// Make newform 1152.2.i.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1152_i();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1152_i_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_1152_2_i_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_1152_2_i_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 := [243, -162, 27, 0, 27, -36, 9, 0, 1, -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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-567, 27, 0, 0, -63, 45, 12, 4, 1, 1], [81, -270, 162, 81, 9, -18, 15, -4, -1, -1], [81, 297, 0, 0, 9, 9, -12, -4, -1, -1], [162, -27, 0, -27, 36, -9, 0, -1, 2, -1], [-567, -297, -648, 162, -63, 45, -24, 28, 7, 7], [1053, -189, 0, 0, 117, -99, -12, -10, 11, -7], [324, -108, -81, 81, 36, -45, 6, 11, -4, -4], [2349, -459, -324, 162, 369, -171, -60, 22, 19, -17]]; Rf_basisdens := [1, 1, 486, 243, 162, 81, 486, 486, 243, 486]; 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_1152_i();" function MakeCharacter_1152_i() N := 1152; order := 3; char_gens := [127, 901, 641]; v := [3, 3, 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_1152_i_Hecke();" function MakeCharacter_1152_i_Hecke(Kf) N := 1152; order := 3; char_gens := [127, 901, 641]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, -1, 1, 1, 1, 1, 0, 0, 1, 0], [0, 1, 0, 0, -1, -1, -1, 0, 0, 1], [0, 0, 1, 0, -1, 1, 0, 0, 0, 0], [-1, 1, 0, 1, 0, 0, 0, -1, 0, 1], [-2, 0, 0, 0, 1, -1, 0, -2, 0, 1], [0, 0, 1, 0, 0, -1, -1, 0, 1, 0], [-1, 1, 1, -2, -2, -2, -2, -1, -2, 2], [0, -2, -1, 0, 2, 1, 1, 2, 3, 0], [-3, 3, 0, 1, -1, 1, 0, -1, 0, 2], [0, 4, -2, 0, -1, -3, 0, -4, -1, 0], [2, 1, -2, -2, -1, -1, -2, 0, -2, 2], [-1, 1, 1, -1, 1, 1, 2, 2, -1, -2], [1, -3, 0, 1, 1, -3, 0, -1, 0, 0], [0, 3, 2, 0, -2, -3, -2, -3, -4, 0], [3, 3, -3, -2, -6, -6, -2, -3, -2, 2], [0, -2, 4, 0, 2, 1, 1, 2, 0, 0], [0, 3, 0, 2, -2, 4, 0, 3, 0, -1], [-3, -1, 0, -1, 0, -2, 0, -1, 0, 1], [-1, -1, 1, -3, 3, 3, 3, 2, -3, -3], [-2, 2, 2, 0, -1, -1, 1, 1, 0, -1], [3, 5, 0, 1, -3, 5, 0, 3, 0, 0], [4, -1, -4, -1, 4, 4, 0, 3, -1, 0], [2, -6, -2, -1, 2, 2, 0, -4, -1, 0], [0, -3, -3, 0, 2, -1, -2, 3, -3, 0], [-6, 4, 0, 2, 1, 3, 0, -2, 0, 1], [-3, -1, 0, 1, -3, 1, 0, 5, 