// Make newform 4730.2.a.w in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4730_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_4730_2_a_w();". // 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_4730_2_a_w();". // 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, 2, -28, -6, 41, 7, -12, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-8, 2, 24, -1, -10, 0, 1, 0], [-2, -16, -4, 39, 7, -12, -1, 1], [4, 12, -8, -39, -5, 12, 1, -1], [4, -16, -6, 39, 7, -12, -1, 1], [2, -26, -6, 41, 7, -12, -1, 1], [8, -8, -32, 26, 17, -10, -2, 1]]; Rf_basisdens := [1, 1, 2, 2, 2, 2, 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_4730_a();" function MakeCharacter_4730_a() N := 4730; order := 1; char_gens := [947, 431, 1981]; v := [1, 1, 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; function MakeCharacter_4730_a_Hecke(Kf) return MakeCharacter_4730_a(); 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, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 1, 1, -1, 1], [0, 0, 1, 0, -1, 1, 0, 1], [-2, 0, 0, 0, 0, 0, 0, -1], [1, -1, -1, -2, 0, -1, 0, 0], [-1, 0, -1, 0, 0, -1, -1, -1], [-2, -1, -1, -1, 1, 1, -1, 1], [2, 1, -1, 1, -1, 0, -1, -1], [-1, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 0, -4, -1, 0, 0], [-1, 0, 1, 1, 2, 1, -1, 1], [0, 0, 1, 0, 2, 3, 0, 1], [1, 1, 1, 0, 0, -1, -2, 0], [-3, -1, 2, -1, -1, -2, 0, 1], [-2, 1, 2, -1, -1, 2, 3, -1], [-1, 1, -1, 0, 0, 0, -2, -1], [-1, 1, -2, 2, 2, -1, -2, 0], [-1, -1, -2, 0, 0, -1, 2, 0], [1, -3, -2, 1, -1, -1, 2, -2], [-3, -3, 1, 2, -2, -4, 0, 1], [3, -1, 3, -1, -3, -5, 2, -1], [1, -6, 0, 0, -1, 0, 1, 1], [-2, 2, 0, 3, 2, -2, -2, 0], [2, -5, 2, -1, 0, -2, 1, 0], [-3, -3, 1, -1, 0, 0, 2, 1], [-2, 1, -4, 2, 2, -2, -1, 0], [-2, -1, -3, -1, 3, 1, -1, 1], [-5, -1, -2, -1, -3, -2, 4, 0], [0, -1, 2, 0, 2, 2, 1, 0], [4, -1, 4, 0, 0, 1, 3, -1], [-1, 3, 0, 1, -1, 4, 0, 0], [-6, -1, 3, 3, 1, 1, 1, 1], [-5, 3, 4, 0, -1, 1, -2, 0], [3, -5, 1, 2, -2, -5, 2, -1], [2, 3, 1, -3, 5, 4, -1, 2], [-1, 2, -2, 3, 3, 4, -3, 1], [3, 3, -1, -2, -2, 0, 2, 1], [5, -1, -2, 2, 3, -2, -2, 3], [1, 0, 2, 0, -8, -4, 1, -1], [0, -5, 3, 