// Make newform 6018.2.a.q in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6018_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_6018_2_a_q();". // 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_6018_2_a_q();". // 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 := [1, -7, 10, 8, -8, -1, 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], [3, -10, -7, 8, 1, -1], [-6, 10, 8, -8, -1, 1], [-10, 32, 16, -24, -2, 3], [-11, 37, 16, -25, -2, 3]]; Rf_basisdens := [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_6018_a();" function MakeCharacter_6018_a() N := 6018; order := 1; char_gens := [4013, 1771, 1123]; 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_6018_a_Hecke(Kf) return MakeCharacter_6018_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], [-1, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1], [0, 1, -1, 1, -1, 0], [0, -1, 1, -1, 1, -1], [-1, 0, 0, 0, 0, 0], [0, 0, 1, 2, 0, -1], [-1, -2, 0, 0, 1, 1], [-1, -2, -1, -1, -1, -2], [-1, -1, -2, 0, 0, 0], [-4, 2, -2, 1, -2, 0], [-1, -2, 2, 2, 1, 1], [-3, -2, 2, 0, 1, -1], [-3, -2, 1, -2, 1, -2], [4, 1, 1, 0, 1, 2], [1, 0, 0, 0, 0, 0], [0, 2, -1, 4, -4, 0], [-2, 3, 1, 0, 1, -5], [1, -3, 5, -1, 2, 0], [-6, 1, 0, 1, -1, 1], [-6, 4, -1, 0, -4, -1], [3, 0, 4, -1, -1, -2], [-2, 1, -4, 0, -3, -2], [-1, -1, -3, -3, 0, 3], [8, -4, 0, -1, 0, 3], [1, 2, -3, 1, 1, 2], [1, 3, 1, 0, 2, 0], [-4, 3, 0, -1, 3, 2], [-5, -4, 2, 1, 1, 2], [-1, -3, 1, 0, 6, 2], [2, 4, -2, 0, 2, 4], [-5, -6, 1, -4, 1, 0], [0, 3, 1, -3, 1, 0], [-3, 3, -2, 2, 0, 3], [-2, 1, -1, 3, -1, 1], [3, -2, 1, 2, -1, 1], [-3, 3, -5, 1, 0, 3], [0, 1, -2, -5, -5, 2], [0, 6, -5, 1, 0, 4], [-9, 1, 0, -4, -2, -5], [-10, -1, 2, 2, 5, -6], [-4, -2, 4, -3, 4, -9], [-1, 5, 0, 0, -4, 1], [-6, -2, -4, -6, 2, -3], [6, -4, -3, 2, -4, 1], [-10, -3, 1, -3, -3, 0], [1, 3, -2, 8, -2, -1], [-1, -2, -3, 0, -1, 2], [4, 1, -1, 9, -5, 5], [-6, 0, 1, 0, 4, 3], [2, 2, 2, 1, 0, 8], [-9, -1, 5, -6, 2, -4], [5, 2, -8, 6, -7, 5], [0, 1, 0, -6, -1, 4], [-5, -4, 2, -8, 3, 2], [-4, -2, 3, 4, -2, 1], [-5, -5, 5, -4, 4, -8], [-13, 4, 7, 3, 3, -5], [-2, 0, 1, 4, 0, -1], [-3, 6, -3, 1, -1, -5], [8, 7, -5, 9, -3, 3], [-3, -5, 9, -3, 4, -1], [0, -2, 4, 6, 6, -5], [0, 0, -3, 2, -6, 9], [-4, 1, 3, 2, 7, -6], [-2, -2, 0, 3, -2, -3], [-5, 1, 2, -5, 10, -2], [-11, 9, 1, 1, -2, -5], [0, 1, 3, 4, 1, 4], [-6, 4, -1, -3, -6, -4], [13, 6, -3, 