0, -2], [0, -3, 5, 0, -1, 3, -1, 3, 4, 0], [4, -4, 0, 0, 4, -4, 0, -4, 0, 0], [0, 4, -6, 0, 1, -2, 3, -4, 0, 0], [0, 7, 0, -1, -2, -2, 0, 5, -1, 0], [0, -3, 8, 0, 2, -3, -4, 3, 0, 0], [0, -6, -5, 0, 3, 3, 0, 6, 2, 0], [-3, 3, 3, 1, -1, -1, -3, 2, 1, 3], [0, 2, 6, 0, -6, 4, 0, -2, -3, 0], [-2, -4, 0, -6, 2, -2, 0, 0, 0, -2], [0, 7, 1, 0, -4, -3, 0, -7, -3, 0], [4, 0, -4, 1, 4, 4, 6, 4, 1, -6], [2, 0, 0, 0, 2, 2, 0, 0, 0, -2], [-3, -1, 0, -3, 1, 3, 0, 3, 0, -4], [-5, 3, 5, 1, -3, -3, -2, 0, 1, 2], [0, -4, 8, 0, 5, -1, 0, 4, 9, 0], [0, -6, 0, -2, 0, -6, 0, 0, 0, 0], [8, -3, 0, -2, 2, -4, 0, -3, 0, 1], [0, -1, 2, 0, 5, -2, 2, 1, 0, 0], [-9, -3, 9, 1, 1, 1, -4, -2, 1, 4], [-10, -1, 10, 4, 3, 3, 1, 2, 4, -1], [0, 2, 8, 0, 0, -4, -2, -2, -1, 0], [-4, 4, 0, -2, -1, 5, 0, 2, 0, -1], [0, 2, -7, 0, 0, -1, 1, -2, -3, 0], [5, -2, -5, 4, 8, 8, 2, 6, 4, -2], [4, -2, 0, -4, -3, 3, 0, 8, 0, -5], [0, 0, -9, 0, 4, -2, 2, 0, -2, 0], [3, 7, -3, -3, -9, -9, -3, -2, -3, 3], [-3, -3, 0, 1, 5, 3, 0, 1, 0, -6], [10, 4, 0, 4, -2, 2, 0, 0, 0, 2], [6, 0, -6, 7, 6, 6, 2, 6, 7, -2], [-6, 2, 6, 5, 2, 2, 1, 4, 5, -1], [0, 2, 0, 0, 3, 0, 5, -2, 4, 0], [0, -6, 8, 0, 2, 4, 0, 6, -1, 0], [-10, 0, 0, -2, -1, 1, 0, 2, 0, -1], [0, -4, 15, 0, 4, -1, -1, 4, -3, 0], [11, 0, -11, -2, -4, -4, 2, -4, -2, -2], [3, 7, -3, -2, -8, -8, -4, -1, -2, 4], [2, -2, -2, -4, 3, 3, 2, 1, -4, -2], [0, 2, 7, 0, 4, -2, 4, -2, -6, 0], [0, 0, 10, 0, 4, -5, -1, 0, 0, 0], [0, -6, 0, 1, -2, -2, -6, -8, 1, 6], [2, -5, -2, 1, 2, 2, 4, -3, 1, -4], [-8, 2, 0, -6, -10, 2, 0, 10, 0, 0], [-15, 1, 15, 1, -1, -1, -3, 0, 1, 3], [0, 2, 3, 0, 7, -5, 4, -2, -2, 0], [-12, 1, 0, -4, -1, 1, 0, 1, 0, 0], [0, 1, -11, 0, -4, -3, -6, -1, -7, 0], [6, 6, -6, -1, -6, -6, -6, 0, -1, 6], [-11, -3, 0, -7, -1, -1, 0, 3, 0, -2], [0, -2, -2, 0, 1, 5, 4, 2, -5, 0], [0, 2, 7, 0, -4, -4, -6, -2, -2, 0], [0, -1, -12, 0, -5, 0, -6, 1, -4, 0], [5, -1, -5, -4, 2, 2, -2, 1, -4, 2], [6, -4, 0, 4, 8, -4, 0, -8, 0, 0], [-7, 1, 0, -1, 2, 4, 0, 1, 0, -3], [-11, 7, 11, 5, -1, -1, 5, 6, 5, -5], [4, -5, -4, -10, -1, -1, 2, -6, -10, -2], [-3, -1, 0, -1, -2, -4, 0, -1, 0, 3], [10, 1, -10, -11, -8, -8, 0, -7, -11, 0], [-5, -1, 5, 10, 6, 6, 2, 5, 10, -2], [0, 6, -13, 