// Make newform 1183.2.e.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1183_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1183_e_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_1183_2_e_f();". // 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_1183_2_e_f();". // 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, 16, 64, 28, 58, 18, 41, 7, 8, -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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [30518, 2376, 9416, 84549, 6908, 880, -5913, -220, 176, -364], [-22604, -98928, -392048, -39747, -287624, -36640, -87336, 9160, -7328, -983], [-348592, -132894, -526654, 25194, -386377, -49220, -109053, 12305, -9844, -1159], [-180842, -41094, -162854, -129531, -119477, -15220, -3978, 3805, -3044, 916], [-3988, -20316, -46906, 39066, -152578, 31195, -34617, 26720, -9541, 2249], [-2692, -957744, -258154, -871206, -272902, -637445, -107253, -121720, 14531, -15259], [532, 1362924, 255184, 981276, 92092, 719570, 96813, 145570, -21776, 18139], [-136, 50058, 11648, 53832, 2993, 39580, 3726, 9041, -1552, 1058]]; Rf_basisdens := [1, 1, 118350, 118350, 118350, 59175, 78900, 236700, 118350, 4734]; 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_1183_e();" function MakeCharacter_1183_e() N := 1183; order := 3; char_gens := [339, 1016]; v := [1, 3]; // 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_1183_e_Hecke();" function MakeCharacter_1183_e_Hecke(Kf) N := 1183; order := 3; char_gens := [339, 1016]; char_values := [[0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [1, 0, 0, 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, -1, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, -1, 0, -1, 0, 0, 1], [2, 0, 0, 1, 0, 0, -1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 2, 2, 0, 1, 1, 0, -1, -1, -1], [0, 2, 0, 0, 0, 0, -1, 2, -1, -1], [0, 0, 0, 0, 0, 0, 0, -2, 0, -1], [-1, 0, -2, 0, -1, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 1, 2, 0, 2, -2, -1], [0, 2, 0, 0, 0, 0, 2, 0, 0, 1], [-4, 0, -2, 2, -3, 1, 0, 0, 0, 0], [0, 0, 2, 2, -1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, -4], [-3, 4, 4, 2, 2, 0, -2, 3, 0, -2], [2, 0, 0, 1, 0, 0, -1, -2, 0, 0], [0, 0, 0, 0, 0, 0, -4, 3, 0, 2], [4, -2, -2, -3, -1, -3, 3, -4, 3, 1], [-2, 0, 4, -1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -1, -2, 0, 1, 0], [0, -2, 0, 0, 0, 0, 2, 0, -1, -2], [-2, 0, 2, 0, 1, 3, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -2, 0, -2, 3], [2, 0, 2, 2, -1, 1, 0, 0, 0, 0], [2, 0, 0, 0, -2, 3, 0, -2, -3, 2], [0, 8, 0, 0, 0, 0, 2, 0, -2, -3], [0, -2, 0, 0, 0, 0, 0, -12, 1, 2], [0, 6, 6, -2, 1, 0, 2, 0, 0, -1], [1, 0, 2, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, -2, -2, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -2, 4, 3, 2], [6, 4, 4, -2, 6, -1, 2, -6, 