// Make newform 882.2.e.n in Magma, downloaded from the LMFDB on 29 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_n();". // 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_n();". // 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 := [4, 0, -2, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]; Rf_basisdens := [1, 1, 2, 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, 1, -1], [0, -1, 1, -1], [0, 0, 0, 0], [-2, 0, 2, 0], [0, 2, 0, -4], [0, 0, 2, 0], [5, -1, -5, 2], [0, 0, 1, 0], [0, 2, 2, 2], [-6, 0, 0, 0], [-2, -4, 2, 8], [0, 4, 0, -8], [0, -2, -2, -2], [0, -8, 0, 4], [0, 2, 6, 2], [2, 0, 0, 0], [-9, 2, 0, -1], [-8, -4, 0, 2], [5, -4, 0, 2], [0, -2, -2, -2], [3, -4, 0, 2], [0, 0, 2, 0], [-12, -2, 12, 4], [0, 2, -2, 2], [-5, -5, 5, 10], [0, 0, 14, 0], [12, 0, -12, 0], [0, -6, 2, -6], [11, -2, -11, 4], [-3, 0, 0, 0], [0, 1, -11, 1], [2, -4, -2, 8], [7, -1, -7, 2], [0, 0, 6, 0], [5, 0, -5, 0], [1, 6, 0, -3], [10, 4, -10, -8], [4, -6, -4, 12], [8, -4, 0, 2], [0, -6, -6, -6], [3, 6, 0, -3], [9, -4, 0, 2], [-13, -4, 0, 2], [2, 12, 0, -6], [0, -2, 2, -2], [-8, -2, 8, 4], [0, 2, -16, 2], [-3, -1, 3, 2], [0, 5, -11, 5], [7, 0, -7, 0], [3, -4, -3, 8], [-4, 2, 4, -4], [-15, -2, 0, 1], [0, -4, 18, -4], [-21, -2, 21, 4], [0, -3, -11, -3], [12, -2, -12, 4], [4, 6, -4, -12], [0, 0, 19, 0], [23, 2, 0, -1], [15, -5, -15, 10], [-13, -10, 0, 5], [-16, -12, 0, 6], [-10, 12, 0, -6], [6, 12, 0, -6], [-10, 12, 0, -6], [-6, 12, 6, -24], [0, 16, 0, -8], [0, -2, 16, -2], [-6, 0, 6, 0], [-1, 4, 1, -8], [-4, -4, 4, 8], [0, 2, 2, 2], [-2, 20, 0, -10], [0, -2, 2, -2], [20, 2, -20, -4], [24, 6, -24, -12], [0, 2, 15, 2], [4, 8, 0, -4], [27, 1, -27, -2], [0, -2, -18, -2], [0, -8, -12, -8], [-2, 8, 0, -4], [-12, 8, 0, -4], [-10, -4, 0, 2], [1, 16, 0, -8], [7, -20, 0, 10], [0, -3, 13, -3], [11, -6, -11, 12], [10, 0, -10, 0], [-10, 8, 10, -16], [0, 12, 7, 12], [-6, -4, 6, 8], [0, 12, -4, 12], [0, 20, 0, -10], [0, 2, -12, 2], [0, 6, -24, 6], [-7, -3, 7, 6], [-6, 10, 6, -20], [0, 8, -12, 8], [0, 2, 8, 2], [13, 22, 0, -11], [-30, 0, 0, 0], [24, 8, 0, -4], [0, -8, 4, -8], [0, 7, -1, 7], [0, 6, 0, -12], [-24, -8, 0, 4], [0, 6, 2, 6], [0, 6, 6, 6], [0, 6, 0, 6], [-14, 12, 14, -24], [13, 7, -13, -14], [23, 4, 0, -2], [-17, 10, 17, -20], [-10, -12, 10, 24], [0, -6, -36, -6], [0, -4, 0, -4], [0, 6, 10, 6], [-7, 2, 0, -1], [0, 8, -11, 8], [0, -12, 0, 6], [0, -4, -42, -4], [-29, -18, 0, 9], [22, -24, 0, 12], [8, 16, 0, -8], [0, 4, 0, -8], [0, -10, 16, -10], [0, -1, -15, -1], [0, -8, 6, -8], [-36, 0, 36, 0], [0, 8, -21, 8], [-6, -24, 0, 12], [0, 0, 2, 0], [10, -18, -10, 36], [0, 11, 7, 11], [-18, 24, 0, -12], [9, -11, -9, 22], [0, -4, 26, -4], [2, 0, 0, 0], [20, -16, 0, 8], [16, -24, 0, 12], [-12, -20, 0, 10], [0, -6, 16, -6], [0, 2, -40, 2], [0, 13, -7, 13], [-30, -2, 30, 4], [0, 0, -10, 0], [-7, -2, 7, 4], [0, -10, 2, -10], [0, -16, 0, 8], [-10, -8, 0, 4], [0, 4, -24, 4], [12, -6, -12, 12], [0, -20, -3, -20], [11, 6, -11, -12], [0, -28, 0, 14], [-26, -16, 0, 8], [-1, 2, 0, -1], [28, 20, 0, -10], [26, -24, 0, 12], [-5, -8, 5, 16], [-27, 11, 27, -22], [-18, -16, 0, 8], [6, -16, -6, 32], [0, 4, -8, 4], [35, 7, -35, -14]]; 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_n();". // 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_n(: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_n();". // 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_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_882_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![25, 10, 9, -2, 1]>,<11,R![4, 2, 1]>,<13,R![576, 0, 24, 0, 1]>],Snew); return Vf; end function;