// Make newform 810.3.j.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_810_j();" // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_810_3_j_g();". // 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_810_j();" function MakeCharacter_810_j() N := 810; order := 6; char_gens := [731, 487]; 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_810_j_Hecke(Kf) return MakeCharacter_810_j(); end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_810_3_j_g();". // 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_810_3_j_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_810_j(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<7,R![8386816, -33709440, 34859232, 41415120, 8604420, -3380928, -139180, 97800, 5598, -1368, -66, 12, 1]>,<17,R![1460109722500, 0, -1040615269800, 0, 23433607236, 0, -199784716, 0, 793137, 0, -1464, 0, 1]>],Snew); return Vf; end function;