// Make newform 2240.2.a.z in Magma, downloaded from the LMFDB on 29 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_z();". // 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_z();". // 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], [3], [-1], [1], [5], [5], [-7], [2], [-2], [-7], [4], [6], [-12], [2], [1], [0], [4], [-4], [-8], [0], [6], [-3], [4], [0], [13], [18], [13], [-18], [-5], [6], [-12], [6], [-8], [-18], [22], [-19], [-10], [14], [-3], [7], [4], [-8], [-13], [-8], [8], [-4], [-5], [-19], [-9], [-4], [-24], [9], [-22], [-6], [18], [-30], [-26], [12], [-2], [19], [25], [-13], [17], [-34], [-1], [-6], [12], [34], [-26], [-18], [5], [0], [-7], [-16], [20], [12], [15], [23], [-15], [-14], [24], [3], [-25], [14], [-26], [-6], [-9], [-8], [0], [-36], [-11], [34], [22], [-9], [1], [3], [14], [22], [44], [-7], [28], [16], [-4], [-22], [-4], [-37], [0], [27], [15], [10], [3], [30], [6], [-26], [7], [18], [-21], [24], [-38], [-15], [0], [0], [29], [-12], [-44], [-35], [25], [-14], [-40], [33], [-51], [24], [-3], [40], [18], [34], [21], [-17], [39], [-39], [10], [-1], [28], [-6], [20], [-34], [22], [6], [4], [48], [-22], [0], [40], [16], [18], [-8], [-55], [-32], [13], [-28], [-40], [0], [6], [16], [26], [3], [32], [3], [5], [34], [40], [-22], [35], [-58], [38], [12], [-55], [-17], [28], [44], [56], [48], [-26], [-38], [-16], [34], [-44], [-29], [65], [52], [14], [66], [7], [40], [20], [35], [-9], [-12], [-7], [-43], [-54], [10], [58], [18], [-42], [19], [-2], [32], [6], [28], [52], [-2], [-4], [-9], [10], [-24], [-2], [68], [-54], [38], [-50], [34], [-11], [-42], [-33], [50], [29], [-1], [-67], [28], [-40], [-30], [-64], [-47], [39], [-27], [-48], [48], [-73], [42], [-16], [8], [1], [-46], [-13], [-22], [16], [-2], [-37], [62], [-30], [-50], [24], [56], [-26], [33], [58], [12], [21], [3], [22], [-50], [-24], [-47], [-10], [62], [-25], [30], [24], [50], [40], [-52], [69], [44], [2], [41], [20], [-2], [-79], [-18], [21], [-27], [-16], [50], [47], [79], [26], [-30], [48], [0], [-25], [69], [-24], [12], [-46], [-72], [38], [-3], [-76], [51], [33], [6], [-85], [40], [39], [58], [-54], [-40], [48], [52], [40], [-51], [78], [14], [14], [40], [-58], [39], [-47], [-59], [82], [32], [54], [-20], [-50], [48], [23], [4], [8], [60], [82], [44], [10], [65], [68], [-93], [91], [-66], [-26], [-15], [-38], [36], [-46], [-85], [50], [-28], [26], [56], [-30], [15], [-4], [12], [23], [-55], [-17], [58], [6], [50], [22], [-46], [52], [28], [56], [-63], [-84], [70], [-57], [74], [5], [-87], [28], [-76], [69], [-18], [46], [24], [59], [11], [6], [-1], [70], [82], [-96], [18], [0], [-59], [-65], [-51], [84], [31], [1], [75], [-21], [68], [23], [-16], [-64], [-48], [-83], [-66], [0], [24], [-98], [21], [60], [90], [-22], [-57], [68], [60], [-11], [-6], [46], [-52], [22], [-24], [7], [52], [54], [69], [51], [38], [46], [-100], [-81], [-57], [-36], [42], [-80], [-52]]; 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_z();". // 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_z(: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_z();". // 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_z( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2240_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-3, 1]>,<11,R![-5, 1]>,<13,R![-5, 1]>,<19,R![-2, 1]>],Snew); return Vf; end function;