// Make newform 729.2.c.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_729_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_729_c_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_729_2_c_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_729_2_c_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, -27, 108, -258, 420, -504, 463, -330, 186, -80, 27, -6, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -33, 64, -72, 62, -34, 16, -4, 1, 0, 0, 0], [9, -58, 145, -224, 249, -199, 127, -58, 22, -5, 1, 0], [-20, 104, -256, 395, -448, 367, -239, 112, -43, 10, -2, 0], [-25, 132, -314, 465, -509, 401, -255, 116, -44, 10, -2, 0], [-71, 500, -1562, 2948, -3892, 3852, -2888, 1715, -761, 270, -61, 11], [85, -613, 1934, -3671, 4884, -4853, 3659, -2177, 970, -344, 78, -14], [124, -876, 2750, -5222, 6936, -6900, 5198, -3097, 1379, -490, 111, -20], [-257, 1785, -5589, 10648, -14172, 14157, -10689, 6396, -2853, 1019, -231, 42], [-412, 2871, -9002, 17165, -22873, 22863, -17277, 10342, -4616, 1649, -374, 68], [595, -4153, 13028, -24840, 33097, -33072, 24987, -14949, 6671, -2381, 540, -98]]; Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]; 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_729_c();" function MakeCharacter_729_c() N := 729; order := 3; char_gens := [2]; v := [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_729_c_Hecke();" function MakeCharacter_729_c_Hecke(Kf) N := 729; order := 3; char_gens := [2]; char_values := [[-1, 0, 0, 0, 0, 0, 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 := [[0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 1, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 0, -2, 0, 0], [-1, -1, 0, -2, 0, -1, -2, -1, 0, 1, 0, 0], [-1, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0], [3, 1, 0, 1, 0, -1, 1, 1, -2, 2, 2, 0], [0, 0, 0, 0, 0, 0, 2, 0, -1, -3, -2, 0], [0, -2, 0, 0, -1, 0, 0, -2, 2, 0, -2, -1], [0, 2, -2, -1, -1, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, -1, 3, -1, -1, 0, -1, 3, 1, 3], [0, 0, 0, 0, 0, 0, -2, -3, -1, 1, 1, -1], [0, 0, 0, 0, 0, 0, 1, 1, -2, -4, -1, -3], [0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [4, 2, -3, 0, -3, -1, 0, 2, -3, 4, 0, -3], [0, 0, 0, 0, 0, 0, -5, 0, 0, 1, 2, -1], [3, 4, 1, 3, -1, 5, 3, 4, 1, 0, 0, -1], [-4, 0, -2, 1, 2, 0, 0, 0, 0, 0, 0, 0], [-1, 3, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 1, -6, 2, 6, 0], [0, 0, 0, 0, 0, 0, -4, -3, 5, 0, -2, 3], [-2, 0, -1, -1, 4, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 3, -1, 1, -2, -1], [0, 0, 0, 0, 0, 0, -2, 1, 4, -1, -1, 6], [2, -1, 4, 1, -3, -2, 1, -1, -1, 1, 5, -3], [-3, -6, -3, -3, 3, -3, 0, 0, 0, 0, 0, 0], [-3, 2, 1, -4, 2, 3, 0, 0, 0, 0, 0, 0], [-2, -1, -6, 3, 3, -1, 3, -1, 0, -5, -6, 3], [0, -1, -8, -1, -4, -3, 0, 0, 0, 0, 0, 0], [3, 1, -3, -2, 0, 5, -2, 1, 4, 5, -7, 0], [0, 0, 0, 0, 0, 0, -1, 3, -1, 6, -2, -6], [0, -2, 0, 0, 5, -3, 0, -2, -1, 0, 1, 5], [-1, 6, -9, 2, -3, -1, 2, 6, -7, -3, -2, -3], [0, 0, 0, 0, 0, 0, 5, 1, 7, -4, 1, 3], [-2, -3, -6, -4, 1, -8, -4, -3, -5, 2, -1, 1], [6, 2, 4, 5, -4, 3, 0, 0, 0, 0, 0, 0], [-2, -1, 3, -9, 3, 2, -9, -1, 3, 7, 0, 3], [0, 0, 0, 0, 0, 