// Make newform 3344.2.a.z in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3344_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_3344_2_a_z();". // 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_3344_2_a_z();". // 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, -10, 25, 12, -9, -2, 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], [1, -5, -1, 1, 0, 0], [-6, 11, 10, -7, -2, 1], [-12, 9, 12, -7, -2, 1], [22, -5, -24, 5, 4, -1]]; Rf_basisdens := [1, 1, 1, 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_3344_a();" function MakeCharacter_3344_a() N := 3344; order := 1; char_gens := [2927, 837, 2433, 705]; v := [1, 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_3344_a_Hecke(Kf) return MakeCharacter_3344_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 := [[0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, -1, 0, 1, -2, 0], [1, -2, -1, 1, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 1, 1, 2, 1, 1], [-1, 0, 0, 1, 1, 1], [-1, 0, -1, -1, -1, 1], [0, 1, 2, -1, 3, 1], [-1, -1, 1, -2, 4, 0], [3, 1, 2, 1, 1, 0], [2, -2, 0, 2, -2, 0], [-1, 2, 1, 3, 0, 0], [5, -1, -1, 2, 1, -1], [2, 0, 0, -2, 2, 0], [4, -2, -2, 0, -1, -1], [2, 1, 1, 1, -2, -2], [-3, 2, -1, 1, -2, -2], [6, -4, 0, 0, 0, -2], [3, 1, -2, 1, 1, 0], [-4, 3, 0, 1, 1, 3], [0, 1, 0, 5, -1, -1], [-2, -2, 0, -2, 4, -2], [4, 0, 3, 4, -1, 1], [5, 0, 1, -3, 2, 2], [1, 6, 1, 3, -2, 2], [0, 3, 2, 3, 1, -1], [4, 2, 4, 2, 0, 2], [-3, 0, -3, -5, 0, -1], [-2, -1, -2, -3, -3, 0], [9, -1, 2, 3, -1, 2], [2, 4, 4, -2, 0, -2], [0, 4, -2, 0, 4, 2], [7, -2, -2, 4, -6, -5], [6, 2, 0, 4, -2, -4], [6, 2, 2, 2, 4, 4], [-6, 2, 1, -2, 5, 1], [-3, -2, -1, -3, -1, -1], [8, -1, -2, -3, 3, -1], [-5, 7, -1, 0, 3, 3], [4, -2, -3, 2, -3, -3], [0, 0, 0, 8, -2, 0], [-3, -2, 1, -3, 4, -2], [9, 4, 3, -3, 2, -2], [-10, -5, 0, -1, 3, 1], [15, 0, -1, 3, -6, -2], [9, -1, -1, -4, 1, -2], [-2, -4, 4, 0, -2, 0], [4, 1, 5, -2, 3, -2], [6, 0, 5, 2, 3, 3], [-8, 7, 2, 3, 5, 5], [-4, -6, -4, 0, -6, 2], [-3, 3, 6, 1, -1, 2], [-10, 0, 0, -2, 0, 2], [-2, 0, -4, 4, -4, 3], [-2, 2, 0, 6, -4, 4], [0, -4, -5, 0, -11, 1], [1, 0, -1, 1, 4, 3], [-10, 3, 0, -1, 4, 2], [-4, 0, 0, 6, 0, 2], [-11, 4, 3, 3, 2, -2], [4, 0, -3, 3, 2, 3], [-1, -5, -5, 0, -9, 3], [9, 5, 4, -2, 0, 0], [-3, 9, 3, -2, 2, -2], [10, -8, -2, 2, -8, 0], [-2, -6, -2, 8, -6, 6], [-15, 7, 