// Make newform 2646.2.f.n in Magma, downloaded from the LMFDB on 19 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2646_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2646_f_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_f_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_2646_2_f_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 := [3, -12, 19, -15, 10, -3, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0], [2, -1, 1, 0, 0, 0], [6, -18, 5, -8, 1, -1], [3, -5, 6, -2, 1, 0], [9, -21, 19, -16, 5, -2], [-9, 30, -22, 19, -5, 2]]; Rf_basisdens := [1, 1, 3, 1, 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_f();" function MakeCharacter_2646_f() N := 2646; order := 3; char_gens := [785, 1081]; v := [2, 3]; // 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_f_Hecke();" function MakeCharacter_2646_f_Hecke(Kf) N := 2646; order := 3; char_gens := [785, 1081]; char_values := [[-1, 0, 0, 0, 1, 0], [1, 0, 0, 0, 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 := [[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 2, -1], [0, 0, 0, 0, 0, 0], [0, 1, -2, -2, 0, -1], [1, 0, 0, 0, -1, 1], [0, -2, 0, -2, 0, 0], [-1, 1, 0, -1, 0, 0], [3, 0, -1, 0, -3, 1], [0, -3, 1, 1, 1, 3], [5, 0, 1, 0, -5, 2], [3, -2, 0, 2, 0, 0], [-4, 0, -1, 0, 4, -1], [0, -2, -1, -1, 5, 2], [0, 2, 1, 1, 2, -2], [2, -4, 0, 1, 0, 0], [0, 0, -1, 0, 0, -5], [0, -5, 3, 3, -2, 5], [3, 0, -3, 0, -3, 1], [-1, 1, 0, 3, 0, 0], [-9, 1, 0, -3, 0, 0], [0, -3, -4, -4, 0, 3], [0, -1, 2, 2, 3, 1], [4, 1, 0, 2, 0, 0], [0, 0, 2, 2, 10, 0], [0, 3, -4, -4, -4, -3], [2, 0, -3, 0, -2, -3], [-6, 0, 0, -3, 0, 0], [2, -2, 0, 1, 0, 0], [2, 0, -3, 0, -2, 4], [-4, -4, 0, 5, 0, 0], [-4, 0, 4, 0, 4, 6], [0, 0, 1, 1, -5, 0], [10, 0, 1, 0, -10, 3], [0, 0, 3, 0, 0, 3], [0, 5, -1, -1, 13, -5], [8, 0, 3, 0, -8, -6], [7, -7, 0, -2, 0, 0], [6, 0, -1, 0, -6, 7], [0, -4, -3, -3, -11, 4], [0, 3, 0, -3, 0, 0], [-17, 1, 0, -5, 0, 0], [0, -5, 2, 2, -11, 5], [9, 0, 2, 0, -9, 5], [-9, 2, 0, 2, 0, 0], [-6, -1, 0, 1, 0, 0], [11, 0, 4, 0, -11, 4], [0, 2, -2, -2, 7, -2], [0, -8, 5, 5, -8, 8], [4, 0, -2, 0, -4, 6], [9, -1, 0, 5, 0, 0], [1, 0, 4, 0, -1, -8], [0, -4, -8, -8, -5, 4], [-17, 1, 0, 2, 0, 0], [4, 0, -8, 0, -4, -5], [0, -1, 4, 4, -10, 1], [-6, 2, 0, -10, 0, 0], [0, -9, 0, 5, 0, 0], [0, 3, 0, 0, 4, -3], [0, -8, 1, 1, 0, 8], [7, 0, -5, 0, -7, -3], [11, 0, 2, 0, -11, 8], [-1, 3, 0, 1, 0, 0], [3, 0, 2, 0, -3, 7], [0, -8, -2, -2, -10, 8], [0, 6, -9, -9, 6, -6], [0, 6, -3, -3, -11, -6], [-17, 0, 2, 0, 17, -8], [-16, 0, 8, 0, 16, -1], [0, -1, 9, 9, 0, 1], [0, 9, -3, -3, -3, -9], [4, 2, 0, -9, 0, 0], [0, -1, -2, -2, 28, 1], [-21, 0, 3, 0, 21, -2], [-3, 5, 0, -12, 0, 0], [11, 0, 3, 0, -11, -2], [0, 3, -8, -8, -11, -3], [-14, -2, 0, -2, 0, 0], [4, 0, -5, 0, -4, -11], [10, 0, 2, 0, -10, 5], [2, 0, -4, 0, -2, -10], [0, 8, 6, 6, 14, -8], [1, -8, 0, 2, 0, 0], [-4, -5, 0, 8, 0, 0], [0, -5, 