// Make newform 2352.2.a.n in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2352_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_2352_2_a_n();". // 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_2352_2_a_n();". // 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_2352_a();" function MakeCharacter_2352_a() N := 2352; order := 1; char_gens := [1471, 1765, 785, 2257]; 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_2352_a_Hecke(Kf) return MakeCharacter_2352_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], [1], [-3], [0], [-3], [4], [0], [-4], [0], [9], [-1], [8], [0], [10], [-6], [-3], [3], [10], [10], [6], [-2], [1], [-9], [-6], [1], [18], [8], [3], [14], [0], [-5], [-9], [18], [2], [18], [1], [4], [16], [6], [-18], [-12], [-8], [0], [-19], [6], [20], [-14], [-19], [-27], [4], [-24], [24], [1], [27], [-6], [6], [-21], [11], [8], [6], [14], [-33], [8], [24], [31], [9], [-20], [-7], [-12], [-26], [-24], [-30], [-19], [8], [-8], [18], [-6], [4], [24], [25], [0], [-22], [-12], [34], [35], [33], [12], [-1], [-30], [-8], [36], [-18], [-41], [33], [-2], [-12], [3], [18], [-4], [26], [-8], [3], [39], [-36], [34], [-23], [21], [-24], [18], [-11], [-7], [-16], [-6], [-34], [7], [-30], [-34], [-18], [3], [24], [-14], [29], [33], [-33], [8], [-15], [-10], [-18], [-13], [10], [-50], [-42], [7], [38], [-12], [19], [-6], [50], [33], [0], [2], [-3], [40], [-15], [4], [-24], [10], [42], [50], [-6], [32], [6], [-32], [-24], [-8], [-6], [-8], [6], [-35], [9], [0], [-6], [1], [-39], [42], [36], [13], [-14], [-49], [33], [27], [58], [60], [5], [41], [36], [-2], [15], [-16], [-26], [49], [45], [-34], [18], [54], [-3], [-34], [-22], [-13], [12], [31], [51], [-20], [6], [33], [54], [-10], [62], [-30], [42], [-21], [8], [-14], [-26], [48], [-33], [-16], [27], [24], [-58], [-46], [9], [1], [69], [-36], [-2], [-5], [0], [-18], [30], [-40], [56], [-30], [-11], [-15], [-64], [-42], [36], [-13], [36], [74], [-16], [49], [6], [-58], [60], [-35], [15], [-60], [-72], [-12], [56], [5], [-70], [-12], [-60], [-25], [60], [-32], [-24], [2], [54], [-30], [-2], [-69], [-51], [-22], [-52], [3], [-29], [16], [9], [-62], [-44], [6], [-70], [-81], [12], [4], [39], [16], [34], [43], [-17], [-5], [-16], [3], [-16], [38], [48], [-54], [37], [12], [-20], [50], [24], [-25], [-9], [-7], [12], [-6], [45], [-36], [-60], [-4], [9], [80], [-42], [36], [-46], [58], [39], [7], [-63], [-2], [50], [-39], [58], [-6], [14], [6], [30], [60], [-32], [-48], [-38], [-21], [-48], [-53], [-24], [-64], [-46], [-33], [16], [48], [37], [-62], [-22], [-18], [75], [2], [-81], [32], [-24], [10], [-12], [20], [-54], [-41], [-19], [-82], [-36], [87], [-77], [9], [-9], [10], [46], [-6], [15], [-4], [11], [-42], [-55], [-10], [-42], [-30], [-45], [12], [-6], [-46], [-24], [30], [-12], [56], [35], [42], [-56], [-7], [51], [-64], [66], [9], [80], [-16], [24], [-12], [-11], [-66], [-62], [6], [30], [16], [-84], [50], [42], [-32], [4], [4], [-12], [-55], [6], [9], [-58], [-78], [23], [-79], [-42], [-80], [-33], [76], [48], [4], [6], [-21], [29], [62], [90], [26], [-81], [1], [-21], [-9], [-26], [-70], [75], [84], [56], [-72], [-6], [51], [106], [78], [-33], [70], [-78], [27], [48], [38], [-12]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2352_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2352_2_a_n();". // 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_2352_2_a_n(:prec:=1) chi := MakeCharacter_2352_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_2352_2_a_n();". // 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_2352_2_a_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2352_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![3, 1]>,<11,R![3, 1]>,<13,R![-4, 1]>,<17,R![0, 1]>],Snew); return Vf; end function;