3, -4, -1, 1], [-6, 0, 0, -8, 0, -3], [0, -5, 4, -3, 5, -1], [-1, 8, 2, 1, 1, 1], [8, 0, 0, -2, 3, -5], [1, -6, -5, -9, -3, 3], [5, -4, 2, 6, -8, -5], [9, 4, 1, -7, 0, 1], [-10, 2, 4, 0, 4, -4], [14, 0, 5, 2, 1, 1], [2, 2, 4, 4, 6, 0], [-9, 4, 3, -7, 6, 4], [6, -2, 0, 4, 0, 4], [-4, 7, 2, 1, 5, 5], [-16, 4, 2, -4, 2, -4], [0, -3, 2, 5, 1, -1], [4, 3, 0, -7, 5, -5], [1, 0, 1, 7, -4, 6], [10, 0, -2, -4, 8, -4], [4, 1, 2, -1, -7, -5], [2, 5, 4, 11, 1, 3], [-15, -4, -3, -11, 2, 3], [-6, -7, 1, 5, 0, -4], [13, 2, -3, -7, 6, 1], [10, -4, -6, -4, 4, -4], [-6, -7, -1, -4, -1, 0], [4, -3, 8, -3, 5, 1], [-4, -7, -4, 5, 3, 3], [7, 2, 3, -9, 12, -4], [18, -6, 0, 2, -2, 0], [12, -6, -6, 0, -2, 2], [0, -4, 8, 4, 2, -2], [12, -8, 2, 14, -6, 0], [-9, -7, -1, 0, 6, -2], [13, 4, 3, -3, 12, -1], [4, -7, -6, -5, -5, -12], [-6, -2, 0, 4, -2, 6], [2, 6, 0, -4, 2, 0], [-18, 6, 3, -4, 9, 7], [-7, 5, 5, -2, 8, -4], [10, -2, -2, -2, -6, -4], [-2, -6, 2, -6, 8, -2], [-15, 4, -1, 1, 2, 3], [0, -3, 0, -7, 1, -5], [-12, -1, 0, 1, -3, -5], [4, 11, 0, 5, -7, 7], [0, -5, -4, -11, -1, 1], [-5, 15, -1, 2, -1, 5], [-27, 3, -1, -8, -3, -2], [3, 2, 7, 5, -6, -2], [5, 5, 5, 10, 3, -3], [1, 1, -3, -18, 4, 4], [-5, 0, 0, 9, -5, 7], [16, -4, -2, 8, -10, 2], [-4, 15, 2, 3, 1, 1], [-4, 0, 4, 8, 2, -4], [-1, -11, -1, -6, -1, 4], [7, -11, -5, 10, -3, -3], [-7, 15, 5, -4, 9, -5], [-18, -4, 2, 8, -8, 6], [-15, -9, 2, -5, -1, 4], [14, -2, 2, 14, -4, -2], [-6, -7, -4, -11, -3, -9], [23, 1, 4, 7, 7, 6], [-13, 6, -1, 7, 0, 4], [17, 8, 1, 5, 0, -8], [-13, -12, 1, 3, -4, 2], [7, 12, 1, 9, 2, -2], [-13, -3, 3, 6, -9, -1], [7, 4, 3, -1, 4, -10], [19, -2, 5, 11, -8, 2], [-24, -4, -2, 10, -16, 8], [-1, -3, -1, 8, -15, -3], [9, -4, 1, -9, 8, -2], [11, -15, -7, 4, 1, -7], [-6, -7, 1, -11, 12, 0], [-2, 4, 10, 2, 6, 0], [-15, 1, -5, -2, 0, 0], [-10, 5, 4, 1, 9, -5], [-1, 5, -5, 6, -10, 6], [-33, -6, -2, -5, 7, 3], [-19, 1, 1, -6, 17, 4], [-22, 10, 2, 0, 0, 8], [26, 4, 0, 4, 4, -6], [7, -2, 9, 9, -6, 2], [22, -4, 4, 6, 10, 6], [-6, -3, -5, -12, -1, 2], [-10, 4, 2, 12, -4, -1], [-15, -4, -1, -1, 2, 8], [19, 6, 3, -3, 10, -6], [0, -1, -4, 1, -13, -9], [-22, -4, -2, -6, -2, -14], [-8, 4, 8, -4, 12, -4], [-10, 7, 3, 1, 6, 2], [-4, -7, -6, -9, -3, 7], [3, -8, 5, -5, -7, -5], [11, -5, -13, 0, -2, -2], [22, -12, 4, -4, 4, 0], [-9, -1, 1, 6, -2, 2], [1, -9, 3, -4, 13, -2], [5, 0, -5, 