// Make newform 4014.2.a.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4014_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_4014_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_4014_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_4014_a();" function MakeCharacter_4014_a() N := 4014; order := 1; char_gens := [893, 2233]; v := [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_4014_a_Hecke(Kf) return MakeCharacter_4014_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], [3], [0], [-2], [-2], [4], [2], [-2], [0], [8], [3], [-2], [6], [9], [-6], [14], [10], [-3], [10], [13], [-15], [-7], [4], [2], [-10], [19], [6], [-19], [-6], [-8], [17], [-17], [-14], [-2], [3], [-10], [-5], [-16], [23], [17], [-3], [-6], [-22], [0], [4], [0], [-1], [-17], [-16], [10], [15], [-17], [19], [-18], [-12], [5], [-16], [-8], [20], [6], [18], [7], [12], [16], [18], [-28], [2], [-20], [33], [-14], [12], [-4], [-14], [-14], [4], [-20], [8], [22], [-18], [12], [28], [24], [-1], [-3], [3], [30], [0], [-28], [-26], [24], [-24], [-34], [-42], [16], [30], [-36], [-21], [20], [-10], [-44], [-13], [-26], [11], [35], [-7], [34], [31], [-16], [-40], [-3], [-38], [-14], [21], [1], [25], [20], [8], [-26], [-9], [42], [-34], [2], [-1], [-5], [24], [10], [-31], [2], [34], [29], [-27], [2], [-34], [21], [-10], [-26], [-15], [-10], [-39], [-20], [-30], [-17], [-36], [0], [-40], [46], [-26], [-3], [12], [-2], [12], [-15], [-27], [-22], [-17], [-24], [0], [24], [-10], [12], [15], [40], [16], [31], [12], [4], [7], [22], [-14], [-24], [46], [-27], [-28], [-36], [42], [-7], [0], [14], [-38], [37], [7], [-25], [47], [-21], [-18], [59], [6], [65], [11], [14], [6], [24], [-20], [1], [48], [31], [-1], [33], [66], [-25], [52], [-14], [-24], [-44], [-4], [20], [26], [55], [27], [-30], [43], [-36], [-40], [60], [10], [-14], [-53], [19], [14], [-27], [-56], [21], [64], [-57], [-28], [-57], [24], [-44], [-60], [14], [50], [56], [-50], [-1], [-32], [38], [-21], [64], [6], [12], [-59], [53], [5], [54], [-64], [-72], [70], [-14], [-10], [-37], [77], [-20], [-76], [3], [-1], [29], [-12], [38], [-40], [-7], [25], [6], [-62], [46], [48], [-24], [9], [-46], [38], [20], [55], [-28], [12], [12], [11], [22], [-20], [-47], [60], [39], [8], [12], [28], [-48], [24], [-79], [-15], [25], [-38], [-31], [52], [-21], [-8], [22], [34], [8], [-61], [-30], [26], [-18], [-28], [-13], [8], [10], [-8], [14], [-86], [13], [-40], [19], [56], [-10], [-28], [76], [37], [78], [58], [-46], [41], [62], [-55], [59], [0], [-54], [82], [32], [28], [25], [-33], [-32], [66], [-30], [-94], [40], [-83], [-46], [-42], [70], [-7], [89], [-21], [45], [-26], [39], [57], [82], [-79], [65], [-78], [75], [-12], [-39], [-26], [76], [26], [-21], [-5], [3], [-88], [28], [-58], [28], [42], [-6], [76], [-51], [94], [-15], [9], [34], [20], [-63], [24], [12], [36], [1], [-44], [-6], [29], [-1], [66], [-19], [-9], [-59], [91], [94], [-58], [78], [-56], [-74], [-54], [-18], [-12], [66], [93], [-52], [12], [-86], [78], [59], [-27], [48], [66], [90], [64], [18], [-18], [-68], [65], [12], [3], [88], [32], [-17], [98], [-75], [-60], [-45], [59], [6], [-87], [-25], [54], [-16], [78], [45], [-38], [-46], [-24]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4014_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4014_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_4014_2_a_g(:prec:=1) chi := MakeCharacter_4014_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_4014_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_4014_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4014_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-3, 1]>,<7,R![0, 1]>,<11,R![2, 1]>],Snew); return Vf; end function;