
// Make newform 503.2.a.f in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_503_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_503_2_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_503_a();"
function MakeCharacter_503_a()
    N := 503;
    order := 1;
    char_gens := [5];
    v := [1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_503_a_Hecke(Kf)
    return MakeCharacter_503_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_503_2_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_503_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_503_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1583, -6437, 24760, 115872, -120174, -725019, 236065, 2081504, -283355, -3137915, 297701, 2799707, -278156, -1589003, 189834, 595540, -86017, -149131, 25321, 24652, -4772, -2577, 554, 154, -36, -4, 1]>,<3,R![-113513, -219997, 2752584, 3683025, -13187700, -17328663, 25434027, 34013592, -27001213, -36123133, 18323065, 23403122, -8635730, -9813106, 2939963, 2735786, -726005, -509651, 127340, 62449, -15259, -4814, 1175, 211, -52, -4, 1]>],Snew);
    return Vf;
end function;