11, -16, -14], [7, 4, 1, -11, 12, 2], [-11, 17, -2, -1, 5, 0], [-9, 10, 5, 9, 2, -4], [-6, 1, 4, -3, 9, 3], [-7, -6, -8, -16, -2, -9], [13, -10, -9, -1, -18, -1], [8, 1, -2, -11, 3, -7], [28, -14, -2, 0, -6, 0], [14, 0, 4, 6, -6, 4], [-7, -10, 1, -9, 8, -10], [3, 4, -5, -5, 2, -4], [4, -1, 0, -7, 11, 1], [10, -14, -9, 0, -5, -5], [-15, -3, -5, -14, 5, 3], [0, -4, -6, 6, -4, 0], [5, -4, -7, -11, -4, 8], [-16, 14, 1, 4, 5, 7], [4, -2, 1, 6, -9, -9], [-6, 19, 5, -8, -3, -2], [20, 1, 0, -3, 11, 3], [-3, 2, 3, -1, 14, -5], [-4, 7, -3, -17, 8, -4], [-29, -2, 7, -5, 2, 8], [19, -20, -3, -9, -8, -9], [-22, 5, -2, -3, -7, 3], [9, 4, 11, 7, 8, 8], [-17, -3, -1, -2, -5, 6], [9, -2, 5, 3, -2, 0], [9, 0, 1, -7, 10, 2], [17, -11, -3, 0, -14, -16], [18, -14, 2, 4, -10, -4], [1, 14, -3, -7, 14, 0], [28, 8, 5, 4, -3, 5], [0, -15, 6, -1, -1, -3], [34, -17, -4, 9, -15, -1], [-17, 1, 3, 4, 7, 7], [-13, 8, -5, 3, -16, -2], [-11, 16, 5, 7, 8, 2], [-3, -8, -10, 3, -7, 3], [7, 6, 11, -9, 24, -6], [13, -4, -8, 0, -12, 1], [-12, 5, -4, 9, -3, -1], [14, 1, 0, -9, 4, -18], [12, -14, 0, -4, -6, -6], [-18, 5, -4, 1, -17, 5], [-1, 2, -1, 1, -8, 12], [14, 2, 10, -16, 16, 4], [11, -7, -15, 2, -16, 8], [13, 1, -3, 4, 3, 6], [-1, -5, -11, -8, 0, -10], [-14, 6, 2, 0, 4, 8], [-10, -5, 2, 19, -17, 3], [-11, -5, 1, 4, 7, 3], [-7, -2, -5, -5, 6, 2], [26, -6, -4, -4, 0, -4], [10, 0, 4, 8, 12, 2], [14, -6, 3, 8, -7, 7], [22, -16, -4, 0, -6, 4], [31, -5, 10, -9, 11, 2], [41, -4, 2, 6, -4, -5], [7, 8, 5, 3, 2, 12], [0, 12, -6, -8, 4, 2], [11, 1, -7, 0, 4, 8], [30, -5, -2, 5, -5, -9], [0, 6, -2, 12, -14, -11], [1, 11, 7, 6, -5, 10], [-4, 8, 6, -10, 2, -18], [-38, 7, 10, -1, 1, 9], [3, -10, -7, 9, -1, 5], [-12, 13, 3, -11, 4, 0], [-5, 4, 5, 3, 8, 3], [-1, 6, 1, -17, 22, 0], [39, -9, -5, 4, -9, 1], [38, -6, -6, 4, -16, -4], [-27, 4, -7, 9, -14, 3], [-2, -12, -8, 4, -2, 6], [-12, 10, -1, 0, -5, 1], [28, -2, -2, 4, 8, 10], [7, -16, -9, 11, -2, 6], [-10, -8, -8, -14, -2, -4], [25, 0, -3, 1, -14, -10], [7, -3, -1, 2, -9, 5], [4, 4, 10, 8, -12, 0], [0, 8, 0, -14, 10, 8], [-9, 7, -6, 3, -5, -4], [7, -5, 2, 15, -15, -6], [-6, -8, 0, 4, 0, -6], [-2, -17, 4, -7, 1, 3], [26, 4, 6, -8, 2, 2], [-2, -8, -6, -16, 4, -2], [-32, -10, -8, -6, 0, 6], [4, -2, -2, -8, 12, -6], [7, -2, 4, -1, 3, -13], [-27, 5, 2, -9, 11, 14], [-11, -2, 3, 3, -22, -2], [13, 9, -7, 0, 