// Make newform 882.2.h.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_882_h();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_882_h_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_h_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_h_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 := [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_h();" function MakeCharacter_882_h() N := 882; order := 3; char_gens := [785, 199]; v := [1, 2]; // 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_h_Hecke();" function MakeCharacter_882_h_Hecke(Kf) N := 882; order := 3; char_gens := [785, 199]; char_values := [[-1, 0, 1, 0], [0, 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, 1, 0], [-1, 1, 1, 0], [1, -2, 0, 1], [0, 0, 0, 0], [2, 0, 0, 0], [0, -2, 0, 4], [-2, 0, 2, 0], [0, 1, -5, 1], [-1, 0, 0, 0], [0, 2, 2, 2], [0, 0, -6, 0], [0, -4, -2, -4], [0, -4, 0, 8], [0, -2, -2, -2], [0, -4, 0, 8], [6, 2, -6, -4], [0, 0, 2, 0], [-9, 1, 9, -2], [0, 2, 8, 2], [5, -4, 0, 2], [2, 2, -2, -4], [-3, 2, 3, -4], [0, 0, -2, 0], [0, 2, 12, 2], [0, -2, 2, -2], [-5, -10, 0, 5], [14, 0, 0, 0], [0, 0, 12, 0], [2, -6, -2, 12], [11, -2, -11, 4], [-3, 0, 0, 0], [-11, 2, 0, -1], [-2, 8, 0, -4], [-7, 1, 7, -2], [-6, 0, 0, 0], [-5, 0, 0, 0], [0, 3, 1, 3], [0, 4, 10, 4], [-4, 6, 4, -12], [8, -2, -8, 4], [-6, -6, 6, 12], [-3, -6, 0, 3], [-9, 2, 9, -4], [0, 2, 13, 2], [2, 12, 0, -6], [-2, 2, 2, -4], [-8, -2, 8, 4], [0, -2, 16, -2], [-3, -2, 0, 1], [-11, 10, 0, -5], [0, 0, 7, 0], [3, -4, -3, 8], [-4, 4, 0, -2], [15, 2, 0, -1], [18, -8, 0, 4], [21, 4, 0, -2], [11, 3, -11, -6], [0, 2, -12, 2], [-4, -12, 0, 6], [0, 0, 19, 0], [0, 1, 23, 1], [-15, 5, 15, -10], [13, 10, 0, -5], [0, -6, -16, -6], [-10, 6, 10, -12], [-6, -6, 6, 12], [10, -6, -10, 12], [-6, 12, 6, -24], [0, -8, 0, -8], [0, 2, -16, 2], [-6, 0, 0, 0], [0, 4, -1, 4], [-4, -8, 0, 4], [-2, -4, 0, 2], [-2, 20, 0, -10], [2, -4, 0, 2], [-20, -4, 0, 2], [0, -6, -24, -6], [-15, -4, 0, 2], [0, 4, 4, 4], [-27, -1, 27, 2], [0, -2, -18, -2], [-12, -8, 12, 16], [2, -8, 0, 4], [-12, 4, 12, -8], [10, 2, -10, -4], [1, 16, 0, -8], [-7, 10, 7, -20], [0, 3, -13, 3], [11, -6, -11, 12], [0, 0, -10, 0], [-10, 16, 0, -8], [7, 12, -7, -24], [-6, -4, 6, 8], [4, -24, 0, 12], [0, -20, 0, 10], [-12, 4, 0, -2], [24, -6, -24, 12], [0, 3, 7, 3], [0, 10, -6, 10], [0, 8, -12, 8], [8, 2, -8, -4], [0, 11, 13, 11], [30, 0, -30, 0], [0, -4, -24, -4], [-4, 8, 4, -16], [0, -7, 1, -7], [0, -6, 0, -6], [0, 4, 24, 4], [0, -6, -2, -6], [6, 12, 0, -6], [0, 6, 0, -12], [-14, 12, 14, -24], [13, 14, 0, -7], [23, 4, 0, -2], [17, -20, 0, 10], [10, 12, -10, -24], [36, 6, -36, -12], [0, 8, 0, -4], [0, 6, 10, 6], [0, 1, -7, 1], [0, 8, -11, 8], [0, -6, 0, 12], [-42, -4, 42, 8], [-29, -9, 29, 18], [22, -24, 0, 12], [-8, -8, 8, 16], [0, -4, 0, -4], [0, 10, -16, 10], [-15, -2, 0, 1], [6, -8, -6, 16], [-36, 0, 36, 0], [21, -16, 0, 8], [-6, -24, 0, 12], [2, 0, 0, 0], [-10, 18, 10, -36], [-7, -11, 7, 22], [0, 12, -18, 12], [-9, 11, 9, -22], [26, -4, -26, 8], [-2, 0, 0, 0], [-20, 8, 20, -16], [0, 12, -16, 12], [-12, -20, 0, 10], [-16, 6, 16, -12], [0, -2, 40, -2], [0, -13, 7, -13], [-30, -4, 0, 2], [-10, 0, 0, 0], [0, -2, -7, -2], [-2, 20, 0, -10], [0, 16, 0, -8], [-10, -8, 0, 4], [-24, 8, 0, -4], [-12, 12, 0, -6], [0, -20, -3, -20], [0, 6, 11, 6], [0, -14, 0, 28], [26, 16, 0, -8], [0, 1, -1, 1], [-28, -10, 28, 20], [26, -24, 0, 12], [-5, -8, 5, 16], [0, -11, 27, -11], [0, 8, 18, 8], [6, -32, 0, 16], [-8, 4, 8, -8], [35, 14, 0, -7]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_882_h_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_882_2_h_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_h_k(:prec:=4) chi := MakeCharacter_882_h(); 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_h_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_h_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_882_h(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-5, -2, 1]>,<11,R![-2, 1]>,<13,R![576, 0, 24, 0, 1]>],Snew); return Vf; end function;