// Make newform 3330.2.a.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3330_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_3330_2_a_i();". // 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_3330_2_a_i();". // 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_3330_a();" function MakeCharacter_3330_a() N := 3330; order := 1; char_gens := [371, 667, 631]; 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_3330_a_Hecke(Kf) return MakeCharacter_3330_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 := [[-1], [0], [1], [0], [-2], [-2], [2], [-8], [4], [10], [2], [1], [-8], [-6], [-6], [0], [4], [-4], [-16], [-4], [6], [10], [-16], [-2], [-8], [18], [-4], [8], [-4], [-2], [-8], [12], [-22], [-12], [6], [-8], [-2], [-18], [-4], [-8], [0], [14], [8], [-4], [16], [-26], [20], [24], [-28], [26], [10], [-16], [-2], [12], [-22], [-10], [10], [8], [-14], [-26], [18], [-12], [12], [0], [-28], [-24], [-12], [-34], [12], [26], [14], [-32], [-16], [10], [-20], [-8], [-10], [14], [-14], [10], [2], [-32], [-24], [14], [6], [24], [30], [0], [18], [36], [-36], [24], [-12], [-26], [-20], [12], [22], [36], [34], [24], [26], [42], [12], [14], [-12], [-32], [-36], [22], [-24], [26], [28], [-14], [-30], [-28], [-46], [8], [14], [-8], [-30], [30], [16], [-38], [12], [4], [28], [50], [-12], [4], [0], [38], [12], [-10], [20], [10], [40], [2], [36], [32], [-34], [2], [44], [30], [-32], [-36], [20], [12], [6], [-42], [0], [-18], [-30], [0], [14], [30], [-54], [-16], [42], [28], [-34], [-18], [-44], [26], [32], [38], [62], [-18], [-38], [-6], [-2], [30], [4], [12], [0], [14], [4], [10], [-36], [54], [48], [-46], [24], [12], [10], [2], [30], [-6], [-38], [-14], [-14], [-40], [4], [-8], [-48], [-54], [-12], [46], [54], [14], [-2], [-12], [-14], [0], [26], [-46], [18], [-54], [-56], [40], [-18], [-44], [32], [30], [32], [20], [24], [-6], [68], [42], [-12], [36], [-46], [52], [72], [24], [-20], [28], [46], [52], [-48], [-36], [-34], [52], [-36], [-16], [-56], [14], [-10], [-18], [-48], [32], [4], [-68], [-16], [-32], [-54], [-40], [8], [24], [28], [36], [50], [-12], [-18], [-50], [-10], [-30], [62], [-12], [0], [-68], [-24], [-36], [46], [-34], [-42], [-68], [66], [66], [4], [-2], [12], [10], [-32], [18], [-30], [64], [60], [-8], [10], [8], [-78], [-42], [-26], [-30], [2], [64], [40], [72], [54], [38], [18], [-20], [-70], [-42], [2], [-50], [-32], [36], [-36], [16], [-56], [60], [-40], [-68], [-32], [-44], [20], [-76], [16], [-22], [26], [30], [-48], [-16], [-4], [-38], [46], [-72], [10], [42], [-64], [-30], [30], [-8], [-58], [14], [-68], [34], [36], [4], [22], [18], [10], [36], [-32], [52], [-38], [26], [66], [24], [74], [-42], [-54], [-50], [-30], [12], [-34], [-24], [24], [48], [-4], [-74], [46], [-60], [-74], [66], [-96], [80], [10], [40], [82], [60], [-8], [-20], [84], [12], [6], [-24], [42], [-60], [64], [30], [-30], [-22], [46], [14], [60], [-10], [-18], [66], [-98], [30], [44], [2], [-16], [18], [60], [-94], [-86], [-44], [90], [42], [12], [-54], [-24], [-46], [14], [36], [20], [26], [90], [-26], [16], [42], [78], [-64], [-78], [12], [28], [-42], [-46], [-60], [-12], [16], [-20], [-30], [4], [48], [90], [0], [-66], [-22], [-6], [-54], [2], [8], [44], [-84], [60], [0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3330_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3330_2_a_i();". // 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_3330_2_a_i(:prec:=1) chi := MakeCharacter_3330_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_3330_2_a_i();". // 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_3330_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3330_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![0, 1]>,<11,R![2, 1]>,<13,R![2, 1]>,<17,R![-2, 1]>],Snew); return Vf; end function;