// Make newform 2736.2.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2736_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_2736_2_a_l();". // 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_2736_2_a_l();". // 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_2736_a();" function MakeCharacter_2736_a() N := 2736; order := 1; char_gens := [1711, 2053, 1217, 1009]; 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_2736_a_Hecke(Kf) return MakeCharacter_2736_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 := [[0], [0], [0], [0], [-2], [-2], [4], [-1], [-2], [6], [-4], [2], [-2], [-4], [-10], [6], [-4], [2], [-4], [-4], [-6], [-8], [2], [6], [-10], [4], [0], [-12], [-2], [6], [-12], [6], [4], [-12], [-20], [8], [2], [4], [8], [10], [12], [-2], [10], [-6], [-8], [0], [-12], [-16], [-24], [2], [-16], [-6], [-18], [18], [-22], [10], [-14], [16], [18], [-6], [12], [-14], [24], [30], [-2], [18], [-8], [-18], [22], [22], [-4], [-14], [8], [2], [0], [36], [16], [38], [-10], [-26], [6], [22], [-12], [2], [16], [-18], [34], [-14], [-28], [8], [-6], [-30], [28], [30], [-4], [-30], [-14], [-18], [-4], [14], [32], [12], [-16], [42], [4], [22], [-22], [8], [16], [2], [12], [26], [36], [-4], [32], [30], [-20], [-18], [-12], [-40], [10], [-6], [30], [-36], [36], [20], [-26], [30], [32], [-18], [44], [-44], [-40], [26], [-52], [-10], [26], [28], [42], [48], [-24], [-4], [-16], [-12], [30], [-4], [6], [46], [-20], [36], [-22], [-52], [44], [-24], [-52], [-40], [16], [-8], [-38], [-30], [38], [-6], [32], [0], [-18], [8], [-20], [26], [-10], [36], [-28], [-38], [46], [-10], [-36], [-16], [-52], [-60], [44], [14], [-16], [48], [-50], [-14], [14], [-44], [38], [-4], [-2], [-2], [-34], [6], [8], [-42], [-50], [6], [2], [-66], [0], [42], [6], [4], [-50], [-58], [54], [12], [-40], [36], [-36], [-8], [22], [24], [-48], [4], [4], [-46], [-32], [38], [-20], [-40], [-66], [-32], [-50], [16], [-8], [-6], [-38], [-20], [52], [62], [-30], [8], [-36], [14], [-20], [6], [50], [-24], [-42], [36], [44], [20], [40], [-54], [18], [18], [-32], [24], [-56], [-18], [-14], [-48], [-50], [-26], [64], [-66], [-42], [16], [-18], [38], [-64], [-72], [-42], [-6], [72], [-4], [54], [-40], [-64], [0], [-22], [20], [6], [-64], [22], [-48], [10], [10], [10], [42], [-60], [72], [2], [10], [20], [14], [30], [58], [56], [42], [20], [34], [42], [-36], [-46], [48], [-64], [-48], [0], [84], [70], [30], [56], [-84], [-12], [38], [-48], [-74], [-74], [-42], [34], [-16], [-58], [72], [-18], [78], [14], [0], [26], [60], [20], [58], [22], [-16], [4], [6], [-80], [-28], [-12], [28], [50], [46], [-88], [-42], [60], [2], [-54], [-18], [22], [80], [-62], [20], [-2], [20], [14], [-48], [42], [-36], [64], [-60], [68], [-18], [8], [60], [-70], [-34], [42], [42], [36], [-36], [62], [-20], [12], [84], [68], [2], [-68], [52], [-38], [18], [16], [22], [78], [56], [-2], [32], [-42], [34], [-32], [-26], [-18], [64], [-8], [24], [-12], [68], [-32], [50], [-4], [-40], [-50], [-34], [-18], [68], [12], [94], [-44], [-54], [-92], [-16], [-22], [24], [-40], [-22], [26], [8], [46], [-14], [16], [54], [2], [-72], [48], [68], [58], [-60], [-48], [100], [-52], [-28], [-82], [-46], [14], [64], [50], [42], [68], [24], [-28], [-2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2736_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2736_2_a_l();". // 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_2736_2_a_l(:prec:=1) chi := MakeCharacter_2736_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_2736_2_a_l();". // 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_2736_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2736_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![0, 1]>,<7,R![0, 1]>,<11,R![2, 1]>,<13,R![2, 1]>],Snew); return Vf; end function;