// Make newform 6040.2.a.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6040_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_6040_2_a_l();". // 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_6040_2_a_l();". // 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, 3, 10, -27, -17, 32, 9, -11, -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], [0, 1, 0, 0, 0, 0, 0, 0, 0], [31, -89, -181, 141, 164, -47, -41, 4, 3], [-23, 22, -106, -100, 192, 68, -74, -8, 7], [-3, -10, 27, 17, -32, -9, 11, 1, -1], [70, -24, -532, -67, 580, 70, -184, -9, 16], [-59, -34, 555, 242, -734, -189, 268, 23, -25], [-71, -18, 771, 122, -917, -123, 304, 16, -27], [160, 38, -1606, -389, 1894, 329, -636, -41, 57]]; Rf_basisdens := [1, 1, 13, 13, 1, 13, 13, 13, 13]; 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_6040_a();" function MakeCharacter_6040_a() N := 6040; order := 1; char_gens := [1511, 3021, 2417, 761]; 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_6040_a_Hecke(Kf) return MakeCharacter_6040_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, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0, 0, 0, 0], [-1, -1, 0, -1, 0, 0, 0, 1, 0], [-1, -1, 1, 1, -1, 2, 0, -1, -1], [-1, 0, 0, -1, 0, 0, -1, 0, 0], [-1, 0, 0, 1, 0, 1, 0, 0, 0], [-1, -1, 0, 0, -2, -1, 1, 2, 2], [-1, 2, -1, 0, 0, 0, 1, 0, 1], [1, 0, 0, 1, -1, 0, -1, 0, -1], [-1, 0, 0, -1, 1, -2, 0, 4, 1], [-1, 1, -1, 0, 3, -2, 1, 1, 2], [-1, 2, 0, 0, -3, -1, 0, 2, 1], [4, 2, -3, 0, 2, -2, 2, -1, -2], [-4, -2, -2, 0, 1, -1, -1, 0, 1], [2, -3, -1, 1, -1, 0, 0, 0, -1], [-2, 3, 1, 1, 0, 0, 1, -1, 1], [0, -3, -1, 1, -1, 3, -1, -3, -3], [-4, -2, 2, -1, -1, 1, -2, 3, 2], [-3, -1, -1, 2, 1, 3, 0, -3, -3], [1, -1, 5, 1, -1, 4, -1, -1, -2], [-1, 0, 3, -1, -2, 1, -1, -2, -2], [-1, -1, 0, 0, -4, 0, 1, -1, -2], [-4, -1, -1, 1, 0, 1, -3, -3, -3], [-3, 1, 2, -3, 1, -3, 1, 4, 2], [2, -2, -3, 1, 1, -3, 3, 1, 1], [-2, 2, -4, 0, 3, -5, 1, 3, 3], [-2, 1, -1, 1, 1, 0, -2, -3, -2], [-2, -3, -3, 0, 0, 1, 3, 1, -1], [0, 3, 4, 0, 0, 0, -1, 2, -1], [-2, 4, 1, 1, -1, 1, -4, -4, 0], [-5, -1, 3, -3, -3, -2, 0, 4, -1], [-2, -3, 2, -1, 2, 