/*
  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![-49, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {9, 3, 3}>;

primesArray := [
[4, 2, 2],
[7, 7, w - 7],
[7, 7, w + 6],
[9, 3, 3],
[19, 19, w + 5],
[19, 19, w - 6],
[23, 23, w + 8],
[23, 23, -w + 9],
[25, 5, 5],
[29, 29, -w - 4],
[29, 29, w - 5],
[37, 37, -w - 3],
[37, 37, w - 4],
[41, 41, -w - 9],
[41, 41, w - 10],
[43, 43, -w - 2],
[43, 43, w - 3],
[47, 47, -w - 1],
[47, 47, w - 2],
[53, 53, 2*w - 13],
[53, 53, -2*w - 11],
[59, 59, -4*w - 25],
[59, 59, -4*w + 29],
[61, 61, -w - 10],
[61, 61, w - 11],
[83, 83, -w - 11],
[83, 83, w - 12],
[97, 97, 2*w - 11],
[97, 97, -2*w - 9],
[101, 101, 3*w - 20],
[101, 101, -3*w - 17],
[107, 107, -w - 12],
[107, 107, w - 13],
[109, 109, -5*w - 31],
[109, 109, -5*w + 36],
[121, 11, -11],
[127, 127, 2*w - 19],
[127, 127, -2*w - 17],
[137, 137, -3*w - 16],
[137, 137, 3*w - 19],
[157, 157, -3*w - 23],
[157, 157, 3*w - 26],
[163, 163, -4*w + 27],
[163, 163, 4*w + 23],
[169, 13, -13],
[173, 173, -6*w - 37],
[173, 173, -6*w + 43],
[181, 181, 2*w - 5],
[181, 181, -2*w - 3],
[191, 191, -w - 15],
[191, 191, w - 16],
[193, 193, 2*w - 3],
[193, 193, -2*w - 1],
[197, 197, 2*w - 1],
[223, 223, -w - 16],
[223, 223, w - 17],
[233, 233, -3*w - 13],
[233, 233, 3*w - 16],
[239, 239, -5*w + 34],
[239, 239, -5*w - 29],
[251, 251, 5*w - 41],
[251, 251, 5*w + 36],
[257, 257, -w - 17],
[257, 257, w - 18],
[289, 17, -17],
[293, 293, -w - 18],
[293, 293, w - 19],
[311, 311, -3*w - 10],
[311, 311, 3*w - 13],
[313, 313, -3*w - 26],
[313, 313, 3*w - 29],
[331, 331, -w - 19],
[331, 331, w - 20],
[347, 347, 4*w - 23],
[347, 347, -4*w - 19],
[353, 353, 3*w - 11],
[353, 353, -3*w - 8],
[379, 379, 2*w - 25],
[379, 379, -2*w - 23],
[401, 401, 3*w - 8],
[401, 401, -3*w - 5],
[409, 409, 5*w - 43],
[409, 409, 5*w + 38],
[419, 419, -5*w - 26],
[419, 419, 5*w - 31],
[431, 431, 3*w - 5],
[431, 431, -3*w - 2],
[433, 433, -7*w + 48],
[433, 433, -7*w - 41],
[443, 443, 3*w - 2],
[443, 443, 3*w - 1],
[449, 449, -9*w - 55],
[449, 449, -9*w + 64],
[457, 457, -w - 22],
[457, 457, w - 23],
[479, 479, 2*w - 27],
[479, 479, -2*w - 25],
[487, 487, -3*w - 29],
[487, 487, 3*w - 32],
[491, 491, -5*w - 39],
[491, 491, 5*w - 44],
[499, 499, 4*w - 19],
[499, 499, -4*w - 15],
[503, 503, -w - 23],
[503, 503, w - 24],
[521, 521, 11*w - 86],
[521, 521, -7*w + 47],
[557, 557, 7*w + 51],
[557, 557, 7*w - 58],
[563, 563, -4*w - 13],
[563, 563, 4*w - 17],
[569, 569, -10*w - 61],
[569, 569, -10*w + 71],
[587, 587, 2*w - 29],
[587, 587, -2*w - 27],
[601, 601, -w - 25],
[601, 601, w - 26],
[607, 607, -7*w - 39],
[607, 607, 7*w - 46],
[613, 613, 3*w - 34],
