/*
  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![11, 6, -8, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3}>;

primesArray := [
[9, 3, 1/3*w^3 - 8/3*w - 2/3],
[9, 3, -w + 1],
[11, 11, w],
[11, 11, -1/3*w^3 + 2/3*w + 5/3],
[11, 11, 1/3*w^3 - 8/3*w + 1/3],
[11, 11, -2/3*w^3 + 13/3*w + 4/3],
[16, 2, 2],
[25, 5, 2/3*w^3 - 10/3*w - 1/3],
[29, 29, -w - 3],
[29, 29, -1/3*w^3 + 8/3*w - 10/3],
[31, 31, w^3 - 6*w + 1],
[31, 31, w^3 - 6*w - 2],
[41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3],
[41, 41, 2/3*w^3 - 13/3*w + 5/3],
[59, 59, 2/3*w^3 - 13/3*w + 8/3],
[59, 59, w^3 + w^2 - 6*w - 4],
[79, 79, 2/3*w^3 + w^2 - 10/3*w - 19/3],
[79, 79, 1/3*w^3 + w^2 - 2/3*w - 17/3],
[89, 89, 4/3*w^3 - 23/3*w - 5/3],
[89, 89, 1/3*w^3 + 2*w^2 - 5/3*w - 32/3],
[101, 101, 5/3*w^3 + w^2 - 31/3*w - 16/3],
[101, 101, 2/3*w^3 - w^2 - 4/3*w + 8/3],
[109, 109, -w^3 - 2*w^2 + 7*w + 7],
[109, 109, -1/3*w^3 + 2*w^2 + 2/3*w - 16/3],
[131, 131, w^3 - w^2 - 7*w + 5],
[131, 131, -2/3*w^3 - w^2 + 4/3*w + 13/3],
[139, 139, -2/3*w^3 - w^2 + 13/3*w + 4/3],
[139, 139, w^2 - w - 7],
[151, 151, -5/3*w^3 + 28/3*w - 5/3],
[151, 151, -1/3*w^3 + w^2 + 2/3*w - 16/3],
[151, 151, w^3 + w^2 - 6*w - 3],
[151, 151, -w^3 + w^2 + 7*w - 6],
[179, 179, 5/3*w^3 + w^2 - 28/3*w - 19/3],
[179, 179, -1/3*w^3 + w^2 + 11/3*w - 4/3],
[181, 181, -4/3*w^3 - 2*w^2 + 20/3*w + 29/3],
[181, 181, 1/3*w^3 + w^2 - 2/3*w - 20/3],
[181, 181, w^3 + w^2 - 5*w - 7],
[181, 181, 5/3*w^3 + w^2 - 28/3*w - 16/3],
[191, 191, 2*w - 1],
[191, 191, 2/3*w^3 - 16/3*w - 1/3],
[199, 199, 1/3*w^3 - 11/3*w + 4/3],
[199, 199, 1/3*w^3 - 11/3*w - 2/3],
[211, 211, -2/3*w^3 - w^2 + 19/3*w + 16/3],
[211, 211, 2/3*w^3 + w^2 - 19/3*w - 7/3],
[229, 229, 1/3*w^3 - 11/3*w + 1/3],
[239, 239, -5/3*w^3 - w^2 + 25/3*w + 19/3],
[239, 239, -4/3*w^3 + w^2 + 20/3*w - 7/3],
[251, 251, -2/3*w^3 + w^2 + 16/3*w - 14/3],
[251, 251, -1/3*w^3 - w^2 - 1/3*w + 14/3],
[269, 269, 2/3*w^3 + w^2 - 7/3*w - 19/3],
[269, 269, 1/3*w^3 + 2*w^2 - 2/3*w - 32/3],
[281, 281, w^3 - w^2 - 5*w + 1],
[281, 281, 4/3*w^3 + w^2 - 20/3*w - 23/3],
[289, 17, 2/3*w^3 + w^2 - 16/3*w - 4/3],
[289, 17, 1/3*w^3 + w^2 - 11/3*w - 20/3],
[331, 331, 1/3*w^3 + 2*w^2 - 8/3*w - 23/3],
[331, 331, 2/3*w^3 + 2*w^2 - 13/3*w - 28/3],
[349, 349, 4/3*w^3 - 17/3*w + 1/3],
[349, 349, -5/3*w^3 + 28/3*w - 2/3],
[361, 19, -1/3*w^3 + 5/3*w + 14/3],
[361, 19, 4/3*w^3 - 20/3*w - 5/3],
[379, 379, 4/3*w^3 + 2*w^2 - 23/3*w - 41/3],
[379, 379, -5/3*w^3 + 31/3*w - 5/3],
[389, 389, 5/3*w^3 + w^2 - 25/3*w - 16/3],
[389, 389, -4/3*w^3 + w^2 + 20/3*w - 10/3],
[401, 401, -1/3*w^3 - 3*w^2 + 5/3*w + 26/3],
