// Make newform 768.2.f.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_768_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_768_f_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_768_2_f_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_768_2_f_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, 3, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [1, 1, 1, 0], [0, 4, 0, 2], [-1, 1, -1, 0]]; Rf_basisdens := [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_768_f();" function MakeCharacter_768_f() N := 768; order := 2; char_gens := [511, 517, 257]; 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_768_f_Hecke();" function MakeCharacter_768_f_Hecke(Kf) N := 768; order := 2; char_gens := [511, 517, 257]; char_values := [[-1, 0, 0, 0], [-1, 0, 0, 0], [-1, 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 := [[0, 0, 0, 0], [1, 0, 0, -1], [-2, -1, 0, 1], [0, 1, 0, 1], [0, 1, 2, 1], [0, 2, 1, 2], [0, 2, 0, 2], [-2, 1, 0, -1], [0, 2, 0, -2], [6, -1, 0, 1], [0, -1, -2, -1], [0, 2, 3, 2], [0, 2, 2, 2], [-6, -1, 0, 1], [8, 0, 0, 0], [-2, -1, 0, 1], [0, -1, 0, -1], [0, -2, 1, -2], [-6, -3, 0, 3], [-8, -6, 0, 6], [2, 0, 0, 0], [0, 3, -2, 3], [0, 1, -2, 1], [0, 0, 2, 0], [-2, 2, 0, -2], [-2, 3, 0, -3], [0, 5, 4, 5], [0, -5, -4, -5], [0, -2, -3, -2], [0, 0, 4, 0], [0, -1, -6, -1], [0, -1, 4, -1], [0, -2, 2, -2], [6, 3, 0, -3], [6, 3, 0, -3], [0, -3, -8, -3], [0, 2, 1, 2], [-2, 5, 0, -5], [16, -2, 0, 2], [14, 7, 0, -7], [0, -5, 0, -5], [0, -2, -9, -2], [8, 4, 0, -4], [-2, -8, 0, 8], [6, -1, 0, 1], [0, 5, 0, 5], [10, 1, 0, -1], [0, -5, -2, -5], [0, 5, -2, 5], [0, -2, 7, -2], [0, -4, -10, -4], [-8, 0, 0, 0], [-2, 0, 0, 0], [0, -3, -10, -3], [0, 4, 8, 4], [8, 2, 0, -2], [-10, -5, 0, 5], [0, 3, -2, 3], [0, 2, -5, 2], [0, 4, 10, 4], [22, -1, 0, 1], [22, -1, 0, 1], [-14, -11, 0, 11], [8, -2, 0, 2], [-2, 2, 0, -2], [-2, -9, 0, 9], [14, 15, 0, -15], [10, 10, 0, -10], [0, -3, -10, -3], [0, 2, 13, 2], [0, 8, 0, 8], [-24, -6, 0, 6], [0, -1, -2, -1], [0, 6, -5, 6], [2, 7, 0, -7], [-24, -4, 0, 4], [-26, -5, 0, 5], [0, 6, 5, 6], [0, -10, -8, -10], [22, 8, 0, -8], [0, 5, 2, 5], [0, 6, 7, 6], [-8, 0, 0, 0], [14, 2, 0, -2], [0, 5, 8, 5], [0, -3, 10, -3], [0, 6, -4, 6], [22, 0, 0, 0], [-2, -5, 0, 5], [0, -1, 18, -1], [0, 1, -6, 1], [16, -8, 0, 8], [0, -7, -12, -7], [0, -13, -8, -13], [26, 1, 0, -1], [24, -2, 0, 2], [22, 3, 0, -3], [0, -10, -6, -10], [18, -5, 0, 5], [0, -10, -7, -10], [6, 1, 0, -1], [6, 7, 0, -7], [0, -15, -6, -15], [0, 2, 22, 2], [6, -9, 0, 9], [26, 2, 0, -2], [0, 11, 4, 11], [0, 4, 12, 4], [0, -6, 0, 6], [-14, -16, 0, 16], [0, -1, 18, -1], [0, -14, -1, -14], [0, 8, 6, 8], [-18, -9, 0, 9], [0, 9, 4, 9], [0, -14, -4, -14], [22, -7, 0, 7], [16, 14, 0, -14], [-18, 3, 0, -3], [0, 3, -16, 3], [0, -14, -17, -14], [-18, -6, 0, 6], [-18, 3, 0, -3], [0, 1, 6, 1], [-26, -3, 0, 3], [-18, 3, 0, -3], [0, 2, -5, 2], [16, 12, 0, -12], [0, 5, 4, 5], [0, -6, -11, -6], [2, 9, 0, -9], [0, 6, 0, -6], [0, 11, 2, 11], [0, 6, -1, 6], [0, -18, -10, -18], [30, 16, 0, -16], [14, -5, 0, 5], [22, 5, 0, -5], [-34, -13, 0, 13], [0, -6, 2, -6], [-46, -5, 0, 5], [22, 23, 0, -23], [0, 9, 20, 9], [0, -9, -4, -9], [0, 2, 1, 2], [-24, -6, 0, 6], [0, 14, 11, 14], [0, 10, -10, 10], [18, -1, 0, 1], [8, 12, 0, -12], [0, -6, 13, -6], [0, 0, 12, 0], [-2, 1, 0, -1], [-8, -10, 0, 10], [-38, -1, 0, 1], [-24, -8, 0, 8], [0, -15, -20, -15], [0, -6, -20, -6], [2, 10, 0, -10], [-2, 3, 0, -3], [0, -13, -4, -13], [0, 10, 6, 10], [0, 1, 4, 1], [0, 17, -2, 17], [0, 10, 24, 10], [24, -2, 0, 2], [0, -5, -22, -5], [0, 10, 3, 10]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_768_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_768_2_f_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_768_2_f_e(:prec:=4) chi := MakeCharacter_768_f(); 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_768_2_f_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_768_2_f_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_768_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, 2, 1]>,<19,R![4, 6, 1]>],Snew); return Vf; end function;