// Make newform 4018.2.a.s in Magma, downloaded from the LMFDB on 29 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_s();". // 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_s();". // 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], [3], [1], [0], [-2], [0], [3], [8], [-4], [-5], [3], [10], [1], [-5], [-6], [-9], [10], [-13], [-2], [9], [-4], [-11], [14], [1], [-7], [10], [-13], [3], [-2], [9], [2], [8], [12], [-14], [3], [19], [-4], [-4], [0], [1], [-16], [14], [-11], [12], [-12], [24], [-12], [-21], [-25], [-20], [10], [-12], [10], [0], [15], [-16], [10], [-20], [26], [2], [-18], [-14], [28], [-18], [22], [-18], [-32], [-19], [12], [-14], [24], [24], [17], [-18], [37], [22], [-2], [-20], [-25], [10], [-28], [-19], [-16], [0], [-34], [-25], [-33], [-18], [21], [16], [-6], [10], [12], [37], [10], [0], [36], [38], [-20], [10], [2], [33], [-4], [-25], [-44], [-46], [21], [43], [-44], [-21], [43], [-16], [30], [24], [30], [-30], [7], [3], [3], [44], [38], [30], [-6], [40], [-13], [16], [3], [8], [28], [-13], [19], [-12], [24], [-5], [-6], [14], [38], [52], [-21], [54], [0], [-4], [-51], [30], [-11], [-14], [21], [38], [8], [-32], [-32], [0], [-26], [-20], [19], [2], [3], [50], [-7], [-18], [-4], [30], [-47], [-13], [54], [3], [40], [-4], [-19], [36], [-30], [49], [-60], [19], [-4], [-49], [-40], [-58], [28], [22], [-46], [-42], [-26], [22], [54], [17], [-58], [-32], [-4], [10], [64], [30], [-4], [43], [33], [-4], [40], [-18], [63], [-62], [54], [28], [8], [-18], [-49], [-60], [-69], [52], [30], [-32], [38], [-42], [-40], [22], [24], [-48], [-23], [-18], [3], [72], [31], [35], [-32], [-32], [28], [23], [-62], [-2], [8], [-60], [-44], [31], [2], [-9], [-56], [-46], [1], [10], [9], [-14], [-16], [1], [52], [-4], [-28], [29], [0], [-67], [33], [30], [30], [-34], [40], [56], [-18], [17], [61], [-4], [28], [43], [12], [2], [66], [-28], [-32], [-34], [79], [42], [-40], [-2], [-76], [19], [52], [-4], [7], [-76], [52], [-2], [38], [36], [24], [-9], [63], [42], [-55], [3], [40], [-82], [38], [35], [33], [-4], [52], [42], [-26], [-74], [78], [42], [29], [-28], [36], [38], [68], [16], [-25], [37], [12], [-14], [-53], [59], [36], [19], [31], [-51], [16], [38], [-42], [40], [35], [72], [38], [10], [14], [65], [-9], [50], [-4], [1], [31], [30], [45], [-72], [49], [59], [47], [28], [-33], [64], [-49], [36], [19], [51], [14], [9], [-88], [-75], [10], [-60], [-14], [22], [1], [19], [-47], [-32], [3], [-14], [15], [80], [66], [-40], [-26], [1], [-72], [-81], [-53], [38], [-70], [-86], [23], [54], [71], [-32], [30], [-32], [17], [-4], [30], [-46], [-6], [-21], [38], [-61], [-26], [47], [-35], [-81], [-37], [10], [-11], [42], [44], [8], [-2], [8], [-102], [68], [-60], [14], [-75], [33], [85], [-46], [-88], [-6], [3], [50], [54], [-68], [3], [-90], [-34], [-4], [-96], [80], [-82], [-55], [24], [52], [-21], [15], [-58], [78], [58], [-56], [-28], [10], [24], [72], [52], [-4]]; 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_s();". // 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_s(: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_s();". // 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_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4018_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-3, 1]>,<5,R![-1, 1]>,<11,R![2, 1]>],Snew); return Vf; end function;