0, -4, -9, -7, -6, 1, 0], [14, 0, 0, 12, 3, -3, 0, -6, 0, 3], [-15, -3, 15, 3, -1, -1, 3, -4, 3, -3], [0, -10, 0, 2, 2, -10, 0, -2, 0, 0], [0, 1, 10, 0, 4, -1, 4, -1, 0, 0], [0, 5, 16, 0, -12, 3, -4, -5, -2, 0], [2, -9, 0, -12, 2, -8, 0, -1, 0, -1], [0, 12, -3, 0, -7, -7, -2, -12, -12, 0], [4, 2, 0, 2, 1, -3, 0, -6, 0, 5], [-2, 1, 0, -10, -6, -2, 0, 3, 0, 3], [0, 2, 0, 2, -2, -4, 0, -4, 0, 6], [14, 5, -14, -10, -9, -9, -5, -4, -10, 5], [-7, 5, 0, -7, -9, 7, 0, 11, 0, -2], [0, -6, -6, 0, 0, 3, -3, 6, 6, 0], [3, 4, -3, -10, -12, -12, -4, -8, -10, 4], [0, 2, 6, 0, -2, 1, 1, -2, -6, 0], [-13, 1, 0, 9, 2, -2, 0, -5, 0, 3], [24, -1, -24, -4, -1, -1, 2, -2, -4, -2], [1, -1, 0, -13, -9, -11, 0, -1, 0, 10], [0, -9, -5, 0, 14, 3, 8, 9, 3, 0], [0, -3, 0, 15, 2, 2, 0, -1, 15, 0], [0, -7, 3, 0, -4, 9, -2, 7, -3, 0], [-7, -3, 0, -3, 7, -5, 0, -9, 0, 2], [0, -4, 6, 0, 1, -1, -4, 4, -1, 0], [10, 13, -10, 2, -13, -13, -3, 0, 2, 3], [12, 6, 0, 6, -2, 14, 0, 10, 0, -8], [-3, -10, 3, 4, 4, 4, -4, -6, 4, 4], [0, 4, 4, 0, -4, -5, -5, -4, 4, 0], [-16, 2, 0, 2, -2, 2, 0, 2, 0, 0], [0, 10, 6, 0, 2, -6, 6, -10, -3, 0], [-10, -2, 10, 0, -9, -9, -4, -11, 0, 4], [0, -12, 12, 0, 2, 8, -2, 12, 7, 0], [2, -2, -2, -7, 0, 0, 1, -2, -7, -1], [10, 4, -10, -1, -6, -6, 0, -2, -1, 0], [0, -6, 0, -8, 3, -7, 0, -4, 0, 1], [0, 12, 0, -12, -17, -17, -3, -5, -12, 3], [-3, 5, 0, 5, 5, -1, 0, -11, 0, 6], [2, -4, -2, 7, 0, 0, -6, -4, 7, 6], [20, -12, 0, 0, 6, -10, 0, -4, 0, -2], [-9, 13, 9, -11, -9, -9, -1, 4, -11, 1], [0, -6, 17, 0, 3, 3, 0, 6, 4, 0], [-10, 5, 0, 6, -3, 7, 0, 5, 0, -2], [0, 3, 19, 0, -10, 7, 0, -3, -5, 0], [0, 1, 5, 0, -8, 5, -2, -1, 1, 0], [-18, 0, 0, 8, 8, -4, 0, -12, 0, 4], [0, -5, 14, 0, 4, 10, 9, 5, -1, 0], [0, 11, 3, 0, -9, -7, -5, -11, -6, 0], [-23, 3, 0, 1, 1, 3, 0, -1, 0, 0], [0, -4, -2, 0, 5, 2, 3, 4, 0, 0], [0, 0, -19, 0, -1, -1, -2, 0, 0, 0], [1, -13, 0, 7, 6, -4, 0, 3, 0, -9], [-10, 4, 0, 0, -1, 9, 0, 6, 0, -5], [-23, 3, 23, 0, -2, -2, 0, 1, 0, 0], [0, 18, -13, 0, -12, -5, 1, -18, -5, 0], [-18, -4, 0, -4, -2, -2, 0, 4, 0, -2], [-7, -5, 0, 1, 11, -5, 0, -11, 0, 0], [15, 5, -15, -9, -15, -15, 0, -10, -9, 0], [-3, -15, 3, 8, 14, 14, 6, -1, 8, -6], [12, -4, -12, 3, 8, 8, 