/*
  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![-3, -7, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {19, 19, -w^2 - w + 4}>;

primesArray := [
[3, 3, w],
[3, 3, w + 1],
[3, 3, w + 2],
[8, 2, 2],
[11, 11, -w^2 + 5],
[13, 13, w^2 - w - 7],
[17, 17, -w^2 + w + 4],
[19, 19, -w^2 - w + 4],
[29, 29, 2*w^2 - 2*w - 11],
[31, 31, w^2 - 2*w - 4],
[37, 37, -2*w^2 + 3*w + 7],
[41, 41, w^2 - 2],
[59, 59, 2*w^2 - 13],
[61, 61, w^2 + w - 10],
[67, 67, 2*w^2 - w - 11],
[73, 73, -w^2 - 1],
[83, 83, w^2 - w - 10],
[89, 89, -w^2 + 4*w - 2],
[97, 97, w^2 - 2*w - 7],
[97, 97, -w^2 - 4*w - 5],
[97, 97, 2*w^2 + w - 8],
[101, 101, -4*w^2 + 2*w + 25],
[103, 103, w^2 + w - 7],
[107, 107, -3*w - 4],
[109, 109, 5*w^2 - 3*w - 31],
[121, 11, 2*w^2 - 3*w - 4],
[125, 5, -5],
[137, 137, -5*w^2 + 3*w + 34],
[139, 139, w^2 + 2*w - 4],
[151, 151, 3*w^2 - 23],
[151, 151, w^2 - 3*w - 5],
[151, 151, 3*w^2 - 3*w - 17],
[157, 157, 4*w^2 - 9*w - 8],
[157, 157, w^2 + 3*w - 2],
[157, 157, 2*w^2 - w - 2],
[167, 167, 2*w^2 + w - 11],
[167, 167, -w^2 + 11],
[167, 167, -3*w^2 + 19],
[169, 13, w^2 - 4*w - 4],
[173, 173, 2*w^2 - 3*w - 10],
[179, 179, -3*w + 7],
[191, 191, -w^2 + 5*w - 5],
[193, 193, -w^2 + w - 2],
[199, 199, 3*w - 2],
[211, 211, w^2 - 3*w - 11],
[223, 223, 3*w^2 - 20],
[223, 223, w^2 - 5*w - 4],
[223, 223, 2*w^2 - w - 8],
[229, 229, 4*w^2 - 3*w - 26],
[229, 229, 2*w^2 + w - 20],
[229, 229, 2*w^2 - 2*w - 5],
[239, 239, -6*w^2 + 3*w + 38],
[241, 241, -w^2 + 4*w - 5],
[257, 257, -2*w^2 + 5*w - 1],
[263, 263, -2*w^2 - w + 17],
[269, 269, -w^2 - w + 13],
[269, 269, 3*w - 4],
[269, 269, -4*w^2 + w + 31],
[271, 271, 2*w^2 - 7],
[271, 271, 3*w - 5],
[271, 271, w^2 - 3*w - 8],
[289, 17, 2*w^2 + w - 5],
[307, 307, 2*w^2 - w - 5],
[311, 311, 3*w^2 - 3*w - 19],
[311, 311, -7*w^2 + 13*w + 19],
[311, 311, w^2 - 2*w - 13],
[313, 313, w^2 + 2*w - 7],
[317, 317, -5*w^2 + w + 35],
[331, 331, -w^2 + 3*w - 4],
[337, 337, 3*w^2 - 3*w - 13],
[343, 7, -7],
[347, 347, 2*w^2 - 3*w - 13],
[349, 349, 3*w^2 - 3*w - 20],
[359, 359, -w^2 + 5*w - 2],
[361, 19, -2*w^2 + 7*w - 1],
[389, 389, -2*w^2 + w - 1],
[397, 397, w^2 - 6*w + 10],
[401, 401, 6*w^2 - 3*w - 44],
[419, 419, -3*w^2 + 6*w + 10],
[419, 419, 4*w^2 - 2*w - 31],
[419, 419, -4*w^2 + 4*w + 19],
[421, 421, 6*w^2 - 12*w - 13],
[431, 431, 4*w^2 - 7*w - 10],
[433, 433, w^2 - 4*w - 7],
[443, 443, 2*w^2 + 4*w - 5],
[443, 443, 5*w^2 - 11*w - 11],