2, -1, 0, -3], [-3, 0, -2, -2, 0, -4, 2, -1, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 1, -3, -2, 2, -1, 3, 4, 3], [5, 3, 3, 1, 6, -2, 0, -4, -1], [-2, -1, 5, -3, -2, 1, -6, 6, 2], [0, 2, 4, -1, -1, -4, -1, 2, -3], [2, 3, 6, 0, -2, 4, -1, 0, 3], [-3, -3, -1, 2, 0, 2, -5, -6, -2], [2, -5, -4, 2, 2, 1, 1, -8, -3], [-4, 1, 9, 0, -2, 1, -5, 3, 3], [0, -7, -3, 1, -3, -1, 2, 5, 2], [-7, -2, -4, 0, 2, 4, 1, -2, 2], [4, 6, -7, 2, 7, -2, 4, -4, 0], [-3, -5, 0, -1, -4, -4, -2, 0, -1], [4, 3, -9, 2, 3, -3, 7, 4, 3], [-4, 4, -1, 1, 5, -5, -3, -4, -2], [4, 0, 1, -1, 4, -2, 2, 2, 6], [-7, 5, -4, -1, 1, 0, -2, -8, -6], [-5, 1, -1, 1, -2, 0, 5, 3, -3], [-3, -1, 0, -5, 3, -5, 1, 5, 3], [-10, -2, 1, -1, -4, 2, 3, 3, 3], [2, 7, 4, -4, 6, -3, -3, -3, 1], [5, -1, -4, 1, 0, -2, 1, -8, -1], [-2, -8, -1, -2, -5, 3, 1, -1, 1], [5, 9, 5, -2, 3, -3, 0, -2, 3], [-10, -2, 3, -2, -5, 2, -4, -2, 0], [5, 1, 4, -1, 5, -1, -4, -2, 1], [-5, -2, -3, 0, -4, 2, -2, -4, -5], [0, 6, 7, -1, -3, 5, 2, 6, 0], [-8, 0, 2, -6, 3, -2, -1, 3, 0], [-2, 0, -3, 2, 3, -1, -3, -7, 1], [-4, 0, -3, 3, 0, 0, 6, 4, 4], [-3, 2, -4, -1, 3, 1, -3, -5, -4], [-4, -1, 1, 6, 2, 3, -3, -4, -3], [4, 2, 2, 1, 7, -3, -2, 3, 4], [-2, 9, -1, 3, 0, -3, 1, -4, 2], [-2, -1, 1, -1, -1, 1, 4, 2, 4], [8, -3, -6, 4, 2, -4, 5, 0, 1], [-3, 1, -1, 3, -3, 5, 0, -10, -1], [-10, 7, 3, -3, 2, 4, -5, -2, 3], [3, -9, 2, -3, 3, 2, 0, 1, 1], [-3, 0, 7, 1, -3, 9, -9, -8, -2], [12, 0, 8, 3, -10, 1, 1, -1, -3], [2, -5, 1, -3, -1, 0, -3, -6, -5], [-5, 3, -3, 3, 0, -5, -2, 1, 5], [-3, 4, -4, -5, 0, -4, 7, 6, 4], [-2, 3, -4, -1, -1, -12, 2, 10, 2], [-5, 4, 4, -2, 2, 4, -1, -2, -6], [-5, -3, 2, 0, 2, -4, 4, 0, 1], [0, 3, 2, -1, -6, 3, 5, 2, 5], [-5, 7, 4, -3, -4, 0, 4, 3, 10], [1, -3, -2, -3, -2, 2, 5, 1, -2], [0, -5, 0, 2, 1, 10, -7, -17, -5], [-8, -6, -1, 0, 0, 3, 3, 0, -1], [-6, -4, 1, -7, 1, -3, 4, 6, 6], [4, 4, -6, 1, 4, -3, 6, -6, 6], [9, 1, 1, 0, -2, -1, 3, 2, 4], [3, -3, 2, 3, -4, 0, 8, 6, 3], [-1, -5, -3, 5, -5, 9, -2, -8, -3], [4, 3, 0, 8, -4, 4, 5, -4, -3], [6, -1, 4, -4, 6, -7, -1, 8, -1], [-2, -1, 3, 3, -7, 9, -1, -16, -3], [-6, -8, 0, 0, 