3, 4, 3, -3], [0, 2, 18, 0, -10, 1, -7, -2, 0, 0], [12, -2, 0, 6, -2, 2, 0, 6, 0, -4], [0, 2, 5, 0, 5, -3, 4, -2, 4, 0], [-17, -5, 0, 5, 5, 5, 0, 5, 0, -10], [0, 6, 0, -2, -15, 7, 0, 16, 0, -1], [0, -8, 13, 0, 14, -15, -9, 8, -1, 0], [8, -15, -8, 12, 11, 11, 4, -4, 12, -4], [-19, 3, 19, -11, -5, -5, -5, -2, -11, 5], [-14, 20, 0, 4, -10, 14, 0, 4, 0, 6], [-6, -2, 6, 9, 4, 4, -3, 2, 9, 3], [-3, -11, 0, -1, 11, 1, 0, 1, 0, -12], [0, -8, 14, 0, 4, 0, -4, 8, 9, 0], [-28, -5, 28, -2, 11, 11, 3, 6, -2, -3], [1, -7, 0, -5, 2, -12, 0, -7, 0, 5], [0, -12, -3, 0, 2, 5, -5, 12, -5, 0], [-6, -13, 0, 6, 6, -2, 0, 5, 0, -11], [0, 6, -5, 0, 10, -8, 8, -6, 6, 0], [17, -3, -17, -3, -1, -1, -9, -4, -3, 9], [-16, 2, 0, -10, -4, -8, 0, -6, 0, 10], [6, 4, -6, -9, -2, -2, 6, 2, -9, -6], [-3, 3, 0, 7, -1, 9, 0, 7, 0, -6], [0, 2, -25, 0, 1, -1, 2, -2, 6, 0], [14, -5, -14, 10, -1, -1, -8, -6, 10, 8], [0, -4, -7, 0, 7, -1, 2, 4, 8, 0], [0, 6, 5, 0, -16, -1, -11, -6, -5, 0], [2, 7, -2, -3, -4, -4, 4, 3, -3, -4], [0, 15, 1, 0, 4, -7, 12, -15, -5, 0], [0, -13, -1, 0, -7, 9, -11, 13, 6, 0], [-10, 5, 10, 10, -1, -1, -2, 4, 10, 2], [9, 5, 0, -7, -13, 11, 0, 19, 0, -6], [6, 2, 0, 2, -4, -8, 0, -6, 0, 10], [-8, -4, 8, -3, 6, 6, -6, 2, -3, 6], [5, -15, -5, 7, 5, 5, 2, -10, 7, -2], [-14, 5, 14, 6, -1, -1, 4, 4, 6, -4], [0, -6, -4, 0, 4, 8, 6, 6, -5, 0], [-13, 1, 0, -11, -2, -6, 0, -5, 0, 7], [0, -9, -5, 0, 6, -1, -4, 9, -3, 0], [-31, 9, 31, 0, 4, 4, -4, 13, 0, 4], [-10, 4, 0, 0, 10, 10, 0, -4, 0, -6], [-18, -10, 18, 14, 19, 19, 9, 9, 14, -9], [0, 0, 1, 0, 4, -14, -10, 0, -4, 0], [22, 6, 0, 2, -8, 0, 0, 2, 0, 6], [-11, -3, 0, 3, 4, -6, 0, -7, 0, 3], [2, -1, -2, 2, -9, -9, 3, -10, 2, -3], [0, -10, 0, -12, 7, -11, 0, -8, 0, 1], [24, -2, -24, -3, 2, 2, 10, 0, -3, -10], [10, -4, 0, -8, 7, -3, 0, -6, 0, -1], [0, -10, 13, 0, 6, 4, 0, 10, 16, 0], [0, -10, -12, 0, -1, 5, -6, 10, 1, 0], [-24, -14, 24, 5, 18, 18, 8, 4, 5, -8], [-9, 6, 9, 2, 6, 6, 8, 12, 2, -8], [-24, 8, 0, 12, -8, 8, 0, 8, 0, 0], [0, -6, -6, 0, -4, 6, -4, 6, -3, 0], [3, 5, -3, -9, -3, -3, 2, 2, -9, -2], [0, 4, 12, 0, -8, -2, -6, -4, -1, 0], [-5, 6, 5, 6, -6, -6, -4, 0, 6, 4], [-10, 4, 0, 20, 3, 13, 0, 6, 0, -9], [21, -23, 0, 5, 13, -13, 