6, 6, -11, 5], [0, -1, 0, 0, -14, 1], [-2, -13, 0, -6, 0, 0], [0, -9, 1, 1, -19, 9], [0, 4, -5, -5, -3, -4], [7, 0, 1, 0, -7, -13], [-12, -4, 0, -7, 0, 0], [0, -3, -5, -5, 16, 3], [8, 9, 0, 3, 0, 0], [-5, 0, 13, 0, 5, -5], [10, 0, 7, 0, -10, 2], [8, 11, 0, -8, 0, 0], [-5, 0, -2, 0, 5, -14], [18, 0, 0, 15, 0, 0], [10, 9, 0, -6, 0, 0], [16, 7, 0, 9, 0, 0], [0, 15, 2, 2, 12, -15], [-1, 15, 0, -7, 0, 0], [11, 0, -3, 0, -11, 10], [0, -6, -6, -6, 3, 6], [9, 0, -2, 0, -9, 9], [12, -11, 0, 6, 0, 0], [0, -11, 0, 0, 14, 11], [-21, 6, 0, 6, 0, 0], [29, 0, -4, 0, -29, 5], [0, 6, -3, -3, 8, -6], [14, 0, 1, 0, -14, -10], [5, 11, 0, -1, 0, 0], [0, 0, -1, 0, 0, -8], [0, -4, -3, -3, -6, 4], [-20, -10, 0, -5, 0, 0], [0, -12, -1, -1, -7, 12], [8, 0, 3, 0, -8, -9], [-33, 0, 0, -3, 0, 0], [22, 0, 1, 0, -22, 10], [0, -2, -11, -11, -9, 2], [-2, 0, -8, 0, 2, -3], [0, -4, 4, 4, -9, 4], [0, -9, -3, -3, 9, 9], [11, -7, 0, -5, 0, 0], [0, 9, 3, 3, 5, -9], [30, 7, 0, 1, 0, 0], [0, 4, 5, 5, 8, -4], [-17, 1, 0, 2, 0, 0], [0, -18, -1, -1, -14, 18], [-17, 0, -3, 0, 17, -14], [-8, -17, 0, -6, 0, 0], [9, 0, 0, 0, -9, -6], [-36, 0, -2, 0, 36, -9], [8, 17, 0, 8, 0, 0], [15, 0, 4, 0, -15, 17], [8, 0, 2, 0, -8, 13], [-22, -11, 0, -6, 0, 0], [-13, 0, -5, 0, 13, 2], [-16, 0, 5, 0, 16, -4], [14, -9, 0, -1, 0, 0], [4, -7, 0, -13, 0, 0], [0, -4, 3, 3, 10, 4], [-8, 0, 13, 0, 8, -7], [-2, 7, 0, -7, 0, 0], [-2, 5, 0, -13, 0, 0], [0, 3, -8, -8, 10, -3], [0, 4, -7, -7, 21, -4], [0, 5, 4, 4, -1, -5], [23, 0, 0, 0, -23, 15], [-8, 11, 0, -3, 0, 0], [9, 0, 6, 0, -9, -2], [32, 8, 0, -5, 0, 0], [-14, -10, 0, -14, 0, 0], [32, 0, -1, 0, -32, 2], [0, 7, 7, 7, 16, -7], [0, 14, -2, -2, 5, -14], [-10, 4, 0, -11, 0, 0], [0, 4, 5, 5, -11, -4], [-7, 7, 0, 14, 0, 0], [10, 0, 6, 0, -10, -4], [0, 12, 10, 10, -5, -12], [0, -10, 0, 8, 0, 0], [-1, 0, 10, 0, 1, 1], [-13, 2, 0, 4, 0, 0], [10, 0, 4, 0, -10, 7], [0, -14, -1, -1, 13, 14], [1, 8, 0, -5, 0, 0], [0, 13, 8, 8, -30, -13], [10, -4, 0, 19, 0, 0], [7, 0, -8, 0, -7, 10], [0, 13, -9, -9, -13, -13], [17, 0, -13, 0, -17, -11], [0, 0, -9, 0, 0, 15], [0, 16, -11, -11, -8, -16], [-17, 0, 15, 0, 17, 4], [24, 0, -22, 0, -24, -8], [0, 20, -12, -12, -1, -20], [2, -5, 0, 21, 0, 0], [-7, -14, 0, -7, 0, 0], [0, -15, -3, -3, 14, 15], [0, -13, 0, 0, -7, 13], [0, 6, 7, 7, -26, -6], [-28, 0, -10, 0, 28, -9], [29, 1, 0, 3, 0, 0], [-48, 0, -2, 0, 48, -4], [0, 2, 13, 13, 5, -2], [-21, 0, 0, -8, 0, 0], [0, 3, 15, 15, 17, -3], [-8, 0, -13, 0, 8, -5], [3, -6, 0, 0, 0, 0], [4, 5, 0, 20, 0, 0], [0, -12, 18, 18, 0, 12], [-10, 16, 0, 1, 0, 0], [0, -12, -14, -14, 19, 12], [9, 30, 0, 0, 0, 0], [-11, 0, 11, 0, 11, -9], [-12, 0, -1, 0, 12, 8], [0, 1, -6, -6, 18, -1], [0, -9, 12, 12, -9, 9], [-19, 5, 0, -5, 0, 0], [-17, 0, 13, 0, 17, 10], [0, -7, 8, 8, -1, 7], [-5, 0, 11, 0, 5, 11], [0, 12, -8, -8, -27, -12], [-1, -25, 0, 1, 0, 0], [19, 22, 0, -10, 0, 0], [18, 2, 0, 10, 0, 