// Make newform 4410.2.a.bk in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4410_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_4410_2_a_bk();". // 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_4410_2_a_bk();". // 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_4410_a();" function MakeCharacter_4410_a() N := 4410; order := 1; char_gens := [3431, 2647, 1081]; 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_4410_a_Hecke(Kf) return MakeCharacter_4410_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], [1], [0], [2], [2], [2], [6], [-4], [0], [-2], [2], [10], [-8], [-8], [-2], [4], [8], [-4], [6], [-2], [-8], [4], [-10], [18], [-2], [16], [0], [10], [-2], [12], [-12], [-14], [14], [20], [0], [-10], [8], [12], [-6], [10], [20], [-6], [-6], [-6], [-2], [-4], [28], [-12], [16], [-26], [-26], [-20], [-20], [6], [-24], [-22], [10], [2], [20], [-8], [18], [4], [-8], [-14], [6], [-4], [34], [-4], [16], [-30], [34], [-4], [-18], [-28], [-28], [36], [26], [-32], [-8], [-12], [26], [-18], [-22], [-26], [-20], [0], [30], [18], [-20], [-28], [-8], [-20], [18], [4], [-36], [18], [-18], [20], [2], [28], [6], [-24], [40], [36], [34], [-40], [10], [26], [4], [-4], [-14], [-18], [-18], [16], [-12], [-12], [28], [-46], [-34], [16], [-26], [2], [-36], [-46], [-32], [-42], [-48], [-28], [-30], [36], [24], [-8], [-2], [6], [-44], [18], [12], [50], [44], [-34], [0], [-52], [32], [44], [0], [2], [-42], [-22], [-36], [18], [2], [-20], [-8], [4], [-14], [8], [30], [-18], [14], [-36], [14], [32], [-44], [-42], [56], [0], [-54], [34], [34], [42], [-28], [-18], [-30], [10], [54], [20], [-52], [-52], [-44], [-56], [-60], [-26], [18], [-16], [6], [26], [4], [-18], [48], [18], [24], [-12], [18], [52], [18], [34], [46], [46], [-36], [48], [14], [-10], [-20], [-60], [18], [34], [-48], [-44], [10], [22], [42], [-56], [20], [56], [-44], [-4], [2], [0], [50], [6], [14], [36], [16], [-32], [-2], [30], [18], [-40], [-6], [-66], [38], [56], [20], [20], [16], [40], [-54], [-6], [8], [-24], [70], [24], [2], [22], [-16], [12], [20], [-20], [-36], [-30], [66], [-20], [-8], [-62], [34], [18], [-24], [-38], [42], [36], [-20], [32], [54], [58], [-6], [16], [30], [36], [-30], [-4], [16], [-46], [32], [58], [36], [32], [-50], [-38], [52], [72], [72], [-32], [-4], [68], [-14], [-50], [22], [-50], [-34], [-32], [-60], [46], [-44], [-2], [-30], [42], [-18], [76], [68], [70], [-22], [-16], [-60], [20], [-70], [-72], [72], [54], [-10], [40], [-8], [-40], [-60], [12], [-76], [-84], [50], [74], [-32], [-62], [-10], [-30], [-16], [34], [28], [44], [-56], [-20], [-26], [-34], [-22], [50], [0], [68], [-24], [66], [30], [28], [52], [-26], [-42], [82], [0], [-46], [54], [56], [-76], [-72], [82], [-18], [70], [24], [-80], [10], [42], [-16], [-36], [6], [64], [6], [-30], [48], [-34], [-52], [-10], [30], [-32], [58], [66], [90], [16], [-36], [-38], [18], [4], [-22], [-14], [70], [62], [-58], [-50], [-28], [-22], [38], [12], [-16], [-62], [44], [48], [22], [-2], [14], [28], [30], [-78], [82], [10], [20], [104], [12], [30], [-16], [-10], [70], [-6], [18], [32], [56], [36], [14], [62], [36], [-68], [-94], [10], [-82], [-60], [50], [8], [-60], [34], [32], [-36], [-6], [50], [104], [56], [10], [48]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4410_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4410_2_a_bk();". // 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_4410_2_a_bk(:prec:=1) chi := MakeCharacter_4410_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_4410_2_a_bk();". // 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_4410_2_a_bk( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4410_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-2, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![-6, 1]>,<29,R![0, 1]>,<31,R![2, 1]>],Snew); return Vf; end function;