// Make newform 4719.2.a.k in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4719_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_4719_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_4719_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_4719_a();" function MakeCharacter_4719_a() N := 4719; order := 1; char_gens := [1574, 3511, 4357]; v := [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_4719_a_Hecke(Kf) return MakeCharacter_4719_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], [-2], [0], [0], [-1], [6], [4], [-8], [10], [0], [6], [-10], [-4], [8], [-10], [-12], [-14], [-12], [0], [6], [-8], [-12], [2], [-14], [2], [8], [-12], [-14], [-14], [8], [12], [18], [12], [10], [0], [-2], [-12], [0], [-6], [20], [-10], [0], [-18], [-6], [-24], [20], [16], [20], [-10], [-18], [-8], [-2], [-20], [18], [-8], [-2], [8], [10], [6], [-4], [-6], [20], [-8], [-6], [22], [-20], [-34], [20], [2], [-6], [16], [-16], [-38], [-20], [-8], [-10], [14], [-22], [-10], [-12], [22], [-24], [-14], [16], [-4], [-22], [6], [2], [16], [-12], [24], [8], [-12], [-28], [-24], [6], [42], [28], [34], [4], [18], [44], [-18], [-4], [2], [-12], [30], [-8], [6], [-8], [-38], [18], [-20], [24], [-30], [36], [-8], [-34], [-20], [-10], [-2], [18], [36], [-44], [10], [-10], [32], [-8], [34], [20], [48], [-32], [22], [54], [-18], [30], [-28], [-18], [-2], [-20], [-6], [-8], [-20], [-34], [0], [-22], [-18], [-20], [24], [18], [50], [-44], [-24], [-4], [-16], [-32], [10], [-10], [2], [-36], [46], [-16], [-36], [42], [48], [48], [-22], [-2], [38], [12], [-2], [-24], [6], [-16], [10], [28], [30], [16], [2], [-48], [28], [38], [-42], [8], [54], [2], [-12], [-26], [40], [-62], [-4], [-28], [6], [-36], [-38], [30], [-34], [-2], [24], [34], [8], [-58], [-18], [20], [-18], [16], [-20], [-10], [-52], [14], [-10], [56], [36], [32], [-54], [8], [-10], [48], [-42], [58], [16], [-6], [32], [28], [26], [-54], [64], [16], [12], [-50], [68], [-24], [-50], [28], [40], [34], [42], [60], [-8], [60], [60], [40], [78], [-34], [24], [16], [12], [-36], [0], [2], [-66], [48], [10], [42], [-12], [54], [-4], [22], [22], [-40], [-36], [-6], [-14], [-54], [-28], [22], [-30], [-20], [34], [46], [-76], [42], [-24], [-34], [8], [4], [2], [-58], [60], [-16], [24], [8], [-70], [44], [0], [50], [26], [-8], [22], [30], [4], [22], [-68], [34], [-54], [-48], [-74], [-12], [-28], [-10], [66], [-56], [36], [44], [18], [44], [-82], [16], [10], [80], [-82], [-10], [-28], [64], [-26], [-84], [72], [50], [-26], [-28], [58], [-54], [-32], [-26], [-14], [4], [-20], [-48], [34], [50], [30], [72], [36], [-68], [4], [-34], [-34], [58], [-56], [86], [-22], [-70], [40], [6], [-84], [-26], [92], [-24], [22], [4], [-38], [6], [-40], [-70], [-82], [-72], [-52], [-42], [-64], [-38], [30], [-48], [28], [-12], [-86], [-30], [16], [-58], [-60], [76], [8], [66], [64], [-66], [-20], [80], [14], [-26], [-42], [-26], [66], [16], [-66], [-28], [48], [0], [-42], [-52], [-16], [-46], [-2], [-60], [-12], [64], [22], [-24], [42], [-68], [-54], [66], [-46], [-72], [-30], [-54], [0], [62], [-34], [68], [-4], [-18], [18], [-36], [4], [6], [-66], [72], [-8], [-38], [8], [-50], [-6], [-88], [100], [10], [102], [36], [-42], [12], [24]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4719_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4719_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_4719_2_a_k(:prec:=1) chi := MakeCharacter_4719_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_4719_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_4719_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4719_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, 1]>,<5,R![2, 1]>,<7,R![0, 1]>],Snew); return Vf; end function;