/*
  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, 1, -3, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {169,13,-3*w^3 + 2*w^2 + 6*w}>;

primesArray := [
[11, 11, -w^3 + 2*w^2 + w - 3],
[11, 11, w^3 - 3*w],
[16, 2, 2],
[19, 19, -w^3 + 2*w + 2],
[19, 19, 2*w^3 - 3*w^2 - 4*w + 2],
[25, 5, 2*w^3 - 2*w^2 - 4*w + 1],
[29, 29, w^3 - w^2 - 4*w + 1],
[31, 31, w^3 - 4*w + 1],
[31, 31, -w^2 + 2*w + 3],
[41, 41, 2*w^2 - w - 3],
[41, 41, -w^3 + 3*w^2 + w - 4],
[49, 7, 2*w^3 - 3*w^2 - 5*w + 2],
[49, 7, w^2 + w - 3],
[61, 61, 2*w^3 - 3*w^2 - 4*w],
[61, 61, -3*w^3 + 4*w^2 + 7*w - 3],
[79, 79, 2*w^3 - 4*w^2 - 3*w + 2],
[79, 79, w^3 + w^2 - 3*w - 5],
[81, 3, -3],
[89, 89, -3*w^3 + 4*w^2 + 5*w - 3],
[89, 89, 3*w^3 - 2*w^2 - 7*w],
[101, 101, -3*w^3 + 5*w^2 + 7*w - 5],
[101, 101, -2*w^2 - w + 4],
[109, 109, 3*w^3 - 5*w^2 - 5*w + 3],
[109, 109, -2*w^3 + 7*w + 1],
[109, 109, 4*w^3 - 5*w^2 - 9*w + 3],
[109, 109, -2*w^3 + 5*w + 4],
[121, 11, 3*w^3 - 3*w^2 - 6*w + 1],
[131, 131, -3*w^3 + 2*w^2 + 8*w - 2],
[131, 131, w^3 + w^2 - 5*w],
[139, 139, -w^3 + 2*w^2 + 2*w - 6],
[139, 139, 2*w^3 - 5*w^2 - 2*w + 6],
[139, 139, 3*w^3 - 5*w^2 - 6*w + 4],
[139, 139, -2*w^3 + 2*w^2 + 7*w - 2],
[149, 149, -3*w^3 + 4*w^2 + 7*w - 2],
[149, 149, w^3 - w^2 - 3*w - 3],
[149, 149, 3*w^3 - 2*w^2 - 9*w + 2],
[149, 149, w - 4],
[169, 13, -4*w^3 + 5*w^2 + 8*w - 4],
[169, 13, 3*w^3 - 2*w^2 - 6*w],
[179, 179, -4*w^3 + 3*w^2 + 10*w - 3],
[179, 179, -4*w^3 + 4*w^2 + 9*w - 3],
[179, 179, -2*w^3 + w^2 + 8*w - 1],
[179, 179, 4*w^3 - 3*w^2 - 8*w - 2],
[191, 191, -w^3 + 4*w^2 - 6],
[191, 191, -4*w^3 + 4*w^2 + 9*w - 2],
[211, 211, 2*w^3 - w^2 - 8*w],
[211, 211, -w^3 + 4*w^2 + w - 5],
[229, 229, w^3 - 3*w^2 + w + 5],
[229, 229, 2*w^3 - 7*w],
[251, 251, 3*w^3 - 4*w^2 - 7*w + 1],
[251, 251, w^3 - w - 4],
[269, 269, 3*w^3 - 6*w^2 - 3*w + 7],
[269, 269, 3*w^3 - 9*w - 2],
[271, 271, -3*w^3 + w^2 + 10*w],
[271, 271, 4*w^3 - 5*w^2 - 9*w + 2],
[281, 281, -3*w^3 + w^2 + 11*w - 1],
[281, 281, -2*w^2 + 5*w + 4],
[281, 281, -3*w^3 + 3*w^2 + 10*w - 1],
[281, 281, 3*w^3 - 4*w^2 - 7*w - 1],
[311, 311, -w^3 + 7*w + 2],
[311, 311, 3*w^2 - 7],
[331, 331, w^3 + 2*w^2 - 3*w - 7],
[331, 331, w^3 - w^2 - w + 5],
[359, 359, -4*w^3 + 5*w^2 + 6*w - 4],
[359, 359, -5*w^3 + 4*w^2 + 12*w - 2],
[361, 19, -4*w^3 + 4*w^2 + 8*w - 3],
[379, 379, -3*w^3 + 10*w + 2],
