// Make newform 1764.2.j.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1764_j();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1764_j_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_1764_2_j_e();". // 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_1764_2_j_e();". // 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], [0, 1, 0, 0, 0, 0], [6, 0, 1, 5, -1, 1], [0, -3, 2, -5, 1, -1], [6, -21, 19, -16, 5, -2], [-9, 33, -22, 19, -5, 2]]; Rf_basisdens := [1, 1, 3, 3, 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_1764_j();" function MakeCharacter_1764_j() N := 1764; order := 3; char_gens := [883, 785, 1081]; v := [3, 1, 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_1764_j_Hecke();" function MakeCharacter_1764_j_Hecke(Kf) N := 1764; order := 3; char_gens := [883, 785, 1081]; char_values := [[1, 0, 0, 0, 0, 0], [-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, 0, 0], [1, -1, -1, 0, 0, 0], [0, -1, -1, -1, 0, 1], [0, 0, 0, 0, 0, 0], [2, -3, 0, 2, -1, -1], [1, -2, 0, -2, 1, 0], [-2, 1, -1, -3, 1, 0], [-1, 1, 3, 1, 1, 0], [-4, 1, 2, 1, -4, -2], [-1, 4, 0, -1, 2, -2], [0, 3, 3, 3, 0, -3], [0, -2, -3, 1, -2, 0], [-1, -1, -3, -1, -1, 3], [-4, 5, 0, -4, 4, 3], [-1, -1, 0, -1, -7, 3], [1, 1, 0, -2, 1, 0], [11, 1, 2, 1, 11, -2], [1, -8, 0, 1, -5, 6], [-3, -2, -3, -2, -3, 3], [6, -3, -8, -2, -3, 0], [1, -1, -3, -1, -1, 0], [1, -5, 0, 1, -5, 3], [1, -3, 0, 1, -8, 1], [-1, -4, -3, 5, -4, 0], [-5, 4, 0, -5, 6, 6], [1, -1, 0, 1, -5, -1], [-4, -3, 0, -3, -4, 0], [9, 0, 6, 6, 0, 0], [1, 3, 9, 3, 3, 0], [0, -2, -8, -2, 0, 8], [-3, 6, 9, -3, 6, 0], [1, -3, -4, -3, 1, 4], [-4, 1, 0, -4, -4, 7], [2, -2, -3, -2, 2, 3], [-3, -3, -3, -3, -3, 3], [-3, 0, 0, -3, 4, 6], [-4, 0, -3, 0, -4, 3], [4, -1, -3, -1, -1, 0], [-1, 4, 2, 4, -1, -2], [6, -5, 0, 6, -12, -7], [3, -3, -3, 3, -3, 0], [0, 6, 3, -9, 6, 0], [-2, 7, 0, -2, 10, -3], [5, 1, 6, 1, 5, -6], [-4, -1, -8, -6, -1, 0], [3, -7, -12, 2, -7, 0], [15, -4, 0, -4, 15, 0], [4, -8, 0, 4, -13, 0], [1, -2, 0, 1, -20, 0], [11, 4, 0, 4, 11, 0], [-10, 8, 1, -15, 8, 0], [-5, -2, -12, -2, -5, 12], [7, -8, 0, 7, 5, -6], [14, -7, -3, 11, -7, 0], [-2, 4, -6, 4, -2, 6], [-1, 11, 0, -1, -1, -9], [-10, -1, 4, 6, -1, 0], [-13, 2, 3, -1, 2, 0], [3, -3, 0, 3, 8, -3], [-1, 3, 0, -1, -22, -1], [2, 1, 3, 1, 2, -3], [-7, -13, -12, -13, -7, 12], [0, 7, 9, -5, 7, 0], [11, -5, 5, -5, 11, -5], [-9, 18, 0, -9, 8, 0], [0, 3, 0, 0, 9, -3], [3, 3, 0, 3, -4, -9], [-15, 3, -6, 3, -15, 6], [8, 8, 15, 8, 8, -15], [2, -4, 0, 2, 22, 0], [-3, -6, 0, -3, 3, 12], [18, -9, -1, 17, -9, 0], [4, -11, 0, 4, -10, 3], [12, -5, -9, -5, 12, 9], [20, -5, 12, 22, -5, 0], [6, 1, -5, 1, 6, 5], [-1, 7, 0, -1, -19, -5], [0, 6, 3, -9, 6, 0], [19, 7, 9, 7, 19, -9], [-6, 4, 6, 4, -6, -6], [-10, 5, 0, 5, -10, 0], [-7, 11, 0, -7, 5, 3], [-22, 5, -6, -16, 5, 0], [-25, 4, 6, -2, 4, 0], [1, 4, 0, 1, 4, -6], [6, -14, 0, 6, 6, 2], [-18, -3, -7, -1, -3, 0], [-1, 11, 0, -1, -3, -9], [-7, 3, 0, -7, 20, 