1, -3, -7, -4, -1], [-4, 4, 2, -1, 2, -7, -3, 8, 4], [2, 3, -11, 2, 5, -11, 12, 8, 10], [0, -8, 6, 0, -6, 9, -1, -6, -6], [8, -6, 6, 6, -6, 11, -9, -11, -7], [-8, -9, 3, 0, -2, 2, -5, 2, 5], [-10, 0, 7, -5, 3, 8, 3, -1, 3], [-12, 4, -2, 2, -3, 6, 1, -2, 1], [0, 6, -2, -2, 3, -5, 7, 1, 0], [0, -7, -6, 3, -6, -2, -4, 3, -4], [11, 6, -6, 6, 3, -3, 0, -6, -5], [1, 2, -10, 4, 4, -6, 5, 3, 0], [-3, -3, -3, -4, -6, 5, -7, -8, -6], [-14, 1, 6, -6, 3, 0, -7, 6, 6], [2, -8, 2, 0, -11, 5, 6, 5, 2], [3, -3, -11, -2, 1, -4, 7, 9, 6], [-2, 1, 2, 0, 6, 4, 3, -3, -9], [-10, 1, 1, 3, -15, 4, -2, -5, -10], [-11, 5, 5, -5, -8, 3, 0, 3, -1], [-8, -11, -10, 4, 1, 0, -1, -4, -6], [0, -4, 5, -7, -1, 7, -6, -4, -2], [11, 12, -6, 1, 12, -12, 1, 2, 14], [-8, -13, 6, 0, 1, 7, -9, -9, -3], [7, 7, -7, 3, 3, -5, 1, 0, 11], [5, -15, 10, -1, -4, 4, -5, 11, 4], [-8, 7, -1, -5, 2, -4, -2, 14, 4], [2, 14, 4, 4, 9, 10, 3, -12, -3], [-8, 1, -6, 7, 6, 4, -3, -13, -5], [-10, -2, -2, 2, 0, 1, -2, -2, -2], [-3, 12, -7, 2, 5, -8, 12, 13, 5], [14, -1, -4, -2, 4, -10, 6, 12, 1], [1, 2, 5, 1, 4, -2, 3, -1, 4], [8, 9, 1, -7, 7, -5, -5, -4, 5], [4, 6, -4, -5, 10, -6, 2, 9, 14], [5, -4, -17, 7, 7, 0, 3, -6, -4], [17, 9, 1, 4, 6, 7, 7, -6, 2], [-4, -3, 10, -5, -8, -6, -8, 11, -4], [-2, -2, 8, 3, -13, 10, -9, -8, -7], [-11, 2, 1, -4, 1, 3, 1, 4, -2], [-1, 8, 13, 0, 2, 6, -5, -7, -8], [9, -3, 2, 1, 10, 6, -6, -8, -13], [13, -6, -12, -2, 3, 0, 2, -2, -5], [-7, 1, -1, 5, -12, 2, 6, 5, 7], [-15, -4, 7, -2, -4, 3, -4, 0, 1], [-2, -8, 2, 0, 3, -3, -5, -4, -1], [-7, 10, 4, 0, 2, 1, 6, 7, 9], [13, 1, -4, 3, 0, 3, -5, -13, -6], [9, 2, 7, -4, -7, -3, 5, 12, 4], [-8, -4, -4, 2, -7, 9, -2, -4, -1], [2, 0, 1, -6, 6, -5, -5, -2, -3], [1, 2, -10, -3, 10, 0, 4, 3, -7], [4, 7, -1, 7, 4, 11, 1, -8, -5], [2, 0, 6, -11, -1, -9, -2, 19, 12], [-3, -2, -4, 0, 3, -4, 5, -9, -12], [3, -6, -7, 6, -13, 8, 3, 3, -2], [0, 5, -2, -5, -12, -8, 3, 0, 1], [19, -10, 6, 6, -8, 12, -6, -5, -11], [10, -2, 0, -8, 4, -4, 7, 6, -1], [5, 4, 4, -12, 2, -7, -3, 20, 16], [-10, -2, -1, 5, -1, 1, -9, -2, -2], [5, 7, -3, 0, 15, 1, 1, -7, 2], [7, -5, -15, 1, 9, -3, 2, -6, -5], [-5, -8, -2, -3, 