// Make newform 847.2.a.e in Magma, downloaded from the LMFDB on 29 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_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_847_2_a_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 := [-3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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], [0, -1], [-1, 2], [1, 0], [0, 0], [-2, -2], [-5, 1], [3, 0], [-5, 1], [-7, 1], [1, 0], [4, -4], [-7, 0], [-4, 1], [5, -3], [-6, -3], [9, -1], [2, 1], [-3, 5], [-2, -1], [5, 0], [6, 1], [-3, 0], [6, -9], [1, -2], [3, 1], [-12, -1], [-17, 2], [8, 0], [-13, 1], [-2, 0], [6, 0], [6, -7], [-17, -2], [-2, 3], [-9, 4], [-2, 4], [-11, 6], [-4, 4], [-1, 1], [-11, 2], [-16, -4], [-13, -6], [-14, 7], [6, 2], [1, 8], [-4, -5], [7, 3], [5, -5], [2, -2], [-18, 0], [-1, -11], [-12, 5], [9, -2], [18, -5], [12, -14], [0, 7], [17, 5], [0, -13], [-11, 2], [-9, 2], [9, -6], [-11, 12], [11, -2], [13, 1], [15, 1], [10, 6], [-21, 4], [-5, -2], [7, -1], [12, -3], [4, 13], [10, -12], [4, 7], [-2, -6], [11, 12], [6, -6], [21, 4], [-12, -4], [-16, -2], [-20, -8], [-8, -6], [-13, 7], [-10, 17], [-7, -9], [-3, 13], [15, 11], [-5, -7], [-15, 3], [-17, -6], [21, -15], [1, 12], [11, -6], [15, -1], [-4, 3], [-24, 8], [20, 1], [-11, 8], [10, 15], [37, -2], [0, 13], [25, -14], [-2, 1], [0, 0], [11, -16], [-7, 4], [-9, -2], [-18, -2], [-24, 4], [-8, 6], [-14, 15], [-13, 20], [-6, 12], [26, 6], [15, -20], [-15, -6], [31, -14], [6, 2], [7, -6], [23, -12], [-16, -1], [-10, 8], [-8, -1], [18, -1], [6, -10], [15, 7], [0, 11], [-11, 14], [8, 7], [16, -4], [27, 7], [-15, -13], [-20, 12], [-10, -6], [-42, 0], [14, 11], [3, 12], [34, -5], [-1, -12], [-15, 3], [4, 16], [9, 8], [26, 4], [28, -6], [34, -9], [33, 4], [25, 9], [13, -21], [-23, -1], [13, 8], [-1, -20], [9, 11], [13, -9], [-28, -7], [5, -2], [-9, -18], [9, -28], [0, -12], [-27, -2], [-19, -8], [-4, 22], [-17, 10], [-29, -5], [-34, 12], [17, 10], [-30, 9], [-23, -9], [-48, 1]]; 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_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_847_2_a_e(: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_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_847_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_847_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-3, 1, 1]>,<3,R![-3, 1, 1]>],Snew); return Vf; end function;