// Make newform 112.4.a.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_112_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_112_4_a_f();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_112_4_a_f();". // 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 ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_112_a();" function MakeCharacter_112_a() N := 112; order := 1; char_gens := [15, 85, 17]; 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_112_a_Hecke(Kf) return MakeCharacter_112_a(); end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 4; raw_aps := [[0], [2], [16], [7], [8], [28], [54], [110], [-48], [-110], [-12], [-246], [182], [-128], [-324], [-162], [-810], [-488], [-244], [768], [-702], [-440], [1302], [730], [294], [-688], [-1388], [-244], [90], [1318], [1776], [1118], [2274], [210], [-2010], [-1112], [124], [-2008], [-2884], [2228], [820], [3892], [5048], [-2962], [3334], [-1860], [4268], [5432], [2046], [-2980], [4458], [-4440], [3302], [-1582], [2354], [3872], [180], [-2032], [-5426], [842], [3782], [-4312], [-2674], [3768], [2438], [-3186], [-8672], [814], [-9344], [-5180], [12178], [-440], [9816], [-442], [3960], [-6708], [-13350], [-1356], [6222], [5150], [-2310], [1262], [4488], [17038], [-16200], [8772], [2130], [10534], [-9268], [9392], [10806], [-4940], [5216], [-4412], [-19060], [-12768], [-5500], [-7338], [17582], [-1618], [-16144], [4654], [-10078], [-5930], [19048], [-14366], [3626], [-1062], [10200], [-25158], [-25664], [19018], [17334], [-18730], [6928], [16302], [-4718], [21436], [4458], [26640], [7432], [58], [-21516], [-18108], [10078], [18762], [6810], [-4860], [13636], [2088], [5160], [28152], [16808], [21674], [7422], [13790], [-6232], [1766], [1204], [-7050], [-23282], [10142], [9192], [46716], [11240], [-700], [-37492], [28894], [2770], [-17688], [-33566], [-16758], [-11468], [50356], [8716], [-7632], [23080], [45110], [16674], [43832], [736], [38138], [-26224], [-18762], [38394], [-5388], [-25472], [-17096]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_112_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_112_4_a_f();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_112_4_a_f(:prec:=1) chi := MakeCharacter_112_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_112_4_a_f();". // 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_112_4_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_112_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<3,R![-2, 1]>,<5,R![-16, 1]>],Snew); return Vf; end function;