14, -3, -1, -6, -2], [-2, -13, 9, 2, -13, 8, -10, 13, -6], [-9, 5, 10, 1, -3, 0, -6, 14, 9], [-11, 3, 22, -2, -14, 5, -2, 14, 9], [24, -11, -14, -3, 9, -5, 9, 8, 3], [-19, 8, 10, 3, -12, 5, 0, -1, 7], [5, 2, -7, 8, 13, -2, 10, -5, 3], [-2, -11, -1, 1, -6, 3, 4, 6, 0], [2, -10, 5, 1, -4, 11, -8, -16, -7], [21, -3, -3, 3, 10, 1, -1, 4, -8], [-11, -12, -3, -3, 4, 13, -5, -3, 4], [-3, 8, 1, -8, 6, -8, -8, 11, -5], [-1, 2, 5, 7, -13, 6, -5, 4, -2], [-5, -9, -1, 2, -14, -2, 6, 11, 3], [-11, 9, -11, 0, -1, -6, 8, 15, 2], [6, 3, -16, 1, 15, -2, -1, -17, -9], [-1, -3, -7, 6, -3, 6, 1, -7, -12], [-4, 4, 6, 2, -9, -5, -1, -2, 9], [-6, -6, 13, -9, -5, 3, 0, 13, 11], [-1, -4, 6, 1, 0, 17, 0, -12, -8], [3, 15, 2, 4, 1, 13, 6, -12, -5], [13, 11, -5, -3, 17, -2, -1, -17, -2], [11, -20, -2, 3, -7, 8, -2, 1, -5], [-15, -5, -1, -1, 0, -2, 2, -10, -2], [19, 8, 5, 2, 7, 1, -7, 6, -4], [11, -3, 7, 0, -12, -10, 3, 20, 14], [-5, -8, -4, 1, 2, 11, -8, -6, -1], [11, 11, -7, 4, 4, -4, 11, 4, 8], [18, 1, 10, 0, -5, 2, -3, -3, -9], [13, -18, 4, 2, -12, 7, 5, 16, 0], [0, -6, -10, 9, -6, 8, 5, -12, -16], [-13, 3, 7, 3, -9, 3, 7, 12, -1], [4, -5, 8, 1, 2, -4, -1, 1, -5], [-9, -4, -4, -13, -2, -8, 0, 11, 7], [-1, 5, 11, -7, 2, -1, -5, 6, 4], [-10, -11, -2, -2, -11, 3, -5, 11, 3], [4, -10, 5, -9, -9, 9, 1, 0, -7], [19, 2, -3, 1, 0, 3, 8, 8, 3], [0, 18, -7, 11, 5, 1, 2, -13, 3], [9, 6, -5, -4, 1, 1, 1, 0, -8], [4, 16, -4, -4, -1, -7, -1, -2, -3], [3, -5, 6, -1, -4, 10, 5, -1, 12], [8, -9, 9, -2, -2, 2, -1, 2, -4], [7, 5, -13, 3, 7, -13, 12, 2, 1], [12, 4, 2, -12, 8, -11, -12, 6, 12], [-2, 2, 2, -1, 2, 1, 1, 7, -7], [4, -15, -16, 0, -1, -14, 7, 7, 3], [1, 6, -12, -9, 8, -10, -6, 7, 3], [13, -10, 0, 8, 3, 1, 8, 16, -1], [-5, -18, 2, 5, -2, 11, -4, -14, -9], [-22, -16, 2, -5, -9, 5, -3, 9, -4], [-6, -1, -12, -2, 5, 9, 4, -18, -6], [-9, 1, 13, 6, -14, 4, -12, -2, -1], [6, 10, -6, -6, 5, -5, -8, 7, -2], [19, 1, 19, 0, -1, -5, -16, 0, 7], [10, -13, -7, -6, 2, -9, 0, 12, 2], [-3, 8, 0, 2, 6, 13, 4, 1, 5], [-8, -18, 4, 3, -1, 9, -7, 4, -3], [8, 1, 17, -7, -6, -9, -3, 14, 5], [-4, -6, -5, -11, -2, -17, -1, 15, 