1, -6], [-4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, -2, -4, 1, 2], [-10, -2, -2, 3, -5, 1, -3, 10, -1, 5], [7, -2, -2, 2, -1, -1, -2, -7, 1, 1], [0, 2, 0, 0, 0, 0, 3, 4, -3, -1], [-6, 0, -8, -3, 1, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 1, 2, -2], [-6, 0, 0, 0, 3, -2, 0, 6, 2, -3], [3, 0, 6, -2, 1, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -2, -6, -1, 0], [4, 2, 2, 2, 1, -4, -2, -4, 4, -1], [0, 0, -2, 0, -5, 1, 0, 0, 0, 0], [-12, -4, -4, -2, 5, 0, 2, 12, 0, -5], [0, 0, -2, -6, -3, -1, 0, 0, 0, 0], [-2, 0, 10, -1, 8, -1, 0, 0, 0, 0], [-2, 4, 4, -4, 1, 2, 4, 2, -2, -1], [0, -2, 0, 0, 0, 0, 0, 8, 2, -7], [0, -4, 0, 0, 0, 0, 2, 1, 2, 4], [-8, 0, -2, 1, -4, -5, 0, 0, 0, 0], [-2, 6, 6, -4, 5, 0, 4, 2, 0, -5], [-14, 0, -2, 4, -1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, -6, 1, -4], [-8, 0, 0, 0, 5, 2, 0, 8, -2, -5], [7, -6, -6, 2, -3, -3, -2, -7, 3, 3], [0, -2, 0, 0, 0, 0, 1, -4, 1, 1], [5, 8, 8, -4, 2, -2, 4, -5, 2, -2], [-2, 0, -2, -4, 7, 3, 0, 0, 0, 0], [10, -6, -6, 0, 0, 3, 0, -10, -3, 0], [8, 0, -6, -4, 5, -9, 0, 0, 0, 0], [0, 0, 6, 5, -4, 3, 0, 0, 0, 0], [0, -4, -4, 4, -5, -4, -4, 0, 4, 5], [0, 4, 0, 0, 0, 0, -2, 0, -3, -4], [0, 0, 0, 0, 0, 0, 6, -8, 5, 2], [0, 0, 0, 0, 0, 0, -4, -6, -4, 9], [-7, 0, -2, -2, -3, 6, 0, 0, 0, 0], [-16, 4, 4, 0, 5, -2, 0, 16, 2, -5], [2, 0, 2, -2, 3, 5, 0, 0, 0, 0], [8, 2, 2, 4, -1, -4, -4, -8, 4, 1], [0, 4, 0, 0, 0, 0, 8, -2, -4, -7], [-14, 2, 2, -4, -6, -3, 4, 14, 3, 6], [0, 2, 0, 0, 0, 0, -6, 7, -5, 7], [0, 0, -10, 2, -11, -1, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 2, 0, -2, 3], [-5, -6, -6, 4, 7, 1, -4, 5, -1, -7], [0, 4, 0, 0, 0, 0, -6, 4, -5, 0], [0, 6, 0, 0, 0, 0, 0, 4, -4, -1], [-10, -4, -4, -6, 2, -1, 6, 10, 1, -2], [-6, 0, 0, -4, 4, -2, 0, 0, 0, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 0, 0, 2, -5, 0, -2, 12, 0, 5], [-7, 0, -4, 4, -2, 1, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, -2, 20, -3, 4], [0, 10, 0, 0, 0, 0, 2, -22, 1, -2], [8, 0, 4, -4, 6, 4, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 2, 9], [8, 0, 2, -2, 11, 3, 0, 0, 0, 0], [-2, 0, -2, 4, 3, -3, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 2, -8, 1, 0], [10, -8, -8, 7, -4, -6, -7, -10, 6, 4], [-12, -12, -12, -5, -2, -6, 5, 12, 6, 2], [10, 0, -4, -8, 6, -4, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, -6, -4, -3], [2, 0, -10, 2, 1, -7, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, -2, 16, 4, 1], [8, -4, -4, -10, 4, -3, 10, -8, 3, -4], [0, 8, 0, 0, 0, 0, 4, -14, 6, -3], [0, 4, 0, 0, 0, 0, 0, -12, 3, -2], [-6, 0, -14, 4, -7, -5, 0, 0, 0, 0], [-2, 8, 8, 4, 0, 7, -4, 2, -7, 0], [8, 10, 10, 4, -8, -3, -4, -8, 3, 8], [0, 8, 0, 0, 0, 0, 6, 17, -4, 2], [-2, -4, -4, -6, -1, -4, 6, 2, 4, 1], [2, -2, -2, 6, -17, 4, -6, -2, -4, 17], [10, 0, -8, 5, 