// Make newform 1800.2.k.q in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1800_k();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1800_k_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_1800_2_k_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_1800_2_k_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 := [16, -16, 0, -4, 9, -2, 0, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, -4, -6, 3, 2, 0, 0, -1], [56, -20, -12, -21, 18, 4, 2, -5], [-64, 24, 10, 25, -18, -4, -4, 5], [-36, 10, 8, 15, -10, -2, -2, 3], [36, -8, -12, -15, 10, 4, 2, -3], [44, -10, -9, -18, 12, 4, 3, -4], [-80, 26, 18, 31, -22, -6, -4, 7]]; Rf_basisdens := [1, 8, 8, 8, 4, 4, 4, 4]; 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_1800_k();" function MakeCharacter_1800_k() N := 1800; order := 2; char_gens := [1351, 901, 1001, 577]; v := [2, 1, 2, 2]; // 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_1800_k_Hecke();" function MakeCharacter_1800_k_Hecke(Kf) N := 1800; order := 2; char_gens := [1351, 901, 1001, 577]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 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, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 1, 0, 0, 1, 0], [1, 0, 0, 1, -1, 1, 0, 0], [1, -1, -1, 0, -1, 1, -1, 0], [0, 1, 0, 0, 1, 0, -2, -1], [-1, -2, -3, -1, -1, -1, 0, -2], [1, -2, 0, 3, 3, 1, 0, -2], [0, 1, 4, 2, -1, 0, 2, -1], [1, 1, 0, 1, -2, -2, -1, 0], [0, 0, 0, 2, -2, 0, 2, -2], [0, -1, 0, 2, -3, -2, 2, 1], [-1, 0, -1, -3, 3, -1, -2, 2], [0, 0, 0, 0, 4, 2, -2, -2], [0, 3, 2, 0, 3, 0, 0, 3], [-4, 2, 2, 0, 2, -4, 4, -2], [1, -1, -5, 2, -3, 1, 1, -2], [-3, 2, -1, 1, 1, -3, 4, -2], [5, -2, 0, 3, -1, -1, 2, 0], [2, 0, 0, -2, 6, 4, -2, -2], [-2, -2, 0, 6, 4, 0, -2, -4], [2, -2, 2, 2, -4, 2, 0, -2], [0, -2, 0, 0, 6, 4, 0, -2], [1, -2, 0, 6, 4, 0, -2, -4], [0, 3, -2, 0, 3, 0, 0, 3], [-8, 2, 0, 0, -2, -2, -2, 0], [-3, -2, -6, 1, -3, -3, 4, -6], [-1, 3, -7, 0, 3, -1, 1, 2], [-4, 2, 0, -4, -2, 0, 0, 2], [4, 4, 0, -6, 2, 2, -4, 0], [1, -2, -2, 1, -3, 1, 0, -2], [-8, -2, 0, 2, 4, 2, 0, -2], [-2, -2, 4, 0, -2, -2, 2, -4], [0, -2, -14, 2, -4, 0, 2, -4], [-3, -1, 0, 1, -8, -4, 5, 4], [-7, -3, 3, -4, 1, -7, 3, -6], [-5, 0, 3, -3, 3, -5, 2, -2], [-5, 0, 0, -3, 7, 