// Make newform 6018.2.a.e in Magma, downloaded from the LMFDB on 30 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6018_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_6018_2_a_e();". // 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_6018_2_a_e();". // 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_6018_a();" function MakeCharacter_6018_a() N := 6018; order := 1; char_gens := [4013, 1771, 1123]; 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_6018_a_Hecke(Kf) return MakeCharacter_6018_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], [1], [2], [-1], [-2], [-2], [1], [-7], [-5], [4], [10], [8], [-3], [-6], [4], [9], [-1], [-2], [-10], [-6], [7], [0], [-9], [-11], [-5], [-6], [-1], [3], [-4], [-14], [-4], [-10], [0], [-20], [12], [17], [23], [-20], [-24], [2], [-19], [-6], [6], [0], [-22], [-5], [-8], [-4], [-12], [1], [-6], [-4], [4], [6], [-18], [-9], [-3], [32], [-9], [6], [11], [6], [-5], [30], [26], [-24], [21], [25], [12], [27], [-9], [-21], [-20], [36], [30], [1], [30], [-2], [-22], [18], [18], [17], [-39], [-39], [-32], [33], [15], [10], [-1], [-8], [24], [-42], [1], [-2], [34], [21], [38], [-39], [-36], [-20], [-42], [-30], [-29], [-15], [-13], [22], [-35], [-20], [13], [15], [17], [-11], [21], [-8], [24], [17], [34], [27], [26], [-9], [-4], [15], [-12], [-12], [49], [4], [-2], [-32], [28], [-24], [20], [16], [50], [2], [-30], [-14], [14], [-28], [-12], [-14], [7], [11], [38], [-15], [6], [-8], [-6], [50], [22], [-28], [31], [-36], [-41], [-11], [28], [36], [-32], [6], [32], [-7], [-7], [10], [-1], [18], [-42], [35], [-56], [61], [22], [-2], [14], [61], [-60], [-20], [-14], [27], [4], [19], [-56], [0], [-10], [-3], [-58], [6], [-23], [-44], [-54], [-18], [3], [53], [-14], [-20], [-20], [-27], [32], [-24], [4], [-14], [-12], [-44], [18], [64], [0], [-50], [-40], [60], [-32], [15], [46], [36], [50], [-19], [-53], [46], [-20], [19], [52], [0], [64], [1], [-5], [36], [73], [-64], [-27], [-1], [-14], [9], [47], [35], [-39], [7], [35], [65], [68], [-25], [-34], [-19], [-30], [-15], [-39], [-40], [42], [-22], [51], [6], [20], [56], [5], [39], [-8], [8], [72], [0], [25], [-24], [71], [28], [-31], [-27], [-64], [-24], [73], [-1], [-65], [36], [46], [-27], [-54], [-1], [16], [-20], [32], [-4], [-37], [16], [10], [74], [-20], [-52], [-55], [32], [52], [-42], [-44], [-8], [1], [36], [51], [28], [54], [-57], [-28], [20], [-58], [-45], [-50], [-53], [40], [-12], [-7], [-48], [46], [-76], [12], [2], [18], [10], [24], [-25], [14], [-2], [-30], [-14], [-28], [-46], [-11], [-68], [62], [-68], [-84], [50], [35], [16], [-6], [-28], [-32], [-56], [32], [-29], [18], [-82], [-40], [28], [-44], [-79], [-90], [-18], [-71], [-62], [32], [21], [37], [-72], [26], [-78], [14], [44], [93], [61], [62], [-52], [86], [-4], [34], [-42], [-18], [60], [-75], [-96], [-26], [-57], [87], [10], [-47], [64], [-3], [16], [11], [-58], [28], [67], [35], [-15], [47], [-46], [0], [-6], [34], [-26], [-38], [61], [-31], [-6], [68], [72], [-80], [-52], [-88], [60], [-53], [37], [-53], [-8], [82], [13], [74], [-42], [-48], [29], [13], [-75], [28], [69], [-3], [0], [-19], [44], [48], [-97], [86], [74], [-15], [10], [-79], [78], [0], [-98], [14], [-96], [72], [-103], [18], [12], [-86], [-43], [-100], [43], [-60], [-24]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6018_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6018_2_a_e();". // 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_6018_2_a_e(:prec:=1) chi := MakeCharacter_6018_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_6018_2_a_e();". // 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_6018_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6018_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 1]>,<7,R![1, 1]>],Snew); return Vf; end function;