// Make newform 2646.2.h.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2646_h();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2646_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_2646_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_2646_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 := [9, -3, -2, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [-3, -2, 2, 1], [0, 5, 1, -1], [-9, 2, 1, 2]]; Rf_basisdens := [1, 6, 3, 3]; 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_2646_h();" function MakeCharacter_2646_h() N := 2646; order := 3; char_gens := [785, 1081]; 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_2646_h_Hecke();" function MakeCharacter_2646_h_Hecke(Kf) N := 2646; order := 3; char_gens := [785, 1081]; char_values := [[-1, 1, 0, 0], [0, -1, 0, 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, 1, 0, 0], [0, 0, 0, 0], [-1, 0, 1, 0], [0, 0, 0, 0], [-2, 0, -1, 0], [-2, 2, 0, 0], [-2, 1, -1, 1], [0, -5, 0, 0], [5, 0, 1, 0], [0, -4, 0, 2], [0, -2, 0, 0], [0, -2, 0, 0], [8, -7, 1, -1], [0, 1, 0, -3], [0, 0, 0, 0], [-2, 4, 2, -2], [0, -3, 0, 3], [7, -4, 3, -3], [0, -5, 0, -3], [-3, 0, -3, 0], [-2, 5, 3, -3], [-5, 2, -3, 3], [0, 8, 0, -4], [0, 10, 0, -2], [0, 1, 0, -3], [-5, 0, -1, 0], [-10, 0, 0, 0], [0, -5, 0, 1], [-14, 14, 0, 0], [-1, 2, 1, -1], [-7, 0, -3, 0], [5, 0, 1, 0], [-14, 0, -1, 0], [7, -1, 6, -6], [-10, 0, -2, 0], [-1, 0, -3, 0], [0, -2, 0, 3], [0, 4, 0, -6], [8, -4, 4, -4], [-6, 6, 0, 0], [10, -8, 2, -2], [11, 0, 3, 0], [-9, 12, 3, -3], [0, 7, 0, 0], [6, 0, 0, 0], [10, -10, 0, 0], [16, -16, 0, 0], [0, 4, 0, 0], [19, 0, 2, 0], [-13, 0, -3, 0], [0, -5, 0, -2], [-7, 2, -5, 5], [8, 0, -3, 0], [9, 0, 0, 0], [-10, 0, -5, 0], [-11, 0, -1, 0], [-5, 4, -1, 1], [0, -2, 0, 6], [8, 0, 6, 0], [0, 14, 0, -1], [0, -20, 0, 3], [-7, 8, 1, -1], [-13, 0, 0, 0], [0, -6, 0, -6], [10, -13, -3, 3], [-6, 6, 0, 0], [-2, 8, 6, -6], [-2, -1, -3, 3], [0, -3, 0, 3], [0, 22, 0, 0], [10, 0, -1, 0], [0, 10, 0, -5], [8, 0, 6, 0], [-10, 0, 0, 0], [2, 0, 3, 0], [26, 0, -2, 0], [16, 0, 8, 0], [0, 22, 0, 0], [5, 0, -2, 0], [0, 1, 0, -9], [11, -4, 7, -7], [0, -20, 0, 6], [-20, 16, -4, 4], [-10, 0, 3, 0], [-8, 8, 0, 0], [-30, 33, 3, -3], [-33, 0, 0, 0], [-5, -7, -12, 12], [0, 14, 0, 5], [13, -16, -3, 3], [0, 7, 0, -11], [-26, 0, 2, 0], [-5, 14, 9, -9], [-22, 23, 1, -1], [8, 0, -3, 0], [12, 0, -6, 0], [2, 0, -8, 0], [-8, 13, 5, -5], [0, 28, 0, -3], [0, -8, 0, -6], [0, -11, 0, 3], [16, -20, -4, 4], [0, 3, 0, 0], [6, -9, -3, 3], [0, 1, 0, -9], [-20, 17, -3, 3], [0, 5, 0, -4], [0, 10, 0, -2], [0, 24, 0, 0], [0, -17, 0, 3], [2, 0, -6, 0], [16, -22, -6, 6], [2, -7, -5, 5], [5, 0, -12, 0], [-7, 0, 9, 0], [13, 0, 14, 0], [4, 5, 9, -9], [-8, 4, -4, 4], [20, 0, -2, 0], [0, -16, 0, 2], [0, -20, 0, 3], [0, 10, 0, 3], [-6, -6, -12, 12], [-14, 1, -13, 13], [13, -16, -3, 3], [18, 0, -6, 0], [-44, 44, 0, 0], [0, -2, 0, -2], [0, 22, 0, -6], [-31, 0, 3, 0], [-2, -1, -3, 3], [20, -10, 10, -10], [-7, 0, 3, 0], [26, 0, -6, 0], [-22, 0, 4, 0], [10, -10, 0, 0], [-23, 28, 5, -5], [0, 4, 0, 0], [-25, 20, -5, 5], [10, -5, 5, -5], [32, 0, -3, 0], [-24, 18, -6, 6], [0, -2, 0, -6], [-12, 0, 0, 0], [34, -28, 6, -6], [0, -4, 0, -4], [0, 