// Make newform 2005.2.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2005_a();" // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_2005_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_2005_a();" function MakeCharacter_2005_a() N := 2005; order := 1; char_gens := [402, 1206]; v := [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_2005_a_Hecke(Kf) return MakeCharacter_2005_a(); end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_2005_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_2005_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2005_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![63, -392, -1542, 8582, 17109, -70969, -102539, 300934, 363628, -733005, -808981, 1084091, 1167440, -1004451, -1114852, 590210, 715044, -216111, -310895, 44746, 91540, -2823, -17935, -935, 2235, 256, -160, -26, 5, 1]>,<11,R![-172526976, -2242901952, -4176889344, 44673534688, 213586779777, 254911732777, -261181248313, -828008567125, -404770325058, 544564837427, 685539822150, 95264570373, -247647424829, -141323029077, 11227665163, 34483845950, 9022374734, -2296324533, -1665265404, -190854937, 86096829, 30039522, 1362523, -1007034, -213562, -6848, 3324, 561, 38, 1]>],Snew); return Vf; end function;