1, -5, -2, 1, -4], [-6, -3, -2, 3, 0, -3, 3, -3], [1, 2, 0, 2, 2, 6, -1, 0], [-6, 6, -1, -1, 7, 4, -2, 3], [-2, -4, -2, 2, 0, -4, 2, 2], [3, -1, -7, -6, 2, 0, 2, -1], [5, 0, -6, 3, 3, 3, -3, 2], [0, 8, -2, -1, 0, 2, -2, 0], [-3, 1, 3, -1, -1, 3, 0, 4], [-3, 8, 0, -4, 3, 0, -3, -2], [-1, -3, -1, -1, 4, -2, -2, -1], [0, -2, -1, 0, -1, -5, 2, -1], [-1, 0, 2, 5, -3, 0, -1, 0], [0, 3, -1, -1, 3, 2, -1, 1], [-2, -2, 3, 6, 1, 4, 2, 1], [2, 3, -3, -1, 1, 0, -3, -1], [-6, 0, -2, 1, 0, -4, -4, 0], [2, 2, 6, -6, -2, 4, 0, 1], [-3, 2, -3, -3, -1, -3, 1, -1], [-4, 2, -2, -2, 6, -1, -2, 2], [1, -3, -1, 5, 2, -8, 0, -1], [-4, 6, -2, -3, -5, 0, -2, -2], [-16, 5, 1, 5, 1, -6, 1, 0], [-8, 4, -7, 0, 5, 3, -4, 0], [-6, -2, 2, 4, 4, 2, -2, 4], [2, 7, -1, -1, 5, 2, -1, -2], [-9, 2, 2, 5, 3, -2, 1, 0], [-3, 3, 3, 0, -4, -6, 0, 1], [-13, -1, 1, -5, -2, 0, 2, -4], [0, 3, 3, 4, 4, 7, -7, 1], [-4, 0, -1, -2, 0, 1, -4, -5], [-10, 0, -1, 3, -7, -2, 0, -1], [-5, 2, 4, -4, -2, 7, -1, 4], [-8, -2, -2, 1, 1, 3, 2, 4], [-1, 8, 1, -1, 0, 3, 5, -1], [2, -5, 3, 2, -5, -8, 5, 0], [-10, 1, -2, -2, -2, 2, 1, -1], [-7, 5, -1, 3, 0, 0, 2, 3], [6, 8, 1, 2, -2, -1, -4, 1], [-6, 0, -2, 0, 2, 4, -6, 2], [-9, -8, 4, -2, -7, 2, 3, 4], [-3, 5, -6, 1, 3, 2, -2, -1], [6, -6, -2, 6, 0, -6, 2, 2], [-17, 3, 0, -3, -1, 1, 6, -2], [2, 5, -4, 3, 2, 3, -1, -2], [-22, 0, 0, 0, -2, 2, 2, 0], [-10, 4, 1, -6, 3, 1, 2, -3], [-3, -7, 7, 2, -4, 2, 4, 3], [-10, 3, 0, 1, -1, 1, 1, 1], [-8, -2, -1, 2, -4, -2, -6, 1], [-14, 7, -8, 0, 4, 0, -1, 1], [-1, -7, -3, 7, -2, -8, 4, -5], [-2, 0, 6, 2, 0, 2, -2, -2], [8, -4, 0, 0, 2, 0, 6, -2], [-3, -3, 6, 3, -1, -6, 2, -6], [-2, 5, -4, -3, 7, 6, -5, 6], [-17, 2, 3, -1, 1, -3, -1, 1], [4, -1, 3, -5, -3, 0, 1, 0], [6, -7, 3, -1, 5, -4, 1, 1], [8, 3, 3, -3, -7, -4, 3, -1], [3, 8, 6, -2, -3, 5, 3, -2], [-8, -3, 4, 3, -3, 3, -9, 2], [-9, 4, 2, 4, -6, 2, -1, -4], [-1, 5, -3, 2, 2, 4, -2, -1], [1, -2, -5, -2, 8, 1, 3, 1], [-5, -1, 1, -3, -7, -1, 8, 0], [3, -12, -2, 2, 3, 5, -1, 4], [2, 2, -6, -4, 2, -2, -4, -5], [0, -3, -1, 3, 1, 2, 1, -3], [9, -8, -2, 2, 1, -8, 1, -2], [-7, -10, 3, -1, -5, -6, 3, 2], [-1, -1, 4, -7, -5, 1, 