5, -9], [-6, -4, 5, -17, 26, -5], [19, 2, 3, -13, -2, -8], [-14, 6, -2, 4, 6, -4], [-5, -1, 1, 4, -7, -5], [21, 1, 2, -2, -14, -2], [-16, -6, -6, -6, 2, 7], [8, 7, 13, -4, 9, -10], [2, 8, -3, 2, -9, 11], [-25, -1, -1, -16, 9, 5], [-2, 5, 0, 11, -1, 15], [13, -12, 4, 3, 5, 1], [2, 2, -7, 4, -1, 3], [-3, 9, 6, 5, 1, -6], [-6, 6, -2, -8, 0, 10], [5, 9, 15, 8, 9, 5], [14, -15, -7, -2, 7, -16], [-9, 2, 0, -21, -1, 7], [-3, 7, 0, -11, 3, 16], [12, -9, 4, 3, 7, 5], [22, -30, 0, 4, -10, 1], [1, 3, 9, -6, 16, 10], [-1, -1, -3, -8, 5, -17], [2, 6, -7, 4, -11, 1], [39, 15, 3, 0, 1, 0], [12, -12, 8, -12, 20, -2], [-3, -25, -3, -4, 10, -10], [-12, 14, 4, -6, -2, 10], [2, -2, 3, 6, 11, 19], [8, 2, 6, -2, 22, -2], [13, -2, -1, -13, -1, 1], [-24, -7, 2, 1, 3, 5], [-1, 4, -7, 11, -4, 2], [3, 7, -8, 5, 3, 6], [2, 8, -2, 0, 6, 18], [-12, 5, -6, 15, 4, 6], [-5, 6, -5, 9, -12, -4], [7, 2, -6, 7, 3, 9], [28, 1, 6, 3, 11, -9], [-8, 13, 2, 3, 3, 13], [-20, 16, 16, -2, 19, 7], [27, -11, -17, -4, -13, -1], [12, -10, 8, -6, 16, 6], [-22, 10, -4, -2, 6, 0], [-7, 10, 7, 17, 8, 13], [-1, 1, -8, -5, 11, -4], [1, -7, -5, -10, -6, 6], [10, 9, 3, 19, 0, 4], [13, 14, 3, -7, 4, -7], [-4, 8, 0, 0, 6, 14], [-6, 5, 6, 5, 9, 23], [26, -16, 0, 0, 4, 4], [22, -23, -3, 6, 1, -4], [26, 2, 0, -12, -2, 4], [15, -10, -5, -1, 4, 4], [3, 13, 13, 12, 9, -1], [3, 10, 3, -19, 2, -10], [-6, 13, -8, 9, 5, 11], [-29, 2, -7, -13, 8, -2], [20, -5, -2, -13, -1, -1], [32, -12, 4, 10, 7, 11], [17, 5, 2, 6, 6, 2], [0, 2, -2, 12, 2, 10], [-14, 10, 6, -14, 12, 0], [-24, -6, -10, -12, 0, -14], [13, 8, 7, -1, -12, -14], [12, -1, -1, 6, -3, 16], [2, -8, -2, -2, -8, 4], [11, 15, -2, -1, -1, -8], [-10, -27, 4, -11, -3, -1], [11, 9, 7, -12, 15, -1], [-5, -3, -1, 12, 10, 14], [-45, 3, 5, 6, -13, 9], [24, -7, 5, 14, 3, -4], [-9, 3, -11, 0, 9, 13], [-10, -10, -7, -11, -8, -7], [-8, 7, 8, -11, 10, 16], [0, -1, -5, 13, -12, 8], [-3, 17, -1, -2, -13, -3], [-10, -12, -14, -20, 6, -4], [-1, 15, 9, -6, 13, 0], [-16, 14, 4, 8, -13, 7], [-30, 24, 4, -16, 6, -10], [-8, 9, -2, 1, -17, -7], [30, 4, -10, 2, 0, 10], [18, -9, 2, -23, -3, -7], [5, -14, -21, -15, -6, -1], [-2, -18, 6, 12, -12, 10], [15, -19, -13, -2, -16, 6], [0, -4, -1, 4, 5, -11], [-23, 23, 5, -22, 11, 9], [-13, -25, 0, -7, 7, -2], [15, -4, -3, 13, 0, -16], [-37, 26, 7, 9, 17, 15], [-10, 8, -2, 0, -10, 2], [-6, 2, 4, 8, 8, -4], [18, 