// Make newform 4840.2.a.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4840_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_4840_2_a_i();". // 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_4840_2_a_i();". // 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_4840_a();" function MakeCharacter_4840_a() N := 4840; order := 1; char_gens := [3631, 2421, 1937, 4721]; 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_4840_a_Hecke(Kf) return MakeCharacter_4840_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], [0], [6], [-3], [5], [-2], [5], [5], [-1], [2], [-12], [-2], [-13], [2], [-1], [16], [15], [-10], [-2], [14], [9], [-16], [10], [-16], [4], [-10], [-16], [8], [-7], [12], [4], [15], [-18], [7], [-1], [-1], [6], [4], [22], [12], [-7], [24], [-1], [1], [26], [-18], [6], [-11], [2], [-18], [-10], [-22], [13], [20], [-24], [-16], [-18], [-10], [-6], [22], [21], [-30], [-3], [-20], [-29], [2], [-10], [-18], [-24], [-12], [-14], [-2], [-14], [-2], [22], [-21], [32], [-4], [-36], [8], [4], [-16], [12], [-6], [-33], [-5], [6], [27], [-20], [-12], [13], [-8], [0], [8], [-26], [4], [-15], [36], [30], [18], [-36], [19], [-14], [-39], [22], [-27], [-14], [21], [10], [12], [0], [15], [-9], [7], [-6], [-19], [-49], [-22], [1], [0], [-19], [34], [31], [46], [1], [-26], [32], [8], [-33], [-7], [-2], [30], [-10], [-9], [48], [34], [6], [-1], [-50], [44], [52], [4], [0], [-22], [21], [-22], [-46], [-2], [50], [5], [40], [-29], [3], [32], [59], [-10], [41], [41], [-29], [-55], [44], [-8], [-14], [16], [-8], [6], [-9], [21], [-22], [-10], [-55], [-24], [-53], [4], [16], [-43], [-5], [10], [33], [-15], [-47], [38], [30], [-20], [-44], [18], [36], [-56], [-20], [-46], [-52], [-24], [34], [-24], [2], [-30], [-35], [5], [-2], [-7], [4], [24], [-33], [40], [0], [-18], [4], [47], [-60], [64], [17], [60], [27], [37], [58], [-24], [-47], [-50], [-14], [-3], [-4], [56], [63], [66], [27], [0], [-36], [-6], [31], [70], [10], [44], [-43], [19], [-30], [48], [-23], [57], [-12], [32], [-26], [2], [62], [-48], [-30], [-15], [-11], [34], [0], [-42], [19], [4], [-39], [72], [-54], [-18], [-37], [-24], [28], [35], [46], [-60], [20], [20], [25], [-44], [-18], [-6], [29], [-4], [-14], [77], [-26], [24], [42], [-14], [27], [47], [4], [24], [55], [-68], [-3], [14], [58], [-29], [10], [48], [-12], [41], [63], [-30], [-22], [-16], [-87], [20], [-34], [-65], [32], [-40], [36], [-22], [40], [-28], [16], [-79], [8], [20], [30], [-36], [15], [-33], [-10], [-10], [-12], [-22], [44], [7], [-58], [-70], [50], [-45], [19], [30], [44], [-69], [-72], [-79], [42], [44], [-38], [69], [3], [58], [93], [-19], [43], [-78], [-42], [-7], [-65], [-29], [-45], [80], [17], [-60], [61], [-23], [2], [20], [45], [31], [75], [-15], [39], [54], [60], [-82], [60], [-12], [-81], [-17], [24], [98], [-29], [-44], [24], [80], [-56], [-9], [-42], [-10], [-60], [-78], [13], [20], [31], [36], [62], [44], [59], [-16], [0], [96], [49], [24], [18], [29], [29], [-24], [64], [8], [-86], [62], [-9], [-62], [55], [-95], [86], [68], [12], [6], [6], [29], [2], [21], [10], [42], [48], [-12], [-20], [78], [-12], [12], [16], [-58], [51], [-68], [4], [76], [-91], [-38], [29], [-31], [20], [40], [2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4840_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4840_2_a_i();". // 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_4840_2_a_i(:prec:=1) chi := MakeCharacter_4840_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_4840_2_a_i();". // 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_4840_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4840_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-3, 1]>,<7,R![1, 1]>,<13,R![-6, 1]>],Snew); return Vf; end function;