0, -3, 0, -10], [-4, -2, 0, -10, -6, -6, 0, 2, 0, 4], [0, -5, -22, 0, 4, 3, 2, 5, 6, 0], [3, -6, -3, -6, -8, -8, -8, -14, -6, 8], [0, -3, -14, 0, 8, -9, -4, 3, 2, 0], [-1, -3, 0, -11, -8, -16, 0, -5, 0, 13], [0, 4, -13, 0, 1, -9, -4, -4, -6, 0], [-23, 5, 23, 9, 1, 1, -10, 6, 9, 10], [8, 0, -8, -2, 15, 15, 6, 15, -2, -6], [0, 13, -5, 0, -16, 5, 2, -13, 3, 0], [1, 2, -1, -4, 6, 6, 6, 8, -4, -6], [0, -6, -1, 0, 2, 13, 9, 6, 7, 0], [-6, 17, 6, -15, -18, -18, 2, -1, -15, -2], [-4, -2, 0, 6, 24, 0, 0, -22, 0, -2], [0, -8, -19, 0, 19, -3, 8, 8, 8, 0], [0, 6, -21, 0, -3, -1, 2, -6, 2, 0], [0, 17, 9, 0, 4, -19, 2, -17, -5, 0], [0, 14, 6, 0, -7, -15, -8, -14, -13, 0], [2, -8, 0, 0, 4, 12, 0, 16, 0, -20], [0, 5, -2, 0, -10, 5, 0, -5, 12, 0], [6, 2, -6, -13, -22, -22, -6, -20, -13, 6], [-31, -1, 31, 0, 4, 4, -3, 3, 0, 3], [-22, 0, 0, -16, -4, -4, 0, 0, 0, 4], [11, -21, -11, 3, 15, 15, 3, -6, 3, -3], [24, 4, -24, -8, -1, -1, -2, 3, -8, 2], [0, 14, -10, 0, -4, -10, 0, -14, -7, 0], [26, 12, 0, 2, -6, 10, 0, 4, 0, 2], [0, -8, 31, 0, 8, -9, -9, 8, 3, 0], [0, 4, -1, 0, 10, -10, 4, -4, -14, 0], [-2, -6, 2, -6, 15, 15, 6, 9, -6, -6], [-3, 5, 3, -1, 5, 5, 6, 10, -1, -6], [0, 5, 0, 0, -10, 20, 15, -5, 1, 0], [33, 1, 0, -11, -1, -7, 0, -7, 0, 8], [0, 11, -22, 0, -6, -7, -2, -11, 0, 0], [0, 8, 0, -8, -12, 16, 0, 20, 0, -8], [-6, 5, 6, 14, 15, 15, 4, 20, 14, -4], [30, -12, 0, -8, 2, -6, 0, 4, 0, -6], [0, -3, 17, 0, 8, 1, 6, 3, -9, 0], [9, 9, 0, 5, 5, -3, 0, -17, 0, 12], [0, 8, -14, 0, -13, 7, 2, -8, -5, 0], [-1, -11, 1, 9, 21, 21, 22, 10, 9, -22], [20, -8, 0, -4, 0, -16, 0, -8, 0, 8], [0, 2, 20, 0, -10, 9, 1, -2, 6, 0], [0, 17, -40, 0, -9, -12, -4, -17, -8, 0], [12, -2, 0, 22, 12, 12, 0, 2, 0, -14], [0, 6, 17, 0, -7, -1, -2, -6, 12, 0], [17, 7, 0, -13, 0, 2, 0, -5, 0, 5], [0, 4, 39, 0, -6, 1, -1, -4, -1, 0], [-7, -9, 7, 0, 20, 20, 3, 11, 0, -3], [32, -2, -32, -7, -4, -4, -10, -6, -7, 10], [2, 7, 0, -2, -21, 1, 0, 15, 0, 6], [9, -1, 0, 9, 9, -3, 0, -11, 0, 2], [-24, -8, 24, -6, -1, -1, -6, -9, -6, 6], [-40, -8, 0, -2, 2, -4, 0, 2, 0, -4], [-5, -7, 0, -7, 4, -20, 0, -17, 0, 13], [-3, -17, 3, 11, 19, 19, 9, 2, 11, -9], [16, 9, -16, -10, -5, -5, -6, 4, -10, 6], [0, -4, -4, 0, -20, 