/*
  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 | {1, 1, 1}>;

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^3 - 13*x - 13;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 2/5*e^2 - 8/5*e - 19/5, -2/5*e^2 - 2/5*e + 24/5, -1/5*e^2 + 4/5*e + 32/5, -3/5*e^2 + 2/5*e + 16/5, -2/5*e^2 + 8/5*e + 9/5, -2/5*e^2 + 8/5*e + 24/5, 4/5*e^2 - 6/5*e - 38/5, -3/5*e^2 + 7/5*e + 46/5, 6/5*e^2 - 14/5*e - 57/5, 3/5*e^2 - 7/5*e - 16/5, 4/5*e^2 - 6/5*e + 2/5, 2/5*e^2 - 3/5*e - 14/5, 1/5*e^2 - 4/5*e - 22/5, -3/5*e^2 - 3/5*e - 4/5, 2*e - 2, -6/5*e^2 + 14/5*e + 17/5, -7/5*e^2 - 7/5*e + 64/5, 2/5*e^2 - 8/5*e - 24/5, 2*e^2 - 4*e - 18, -4/5*e^2 + 11/5*e + 38/5, -4/5*e^2 + 16/5*e + 18/5, 2/5*e^2 + 7/5*e - 24/5, -18/5*e^2 + 17/5*e + 136/5, -2*e^2 + e + 14, -2*e^2 + 4*e + 21, -6/5*e^2 + 24/5*e + 42/5, 1/5*e^2 + 16/5*e - 32/5, -1/5*e^2 - 6/5*e - 38/5, 4*e + 1, -1/5*e^2 - 1/5*e - 8/5, -2*e + 14, 18/5*e^2 - 42/5*e - 176/5, 2*e + 2, 14/5*e^2 - 16/5*e - 118/5, -1/5*e^2 + 9/5*e + 32/5, -4/5*e^2 + 6/5*e + 118/5, -4/5*e^2 + 16/5*e + 58/5, 14/5*e^2 - 31/5*e - 128/5, 2*e, 22/5*e^2 - 28/5*e - 154/5, 3*e^2 - e - 22, 2/5*e^2 - 8/5*e + 86/5, -13/5*e^2 - 3/5*e + 126/5, 12/5*e^2 - 28/5*e - 94/5, 2*e^2 - 8*e - 10, 13/5*e^2 - 27/5*e - 56/5, -7/5*e^2 + 33/5*e + 54/5, 2*e^2 - 4*e - 26, 6/5*e^2 - 29/5*e - 102/5, 8/5*e^2 + 8/5*e - 126/5, -22/5*e^2 + 8/5*e + 204/5, 4/5*e^2 + 24/5*e + 2/5, -6*e^2 + 8*e + 50, 14/5*e^2 + 4/5*e - 168/5, -13/5*e^2 + 17/5*e + 156/5, -12/5*e^2 - 2/5*e + 64/5, 6/5*e^2 + 26/5*e - 82/5, 22/5*e^2 - 28/5*e - 149/5, -12/5*e^2 + 8/5*e + 54/5, 2/5*e^2 + 2/5*e - 89/5, 4/5*e^2 + 24/5*e + 2/5, 4/5*e^2 + 4/5*e - 128/5, -8/5*e^2 + 2/5*e + 26/5, -2*e^2 - 4*e + 26, -4*e^2 + 6*e + 42, 14/5*e^2 + 4/5*e - 158/5, 4*e^2 - 6*e - 43, 1/5*e^2 - 29/5*e - 52/5, 2/5*e^2 + 17/5*e - 64/5, -2/5*e^2 + 8/5*e + 24/5, 6/5*e^2 - 14/5*e - 97/5, -19/5*e^2 + 41/5*e + 98/5, -e^2 + 2*e + 8, -21/5*e^2 + 29/5*e + 202/5, 16/5*e^2 - 34/5*e - 122/5, 22/5*e^2 - 8/5*e - 174/5, -2*e^2 - 2*e + 28, 2*e^2 - 8*e - 30, -2*e^2 + 4*e + 42, -4/5*e^2 + 46/5*e + 8/5, -16/5*e^2 + 44/5*e + 102/5, 4/5*e^2 + 4/5*e - 128/5, -22/5*e^2 + 28/5*e + 114/5, -14/5*e^2 + 16/5*e + 108/5, 2*e + 18, 