5, -2, -2], [-6, -1, 0, -4, 3, -6, 2, -3], [0, -4, -2, -4, 0, 0, -4, 0], [1, 1, 7, 2, -1, 1, 1, 0], [-7, 0, 0, 3, -3, -3, 0, 0], [1, 0, 0, -6, 2, 4, 2, 2], [-4, 2, 10, 2, 0, -4, 6, -4], [1, -5, 0, 11, 2, -2, 1, -4], [1, 0, 3, -1, 1, 1, -2, 2], [7, -3, 0, 7, -4, -4, 3, 0], [-1, 4, 10, 3, 1, -1, 4, 0], [3, 5, 17, 6, -1, 3, 3, 2], [0, 7, 0, -10, 1, 2, -6, 1], [9, 2, 0, 3, 3, -1, -6, -4], [5, 6, 0, -10, -4, 0, -2, 4], [0, -2, 4, -4, 2, 0, -4, 2], [-2, 0, 0, 2, 6, 2, -4, -4], [5, 6, 0, -5, -1, -1, -6, 0], [2, 1, 16, -4, 5, 2, -6, 7], [0, -2, 0, 4, 6, 2, -2, -4], [1, -5, -17, -4, -1, 1, -5, 0], [-6, -3, 0, -2, -1, 2, 6, 3], [-3, 0, 7, 3, -3, -3, 6, -6], [2, 0, -14, 4, -4, 2, 2, -2], [3, 0, 1, 1, -1, 3, -2, 2], [10, 2, 0, -6, -4, 0, 2, 4], [3, 2, 0, -4, -10, -4, 4, 6], [-2, 0, -8, -2, 2, -2, 0, 0], [2, 0, 16, -2, 2, 2, -4, 4], [-7, 2, 0, -4, -2, 0, 0, 2], [6, 6, 14, 2, 4, 6, -4, 10], [4, 4, -12, 4, 0, 4, 0, 4], [2, -9, 0, 10, 5, 2, 6, -3], [-11, 4, 0, -5, 5, 3, -6, -2], [11, -3, 0, 1, 2, 2, 3, 0], [-3, 5, 11, 2, 3, -3, 5, 0], [-5, 0, -3, -7, 7, -5, -2, 2], [-4, -2, 0, 12, -6, -8, 0, -2], [2, 5, -14, 2, 3, 2, 0, 5], [-1, -1, 17, -6, 5, -1, -5, 4], [2, -1, 0, 8, 7, 0, -6, -7], [-3, 4, 0, -14, -2, 4, 2, 6], [1, -2, -2, 5, -7, 1, 4, -6], [0, 4, 8, -6, 10, 0, -6, 10], [-17, -2, 0, 1, 5, 3, 0, -2], [-5, 2, 0, 0, 2, 0, -4, -2], [-11, -3, 0, 9, 2, -2, -1, -4], [2, 6, 0, 2, 4, 2, 0, 6], [-6, 2, 0, -10, -4, 2, 4, 6], [10, -2, 0, 6, -4, -4, 2, 0], [2, 1, 6, 2, -1, 2, 0, 1], [12, -2, 0, 4, 6, 2, -2, -4], [-3, -2, -20, -3, 1, -3, 0, -2], [10, -12, 0, 14, 2, 0, 10, -2], [-9, -7, 0, 7, 8, 4, 3, -4], [-2, 8, -8, 2, 6, -2, 4, 4], [-5, -2, 3, -9, 7, -5, -4, 2], [-9, -8, 0, 9, -5, -3, 10, 2], [-8, 2, 18, 2, 0, -8, 10, -8], [0, -9, 0, 8, 15, 8, 2, -7], [-3, -6, -11, 1, -7, -3, 4, -10], [-7, 3, -9, -4, 7, -7, 3, 0], [10, 2, 0, 0, 2, 10, -10, 12], [6, -2, -22, 4, -6, 6, -2, 0], [-5, 2, 0, -1, 3, -5, 4, -2], [6, 5, 0, -10, -17, -6, 6, 11], [7, -6, -15, 3, -9, 7, -4, -2], [7, 6, 0, -2, -4, -4, -6, 0], [3, -8, -18, -9, 1, 3, -12, 4], [-2, -5, 0, 4, 7, 4, 2, -3], [-14, -10, 0, 10, -12, -6, 16, 6], [3, 0, 0, -2, -10, -4, 6, 6], [14, -2, 0, -2, -4, 0, 6, 4], [-4, -2, 12, -6, 4, -4, -2, 0], [-12, -3, 0, 6, -9, -6, 6, 3], [-5, -4, -7, -7, 