2], [-1, 9, -7, 3, -14, 7, 3, -6, -2], [6, 1, -20, -1, 9, -9, 3, -16, -11], [13, -6, -17, 4, 6, 1, 10, -1, 4], [-3, 0, 20, 0, -2, 17, -26, -13, -5], [4, 5, -8, 2, 6, 9, 0, -12, -1], [-9, 7, 4, -7, 0, 0, 4, -2, -1], [23, 14, 1, -11, 12, -19, 0, 10, 3], [7, -10, 10, -10, -7, 2, -6, 20, 11], [14, -3, -4, 6, -3, -6, 3, -5, -5], [-1, -5, 5, 9, 5, 1, 5, 7, 4], [4, -14, -8, 3, 0, 8, -1, -15, -1], [13, 3, -11, 6, 4, -3, 11, -4, 0], [9, 15, 2, -6, -3, -11, 5, -8, 8], [-2, 17, 6, -2, 11, -2, -9, -10, -6], [6, 10, -15, 1, 26, -6, -6, -23, -6], [-2, 12, 16, 3, 3, -5, -1, -4, -1], [24, 21, 1, -1, 20, -6, 10, -14, 0], [5, 1, 9, -1, -1, -4, -7, -1, -6], [-10, -16, 9, -1, -13, 1, -2, 12, 4], [2, 2, -7, -1, -5, -7, -4, 7, 4], [-14, -5, -6, -5, 2, 4, 10, 1, -4], [-5, 20, -3, 2, 2, -5, 5, 0, 11], [7, 10, 8, 1, 9, -16, 5, 14, 2], [-9, -3, -8, 5, -6, 3, -5, -18, -9], [12, -5, 13, 10, -10, 20, -2, -23, -6], [24, 8, 5, 0, -1, -2, 1, -10, -2], [-7, -3, 21, 4, -8, 23, -8, -25, -11], [5, -5, -12, -4, 13, -17, -3, 1, -5], [5, 2, 13, 4, -12, 2, -3, 10, 2], [1, 8, 9, 1, 3, -11, 0, 7, 5], [9, -8, 9, 2, -6, 7, -12, -5, -8], [-17, 9, -11, -3, -9, -7, -3, -2, -4], [27, 15, -2, 7, 12, 1, 10, -13, 5], [4, 2, 0, 5, 7, -2, 10, 8, 3], [4, -3, -17, 3, 8, 4, 9, -13, -9], [-3, 12, 4, 9, -3, 9, 3, -2, 5], [-23, -10, 4, -12, -10, -5, 0, 9, 11], [-6, -1, -6, 11, -5, 18, 5, -19, -9], [1, -7, -6, 4, -1, -3, 4, -7, -3], [17, -4, -14, 2, 13, -4, -4, -7, -11], [2, 9, 10, -6, 0, -6, -1, 10, 1], [-33, -12, 10, 1, -22, 19, 0, -5, -3], [-10, 7, -15, -5, 8, -8, 9, -1, 15], [15, -4, 3, 7, 4, 0, 2, 0, 3], [3, -6, -1, 7, 0, 10, -13, -12, -5], [3, 14, -7, -4, -2, -14, 1, -1, 0], [0, -4, 1, 2, 12, -3, 3, -2, -5], [-12, -10, 11, 1, -8, 5, -5, 12, 3], [16, 11, -1, -3, -2, -6, 0, -4, -2], [-1, -1, -9, 4, 9, 5, 12, -16, -7], [-4, -6, 0, 0, 0, -6, -10, 2, 6], [-17, 0, 20, 6, -17, 14, -6, 2, -3], [21, -8, 1, 12, -14, 8, 1, -5, -4], [15, 3, -9, 6, -2, 4, 0, 11, 3], [15, -16, 13, 3, -10, 11, -4, 12, 3], [5, -11, -18, -4, 3, -14, -2, 0, -4], [10, -1, 4, 11, -5, 10, 14, -16, 6], [0, -1, 18, 7, -3, 17, -17, -14, -19], [-7, 8, -4, 4, 