0, 0, 2, 3, 4, -9, 3], [-1, 6, -8, -2, -1, 0, 0, 0, 0, 0, 0, 0], [3, 2, 10, 5, 5, 15, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 6, 2, -9, 4, 3], [-2, 3, -5, -7, -2, -3, -7, 3, -6, 5, 1, -2], [7, -3, -7, 5, -5, -6, 0, 0, 0, 0, 0, 0], [-3, -4, 4, -7, 5, 3, 0, 0, 0, 0, 0, 0], [-8, 3, 3, -7, 1, 7, -7, 3, 4, -1, -1, 1], [0, 0, 0, 0, 0, 0, 9, 5, 10, 0, -7, -2], [0, 0, 0, 0, 0, 0, -3, 5, 0, -2, 9, 3], [-4, -1, 12, 1, 6, 8, 1, -1, 9, -5, 3, 6], [2, 0, 7, -5, 5, -6, 0, 0, 0, 0, 0, 0], [12, -2, 0, 4, 0, 2, 4, -2, 4, 8, -4, 0], [0, 0, 0, 0, 0, 0, 8, -5, 3, -4, -6, -3], [-5, 3, -7, 5, -2, 6, 0, 0, 0, 0, 0, 0], [-1, -6, -3, -7, -3, -10, -7, -6, -4, 6, 1, -3], [0, 0, 0, 0, 0, 0, -5, -8, -2, -1, 2, 0], [-6, 0, 9, -6, 6, 0, 0, 0, 0, 0, 0, 0], [-1, 3, -6, 3, -3, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, -1, -5, -3, 5, -8], [0, 0, 0, 0, 0, 0, -3, -4, -3, -2, -3, -6], [-4, 5, -6, 1, -3, -7, 1, 5, -12, -5, 6, -3], [-3, -2, -6, 1, -3, -13, 1, -2, -11, -4, 5, -3], [-4, 0, -3, 3, 9, 0, 0, 0, 0, 0, 0, 0], [14, 3, 9, 5, 0, 8, 5, 3, 5, 9, 4, 0], [0, 0, 0, 0, 0, 0, 1, 6, 2, 4, 7, -4], [0, 0, 0, 0, 0, 0, 1, 1, -8, 5, 2, -3], [0, 0, 0, 0, 0, 0, -3, -1, -5, 6, -7, -2], [2, 2, -12, 4, -9, 2, 4, 2, 0, -2, -12, -9], [3, 1, 12, 1, 9, 14, 1, 1, 13, 2, -1, 9], [0, 0, 0, 0, 0, 0, 8, 13, -9, -4, 6, -6], [0, 0, 0, 0, 0, 0, -4, 3, -19, 12, 4, 0], [-8, 0, 14, -1, 1, 12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 4, -11, -4, 1, -9], [1, 3, -3, 2, -5, 7, 2, 3, 4, -1, -7, -5], [0, -1, -2, -7, 5, -12, 0, 0, 0, 0, 0, 0], [10, 2, -3, 3, 0, -7, 3, 2, -9, 7, 6, 0], [0, 0, 0, 0, 0, 0, 9, -4, 0, 1, -6, 6], [3, -4, -2, 5, -4, -12, 0, 0, 0, 0, 0, 0], [-3, 7, 6, -5, -3, -1, -5, 7, -8, 2, 14, -3], [-6, 1, 6, 6, 8, 12, 6, 1, 11, -12, -5, 8], [-1, 0, 12, -1, 6, 11, -1, 0, 11, 0, 1, 6], [0, 0, 0, 0, 0, 0, 4, -6, 8, 7, -17, -1], [-6, 0, 6, 6, -3, 18, 0, 0, 0, 0, 0, 0], [9, -4, -2, 11, 5, 9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, -9, -5, 11, 2], [0, 0, 0, 0, 0, 0, -6, -7, 0, 13, 6, 0], [11, 6, 1, 13, -4, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, -5, 3, 5, 6, 9], [0, 0, 0, 0, 0, 0, 8, -9, 11, -3, -8, 3], [-6, -8, 0, -3, -7, -6, -3, -8, 2, -3, -2, -7], [-8, 0, -10, -4, 1, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 10, -14, -10, 17, -3], [-9, -1, 10, 2, 5, 12, 0, 0, 0, 0, 0, 0], [7, 2, 3, 6, -6, 17, 6, 2, 15, 1, -12, -6], [3, -2, 7, -6, 8, 5, -6, -2, 7, 9, 0, 8], [-4, -3, 10, 4, 11, 9, 0, 0, 0, 0, 0, 0], [-6, -14, 6, -5, 9, -7, -5, -14, 7, -1, -1, 9], [-8, -6, 5, -19, 1, -3, 0, 0, 0, 0, 0, 0], [2, 6, 3, -3, -9, 3, 0, 0, 0, 0, 0, 0], [0, -7, -2, 2, -13, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 12, -8, 2, 5], [8, -3, 4, -11, -1, 6, 0, 0, 0, 0, 0, 0], [-9, -5, 0, 1, -15, -1, 1, -5, 4, -10, -4, -15], [0, 0, 0, 0, 0, 0, -7, 6, -10, 9, 10, 6], [-6, 1, -3, -12, -4, -9, -12, 1, -10, 6, 7, -4], [-10, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 4, 13, 5, 5, -3], [15, -6, -6, 0, -6, -21, 0, 0, 0, 0, 0, 0], [10, -4, 15, 6, 0, 11, 6, -4, 15, 4, 0, 0], [0, 0, 0, 0, 0, 0, 6, 5, 1, 6, -1, 10], [11, -7, -3, 4, -3, -1, 4, -7, 6, 7, -9, -3], [5, -6, -12, 12, -3, 3, 0, 0, 