
// Make newform 135.2.k.b in Magma, downloaded from the LMFDB on 01 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_135_k();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_135_2_k_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_135_k();"
function MakeCharacter_135_k()
    N := 135;
    order := 9;
    char_gens := [56, 82];
    v := [2, 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_135_k_Hecke( : Kf := Kf)
    return MakeCharacter_135_k();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_135_2_k_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_135_2_k_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_135_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![29241, -466317, 3094767, -12791466, 47531934, -130860198, 160429788, 3874608, 12843387, 26849553, 10877391, 38023068, 500720488, -66372606, 13522041, 57446740, 324157914, -72317484, 182130030, -34822377, 30137607, -21766812, 23020785, -9454437, 7550580, -332010, -423216, 464688, 54666, -18657, 122169, -7848, 1278, 1191, -387, 153, 402, -36, 3, 7, 0, 0, 1]>],Snew);
    return Vf;
end function;