10, -1, -4, -28, -5], [7, -1, -10, 3, 1, 0, 16, 3, -3], [-4, 3, -1, -3, 10, -10, 8, 20, 14], [10, 8, 4, -6, 7, 4, 6, -3, 6], [-28, 9, -1, 2, -9, 3, 1, -22, -5], [16, -2, 6, 2, 6, 8, -4, -2, -8], [-12, 6, 3, 2, 0, 21, -6, -24, -12], [1, 17, 5, 10, 12, 17, -1, -20, -8], [8, 10, 11, -9, -5, -15, 4, 10, 6], [-12, 13, 3, 2, -5, -12, -2, 7, -2], [10, -17, 2, -1, -2, 13, 2, -14, -16], [-3, -5, 0, 9, 11, 13, 1, -21, -10], [-9, 1, -7, -13, -4, -2, 11, 10, 6], [9, 5, 20, 7, 0, 18, -10, -12, -9], [8, -11, -5, 3, 11, 6, 7, -30, -13], [29, 3, -3, 10, -1, 2, 7, 1, -6], [-2, -16, -18, 5, -10, -3, 18, 5, -6], [-18, -2, -8, -1, 14, -11, -7, -7, -7], [-13, 22, -11, 10, 11, -6, 1, -10, 4], [-7, 7, -10, -3, 7, -12, 0, 10, 22], [-4, 5, -13, -9, 20, -5, -5, 15, 11], [-19, -21, 12, 1, -9, 23, -11, -14, -8], [-6, -20, -5, 5, 13, 19, -10, -11, -25], [-8, -1, -14, 1, 12, 13, 5, -6, -7], [2, -5, 20, -6, -11, 7, -15, 12, -1], [8, 11, 18, 8, -6, 24, -11, -26, -13], [-13, 2, -4, -6, -1, -11, 2, 13, 18], [-5, 14, -5, -1, -2, -20, 1, 5, -8], [13, 21, 10, 3, 12, -11, 2, -8, 7], [-15, 15, -6, 11, 12, -9, 10, 5, 7], [-4, 1, 3, -16, 2, -6, 0, 16, 6], [24, 10, -23, 13, 16, -17, 7, -4, 1], [0, 18, 5, -2, 12, 1, 9, -9, 2], [24, -13, 2, 10, 8, 10, -8, -17, -20], [23, -10, 3, -15, 2, -6, 1, -1, -4], [28, 6, -3, -11, 3, -4, 7, 15, 3], [-23, 1, 19, -4, -7, 13, -3, -3, 4], [2, 18, -21, -2, 4, -27, 12, 11, 12], [0, 14, -12, -2, 13, -31, 13, 28, 21], [13, -6, -7, 2, 8, -8, -2, -2, 9], [-7, 0, -7, -12, -2, -12, 0, 15, 14], [-36, 2, -9, -10, 20, -2, -1, 11, 4], [1, -1, -11, -1, 5, -2, 7, 6, 20], [1, 13, -6, -14, 17, -24, 10, 23, 21], [7, -10, 20, -13, -5, -6, -17, 19, 14], [-12, 7, 13, -14, -13, -11, 1, 25, 9], [15, -5, -11, 4, -2, -9, 10, 15, 21], [20, -1, 7, -2, 3, 3, 7, 2, 5], [-26, -17, -4, -6, 1, 2, -10, 12, 8], [-19, -17, 4, 11, -24, 20, 9, -17, -2], [-19, 14, 1, 5, -16, -1, 3, 1, -2], [-15, 6, 0, 5, -14, -2, -8, -5, 5], [-24, -8, -13, -1, 14, 0, 7, -1, 3], [10, -16, -13, 12, -11, 17, -3, -27, -15], [2, -21, 1, -10, -7, -4, -13, 14, -3], [-26, -2, -4, -5, -11, -3, 6, 3, 4], [-26, 6, -8, 15, 2, -1, -8, -7, -7], [-11, -9, 5, -6, -14, -3, -9, 