// Make newform 4410.2.a.be in Magma, downloaded from the LMFDB on 29 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_be();". // 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_be();". // 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], [-6], [-6], [0], [4], [0], [8], [-2], [4], [-10], [-6], [-2], [-10], [-4], [-14], [14], [-8], [-6], [-8], [-8], [-18], [-2], [-14], [12], [-4], [6], [-6], [4], [12], [-10], [-8], [0], [-8], [14], [2], [-18], [-6], [-2], [-10], [20], [22], [-6], [18], [12], [-16], [20], [14], [10], [4], [28], [-20], [-8], [32], [22], [18], [-8], [-6], [-28], [22], [20], [-12], [-26], [-10], [-20], [-26], [28], [-2], [4], [24], [4], [-8], [16], [-14], [-24], [14], [-2], [16], [12], [-34], [0], [-18], [-10], [12], [10], [38], [-2], [8], [0], [-36], [12], [22], [-32], [6], [34], [-2], [44], [18], [30], [-2], [24], [-26], [-4], [14], [32], [-16], [-44], [-12], [40], [36], [-42], [-20], [-40], [-18], [4], [-46], [-14], [6], [18], [-14], [-22], [24], [-40], [-24], [2], [-16], [32], [46], [-40], [0], [40], [-40], [-6], [-40], [-26], [-20], [18], [18], [16], [0], [-24], [24], [-22], [-44], [22], [0], [-16], [32], [-32], [18], [14], [26], [-10], [-12], [-8], [42], [-30], [2], [48], [6], [-48], [-4], [-54], [-42], [8], [-42], [46], [18], [-6], [-2], [-4], [54], [-22], [42], [-56], [-24], [-24], [-6], [-20], [12], [-16], [24], [-40], [-18], [-44], [-4], [-38], [0], [-2], [-36], [0], [6], [8], [-40], [-34], [4], [0], [10], [-4], [-38], [-22], [-64], [4], [22], [50], [-16], [-50], [-8], [2], [22], [32], [24], [-44], [0], [60], [-50], [24], [18], [58], [10], [10], [-28], [64], [18], [-24], [-12], [-32], [46], [-12], [44], [24], [22], [68], [42], [-64], [18], [-34], [44], [4], [-28], [-64], [-46], [-52], [24], [-8], [60], [4], [32], [0], [-50], [8], [4], [-66], [6], [38], [-44], [30], [22], [64], [4], [50], [2], [12], [8], [-12], [-18], [-14], [58], [-42], [-10], [14], [8], [46], [-4], [12], [-22], [-10], [-36], [6], [64], [-30], [50], [20], [24], [54], [6], [6], [2], [12], [36], [-6], [-84], [-28], [-14], [2], [-78], [36], [28], [-70], [-38], [24], [56], [68], [66], [-68], [-38], [-48], [-80], [38], [24], [-74], [-2], [48], [0], [12], [-16], [-10], [-46], [-80], [-22], [-78], [-40], [-18], [52], [44], [-4], [64], [-26], [-2], [-46], [-26], [44], [-32], [52], [-46], [20], [24], [-52], [56], [54], [-6], [0], [58], [2], [-50], [82], [12], [-26], [0], [14], [56], [-88], [10], [-44], [8], [-52], [18], [64], [8], [-6], [-48], [-66], [-28], [78], [-86], [-4], [-30], [-62], [0], [48], [-36], [-38], [8], [36], [-40], [78], [70], [-62], [-34], [-86], [-44], [82], [16], [-30], [-64], [70], [46], [42], [-34], [30], [62], [-76], [-24], [-46], [62], [50], [-24], [84], [26], [-34], [92], [84], [-6], [22], [40], [-102], [-28], [60], [34], [-42], [-84], [56], [78], [18], [-60], [-44], [36], [-78], [48], [42], [-88], [-12], [-34], [70], [24], [-6], [68], [-20]]; 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_be();". // 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_be(: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_be();". // 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_be( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4410_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![6, 1]>,<13,R![6, 1]>,<17,R![0, 1]>,<19,R![-4, 1]>,<29,R![-8, 1]>,<31,R![2, 1]>],Snew); return Vf; end function;