1, -6, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, -12, 12, -1, -2], [-6, 6, 6, 2, -2, 11, -2, 6, -11, 2], [13, 0, -6, 8, 5, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 8, 14, -7, 0], [-8, 4, 4, -4, 6, 7, 4, 8, -7, -6], [-4, 0, -2, 4, 7, -7, 0, 0, 0, 0], [16, -12, -12, 6, -5, -4, -6, -16, 4, 5], [6, 0, 6, 4, -9, 1, 0, 0, 0, 0], [0, 0, 0, 6, -18, 7, -6, 0, -7, 18], [10, 0, 8, -7, -3, -6, 0, 0, 0, 0], [0, -2, -2, 10, 6, 1, -10, 0, -1, -6], [0, -4, 0, 0, 0, 0, -6, -4, -1, -10], [-4, 0, 16, 4, 4, 8, 0, 0, 0, 0], [-12, 2, 2, -2, -9, 2, 2, 12, -2, 9], [-12, 0, -8, 10, -10, -2, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, -8, -1, -4, 14], [28, 12, 12, 2, 3, 4, -2, -28, -4, -3], [0, 16, 0, 0, 0, 0, -5, -6, -4, 0], [2, 0, -4, -4, 10, 2, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, -2, -8, -7, -2], [0, -8, 0, 0, 0, 0, 0, 18, 4, -1], [6, 0, 14, 2, 5, 1, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, 2, 14, -4, -1], [10, 4, 4, -8, 1, 2, 8, -10, -2, -1], [-6, 0, 10, 11, -4, 11, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, -6, 0, -7], [11, 0, 0, -2, 2, -7, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 4, 4, 3, -6], [2, 0, 4, -4, -2, 6, 0, 0, 0, 0], [-4, 6, 6, 4, 3, 0, -4, 4, 0, -3], [26, 10, 10, 1, -3, -3, -1, -26, 3, 3], [8, 0, -8, -10, -6, -2, 0, 0, 0, 0], [3, 2, 2, -4, -3, -5, 4, -3, 5, 3], [26, 0, -6, 0, 5, -5, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, -2, -2, -7, 8], [12, 8, 8, -4, 1, -10, 4, -12, 10, -1], [-28, 0, 4, 1, -5, 2, 0, 0, 0, 0], [-1, 12, 12, -2, 10, 6, 2, 1, -6, -10], [-20, 0, -2, 7, 2, 7, 0, 0, 0, 0], [0, 0, 12, -2, 12, 4, 0, 0, 0, 0], [3, 8, 8, 12, 8, 6, -12, -3, -6, -8], [0, -6, 0, 0, 0, 0, 4, 20, 3, -8], [0, -8, 0, 0, 0, 0, -16, 18, -2, 15], [0, 14, 0, 0, 0, 0, -2, 16, 2, -17], [0, 0, 12, 14, 4, 4, 0, 0, 0, 0], [16, 0, -2, 4, 7, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -16, 14, -4, 11], [-2, 2, 2, 0, -4, -13, 0, 2, 13, 4], [-16, 0, -2, 2, -3, -5, 0, 0, 0, 0], [0, 16, 0, 0, 0, 0, 8, -10, 4, -13], [0, 2, 0, 0, 0, 0, 2, 22, 4, 7], [-9, 0, 0, -12, 8, 1, 0, 0, 0, 0], [-12, 10, 10, 6, -5, 6, -6, 12, -6, 5], [0, 0, 0, 0, 0, 0, 7, 16, -2, -8], [-17, 0, -4, 10, -12, -1, 0, 0, 0, 0], [-8, 0, 0, -11, 5, 8, 0, 0, 0, 0], [0, -18, 0, 0, 0, 0, -2, 6, -5, 8], [-6, -8, -8, 0, 10, -3, 0, 6, 3, -10], [0, 6, 6, 1, 5, 7, -1, 0, -7, -5], [8, 4, 4, -4, 7, 0, 4, -8, 0, -7], [-3, -8, -8, -14, 4, 2, 14, 3, -2, -4], [-12, 0, 8, 4, 8, -6, 0, 0, 0, 0], [0, 20, 0, 0, 0, 0, -6, -13, -10, -10], [10, -2, -2, 11, 1, -1, -11, -10, 1, -1], [-18, 0, -20, 2, 0, 10, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, -6, -14, 5], [2, 14, 14, 0, 5, 8, 0, -2, -8, -5], [-12, -8, -8, 0, -7, -6, 0, 12, 6, 7], [-29, 0, 0, 8, -12, 5, 0, 0, 0, 0], [8, 0, -10, -7, 8, -5, 0, 0, 0, 0], [2, 2, 2, 4, 5, 