3, -5, -2, -2], [1, 7, 0, -7, 8, 4, -11, -4], [14, 2, 0, -10, 12, 10, -4, -2], [-2, -6, 4, -8, 2, -2, -6, 0], [-20, -4, 0, 8, -12, -8, 8, 4], [2, 8, 22, 12, -4, 2, 10, -2], [-7, 2, 6, 1, 1, -7, 8, -6], [-4, -6, -4, -2, -4, -4, 2, -8], [14, 0, 0, -2, -18, -8, 10, 10], [4, 1, 16, 2, -1, 4, -2, 3], [2, 0, 16, -6, 6, 2, -8, 8], [2, 0, 4, 6, -6, 2, 4, -4], [4, 7, 24, 6, 1, 4, 2, 5], [-7, -3, 19, -4, 1, -7, 3, -6], [19, 0, 0, -3, -5, -1, 4, 4], [-1, 3, 0, 1, -12, -8, 1, 4], [-4, 0, -20, 0, 0, -4, 4, -4], [12, 0, -4, 4, -4, 12, -8, 8], [-10, 0, 0, 2, -10, -6, 4, 4], [-4, -4, 0, 2, 2, 2, 4, 0], [-5, -3, -27, -2, -1, -5, 3, -6], [-12, 0, 0, 8, -8, -8, 0, 0], [3, -6, 0, 8, -6, -4, 8, 2], [-2, -9, -18, -10, 1, -2, -8, -1], [3, 0, 15, 5, -5, 3, 2, -2], [6, 2, -4, -2, 4, 6, -8, 10], [-8, -5, 0, 2, 9, 6, 2, -3], [-9, 6, 23, 3, 3, -9, 12, -6], [6, -4, -22, 8, -12, 6, 2, -6], [-9, -3, 0, -5, 0, 4, 7, 4], [3, 10, 30, 11, -1, 3, 8, 2], [8, -2, -24, 2, -4, 8, -6, 4], [3, 6, 0, 5, 1, -5, -12, -6], [3, -7, 5, -6, -1, 3, -9, 2], [26, 4, 0, -14, -6, 2, 4, 8], [6, 0, -12, 10, -10, 6, 4, -4], [16, 2, 0, -12, 6, 8, 0, 2], [1, -9, -5, 2, -11, 1, 1, -10], [-26, 10, 0, -18, -4, 2, -4, 6], [-3, 10, 13, 9, 1, -3, 12, -2], [-18, 16, 0, -22, -6, 0, -10, 6], [0, -4, 4, 0, -4, 0, 0, -4], [-10, -4, 0, 10, -2, -4, 2, -2], [-9, 1, 0, -7, -18, -6, 11, 12], [-8, -1, 0, 4, -5, -4, 2, 1], [11, -6, 0, -4, 6, 8, 8, 2], [-14, -8, 2, -4, -4, -14, 10, -18], [5, -4, 8, 5, -9, 5, 0, -4], [0, 4, 0, -8, -12, -4, 4, 8], [-14, 2, 0, 6, -8, -8, -2, 0], [0, -4, -38, -4, 0, 0, -4, 0], [0, -12, 0, 22, -2, -6, 8, -4], [-18, 4, 0, -10, 6, 6, -4, 0], [-25, -7, 0, 7, 8, 4, 3, -4], [-4, -8, -28, -12, 4, -4, -8, 0], [-13, -8, 0, 16, -16, -12, 12, 4], [-4, 8, -6, -2, 10, -4, 2, 6], [3, -6, -8, -5, -1, 3, -8, 2], [-9, 7, -7, 2, 5, -9, 11, -4], [22, 6, 0, -22, 16, 16, -6, 0], [9, -18, 0, 24, -10, -8, 20, 2], [23, -3, 0, -1, 4, 4, 3, 0], [10, -6, 0, 18, 0, -6, 0, -6], [1, -2, -51, -3, 1, 1, -4, 2], [-8, 9, -4, -6, 15, -8, 2, 7], [3, 15, 0, -29, 6, 10, -11, 4], [-9, 5, -11, 6, -1, -9, 15, -10], [9, 3, 0, 5, -16, -12, 1, 4], [-14, -8, -4, -10, 2, -14, 4, -12], [7, -1, 25, 2, -3, 7, -5, 4], [-14, -2, 0, 0, -2, 0, 4, 2], [30, -4, 0, 18, 