11, 1, 3, -8, 0], [-13, 22, 7, 3, 2, 14], [-2, 15, 0, -7, 15, 3], [2, -19, 8, -11, 13, 11], [4, 10, 8, 6, -8, -4], [-2, 12, -4, 10, -4, -6], [44, 4, 1, 12, 5, -7], [-26, -15, -2, -5, 3, 5], [-14, 5, 8, -9, 20, 18], [63, -8, -13, -1, -2, -10], [-19, -14, -2, 13, -7, 13], [20, -2, -8, -2, 2, -10], [-6, 7, 8, 13, 9, 19], [-12, -1, -4, -13, 14, 4], [13, -6, 1, 7, -20, -19], [-14, 22, 8, -12, -4, -12], [9, 6, 3, 3, -4, 0], [-26, -16, -2, 10, -2, -6], [-20, 18, 12, -6, 6, -8], [21, -21, 5, 24, -13, -2], [16, -10, -2, -8, 6, -3], [0, -6, 2, 26, -16, -2], [37, 16, 5, -1, 20, -14], [10, -5, 4, 23, -17, 5], [3, 2, 5, -15, 14, 2], [7, -14, 6, 12, 6, 1], [-23, 0, -9, -3, -12, -21], [0, 5, 5, -1, 14, -2], [0, -11, -2, -9, 9, -11], [-9, -11, -11, -6, -9, 3], [-12, 5, 10, -9, 5, 9], [-6, -11, -6, 7, -11, 21], [-3, 14, -3, 7, -13, -11], [10, -6, -5, 4, -19, -11], [-26, 20, 4, 2, 20, 16], [2, -16, 2, -12, 10, -20], [19, -25, 5, -6, 0, -2], [-12, -8, 0, 6, -14, 12], [-21, 9, 5, -10, 6, 8], [11, -2, 2, -14, 26, 1], [16, -2, -16, 0, 4, 0], [-26, 16, -10, -8, -10, -10], [2, -4, -1, -16, 17, 5], [-37, 20, 3, -9, 22, 2], [-63, 4, -3, -3, -4, -3], [-22, 14, 2, 0, 8, 12], [-5, 10, -7, 13, -26, -2], [-12, -5, -4, -7, -19, -25], [8, -12, 12, -4, 14, 6], [-4, 18, 4, -2, 14, 10], [-51, 6, 7, -13, 18, 6], [25, -29, -14, 7, -17, 12], [-11, -10, 9, 15, -2, 6], [20, 7, -6, 11, -11, 11], [22, 8, -4, 8, -10, -16], [13, -2, -5, -17, 1, -21], [-20, -12, -8, -6, 0, 11], [4, 10, 4, -12, -2, -2], [-16, -15, 0, -7, -7, 1], [21, 16, -7, -11, 12, 0], [25, -22, -5, -17, -6, 0], [7, -18, -1, -13, 2, -20], [2, -10, 4, -16, 0, -8], [43, -15, 1, 4, -1, 9], [2, -2, 8, 14, -7, 17], [3, 0, 3, 5, -30, -10], [36, -5, -4, 17, -3, -15], [-20, -17, -15, -8, -15, 4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3344_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3344_2_a_z();". // 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_3344_2_a_z(:prec:=6) chi := MakeCharacter_3344_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_3344_2_a_z();". // 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_3344_2_a_z( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3344_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, -36, 2, 24, -4, -4, 1]>,<5,R![-2, -11, 41, -23, -13, 3, 1]>,<7,R![58, -101, 17, 39, -13, -3, 1]>],Snew); return Vf; end function;