8, -4, -2, -8, -5], [-6, 0, 0, -2, 22, -6, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, 0, 5, 11, 15], [0, -8, 0, 0, 0, 0, -5, 10, 6, 2], [-10, 0, -2, 8, 3, 13, 0, 0, 0, 0], [-9, 0, 8, 2, 2, 1, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 4, 40, -3, 0], [20, -6, -6, -3, -17, 1, 3, -20, -1, 17], [-6, -4, -4, -8, 4, -7, 8, 6, 7, -4], [11, -10, -10, -12, 1, 3, 12, -11, -3, -1], [4, 0, 0, -6, 13, -10, 6, -4, 10, -13], [0, 10, 0, 0, 0, 0, -2, 16, -6, -1], [-16, 6, 6, -3, 13, -5, 3, 16, 5, -13], [0, -10, 0, 0, 0, 0, 0, 35, 11, 9], [-10, 0, -14, 13, -14, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, -2, 2, -19], [0, -4, 0, 0, 0, 0, 0, -18, -7, 10], [-12, 10, 10, 14, 2, 9, -14, 12, -9, -2], [-19, -8, -8, 6, -22, -8, -6, 19, 8, 22], [12, -4, -4, -8, 12, -7, 8, -12, 7, -12], [0, -4, 0, 0, 0, 0, 10, -20, 3, -6], [24, 0, 12, -6, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -6, 20, 5, 8], [6, -22, -22, 4, -11, -12, -4, -6, 12, 11], [14, 0, -6, 2, -5, 7, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, -4, -14, 2, -5], [14, 24, 24, 8, 0, 5, -8, -14, -5, 0], [-4, 0, 2, 10, 11, 9, 0, 0, 0, 0], [-2, -4, -4, -8, 12, -3, 8, 2, 3, -12], [0, -14, 0, 0, 0, 0, 1, -18, 5, 3], [0, -10, 0, 0, 0, 0, -2, 12, -7, 0], [-18, 0, 0, -12, 12, 0, 0, 0, 0, 0], [14, 2, 2, -12, 16, 3, 12, -14, -3, -16], [0, 12, 0, 0, 0, 0, -2, -15, -2, 12], [21, 0, 0, 12, 0, 3, 0, 0, 0, 0], [-2, 0, 12, -2, 12, 4, 0, 0, 0, 0], [0, 22, 0, 0, 0, 0, -5, 16, 3, -15], [6, -16, -16, 0, -19, -2, 0, -6, 2, 19], [0, -18, 0, 0, 0, 0, -8, 4, 4, 15], [12, -14, -14, -4, -6, 7, 4, -12, -7, 6], [-1, 8, 8, 10, -10, 10, -10, 1, -10, 10], [0, -10, 0, 0, 0, 0, 15, 8, 1, -13], [-10, 0, -2, -6, -19, -9, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 2, -7, 6, -8], [-14, 0, 2, 16, 1, -5, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, -4, -40, 4, -1], [0, 8, 0, 0, 0, 0, 8, -22, 14, -1], [-12, 0, -10, 6, 5, -1, 0, 0, 0, 0], [15, 0, -8, -4, 4, -15, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 10, 8, -9, -6], [8, 4, 4, 0, -13, -8, 0, -8, 8, 13], [0, 2, 0, 0, 0, 0, 2, 20, 9, 4], [0, -2, 0, 0, 0, 0, -11, -22, 1, -1], [-27, -10, -10, 2, 7, 5, -2, 27, -5, -7], [12, 0, 0, 10, -15, 4, -10, -12, -4, 15], [22, 0, 12, -9, 21, 12, 0, 0, 0, 0], [27, -2, -2, 14, -7, -3, -14, -27, 3, 7], [-12, 0, 0, -10, 6, 8, 0, 0, 0, 0], [2, 4, 4, 8, 3, 0, -8, -2, 0, -3], [0, -14, 0, 0, 0, 0, -6, -14, 4, 3], [0, 0, 0, 0, 0, 0, 2, -4, 1, -10], [0, 0, -18, -2, -7, -1, 0, 0, 0, 0], [-4, 0, 14, -10, 1, -1, 0, 0, 0, 0], [24, 16, 16, -3, 4, 6, 3, -24, -6, -4], [0, 0, 0, 0, 0, 0, -16, -12, 0, 17], [22, 12, 12, -6, 21, 6, 6, -22, -6, -21], [0, -4, 0, 0, 0, 0, -6, -12, 11, 0], [26, 0, 4, -10, 24, -2, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 8, 2, 7, 10], [10, 0, -16, -7, -15, -6, 0, 0, 0, 0], [-20, 4, 4, -5, 8, 