7, 1, 6], [0, -3, -3, -7, -1, -1], [-2, -4, 3, -3, -2, 0], [-5, -7, -6, -4, 2, 1], [13, -2, -6, 3, -7, 3], [-5, -9, 7, -2, -4, -5], [-8, -1, 2, -5, 7, 0], [-2, 3, 6, 6, -1, -2], [-6, -5, 0, -11, 1, -3], [4, 5, -1, 5, 3, -2], [-12, 2, -4, -2, 2, -3], [2, 1, -2, 6, -3, -4], [6, -13, 1, -10, 3, 1], [9, 10, 0, 3, -3, 8], [-4, 0, -7, -6, 0, -1], [-1, -8, 7, -4, -1, -6], [8, -6, 2, -1, -2, 7], [1, 1, -1, -7, 2, -4], [5, -4, 1, -1, -5, 0], [-5, 2, -2, -2, -5, 3], [5, 2, 3, 0, 9, -3], [6, -2, 0, -2, -2, 4], [8, 13, -7, 9, -1, 2], [-16, 8, 7, 6, 0, -3], [-7, -2, 12, 1, 1, -4], [-1, 5, -5, 2, 8, 0], [-2, 6, 2, 0, 2, -8], [3, 2, -8, 4, 3, 7], [1, 9, 1, -1, -4, 4], [-6, 6, 0, -1, 2, 5], [5, 1, 4, 9, 2, 1], [8, -3, -11, -12, 1, 1], [6, 5, -1, -3, -13, 4], [12, -1, 3, -2, 3, -1], [-5, -12, -7, -9, 5, -5], [-4, 1, -10, 1, -1, 7], [9, 7, 0, 1, 2, 4], [12, 3, -4, 4, 1, -1], [-3, 3, 2, 8, 0, -1], [4, -3, -7, 6, -13, -1], [1, 13, 0, 4, -4, 10], [9, 3, -6, 3, -2, 6], [17, -1, 11, -2, 2, 2], [4, -5, 0, 0, -5, 0], [8, -6, -6, 2, -4, 6], [0, 6, -9, -7, 0, 0], [-2, -6, 17, -9, 4, -11], [-9, -1, 8, 11, 4, -5], [-4, 0, -10, -8, 0, -6], [23, -5, 1, 2, -2, 4], [7, -15, 2, -7, 6, 4], [-1, -2, 6, -9, 3, -5], [-6, 8, 6, 0, 10, 4], [1, -6, -3, 9, -9, 3], [9, 7, -7, -2, -4, -4], [1, 8, 0, 5, -11, 6], [8, 7, -6, 1, -1, -3], [18, 3, 0, -1, -5, 11], [7, -5, 9, -2, 2, 2], [10, -12, 1, -13, 8, 0], [2, -8, -6, 9, 0, 4], [12, -7, -5, -2, -3, -2], [-22, -4, -2, -5, -2, -11], [-8, 10, 6, -3, 2, -5], [4, -6, 5, 6, -4, -5], [5, 6, -4, -4, 3, -3], [4, -8, 7, 2, 4, 3], [-11, 1, 10, -10, 10, -9], [-2, -15, 14, 0, 9, 4], [-1, -9, 19, -9, 10, -9], [-16, -14, 8, -7, 0, -13], [0, -8, -5, -8, 8, 4], [-6, 0, 8, 4, -6, 4], [-19, 9, 2, -7, 4, 2], [-10, 5, 6, -2, -7, -4], [-17, 7, -4, 3, -2, 0], [3, 6, 3, 2, -7, 8], [-7, 6, -4, 6, -11, 13], [1, 2, 4, 4, -3, 3], [2, 5, -10, 16, -5, 6], [-3, -2, 3, -5, -3, -13], [15, -10, 13, -12, 7, 0], [-2, -4, -5, 2, -16, -1], [3, -10, 16, 9, 13, -6], [6, 9, -5, -1, 1, 3], [-29, 19, -1, 4, -10, -1], [-19, 17, -4, 15, 0, -4], [-13, 2, -8, -3, 3, 8], [-1, 1, 14, 5, 4, 2], [9, 1, -14, -5, -6, 4], [-3, 7, -4, 4, 0, -1], [9, 0, -1, 13, -13, 6], [-12, -4, 5, -3, 6, 12], [7, -4, -1, -12, 15, 0], [6, 9, -15, 11, -5, 12], [-15, 14, -2, -5, -13, -5], [11, 15, -2, -1, -10, -2], [-19, 15, 2, -4, -8, -5], [-7, -15, 10, -2, 4, -11], [28, -2, -1, -5, -4, 8], [-7, 21, -9, 