// Make newform 243.2.c.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_243_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_243_c_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_243_2_c_e();". // 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_243_2_c_e();". // 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 poly := [1, 0, 0, -1, 0, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1], [0, 1, 1, 0, -1, 0], [0, 0, 0, 0, 1, -1], [0, 1, 0, 0, -1, -1]]; Rf_basisdens := [1, 1, 1, 1, 1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_243_c();" function MakeCharacter_243_c() N := 243; order := 3; char_gens := [2]; v := [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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_243_c_Hecke();" function MakeCharacter_243_c_Hecke(Kf) N := 243; order := 3; char_gens := [2]; char_values := [[0, -1, 0, 0, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0], [0, -2, 0, -1, -1, 1], [1, -1, -2, 2, 0, -1], [-1, 1, -3, 3, 0, -2], [0, 1, -1, -1, -1, 1], [3, 0, 0, 0, 0, 0], [-1, 0, 0, 3, 0, 0], [0, -2, -3, 2, 2, -2], [-4, 4, 3, -3, 0, 1], [0, 4, 2, -1, -1, 1], [-1, 0, 0, -3, -3, 0], [0, 1, -3, 5, 5, -5], [4, -4, 1, -1, 0, 2], [2, -2, 0, 0, 0, -5], [6, 0, 0, -3, 0, 0], [0, 7, 0, -1, -1, 1], [-2, 2, -5, 5, 0, -1], [0, -2, -1, 5, 5, -5], [-3, 0, 0, -6, -9, 0], [2, 0, 0, 3, 6, 0], [-2, 2, 4, -4, 0, 5], [2, -2, -3, 3, 0, 1], [0, 0, 0, 3, 9, 0], [-5, 5, 4, -4, 0, 8], [-4, 4, -3, 3, 0, 4], [0, -5, -4, -4, -4, 4], [3, 0, 0, 3, 0, 0], [-10, 0, 0, -3, -3, 0], [0, -5, 6, -1, -1, 1], [-1, 0, 0, 0, 3, 0], [0, 4, -6, -7, -7, 7], [-7, 7, 6, -6, 0, 1], [0, -5, 5, -7, -7, 7], [0, -5, 9, -4, -4, 4], [-8, 8, -5, 5, 0, -4], [0, 7, -10, 11, 11, -11], [-1, 0, 0, -9, 3, 0], [0, 1, 3, -1, -1, 1], [-7, 7, 3, -3, 0, 7], [-3, 0, 0, -6, 0, 0], [-1, 0, 0, -9, -6, 0], [11, -11, -3, 3, 0, 1], [0, 10, -4, -1, -1, 1], [12, 0, 0, 6, 0, 0], [-1, 0, 0, 0, -6, 0], [0, 7, -4, -1, -1, 1], [1, -1, 4, -4, 0, -7], [11, -11, 3, -3, 0, -2], [0, 10, 8, -10, -10, 10], [3, 0, 0, 0, 9, 0], [0, 16, 6, 2, 2, -2], [-2, 2, 1, -1, 0, -10], [-15, 0, 0, -3, -9, 0], [0, 1, 6, 5, 5, -5], [-13, 13, -3, 3, 0, -5], [9, 0, 0, 9, 0, 0], [-7, 0, 0, 3, -3, 0], [1, -1, -17, 17, 0, -13], [2, -2, 12, -12, 0, 16], [0, 1, -7, -7, -7, 7], [0, -8, 9, -16, -16, 16], [11, 0, 0, -15, -12, 0], [0, -2, -6, -4, -4, 4], [-5, 5, 1, -1, 0, 2], [-25, 25, 0, 0, 0, 4], [1, -1, 4, -4, 0, 20], [0, -8, -10, 8, 8, -8], [0, -11, 6, -7, -7, 7], [-20, 20, 4, -4, 0, -4], [17, -17, -9, 9, 0, -8], [15, 0, 0, 0, 9, 0], [19, -19, -5, 5, 0, -4], [0, -11, 11, 5, 5, -5], [-1, 0, 0, 3, 18, 0], [0, -5, -3, -10, -10, 10], [-4, 4, -9, 9, 0, -11], [8, 0, 0, -15, -9, 0], [0, 4, -6, 2, 2, -2], [0, 