// Make newform 6048.2.a.g 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_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_6048_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_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], [-1], [-1], [2], [4], [3], [4], [0], [8], [8], [-11], [-7], [11], [-11], [-2], [-11], [-6], [-4], [4], [12], [-1], [3], [10], [4], [-18], [12], [4], [11], [14], [-7], [20], [0], [14], [-18], [9], [8], [25], [1], [-18], [-18], [2], [18], [15], [-16], [26], [-8], [16], [12], [2], [24], [6], [-8], [3], [14], [12], [-9], [6], [13], [-12], [2], [9], [0], [-15], [-2], [-12], [-1], [23], [28], [-32], [-29], [14], [-6], [-1], [25], [-23], [-30], [-8], [-2], [4], [-13], [38], [-24], [10], [-26], [-24], [40], [-10], [-39], [5], [-4], [-9], [-8], [20], [-15], [21], [-21], [3], [-6], [25], [27], [32], [-12], [20], [-5], [30], [-12], [-21], [46], [-32], [38], [-6], [12], [36], [29], [10], [-18], [0], [10], [6], [14], [26], [-26], [-12], [34], [42], [-29], [1], [-12], [2], [-20], [-36], [-16], [27], [21], [-50], [21], [-12], [18], [46], [-2], [-16], [37], [-22], [-4], [-11], [-30], [37], [14], [10], [-47], [30], [-43], [-27], [-47], [42], [-3], [-3], [-52], [-41], [48], [36], [-24], [11], [2], [33], [-25], [24], [61], [1], [24], [12], [2], [43], [16], [43], [-52], [6], [-26], [-50], [11], [-49], [-14], [23], [42], [-59], [6], [20], [-10], [-33], [-44], [-46], [17], [13], [58], [39], [10], [7], [-46], [-17], [30], [-70], [-46], [-24], [17], [21], [-50], [-56], [-32], [-10], [-67], [-42], [-69], [-9], [-65], [-60], [35], [-50], [2], [12], [11], [50], [42], [-17], [15], [41], [33], [0], [10], [-16], [-14], [-16], [55], [-54], [26], [8], [-10], [-6], [48], [41], [6], [6], [54], [-18], [34], [-19], [-20], [-45], [-71], [16], [69], [50], [-78], [60], [29], [24], [-69], [-32], [-18], [-38], [4], [-66], [50], [-26], [-42], [-44], [-8], [-58], [-43], [-18], [-64], [16], [32], [-69], [-26], [36], [-82], [11], [-5], [-5], [-33], [-37], [49], [36], [40], [60], [-23], [32], [-62], [-35], [66], [-45], [-18], [7], [51], [45], [4], [39], [-57], [50], [10], [32], [-16], [74], [-59], [-2], [0], [30], [18], [26], [-36], [16], [74], [61], [-78], [-50], [-39], [-16], [-86], [-48], [-54], [-31], [17], [-20], [-42], [-26], [-81], [18], [62], [-86], [-26], [-22], [80], [91], [-83], [-67], [30], [-71], [-94], [42], [59], [-2], [-41], [-32], [38], [-36], [-24], [23], [-5], [-10], [-16], [29], [40], [-18], [-45], [51], [-72], [69], [-76], [26], [-17], [93], [32], [-94], [40], [-59], [27], [31], [7], [-36], [-20], [-12], [24], [22], [38], [-60], [50], [44], [-78], [-48], [81], [44], [52], [-6], [2], [-88], [24], [14], [53], [68], [59], [-38], [-24], [74], [56], [-90], [80], [15], [-92], [44], [-8], [-66], [65], [33], [-38], [-50], [-7], [-58], [0], [76], [-32], [80], [90], [-39], [67], [-14], [50], [52], [-77], [-44], [-82], [-4], [-2], [9], [-70], [29], [-24], [14], [74], [0]]; 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_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_6048_2_a_g(: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_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_6048_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6048_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![1, 1]>,<11,R![-2, 1]>,<13,R![-4, 1]>,<17,R![-3, 1]>],Snew); return Vf; end function;