11], [-4, -15, -6, -15, -4, 6], [14, -10, -17, 3, -10, 0], [5, -5, 0, 5, -19, -5], [14, -3, -6, 0, -3, 0], [28, 1, 3, 1, 28, -3], [-16, -6, -6, -6, -16, 6], [7, 4, 9, 1, 4, 0], [16, 1, 12, 1, 16, -12], [-3, 12, 6, -18, 12, 0], [10, 3, 6, 0, 3, 0], [5, 8, 6, -10, 8, 0], [7, -2, 0, 7, -5, -12], [-10, 11, 2, -20, 11, 0], [21, 19, 13, 19, 21, -13], [-12, 18, 0, -12, -3, 6], [-11, 7, 0, 7, -11, 0], [1, 5, 18, 8, 5, 0], [6, -1, 0, 6, -3, -11], [-3, 3, 15, 9, 3, 0], [2, 2, 0, 2, 2, 0], [0, -3, 0, 0, -23, 3], [3, 0, -6, 0, 3, 6], [-15, 6, 12, 0, 6, 0], [23, 1, 5, 1, 23, -5], [-13, 26, 0, -13, 13, 0], [-17, 2, 9, 5, 2, 0], [-5, 2, 0, -5, -29, 8], [-13, -15, -12, -15, -13, 12], [-27, -3, -9, -3, -3, 0], [-17, -8, -15, -8, -17, 15], [-13, 21, 0, -13, -1, 5], [23, -5, 0, -5, 23, 0], [-2, 7, 0, -2, -15, -3], [-3, 12, 0, -3, -18, -6], [4, -2, 18, 22, -2, 0], [0, 3, 0, 0, -11, -3], [-14, -2, 2, 6, -2, 0], [-10, 8, 0, -10, 16, 12], [-16, 11, 18, -4, 11, 0], [-8, 19, 0, -8, 18, -3], [-23, -2, -12, -2, -23, 12], [8, -1, -6, -4, -1, 0], [9, -12, -9, -12, 9, 9], [28, 4, 9, 4, 28, -9], [18, -15, -21, 9, -15, 0], [-2, 2, 16, 2, -2, -16], [-14, 9, 6, 9, -14, -6], [-21, 0, -8, -8, 0, 0], [12, -3, -15, -3, 12, 15], [11, -4, -3, -4, 11, 3], [-16, -1, -19, -17, -1, 0], [-7, -15, -12, 18, -15, 0], [9, -5, 0, 9, -15, -13], [-16, 0, 0, 0, -16, 0], [-10, 5, 24, 14, 5, 0], [11, -3, -3, 3, -3, 0], [14, -20, 0, 14, 5, -8], [11, -13, 0, 11, -9, -9], [-7, 14, 0, -7, -13, 0], [-25, 6, 6, 6, -25, -6], [24, 3, -7, -13, 3, 0], [-27, -14, -15, -14, -27, 15], [28, -2, 3, 7, -2, 0], [-14, -4, -15, -7, -4, 0], [-4, -13, -6, -13, -4, 6], [12, 0, 0, 12, -21, -24], [5, -22, 0, 5, -4, 12], [25, 6, 9, -3, 6, 0], [7, -32, 0, 7, -32, 18], [26, -5, 9, 19, -5, 0], [12, -1, -13, -1, 12, 13], [5, -14, 0, 5, 29, 4], [-10, 8, 16, 0, 8, 0], [15, 2, -3, 2, 15, 3], [10, 7, 15, 1, 7, 0], [-5, -8, 0, -8, -5, 0], [-5, -2, 0, -5, -17, 12], [-23, -7, -15, -1, -7, 0], [-4, -4, 0, -4, 42, 12], [4, -13, -21, 5, -13, 0], [28, 22, 15, 22, 28, -15], [9, 14, 0, 9, 0, -32], [13, 27, 27, 27, 13, -27], [-3, 6, 6, 6, -3, -6], [0, 0, 0, 0, 33, 0], [1, -20, -12, -20, 1, 12], [11, 19, 23, 19, 11, -23], [8, 11, 0, 8, -4, -27], [0, 0, 25, 25, 0, 0], [-31, 1, -12, -14, 1, 0], [12, -21, 0, 12, -32, -3], [-7, 8, 0, -7, -19, 6], [-13, 25, 0, -13, 17, 1], [-32, -10, -12, -10, -32, 12], [-27, 9, 13, -5, 9, 0], [-2, 5, 22, 5, -2, -22], [-7, 11, 0, -7, -37, 3], [-20, -2, 3, 7, -2, 0], [3, -3, 0, 3, 10, -3], [15, 15, 9, 15, 15, -9], [-27, 3, 3, -3, 3, 0], [2, -21, -18, 24, -21, 0], [12, -15, 0, 12, -18, -9], [-26, 9, -6, -24, 9, 0], [2, -5, 0, 2, -25, 1], [24, -6, 9, 21, -6, 0], [2, -15, -14, -15, 2, 14], [2, -2, 3, -2, 2, -3], [7, -29, 0, 7, -19, 15], [-15, 21, 0, -15, -12, 9], [7, 10, -6, -26, 10, 0], [-5, -8, -18, -8, -5, 18], [6, 18, 0, 6, 3, -30], [-27, -5, -18, -5, -27, 18], [-10, 5, 0, -10, -19, 15], [-29, 4, 3, -5, 4, 0], [-7, -16, -21, 11, -16, 0], [12, -10, 0, 20, -10, 0], [-26, 7, 0, 7, -26, 0], [2, -16, 