
// Make newform 1323.2.i.d in Magma, downloaded from the LMFDB on 12 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1323_i();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1323_2_i_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_1323_i();"
function MakeCharacter_1323_i()
    N := 1323;
    order := 6;
    char_gens := [785, 1081];
    v := [1, 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_1323_i_Hecke( : Kf := Kf)
    return MakeCharacter_1323_i();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1323_2_i_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_1323_2_i_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1323_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![49, 0, 3828, 0, 39240, 0, 151416, 0, 290826, 0, 319272, 0, 216050, 0, 93876, 0, 26613, 0, 4884, 0, 558, 0, 36, 0, 1]>],Snew);
    return Vf;
end function;