// Make newform 1323.2.o.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1323_o();" // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_1323_2_o_e();". // 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_o();" function MakeCharacter_1323_o() N := 1323; order := 6; char_gens := [785, 1081]; v := [1, 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_1323_o_Hecke(Kf) return MakeCharacter_1323_o(); end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_1323_2_o_e();". // 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_o_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1323_o(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![49, 504, 678, -10800, 14130, 33444, -42660, -70116, 112545, 34488, -101154, -9612, 61340, 0, -24510, 540, 7227, -108, -1434, 12, 207, 0, -18, 0, 1]>,<5,R![2401, 0, 3040548, 0, 3795958686, 0, 68412222392, 0, 859511233110, 0, 5059172837124, 0, 20837133890780, 0, 52794932235564, 0, 95399108748591, 0, 103567280273012, 0, 79686691764444, 0, 42196954289376, 0, 16657040868748, 0, 4979579304444, 0, 1167814077690, 0, 216703289260, 0, 32327781627, 0, 3876266316, 0, 375560984, 0, 29110188, 0, 1793667, 0, 85168, 0, 3024, 0, 72, 0, 1]>],Snew); return Vf; end function;