// Make newform 2240.2.a.k in Magma, downloaded from the LMFDB on 19 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2240_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_2240_2_a_k();". // 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_2240_2_a_k();". // 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_2240_a();" function MakeCharacter_2240_a() N := 2240; order := 1; char_gens := [1471, 1541, 897, 1921]; 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_2240_a_Hecke(Kf) return MakeCharacter_2240_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], [1], [1], [3], [-5], [3], [-2], [-6], [-3], [-4], [-2], [-12], [10], [9], [-12], [0], [-8], [4], [0], [2], [-1], [-12], [-12], [-1], [-6], [5], [-6], [7], [6], [-16], [6], [-12], [-14], [6], [-1], [-14], [-2], [-3], [9], [-12], [-20], [9], [-4], [0], [-16], [13], [-19], [3], [4], [24], [-21], [-10], [-18], [30], [6], [6], [-16], [10], [3], [13], [21], [-11], [18], [-19], [18], [28], [14], [18], [-26], [15], [24], [17], [4], [-20], [12], [3], [25], [-15], [14], [12], [-17], [21], [2], [26], [18], [-9], [8], [24], [32], [-15], [-30], [38], [-15], [31], [27], [6], [-42], [-20], [-11], [-8], [24], [-36], [18], [4], [-7], [24], [-39], [45], [-10], [-13], [-2], [42], [-26], [29], [-30], [-41], [-24], [6], [15], [-32], [-28], [-45], [24], [28], [9], [-35], [-30], [8], [31], [43], [-12], [23], [16], [30], [14], [-21], [-5], [-15], [-15], [-2], [27], [-4], [-54], [52], [-42], [10], [-30], [4], [24], [-50], [0], [-20], [-24], [-26], [-24], [11], [36], [47], [0], [0], [0], [-22], [48], [54], [-21], [56], [37], [-43], [6], [-24], [-50], [-15], [-46], [50], [-12], [55], [51], [-52], [-8], [-28], [-12], [-14], [6], [-36], [-42], [16], [-17], [-31], [48], [2], [-30], [25], [0], [60], [9], [-49], [-44], [3], [-3], [18], [26], [-50], [-46], [-54], [-3], [2], [60], [-42], [-32], [-52], [30], [-28], [27], [18], [-28], [50], [60], [42], [-6], [22], [-22], [-51], [2], [75], [10], [-9], [-51], [-19], [-36], [-68], [-38], [-40], [9], [19], [-3], [32], [48], [-15], [-54], [-24], [-56], [-31], [-38], [33], [54], [56], [-18], [-59], [-18], [-50], [42], [-36], [8], [-6], [39], [-38], [28], [27], [-7], [-10], [78], [4], [7], [18], [46], [27], [-6], [4], [-18], [40], [52], [71], [-4], [-58], [-7], [-12], [46], [65], [-18], [-27], [-49], [48], [10], [-77], [69], [38], [-42], [-28], [-36], [75], [-39], [-24], [12], [-26], [0], [26], [-21], [-60], [55], [-85], [-78], [65], [48], [-23], [2], [-6], [-32], [48], [16], [72], [-39], [-18], [-86], [-90], [20], [-66], [-27], [-37], [69], [82], [32], [54], [20], [-66], [20], [25], [-44], [72], [36], [10], [-36], [-82], [21], [-20], [63], [-65], [66], [14], [-7], [-26], [48], [-42], [17], [18], [-36], [-38], [-44], [66], [57], [-32], [-16], [3], [89], [-5], [-18], [-42], [42], [-6], [-6], [64], [0], [48], [63], [52], [98], [-39], [-70], [29], [-33], [16], [48], [-39], [86], [-2], [72], [9], [-79], [6], [53], [-6], [78], [-52], [-66], [40], [21], [-19], [-77], [-20], [15], [-55], [-27], [-75], [-44], [-27], [-40], [20], [-24], [67], [-18], [4], [-84], [-94], [-33], [-12], [-70], [-26], [-33], [76], [36], [71], [18], [18], [28], [-58], [-24], [-75], [-28], [-54], [-51], [-33], [-38], [-78], [12], [-43], [9], [-72], [-54], [16], [-48]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2240_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2240_2_a_k();". // 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_2240_2_a_k(:prec:=1) chi := MakeCharacter_2240_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_2240_2_a_k();". // 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_2240_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2240_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, 1]>,<11,R![-3, 1]>,<13,R![5, 1]>,<19,R![2, 1]>],Snew); return Vf; end function;