// Make newform 4018.2.a.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4018_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_4018_2_a_l();". // 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_4018_2_a_l();". // 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_4018_a();" function MakeCharacter_4018_a() N := 4018; order := 1; char_gens := [493, 785]; 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_4018_a_Hecke(Kf) return MakeCharacter_4018_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], [-2], [2], [0], [-4], [-1], [0], [6], [-2], [-5], [-4], [-2], [-1], [-1], [8], [-10], [-1], [-14], [10], [-15], [-7], [0], [-9], [0], [4], [10], [10], [13], [-7], [-3], [4], [-12], [-18], [-4], [1], [19], [15], [-23], [-5], [-24], [-10], [-2], [5], [-10], [-2], [3], [0], [-26], [18], [21], [-16], [16], [5], [9], [2], [-31], [-16], [-20], [8], [14], [1], [-3], [-9], [-13], [-16], [-2], [12], [31], [-24], [26], [-21], [14], [18], [16], [-29], [15], [0], [-26], [25], [-35], [33], [-30], [26], [-2], [7], [-15], [11], [26], [20], [-7], [12], [7], [22], [-13], [36], [16], [-19], [42], [36], [8], [-4], [27], [-16], [-3], [-42], [-6], [-18], [38], [24], [-2], [-32], [-6], [-3], [29], [30], [-6], [-24], [-34], [15], [22], [18], [40], [-12], [-4], [30], [-44], [35], [31], [43], [42], [-15], [-36], [5], [31], [50], [34], [14], [-40], [14], [-50], [-33], [-52], [-32], [-54], [30], [29], [38], [53], [13], [36], [-52], [7], [-20], [-13], [23], [40], [23], [38], [-30], [-36], [45], [-9], [31], [-2], [-2], [-44], [-19], [35], [25], [-6], [-54], [-22], [48], [-15], [-4], [-10], [18], [-22], [-19], [-18], [38], [53], [-51], [9], [-24], [-52], [-22], [-8], [20], [-48], [31], [40], [-17], [-24], [-21], [-1], [40], [-42], [-36], [21], [-60], [20], [-1], [-42], [30], [-29], [-24], [20], [-9], [-67], [6], [27], [22], [-12], [-51], [29], [61], [-30], [-67], [-54], [10], [30], [-54], [43], [49], [58], [-51], [-2], [-28], [-69], [58], [62], [0], [15], [-16], [-43], [-64], [-31], [-11], [29], [54], [-70], [-31], [-4], [-14], [74], [10], [-2], [-75], [-12], [-34], [-5], [16], [54], [-33], [-67], [45], [68], [-27], [2], [76], [-66], [47], [-17], [-25], [27], [-39], [49], [1], [21], [67], [17], [54], [-68], [38], [-50], [-54], [35], [-32], [24], [-31], [4], [66], [-8], [50], [80], [-38], [-30], [0], [52], [-86], [36], [-18], [-1], [23], [49], [-40], [-36], [-6], [-56], [-15], [33], [-40], [-48], [-4], [-18], [-72], [40], [27], [-70], [-21], [42], [-18], [-51], [-34], [65], [-36], [60], [56], [18], [-27], [-54], [-14], [-48], [19], [18], [58], [66], [-39], [-50], [-25], [-42], [46], [-92], [0], [-32], [-33], [21], [20], [74], [77], [-62], [-18], [-35], [-20], [59], [20], [-60], [31], [-84], [-86], [47], [75], [-14], [1], [-6], [60], [37], [-23], [68], [-38], [-78], [-12], [-36], [-18], [52], [3], [8], [74], [-28], [4], [-52], [6], [82], [-97], [-13], [78], [2], [34], [-64], [-66], [-33], [52], [-18], [-4], [-54], [65], [-38], [45], [-72], [-60], [21], [0], [-33], [-10], [-54], [-99], [-30], [28], [30], [12], [-6], [-82], [6], [-40], [-37], [84], [98], [14], [36], [-19], [-48], [40], [10], [-60], [-30], [18], [-77], [-2], [-38], [30], [-23], [-18], [54], [1], [8], [-42], [-20], [80], [33]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4018_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4018_2_a_l();". // 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_4018_2_a_l(:prec:=1) chi := MakeCharacter_4018_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_4018_2_a_l();". // 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_4018_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4018_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![2, 1]>,<5,R![-2, 1]>,<11,R![4, 1]>],Snew); return Vf; end function;