10, 0], [11, 1, -6, -1, 5, 3, -4, -5], [-2, -2, 1, 3, 13, 10, -2, 7], [-1, 1, -2, -5, -5, 1, -4, 2], [-2, -1, 2, -4, -2, 9, -3, 4], [-3, 3, 3, 2, 6, 4, 0, 5], [-5, 6, 0, -5, 5, 13, 1, 0], [-14, 9, -5, -4, 8, 13, -3, 5], [-12, 5, 3, 3, 1, 1, 1, -2], [-2, 5, -6, -4, 6, 0, -3, 5], [9, -5, 9, 1, 0, 3, 4, 1], [5, 14, -6, -5, 3, 8, -1, 0], [2, -6, 6, 6, 2, 1, 0, 2], [-19, -9, -3, 2, -4, -10, 0, -1], [9, 7, 2, -2, -9, -6, -2, -6], [-9, 7, 3, -5, -9, -3, 0, -5], [8, -10, -1, 6, 2, -10, 2, 3], [2, 14, -2, 5, -1, 0, -4, -3], [-2, 0, -6, -2, 0, -6, 6, -6], [8, 3, -6, -6, -4, 0, 3, -7], [0, -4, 0, 0, 2, 6, 2, 6], [18, -3, -3, 3, -1, -7, -1, -1], [-13, -3, 5, -7, -4, 2, 10, -3], [7, -5, -1, 3, 3, 5, -10, 3], [8, 3, -3, -1, 1, 4, -9, 1], [4, -8, -1, 4, 6, -2, 4, -5], [13, 3, -1, -3, -1, -5, 8, -1], [6, -10, 2, 2, -6, 0, 10, -7], [9, -9, 1, -2, -14, -8, 0, -5], [-3, -1, -8, 5, 5, -4, 0, 2], [17, 0, 2, -1, 1, 10, -3, 5], [6, -2, -8, 4, -4, -2, -8, 3], [-2, -10, -7, 8, -2, -12, 4, -8], [-6, -1, 5, -1, -5, 4, 9, 0], [11, -7, -1, -10, -4, 4, 4, 3], [-8, -3, -3, -1, -3, -2, -5, 2], [4, -2, 8, 7, -7, -11, -2, 2], [-5, 5, -2, -3, 5, 1, -8, 5], [5, 7, 1, 5, 0, -8, -4, -1], [-9, 11, -2, -3, 11, 5, -6, 5], [-1, 2, 1, 7, -8, -3, -7, 0], [7, -4, -7, 3, 3, -9, -3, -3], [8, -5, 15, 5, -9, -7, 3, -1], [4, 5, 10, -8, -4, 8, 3, 0], [5, 13, 1, 1, -4, -7, -6, 1], [-8, 11, -3, -1, -5, -3, -9, 2], [16, 12, -7, -3, 9, 1, 4, -5], [8, 9, 6, 2, 6, 9, -7, 2], [9, -1, 5, 3, -16, -10, 2, -1], [13, -3, -1, -2, 8, 2, 0, -1], [5, -1, 4, 7, 3, 8, 2, -2], [-19, -4, -6, -1, 1, -6, -3, -3], [-18, 3, 2, 5, 5, 3, -3, 2], [-1, 4, 8, -3, -1, 6, 9, 1], [3, 10, 1, -5, -1, 10, 1, -3], [-5, -2, -8, 8, -1, -1, 1, 0], [5, -4, -2, -2, 12, 5, -5, 6], [-1, 1, 2, -11, 3, 10, 6, 0], [-4, 1, 1, -7, 13, 9, 3, 2], [-8, 1, -1, 2, -7, 0, -5, -6], [8, -11, -6, -1, -5, -12, 7, -6], [-16, 0, 4, 4, -8, -4, -4, 0], [-8, 1, -10, 3, 1, -7, 1, 2], [10, -12, 4, -2, 0, 9, -2, 8], [-6, 11, 0, -2, 6, -3, -3, 5], [-21, -9, -5, -2, 4, 2, 6, -3], [21, -4, 3, -6, 2, 7, -1, -3], [-10, 5, 11, -9, -3, 7, 5, -3], [10, 0, -5, -1, 1, 3, 2, 3], [9, 15, 4, 7, -5, 2, -2, 0], [4, 