// Make newform 819.2.dl.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_819_dl();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_819_dl_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_819_2_dl_e();". // 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_819_2_dl_e();". // 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 := [9, 0, -93, 0, 853, 0, -1050, 0, 952, 0, -334, 0, 85, 0, -11, 0, 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, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-40872159, 0, -690027, 0, 6321845, 0, -17193085, 0, 6411671, 0, -1852321, 0, 246060, 0, -24498, 0], [-23829231, 0, 218482087, 0, -377649654, 0, 236538400, 0, -78484018, 0, 17340370, 0, -2170865, 0, 172099, 0], [-20454191, 0, -2790528, 0, 25566080, 0, -66564370, 0, 25929344, 0, -7490944, 0, 1000291, 0, -99072, 0], [0, 55035781, 0, 690027, 0, -6321845, 0, 17193085, 0, -6411671, 0, 1852321, 0, -246060, 0, 24498], [-5618970, 0, 441286822, 0, -514968195, 0, 494434675, 0, -174659083, 0, 45125185, 0, -5861765, 0, 539569, 0], [23341327, 0, 2881788, 0, -26402180, 0, 67022271, 0, -26777324, 0, 7735924, 0, -1048444, 0, 102312, 0], [46491129, 0, -426433173, 0, 508646350, 0, -477241590, 0, 168247412, 0, -43272864, 0, 5615705, 0, -515071, 0], [366555, 0, -3361729, 0, 4460568, 0, -3722089, 0, 1287475, 0, -317920, 0, 40712, 0, -3598, 0], [0, -61326350, 0, -3480555, 0, 31887925, 0, -83757455, 0, 32341015, 0, -9343265, 0, 1246351, 0, -123570], [0, -48109836, 0, 441286822, 0, -514968195, 0, 494434675, 0, -174659083, 0, 45125185, 0, -5861765, 0, 539569], [0, -11237940, 0, 861328211, 0, -1029936390, 0, 988869350, 0, -349318166, 0, 90250370, 0, -11723530, 0, 1079138], [0, 19219314, 0, -176278948, 0, 219868809, 0, -196608895, 0, 68898667, 0, -17480377, 0, 2261795, 0, -205171], [0, -167644965, 0, 1094093084, 0, -1313214930, 0, 1125399698, 0, -390741062, 0, 98087264, 0, -12639109, 0, 1137374], [0, -359333319, 0, -12180871093, 0, 15781577445, 0, -15847248841, 0, 5653350919, 0, -1470474691, 0, 191553668, 0, -17689120]]; Rf_basisdens := [1, 1, 14163622, 42490866, 14163622, 14163622, 42490866, 7081811, 14163622, 95271, 14163622, 42490866, 21245433, 3862806, 11588418, 127472598]; 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_819_dl();" function MakeCharacter_819_dl() N := 819; order := 6; char_gens := [92, 703, 379]; v := [6, 4, 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_819_dl_Hecke();" function MakeCharacter_819_dl_Hecke(Kf) N := 819; order := 6; char_gens := [92, 703, 379]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1], [-1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -2, -1, 0, 0, 0], [1, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 2, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1], [0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -3, 0, 0, 2, -1], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, -1, -2, -1, 1, -2], [2, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 3, 1, -2, 1], [3, 0, 0, 0, 0, 0, -3, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 1, 0, 0, 0, 0, -3, 0, 0, 0, 1, 1], [0, 0, 2, 0, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, 0, 0, -1, -4, -3, -1, 0, 0], [0, 2, 0, 0, 0, -2, 0, 0, 0, 0, -3, 0, 0, 0, 2, 2], [-1, 0, 0, 1, -1, 0, 3, -1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, -1, 0, 0, 0, 0, -1, 7, 2, -1, -1, 2], [0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, -2, -2, -2, 1], [0, -2, 0, 0, 0, 1, 0, 0, 0, 0, 1, -3, 0, 1, 1, -2], [-1, 0, 0, 3, 0, 0, 1, 0, 2, -1, 0, 0, 0, 0, 0, 0], [-1, 0, 2, 1, -1, 0, 5, 2, 2, -2, 0, 0, 0, 0, 0, 0], [-5, 0, -2, 5, -5, 0, 3, -1, -2, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 1, 1], [-3, 0, -4, 0, -5, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, -2, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 6, -1, 1, 0, 5, 1, 6, -1, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, -4, 0, 0, 0, 0, -1, 0, 0, 0, 2, 2], [4, 0, -4, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -4, 2, 3, 0, 0], [0, -4, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, 0, 2, 0, 0, -7, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, -7, 