25, 8], [-6, -3, -1, 7, 2, 3, -8, 0, -10], [-29, 15, 21, -10, -3, 16, -7, -3, 12], [-5, -11, 15, 7, -10, 17, -13, 1, 3], [0, 4, 12, 1, 8, -1, -6, 4, 8], [-32, 9, 11, 3, -15, 22, -6, -20, -1], [16, 12, 2, -3, -1, -30, 9, 8, 3], [19, -13, 14, -6, -4, 1, -3, 2, 2], [-8, 4, 8, 6, -2, -15, -7, 21, 27], [-7, 17, 3, -1, -2, -4, 3, 17, 23], [16, 6, -7, 4, 2, 2, 17, 4, -9], [2, -4, -8, 12, -8, 2, 7, 3, 11], [-26, -17, 12, -2, -22, 0, -1, -10, 1], [11, -5, -6, 0, -12, -2, -5, 18, 3], [24, 24, 0, 3, 8, -21, 7, -2, 11], [-24, 7, -10, -17, 23, -12, 3, 17, 9], [6, 2, -14, 4, 12, -11, 17, 10, 9], [13, 3, -18, -4, 2, -23, 14, 24, 21], [-33, 12, -7, -2, 10, 11, 1, -22, 0], [-12, -3, 19, -5, -17, 8, -9, -3, -2], [-5, -5, 6, -4, 5, 13, -19, -10, -4], [3, -7, 10, 2, 6, 16, -17, -18, -27], [18, 6, 1, 4, 5, -15, 12, -2, -2], [11, -12, 0, -10, -2, -21, -12, 7, -7], [-34, 14, 7, -4, 4, -1, -9, -6, -15], [-20, 23, -17, 7, 17, -4, -2, -15, 2], [-9, -6, -6, 5, 3, 6, 13, 14, -8], [-3, -2, -9, -8, -5, -20, -4, 13, 3], [-10, 12, 17, 10, -2, 3, -9, 0, -5], [7, -3, 0, 10, -11, 18, -10, -23, -17], [-31, 8, 27, 5, -17, 12, -11, -7, -7], [25, 6, 10, 5, -18, -9, 4, 5, -1], [10, -8, -12, -2, 8, 0, 9, -6, -11], [-31, -12, 10, 0, -31, 0, -7, 18, 12], [8, -35, -23, 4, -11, -13, 7, 13, 1], [-21, 4, -5, -17, 13, 0, 1, 15, 1], [-38, -5, -5, -4, -5, 13, 6, 1, 2], [-15, -4, 7, -11, 12, 1, 9, 8, -5], [-21, -19, 21, -15, -27, -8, -5, 21, 10], [11, 10, 12, -3, 9, -4, -14, 14, -7], [10, -8, 19, -5, -6, -6, -12, 17, 4], [7, -10, 3, -16, 6, 11, 2, -5, 7], [-16, -3, 3, 1, -17, -1, -8, 33, 12], [-17, -5, 6, -7, -20, 4, -3, 25, 2], [20, 5, -26, 6, 12, -5, 2, -11, -6], [12, -6, -13, 1, -9, -7, 22, 5, -1], [-9, 2, -9, 3, 25, 1, 0, 13, 5], [-7, -21, -6, -8, -2, -3, 1, 23, 0], [6, -4, 6, 16, -15, 14, -14, -20, -4], [27, 13, 7, 9, 3, 4, -1, -19, -4], [-5, -11, 7, 5, 6, 15, 1, 12, -2], [0, 10, 5, 4, 16, 21, 3, -12, -13], [-10, -17, -5, 5, 2, -1, 8, 16, 4], [8, 22, 0, 1, 15, -17, -5, 6, 3], [4, -3, 7, -9, -12, -1, 1, 17, -5], [40, 13, 2, 7, 18, -18, 5, -4, -5], [-21, 5, 11, 2, -23, -1, 6, 25, 17], [-16, -15, 7, 6, 12, 22, -9, -15, -17], [6, 