[401, 401, 1/3*w^3 + w^2 - 11/3*w - 23/3],
[401, 401, -2/3*w^3 - 3*w^2 + 10/3*w + 52/3],
[401, 401, 2/3*w^3 + w^2 - 16/3*w - 1/3],
[409, 409, -2*w^3 - w^2 + 11*w + 2],
[409, 409, -5/3*w^3 - 2*w^2 + 28/3*w + 37/3],
[409, 409, 1/3*w^3 - 11/3*w - 17/3],
[409, 409, 5/3*w^3 + 3*w^2 - 25/3*w - 37/3],
[419, 419, -1/3*w^3 + 3*w^2 - 1/3*w - 37/3],
[419, 419, 2*w^3 + 3*w^2 - 12*w - 13],
[421, 421, -1/3*w^3 + 3*w^2 + 2/3*w - 31/3],
[421, 421, -5/3*w^3 - w^2 + 25/3*w + 13/3],
[421, 421, -w^3 + 7*w - 4],
[421, 421, -4/3*w^3 + w^2 + 20/3*w - 13/3],
[439, 439, -4/3*w^3 + w^2 + 14/3*w - 7/3],
[439, 439, 4/3*w^3 - 17/3*w - 5/3],
[439, 439, -7/3*w^3 - w^2 + 41/3*w + 17/3],
[439, 439, 5/3*w^3 - 28/3*w - 4/3],
[449, 449, w^3 + 3*w^2 - 6*w - 17],
[449, 449, 1/3*w^3 + 3*w^2 - 8/3*w - 26/3],
[449, 449, -4/3*w^3 + 26/3*w + 17/3],
[449, 449, 7/3*w^3 + w^2 - 38/3*w - 8/3],
[461, 461, -5/3*w^3 + w^2 + 28/3*w - 14/3],
[461, 461, -2/3*w^3 - 2*w^2 + 10/3*w + 43/3],
[461, 461, -2*w^3 - w^2 + 12*w + 4],
[461, 461, -w^3 + 8*w - 5],
[479, 479, w^3 + w^2 - 7*w - 2],
[479, 479, -4/3*w^3 - w^2 + 23/3*w + 5/3],
[499, 499, -4/3*w^3 - 2*w^2 + 23/3*w + 17/3],
[499, 499, -4/3*w^3 - w^2 + 29/3*w + 8/3],
[521, 521, 2*w^3 - 11*w + 2],
[521, 521, 4/3*w^3 - w^2 - 14/3*w + 1/3],
[569, 569, w^3 + 3*w^2 - 6*w - 9],
[569, 569, 4/3*w^3 + w^2 - 26/3*w - 32/3],
[569, 569, -w^3 + 6*w - 6],
[569, 569, -1/3*w^3 - 3*w^2 + 8/3*w + 50/3],
[571, 571, 2/3*w^3 + 3*w^2 - 10/3*w - 46/3],
[571, 571, 2*w^2 - 15],
[599, 599, -w^3 + 2*w^2 + 4*w - 10],
[599, 599, -w^2 - 3*w + 8],
[601, 601, 5/3*w^3 + 3*w^2 - 28/3*w - 34/3],
[601, 601, 1/3*w^3 - 3*w^2 - 2/3*w + 43/3],
[619, 619, 2/3*w^3 - w^2 - 13/3*w + 2/3],
[619, 619, 2/3*w^3 + w^2 - 7/3*w - 25/3],
[619, 619, 2*w^3 + w^2 - 11*w - 6],
[619, 619, -2/3*w^3 + 13/3*w - 17/3],
[631, 631, 2/3*w^3 + 2*w^2 - 19/3*w - 31/3],
[631, 631, w^3 + 2*w^2 - 8*w - 6],
[631, 631, 2/3*w^3 + 2*w^2 - 4/3*w - 31/3],
[631, 631, -2/3*w^3 + 2*w^2 + 16/3*w - 23/3],
[659, 659, -2/3*w^3 + 1/3*w + 16/3],
[659, 659, 5/3*w^3 - 34/3*w - 13/3],
[661, 661, w^3 - 8*w + 2],
[661, 661, 4/3*w^3 + w^2 - 23/3*w + 1/3],
[661, 661, 5/3*w^3 + w^2 - 31/3*w - 10/3],
[661, 661, -3*w - 1],
[691, 691, -4/3*w^3 + 29/3*w + 2/3],
[691, 691, 1/3*w^3 + 4/3*w - 5/3],
[691, 691, -5/3*w^3 + w^2 + 25/3*w - 5/3],
[691, 691, 2*w^3 + w^2 - 10*w - 7],
[701, 701, -2/3*w^3 - 3*w^2 + 19/3*w + 46/3],
[701, 701, w^3 - 2*w^2 - 6*w + 8],
[701, 701, 4/3*w^3 + 2*w^2 - 17/3*w - 29/3],
[701, 701, 5/3*w^3 + 3*w^2 - 28/3*w - 52/3],
[709, 709, 1/3*w^3 - 14/3*w + 10/3],
[709, 709, -2/3*w^3 + 19/3*w + 7/3],
[719, 719, 3*w^2 - w - 10],
[719, 719, -4/3*w^3 - 3*w^2 + 23/3*w + 47/3],