18/5*e^2 - 62/5*e - 186/5, 18/5*e^2 - 22/5*e - 156/5, -22/5*e^2 + 8/5*e + 144/5, -26/5*e^2 + 34/5*e + 202/5, 24/5*e^2 - 41/5*e - 118/5, 38/5*e^2 - 57/5*e - 326/5, 3*e^2 - 8*e - 38, 12/5*e^2 - 8/5*e - 164/5, 21/5*e^2 - 64/5*e - 172/5, 4*e + 18, -2*e^2 - 2*e + 32, 8/5*e^2 - 62/5*e - 101/5, 3*e^2 - 2*e - 38, -18/5*e^2 + 22/5*e + 186/5, -8/5*e^2 + 32/5*e - 39/5, 2*e^2 - 6*e + 2, 8/5*e^2 - 32/5*e - 6/5, -3*e^2 + 14*e + 18, 24/5*e^2 - 6/5*e - 128/5, 12/5*e^2 - 18/5*e - 89/5, -4*e^2 + 4*e + 28, -19/5*e^2 + 31/5*e + 138/5, -4*e^2 + 12*e + 42, 18/5*e^2 + 8/5*e - 136/5, 44/5*e^2 - 66/5*e - 328/5, -28/5*e^2 + 57/5*e + 226/5, -14/5*e^2 + 11/5*e - 52/5, 8/5*e^2 - 22/5*e - 36/5, -2*e^2 - 2*e + 32, -2/5*e^2 + 18/5*e - 146/5, 6/5*e^2 - 4/5*e + 98/5, 6*e^2 - 12*e - 46, 7/5*e^2 - 13/5*e - 54/5, -32/5*e^2 + 18/5*e + 244/5, 8/5*e^2 - 22/5*e - 11/5, 13/5*e^2 - 42/5*e - 196/5, 16/5*e^2 - 64/5*e - 222/5, -2*e^2 - 2*e + 18, -23/5*e^2 + 32/5*e + 246/5, -18/5*e^2 + 62/5*e + 266/5, -12/5*e^2 - 32/5*e + 164/5, 4/5*e^2 - 6/5*e - 208/5, 12/5*e^2 - 38/5*e - 134/5, -14/5*e^2 + 36/5*e + 308/5, 6/5*e^2 - 44/5*e - 142/5, -26/5*e^2 + 4/5*e + 277/5, 16/5*e^2 + 11/5*e - 282/5, -8/5*e^2 - 23/5*e + 86/5, 12/5*e^2 - 13/5*e - 134/5, 4/5*e^2 - 11/5*e - 138/5, -4/5*e^2 - 4/5*e - 42/5, -17/5*e^2 + 23/5*e + 174/5, 16/5*e^2 - 29/5*e - 242/5, -19/5*e^2 + 21/5*e + 218/5, 4*e^2 - 20, 6*e^2 - 10*e - 51, 27/5*e^2 - 68/5*e - 314/5, 1/5*e^2 + 46/5*e - 52/5, 23/5*e^2 - 52/5*e - 96/5, -2/5*e^2 - 2/5*e - 121/5, 3*e^2 - 9*e - 46, -6*e^2 + 3*e + 56, 8/5*e^2 + 8/5*e - 136/5, 4/5*e^2 - 1/5*e - 48/5, 24/5*e^2 - 31/5*e - 88/5, 7*e^2 - 6*e - 34, 2/5*e^2 - 28/5*e - 19/5, -6/5*e^2 + 4/5*e + 62/5, 4*e^2 + 4*e - 30, -14/5*e^2 + 56/5*e + 28/5, -24/5*e^2 + 36/5*e + 258/5, 24/5*e^2 - 36/5*e - 348/5, 16/5*e^2 + 6/5*e - 212/5, -27/5*e^2 + 18/5*e + 164/5, -14/5*e^2 + 26/5*e - 37/5, 8*e^2 - 20*e - 70, 6/5*e^2 - 4/5*e - 152/5, 17/5*e^2 - 43/5*e - 234/5, e^2 - 4*e - 34, -37/5*e^2 + 78/5*e + 404/5, 4/5*e^2 + 64/5*e - 158/5, -6/5*e^2 + 4/5*e + 102/5, -36/5*e^2 + 54/5*e + 382/5, -4*e^2 + 12*e + 45];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();

// 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;