// Make newform 5850.2.a.bj in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5850_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_5850_2_a_bj();". // 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_5850_2_a_bj();". // 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_5850_a();" function MakeCharacter_5850_a() N := 5850; order := 1; char_gens := [3251, 3277, 2251]; 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_5850_a_Hecke(Kf) return MakeCharacter_5850_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], [0], [0], [-2], [-2], [1], [2], [-4], [0], [-4], [8], [-6], [6], [-4], [-8], [2], [-10], [-14], [16], [4], [-8], [-8], [12], [-6], [-12], [-4], [-6], [-4], [-18], [6], [-6], [6], [-14], [4], [-12], [-16], [-10], [8], [12], [14], [18], [-6], [12], [0], [10], [-24], [-4], [14], [12], [-22], [-14], [12], [-18], [-18], [-18], [-8], [20], [16], [2], [18], [-16], [26], [28], [-20], [20], [-18], [-12], [-20], [12], [14], [-34], [-32], [10], [22], [8], [24], [-16], [18], [18], [-26], [14], [34], [-16], [-12], [32], [4], [-30], [28], [-20], [-6], [28], [-36], [26], [22], [36], [-36], [40], [10], [-16], [22], [20], [-14], [-44], [-6], [0], [-28], [36], [34], [-32], [-46], [-18], [2], [46], [-24], [-8], [38], [-32], [36], [-34], [-10], [30], [36], [-10], [12], [-28], [-12], [34], [24], [-34], [-14], [28], [16], [0], [26], [-38], [2], [22], [8], [14], [22], [16], [-24], [-34], [-12], [18], [-40], [18], [34], [20], [36], [-42], [10], [-24], [0], [28], [-16], [-48], [50], [-56], [44], [-36], [6], [-58], [-50], [-18], [56], [-16], [14], [30], [-26], [-42], [50], [32], [8], [32], [58], [32], [-12], [46], [-50], [10], [-34], [26], [-22], [-24], [24], [-34], [8], [-6], [-52], [-28], [36], [4], [-60], [36], [-26], [-34], [22], [42], [36], [-36], [32], [14], [46], [30], [2], [48], [-4], [42], [24], [-20], [0], [34], [-36], [36], [-2], [-22], [58], [0], [2], [22], [8], [6], [-22], [68], [54], [6], [-8], [-10], [-18], [-34], [12], [-8], [42], [-56], [-44], [-2], [-62], [-6], [12], [52], [16], [14], [54], [-18], [12], [-2], [30], [-12], [-24], [-70], [-18], [52], [22], [-54], [-54], [-38], [32], [-78], [40], [-22], [12], [26], [10], [10], [0], [-44], [78], [56], [-74], [-10], [-52], [32], [-56], [-44], [14], [-68], [-10], [6], [78], [0], [-32], [-20], [-6], [52], [-60], [80], [-50], [-56], [-2], [8], [28], [-50], [-18], [-62], [-24], [-8], [14], [18], [44], [-44], [38], [-80], [-36], [16], [-76], [-68], [46], [-60], [-34], [-72], [24], [-54], [-12], [48], [54], [54], [-44], [-24], [54], [0], [88], [-20], [10], [-18], [-10], [-16], [36], [36], [54], [-14], [2], [-8], [-4], [-40], [84], [26], [30], [-46], [-70], [-74], [-30], [24], [-40], [-46], [-46], [-50], [68], [-28], [-82], [-40], [52], [16], [-38], [74], [-6], [32], [-58], [-66], [-28], [-34], [-66], [36], [6], [-24], [-88], [66], [54], [-22], [18], [-20], [-24], [-20], [-16], [-22], [54], [-8], [-84], [-14], [48], [-60], [26], [30], [38], [-8], [-52], [0], [-14], [-68], [-72], [-46], [26], [-66], [24], [40], [40], [0], [-10], [40], [12], [-6], [-98], [-18], [42], [-40], [-48], [-94], [-22], [-24], [2], [48], [-38], [28], [64], [76], [-100], [-72], [58], [18], [48], [48], [-106], [-24], [30], [-48], [2], [-4], [-10], [4], [-72]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5850_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5850_2_a_bj();". // 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_5850_2_a_bj(:prec:=1) chi := MakeCharacter_5850_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_5850_2_a_bj();". // 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_5850_2_a_bj( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5850_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![2, 1]>,<11,R![2, 1]>,<17,R![-2, 1]>,<23,R![0, 1]>,<31,R![-8, 1]>],Snew); return Vf; end function;