15, -2, -3, 3, -7, -5, 2], [3, -17, 2, 2, 1, -12, 12, -2], [-13, -3, -13, -3, 4, 0, -6, 7], [-4, -4, 4, -2, -4, -6, 14, -4], [10, 2, 2, 0, 6, 4, 8, 2], [1, 0, -4, -3, -9, -4, 5, -8], [7, -7, 8, 7, -5, -4, 8, 8], [-8, 0, -3, -2, 8, 3, -8, -7], [26, -4, -5, 6, 3, -6, 4, -3], [-15, -1, -2, -1, -1, -2, 10, 0], [4, 3, 6, 7, -9, -13, -1, 0], [-8, 0, 2, 0, -2, -4, 6, 6], [1, 16, -3, -3, 4, 8, 1, 7], [-9, -2, -4, 0, 6, 11, -7, -1], [-8, 0, -5, -6, -13, -9, 8, -5], [7, -10, -5, 2, -2, -1, -1, 1], [28, -4, 0, 0, -4, -14, 0, -2], [-7, 9, 6, -7, -9, 3, 6, -2], [-8, 15, -3, -1, -3, 2, 13, -2], [-5, 12, 3, -4, 2, 10, -7, 0], [-3, -5, 10, 3, -11, 3, 6, -2], [12, 17, -5, 0, 2, 3, -9, -6], [1, 7, 9, -2, -6, 8, 8, 5], [18, 12, 2, -3, 13, 19, -4, 3], [-3, -1, -3, 1, 4, -8, 2, -3], [-8, 2, -7, -3, -9, -1, 6, -3], [-13, 1, 5, -4, 4, -8, 2, -1], [6, -10, 11, 4, 0, 7, -2, 6], [-1, 3, 4, -5, -3, 7, 6, -5], [8, -8, -10, 0, 10, -4, -8, -2], [-23, -7, 3, 6, -2, -7, 2, -9], [-13, -2, -10, 0, -5, -7, 5, -4], [3, -5, -2, -4, -12, -17, 6, -5], [6, 4, -1, -7, -13, -10, 4, -7], [-3, -1, 9, 1, 6, -5, 0, -3], [-13, 12, 4, -3, -11, 0, -3, -7], [5, -1, 7, 12, 0, 8, 4, 5], [11, -3, 10, 3, -1, 5, 12, 0], [6, -4, 11, 0, -8, -3, 0, 0], [-17, -1, 4, 1, 5, -2, 2, 0], [-5, -9, 3, -5, -9, 6, 6, 3], [-7, 25, 3, 8, -6, 2, 0, -1], [15, -8, -5, 1, 2, -11, -1, 0], [-13, 13, -3, 3, 6, -8, 6, -3], [-15, 5, 5, -3, 6, -2, -8, 7], [-8, 6, 4, -6, 2, 0, -4, 0], [-12, 9, 2, 2, -12, -7, 7, -2], [4, 3, 4, -9, -15, 5, 11, 4], [14, -1, -9, 0, 15, 12, -7, 4], [2, 7, -1, 5, 1, 8, -3, 2], [24, 16, 0, -10, 2, 13, 0, 2], [-9, 7, 4, 9, -7, 7, 2, 2], [8, 4, -12, -6, 6, -2, -4, -9], [-7, 14, -3, -2, -2, 8, -3, 6], [-3, -8, -3, -2, 4, -5, 5, 7], [-11, -3, 4, -10, 1, 11, -2, 4], [-9, 5, 7, -1, 2, -6, -2, 3], [6, 16, -5, 4, 8, 9, -4, 4], [1, 4, 4, 15, -5, 2, 3, 2], [3, -5, 7, 0, 0, 8, -8, 3], [-8, 5, -6, -1, -1, 7, -1, 4], [-3, -3, 11, 2, 4, 4, 2, 1], [9, -14, 14, 4, -15, -8, 3, -6], [-20, -2, -10, -6, 12, 9, -8, 6], [2, 3, -14, -10, 6, -9, 5, -6], [-2, -5, 1, -1, -11, -4, 11, 4], [8, -8, 2, 0, -4, -6, -10, 0], [11, 16, -6, -12, -9, 6, -7, -1], [-1, 13, 6, 1, -1, 