// Make newform 882.2.e.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_882_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_e_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_882_2_e_k();". // 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_882_2_e_k();". // 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 := [9, -3, -2, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -2, 2, 1], [3, 2, 0, -1]]; Rf_basisdens := [1, 1, 6, 2]; 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_882_e();" function MakeCharacter_882_e() N := 882; order := 3; char_gens := [785, 199]; 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_e_Hecke();" function MakeCharacter_882_e_Hecke(Kf) N := 882; order := 3; char_gens := [785, 199]; char_values := [[-1, 0, 1, 0], [-1, 0, 1, 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, 0, 0, 0], [0, -1, 0, 1], [-1, 2, 2, -1], [0, 0, 0, 0], [-1, -1, 2, 2], [2, 0, -2, 0], [-1, 2, -1, -1], [5, 0, -5, 0], [1, -2, 4, 1], [-2, 4, 4, -2], [-2, 0, 0, 0], [-2, 0, 2, 0], [7, 1, -8, -2], [-3, 6, 1, -3], [0, 0, 0, 0], [-2, 4, 4, -2], [0, 3, 3, 3], [7, -3, -3, -3], [8, -3, -3, -3], [3, -3, -3, -3], [-3, 6, 5, -3], [5, -3, -3, -3], [-4, 8, 8, -4], [10, -2, -8, 4], [3, -6, -1, 3], [4, 1, -5, -2], [0, 0, -10, 0], [5, -1, -4, 2], [0, 0, -14, 0], [2, -1, -1, 2], [-7, 3, 3, 3], [-1, 2, -4, -1], [-13, -1, 14, 2], [-1, -6, 7, 12], [-2, 4, -8, -2], [-2, 3, -1, -6], [1, -3, -3, -3], [4, -6, 2, 12], [4, 4, -8, -8], [6, 0, 0, 0], [-2, 4, -8, -2], [-11, 3, 3, 3], [-9, -3, -3, -3], [-7, 0, 0, 0], [-6, 0, 0, 0], [0, 0, -10, 0], [16, 0, -16, 0], [0, 0, -4, 0], [-17, -2, 19, 4], [-3, 6, -10, -3], [5, 2, -7, -4], [2, 5, -7, -10], [11, -3, -8, 6], [9, 0, 0, 0], [5, -10, 5, 5], [-10, -1, 11, 2], [-1, 2, -4, -1], [2, -6, 4, 12], [-2, -6, 8, 12], [1, -2, -14, 1], [-17, -3, -3, -3], [-8, 1, 7, -2], [13, 0, 0, 0], [12, -6, -6, -6], [10, 3, 3, 3], [-6, 0, 0, 0], [2, 6, 6, 6], [1, -3, 2, 6], [0, -3, -3, -3], [0, 0, -22, 0], [-11, 1, 10, -2], [-10, 5, 5, -10], [2, 6, -8, -12], [0, 0, 10, 0], [2, -3, -3, -3], [2, -4, -28, 2], [8, 8, -16, -16], [-22, 0, 22, 0], [-2, 4, 7, -2], [-8, 9, 9, 9], [4, 7, -11, -14], [6, -12, -20, 6], [4, -8, 16, 4], [10, 3, 3, 3], [-8, 0, 0, 0], [-30, -3, -3, -3], [33, 0, 0, 0], [5, -12, -12, -12], [5, -10, 14, 5], [16, -3, -13, 6], [7, -11, 4, 22], [28, -2, -26, 4], [9, -18, -14, 9], [23, -1, -22, 2], [3, -6, -11, 3], [12, 6, 6, 6], [8, -16, -10, 8], [5, -10, -13, 5], [-28, 3, 25, -6], [-8, -6, 14, 12], [3, -6, -11, 3], [4, -8, -20, 4], [-3, 0, 0, 0], [6, 3, 3, 3], [8, -9, -9, -9], [3, -6, 17, 3], [-4, 8, 5, -4], [10, -2, -8, 4], [24, 0, 0, 0], [-3, 6, 17, -3], [-6, 12, 8, -6], [-6, 12, 22, -6], [-7, 5, 2, -10], [17, -12, -5, 24], [-7, -9, -9, -9], [-1, 14, -13, -28], [5, -9, 4, 18], [-4, 8, -4, -4], [-2, 4, 22, -2], [-2, 4, 16, -2], [-17, -3, -3, -3], [3, -6, 10, 3], [6, -12, -12, -12], [13, -26, 1, 13], [13, 3, 3, 3], [-18, -6, -6, -6], [44, 0, 0, 0], [-2, -2, 4, 4], [6, -12, -22, 6], [3, -6, -34, 3], [-3, 6, 1, -3], [-10, -10, 20, 20], [-3, 6, 10, -3], [26, 6, 6, 6], [-4, 8, 26, -4], [-10, 0, 10, 0], [5, -10, -28, 5], [4, 0, 0, 0], [-20, -5, 25, 10], [-5, 10, -5, -5], [-32, -3, -3, -3], [-24, 6, 6, 6], [8, -6, -6, -6], [12, 0, 0, 0], [-6, 12, -28, -6], [-4, 8, -4, -4], [3, -6, -28, 3], [-2, 16, -14, -32], [3, -6, 41, 3], [2, 5, -7, -10], [-18, 36, 16, -18], [0, 6, 6, 6], [-10, -9, -9, -9], [-10, 20, 2, -10], [19, -3, -16, 6], [1, -2, 40, 1], [40, -3, -37, 6], [-24, -6, -6, -6], [10, -12, -12, -12], [-9, -3, -3, -3], [24, 3, 3, 3], [18, 3, 3, 3], [-38, -3, 41, 6], [4, 1, -5, -2], [30, 3, 3, 3], [-44, -2, 46, 4], [12, -24, -20, 12], [2, 3, -5, -6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_882_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_882_2_e_k();". // 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_882_2_e_k(:prec:=4) chi := MakeCharacter_882_e(); 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_882_2_e_k();". // 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_882_2_e_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_882_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![36, 18, 15, -3, 1]>,<11,R![36, -18, 15, 3, 1]>,<13,R![4, -2, 1]>],Snew); return Vf; end function;