6, -4, -6, -10], [-6, -12, -34, -4, -8, -6, 2, -14], [-1, 3, -3, 10, -7, -1, 11, -8], [5, 16, 25, 3, 13, 5, -2, 18], [15, -10, 0, 20, 10, 0, 0, -10], [-7, -4, 0, 3, 9, 5, 0, -4], [3, -8, 0, 18, 18, 4, -6, -14], [11, -8, -2, 3, -11, 11, -8, 0], [9, 0, 17, 11, -11, 9, 2, -2], [-12, 7, 4, -10, 17, -12, 2, 5], [3, -4, 4, -1, -3, 3, -4, 0], [-14, 0, 0, -4, 4, 4, 0, 0], [-2, -8, 0, 6, 6, 4, 6, -2], [12, -8, -4, 0, -8, 12, -12, 4], [-4, -2, 0, -2, 0, 2, 4, 2], [-11, 8, 0, -13, 13, 9, -12, -4], [-8, -9, 6, -4, -5, -8, 4, -13], [-16, 6, 0, -4, -18, -10, 2, 8], [12, 14, 16, 6, 8, 12, -6, 20], [-10, 16, 0, -14, -14, -8, -10, 6], [-8, 0, -4, -8, 8, -8, 0, 0], [-2, 3, -16, 12, -9, -2, 14, -11], [27, 1, 0, -7, -10, -2, 7, 8], [-5, -14, -20, -5, -9, -5, 0, -14], [2, 0, 0, 2, 14, 6, -8, -8], [-3, -4, 29, -5, 1, -3, -2, -2], [-11, 2, 0, -2, 24, 12, -14, -12], [2, -10, -14, -12, 2, 2, -14, 4], [25, 5, 0, -5, -32, -16, 11, 16], [-14, -8, 8, -10, 2, -14, 4, -12], [2, 16, 0, -34, -18, 0, 2, 18], [-10, -16, 0, 22, -10, -8, 18, 2], [11, -5, 0, 19, 10, -2, -7, -12], [-12, 5, 0, -16, -19, -4, 10, 15], [-33, 0, 0, 5, 15, 5, -10, -10], [0, -7, -24, -6, -1, 0, -6, -1], [-1, -7, 9, 0, -7, -1, 1, -8], [-7, -17, 0, 11, 14, 10, 13, -4], [-34, -6, 0, 12, 14, 4, -4, -10], [-12, 2, 0, -8, 14, 10, -6, -4], [13, 2, -10, 9, -7, 13, -4, 6], [-13, -13, 13, -6, -7, -13, 7, -20], [20, -3, 0, 16, 5, -4, -6, -9], [-4, -10, 0, 8, 10, 6, 6, -4], [-17, -1, 0, -13, 6, 10, 5, 4], [9, -4, -20, -7, 3, 9, -16, 12], [-9, -15, -39, -4, -11, -9, 5, -20], [6, -4, 16, 2, -6, 6, -4, 0], [-3, 15, 0, -23, 8, 8, -15, 0], [-10, 8, 0, 4, -12, -12, -8, 0], [6, 8, -16, 6, 2, 6, 0, 8], [20, -4, 0, 20, 16, 0, -12, -16], [-2, -4, 0, 18, -2, -8, -2, -6], [-12, 9, 2, -8, 17, -12, 4, 5], [-8, 14, 22, 4, 10, -8, 12, 2], [25, 10, 0, -13, -21, -9, 2, 12], [9, 6, -4, 5, 1, 9, -4, 10], [-7, 18, -9, 1, 17, -7, 8, 10], [-11, -1, 0, 11, 22, 6, -15, -16], [9, -1, 3, -6, 5, 9, -15, 14], [-10, 0, 0, -4, -4, 0, 4, 4], [-42, 0, 0, -2, 6, 4, -2, -2], [-1, -13, 0, 23, 14, 2, 1, -12], [13, 16, 32, 9, 7, 13, -4, 20], [7, 2, -5, -9, 11, 7, -16, 18], [30, 8, 0, -6, -6, -4, -6, 2], [8, -14, -4, -10, -4, 8, -18, 4], [16, -7, 0, 4, -19, -8, 18, 11], [-52, -2, 0, 4, 10, 4, -4, -6], [-21, -8, 0, 14, 22, 