[379, 379, 2*w^3 - 5*w^2 + 6],
[389, 389, -5*w^3 + 6*w^2 + 10*w - 3],
[389, 389, w^3 + 3*w^2 - 6*w - 7],
[409, 409, -4*w^3 + 4*w^2 + 7*w],
[409, 409, -4*w^2 + 4*w + 5],
[421, 421, -2*w^3 + 2*w^2 + 8*w - 3],
[421, 421, w^3 + 3*w^2 - 6*w - 6],
[431, 431, 4*w^3 - 6*w^2 - 10*w + 5],
[431, 431, 3*w^3 - 5*w^2 - 6*w + 2],
[431, 431, -4*w^3 + 3*w^2 + 8*w - 2],
[431, 431, -5*w^3 + 6*w^2 + 10*w - 6],
[449, 449, 4*w^3 - 5*w^2 - 6*w + 2],
[449, 449, -w^3 + 2*w^2 + w - 7],
[461, 461, 5*w^3 - 4*w^2 - 12*w + 1],
[461, 461, w^3 + 2*w^2 - 2*w - 6],
[479, 479, w^3 + 3*w^2 - 5*w - 7],
[479, 479, 2*w^3 - 6*w^2 - w + 6],
[491, 491, 3*w^2 - 5*w - 6],
[491, 491, -2*w^3 - w^2 + 9*w + 1],
[499, 499, 4*w^3 - 3*w^2 - 12*w + 4],
[499, 499, 4*w^3 - 5*w^2 - 10*w + 2],
[499, 499, -5*w^3 + 5*w^2 + 11*w - 4],
[499, 499, -4*w^3 + 4*w^2 + 7*w - 3],
[509, 509, -5*w^3 + 6*w^2 + 10*w - 4],
[509, 509, -4*w^2 + 3*w + 8],
[509, 509, -3*w^3 + w^2 + 10*w - 1],
[509, 509, 5*w^3 - 6*w^2 - 12*w + 3],
[521, 521, -5*w^3 + 6*w^2 + 10*w - 5],
[521, 521, 2*w^3 - 6*w^2 - w + 7],
[521, 521, w^3 - 5*w^2 + 9],
[521, 521, w^3 + 3*w^2 - 5*w - 6],
[529, 23, 3*w^3 - 5*w^2 - 6*w + 1],
[529, 23, -2*w^3 - w^2 + 8*w + 1],
[541, 541, 3*w^3 - 5*w^2 - 8*w + 3],
[541, 541, -w^3 + 3*w^2 + 4*w - 7],
[569, 569, -w^3 + 7*w - 1],
[569, 569, -3*w^3 + 2*w^2 + 11*w],
[571, 571, w^3 - 4*w - 6],
[571, 571, 4*w^3 - 5*w^2 - 11*w + 2],
[571, 571, -5*w^3 + 5*w^2 + 11*w - 3],
[571, 571, -2*w^3 + 6*w^2 + 2*w - 9],
[599, 599, 3*w^3 - 6*w^2 - 6*w + 4],
[599, 599, 3*w^2 - 8],
[601, 601, -4*w^3 + 2*w^2 + 12*w - 1],
[601, 601, -5*w^3 + 7*w^2 + 11*w - 5],
[619, 619, -w^3 + 5*w^2 - w - 6],
[619, 619, 4*w^2 - 3*w - 7],
[631, 631, 2*w^3 - 3*w^2 - 5*w - 3],
[631, 631, 5*w^3 - 3*w^2 - 13*w + 2],
[631, 631, 2*w^3 + w^2 - 11*w - 2],
[631, 631, 6*w^3 - 6*w^2 - 14*w + 3],
[641, 641, 2*w^2 - w - 9],
[641, 641, -w^3 + 3*w^2 + 2*w - 10],
[659, 659, -4*w^3 + w^2 + 14*w + 1],
[659, 659, 5*w^3 - 6*w^2 - 11*w + 1],
[709, 709, 6*w^3 - 7*w^2 - 14*w + 3],
[709, 709, 6*w^3 - 8*w^2 - 15*w + 7],
[709, 709, 2*w^3 - 5*w^2 - 5*w + 2],
[709, 709, -4*w^3 + 3*w^2 + 11*w - 5],
[739, 739, 5*w^3 - 7*w^2 - 7*w + 5],
[739, 739, -w^3 - 3*w^2 + 6*w + 1],
[751, 751, 5*w^3 - 3*w^2 - 11*w + 1],
[751, 751, 5*w^3 - 6*w^2 - 13*w + 3],
[769, 769, -w^2 + 2*w + 7],
[769, 769, -4*w^3 + 6*w^2 + 11*w - 4],
[809, 809, -5*w^3 + 4*w^2 + 13*w - 5],