4, -7, -1, -1, 1], [0, -14, 0, 14, 14, -14], [4, -4, 16, -16, 0, 5], [3, 0, 0, -24, -9, 0], [-19, 0, 0, -9, -15, 0], [-2, 2, 16, -16, 0, 14], [-16, 16, 12, -12, 0, 7], [-18, 0, 0, -12, 9, 0], [7, -7, 7, -7, 0, 2], [-13, 13, -6, 6, 0, 10], [0, -5, 17, 5, 5, -5], [21, 0, 0, -12, -9, 0], [14, -14, 15, -15, 0, 13], [8, 0, 0, 12, 9, 0], [0, -8, -6, 8, 8, -8], [0, -2, 5, -7, -7, 7], [24, 0, 0, 3, 0, 0], [0, 4, -15, 11, 11, -11], [-27, 0, 0, 3, 0, 0], [2, 0, 0, 3, -12, 0], [17, 0, 0, 15, -3, 0], [16, -16, 7, -7, 0, 5], [15, 0, 0, -15, -9, 0], [0, -14, -15, 11, 11, -11], [2, -2, 15, -15, 0, -8], [0, -23, -13, 2, 2, -2], [2, 0, 0, 3, 15, 0], [17, -17, 12, -12, 0, 22], [6, 0, 0, -21, -9, 0], [0, 1, -6, -1, -1, 1], [19, -19, -5, 5, 0, 2], [0, 10, -16, 20, 20, -20], [11, 0, 0, -6, -3, 0], [0, 7, -12, 11, 11, -11], [13, -13, -23, 23, 0, -7], [8, 0, 0, 12, 27, 0], [5, -5, -12, 12, 0, -14], [0, 13, 17, -10, -10, 10], [3, 0, 0, 3, 9, 0], [0, -23, 15, -19, -19, 19], [-1, 1, 6, -6, 0, -11], [0, -2, 20, -10, -10, 10], [16, -16, -11, 11, 0, -7], [-1, 1, -18, 18, 0, 1], [-9, 0, 0, 12, 9, 0], [13, -13, 13, -13, 0, -19], [15, 0, 0, 24, 0, 0], [1, -1, 13, -13, 0, 2], [-21, 0, 0, 12, 0, 0], [-14, 14, -2, 2, 0, 8], [0, 13, 14, 2, 2, -2], [-19, 0, 0, 0, -15, 0], [0, -23, 0, 14, 14, -14], [0, 25, -13, -10, -10, 10], [-25, 0, 0, 21, 15, 0], [0, 25, -9, -10, -10, 10], [0, 10, 11, -16, -16, 16], [0, 0, 0, 6, -27, 0], [0, -8, -19, 20, 20, -20], [0, 16, -15, -4, -4, 4], [-18, 0, 0, 0, 9, 0], [17, 0, 0, 3, 0, 0], [-13, 13, 9, -9, 0, 16], [0, -17, -4, 5, 5, -5], [-9, 0, 0, 15, 18, 0], [8, 0, 0, 6, 24, 0], [-4, 4, 21, -21, 0, 10], [-17, 17, -11, 11, 0, -7], [17, -17, 3, -3, 0, 4], [0, -29, 14, -10, -10, 10], [3, 0, 0, 21, 9, 0], [0, 10, 5, 5, 5, -5], [3, 0, 0, 18, 0, 0], [-1, 0, 0, 24, 24, 0], [0, 1, -9, 11, 11, -11], [4, -4, 4, -4, 0, 8], [2, -2, 9, -9, 0, -5], [11, 0, 0, 12, 6, 0], [23, -23, 0, 0, 0, -11], [8, 0, 0, -9, -15, 0], [0, 22, 21, -16, -16, 16], [-10, 10, 9, -9, 0, 19], [-3, 0, 0, -18, -18, 0], [0, -23, -4, -13, -13, 13], [24, 0, 0, 15, 18, 0], [0, 16, -18, -1, -1, 1], [14, -14, 0, 0, 0, -2], [2, 0, 0, 0, 0, 0], [-5, 5, 28, -28, 0, 5]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_243_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_243_2_c_e();". // 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_243_2_c_e(:prec:=6) chi := MakeCharacter_243_c(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_243_2_c_e();". // 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_243_2_c_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_243_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![9, 0, 9, 6, 9, 3, 1]>,<7,R![289, -102, 87, -16, 15, -3, 1]>],Snew); return Vf; end function;