// Make newform 4830.2.a.bc in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4830_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_4830_2_a_bc();". // 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_4830_2_a_bc();". // 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_4830_a();" function MakeCharacter_4830_a() N := 4830; order := 1; char_gens := [3221, 967, 2761, 1891]; 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_4830_a_Hecke(Kf) return MakeCharacter_4830_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], [-1], [1], [-6], [-4], [2], [0], [1], [-8], [4], [-6], [-10], [6], [-8], [-2], [0], [-2], [-10], [6], [-10], [8], [0], [8], [4], [-12], [-8], [0], [-14], [2], [-18], [-12], [-14], [-4], [-22], [4], [18], [8], [-12], [2], [16], [22], [8], [10], [6], [8], [-12], [6], [20], [26], [-10], [6], [22], [2], [14], [-16], [16], [-12], [-8], [12], [-2], [-2], [12], [-6], [24], [18], [20], [-20], [-12], [-18], [-14], [0], [8], [-22], [-32], [-24], [6], [20], [0], [-18], [-18], [-10], [8], [4], [-8], [-36], [-26], [32], [-28], [-30], [-24], [16], [18], [28], [-28], [24], [-12], [24], [22], [-38], [20], [-14], [24], [0], [-16], [22], [-12], [-2], [34], [38], [22], [-2], [38], [24], [40], [4], [14], [8], [-30], [-14], [2], [30], [-6], [-36], [-20], [30], [-6], [-18], [-12], [-30], [-20], [0], [48], [-18], [-22], [-6], [6], [-18], [-30], [18], [52], [-52], [-34], [-8], [38], [4], [44], [42], [-12], [-4], [-8], [-32], [24], [-48], [38], [-12], [-56], [-2], [52], [-22], [-36], [34], [-38], [-6], [-10], [-16], [20], [28], [10], [42], [-42], [50], [0], [-16], [-56], [-36], [-28], [44], [44], [-46], [18], [42], [-16], [38], [40], [-10], [8], [-38], [-58], [6], [2], [12], [0], [52], [36], [-2], [54], [26], [46], [-56], [38], [-8], [-56], [30], [-22], [18], [-48], [4], [-70], [20], [38], [52], [-8], [-8], [26], [-54], [46], [18], [-48], [-30], [-50], [32], [-30], [-28], [-12], [22], [-18], [-30], [48], [-56], [-36], [52], [-16], [-42], [-34], [-32], [42], [-42], [-56], [-42], [68], [-68], [-70], [42], [6], [-26], [26], [-2], [-12], [-24], [-2], [-72], [24], [-30], [-42], [-20], [74], [-42], [38], [26], [-40], [-64], [-14], [-74], [26], [-4], [-26], [-48], [6], [74], [-38], [42], [-4], [-64], [38], [2], [44], [-38], [-2], [-14], [4], [-32], [-24], [-74], [32], [-66], [44], [-58], [-32], [-10], [-46], [68], [-46], [-30], [-44], [-30], [-64], [-6], [-48], [-44], [0], [-46], [-80], [12], [76], [70], [-68], [-46], [4], [36], [-24], [26], [76], [52], [-32], [-50], [-44], [-38], [84], [-6], [40], [-68], [8], [-58], [54], [70], [-68], [-68], [48], [66], [-42], [82], [40], [20], [20], [52], [-82], [-6], [-58], [-84], [-4], [-42], [60], [-48], [-14], [-72], [-22], [72], [-28], [42], [44], [2], [80], [4], [-58], [42], [84], [14], [-42], [-56], [30], [-66], [0], [-90], [-60], [38], [30], [32], [62], [-12], [84], [-96], [6], [0], [-52], [36], [12], [-56], [40], [-26], [-18], [-54], [42], [2], [20], [-24], [-20], [52], [86], [72], [-50], [94], [4], [12], [-40], [-44], [-32], [-4], [32], [-48], [-94], [102], [-76], [38], [28], [0], [38], [-54], [-66], [-48], [18], [38], [60], [-4], [-12], [32], [6], [-94], [-2], [-16], [78], [-46], [48], [-16], [-38], [-70], [56], [74], [-52], [70]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4830_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4830_2_a_bc();". // 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_4830_2_a_bc(:prec:=1) chi := MakeCharacter_4830_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_4830_2_a_bc();". // 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_4830_2_a_bc( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4830_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![6, 1]>,<13,R![4, 1]>,<17,R![-2, 1]>,<19,R![0, 1]>,<29,R![8, 1]>],Snew); return Vf; end function;