],
[809, 809, 4*w^3 - 6*w^2 - 28*w - 11],
[827, 827, 4*w^3 - 8*w^2 - 22*w - 1],
[827, 827, 2*w^2 - 6*w - 9],
[829, 829, -w^3 + w^2 + 10*w + 1],
[839, 839, -2*w^3 + 2*w^2 + 16*w + 7],
[877, 877, 6*w^3 - 16*w^2 - 25*w + 13],
[877, 877, w^3 - w^2 - 7*w + 1],
[881, 881, 2*w^3 - 2*w^2 - 15*w - 7],
[881, 881, -4*w^3 + 7*w^2 + 26*w + 9],
[887, 887, 2*w^3 - 4*w^2 - 13*w - 1],
[907, 907, 5*w^3 - 11*w^2 - 26*w - 1],
[907, 907, 5*w^3 - 13*w^2 - 22*w + 9],
[907, 907, w^3 - 5*w^2 + 2*w + 13],
[907, 907, -2*w^3 + 7*w^2 + 5*w - 7],
[911, 911, -3*w^3 + 8*w^2 + 9*w - 5],
[919, 919, -2*w^3 + 7*w^2 + 6*w - 3],
[919, 919, -3*w^3 + 8*w^2 + 15*w - 7],
[947, 947, -2*w^3 + 5*w^2 + 12*w - 9],
[947, 947, -w^2 + w - 3],
[947, 947, 3*w^3 - 6*w^2 - 14*w - 7],
[947, 947, -w^3 + 5*w^2 - w - 17],
[953, 953, 3*w^3 - 9*w^2 - 9*w + 13],
[953, 953, -4*w^3 + 13*w^2 + 11*w - 23],
[967, 967, 8*w^3 - 20*w^2 - 33*w + 19]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 + 3*x^2 - 6*x - 9;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -1, -1/3*e^2 - 2*e - 1, 2/3*e^2 + e - 6, 1/3*e^2 + e - 2, -5/3*e^2 - 3*e + 7, 1, 1/3*e^2 + 2*e - 2, 1/3*e^2 - e - 7, -1/3*e^2 + 1, -1/3*e^2 - 2*e, e^2 - e - 12, -e^2 - e - 2, -1/3*e^2 - 3*e + 4, e^2 + 3*e - 8, e^2 + 5*e - 3, -5/3*e^2 + e + 16, -1/3*e^2 - 3*e + 2, e - 12, 2*e^2 + 3*e - 12, -7/3*e^2 - 6*e + 6, -e^2 - 7*e + 1, -4/3*e^2 - e, -8/3*e^2 - 4*e + 14, 1/3*e^2 + 7*e + 2, 1/3*e^2 + 3*e + 14, 5/3*e^2 + 7*e - 7, 4/3*e^2 + 3*e - 21, 1/3*e^2 - 3*e - 5, 7/3*e^2 + 6*e - 14, -1/3*e^2 - 7*e + 1, e^2 + 4*e - 2, 5/3*e^2 - e - 13, -e + 3, -3*e^2 - 7*e + 15, -1/3*e^2 - 4*e + 4, 10/3*e^2 + 2*e - 25, -e^2 + 3*e + 14, -e^2 - 6*e + 8, 16/3*e^2 + 11*e - 18, -1/3*e^2 - 5*e - 16, 7/3*e^2 + 6*e - 14, -5/3*e^2 - 2*e - 5, e^2 + 5*e + 3, -3*e^2 - 6*e + 3, 14/3*e^2 + 7*e - 16, 3*e - 6, 3*e^2 + 8*e - 18, -13/3*e^2 - 8*e + 23, -e^2 - 4*e - 9, -4*e^2 - 2*e + 33, -4/3*e^2 - 5*e - 4, -10/3*e^2 + e + 28, 5/3*e^2 - 2*e - 15, -10/3*e^2 + 2*e + 37, e^2 - 6*e - 10, 5/3*e^2 + 2*e - 4, 2/3*e^2 - 3*e - 22, -e^2 + 8, 8/3*e^2 + 9*e - 19, -1/3*e^2 - 3, -2/3*e^2 - 22, 5*e^2 + 3*e - 35, 5*e^2 + 7*e - 