// Make newform 4830.2.a.i 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_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_4830_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_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], [4], [2], [-6], [-4], [1], [2], [-8], [6], [-6], [-4], [0], [-6], [-8], [-14], [12], [0], [10], [4], [0], [-10], [2], [-14], [8], [0], [10], [-14], [4], [0], [18], [-12], [10], [-16], [14], [4], [-24], [-14], [0], [-14], [-16], [-6], [-6], [-4], [-20], [-4], [-24], [10], [-10], [0], [10], [-4], [-2], [16], [26], [8], [10], [-2], [20], [-30], [-20], [0], [-6], [-30], [-20], [2], [-12], [-10], [14], [0], [24], [14], [-4], [16], [10], [-6], [-26], [10], [12], [-30], [-16], [2], [24], [-36], [-6], [-6], [2], [4], [-24], [24], [12], [32], [-4], [-40], [-22], [30], [-4], [30], [20], [-22], [24], [-18], [-20], [10], [12], [-18], [-48], [-22], [-12], [-2], [-14], [-4], [20], [30], [4], [16], [-14], [4], [-22], [10], [42], [-36], [-20], [-30], [18], [48], [48], [14], [-28], [24], [52], [-42], [-6], [-30], [50], [-12], [34], [18], [44], [26], [12], [0], [6], [0], [2], [46], [-20], [8], [-22], [-18], [-20], [48], [52], [-16], [28], [-6], [10], [18], [28], [-38], [-28], [-60], [18], [8], [-16], [-62], [18], [42], [-12], [-18], [-56], [-54], [-32], [-34], [20], [-46], [16], [34], [36], [36], [26], [-18], [16], [42], [2], [20], [-22], [32], [34], [36], [60], [50], [-40], [50], [34], [-50], [26], [-24], [-46], [32], [2], [2], [36], [18], [36], [-12], [-30], [-4], [-6], [42], [64], [-8], [16], [-54], [-44], [-14], [48], [-30], [22], [-20], [-6], [56], [44], [-10], [-38], [24], [72], [-32], [26], [4], [60], [-6], [-28], [40], [-38], [18], [24], [24], [16], [44], [36], [38], [-74], [40], [-52], [-60], [-44], [-32], [-34], [-18], [16], [74], [2], [48], [18], [-4], [58], [-22], [-16], [48], [22], [-18], [-18], [60], [50], [22], [-36], [74], [-2], [-44], [-54], [-44], [-14], [-20], [60], [-42], [-38], [68], [48], [28], [-64], [-78], [20], [-8], [-30], [-62], [-40], [-6], [-30], [80], [-34], [-84], [34], [34], [4], [-22], [-56], [44], [58], [-22], [84], [-36], [4], [18], [-12], [-30], [0], [-86], [-16], [10], [-82], [76], [72], [-54], [64], [48], [66], [34], [-28], [-86], [90], [-4], [34], [-38], [-36], [52], [72], [-46], [70], [-62], [-40], [84], [-4], [-4], [50], [-70], [-86], [-64], [26], [50], [18], [28], [18], [-72], [-82], [4], [-64], [82], [20], [-22], [-14], [8], [42], [-58], [-24], [-12], [-18], [-24], [-42], [-14], [-48], [28], [-12], [26], [-62], [16], [-46], [-80], [-44], [-48], [-22], [-28], [-6], [-72], [48], [2], [38], [-46], [66], [66], [28], [-2], [84], [-48], [80], [-38], [-52], [-24], [26], [-6], [96], [-100], [72], [58], [-76], [22], [-36], [-30], [-90], [-50], [40], [-6], [-62], [40], [-34], [50], [44], [0], [106], [26], [-64], [-44], [-22], [74], [8], [-36], [-102], [-72], [18], [54], [-8], [80], [-54], [-78], [88], [26], [-92], [32]]; 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_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_4830_2_a_i(: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_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_4830_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4830_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-4, 1]>,<13,R![-2, 1]>,<17,R![6, 1]>,<19,R![4, 1]>,<29,R![-2, 1]>],Snew); return Vf; end function;