// Make newform 2352.2.a.j in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2352_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_2352_2_a_j();". // 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_2352_2_a_j();". // 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_2352_a();" function MakeCharacter_2352_a() N := 2352; order := 1; char_gens := [1471, 1765, 785, 2257]; v := [1, 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_2352_a_Hecke(Kf) return MakeCharacter_2352_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 := 2; raw_aps := [[0], [-1], [2], [0], [-2], [-4], [6], [-8], [6], [-10], [-4], [6], [-6], [-4], [-8], [2], [4], [-8], [8], [10], [4], [-4], [-12], [-14], [4], [-10], [4], [14], [10], [-14], [4], [12], [-10], [-12], [2], [8], [-16], [-24], [-16], [-2], [-6], [-20], [-6], [-10], [-6], [8], [4], [8], [-4], [20], [14], [-18], [28], [28], [10], [-26], [2], [4], [10], [6], [-16], [-18], [-16], [-16], [24], [18], [4], [14], [18], [0], [6], [-22], [-16], [-26], [36], [0], [-18], [32], [30], [12], [4], [-6], [2], [32], [24], [18], [-6], [-22], [-18], [4], [20], [0], [-16], [-6], [36], [-16], [10], [22], [4], [34], [-16], [-6], [-4], [6], [0], [-8], [4], [6], [-30], [8], [-24], [26], [-2], [-4], [-12], [-26], [0], [48], [-34], [14], [8], [-22], [30], [-22], [20], [-50], [-14], [-48], [-44], [4], [-40], [18], [48], [-14], [-22], [32], [6], [-4], [6], [-6], [52], [2], [44], [30], [4], [24], [-16], [42], [-48], [-18], [18], [-34], [20], [-48], [-28], [50], [-24], [-6], [0], [18], [46], [42], [-4], [-20], [38], [16], [-32], [8], [-2], [6], [54], [-48], [-30], [-6], [56], [-2], [-32], [6], [-44], [-8], [12], [-52], [6], [6], [-26], [22], [38], [36], [10], [-40], [-28], [-58], [-20], [-42], [62], [-58], [-38], [-2], [2], [48], [-22], [-32], [-52], [44], [-12], [-18], [-16], [6], [34], [-32], [30], [-54], [-16], [-60], [40], [-40], [-64], [-66], [-18], [38], [-14], [-36], [-6], [16], [-12], [-58], [26], [-10], [36], [-30], [-54], [8], [32], [-46], [0], [-32], [52], [-38], [-46], [16], [2], [20], [-8], [-46], [-30], [-24], [68], [60], [-16], [-10], [54], [14], [66], [4], [30], [-50], [-10], [-16], [-42], [56], [52], [2], [-4], [-24], [54], [4], [18], [6], [64], [22], [24], [20], [-76], [56], [-4], [-20], [-2], [34], [-50], [-36], [-40], [-44], [-64], [56], [-28], [-78], [2], [18], [44], [-78], [-54], [-20], [-18], [4], [-18], [70], [-16], [50], [-12], [20], [16], [-18], [20], [-66], [0], [62], [58], [44], [-54], [10], [24], [-18], [-18], [-68], [-34], [20], [52], [22], [52], [-26], [40], [2], [54], [-64], [50], [36], [0], [24], [42], [-34], [-10], [78], [60], [-76], [16], [-4], [-14], [-14], [44], [60], [-58], [-42], [82], [-72], [-34], [6], [-60], [-52], [0], [-18], [48], [-58], [-70], [-80], [86], [14], [-48], [-20], [-62], [10], [-58], [-6], [-30], [-34], [68], [18], [-30], [28], [62], [-54], [-56], [58], [26], [-28], [-62], [76], [-10], [-88], [14], [-68], [54], [90], [-72], [2], [76], [-96], [72], [36], [64], [96], [50], [-38], [-74], [8], [-46], [-26], [-24], [-42], [16], [30], [-16], [-62], [-68], [10], [90], [-8], [82], [-102], [52], [36], [56], [-18], [58], [44], [42], [90], [-18], [-28], [-26], [-72], [62], [-80], [-78], [84], [-80], [-26], [18], [30], [-52], [64]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2352_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2352_2_a_j();". // 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_2352_2_a_j(:prec:=1) chi := MakeCharacter_2352_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(2999) - 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_2352_2_a_j();". // 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_2352_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2352_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 1]>,<11,R![2, 1]>,<13,R![4, 1]>,<17,R![-6, 1]>],Snew); return Vf; end function;