[613, 613, -3*w - 31],
[617, 617, -6*w - 31],
[617, 617, 6*w - 37],
[619, 619, 4*w - 15],
[619, 619, -4*w - 11],
[631, 631, 5*w - 27],
[631, 631, -5*w - 22],
[653, 653, -w - 26],
[653, 653, w - 27],
[661, 661, 5*w - 46],
[661, 661, -5*w - 41],
[683, 683, -9*w + 62],
[683, 683, -9*w - 53],
[691, 691, 7*w - 45],
[691, 691, -7*w - 38],
[727, 727, -6*w - 47],
[727, 727, 6*w - 53],
[733, 733, 4*w - 41],
[733, 733, -4*w - 37],
[739, 739, -4*w - 5],
[739, 739, 4*w - 9],
[751, 751, 8*w - 53],
[751, 751, -8*w - 45],
[769, 769, 5*w - 24],
[769, 769, -5*w - 19],
[773, 773, 7*w - 44],
[773, 773, -7*w - 37],
[787, 787, 4*w - 3],
[787, 787, 4*w - 1],
[797, 797, -9*w - 52],
[797, 797, -9*w + 61],
[811, 811, -5*w - 18],
[811, 811, 5*w - 23],
[821, 821, -w - 29],
[821, 821, w - 30],
[827, 827, 2*w - 33],
[827, 827, -2*w - 31],
[829, 829, -10*w + 69],
[829, 829, -10*w - 59],
[839, 839, 5*w - 48],
[839, 839, -5*w - 43],
[853, 853, -7*w - 36],
[853, 853, 7*w - 43],
[881, 881, -w - 30],
[881, 881, w - 31],
[961, 31, -31],
[991, 991, -5*w - 13],
[991, 991, 5*w - 18]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 - 3*x^3 - 23*x^2 + 41*x + 139;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1/11*e^3 - 1/11*e^2 - 14/11*e - 20/11, 1/11*e^3 - 1/11*e^2 - 25/11*e + 13/11, e, -1, -2/11*e^3 - 9/11*e^2 + 39/11*e + 150/11, e^2 - e - 13, -2/11*e^3 + 2/11*e^2 + 39/11*e + 40/11, -1/11*e^3 + 1/11*e^2 + 3/11*e + 53/11, -3/11*e^3 + 3/11*e^2 + 42/11*e - 50/11, -1/11*e^3 + 1/11*e^2 + 14/11*e - 24/11, -1/11*e^3 + 1/11*e^2 + 14/11*e - 24/11, 5/11*e^3 + 6/11*e^2 - 92/11*e - 155/11, 2/11*e^3 - 13/11*e^2 - 6/11*e + 125/11, 5/11*e^3 - 5/11*e^2 - 59/11*e - 1/11, 6/11*e^3 - 6/11*e^2 - 95/11*e + 12/11, -1/11*e^3 + 12/11*e^2 - 8/11*e - 145/11, -4/11*e^3 - 7/11*e^2 + 78/11*e + 135/11, 1/11*e^3 - 1/11*e^2 - 14/11*e - 42/11, 1/11*e^3 - 1/11*e^2 - 14/11*e - 42/11, -2/11*e^3 - 9/11*e^2 + 61/11*e + 139/11, 2/11*e^3 + 9/11*e^2 - 61/11*e - 128/11, -e^2 + 2*e + 19, 3/11*e^3 + 8/11*e^2 - 64/11*e - 71/11, e^2 - e - 8, -2/11*e^3 - 9/11*e^2 + 39/11*e + 205/11, 2/11*e^3 - 13/11*e^2 - 6/11*e + 169/11, 5/11*e^3 + 6/11*e^2 - 92/11*e - 111/11, 9/11*e^3 + 2/11*e^2 - 126/11*e - 191/11, 8/11*e^3 - 19/11*e^2 - 112/11*e + 115/11, -9/11*e^3 - 2/11*e^2 + 126/11*e + 213/11, -8/11*e^3 + 19/11*e^2 + 112/11*e - 93/11, -2/11*e^3 - 20/11*e^2 + 50/11*e + 326/11, 2/11*e^3 + 20/11*e^2 - 50/11*e - 260/11, 2*e^2 - 3*e - 23, -5/11*e^3 - 17/11*e^2 + 103/11*e + 320/11, 10/11*e^3 - 10/11*e^2 - 140/11*e - 112/11, 1/11*e^3 - 1/11*e^2 - 47/11*e + 101/11, -2/11*e^3 + 2/11*e^2 + 61/11*e + 62/11, 7/11*e^3 - 7/11*e^2 - 142/11*e + 47/11, 3/11*e^3 - 3/11*e^2 + 2/11*e - 5/11, 4/11*e^3 - 