19, 4, 2], [19, 8, 1, -9, -13, 1, 11, -5], [20, 4, 1, 12, 2, -6, 4, -3], [-1, 3, 3, -5, -2, 10, 2, -1], [2, 15, 6, -7, -9, 3, 7, 0], [-26, -2, 10, 2, -2, -4, 8, 2], [-14, -2, -1, -4, -7, -9, 12, -6], [0, 5, -4, 1, 1, 5, 3, -2], [6, -16, 0, 4, -8, -4, -8, -2], [2, 0, 2, 4, 2, -7, -4, -2], [22, 14, -4, -4, 6, 16, 0, -5], [12, 9, 12, -7, 5, 7, -3, -2], [11, -12, -6, -5, 11, 2, 3, 1], [-17, -7, -3, 13, -12, -16, 0, -9], [2, -2, -4, -15, -8, 5, 6, 0], [21, -9, 2, -7, 1, 14, -4, 10], [8, -17, -3, -3, 7, -8, -3, -2], [11, 8, 4, -3, -7, -2, 11, 6], [-9, -1, -10, 1, 3, -9, -8, 6], [-13, 2, -2, 6, 6, 7, -13, -2], [7, -3, -12, 5, 3, 2, -18, 4], [11, -6, 6, 0, 4, 10, -3, 15], [-15, 9, 8, -11, -9, 8, -12, 2], [1, -5, -2, -1, -1, 2, 4, 3], [3, -6, -2, 11, 5, -14, 7, -3], [-8, -1, -1, 5, -3, -2, -1, -13], [5, -8, -6, 2, 2, -22, 1, -4], [7, -2, -5, -10, 10, 11, -3, -1], [5, 8, 9, 0, 6, 14, -11, -5], [14, 13, -10, -4, 2, 8, 3, -4], [14, -4, -4, -8, 0, -1, 6, -1], [-1, -17, 0, -1, 7, 0, -8, 6], [5, 9, 3, -4, 10, 7, -4, -2], [9, 19, -1, 2, -6, -4, -10, 7], [-6, 12, -2, 1, 6, 0, -2, 8], [-2, -3, 0, -9, 6, -3, -1, 3], [-20, -2, -4, 4, -16, -18, 4, -12], [12, 8, 0, 2, -8, -2, 4, 4], [-27, -1, 6, -1, -5, 10, 4, -1], [4, -11, -1, -5, -1, 1, 3, 6], [-9, 6, 0, 0, 3, 17, -23, -2], [12, -9, 15, -7, -5, 0, 5, 6], [24, -1, 4, 3, -5, -15, -5, -4], [1, 0, -3, 0, -14, 6, 9, -5], [-2, -16, -4, -2, -8, -9, 4, 0], [-4, -17, 4, -6, 0, 6, 11, 0], [-5, 14, -15, -7, 3, 9, -1, 3], [-10, -12, 1, 4, 6, 2, 2, 4], [5, 3, -13, 6, 10, 2, -6, -7], [16, -17, -3, 0, 11, -8, 1, -3], [-16, 3, -13, 1, 5, 10, 7, -2], [-30, -5, 3, -5, -3, 0, 1, 0], [-6, -20, 1, 4, 7, -4, -12, 3], [6, -3, 5, -1, 9, 6, -5, 4], [0, -10, -1, 6, 17, 4, -8, 13], [-23, 4, -9, -15, 16, 19, -7, 14], [20, -4, 0, 11, -2, -8, 8, 2], [9, -25, -11, -1, 5, -5, 2, -7], [-4, 5, -4, -5, 9, 25, -11, -2], [22, 22, 2, 8, 6, 10, -14, 6], [-16, 1, -3, -4, -2, -16, 5, 1], [-25, -8, -3, 0, 10, 13, 3, 1], [12, -18, -6, -2, 12, -10, -8, -6], [-5, 5, -3, -17, 10, 16, -6, -1], [5, -4, 14, 1, 3, 12, 3, 4], [-3, -8, -8, 3, -1, -6, 1, 0], [14, -1, -2, 11, 0, -12, -5, -9], [5, 9, -9, 0, -8, 