
// Make newform 385.2.o.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_385_o();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_385_2_o_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_385_o();"
function MakeCharacter_385_o()
    N := 385;
    order := 6;
    char_gens := [232, 276, 211];
    v := [3, 5, 3];
    // 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_385_o_Hecke(Kf)
    return MakeCharacter_385_o();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_385_2_o_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_385_2_o_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_385_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![441, 0, 9198, 0, 138987, 0, 858468, 0, 3653922, 0, 8988030, 0, 15652678, 0, 18674472, 0, 16569017, 0, 10838222, 0, 5451354, 0, 2098050, 0, 631988, 0, 147177, 0, 26722, 0, 3657, 0, 372, 0, 25, 0, 1]>],Snew);
    return Vf;
end function;