11, -2, 2], [-29, 10, -4, 5, -3, -5], [-8, -2, 0, -16, 8, 8], [10, 2, -14, -13, -2, 3], [11, -3, 1, -4, 8, -2], [13, 0, -1, -3, 5, 3], [-13, -5, 21, 1, 10, -9], [-14, -3, -5, -9, 5, -2], [-2, 3, -4, -9, 9, -5], [-2, 13, -3, 2, -1, -17], [5, -15, 10, 0, 0, -3], [-2, -9, 5, -17, 5, -7], [-9, -6, -11, -3, 3, 15], [25, -1, -4, -8, 4, 0], [30, 9, -8, 12, 5, 8], [-4, 5, 10, 15, 1, -8], [5, 7, -5, 15, -8, 5], [-2, -5, 1, -1, 1, -4], [14, -9, 9, 14, 7, -3], [-6, -10, 14, -4, 2, -18], [-24, -1, -13, -5, -3, -10], [-11, 12, 3, -3, -15, -5], [16, 17, -5, 0, -3, 14], [6, 8, 1, 7, -8, 10], [4, -2, 3, 11, 6, -2], [-6, -3, 9, 2, 21, -9], [-26, 3, 2, -1, 1, -8], [-6, 3, -14, -8, -5, -12], [-8, 10, -5, -5, -16, -8], [9, 3, -11, 4, -16, -1], [0, -8, -8, 3, -16, -9], [-22, -15, 5, 4, 1, 9], [-7, 7, 0, 1, 2, -8], [-4, 19, -6, 3, -3, -9], [-15, 14, 0, -11, -7, -14], [29, -6, 4, -6, -1, 0], [21, 3, -13, 0, -6, 4], [19, 0, 4, 4, -1, 7], [19, -7, 3, 9, 12, 7], [-24, 16, -10, 0, -10, 2], [-19, 7, -11, 1, -8, 2], [12, -14, 3, 9, 2, 0], [24, 10, -3, -5, 0, -8], [24, 3, 3, -13, 1, 5], [2, 10, 1, -2, -2, 1], [-4, 5, 10, 5, 3, -11], [-4, -9, 4, -5, -9, -1], [-3, 9, -6, 9, 8, 8], [10, 17, 4, 19, -11, -6], [25, -11, 6, 4, 8, -2], [4, 2, 12, -8, 10, 1], [17, 3, -16, -6, -16, 1], [1, -19, 5, -7, 4, -1], [-1, -1, -2, 0, 12, -11], [-12, -15, -1, -1, 11, 7], [18, 11, 2, 13, -7, 1], [21, -7, -12, -14, 6, -5], [-1, 1, -2, 9, -6, 21], [-22, -1, -3, 13, -9, 4], [-17, -4, 15, 0, 1, -11], [4, -11, -12, -1, -1, -3], [6, -5, -12, 15, -11, 12], [5, 9, -8, 16, -18, 3], [31, -5, -12, -3, 2, 2], [10, -6, 12, 4, 10, 5], [3, -19, 5, 10, 8, 2], [-12, -6, -11, -4, -6, 1], [6, -7, 13, -5, 13, -8], [27, -7, 4, 6, 0, 6], [3, 5, -5, -6, -2, -4], [-18, 19, 4, 4, 1, 0], [2, 18, -3, 2, -12, -5], [-16, 30, -2, 8, -8, 2], [1, -26, -2, -12, 3, -1], [11, -15, 6, -5, 0, -19], [-22, 0, -18, -7, 4, 6], [3, -2, -10, 13, -7, 12], [22, -13, -6, -6, -11, 6], [-2, -10, 15, -18, 14, -4], [-14, -7, 8, 6, 9, -10], [-4, -1, 12, -21, 15, -9], [10, -9, -2, -14, -7, 12], [-6, 16, -6, 12, -12, 2], [17, -6, -13, 7, -13, 14], [15, 0, 0, -11, -1, 8], [-25, -8, 8, -8, 1, -9], [-15, 15, 3, 2, -2, 8], [20, 10, 5, 17, -4, -6], [-10, -5, 4, 12, -1, -10], [-1, -7, -8, -2, -6, 2], [11, -4, -6, 12, -3, 21], [12, -4, 4, -3, 14, 7], [-3, -4, -4, 20, -9, 7], [7, 0, -10, 4, 3, -3], [12, -12, -1, 6, 16, 1], [-21, -1, -12, 0, 4, 