1, -2, -5], [4, 11, -4, -11, -3, -3, -13, 6], [10, 5, 11, 9, -11, -6, 5, 6], [18, 7, 6, -3, 7, 11, 5, 9], [3, -15, 8, 3, 5, 15, 0, 4], [-16, 6, -10, 9, 8, -1, 0, -6], [7, -22, -3, 6, -10, -12, 7, -10], [-1, -11, 8, -13, -11, 8, 16, 7], [-1, 15, -1, 8, -2, -11, -2, 5], [7, -2, 2, -12, -12, 17, 3, 6], [-13, 16, -3, -2, -14, -7, -5, 2], [4, -1, 2, 9, 16, -4, -13, 0], [-5, -19, 4, 5, 13, 6, 2, 2], [-21, -5, 8, 3, -1, 4, -2, -8], [2, -7, 8, 4, 0, -1, -5, -6], [9, -1, -8, -3, 11, 1, 4, 17], [1, -10, -6, 10, 12, -3, -3, -2], [20, 5, 7, -11, 15, 11, 1, 3], [-15, 9, 2, -5, 11, 11, -2, 2], [2, 9, 13, 3, -17, -9, 5, -13], [-14, 6, 2, 1, -11, -11, 16, -12], [-3, -10, 4, 0, 0, 19, 5, -4], [-22, -4, -16, -4, 4, -3, -2, 7], [-26, -2, -2, -13, 5, 0, -4, -7], [-6, 5, -10, 21, 5, -8, -5, -8], [-5, -9, 11, -2, -12, -17, 6, -1], [-6, 9, 14, 5, -9, 3, 7, -2], [2, 1, -3, -13, -5, 3, -9, 2], [-23, -7, 4, -1, 13, 0, 4, 0], [-14, 14, -6, 15, 5, -16, -8, 4], [-16, -25, 8, 8, -4, 5, -1, 6], [3, -8, -2, -6, 0, 7, -9, -2], [13, 8, 6, -11, -3, -4, 1, 4], [-7, -11, -16, -11, 11, 3, 4, 0], [-2, 7, 2, -5, -19, -4, 5, -12], [-16, 9, 11, 1, -1, 1, 9, 3], [2, -6, -10, 3, -7, -6, 16, -9], [12, -24, -19, 4, -2, -24, 0, -13], [-11, 4, 2, -8, 14, 7, -7, 5], [-1, 0, 0, 10, 5, -10, -3, -9], [11, -17, 1, 10, -2, -6, 12, -3], [10, -7, 0, -7, 2, 0, 5, 3], [-24, -10, 0, -10, 6, 0, 2, 4], [-12, -16, 5, 2, 8, 2, 4, -1], [30, 3, -9, 3, -3, -3, -7, -10], [29, -10, 6, -2, -2, -1, 1, 6], [24, 15, -7, 6, 2, 7, -15, 5], [-12, -18, 8, -4, -8, -7, -2, -4], [-16, -5, -5, -4, -7, -8, 15, 6], [-14, -13, 6, 14, 6, -18, 3, -3], [-14, 13, 2, -6, 14, 12, 9, 7], [6, -16, 14, -4, -4, 4, 22, 4], [-13, 12, 5, -2, -18, -5, 3, -2], [5, -5, -3, 2, 2, 4, 4, -9], [6, 8, -6, 14, 0, -10, -8, 2], [10, -13, 2, 2, 10, 0, -1, -1], [-11, -7, -2, -2, -2, -13, 0, 0], [-6, -20, -9, 6, 1, -3, 6, 0], [11, -4, 1, 0, 22, 4, -9, 10], [-10, -21, 1, 11, 13, 7, 9, 5], [10, -17, 7, 13, 9, -6, 5, 3], [3, 0, -6, 3, -11, -8, 3, -12], [39, 8, 11, 2, 2, 10, -1, -3], [10, 8, 5, -5, -5, -11, 10, -9], [23, 16, -7, -12, 0, 18, -7, 13], [-38, 14, 11, 2, 2, 19, -2, 6], [6, -6, 3, 