// Make newform 6272.2.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6272_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_6272_2_a_g();". // 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_6272_2_a_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 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_6272_a();" function MakeCharacter_6272_a() N := 6272; order := 1; char_gens := [4607, 3333, 4609]; 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_6272_a_Hecke(Kf) return MakeCharacter_6272_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], [2], [-2], [0], [2], [-2], [2], [2], [-4], [-6], [0], [10], [6], [-6], [-8], [-6], [14], [-2], [-10], [-12], [-14], [8], [-6], [2], [2], [6], [-4], [2], [-6], [2], [-16], [-6], [10], [-10], [18], [-4], [-18], [-2], [-20], [-18], [6], [-2], [16], [-2], [-14], [-4], [22], [0], [18], [14], [-18], [24], [2], [-18], [-18], [-12], [-10], [8], [-6], [-18], [6], [14], [18], [-28], [-10], [-6], [-14], [-14], [18], [-10], [-18], [4], [-8], [10], [2], [0], [10], [6], [30], [-10], [26], [34], [-40], [-30], [36], [-6], [-34], [-6], [-10], [8], [-14], [0], [20], [10], [22], [20], [14], [22], [14], [34], [38], [2], [18], [26], [-38], [-2], [-34], [-18], [12], [-30], [-16], [34], [-2], [46], [44], [30], [42], [12], [42], [6], [-34], [-2], [22], [42], [-6], [2], [-22], [24], [-12], [6], [-18], [-44], [-8], [-46], [-10], [34], [54], [-22], [-18], [-6], [-18], [10], [28], [-22], [14], [-36], [-26], [6], [-50], [32], [-22], [46], [-34], [36], [-38], [24], [-36], [-30], [-46], [38], [14], [58], [-28], [38], [30], [20], [-16], [54], [-34], [-34], [-6], [-42], [20], [-34], [40], [18], [-30], [26], [-52], [46], [32], [-30], [-30], [2], [24], [-18], [-46], [2], [-38], [64], [14], [-54], [14], [-26], [-34], [34], [50], [-14], [-2], [12], [2], [-40], [-10], [14], [30], [-42], [64], [14], [14], [-2], [-34], [62], [44], [22], [12], [-10], [24], [2], [12], [-6], [-46], [-12], [30], [-24], [-62], [-22], [34], [16], [28], [26], [74], [2], [-32], [-2], [38], [-8], [34], [2], [-6], [12], [54], [-10], [-20], [-14], [14], [52], [-64], [-54], [-22], [24], [42], [14], [20], [-46], [38], [-74], [-46], [30], [-42], [-42], [64], [-2], [70], [22], [66], [-6], [58], [-30], [50], [26], [-2], [-50], [-46], [16], [-66], [-76], [10], [18], [-18], [66], [32], [4], [-12], [-34], [22], [-40], [-2], [-6], [4], [-34], [-86], [18], [-66], [-66], [10], [-66], [64], [14], [54], [58], [22], [-30], [24], [6], [26], [-66], [50], [14], [60], [10], [-56], [-30], [30], [86], [52], [18], [-22], [-64], [-66], [-46], [-70], [-22], [-42], [16], [-54], [-62], [30], [-10], [-32], [82], [-62], [-6], [-64], [42], [-70], [30], [-70], [46], [-30], [56], [34], [74], [-42], [20], [-30], [-2], [-2], [18], [-24], [-82], [-14], [-22], [-30], [24], [50], [-14], [48], [6], [30], [-36], [-38], [-78], [-72], [-14], [-14], [-34], [94], [20], [-70], [-58], [-50], [-24], [-54], [52], [-38], [-22], [0], [14], [66], [86], [-58], [-6], [-68], [-78], [-14], [-36], [8], [-58], [82], [-32], [-62], [-42], [-46], [58], [76], [10], [-80], [86], [42], [-54], [54], [30], [-24], [102], [-66], [60], [2], [30], [74], [-70], [30], [42], [-94], [-50], [62], [-98], [32], [-52], [66], [-12], [-62], [-90], [-56], [-66], [-90], [-18], [14], [30], [102], [36]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6272_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6272_2_a_g();". // 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_6272_2_a_g(:prec:=1) chi := MakeCharacter_6272_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_6272_2_a_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_6272_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6272_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, 1]>,<5,R![2, 1]>,<11,R![-2, 1]>,<13,R![2, 1]>,<23,R![4, 1]>,<29,R![6, 1]>],Snew); return Vf; end function;