[739, 739, 5/3*w^3 + w^2 - 34/3*w - 10/3],
[739, 739, 4/3*w^3 + 2*w^2 - 23/3*w - 14/3],
[761, 761, 7/3*w^3 + 2*w^2 - 41/3*w - 26/3],
[761, 761, 2*w^3 + 3*w^2 - 12*w - 15],
[769, 769, -2/3*w^3 + 19/3*w - 8/3],
[769, 769, -4/3*w^3 + 2*w^2 + 17/3*w - 31/3],
[769, 769, 1/3*w^3 - 14/3*w - 5/3],
[769, 769, -1/3*w^3 + 3*w^2 + 2/3*w - 34/3],
[809, 809, -4/3*w^3 + w^2 + 26/3*w - 7/3],
[809, 809, w^3 + w^2 - 3*w - 7],
[811, 811, 4/3*w^3 + w^2 - 17/3*w - 23/3],
[811, 811, -1/3*w^3 - 2*w^2 + 2/3*w + 38/3],
[821, 821, 8/3*w^3 + 3*w^2 - 43/3*w - 34/3],
[821, 821, 2/3*w^3 + 2*w^2 - 7/3*w - 34/3],
[821, 821, -w^3 - w^2 + 4*w + 8],
[821, 821, 2/3*w^3 + 4*w^2 - 7/3*w - 58/3],
[829, 829, 4/3*w^3 + 3*w^2 - 26/3*w - 50/3],
[829, 829, 7/3*w^3 - 41/3*w - 5/3],
[839, 839, 2*w^3 + w^2 - 10*w - 3],
[839, 839, 5/3*w^3 - w^2 - 25/3*w + 17/3],
[841, 29, 5/3*w^3 - 25/3*w - 7/3],
[859, 859, -7/3*w^3 - w^2 + 32/3*w + 29/3],
[859, 859, 2*w^3 + w^2 - 14*w - 6],
[859, 859, 1/3*w^3 + 2*w^2 - 2/3*w - 47/3],
[859, 859, -5/3*w^3 - 2*w^2 + 31/3*w + 49/3],
[881, 881, w^3 + 4*w^2 - 4*w - 20],
[881, 881, 8/3*w^3 + 2*w^2 - 43/3*w - 34/3],
[919, 919, -w^3 + 3*w - 7],
[919, 919, 1/3*w^3 + 3*w^2 - 5/3*w - 35/3],
[919, 919, -5/3*w^3 + 31/3*w - 23/3],
[919, 919, 2/3*w^3 + 3*w^2 - 10/3*w - 43/3],
[929, 929, w^2 - 2*w - 9],
[929, 929, w^3 + w^2 - 7*w + 1],
[961, 31, 2/3*w^3 - 10/3*w - 19/3],
[971, 971, -w^3 - 3*w^2 + 7*w + 16],
[971, 971, -5/3*w^3 + w^2 + 25/3*w - 8/3],
[971, 971, 2*w^3 + w^2 - 10*w - 6],
[971, 971, 1/3*w^3 + 2*w^2 - 8/3*w - 47/3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^9 - 2*x^8 - 43*x^7 + 94*x^6 + 446*x^5 - 1101*x^4 - 616*x^3 + 2034*x^2 + 270*x - 914;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, -233667190/12191019113*e^8 + 74359977/12191019113*e^7 + 9996783760/12191019113*e^6 - 4939271955/12191019113*e^5 - 104955430794/12191019113*e^4 + 69434522490/12191019113*e^3 + 184011546248/12191019113*e^2 - 24209839454/12191019113*e - 41841204404/12191019113, 218071166/12191019113*e^8 - 93936012/12191019113*e^7 - 9220535706/12191019113*e^6 + 5638060361/12191019113*e^5 + 93730337924/12191019113*e^4 - 74524367938/12191019113*e^3 - 139529678404/12191019113*e^2 + 20424327736/12191019113*e + 57592487710/12191019113, -2342550/1741574159*e^8 + 12248927/1741574159*e^7 + 102426737/1741574159*e^6 - 548462391/1741574159*e^5 - 1057674089/1741574159*e^4 + 6145350693/1741574159*e^3 + 817893490/1741574159*e^2 - 10031713225/1741574159*e - 347458392/1741574159, 114260022/12191019113*e^8 + 232846777/12191019113*e^7 - 5366348823/12191019113*e^6 - 8945052609/12191019113*e^5 + 73320161160/12191019113*e^4 + 76189119432/12191019113*e^3 - 331085684898/12191019113*e^2 - 49408931608/12191019113*e + 252342337700/12191019113, -250065040/12191019113*e^8 + 160102466/12191019113*e^7 + 10713770919/12191019113*e^6 - 8778508692/12191019113*e^5 - 112359149417/12191019113*e^4 + 112451977341/12191019113*e^3 + 189736800678/12191019113*e^2 - 106622851142/12191019113*e - 44273413148/12191019113, -57650961/12191019113*e^8 - 325304508/12191019113*e^7 + 2876519211/12191019113*e^6 + 13154610958/12191019113*e^5 - 45826310343/12191019113*e^4 - 122696022356/12191019113*e^3 + 274800109905/12191019113*e^2 + 118075459812/12191019113*e - 213904251473/12191019113, -218962671/12191019113*e^8 + 211702704/12191019113*e^7 + 9193189186/12191019113*e^6 - 10683284277/12191019113*e^5 - 90030632533/12191019113*e^4 + 126664421074/12191019113*e^3 + 83056132921/12191019113*e^2 - 95024724128/12191019113*e - 8367190254/12191019113, 77130744/12191019113*e^8 - 186196012/12191019113*e^7 - 3187460910/12191019113*e^6 + 8393291812/12191019113*e^5 + 28307942095/12191019113*e^4 - 88787516390/12191019113*e^3 + 19346796822/12191019113*e^2 + 61739536860/12191019113*e - 15047365156/12191019113, 235270842/12191019113*e^8 - 284997025/12191019113*e^7 - 9878261970/12191019113*e^6 + 14015322241/12191019113*e^5 + 97312682300/12191019113*e^4 - 165649123088/12191019113*e^3 - 106498319420/12191019113*e^2 + 181464839394/12191019113*e + 78208188504/12191019113, 49818434/1741574159*e^8 + 7595468/1741574159*e^7 - 2186267384/1741574159*e^6 + 76034927/1741574159*e^5 + 24922031101/1741574159*e^4 - 5907529051/1741574159*e^3 - 68048659676/1741574159*e^2 + 9374203847/1741574159*e + 44623862022/1741574159, 228963018/12191019113*e^8 - 387784876/12191019113*e^7 - 9418103386/12191019113*e^6 + 18003605390/12191019113*e^5 + 84092542452/12191019113*e^4 - 196506366016/12191019113*e^3 + 41728778869/12191019113*e^2 + 144280510022/12191019113*e - 126095586404/12191019113, -1, 327601914/12191019113*e^8 - 804649017/12191019113*e^7 - 13654253719/12191019113*e^6 + 36159817168/12191019113*e^5 + 126743952354/12191019113*e^4 - 384125796192/12191019113*e^3 + 2436684214/12191019113*e^2 + 374233260214/12191019113*e - 46180692260/12191019113, -174257033/12191019113*e^8 - 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31775190976/12191019113*e^5 + 485235053351/12191019113*e^4 + 224101992388/12191019113*e^3 - 1370072593684/12191019113*e^2 - 404775950424/12191019113*e + 856218491536/12191019113];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {41, 41, 2/3*w^3 + w^2 - 13/3*w - 10/3}>] := 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;