
// Make newform 1512.2.j.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1512_j();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1512_2_j_d();".
// 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_1512_j();"
function MakeCharacter_1512_j()
    N := 1512;
    order := 2;
    char_gens := [1135, 757, 785, 1081];
    v := [1, 1, 1, 2];
    // 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_1512_j_Hecke(Kf)
    return MakeCharacter_1512_j();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1512_2_j_d();".
// 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_1512_2_j_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1512_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![350464, 0, -4855040, 0, 22661856, 0, -46907536, 0, 50346977, 0, -30803128, 0, 11388476, 0, -2649064, 0, 395238, 0, -37640, 0, 2204, 0, -72, 0, 1]>],Snew);
    return Vf;
end function;