// Make newform 847.2.a.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_847_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_847_2_a_f();". // 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_847_2_a_f();". // 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, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0], [-1, 2]]; Rf_basisdens := [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_847_a();" function MakeCharacter_847_a() N := 847; order := 1; char_gens := [122, 365]; v := [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_847_a_Hecke(Kf) return MakeCharacter_847_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], [1, -1], [-2, 0], [-1, 0], [0, 0], [-1, -1], [1, 1], [-2, -2], [-2, 2], [-4, -2], [-5, 1], [-4, -2], [9, 1], [-8, 0], [5, -1], [4, -2], [1, -1], [5, 1], [10, 2], [-6, -2], [3, -1], [0, 4], [-2, 6], [2, 0], [4, 6], [7, -1], [1, 3], [4, 0], [0, 2], [2, 0], [-12, 4], [-4, -8], [12, 2], [6, -2], [-14, 0], [0, -4], [2, 4], [10, -6], [4, -4], [15, -1], [0, -4], [-12, -6], [-12, 4], [6, -8], [2, 0], [9, -5], [-4, 8], [1, -5], [12, -8], [0, -2], [4, -6], [-8, 8], [-7, 9], [11, -3], [-6, 0], [0, 0], [0, 6], [6, 2], [2, 8], [8, -2], [-12, -8], [5, 9], [0, 4], [-15, 3], [6, 4], [14, 0], [4, -8], [16, -2], [-12, -4], [5, 1], [2, 8], [16, 4], [-3, -9], [-6, 0], [6, 14], [11, -7], [-2, 8], [-18, -8], [14, 4], [11, -9], [9, 7], [0, 10], [-12, 0], [-4, -2], [-6, -2], [-16, -4], [-24, -2], [2, 12], [1, 5], [-10, 2], [-5, 13], [-18, 2], [-14, -10], [0, 0], [6, -2], [-10, -6], [36, 2], [8, 10], [-44, 0], [6, -12], [-28, 0], [30, 4], [-26, 6], [-12, 2], [24, -4], [-24, 2], [7, 9], [39, 3], [-10, -6], [17, 9], [-4, -4], [16, -14], [20, -6], [-9, 9], [-24, 0], [-20, 2], [-9, 9], [7, 13], [34, -4], [0, 8], [-14, -12], [8, 6], [3, 3], [12, -8], [-15, 15], [20, -2], [-6, -4], [-9, -19], [-13, 17], [-11, 1], [-16, -4], [16, -8], [34, -2], [-2, 8], [7, 11], [-25, 3], [-16, -10], [30, 6], [-6, -4], [-30, 4], [8, 12], [-18, -12], [-32, 8], [4, 4], [10, 12], [15, 13], [15, 7], [-13, -1], [7, -15], [-12, 4], [-28, 6], [16, -6], [0, -4], [-18, 10], [18, 2], [38, 2], [-24, -8], [-12, -18], [-5, 7], [17, -11], [-32, 12], [-14, -4], [-4, 8], [-9, 9], [14, 4], [33, -5], [14, 18], [-17, 7]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_847_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_847_2_a_f();". // 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_847_2_a_f(:prec:=2) chi := MakeCharacter_847_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(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_847_2_a_f();". // 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_847_2_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_847_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-5, 0, 1]>,<3,R![-4, -2, 1]>],Snew); return Vf; end function;