20], [4, -18, 14, -8, -4, -4], [19, 5, -4, -9, 4, 2], [16, 22, -11, 1, -8, -9], [-17, 6, 8, -21, 7, -5], [-30, 3, 13, 4, 13, -9], [-17, 1, 2, -7, -2, -12], [5, -1, -19, -7, -8, 9], [-13, -21, 4, -15, 0, -2], [21, 20, 5, -9, -1, -6], [-39, -7, 6, -8, -4, -9], [4, -2, 5, -16, 8, -13], [-22, 13, -20, 16, -3, 2], [-27, 4, 19, -5, 9, -11], [6, -18, 23, -4, 18, -4], [-11, -4, -5, 14, -1, -11], [-22, -8, -4, -11, 10, -5], [-2, -26, 2, -10, 4, 9], [19, -21, -7, -9, 2, 3], [5, 9, -4, 25, -18, 10], [-5, 9, -11, 2, 0, 0], [15, -13, 5, 1, -4, -10], [-15, 2, 5, -20, 11, 1], [-8, 19, 11, -6, -3, -15], [7, 12, -1, 7, -1, 3], [12, 6, 9, 14, -2, 19], [-15, 5, 16, 6, 10, -3], [-21, -24, 14, -14, 1, -19], [-15, 18, -13, -7, -13, 7], [-9, 11, -5, -4, 12, -2], [-24, -1, 10, -15, 5, -1], [-1, -1, 5, -24, 8, 7], [0, 15, 11, -12, 7, 3], [31, 11, -2, 8, 0, -9], [19, -3, 9, 10, 0, -8], [-11, -13, -6, 7, 2, 8], [3, -17, 19, -2, 16, 11], [-11, -7, 10, 1, 18, -6], [9, -6, -3, 6, -11, 30], [-18, -12, -7, -3, 2, 8], [-21, 12, -2, 7, -9, -6], [-4, 19, -10, 3, -17, 0], [34, -14, -2, -8, 0, 0], [-15, 3, 8, 12, -10, 8], [-24, -12, -12, -7, 6, 0], [39, 9, -17, 10, -4, 7], [-5, -3, 4, -17, 18, -18], [-19, 14, -9, -2, -15, 10], [25, -11, -2, -15, 16, -9], [-3, 23, -15, 2, -12, -20], [-16, -3, 11, 3, 11, -6], [8, -15, 9, 0, 1, -2], [17, -19, -6, -8, -6, 9], [29, -3, -8, 11, -16, -4], [-4, -20, 10, -9, 2, 13], [-38, -7, 0, -17, 7, -1], [23, 15, 9, 25, -6, 2], [3, 21, -2, -4, 10, 7], [-5, -17, -6, 9, -8, -2], [-25, -1, -16, 7, 4, -2], [-17, -31, 10, -15, -4, -6], [8, 7, 12, 5, 9, 1], [8, 11, 7, 19, -21, -4], [10, -21, -3, -10, -3, 11], [-18, 1, -16, 9, -13, 10], [5, 18, 8, -6, 5, 1], [8, -12, 7, -20, 16, -16], [-4, 2, 2, -10, -2, 0], [-30, 14, 8, 7, 6, -3], [-38, 12, -1, 2, -2, 5], [-29, -6, -8, -2, 9, 13], [13, 7, 14, -12, -4, 3], [-1, -26, 6, -4, 13, -5], [-40, 10, 11, -6, 18, 7], [28, 4, -15, 18, 2, 21], [4, 9, -14, 19, -7, 17], [7, -4, -3, -16, 1, -4], [-5, -13, 26, -3, 16, 11], [-35, -1, 9, -14, 0, -8], [10, -27, 4, 10, 11, 14], [-8, -9, 7, -18, 13, -2], [-1, 6, -7, 14, -21, 10], [27, -23, 13, 8, 20, 2], [-31, -1, 11, -1, 0, 3], [17, -13, 0, 8, 2, -11], [-26, -5, -9, -3, -7, 4], [-18, -1, -6, 2, -19, -3], [18, 4, 2, 1, 0, -26], [-26, -1, -10, 0, 11, 8], [-20, 0, -5, 10, 2, -9], [-19, -22, -7, -8, 5, -8], [7, -5, 4, -28, 8, -5], [1, 36, -14, 19, -1, 12], [-11, -1, -4, 