// Make newform 6422.2.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6422_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_6422_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_6422_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_6422_a();" function MakeCharacter_6422_a() N := 6422; order := 1; char_gens := [4903, 4733]; 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_6422_a_Hecke(Kf) return MakeCharacter_6422_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], [-3], [1], [2], [0], [1], [-1], [-4], [2], [-4], [3], [10], [-11], [-3], [0], [-6], [6], [-2], [-3], [-4], [-6], [-14], [8], [16], [-6], [-8], [-12], [-9], [-18], [0], [-15], [2], [1], [6], [1], [14], [12], [-24], [-16], [19], [-10], [4], [2], [15], [-16], [-1], [-23], [-10], [7], [-19], [5], [10], [0], [-21], [-6], [30], [-7], [-6], [-10], [-32], [-13], [16], [34], [23], [18], [-6], [5], [-3], [-35], [14], [32], [-18], [-10], [-8], [-7], [36], [-10], [-12], [38], [-5], [-5], [11], [37], [-10], [3], [8], [-26], [13], [-4], [-20], [-15], [8], [29], [-10], [44], [38], [-31], [24], [3], [-27], [3], [-25], [-11], [27], [-22], [38], [46], [-12], [-29], [28], [22], [-18], [0], [3], [30], [-22], [38], [8], [4], [14], [-29], [42], [-44], [-12], [-8], [26], [36], [-26], [-33], [-40], [-1], [-24], [0], [6], [30], [19], [14], [30], [39], [54], [-39], [20], [18], [38], [24], [-5], [6], [24], [-33], [45], [21], [27], [44], [-53], [-44], [-4], [12], [-54], [25], [-56], [25], [11], [-3], [48], [39], [6], [16], [-40], [18], [12], [-47], [32], [44], [-28], [31], [-38], [-14], [-38], [20], [56], [-52], [-38], [40], [-8], [6], [-58], [8], [-10], [48], [-45], [-54], [-43], [-34], [-5], [-2], [-14], [4], [-40], [66], [-45], [-56], [-63], [-69], [-46], [26], [35], [12], [26], [-3], [-66], [30], [16], [-22], [1], [-10], [-12], [-21], [-56], [-37], [-20], [13], [-36], [-1], [49], [-62], [21], [10], [10], [36], [52], [4], [-48], [13], [23], [-27], [-34], [31], [20], [-50], [-20], [5], [-32], [-31], [16], [-56], [11], [16], [-16], [-4], [37], [-28], [-37], [77], [50], [12], [74], [8], [-22], [68], [26], [27], [-49], [46], [-48], [-17], [19], [-48], [-64], [-20], [20], [56], [-64], [44], [-74], [56], [-48], [22], [50], [67], [-18], [-81], [20], [10], [14], [66], [-19], [45], [33], [-18], [8], [-27], [34], [10], [8], [46], [74], [-46], [-27], [80], [-1], [63], [32], [36], [71], [52], [-5], [-54], [-24], [44], [-66], [-1], [-15], [-16], [-61], [39], [-30], [0], [64], [-18], [-12], [22], [58], [37], [-14], [34], [44], [68], [-42], [10], [62], [-40], [26], [18], [-90], [76], [-82], [-12], [-56], [-32], [-5], [19], [-57], [-22], [-11], [55], [-20], [82], [37], [-44], [-6], [58], [-23], [-42], [10], [6], [91], [52], [-69], [-51], [-58], [-50], [-34], [-46], [-20], [-61], [51], [-80], [-81], [-57], [41], [-91], [-64], [-14], [16], [-72], [-70], [-22], [46], [-58], [66], [-24], [68], [-54], [-82], [-85], [-83], [2], [64], [-44], [-52], [21], [-96], [-87], [27], [31], [-15], [-85], [76], [-35], [-85], [-22], [-3], [-78], [73], [-68], [-93], [-60], [-34], [-30], [-41], [-12], [-79], [69], [-31], [90], [-39], [44], [58], [-16], [46], [-58], [-60], [-55], [86], [-6], [91], [-84], [74], [32]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6422_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6422_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_6422_2_a_g(:prec:=1) chi := MakeCharacter_6422_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_6422_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_6422_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6422_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 1]>,<5,R![3, 1]>,<7,R![-1, 1]>],Snew); return Vf; end function;