
// Make newform 405.3.l.o in Magma, downloaded from the LMFDB on 05 February 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_405_l();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_405_3_l_o();".
// 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_405_l();"
function MakeCharacter_405_l()
    N := 405;
    order := 12;
    char_gens := [326, 82];
    v := [4, 9];
    // 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_405_l_Hecke(Kf)
    return MakeCharacter_405_l();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_405_3_l_o();".
// 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_405_3_l_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_405_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![815730721, 0, 0, 0, -1699293817, 0, 0, 0, 3275303905, 0, 0, 0, -538784734, 0, 0, 0, 72882286, 0, 0, 0, -1891294, 0, 0, 0, 37825, 0, 0, 0, -217, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;