// Make newform 6048.2.a.x in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6048_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_6048_2_a_x();". // 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_6048_2_a_x();". // 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_6048_a();" function MakeCharacter_6048_a() N := 6048; order := 1; char_gens := [4159, 3781, 3809, 2593]; v := [1, 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_6048_a_Hecke(Kf) return MakeCharacter_6048_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 := [[0], [0], [3], [1], [-1], [2], [2], [-3], [-1], [-2], [-5], [7], [7], [8], [4], [-4], [4], [4], [2], [3], [0], [4], [2], [15], [4], [6], [-5], [-4], [1], [6], [-2], [0], [14], [4], [6], [22], [6], [-22], [-22], [-3], [12], [-6], [21], [-10], [18], [-25], [10], [21], [-22], [4], [18], [-12], [-28], [20], [-5], [-13], [15], [-20], [-13], [12], [-28], [22], [17], [-18], [-26], [4], [-12], [-21], [3], [14], [-15], [32], [-17], [-27], [8], [-10], [36], [2], [-10], [16], [30], [15], [5], [-2], [0], [-3], [-18], [25], [17], [30], [24], [-12], [0], [-37], [-16], [2], [10], [-11], [29], [15], [-4], [-18], [-26], [-6], [22], [-14], [42], [-3], [-39], [14], [-12], [11], [8], [-41], [28], [-14], [-11], [36], [8], [-33], [30], [-6], [33], [9], [-8], [-6], [29], [36], [-16], [32], [38], [-41], [0], [-2], [-6], [-54], [45], [28], [-35], [26], [11], [-40], [-26], [-51], [0], [-34], [12], [-21], [17], [12], [-38], [-31], [16], [46], [-14], [-8], [4], [-18], [-28], [-17], [-43], [-26], [-22], [6], [-44], [-24], [-26], [-34], [-23], [-35], [24], [-12], [24], [-9], [-13], [1], [-44], [-54], [17], [-44], [62], [18], [-59], [2], [9], [-51], [10], [36], [-41], [-60], [-34], [-36], [14], [-34], [47], [45], [-3], [38], [15], [46], [42], [37], [-20], [-52], [6], [14], [8], [-3], [-12], [20], [-23], [-65], [40], [48], [28], [-8], [-22], [-27], [39], [-40], [-58], [-16], [30], [-12], [2], [-5], [-53], [-35], [51], [59], [-3], [15], [-26], [-74], [-35], [-26], [-6], [-42], [-68], [70], [-21], [39], [9], [17], [31], [-4], [-61], [-54], [-42], [-3], [-2], [26], [-40], [50], [-31], [-11], [-59], [67], [26], [48], [12], [-7], [46], [4], [-7], [31], [34], [6], [-54], [42], [36], [-28], [34], [46], [4], [-47], [21], [34], [46], [38], [-4], [-46], [-14], [-56], [37], [15], [29], [4], [61], [-45], [50], [-72], [-66], [2], [61], [5], [49], [-6], [26], [-19], [10], [42], [-70], [37], [-86], [-29], [-47], [4], [19], [-3], [6], [-60], [-88], [34], [-79], [-4], [58], [15], [-4], [20], [12], [57], [37], [-44], [0], [-68], [-22], [35], [8], [34], [31], [78], [63], [-74], [22], [-56], [23], [-69], [-32], [5], [-67], [90], [34], [42], [40], [-60], [-56], [-42], [78], [22], [47], [-22], [-12], [9], [-67], [-39], [72], [12], [60], [-72], [-38], [-57], [48], [27], [20], [-59], [67], [-58], [7], [73], [65], [-68], [52], [-39], [-53], [-40], [-56], [46], [41], [78], [-95], [-50], [86], [44], [-19], [36], [-14], [-100], [20], [76], [-10], [23], [-87], [32], [12], [79], [-41], [69], [-52], [-2], [-12], [90], [54], [43], [95], [1], [-73], [-6], [25], [-76], [-12], [-8], [-16], [-68], [26], [-33], [-87], [-12], [-18], [6], [16], [-76], [-3], [36], [-54], [82], [52], [-100], [39], [90]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6048_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6048_2_a_x();". // 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_6048_2_a_x(:prec:=1) chi := MakeCharacter_6048_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_6048_2_a_x();". // 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_6048_2_a_x( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6048_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-3, 1]>,<11,R![1, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>],Snew); return Vf; end function;