27, -4/3*e^2 - 7*e - 4, -13/3*e^2 - 9*e + 14, 1/3*e^2 - 3*e + 5, -2*e - 27, -3*e^2 - 4*e + 9, 1/3*e^2 + e - 5, -4/3*e^2 - 12*e + 5, 8/3*e^2 + 10*e + 1, -2*e^2 + 6*e + 28, e^2 - e - 5, 1/3*e^2 + e + 22, -2*e^2 - 2*e - 2, -e^2 - 5*e - 16, 1/3*e^2 - e - 27, 8/3*e^2 + 10*e + 1, 14/3*e^2 + 5*e - 25, 22/3*e^2 + 17*e - 29, -5/3*e^2 + 2*e + 14, 10/3*e^2 + 9*e + 2, 10/3*e^2 + 12*e - 3, -7/3*e^2 - 6*e + 8, -5/3*e^2 - 7*e + 13, -10*e - 18, -5*e^2 - 4*e + 21, -1/3*e^2 + 4*e - 8, 2/3*e^2 - 3*e - 4, e^2 + 12*e + 6, -4*e^2 - 5*e + 13, 10/3*e^2 + 6*e, 5/3*e^2 + 6*e + 11, -4/3*e^2 - 8*e - 13, 4*e^2 + 5*e - 24, 13/3*e^2 + 6*e - 18, -5*e^2 - 14*e + 7, 4*e^2 + 14*e - 15, 11/3*e^2 + 16*e - 13, 14/3*e^2 + 2*e - 39, e^2 - 3*e - 29, 3*e^2 + 18*e - 10, e^2 + 11*e + 9, -4/3*e^2 + 4*e + 5, -8/3*e^2 - 10*e + 7, -5*e^2 - 9*e + 25, 6*e^2 + 14*e - 9, -2/3*e^2 - 11, e^2 - 4*e - 30, -5/3*e^2 - 4*e - 6, -2*e^2 - 7*e + 34, -17/3*e^2 - 8*e + 20, -7/3*e^2 - 2*e - 17, 11/3*e^2 + 13*e + 7, -e^2 - e + 3, -2*e^2 + e + 18, 7/3*e^2 + 11*e + 5, -5/3*e^2 - 10*e - 16, -5/3*e^2 + 5*e + 23, -4/3*e^2 + 4*e + 15, -23/3*e^2 - 18*e + 32, 3*e^2 + 5*e - 11, 3*e^2 - 2*e - 24, -2*e^2 - 3*e - 6, -4*e^2 - 11*e + 24, -4*e^2 - 13*e + 15, -e^2 - 4*e, 5*e^2 + 14*e - 22, -5*e^2 - 4*e + 18, -11/3*e^2 - 5*e + 28, 26/3*e^2 + 21*e - 32, 16/3*e^2 + 12*e - 8, -1/3*e^2 - 11*e - 8, -1/3*e^2 + 2*e - 18, 5/3*e^2 + 6*e - 44, 7/3*e^2 - e - 7, 14/3*e^2 + 5*e - 26, -10/3*e^2 - 4*e + 22, 6*e^2 + 11*e - 24, -20/3*e^2 - 20*e + 38, 5/3*e^2 - 3*e - 16, 4*e^2 + 17*e, 2/3*e^2 - 6*e - 19, -19/3*e^2 - 8*e + 29, -2*e^2 - 2*e - 15, -5*e^2 - 12*e + 25, -14/3*e^2 - 13*e + 10, 7*e + 28, -1/3*e^2 + 14*e + 6, 1/3*e^2 + 9*e - 5, 8*e^2 + 16*e - 21, -23/3*e^2 - 15*e + 43, 3*e^2 + 15*e + 13, -2/3*e^2 - e - 22, -2*e^2 - 15*e + 14, 5/3*e^2 + 1, e^2 + 18*e + 3, -6*e^2 - 12*e - 8, -e^2 + 4*e + 16, -17/3*e^2 - 2*e + 46, 3*e^2 - 15, -6*e^2 - 9*e + 21, -13/3*e^2 - 16*e + 41, -7*e - 6, 3*e^2 + 4*e - 3, 5*e^2 + 12*e - 32];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {2, 2, w}>] := -1;
ALEigenvalues[ideal<ZF | {5, 5, 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;