0, 0, 0, 0], [-6, 16, -9, 13, 0, 11, 13, 16, -5, -19, -4, 0], [0, 0, 0, 0, 0, 0, 3, 5, -7, -12, 9, -2], [3, 2, -11, 5, -10, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 19, 1, 4, 8, -1, -6], [2, 8, -11, 1, 3, -8, 1, 8, -16, 1, 5, 3], [-12, 6, -3, -15, 3, 3, 0, 0, 0, 0, 0, 0], [-26, 2, -3, -6, -9, -16, -6, 2, -18, -20, 15, -9], [0, 0, 0, 0, 0, 0, 15, 5, 12, -5, -9, -12], [-12, 1, 7, -9, 2, 8, -9, 1, 7, -3, 0, 2], [0, 0, 0, 0, 0, 0, -13, -14, -9, 20, 9, -6], [0, 0, 0, 0, 0, 0, 5, 6, 14, -3, -14, -3], [28, -6, 2, -1, 10, 9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -17, 0, -4, 13, 16, 8], [-6, -6, 0, -12, 6, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, -7, -5, 12, 8, -2], [-16, -12, 1, -17, -13, -18, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, -1, 11, -9, -9, -11], [21, 1, -9, 6, -4, 6, 6, 1, 5, 15, -14, -4], [12, -7, -11, 5, -4, -18, 0, 0, 0, 0, 0, 0], [-10, -3, 0, -16, 6, -19, -16, -3, -16, 6, 16, 6], [-8, -3, 9, 2, -2, 4, 2, -3, 7, -10, 2, -2], [-16, 6, -12, -9, -12, -24, 0, 0, 0, 0, 0, 0], [-2, -1, 0, -12, 12, -1, -12, -1, 0, 10, 0, 12], [-10, 2, -18, 7, -6, -1, 7, 2, -3, -17, -15, -6], [17, 12, 1, 4, -1, 0, 0, 0, 0, 0, 0, 0], [13, 12, 4, 14, -8, 24, 14, 12, 12, -1, -8, -8], [8, 0, -3, 11, -3, 14, 11, 0, 14, -3, -17, -3], [3, -12, 3, 6, -3, 0, 0, 0, 0, 0, 0, 0], [3, 2, 4, 8, 5, 18, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 17, -12, 4, 12, 0], [2, -7, 12, 13, -6, 8, 13, -7, 15, -11, -3, -6], [2, -6, -5, -2, 14, -3, 0, 0, 0, 0, 0, 0], [-18, -10, 4, -1, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7, -9, -13, -12, 13, -9], [0, 0, 0, 0, 0, 0, 5, -14, 10, -13, 1, -3], [0, 0, 0, 0, 0, 0, 7, -11, 16, -1, -16, 6], [7, -3, -12, 14, -5, 10, 14, -3, 13, -7, -25, -5], [3, 0, 18, 6, -3, 6, 0, 0, 0, 0, 0, 0], [8, 5, 6, 16, 6, 11, 16, 5, 6, -8, 0, 6], [2, 6, 7, 7, 5, 24, 0, 0, 0, 0, 0, 0], [6, 14, -8, 11, 2, 12, 0, 0, 0, 0, 0, 0], [6, -5, -3, -14, -12, -13, -14, -5, -8, 20, 5, -12], [0, 0, 0, 0, 0, 0, 10, 6, -10, 13, 10, 2], [0, 0, 0, 0, 0, 0, 1, -11, 10, -25, -1, 0], [-4, 0, -18, -12, -12, -18, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, -4, 3, -8, -12, 3], [15, 11, -8, 2, 8, 12, 0, 0, 0, 0, 0, 0], [15, -8, 9, 10, 6, 2, 10, -8, 10, 5, -1, 6], [0, 0, 0, 0, 0, 0, 14, -3, 14, 6, 4, -3], [-8, -6, 8, -4, 1, -12, 0, 0, 0, 0, 0, 0], [5, 5, -2, -11, 3, -2, -11, 5, -7, 16, 5, 3], [-3, 3, -15, 18, -6, 6, 0, 0, 0, 0, 0, 0], [4, 8, 9, 12, 6, 17, 12, 8, 9, -8, 0, 6], [0, 0, 0, 0, 0, 0, -12, 2, -33, 7, 12, -6], [20, -15, -3, 6, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, -1, -19, -12, 0, -2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_729_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_729_2_c_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_729_2_c_e(:prec:=12) chi := MakeCharacter_729_c(); 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_729_2_c_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_729_2_c_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_729_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![9, -27, 81, -72, 117, -63, 123, -45, 45, -15, 12, -3, 1]>],Snew); return Vf; end function;