18, -6, 4, -13, 0], [17, 3, 0, -11, -17, 5, 0, 19, 0, -2], [0, 15, 10, 0, -2, -18, -5, -15, -5, 0], [-2, 11, 2, 2, -13, -13, -19, -2, 2, 19], [9, 3, 0, 5, 1, -13, 0, -17, 0, 16], [33, 12, -33, 8, -10, -10, -8, 2, 8, 8], [0, 4, -4, 0, 18, -7, 15, -4, 0, 0], [0, -4, 51, 0, -7, 5, -6, 4, 2, 0], [0, 12, -3, 0, -7, -7, -2, -12, -24, 0], [0, -18, 0, 2, 16, -16, 0, -14, 0, -2], [-1, 18, 1, -6, -4, -4, 0, 14, -6, 0], [0, -7, -3, 0, 8, -3, -2, 7, 21, 0], [0, -12, 0, 0, 3, -11, -20, 12, -7, 0], [-16, 4, 0, -12, -4, 4, 0, 4, 0, 0], [0, 2, 28, 0, -1, 0, 1, -2, 12, 0], [-1, -3, 1, -6, 14, 14, 8, 11, -6, -8], [25, 5, 0, -9, -4, -6, 0, -7, 0, 11], [-12, 5, 12, -22, -21, -21, -11, -16, -22, 11], [0, 13, 9, 0, -12, -11, -10, -13, -3, 0], [0, -13, 31, 0, 4, 17, 8, 13, 1, 0], [5, 1, -5, 21, 11, 11, -1, 12, 21, 1], [-22, 0, 22, 3, -4, -4, 6, -4, 3, -6], [0, -29, 26, 0, 18, 13, 2, 29, 22, 0], [-16, 3, 0, -18, -4, 6, 0, 7, 0, -3], [11, -9, 0, 1, 5, -15, 0, -11, 0, 6], [0, 2, -40, 0, -14, 8, -4, -2, -13, 0], [23, 5, -23, 5, 9, 9, 10, 14, 5, -10], [-28, 12, 0, 2, -1, 13, 0, 2, 0, -1], [-5, 9, 5, -10, -14, -14, -8, -5, -10, 8], [-4, 17, 0, 0, 7, 13, 0, -11, 0, 4], [0, 12, 7, 0, 6, 2, 20, -12, -6, 0], [0, -2, -24, 0, -8, 7, -3, 2, 18, 0], [-16, 16, 16, -15, -28, -28, -12, -12, -15, 12], [0, -22, 14, 0, 10, 4, -8, 22, 9, 0], [29, 3, -29, -13, -9, -9, -11, -6, -13, 11], [-23, 3, 23, -4, -4, -4, 4, -1, -4, -4], [24, -10, 0, 4, 0, -8, 0, 2, 0, -2], [-22, -6, 22, 6, 3, 3, -3, -3, 6, 3], [0, 4, -7, 0, 0, 6, 10, -4, 4, 0], [1, 5, 0, -19, -23, -5, 0, 13, 0, 10], [-2, 2, 0, 14, 8, 4, 0, -6, 0, -2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1152_i_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1152_2_i_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_1152_2_i_e(:prec:=10) chi := MakeCharacter_1152_i(); 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_1152_2_i_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_1152_2_i_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1152_i(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![36, -162, 717, -198, 328, 18, 117, 4, 12, 0, 1]>,<7,R![4, -2, 33, 48, 240, 138, 129, 0, 24, 4, 1]>],Snew); return Vf; end function;