// Make newform 4029.2.a.k in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4029_a();" // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_4029_2_a_k();". // 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_4029_a();" function MakeCharacter_4029_a() N := 4029; order := 1; char_gens := [2687, 3556, 3163]; 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_4029_a_Hecke(Kf) return MakeCharacter_4029_a(); end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_4029_2_a_k();". // 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_4029_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4029_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-16, 534, -1486, -67130, 197930, 665716, -1828184, -2749536, 7215218, 5980276, -15696141, -7447803, 20973717, 5435973, -18338723, -2141954, 10908523, 187089, -4512781, 250363, 1309226, -144079, -264955, 40284, 36580, -6767, -3280, 693, 172, -40, -4, 1]>,<5,R![-245126, 50960943, 279510163, -500388822, -2519065414, 2618564777, 7749290935, -6973117378, -11863323803, 10452034099, 10476862922, -9667408994, -5670304352, 5850464993, 1874501176, -2393234495, -333718251, 670333192, 6469056, -127984791, 11412047, 16257159, -2793266, -1299128, 331924, 57042, -21949, -751, 767, -33, -11, 1]>],Snew); return Vf; end function;