// Make newform 3330.2.a.q in Magma, downloaded from the LMFDB on 19 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_q();". // 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_q();". // 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], [3], [5], [-2], [7], [-2], [-4], [5], [-7], [1], [-1], [9], [0], [-3], [14], [-7], [-4], [-8], [-12], [4], [10], [14], [1], [0], [8], [-14], [-7], [5], [16], [6], [22], [-3], [6], [10], [-11], [-15], [-20], [-19], [24], [14], [7], [14], [-10], [4], [-13], [9], [7], [-28], [20], [-17], [-2], [0], [-14], [-11], [-16], [20], [16], [-10], [-12], [-9], [6], [-21], [-10], [21], [0], [-22], [12], [-16], [-17], [-34], [-13], [22], [16], [-4], [31], [-34], [-22], [34], [-20], [10], [-21], [-16], [-33], [-42], [12], [-39], [15], [-6], [-3], [24], [-30], [-28], [4], [-24], [20], [-3], [16], [42], [-7], [18], [3], [10], [27], [10], [21], [-46], [24], [-7], [-14], [43], [-6], [5], [-7], [-41], [-43], [-40], [-36], [0], [31], [-8], [42], [47], [-41], [34], [-27], [-34], [-18], [-13], [33], [13], [20], [22], [47], [2], [-33], [-4], [-32], [4], [20], [6], [-32], [-21], [29], [-18], [6], [-39], [-6], [51], [-3], [15], [-49], [-15], [-36], [40], [0], [41], [32], [-6], [-49], [4], [-4], [-53], [-5], [-45], [19], [6], [-35], [30], [38], [-15], [-45], [-40], [22], [26], [-18], [-54], [-9], [-4], [12], [36], [64], [25], [-57], [-6], [7], [49], [34], [40], [-62], [-16], [-66], [36], [42], [-10], [-24], [-4], [2], [12], [50], [-30], [17], [-19], [3], [42], [-14], [26], [36], [-5], [65], [69], [-34], [58], [33], [-33], [-22], [24], [-18], [-57], [-4], [46], [-9], [24], [-28], [46], [-40], [68], [-57], [54], [-7], [-68], [-6], [-2], [-62], [-11], [62], [54], [66], [40], [29], [-32], [-16], [-65], [30], [31], [-16], [12], [-17], [6], [20], [-30], [3], [10], [46], [27], [-22], [66], [33], [-17], [-66], [48], [16], [41], [-18], [31], [-36], [-72], [-68], [2], [18], [-41], [-74], [-27], [18], [25], [-42], [46], [25], [58], [0], [51], [-67], [-24], [20], [62], [49], [21], [-72], [28], [39], [-76], [-26], [21], [29], [14], [82], [-54], [-84], [-68], [-47], [78], [-28], [-64], [40], [-29], [67], [-58], [62], [80], [-32], [60], [3], [2], [-62], [76], [71], [-12], [46], [24], [44], [30], [-21], [-23], [-8], [8], [67], [-77], [66], [-13], [-26], [90], [37], [-36], [22], [26], [-26], [58], [-30], [0], [74], [-54], [-30], [-47], [-30], [6], [20], [-18], [-93], [21], [83], [52], [29], [-36], [10], [-54], [57], [-14], [2], [-4], [10], [-6], [-58], [-58], [42], [51], [90], [66], [6], [12], [34], [-63], [-66], [-13], [37], [88], [60], [-28], [60], [-12], [65], [6], [56], [-8], [-70], [9], [-78], [-34], [19], [-40], [-21], [-24], [-6], [-9], [-12], [68], [-40], [21], [23], [-92], [-66], [77], [46], [54], [-66], [-28], [84], [-15], [-32], [-63], [80], [-69], [-87], [22], [-29], [-36], [38], [33], [24], [-66], [-18], [-28], [84], [9], [-88], [49], [-56], [45], [84], [-45]]; 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_q();". // 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_q(: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_q();". // 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_q( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3330_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-3, 1]>,<11,R![-5, 1]>,<13,R![2, 1]>,<17,R![-7, 1]>],Snew); return Vf; end function;