6, -8, -4], [-15, 6, -7, 0, -1, 19], [10, 14, -3, 13, -12, 14], [-29, 8, -12, 12, -5, 2], [14, -20, -11, -7, 8, 6], [15, 15, -12, 3, -10, 10], [-17, -2, 10, 3, 23, -8], [-8, -5, -20, 5, 13, 11], [19, 11, 11, 19, -10, -1], [-21, 8, -4, 8, 15, -7], [6, -15, -16, -10, 15, 2], [-13, -22, 11, -2, 15, -12], [23, 10, 4, 25, 3, -12], [-13, 4, 16, 0, -3, -14], [-6, -8, 1, -19, 22, 1], [-44, 6, -7, -5, -4, 6], [-6, 11, 2, 2, 1, -14], [2, 12, -1, 4, 0, -15], [-51, 2, -18, 3, -3, -4], [-11, -4, 16, -5, -1, -12], [-32, 24, -2, -5, -24, -7], [7, -9, 13, 1, 24, 8], [21, -24, 17, -21, 15, -7], [20, -21, -4, -2, 13, 8], [18, 3, 5, 4, 7, 11], [15, -25, -1, -20, -8, 10], [-6, 34, -8, 9, -16, 1], [-29, -22, -4, 16, 11, -7], [2, 21, -13, 0, -19, 3], [16, 4, 5, -14, 24, -3], [11, -10, 28, -14, 23, 13], [-11, 15, 15, 30, -16, 4], [-9, 8, 10, -11, 3, -2], [4, 17, -4, -4, -19, -8], [40, -7, -9, 1, -5, 2], [-2, 33, -12, -1, -11, 13], [63, 2, 3, 15, 3, 13], [4, -9, 10, -1, 5, -17], [-30, -16, 2, -2, 16, 0], [-10, 2, 18, -10, -8, -8], [-7, 6, 25, -9, 15, 2], [-28, -15, -8, 3, -1, 5], [24, 34, -2, 17, 2, 19], [4, 7, 7, 21, -11, -4], [-7, -14, -16, -24, 19, -11], [12, -39, -1, -8, 1, -2], [-15, -22, 21, -5, -5, -8], [-14, -4, -12, -20, -6, -1], [5, 12, 17, 14, -3, 4], [43, -8, 2, 0, -7, 11], [0, 17, -1, 16, -7, -9], [-18, 5, 12, 6, 9, -6], [-16, 4, 1, 0, 0, 4], [-27, 8, -12, 13, -7, -2], [12, -6, -1, -7, 12, -2], [-21, 12, -22, 1, -15, -14], [13, -4, 6, 11, -9, -9], [-11, 22, 8, 17, -7, -14], [15, 2, -5, 22, 3, -5], [-4, -24, -3, -8, 2, 29], [50, 2, 1, 30, 6, 8], [-44, 8, -13, 14, -10, 7], [4, -21, -1, -30, 11, -3], [-4, -19, 3, 1, -9, -3], [31, 5, 7, 12, 6, 30], [10, -15, 10, 3, 27, -3], [-27, 9, 1, 11, -18, 4], [21, -11, 14, 19, 6, -12], [2, -6, 14, -3, 28, 3], [9, -23, -15, 5, 10, 11], [26, 5, -9, 5, -11, 13]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6018_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6018_2_a_q();". // 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_6018_2_a_q(:prec:=6) chi := MakeCharacter_6018_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_6018_2_a_q();". // 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_6018_2_a_q( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6018_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![4, 6, -9, -14, 2, 5, 1]>,<7,R![4, 32, 41, -23, -25, 1, 1]>],Snew); return Vf; end function;