8, -6, -14], [-18, -11, 8, -8, -3, -18, 10, -21], [10, 0, 12, 6, -6, 10, -4, 4], [-5, -3, -11, -18, 15, -5, -13, 10], [2, -8, -24, -2, -6, 2, -4, -4], [2, -1, -6, -10, 9, 2, -12, 11], [-22, 4, 0, -16, 4, 8, 0, 4], [-9, 3, 0, -23, -20, 0, 17, 20], [-17, -8, 8, -13, 5, -17, 4, -12], [12, -10, -4, -6, -4, 12, -18, 8], [-5, -7, -19, -8, 1, -5, -3, -4], [-2, -14, 0, 20, 30, 12, -4, -18], [-2, -8, -36, 6, -14, -2, 8, -16], [-12, 12, 30, 10, 2, -12, 22, -10], [18, 10, 0, -34, 4, 14, 0, 10], [0, 12, 44, 16, -4, 0, 16, -4], [8, 6, 2, 10, -4, 8, 2, 4], [-11, 3, 11, -8, 11, -11, 3, 0], [6, -4, 20, 2, -6, 6, -4, 0], [-10, 6, 0, -2, -28, -16, 6, 12], [1, 9, 0, -19, 10, 10, -9, 0], [-5, -10, 0, 10, 16, 8, 2, -8], [-5, -3, 0, 9, 18, 6, -9, -12], [-2, -10, 18, -6, -4, -2, -4, -6], [-11, -5, 3, -12, 7, -11, -1, -4], [-9, 10, 0, -4, 6, 0, -16, -6], [-4, -14, -32, -12, -2, -4, -8, -6], [7, 12, 0, -15, 7, 5, -14, -2], [-10, 18, 0, -34, -16, 0, -2, 16], [-41, -4, 0, 13, 15, 3, -8, -12], [9, 9, 15, 0, 9, 9, -9, 18], [11, -2, 25, 15, -17, 11, 4, -6], [-18, 16, 0, -10, -26, -16, -6, 10], [-53, -8, 0, 12, -4, -4, 8, 0], [-20, -1, 6, 0, -1, -20, 20, -21], [-44, 4, 0, -10, 6, 6, -4, 0], [-22, -9, 0, 20, 11, 0, -2, -11], [16, 12, 6, 10, 2, 16, -6, 18], [-14, 2, 8, -10, 12, -14, 4, -2], [-6, -8, 0, -2, 10, 10, 8, 0], [-6, 2, -56, 6, -4, -6, 12, -10], [-9, 9, 5, 6, 3, -9, 15, -6], [18, -5, 10, 2, -7, 18, -16, 11], [-9, -9, 0, 11, -2, -2, 9, 0], [6, -7, 26, 10, -17, 6, 4, -11], [-11, -8, -16, 5, -13, -11, 16, -24], [-12, 0, 12, -8, 8, -12, 4, -4], [-6, 12, 0, -28, -8, 4, 0, 12], [-12, 2, 8, -8, 10, -12, 4, -2], [27, -1, 0, -3, 12, 8, -3, -4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1800_k_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1800_2_k_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_1800_2_k_q(:prec:=8) chi := MakeCharacter_1800_k(); 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_1800_2_k_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_1800_2_k_q( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1800_k(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![1, -12, -6, 4, 1]>,<11,R![1600, 0, 1344, 0, 336, 0, 32, 0, 1]>,<17,R![-24, 104, -40, 0, 1]>],Snew); return Vf; end function;