15/11*e^2 - 12/11*e + 173/11, 9/11*e^3 + 2/11*e^2 - 170/11*e - 81/11, 4/11*e^3 - 26/11*e^2 - 34/11*e + 305/11, 8/11*e^3 + 14/11*e^2 - 134/11*e - 281/11, 4/11*e^3 - 4/11*e^2 - 56/11*e - 102/11, -7/11*e^3 - 4/11*e^2 + 98/11*e + 217/11, -6/11*e^3 + 17/11*e^2 + 84/11*e - 89/11, 6/11*e^3 - 6/11*e^2 - 106/11*e + 56/11, 4/11*e^3 - 4/11*e^2 - 34/11*e + 30/11, 13/11*e^3 + 20/11*e^2 - 226/11*e - 524/11, 6/11*e^3 - 39/11*e^2 - 40/11*e + 342/11, -3/11*e^3 + 36/11*e^2 - 2/11*e - 314/11, -10/11*e^3 - 23/11*e^2 + 184/11*e + 552/11, 8/11*e^3 - 8/11*e^2 - 112/11*e + 5/11, -10/11*e^3 - 12/11*e^2 + 162/11*e + 376/11, -6/11*e^3 + 28/11*e^2 + 62/11*e - 210/11, 2/11*e^3 - 35/11*e^2 + 27/11*e + 433/11, 10/11*e^3 + 23/11*e^2 - 195/11*e - 420/11, -10/11*e^3 - 1/11*e^2 + 162/11*e + 90/11, -7/11*e^3 + 18/11*e^2 + 76/11*e - 190/11, 3/11*e^3 - 14/11*e^2 + 2/11*e + 72/11, 8/11*e^3 + 3/11*e^2 - 156/11*e - 182/11, 2*e^2 - 5*e - 13, -7/11*e^3 - 15/11*e^2 + 153/11*e + 404/11, -13/11*e^3 + 13/11*e^2 + 182/11*e + 51/11, -1/11*e^3 + 34/11*e^2 - 41/11*e - 552/11, -9/11*e^3 - 24/11*e^2 + 181/11*e + 301/11, 1/11*e^3 - 23/11*e^2 - 3/11*e + 266/11, 4/11*e^3 + 18/11*e^2 - 67/11*e - 333/11, -5/11*e^3 - 17/11*e^2 + 81/11*e + 386/11, -2/11*e^3 + 24/11*e^2 + 17/11*e - 213/11, -7/11*e^3 - 15/11*e^2 + 175/11*e + 228/11, 2/11*e^3 + 20/11*e^2 - 105/11*e - 293/11, -14/11*e^3 + 3/11*e^2 + 174/11*e + 247/11, -15/11*e^3 + 26/11*e^2 + 232/11*e - 85/11, -12/11*e^3 + 23/11*e^2 + 146/11*e - 167/11, -15/11*e^3 + 4/11*e^2 + 232/11*e + 113/11, -7/11*e^3 - 26/11*e^2 + 153/11*e + 547/11, 1/11*e^3 + 32/11*e^2 - 69/11*e - 306/11, 12/11*e^3 + 32/11*e^2 - 223/11*e - 504/11, 3/11*e^3 - 47/11*e^2 + 13/11*e + 655/11, 5/11*e^3 - 16/11*e^2 - 48/11*e + 120/11, 8/11*e^3 + 3/11*e^2 - 134/11*e - 160/11, -20/11*e^3 + 9/11*e^2 + 247/11*e + 290/11, -2*e^3 + 3*e^2 + 31*e - 5, -7/11*e^3 + 18/11*e^2 + 120/11*e - 256/11, -6/11*e^3 - 5/11*e^2 + 62/11*e + 76/11, -13/11*e^3 + 2/11*e^2 + 182/11*e + 73/11, -12/11*e^3 + 23/11*e^2 + 168/11*e - 233/11, -7/11*e^3 + 40/11*e^2 + 65/11*e - 322/11, -13/11*e^3 - 20/11*e^2 + 215/11*e + 557/11, -10/11*e^3 + 21/11*e^2 + 151/11*e - 31/11, -10/11*e^3 - 1/11*e^2 + 129/11*e + 288/11, e^3 - 5*e^2 - 6*e + 53, 23/11*e^3 + 21/11*e^2 - 410/11*e - 537/11, -3/11*e^3 + 3/11*e^2 + 20/11*e + 115/11, -5/11*e^3 + 5/11*e^2 + 92/11*e + 89/11, -12/11*e^3 - 21/11*e^2 + 201/11*e + 735/11, -6/11*e^3 + 39/11*e^2 + 51/11*e - 144/11, -15/11*e^3 - 51/11*e^2 + 287/11*e + 938/11, -2/11*e^3 + 68/11*e^2 - 49/11*e - 807/11, -6/11*e^3 - 60/11*e^2 + 205/11*e + 802/11, e^3 + 5*e^2 - 25*e - 81, -5/11*e^3 - 6/11*e^2 + 103/11*e + 243/11, -1/11*e^3 + 12/11*e^2 - 19/11*e - 24/11, -5/11*e^3 - 6/11*e^2 + 103/11*e - 65/11, -1/11*e^3 + 12/11*e^2 - 19/11*e - 332/11, 14/11*e^3 - 3/11*e^2 - 163/11*e - 269/11, 16/11*e^3 - 27/11*e^2 - 257/11*e + 76/11, -6/11*e^3 - 27/11*e^2 + 117/11*e + 472/11, 3*e^2 - 3*e - 37, 4/11*e^3 + 18/11*e^2 - 34/11*e - 498/11, 4/11*e^3 - 26/11*e^2 - 78/11*e + 140/11, -19/11*e^3 - 47/11*e^2 + 321/11*e + 1172/11, -8/11*e^3 + 74/11*e^2 + 57/11*e - 599/11, -8/11*e^3 + 19/11*e^2 + 156/11*e - 60/11, -5/11*e^3 - 6/11*e^2 + 26/11*e + 298/11, 4/11*e^3 + 40/11*e^2 - 111/11*e - 575/11, -5/11*e^3 - 39/11*e^2 + 125/11*e + 584/11, 3/11*e^3 - 47/11*e^2 - 9/11*e + 534/11, 10/11*e^3 + 34/11*e^2 - 173/11*e - 651/11, -21/11*e^3 + 43/11*e^2 + 250/11*e - 328/11, -27/11*e^3 + 5/11*e^2 + 422/11*e + 232/11, -18/11*e^3 + 18/11*e^2 + 263/11*e - 190/11, -17/11*e^3 + 17/11*e^2 + 227/11*e - 177/11, -16/11*e^3 + 49/11*e^2 + 169/11*e - 329/11, -24/11*e^3 - 9/11*e^2 + 391/11*e + 524/11, -3/11*e^3 - 30/11*e^2 + 174/11*e + 511/11, 12/11*e^3 + 21/11*e^2 - 300/11*e - 251/11, -2/11*e^3 - 9/11*e^2 - 60/11*e + 227/11, -9/11*e^3 + 20/11*e^2 + 214/11*e - 183/11, -14/11*e^3 + 36/11*e^2 + 141/11*e - 105/11, -21/11*e^3 - 1/11*e^2 + 349/11*e + 442/11, -3/11*e^3 + 14/11*e^2 + 86/11*e + 49/11, -e^2 - 4*e + 37, 10/11*e^3 + 45/11*e^2 - 217/11*e - 475/11, -2/11*e^3 - 53/11*e^2 + 105/11*e + 964/11, 2/11*e^3 + 9/11*e^2 - 28/11*e - 106/11, 1/11*e^3 - 12/11*e^2 - 14/11*e + 200/11, -9/11*e^3 - 2/11*e^2 + 71/11*e + 125/11, -13/11*e^3 + 24/11*e^2 + 237/11*e - 246/11, -10/11*e^3 - 78/11*e^2 + 250/11*e + 1234/11, 8/11*e^3 + 80/11*e^2 - 222/11*e - 1084/11, -12/11*e^3 - 76/11*e^2 + 267/11*e + 1307/11, 5/11*e^3 + 83/11*e^2 - 169/11*e - 1024/11, -e^3 + 4*e^2 + 5*e - 32, -23/11*e^3 - 10/11*e^2 + 421/11*e + 449/11, -e^3 + 14*e + 43, -10/11*e^3 + 21/11*e^2 + 140/11*e + 167/11, 23/11*e^3 - 56/11*e^2 - 300/11*e + 332/11, 28/11*e^3 + 5/11*e^2 - 414/11*e - 560/11, 19/11*e^3 - 30/11*e^2 - 288/11*e + 203/11, 18/11*e^3 - 7/11*e^2 - 230/11*e - 129/11, -3/11*e^3 + 25/11*e^2 + 31/11*e - 182/11, -6/11*e^3 - 16/11*e^2 + 95/11*e + 417/11, -7/11*e^3 - 37/11*e^2 + 131/11*e + 536/11, 4*e^2 - 3*e - 59, -4/11*e^3 + 92/11*e^2 - 54/11*e - 1141/11, -2*e^3 - 6*e^2 + 38*e + 107, -12/11*e^3 + 1/11*e^2 + 223/11*e - 68/11, -6/11*e^3 + 17/11*e^2 + 29/11*e - 309/11, 8/11*e^3 + 25/11*e^2 - 189/11*e - 138/11, -2/11*e^3 - 31/11*e^2 + 105/11*e + 689/11, -18/11*e^3 - 4/11*e^2 + 318/11*e - 47/11, -10/11*e^3 + 32/11*e^2 + 74/11*e - 581/11, -2*e^3 + 2*e^2 + 28*e + 2, -17/11*e^3 - 38/11*e^2 + 348/11*e + 758/11, -2/11*e^3 + 57/11*e^2 - 82/11*e - 642/11];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {9, 3, 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;