[809, 809, 2*w^3 + w^2 - 8*w],
[811, 811, -w^3 - 4*w^2 + 7*w + 6],
[811, 811, 3*w^3 - 5*w^2 - 7*w],
[811, 811, -2*w^3 - w^2 + 9*w],
[811, 811, -5*w^3 + 5*w^2 + 9*w - 5],
[821, 821, -w^3 + 3*w^2 + 4*w - 8],
[821, 821, -3*w^3 + 7*w + 6],
[821, 821, -3*w^3 + w^2 + 6*w + 5],
[821, 821, 5*w^3 - 8*w^2 - 9*w + 5],
[829, 829, -w^3 + w^2 + w - 6],
[829, 829, 5*w^3 - 6*w^2 - 9*w + 3],
[839, 839, -4*w^3 + 2*w^2 + 14*w - 3],
[839, 839, 5*w^3 - 7*w^2 - 7*w + 4],
[841, 29, -w^2 + 6*w + 1],
[859, 859, -4*w^3 + 2*w^2 + 13*w - 2],
[859, 859, 5*w^3 - 6*w^2 - 12*w + 2],
[881, 881, -2*w^3 + 4*w^2 + w - 9],
[881, 881, -6*w^3 + 9*w^2 + 12*w - 7],
[911, 911, 5*w^3 - 6*w^2 - 11*w],
[911, 911, -w^3 + 4*w^2 + 4*w - 8],
[929, 929, 5*w^3 - 8*w^2 - 11*w + 7],
[929, 929, -4*w^3 + 13*w + 3],
[929, 929, 3*w^3 - 7*w^2 - w + 8],
[929, 929, -6*w^3 + 5*w^2 + 16*w - 5],
[941, 941, -4*w^3 + w^2 + 11*w],
[941, 941, -2*w^3 - w^2 + 10*w - 1],
[941, 941, -5*w^3 + 5*w^2 + 9*w],
[941, 941, -5*w^2 + 5*w + 6],
[961, 31, -5*w^3 + 5*w^2 + 10*w - 3],
[971, 971, -2*w^3 - 3*w^2 + 8*w + 7],
[971, 971, 3*w^3 - 8*w^2 - 2*w + 9]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^3 + 2*x^2 - 20*x - 8;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-1/2*e + 1, e, -1/4*e^2 - 3/2*e + 3, -1/8*e^2 - 1/2*e + 3/2, -1/4*e^2 + 5, 3/4*e^2 + e - 9, 1/4*e^2 + e - 1, 1/2*e^2 - 6, -1/8*e^2 + 1/2*e + 3/2, -7/8*e^2 - 1/2*e + 25/2, -1/4*e^2 - e + 5, 1/8*e^2 - 3/2*e - 19/2, 1/8*e^2 + 1/2*e + 1/2, -1/8*e^2 + 1/2*e - 5/2, 7/8*e^2 + 7/2*e - 25/2, 1/4*e^2 + 2*e + 3, 1/2*e^2 + 1/2*e - 15, 1/4*e^2 + 3/2*e - 6, -1/4*e^2 - 1/2*e - 4, -e^2 + e + 18, 3/4*e^2 - e - 11, 1/4*e^2 + 1/2*e - 8, 5/8*e^2 + 5/2*e - 3/2, 5/8*e^2 + 1/2*e - 23/2, -5/4*e^2 - 4*e + 15, -2*e - 2, -1/4*e^2 - 3*e + 7, 1/8*e^2 - 1/2*e - 15/2, -7/8*e^2 - 3/2*e + 25/2, -1/4*e^2 + 4*e + 9, 9/8*e^2 + 5/2*e - 19/2, 11/8*e^2 + 3/2*e - 33/2, -1/2*e^2 - 3/2*e + 9, -5/4*e^2 - 3*e + 9, 1/4*e^2 - e + 7, -5/8*e^2 - 9/2*e + 27/2, 5/8*e^2 + 3/2*e - 35/2, -1/2*e^2 - 5/2*e + 17, -1, 4*e - 4, -5/8*e^2 - 7/2*e - 5/2, -11/8*e^2 - 1/2*e + 37/2, -1/4*e^2 - 4*e + 9, -1/2*e^2 + 4*e + 12, 1/4*e^2 - 2*e - 7, 5/4*e^2 + 2*e - 9, -5/4*e^2 - 5/2*e + 14, 1/2*e^2 + 7/2*e - 3, -e^2 - e + 12, 1/4*e^2 - e - 27, -3/4*e^2 + 3, -7/8*e^2 + 3/2*e + 25/2, -1/8*e^2 - 9/2*e - 13/2, -1/4*e^2 + 4*e + 5, e^2 + 3*e - 10, -11/4*e^2 - 