0, -11, -6, 10, -7, 12, -6, 4], [-22, 17, -20, 13, 11, 8, 6, -20, -6], [-14, 5, 2, -5, -4, -7, -1, 4, 1], [-20, -1, -5, 4, 15, 5, -6, -12, -11], [8, 10, 0, 8, -4, 19, -7, -29, 1], [-11, 2, 14, 11, 6, 26, 8, -20, -10], [-4, -16, 1, 13, -2, 8, -1, -7, 2], [-4, 3, -8, 10, -2, 7, 12, -1, 22], [-28, 3, -11, -7, 5, -9, -10, 7, 0], [17, 10, -1, -16, 19, -25, -5, 39, 21], [6, 4, -13, 5, -5, -16, 21, 14, -1], [-21, 12, 16, -7, -26, -1, -7, 22, 2], [12, 5, -14, 12, 2, -13, -2, -17, -8], [28, -17, 16, 13, -15, 27, -15, -19, -19], [-6, -6, 5, 0, -6, -5, -15, 20, 5], [16, 6, 13, 20, -5, 10, 8, -20, -6], [-24, -3, 22, -4, 5, 11, -16, 4, -2], [-17, -13, 1, 15, -13, 15, -5, -3, -14], [5, -8, -2, 16, -10, 4, -4, 3, -9], [-21, -14, 22, 0, -8, 16, -7, 19, 6], [2, 23, -11, 10, 10, 4, 13, -12, -13], [-4, 8, -14, 26, 11, 12, 16, -27, -8], [7, 21, 13, -13, -2, -5, 11, 34, 22], [-19, 23, 15, 7, -27, 3, -12, 2, 3], [25, 9, -10, 7, 16, -19, 7, 11, 14], [-22, -8, 5, 2, -14, -1, 1, -3, -12], [21, 11, 0, 8, -11, 15, -2, -3, 5], [-21, -6, 4, -11, -2, 12, -21, -9, -6], [0, 48, 3, -2, 14, -16, 2, -5, 8], [-8, -24, 17, -6, -1, 19, 3, -6, -4], [-5, -15, -23, 7, -12, 4, 9, 11, 8], [-7, -15, -5, -7, -6, 1, -12, 3, 15], [-25, -24, -17, 7, 3, -11, 11, 2, 4], [-9, -13, -16, -9, 22, -22, -3, 15, 12], [-16, 13, 21, -7, 8, -6, 7, 13, 21], [-19, -8, 5, 0, 5, 6, -5, -2, 4], [28, 23, -29, 9, 18, -27, 6, 7, 5], [-8, 2, -7, 12, 0, 33, -6, -31, -10], [2, -14, -8, 2, 11, 0, 3, 18, 1]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6040_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6040_2_a_l();". // 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_6040_2_a_l(:prec:=9) chi := MakeCharacter_6040_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_6040_2_a_l();". // 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_6040_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6040_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 16, -26, -55, 25, 46, -4, -12, 0, 1]>,<7,R![4, 4, -99, -110, 95, 72, -25, -15, 2, 1]>,<11,R![-1, 10, 116, 361, 429, 98, -116, -20, 6, 1]>],Snew); return Vf; end function;