0], [-35, 0, 4, 0, 35, -5], [0, -16, -11, -11, 5, 16], [-30, 0, 28, 0, 30, 10], [-10, 6, 0, 12, 0, 0], [-43, 0, 13, 0, 43, 0], [0, -16, 6, 6, 5, 16], [0, -3, -1, -1, -7, 3], [34, 0, -5, 0, -34, 1], [0, -3, 9, 9, -7, 3], [8, 0, 8, 0, -8, 15], [0, 4, -18, -18, -43, -4], [18, 8, 0, 11, 0, 0], [-12, 0, 5, 0, 12, 10], [-8, 0, 10, 0, 8, 14], [11, 0, -10, 0, -11, -2], [-33, 0, 0, 0, 33, -12], [18, 2, 0, 15, 0, 0], [14, 0, 17, 0, -14, 8], [0, 3, -6, -6, -35, -3], [0, 0, 16, 16, -14, 0], [28, 11, 0, -9, 0, 0], [0, -18, 4, 4, -9, 18], [0, -9, 12, 12, 3, 9], [23, 0, 15, 0, -23, -1], [7, 2, 0, -19, 0, 0], [-22, 0, -13, 0, 22, -18], [-4, 0, -22, 0, 4, 5], [0, 20, -2, -2, -2, -20], [0, 28, 2, 2, 25, -28], [0, 0, -15, 0, 0, -1], [-24, 9, 0, 12, 0, 0], [21, 0, -6, 0, -21, -3], [-18, 8, 0, -7, 0, 0], [0, -24, 2, 2, -10, 24], [-19, 4, 0, 8, 0, 0], [-7, 0, -18, 0, 7, 3], [-30, 6, 0, 7, 0, 0], [29, 0, 9, 0, -29, 7], [0, 13, -1, -1, 34, -13], [5, -13, 0, 18, 0, 0], [-6, 0, 4, 0, 6, 2], [-8, 0, 0, 0, 8, -15], [21, 30, 0, -15, 0, 0], [7, 0, -2, 0, -7, 32], [5, 8, 0, 25, 0, 0], [-22, 0, 20, 0, 22, -1], [0, -5, -11, -11, -1, 5], [0, -24, 4, 4, -32, 24], [38, -8, 0, 21, 0, 0], [5, 12, 0, -21, 0, 0], [0, -4, 2, 2, -28, 4], [47, -11, 0, 6, 0, 0], [8, -14, 0, -19, 0, 0], [0, 1, -24, -24, -21, -1], [0, 2, 23, 23, 27, -2], [-29, 0, 4, 0, 29, -24], [-4, -30, 0, 4, 0, 0], [-14, 0, 10, 0, 14, 19], [0, -2, 19, 19, 8, 2], [-11, -2, 0, -13, 0, 0], [0, 5, 15, 15, 31, -5], [26, 0, -10, 0, -26, -12], [6, 0, -3, 0, -6, -6], [-25, 0, -8, 0, 25, -7], [-20, 17, 0, 4, 0, 0], [0, 15, -26, -26, -6, -15], [29, 0, 1, 0, -29, -2], [-38, 0, 4, 0, 38, 12], [33, 15, 0, -10, 0, 0], [-9, 0, 17, 0, 9, 7], [0, 23, -18, -18, -1, -23], [16, 0, 0, 2, 0, 0], [0, 19, 8, 8, -38, -19], [39, 0, 9, 0, -39, -9], [-4, 0, -15, 0, 4, -1], [0, 6, -6, -6, -57, -6], [0, -8, 11, 11, -48, 8], [4, 0, 9, 0, -4, -26], [18, -16, 0, -13, 0, 0], [57, 0, 0, 1, 0, 0], [-28, 0, -2, 0, 28, -21], [0, 19, 20, 20, 15, -19], [20, 8, 0, -8, 0, 0], [0, 12, 0, 0, 27, -12], [-4, -10, 0, 10, 0, 0], [12, 0, 9, 0, -12, -6], [3, 0, -7, 0, -3, 19], [0, 2, 13, 13, -18, -2], [-1, 0, 2, 0, 1, 29], [0, 0, 3, 3, -43, 0], [0, -14, 14, 14, -44, 14], [-3, -3, 0, -9, 0, 0], [0, 10, -6, -6, 43, -10], [-32, 0, 13, 0, 32, -21], [-18, -6, 0, 9, 0, 0], [-13, -26, 0, -11, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2646_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2646_2_f_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_2646_2_f_n(:prec:=6) chi := MakeCharacter_2646_f(); 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_f_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_2646_2_f_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2646_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![9, -12, 31, 26, 21, 5, 1]>,<11,R![1089, 858, 643, 92, 27, -1, 1]>,<13,R![9, -9, 15, 0, 7, -2, 1]>],Snew); return Vf; end function;