// Make newform 8015.2.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8015_a();" // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_8015_2_a_l();". // 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_8015_a();" function MakeCharacter_8015_a() N := 8015; order := 1; char_gens := [3207, 4581, 4586]; 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_8015_a_Hecke(Kf) return MakeCharacter_8015_a(); end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_8015_2_a_l();". // 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_8015_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8015_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-42758, -474169, 12241016, 4399105, -667644280, 545374332, 16677087686, -14096028527, -240061378290, 131791760734, 2166424947105, -486157325899, -12878081311615, -297226273365, 53087159403264, 10237311636943, -158940722742200, -48493271989887, 358788761857983, 135354446472340, -628751709975712, -264884788529573, 874918755328548, 389494146315591, -983793319091042, -446944155629008, 906107694041297, 409945333560961, -690791746640699, -305497353390180, 439416697005311, 187101306668744, -234605775096496, -94934567304436, 105566396486875, 40122895321631, -40135290396558, -14169739622374, 12905134589717, 4186434565195, -3507422334369, -1033946851227, 803988373316, 212853527134, -154810698241, -36334909566, 24885837373, 5101636939, -3310008018, -582222606, 359732161, 53093437, -31384856, -3773350, 2142834, 201232, -110189, -7570, 4010, 179, -92, -2, 1]>,<3,R![-82512, 5718384, -71810364, -1611760166, 19808458087, 187993444091, -1120858051165, -7414159685020, 23189400395769, 127642172285097, -238567958990605, -1147283271661558, 1463079506344409, 6094959964075969, -6039323160828695, -21089163988113815, 17955187960736328, 50874844532325019, -39954053126704901, -89706825582509993, 68199283034125005, 119529831496158504, -90932079344587970, -123231561865554149, 96095075972875246, 99946207922108762, -81463514948301001, -64469797616315189, 55949423929459155, 33264788267313245, -31377910595211849, -13729230481537504, 14456083100460817, 4495862295866761, -5493890115069654, -1141347787539293, 1726318309152456, 211149800764795, -448699600957514, -22646471422001, 96307839970999, -1070312817625, -17000451325855, 1108696284154, 2450477739697, -280209394373, -285188426826, 45958056587, 26333637313, -5504943952, -1876077236, 494985612, 98204795, -33266881, -3404316, 1627229, 54202, -54828, 1017, 1139, -67, -11, 1]>],Snew); return Vf; end function;