0, 5, 20, 0, -8], [4, -6, -6, 5, -7, -9, -5, -4, 9, 7], [-16, 0, 22, -4, 3, 7, 0, 0, 0, 0], [-10, 0, -4, -2, 12, 2, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, -10, -10, -3, 4], [14, -8, -8, -9, 8, 2, 9, -14, -2, -8], [19, 0, 22, 12, -1, 12, 0, 0, 0, 0], [3, -22, -22, 14, -15, 1, -14, -3, -1, 15], [0, 8, 0, 0, 0, 0, -4, 18, 0, 5], [25, 18, 18, -4, 7, 7, 4, -25, -7, -7], [-26, 0, 0, -13, 14, 0, 13, 26, 0, -14], [-9, 0, 30, -6, 9, 0, 0, 0, 0, 0], [0, 16, 0, 0, 0, 0, 10, 0, -1, 0], [-4, 0, 0, 2, -9, 14, -2, 4, -14, 9], [36, 0, -4, 10, -8, -2, 0, 0, 0, 0], [28, -6, -6, -4, 5, -6, 4, -28, 6, -5], [10, 0, 16, -16, 24, 4, 0, 0, 0, 0], [24, -6, -6, -4, 7, 12, 4, -24, -12, -7], [0, 22, 0, 0, 0, 0, 4, -22, -1, -18], [-16, 0, 0, 0, -8, -10, 0, 0, 0, 0], [-22, 0, 24, -4, 8, 0, 0, 0, 0, 0], [14, 0, 2, 8, -7, -5, 0, 0, 0, 0], [5, -16, -16, -4, -2, -6, 4, -5, 6, 2], [0, -8, 0, 0, 0, 0, -4, -2, -9, -18], [-6, 20, 20, -5, 12, 2, 5, 6, -2, -12], [-22, 0, 0, 18, 0, 9, -18, 22, -9, 0], [0, 0, 0, 0, 0, 0, 0, 18, -16, -5], [10, 0, 0, 4, -12, -8, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 13, 14, -1, -7], [0, 2, 0, 0, 0, 0, 5, 28, -11, 1], [20, -4, -4, -4, 18, -9, 4, -20, 9, -18], [0, -10, 0, 0, 0, 0, 2, 10, -2, -3], [0, 16, 0, 0, 0, 0, -6, 14, -18, 15], [34, -2, -2, 0, -12, 3, 0, -34, -3, 12], [14, -12, -12, 16, -13, -4, -16, -14, 4, 13], [-2, 0, -8, 20, -8, -6, 0, 0, 0, 0], [18, 0, 6, -2, -3, 7, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, -17, 32, -7, 11], [0, -6, 0, 0, 0, 0, 0, -4, 1, -14], [13, 8, 8, -16, 36, 0, 16, -13, 0, -36], [6, 0, 14, 10, -11, 7, 0, 0, 0, 0], [-4, 6, 6, 9, -3, 3, -9, 4, -3, 3], [-21, 0, 16, 2, 14, 5, 0, 0, 0, 0], [5, -4, -4, 2, -14, 0, -2, -5, 0, 14], [8, -14, -14, 4, -8, -3, -4, -8, 3, 8], [0, 0, 0, 0, 0, 0, -4, 18, -7, 10], [-26, 0, 2, 7, 12, -1, 0, 0, 0, 0], [2, 0, -4, -12, 2, 2, 0, 0, 0, 0], [-9, -6, -6, 4, -13, -15, -4, 9, 15, 13], [0, -22, 0, 0, 0, 0, 2, 14, 15, 16], [5, 0, -18, -8, -9, -2, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 14, -28, 10, -23], [2, 0, 10, -8, 5, 19, 0, 0, 0, 0], [0, 24, 0, 0, 0, 0, 6, -27, 8, 10], [0, 8, 0, 0, 0, 0, 8, -24, -3, -4], [34, -8, -8, 10, -21, 0, -10, -34, 0, 21]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1183_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1183_2_e_f();". // 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_1183_2_e_f(:prec:=10) chi := MakeCharacter_1183_e(); 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_1183_2_e_f();". // 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_1183_2_e_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1183_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![9, 36, 195, -210, 265, -116, 81, -30, 17, -4, 1]>,<3,R![16, -64, 256, -72, 144, -4, 65, 0, 9, 0, 1]>],Snew); return Vf; end function;