28, 0, -3], [-14, 0, -16, 0], [44, 0, 3, 0], [0, -2, 0, -5], [2, 0, 18, 0], [0, 0, -6, 0], [-10, 0, 9, 0], [8, 0, 10, 0], [-16, 0, 3, 0], [0, -40, 0, -1], [0, 40, 0, -3], [24, -30, -6, 6], [-10, 0, -12, 0], [0, 12, 0, -3], [24, -27, -3, 3], [-18, 0, 3, 0], [-41, 38, -3, 3], [0, 4, 0, 1], [0, 33, 0, -3], [46, 0, 2, 0], [-8, 20, 12, -12], [5, 0, 3, 0], [26, 0, 0, 0], [5, 0, 13, 0], [-50, 0, -1, 0], [-11, 2, -9, 9], [-13, 0, 13, 0], [-19, 0, -18, 0], [0, 40, 0, 0], [20, -22, -2, 2], [0, 13, 0, -3], [-20, 22, 2, -2], [-22, 0, 6, 0], [-41, 38, -3, 3], [-11, 20, 9, -9], [0, 5, 0, 11], [58, -58, 0, 0], [0, 37, 0, 1], [0, 24, 0, -9], [-2, 0, 8, 0], [-2, -4, -6, 6], [-49, 0, 3, 0], [-37, 0, -6, 0], [-20, 16, -4, 4], [0, -17, 0, -3], [0, 29, 0, -7], [0, -23, 0, 3], [39, -42, -3, 3], [-2, 7, 5, -5], [0, -3, 0, -9], [0, -23, 0, 6], [40, -34, 6, -6], [0, 29, 0, -7], [0, 10, 0, -2], [0, 6, 0, -12], [0, -26, 0, -12], [17, 0, -15, 0], [-4, 0, -9, 0], [-27, 0, 12, 0], [1, -2, -1, 1], [0, -8, 0, 18], [-4, 0, 1, 0], [0, -37, 0, 11], [0, -23, 0, 18], [0, -41, 0, 6], [14, 8, 22, -22], [0, -38, 0, -6], [3, 0, 3, -3], [0, 30, 0, 6], [22, -13, 9, -9], [0, -26, 0, -3], [28, 0, 11, 0], [0, -14, 0, 22], [-28, 26, -2, 2], [-4, 0, 6, 0], [10, 2, 12, -12], [47, 0, -8, 0], [0, 22, 0, -6], [-16, 11, -5, 5], [0, 10, 0, 6], [24, -6, 18, -18], [37, -32, 5, -5], [-56, 56, 0, 0], [12, -12, 0, 0], [0, 22, 0, -6], [-5, 20, 15, -15], [-29, 38, 9, -9], [0, -33, 0, 3], [0, 4, 0, -15], [-50, 0, -4, 0], [20, 0, 6, 0], [0, 10, 0, -20], [-16, 17, 1, -1], [-6, 0, 6, 0], [-56, 0, -1, 0], [0, -44, 0, 0], [0, 16, 0, 12], [0, -8, 0, -18], [50, -49, 1, -1], [0, -18, -18, 18], [2, 0, 0, 0], [0, -33, 0, 12], [0, 28, 0, -12], [-15, 0, 3, 0], [-20, 8, -12, 12], [0, 13, 0, -5], [0, -12, 0, -15], [0, -26, 0, -6], [10, 0, 8, 0], [0, 25, 0, -5], [-2, -10, -12, 12], [11, 0, -9, 0], [-9, 0, -3, 0], [0, -35, 0, -9], [-16, 0, -18, 0], [0, -43, 0, -1], [0, 22, 0, -12], [-7, 0, 3, 0], [0, 54, 0, 6], [4, -13, -9, 9], [42, 0, 12, 0], [0, -19, 0, 11], [-20, 29, 9, -9], [0, 6, 0, -6], [14, 0, 0, 0], [4, -16, -12, 12], [-16, 0, -9, 0], [-16, 0, -12, 0], [22, -19, 3, -3], [0, -8, 0, 0], [-4, 0, 16, 0], [-58, 0, 6, 0], [0, 61, 0, -3], [-15, 21, 6, -6], [0, -48, 0, 0], [0, -2, 0, 18], [0, 26, 0, 8], [0, -50, 0, -3], [23, 0, 6, 0], [0, -20, 0, 1], [-5, 2, -3, 3], [-52, 50, -2, 2], [2, 0, -18, 0], [-42, 0, 3, 0], [4, 0, 2, 0], [-29, 19, -10, 10], [-13, 0, 10, 0], [-52, 47, -5, 5], [0, 10, 0, 0], [0, -48, 0, -9], [-50, 56, 6, -6], [0, -22, 0, 17], [0, -32, 0, 4], [0, 4, 0, 0], [56, 0, 9, 0], [0, -32, 0, 16], [13, 2, 15, -15]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2646_h_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2646_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_2646_2_h_k(:prec:=4) chi := MakeCharacter_2646_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(1999) - 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_2646_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_2646_2_h_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2646_h(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-6, 3, 1]>,<11,R![-6, 3, 1]>,<13,R![4, 2, 1]>],Snew); return Vf; end function;