4*e + 29, -15/8*e^2 - 13/2*e + 53/2, -3/8*e^2 - 7/2*e + 25/2, -1/4*e^2 + 3*e - 7, -15/8*e^2 - 3/2*e + 61/2, 9/8*e^2 - 3/2*e - 55/2, e^2 + 3*e - 18, -1/8*e^2 - 1/2*e + 15/2, 1/4*e^2 - 2*e - 9, 5/8*e^2 - 3/2*e - 19/2, 7/4*e^2 + 5*e - 35, e^2 - 36, -1/4*e^2 + 5/2*e + 12, -3*e - 16, -e^2 - 5*e + 16, 1/4*e^2 + 2*e - 7, 3/8*e^2 + 5/2*e - 45/2, e^2 - e - 16, -11/8*e^2 - 17/2*e + 33/2, 3/2*e^2 + 3*e - 24, 5/8*e^2 + 5/2*e - 39/2, 3/4*e^2 + 11/2*e - 14, -7/4*e^2 - 8*e + 15, 1/2*e^2 + 3*e + 2, 13/8*e^2 + 15/2*e - 71/2, 5/4*e^2 - e - 29, -e^2 + 3/2*e + 27, -3/2*e^2 + e + 38, -2*e^2 - 17/2*e + 37, 7/8*e^2 + 9/2*e - 5/2, 17/8*e^2 + 15/2*e - 63/2, 15/8*e^2 + 13/2*e - 53/2, e^2 - e - 6, 5/2*e^2 + 4*e - 10, 5/4*e^2 + 3/2*e - 4, -e^2 + e + 30, -e^2 - 8*e + 2, 3/4*e^2 + 3*e + 9, -2*e^2 + e + 38, -3/4*e^2 - 5/2*e + 14, -7/4*e^2 - 3*e + 11, e^2 - 30, 3/4*e^2 + 5*e - 3, -5/4*e^2 - 2*e + 3, 5/4*e^2 + 6*e - 11, -7/4*e^2 - 3*e + 31, 11/4*e^2 + 8*e - 35, -7/4*e^2 + 2*e + 33, 2*e^2 + 5*e - 16, 3/2*e^2 + 5*e - 44, -9/4*e^2 - 9*e + 35, -9/4*e^2 - 7/2*e + 20, -5/4*e^2 + 7/2*e + 30, -3/4*e^2 - 11/2*e + 26, 7/4*e^2 + 7/2*e - 34, -3/4*e^2 - 9*e + 19, 1/2*e^2 + 20, -5/4*e^2 - 10*e + 19, 2*e^2 + 6*e - 12, -3*e^2 + 44, 1/2*e^2 - e - 6, 3/4*e^2 - 3*e - 1, -1/8*e^2 - 1/2*e - 65/2, 15/8*e^2 + 11/2*e - 85/2, 11/8*e^2 + 1/2*e - 25/2, 4*e, 1/2*e^2 + 3*e + 8, 5/2*e^2 + 5/2*e - 37, -3/4*e^2 + 15/2*e + 14, 1/2*e^2 - 3*e - 6, 3/2*e^2 + 8*e - 10, 1/2*e^2 + 11*e - 8, -15/8*e^2 - 9/2*e + 37/2, -1/8*e^2 + 13/2*e + 7/2, 7/8*e^2 + 17/2*e + 11/2, 5/4*e^2 + 7/2*e - 30, e^2 - 6*e - 16, -5/4*e^2 - 4*e + 1, 5/4*e^2 - 6*e - 29, 3/2*e^2 + 4*e - 30, e + 18, e^2 - 4*e - 32, 7/8*e^2 + 17/2*e - 5/2, -5/2*e^2 - 5*e + 18, 1/8*e^2 - 17/2*e - 7/2, 1/8*e^2 - 7/2*e + 9/2, -e^2 - 2*e - 30, -3/4*e^2 + 37, 1/4*e^2 - 2*e - 15, 3/2*e^2 + 2*e - 12, e^2 - 3/2*e - 17, -e^2 - 13/2*e + 27, 3/4*e^2 + e + 9, 1/4*e^2 - 11*e - 13, e^2 + 5*e - 36, 5/4*e^2 + 3*e + 3, -7/4*e^2 + 2*e + 35, 9/4*e^2 + 6*e - 45, -2*e^2 - 2*e + 18, -5/4*e^2 - 3*e + 41, 1/8*e^2 - 13/2*e - 43/2, 5/8*e^2 + 1/2*e - 83/2, e^2 + 16, 9/4*e^2 + 9/2*e - 48, 9/4*e^2 + 3*e - 13, -5/2*e^2 - 21/2*e + 17, -5/4*e^2 + 3*e + 1, 11/8*e^2 + 5/2*e - 93/2, 25/8*e^2 + 23/2*e - 99/2];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {169,13,-3*w^3 + 2*w^2 + 6*w}>] := 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;