4, -12, -21, -2, -8], [3, -10, -7, -5, -17, -20, -3, -5], [11, 5, 6, 14, -10, 1, -4, -14], [25, -6, 6, -8, 1, 8, -11, -3], [-15, -4, 1, -7, 7, -2, 21, -1], [12, -9, 5, -9, -21, -3, 11, -1], [-17, -4, 6, -11, 9, 6, 11, 5], [-16, 5, 13, 3, 7, 14, 1, 2], [-19, -13, 2, -10, 2, -5, 14, 7], [19, 12, 0, 6, -8, -7, -1, -19], [-15, -2, -11, -5, 3, -9, 1, 5], [8, 3, -23, 7, 17, 1, -1, -1], [-16, -1, 0, -12, -2, 2, 21, -7], [-17, 6, -11, 20, 4, -3, -19, -6], [5, 7, 9, 1, 12, 1, -8, -3], [-5, 7, -11, -4, -4, -12, 4, -3], [14, 7, 0, 12, 14, -2, -3, -2], [26, -7, -1, -9, -13, -4, 7, -1], [-9, -8, -2, 5, -5, 12, 9, 3], [-18, -2, 14, -8, -12, 4, -4, 2], [10, 1, 7, 1, -9, 5, 5, -11], [6, -4, 0, -2, -14, 8, -4, 0], [10, -7, 3, 4, -1, -7, 11, 9], [-1, -2, 10, -14, -17, 6, 13, 6], [14, 16, 5, 4, -9, -6, 2, -18], [-2, 7, 3, -16, 16, 13, -1, 7], [-24, -9, 0, -4, 6, 10, 11, -4], [-25, -18, 1, 8, -4, -10, 1, 3], [-2, -7, -14, -4, 8, -2, -15, 2], [-21, -10, -16, -11, 13, 16, -1, 14], [28, -8, 3, 6, 10, 3, -2, 11], [9, -29, -11, 3, -8, -14, 16, -2], [15, 2, 6, 1, 5, 12, -23, 12], [26, 0, -1, 2, 9, 17, -12, 1], [12, 14, -1, -2, 1, 18, -8, -1], [6, -5, 1, 7, 17, 4, -17, -4], [3, -9, 3, 6, 0, -4, -6, -11], [5, 17, -1, -2, -8, 3, -8, 1], [-9, 3, 0, 15, 15, -4, -14, -2], [-1, -5, -12, -3, 11, 11, -10, 12], [-43, 1, -6, 3, 5, -9, -4, 6], [-17, 10, -4, -6, -7, -12, 1, -5], [-26, -21, -6, -1, 3, -9, -1, 6], [3, -11, -12, -7, 15, -4, 2, -2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4730_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4730_2_a_w();". // 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_4730_2_a_w(:prec:=8) chi := MakeCharacter_4730_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(2999) - 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_4730_2_a_w();". // 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_4730_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4730_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![8, 56, 97, 9, -69, -30, 9, 7, 1]>,<7,R![88, -260, 78, 243, -56, -93, -9, 6, 1]>,<13,R![-64, -624, -256, 620, 280, -90, -42, 2, 1]>],Snew); return Vf; end function;