0, 0, 0, 0, 0, 4, 9, -4, -4, 2], [0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 1, -4, -1, 1, 2, -4], [0, 0, -2, 0, 0, 0, -7, 2, -2, -2, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -5, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0], [-9, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, -2, -7, 7, 0, 1, 3, -2, -3, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -3, -3], [0, -6, 0, 0, 0, 1, 0, 0, 0, 0, 1, -7, -2, 1, 1, -2], [13, 0, 0, -5, 0, 0, -13, 0, 4, -2, 0, 0, 0, 0, 0, 0], [6, 0, 2, 0, -2, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 4, 3, -5, 4, 1, -2], [0, 7, 0, 0, 0, 8, 0, 0, 0, 0, 2, 0, 0, 0, -3, -3], [0, 3, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1, 6, 0, -6, 3], [4, 0, 4, -4, 4, 0, -15, 2, 4, -2, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, -1, 0, 0, 0, 0, 2, 3, -7, 2, -1, 2], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 1, 1], [8, 0, -4, 0, -2, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, -2, 3, -3, 0, -3, -3, -2, 3, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -5, 0, 0, 3, 0, 4, 2, 0, 0, 0, 0, 0, 0], [19, 0, 0, 0, 0, 0, -19, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 3, 0, 0, 0, 0, 0, 8, 1, -4, -8, 4], [7, 0, 0, -4, 0, 0, -7, 0, -4, -2, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, -3, 0, 0, 0, 0, 1, 9, 6, 1, -3, 6], [-1, 0, 0, -3, 0, 0, 1, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, -3, 0, 0, 0, 0, -1, 11, 8, -1, -3, 6], [0, 4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 5, -5, 0, -1, 2], [-13, 0, 0, -3, 0, 0, 13, 0, -2, -4, 0, 0, 0, 0, 0, 0], [5, 0, 8, -5, 5, 0, 11, 1, 8, -1, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, -4, 0, 0, 0, 0, 0, 10, 8, 1, -8, 4], [0, -3, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, -6, 0, 6, -3], [5, 0, -2, 0, 7, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-13, 0, 0, 5, 0, 0, 13, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 7, -5, -2, 7, 5, -10], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 1, 1], [0, 1, 0, 0, 0, -7, 0, 0, 0, 0, 0, 11, 8, -10, -2, 1], [-5, 0, 0, 1, 0, 0, 5, 0, -2, 5, 0, 0, 0, 0, 0, 0], [8, 0, 4, -8, 8, 0, -11, 3, 4, -3, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 3, 0, 0, 0, 0, -3, -11, -2, -3, 3, -6], [0, -3, 0, 0, 0, 7, 0, 0, 0, 0, 0, -5, -10, 6, 6, -3], [9, 0, 0, 8, 0, 0, -9, 0, -2, 1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 8, -4, -1, -4, 2], [0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -11, -2, 4, 2, -1], [0, 14, 0, 0, 0, 6, 0, 0, 0, 0, -7, 0, 0, 0, -2, -2], [2, 0, 12, 0, 6, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 6, 0, 2, 6, -2, 4], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 3, 3], [-1, 0, 0, 0, 3, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 2, -1, 1, 0, 13, -1, 2, 1, 0, 0, 0, 0, 0, 0], [-3, 0, -4, 3, -3, 0, -5, -5, -4, 5, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, -6, 0, 0, 0, 0, -10, 12, 14, -10, -6, 12], [0, 1, 0, 0, 0, -3, 0, 0, 0, 0, 0, -7, 4, 2, -2, 1], [0, -2, 0, 0, 0, 1, 0, 0, 0, 0, 1, -3, 0, 1, 1, -2], [0, 8, 0, 0, 0, -1, 0, 0, 0, 0, -3, 9, 4, -3, -1, 2], [7, 0, 10, -7, 7, 0, -1, 1, 10, -1, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 7, 0, 0, 0, -7, -7], [0, 1, 0, 0, 0, -5, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1], [6, 0, -2, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, -11, 0, 4, -6, 3], [-16, 0, -6, 0, -8, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 7, 0, 0, 0, 0, 0, -3, -6, 8, -2, 1], [3, 0, 0, 1, 0, 0, -3, 0, -10, -1, 0, 0, 0, 0, 0, 0], [3, 0, -2, -3, 3, 0, -5, -6, -2, 6, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, -2, 0, 0, 0, 0, 0, 6, -2, 1, 8, -4], [-12, 0, -4, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, -2, 0, 0, 0, 0, 9, 0, 0, 0, -6, -6], [-17, 0, 0, -5, 0, 0, 17, 0, 10, 3, 0, 0, 0, 0, 0, 0], [-14, 0, -6, 14, -14, 0, 19, 0, -6, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -7, 0, 0, -1, 0, 8, -6, 0, 0, 0, 0, 0, 0], [0, -11, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, -3, -3], [0, 4, 0, 0, 0, -6, 0, 0, 0, 0, -9, 10, -1, -9, -6, 12], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 1, 0, 0], [3, 0, 0, -7, 0, 0, -3, 0, 8, 1, 0, 0, 0, 0, 0, 0], [-1, 0, -2, 0, -3, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, -2, 7, -7, 0, -5, -5, -2, 5, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 4, 0, 0, 0, 0, -7, 0, 0, 0, 6, 6], [0, 8, 0, 0, 0, 5, 0, 0, 0, 0, -3, 3, -4, -3, 5, -10], [0, 5, 0, 0, 0, 2, 0, 0, 0, 0, 10, 0, 0, 0, -1, -1], [0, 8, 0, 0, 0, -5, 0, 0, 0, 0, -11, 13, 4, -11, -5, 10], [-7, 0, 0, -1, 0, 0, 7, 0, -8, -3, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, -4, 0, 0, 0, 0, -11, 2, -1, -11, -4, 8], [-13, 0, 0, -5, 0, 0, 13, 0, 8, 1, 0, 0, 0, 0, 0, 0], [1, 0, 10, -1, 1, 0, -1, 1, 10, -1, 0, 0, 0, 0, 0, 0], [-28, 0, -8, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 1, 1], [-20, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 12, 8, -8, 0, 1, 0, 12, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, -6, 0, 0, 0, 0, 6, 0, 0, 0, 3, 3], [0, 3, 0, 0, 0, -8, 0, 0, 0, 0, 0, 9, 11, 11, -6, 3], [-10, 0, -10, 0, -2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, -22, -8, 13, 8, -4], [1, 0, -6, -1, 1, 0, -19, -2, -6, 2, 0, 0, 0, 0, 0, 0], [-20, 0, -6, 0, -16, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, -5, 0, 0, 0, 0, 0, -3, 8, -6, -6, 3], [0, 3, 0, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, 0, -1, -1], [0, 8, 0, 0, 0, -3, 0, 0, 0, 0, 2, 11, 9, 2, -3, 6], [3, 0, 12, -3, 3, 0, 11, -2, 12, 2, 0, 0, 0, 0, 0, 0], [31, 0, 6, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, -12, 0, 0, 0, 0, 0, 8, 16, -9, -8, 4], [0, 14, 0, 0, 0, 2, 0, 0, 0, 0, 4, 12, 6, 4, 2, -4], [0, 1, 0, 0, 0, -6, 0, 0, 0, 0, -12, 0, 0, 0, 7, 7], [0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 12, 0, 0, 0, 2, 2], [12, 0, -4, 0, -6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [25, 0, 0, 2, 0, 0, -25, 0, -8, -1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, -3, 0, 0, 0, 0, -3, 5, -2, -3, -3, 6], [0, -6, 0, 0, 0, 10, 0, 0, 0, 0, 0, -16, -16, 17, 12, -6], [-15, 0, 0, 15, 0, 0, 15, 0, -12, 8, 0, 0, 0, 0, 0, 0], [0, 14, 0, 0, 0, -2, 0, 0, 0, 0, -9, 16, 7, -9, -2, 4], [1, 0, 0, 4, 0, 0, -1, 0, 2, 10, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 5, 0, 0, 0, 0, 12, -9, -1, 12, 5, -10], [0, -4, 0, 0, 0, -2, 0, 0, 0, 0, 10, -2, -6, 10, -2, 4], [3, 0, 0, -12, 0, 0, -3, 0, 8, -6, 0, 0, 0, 0, 0, 0], [-1, 0, 2, 1, -1, 0, 7, -9, 2, 9, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 3, 0, 0, 0, 0, 0, -15, -6, 2, 6, -3], [0, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 5, 4, 0, -14, 7], [8, 0, 2, 0, 16, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, -2, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, -6, 5, -5, 0, 31, -6, -6, 6, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -1, 0, 0, 3, 0, -2, -1, 0, 0, 0, 0, 0, 0], [10, 0, -2, 0, 4, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, -2, 5, -5, 0, 11, 8, -2, -8, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, -1, 0, 0, 0, 0, 0, -23, -2, 4, 6, -3], [3, 0, 14, 0, 7, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 6, 0, 0, 0, 0, 4, 0, 0, 0, -3, -3], [0, 3, 0, 0, 0, 12, 0, 0, 0, 0, 0, 1, -9, -15, -6, 3], [3, 0, 8, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, 0, 0, 0, -7, 8, 13, -7, 0, 0], [-9, 0, 6, 9, -9, 0, 25, 1, 6, -1, 0, 0, 0, 0, 0, 0], [0, 16, 0, 0, 0, -8, 0, 0, 0, 0, 3, 0, 0, 0, 4, 4], [0, 6, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6], [5, 0, 0, 3, 0, 0, -5, 0, 4, -2, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_819_dl_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_819_2_dl_e();". // 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_819_2_dl_e(:prec:=16) chi := MakeCharacter_819_dl(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_819_2_dl_e();". // 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_819_2_dl_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_819_dl(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![9, 0, -93, 0, 853, 0, -1050, 0, 952, 0, -334, 0, 85, 0, -11, 0, 1]>,<19,R![10673289, 0, -7618644, 0, 3674044, 0, -971784, 0, 185725, 0, -19096, 0, 1396, 0, -44, 0, 1]>],Snew); return Vf; end function;