[443, 443, 3*w^2 - 14],
[449, 449, -2*w^2 + 2*w - 1],
[457, 457, w^2 - 6*w - 5],
[457, 457, w^2 + 3*w - 5],
[457, 457, 6*w^2 - 3*w - 37],
[463, 463, w^2 + 5*w - 1],
[467, 467, w^2 - w - 13],
[479, 479, 5*w^2 - 9*w - 16],
[487, 487, -6*w - 5],
[499, 499, 3*w^2 - 3*w - 4],
[499, 499, 3*w^2 + 3*w - 1],
[499, 499, w^2 - 5*w + 8],
[503, 503, 4*w^2 - 5*w - 16],
[509, 509, 2*w^2 - 5*w - 8],
[521, 521, 4*w^2 - w - 22],
[523, 523, w^2 - 5*w - 16],
[547, 547, 5*w^2 - 2*w - 38],
[547, 547, 8*w^2 - 3*w - 58],
[547, 547, 4*w^2 - 29],
[557, 557, 7*w^2 - 4*w - 43],
[557, 557, 3*w - 11],
[557, 557, 2*w^2 - 4*w - 11],
[569, 569, 5*w^2 - 4*w - 32],
[569, 569, 3*w^2 - 3*w - 11],
[569, 569, w^2 + 2*w - 16],
[571, 571, w^2 - 4*w - 10],
[593, 593, 2*w^2 - 6*w - 7],
[601, 601, -w^2 - 3*w + 14],
[607, 607, 3*w^2 + 3*w - 7],
[619, 619, 3*w^2 - 6*w - 11],
[631, 631, -w^2 - w - 5],
[641, 641, 3*w^2 - 3*w - 5],
[647, 647, -w^2 + 6*w - 1],
[653, 653, -5*w^2 + 4*w + 35],
[661, 661, 3*w^2 - 3*w - 10],
[673, 673, w^2 + 4*w - 4],
[677, 677, 5*w^2 - 6*w - 25],
[677, 677, w^2 - 14],
[677, 677, 4*w^2 - 3*w - 20],
[691, 691, 5*w^2 - 2*w - 29],
[701, 701, 5*w^2 - 10*w - 14],
[719, 719, -3*w - 11],
[727, 727, -2*w^2 + 9*w - 8],
[733, 733, -w^2 - 4*w - 8],
[739, 739, w^2 + 3*w - 8],
[743, 743, 2*w^2 - 4*w - 17],
[757, 757, -w^2 + w - 5],
[769, 769, 3*w^2 - 3*w - 7],
[769, 769, -w^2 - 3*w - 7],
[769, 769, 5*w^2 - 3*w - 28],
[773, 773, 7*w^2 - 6*w - 38],
[787, 787, 2*w^2 - w - 20],
[797, 797, -5*w^2 + 14*w - 1],
[821, 821, -w^2 + 6*w - 4],
[823, 823, 7*w^2 - 6*w - 41],
[839, 839, 4*w^2 - 6*w - 11],
[841, 29, 5*w^2 - 4*w - 26],
[857, 857, 3*w^2 - 11],
[863, 863, 2*w^2 + 2*w - 17],
[881, 881, 5*w^2 - 5*w - 29],
[883, 883, -4*w^2 + 8*w + 13],
[907, 907, 4*w^2 - 9*w - 11],
[911, 911, 2*w^2 + 3*w - 10],
[947, 947, 2*w^2 + 5*w - 5],
[953, 953, 2*w^2 - 5*w - 11],
[961, 31, w^2 - 5*w - 13],
[967, 967, 2*w^2 - 3*w - 22],
[967, 967, 5*w^2 - 9*w - 10],
[967, 967, 3*w^2 - 10],
[977, 977, 5*w^2 - 37],
[983, 983, 7*w^2 - 5*w - 40],
[997, 997, -7*w^2 + 5*w + 49]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^8 - 10*x^6 + 24*x^4 - 18*x^2 + 2;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [-e^6 + 9*e^4 - 15*e^2 + 4, e, e^7 - 8*e^5 + 7*e^3 + 4*e, e^6 - 9*e^4 + 15*e^2 - 5, -3*e^7 + 26*e^5 - 38*e^3 + 9*e, 3*e^7 - 27*e^5 + 46*e^3 - 17*e, e^3 - 6*e, -1, -e^7 + 10*e^5 - 24*e^3 + 18*e, -3*e^7 + 25*e^5 - 30*e^3 - e, e^7 - 11*e^5 + 33*e^3 - 33*e, -2*e^7 + 16*e^5 - 16*e^3, 7*e^6 - 61*e^4 + 90*e^2 - 18, e^4 - 5*e^2 - 2, 2*e^7 - 17*e^5 + 23*e^3 - 3*e, -3*e^6 + 25*e^4 - 30*e^2, -8*e^6 + 68*e^4 - 91*e^2 + 6, e^6 - 8*e^4 + 4*e^2 + 14, 4*e^7 - 37*e^5 + 69*e^3 - 31*e, 4*e^6 - 34*e^4 + 46*e^2 - 18, 2*e^7 - 21*e^5 + 54*e^3 - 28*e, -3*e^7 + 32*e^5 - 87*e^3 + 59*e, 4*e^6 - 35*e^4 + 56*e^2 - 16, e^7 - 8*e^5 + 5*e^3 + 19*e, -e^7 + 6*e^5 + 8*e^3 - 15*e, 3*e^7 - 24*e^5 + 22*e^3 + 2*e, -6*e^6 + 54*e^4 - 93*e^2 + 28, -3*e^7 + 27*e^5 - 48*e^3 + 25*e, 2*e^6 - 17*e^4 + 23*e^2 - 6, -4*e^6 + 32*e^4 - 26*e^2 - 12, 5*e^7 - 37*e^5 + 12*e^3 + 45*e, -8*e^7 + 74*e^5 - 141*e^3 + 70*e, -2*e^7 + 16*e^5 - 12*e^3 - 26*e, -6*e^6 + 53*e^4 - 84*e^2 + 22, 9*e^7 - 79*e^5 + 122*e^3 - 37*e, e^6 - 6*e^4 - 6*e^2 + 14, e^6 - 9*e^4 + 16*e^2 - 18, -9*e^6 + 80*e^4 - 131*e^2 + 36, -6*e^7 + 55*e^5 - 104*e^3 + 63*e, -7*e^7 + 57*e^5 - 56*e^3 - 29*e, e^6 - 8*e^4 + 12*e^2 - 6, 12*e^6 - 105*e^4 + 156*e^2 - 40, -e^6 + 10*e^4 - 22*e^2 - 2, 6*e^6 - 49*e^4 + 48*e^2 + 16, -4*e^5 + 37*e^3 - 56*e, -6*e^6 + 56*e^4 - 110*e^2 + 42, -7*e^7 + 63*e^5 - 105*e^3 + 29*e, -3*e^7 + 23*e^5 - 14*e^3 - 11*e, -2*e^7 + 22*e^5 - 64*e^3 + 57*e, -3*e^6 + 26*e^4 - 34*e^2 - 18, -10*e^7 + 84*e^5 - 103*e^3 + 2*e, 9*e^5 - 73*e^3 + 86*e, 16*e^6 - 137*e^4 + 192*e^2 - 28, -8*e^6 + 71*e^4 - 114*e^2 + 18, -13*e^6 + 111*e^4 - 153*e^2 + 34, -e^6 + 4*e^4 + 18*e^2 - 14, 5*e^6 - 47*e^4 + 90*e^2 - 28, -9*e^6 + 80*e^4 - 133*e^2 + 34, 10*e^7 - 94*e^5 + 187*e^3 - 91*e, -8*e^6 + 68*e^4 - 86*e^2 - 2, 3*e^5 - 26*e^3 + 48*e, 5*e^7 - 47*e^5 + 95*e^3 - 58*e, -14*e^7 + 121*e^5 - 172*e^3 + 28*e, 5*e^7 - 50*e^5 + 115*e^3 - 53*e, -6*e^7 + 53*e^5 - 81*e^3 + 13*e, 3*e^6 - 29*e^4 + 62*e^2 - 26, e^4 - 3*e^2 - 6, 15*e^6 - 131*e^4 + 196*e^2 - 44, -20*e^6 + 169*e^4 - 222*e^2 + 20, -13*e^7 + 120*e^5 - 224*e^3 + 103*e, 3*e^4 - 20*e^2 + 6, 4*e^7 - 35*e^5 + 58*e^3 - 37*e, 20*e^7 - 173*e^5 + 249*e^3 - 48*e, 16*e^6 - 135*e^4 + 173*e^2 - 8, -5*e^6 + 43*e^4 - 66*e^2 + 16, -e^6 + 8*e^4 - 12*e^2 - 6, 17*e^6 - 145*e^4 + 199*e^2 - 18, 18*e^6 - 153*e^4 + 204*e^2 - 24, -8*e^7 + 78*e^5 - 171*e^3 + 78*e, 5*e^6 - 46*e^4 + 80*e^2 - 38, -4*e^7 + 36*e^5 - 58*e^3 + 12*e, 3*e^7 - 22*e^5 + 7*e^3 + 18*e, -8*e^7 + 71*e^5 - 116*e^3 + 42*e, 12*e^7 - 112*e^5 + 216*e^3 - 92*e, 18*e^6 - 158*e^4 + 238*e^2 - 46, -2*e^7 + 14*e^5 + 4*e^3 - 40*e, 4*e^7 - 34*e^5 + 44*e^3 + e, -19*e^6 + 159*e^4 - 194*e^2, -4*e^7 + 36*e^5 - 64*e^3 + 37*e, -13*e^6 + 112*e^4 - 160*e^2 + 20, 6*e^5 - 51*e^3 + 74*e, -3*e^6 + 22*e^4 - 9*e^2 - 18, 2*e^6 - 17*e^4 + 18*e^2 - 12, 23*e^7 - 210*e^5 + 378*e^3 - 163*e, -6*e^7 + 64*e^5 - 175*e^3 + 126*e, -7*e^7 + 67*e^5 - 140*e^3 + 65*e, 12*e^7 - 114*e^5 + 234*e^3 - 132*e, 4*e^6 - 29*e^4 + 16*e^2 - 2, 2*e^7 - 23*e^5 + 69*e^3 - 49*e, -26*e^7 + 231*e^5 - 375*e^3 + 124*e, -3*e^7 + 35*e^5 - 116*e^3 + 111*e, 11*e^7 - 92*e^5 + 109*e^3 + 9*e, -3*e^6 + 23*e^4 - 14*e^2 - 6, -3*e^6 + 24*e^4 - 22*e^2 - 22, -16*e^6 + 135*e^4 - 164*e^2 - 20, 3*e^7 - 15*e^5 - 58*e^3 + 126*e, 15*e^6 - 125*e^4 + 155*e^2 - 16, -12*e^7 + 94*e^5 - 65*e^3 - 86*e, -7*e^7 + 62*e^5 - 104*e^3 + 49*e, 14*e^7 - 115*e^5 + 120*e^3 + 44*e, 11*e^6 - 96*e^4 + 140*e^2 - 12, 10*e^7 - 84*e^5 + 106*e^3 - 15*e, -4*e^7 + 46*e^5 - 139*e^3 + 96*e, 12*e^6 - 106*e^4 + 162*e^2 - 30, -14*e^7 + 131*e^5 - 258*e^3 + 127*e, 19*e^7 - 176*e^5 + 333*e^3 - 144*e, -6*e^6 + 52*e^4 - 80*e^2 + 24, -7*e^7 + 57*e^5 - 57*e^3 - 9*e, -13*e^6 + 116*e^4 - 194*e^2 + 74, -6*e^7 + 55*e^5 - 104*e^3 + 62*e, 8*e^7 - 74*e^5 + 139*e^3 - 76*e, 6*e^6 - 49*e^4 + 49*e^2 + 16, e^7 - 15*e^5 + 67*e^3 - 73*e, 5*e^6 - 47*e^4 + 82*e^2 - 14, -16*e^7 + 148*e^5 - 273*e^3 + 102*e, 2*e^5 - 16*e^3 + 32*e, -5*e^7 + 52*e^5 - 130*e^3 + 63*e, -4*e^6 + 32*e^4 - 27*e^2 - 24, 30*e^6 - 260*e^4 + 373*e^2 - 64, 7*e^6 - 60*e^4 + 83*e^2 - 4, -2*e^6 + 16*e^4 - 7*e^2 - 42, -2*e^7 + 11*e^5 + 28*e^3 - 61*e, -10*e^6 + 85*e^4 - 112*e^2 + 30, 10*e^7 - 81*e^5 + 81*e^3 + 24*e, -8*e^6 + 74*e^4 - 130*e^2 + 38, 26*e^7 - 218*e^5 + 271*e^3 - e, 22*e^7 - 196*e^5 + 328*e^3 - 135*e, 9*e^6 - 71*e^4 + 70*e^2 - 12, 2*e^6 - 12*e^4 - 14*e^2 + 10, -6*e^6 + 51*e^4 - 60*e^2 - 14, 14*e^7 - 136*e^5 + 295*e^3 - 168*e, -6*e^7 + 63*e^5 - 165*e^3 + 124*e, 3*e^7 - 28*e^5 + 57*e^3 - 66*e, -28*e^7 + 247*e^5 - 388*e^3 + 108*e, 11*e^6 - 92*e^4 + 119*e^2 - 34, 16*e^7 - 134*e^5 + 168*e^3 - 12*e, 20*e^7 - 181*e^5 + 309*e^3 - 92*e, 10*e^7 - 92*e^5 + 174*e^3 - 96*e, 2*e^6 - 12*e^4 - 12*e^2 + 32, 27*e^6 - 236*e^4 + 347*e^2 - 76, 5*e^7 - 32*e^5 - 33*e^3 + 108*e, -e^7 + 2*e^5 + 37*e^3 - 18*e, -22*e^6 + 183*e^4 - 214*e^2 - 12, -4*e^7 + 27*e^5 + 16*e^3 - 85*e, -9*e^7 + 72*e^5 - 64*e^3 - 49*e, 20*e^6 - 166*e^4 + 196*e^2 - 10, -14*e^7 + 123*e^5 - 196*e^3 + 80*e, -23*e^7 + 210*e^5 - 384*e^3 + 207*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {19, 19, -w^2 - w + 4}>] := 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;