0, 2, 14, 12], [7, 23, 15, 23, 7, -15], [11, -12, -9, 15, -12, 0], [28, 6, -7, 6, 28, 7], [8, -10, 0, 8, -22, -6], [-2, 5, 0, -2, 31, -1], [1, -8, 3, -8, 1, -3], [-9, 30, 0, -9, 34, -12], [-38, -1, -12, -1, -38, 12], [-12, 5, 0, -12, 30, 19], [26, 5, -5, -15, 5, 0], [-31, -17, -10, -17, -31, 10], [23, 21, 6, 21, 23, -6], [-2, 0, -12, 0, -2, 12], [24, -12, -12, -12, 24, 12], [-16, 16, -3, -35, 16, 0], [14, 14, 12, 14, 14, -12], [-3, 15, 0, -3, -10, -9], [8, 19, 0, 8, -7, -35], [9, -9, -10, 8, -9, 0], [-4, 5, 0, -4, 5, 3], [0, 9, 0, 0, -12, -9], [-4, 29, 45, 29, -4, -45], [8, -6, 0, 12, -6, 0], [-26, -10, -6, -10, -26, 6], [38, 2, -3, 2, 38, 3], [10, -2, 0, 10, 26, -18], [10, -23, 0, 10, -32, 3], [-12, 14, 18, 14, -12, -18], [-15, 3, -18, -24, 3, 0], [36, -9, -3, -9, 36, 3], [41, -1, 1, 3, -1, 0], [-8, 17, 0, -8, 25, -1], [26, -11, -24, -2, -11, 0], [-13, 3, -3, 3, -13, 3], [-14, 13, 6, -20, 13, 0], [-6, -11, 4, -11, -6, -4], [-17, 25, 0, -17, 34, 9], [-20, -11, -24, -2, -11, 0], [-5, 17, 19, 17, -5, -19], [28, -18, -21, -18, 28, 21], [-33, 12, 0, -24, 12, 0], [5, 27, 15, 27, 5, -15], [-14, 10, -6, -26, 10, 0], [-4, -7, 0, -7, -4, 0], [-9, 15, 0, -9, 26, 3], [23, -23, 0, 23, -13, -23], [-24, -3, -23, -17, -3, 0], [-25, 9, -3, -21, 9, 0], [15, -21, 0, 15, -27, -9], [-6, -12, -5, 19, -12, 0], [5, 4, -12, -20, 4, 0], [25, -32, 0, 25, -19, -18], [-11, 15, 0, -11, -2, 7], [-7, -11, -9, -11, -7, 9], [15, -2, 12, 16, -2, 0], [4, 4, 0, 4, 4, 0], [6, -6, 0, 6, -10, -6], [65, -7, 3, 17, -7, 0], [12, 1, 0, 12, 0, -25], [37, -7, -9, -7, 37, 9], [30, -15, -12, -15, 30, 12], [21, -18, -33, -18, 21, 33], [-17, -1, 21, 23, -1, 0], [14, -22, 0, 14, 5, -6], [-3, -10, -15, -10, -3, 15], [47, 1, 6, 1, 47, -6], [-1, 8, 0, -16, 8, 0], [14, -5, -10, -5, 14, 10], [-13, 23, 0, -13, 47, 3], [42, -10, 6, 26, -10, 0], [-2, 1, 0, -2, 40, 3], [-21, -30, -30, -30, -21, 30], [-7, 17, 21, 17, -7, -21], [-9, 18, 0, -9, 39, 0], [5, 8, 0, 5, 24, -18], [1, 10, 3, 10, 1, -3], [11, -1, 25, 27, -1, 0], [13, -17, 9, 43, -17, 0], [-35, 15, 2, 15, -35, -2], [-13, 14, 0, -13, 6, 12], [16, -11, -33, -11, -11, 0], [-9, -6, 0, -9, 9, 24], [-2, 4, 24, 16, 4, 0], [36, -12, -9, -12, 36, 9], [-16, 16, 17, 16, -16, -17], [-5, -26, 0, -5, 15, 36], [-28, 17, 9, 17, -28, -9], [3, -18, 0, 3, 16, 12], [-26, 16, 0, -26, 4, 36], [36, 12, 21, -3, 12, 0], [4, -14, 0, 4, 19, 6], [23, 4, 0, 4, 23, 0], [9, -3, 0, 6, -3, 0], [-8, 3, -30, -36, 3, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1764_j_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1764_2_j_e();". // 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_1764_2_j_e(:prec:=6) chi := MakeCharacter_1764_j(); 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_1764_2_j_e();". // 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_1764_2_j_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1764_j(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![1, -4, 17, 2, 5, -1, 1]>],Snew); return Vf; end function;