// Make newform 8041.2.a.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8041_a();" // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_8041_2_a_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_8041_a();" function MakeCharacter_8041_a() N := 8041; order := 1; char_gens := [6580, 2366, 562]; v := [1, 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_8041_a_Hecke(Kf) return MakeCharacter_8041_a(); end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_8041_2_a_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_8041_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8041_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![10656, 197516, -1464434, -27831312, 119802278, 1362387103, -5135838978, -31083368382, 110464177084, 401273696242, -1372518157822, -3257343143456, 10937782750760, 17763643102944, -59928800986132, -68113670132602, 237054326788384, 189556246908581, -702030639654940, -390435709373936, 1600881732395736, 598354946321052, -2873865586641718, -669123539706975, 4133395774451365, 499483009889372, -4829927250083757, -142624367049734, 4636056446227914, -216783047477330, -3686762178157311, 413050011773150, 2444699382525104, -413782693513080, -1357897262362638, 299722579964519, 633553006217094, -170022377830830, -248560945432378, 78032259179556, 81936725411813, -29437871117515, -22629208816363, 9200268222886, 5207298896088, -2389172954330, -989317184292, 515199653576, 152901116191, -91899544751, -18749837383, 13462279309, 1740044375, -1601340445, -108899204, 152104680, 2608777, -11255363, 287742, 624815, -38619, -24463, 2271, 602, -72, -7, 1]>],Snew); return Vf; end function;