// Make newform 5808.2.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5808_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_5808_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_5808_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_5808_a();" function MakeCharacter_5808_a() N := 5808; order := 1; char_gens := [3631, 4357, 1937, 2785]; 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_5808_a_Hecke(Kf) return MakeCharacter_5808_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], [-1], [0], [2], [0], [4], [6], [-4], [-6], [-6], [-8], [-10], [-6], [8], [6], [0], [0], [-8], [4], [-6], [-2], [14], [-12], [-6], [14], [-6], [4], [-12], [4], [18], [14], [-12], [-18], [-4], [6], [-10], [2], [4], [12], [-6], [-24], [-22], [-18], [-14], [-6], [4], [8], [16], [-12], [-22], [18], [-12], [10], [0], [-30], [0], [-24], [2], [16], [6], [8], [6], [20], [18], [26], [12], [4], [-2], [36], [4], [-6], [-12], [-8], [-20], [-20], [-6], [0], [26], [30], [34], [-24], [-10], [0], [26], [-10], [24], [6], [10], [-42], [4], [12], [-24], [-20], [12], [4], [12], [24], [-18], [-16], [-20], [-28], [-18], [-12], [-18], [-28], [-34], [-24], [30], [-30], [22], [14], [16], [-30], [-44], [16], [6], [4], [-6], [-36], [36], [-22], [-14], [-30], [24], [-20], [6], [26], [-30], [28], [4], [8], [36], [-8], [-34], [-18], [34], [0], [32], [0], [42], [8], [18], [-8], [-12], [14], [-18], [-8], [-54], [-20], [-42], [52], [-30], [28], [-36], [-20], [42], [2], [30], [22], [18], [-12], [42], [14], [0], [54], [-30], [-56], [-44], [34], [12], [-12], [50], [-60], [46], [40], [-6], [-4], [36], [2], [4], [-32], [-60], [14], [-6], [-54], [-12], [-56], [4], [-14], [0], [-10], [60], [28], [48], [-12], [66], [22], [-22], [-30], [-24], [54], [2], [26], [46], [48], [-12], [4], [-36], [6], [-44], [10], [-12], [64], [12], [-12], [-10], [38], [-18], [18], [-36], [-56], [-58], [6], [-8], [60], [-44], [-6], [30], [62], [-36], [14], [8], [74], [6], [52], [72], [-46], [6], [-24], [30], [12], [56], [-32], [-70], [-18], [-36], [-20], [36], [-4], [-12], [-32], [-6], [-42], [-10], [6], [-12], [38], [8], [-12], [-14], [-46], [12], [4], [-44], [-18], [4], [36], [-30], [44], [-30], [50], [28], [14], [-58], [70], [-20], [12], [-80], [46], [60], [0], [-20], [12], [28], [68], [-6], [14], [-54], [28], [30], [-48], [-24], [6], [-36], [-44], [-18], [76], [-72], [-60], [-40], [10], [-18], [-46], [12], [-68], [-22], [60], [14], [30], [4], [-36], [84], [18], [4], [-72], [-50], [-60], [72], [2], [-30], [44], [38], [-42], [28], [6], [14], [-20], [28], [-36], [42], [64], [-72], [74], [84], [-88], [36], [-70], [-18], [26], [-34], [26], [-42], [-66], [-56], [24], [-12], [2], [-20], [-36], [48], [-40], [-94], [-24], [62], [40], [42], [18], [60], [-78], [-42], [16], [6], [-30], [36], [4], [26], [30], [38], [-14], [-24], [28], [72], [66], [14], [-10], [84], [-36], [46], [66], [10], [96], [54], [-58], [30], [8], [30], [-80], [-58], [-16], [102], [-58], [-36], [72], [-68], [-66], [-14], [26], [-18], [100], [90], [40], [-42], [-22], [-42], [-66], [2], [38], [66], [4], [-84], [-98], [78], [84], [-28], [-50], [-12], [12], [-20], [54], [36], [12], [-32], [-102], [84], [-10], [24], [60], [-30], [52], [-48]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5808_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5808_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_5808_2_a_l(:prec:=1) chi := MakeCharacter_5808_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_5808_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_5808_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5808_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![0, 1]>,<7,R![-2, 1]>,<13,R![-4, 1]>],Snew); return Vf; end function;