
// Make newform 684.2.z.b in Magma, downloaded from the LMFDB on 27 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_684_z();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_684_2_z_b();".
// 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_684_z();"
function MakeCharacter_684_z()
    N := 684;
    order := 6;
    char_gens := [343, 533, 325];
    v := [3, 3, 4];
    // 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_684_z_Hecke(Kf)
    return MakeCharacter_684_z();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_684_2_z_b();".
// 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_684_2_z_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_684_z();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16384, 0, -327680, 0, 4087808, 0, -32288768, 0, 187995136, 0, -773989888, 0, 2371900160, 0, -4848667520, 0, 6679092480, 0, -3489901280, 0, 1222656336, 0, -276980128, 0, 46184000, 0, -5572760, 0, 510748, 0, -33968, 0, 1664, 0, -52, 0, 1]>],Snew);
    return Vf;
end function;