// Make newform 4680.2.l.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4680_l();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4680_l_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_4680_2_l_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_4680_2_l_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 := [64, 0, 176, 0, 97, 0, 18, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-80, 184, 2, 15, 20, -10, 2, -1], [0, -168, 0, -179, 0, -46, 0, -3], [40, 176, -1, 97, -10, 18, -1, 1], [0, -104, 0, -89, 0, -18, 0, -1], [272, 120, 358, 15, 92, -10, 6, -1], [8, 0, -37, 0, -14, 0, -1, 0], [-272, 120, -358, 15, -92, -10, -6, -1]]; Rf_basisdens := [1, 64, 64, 32, 16, 64, 8, 64]; 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_4680_l();" function MakeCharacter_4680_l() N := 4680; order := 2; char_gens := [3511, 2341, 2081, 937, 1081]; v := [2, 2, 2, 1, 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_4680_l_Hecke();" function MakeCharacter_4680_l_Hecke(Kf) N := 4680; order := 2; char_gens := [3511, 2341, 2081, 937, 1081]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 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, 0], [0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [2, 1, -1, -1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 1, -1, 1, -1, -1, 0, -1], [4, -1, 1, 1, 0, 1, -1, -1], [0, 1, 1, 1, 1, -1, 0, -1], [2, 0, 0, 0, 0, 0, -1, 0], [-4, 0, 0, 0, 0, 0, 2, 0], [0, 0, -2, 0, 0, 2, 0, 2], [0, 0, 0, 0, 0, -1, 0, 1], [0, 1, -1, 1, 2, 1, 0, 1], [0, 1, 3, 1, 0, -2, 0, -2], [0, -2, 0, -2, 0, 0, 0, 0], [-2, -1, 1, 1, 0, 0, 0, 0], [4, 1, -1, -1, 0, 1, 2, -1], [0, 2, -6, 2, 0, 0, 0, 0], [2, 2, -2, -2, 0, -1, 0, 1], [0, 0, 6, 0, 1, -2, 0, -2], [6, 1, -1, -1, 0, 3, 0, -3], [0, 3, -3, 3, 2, 2, 0, 2], [-4, 0, 0, 0, 0, -3, 2, 3], [0, 0, -2, 0, -1, -2, 0, -2], [2, 0, 0, 0, 0, 0, -5, 0], [0, -2, 0, -2, 0, 4, 0, 4], [0, 2, 6, 2, 2, 0, 0, 0], [-2, -1, 1, 1, 0, -1, -1, 1], [0, 1, -1, 1, -1, -3, 0, -3], [0, -3, -3, -3, -2, 3, 0, 3], [0, 2, -2, -2, 0, -2, 3, 2], [0, -1, 7, -1, 0, -2, 0, -2], [-4, 2, -2, -2, 0, -2, 2, 2], [-4, -1, 1, 1, 0, 0, 2, 0], [4, -2, 2, 2, 0, 2, -4, -2], [0, 5, -3, 5, 0, -1, 0, -1], [0, 0, 4, 0, 4, 0, 0, 0], [0, 3, 9, 3, -2, 2, 0, 2], [0, 0, -2, 0, 6, 0, 0, 0], [0, 4, -4, -4, 0, 2, -1, -2], [10, 2, -2, -2, 0, -2, -2, 2], [12, 2, -2, -2, 0, 2, 2, -2], [0, -4, 6, -4, 3, 4, 0, 4], [0, 3, 3, 3, 4, 2, 0, 2], [6, 1, -1, -1, 0, 3, 0, -3], [10, 1, -1, -1, 0, -1, -2, 1], [0, 2, -6, 2, 3, 0, 0, 0], [0, 1, 15, 1, 0, 2, 0, 2], [-10, 5, -5, -5, 0, -3, 3, 3], [0, 3, -3, 3, -3, -1, 0, -1], [-18, 0, 0, 0, 0, -1, 0, 1], [-2, 0, 0, 0, 0, -2, -2, 2], [0, 0, 0, 0, 0, 4, 1, -4], [0, 3, 13, 3, 1, -1, 0, -1], [0, -1, -1, -1, 3, 5, 0, 5], [10, -2, 2, 2, 0, -2, -3, 2], [4, -4, 4, 4, 0, 2, -4, -2], [0, 1, 5, 1, 4, 3, 0, 3], [8, 4, -4, -4, 0, 1, -2, -1], [0, 0, 4, 0, -4, -4, 0, -4], [0, -1, -13, -1, -4, 0, 0, 0], [0, -4, 0, -4, -2, -6, 0, -6], [-4, 0, 0, 0, 0, 2, -2, -2], [0, -1, 9, -1, -4, -3, 0, -3], [0, 3, 3, 3, 0, 2, 0, 2], [8, 5, -5, -5, 0, 1, 5, -1], [0, -1, 1, -1, -8, -3, 0, -3], [0, 2, -2, 2, 6, -2, 0, -2], [-18, -3, 3, 3, 0, -1, -5, 1], [0, 3, 3, 3, 0, 4, 0, 4], [10, 4, -4, -4, 0, 5, 0, -5], [0, 8, -6, 8, 4, 2, 0, 2], [0, 2, 12, 2, 2, -6, 0, -6], [4, -1, 1, 1, 0, 1, 5, -1], [0, 3, -19, 3, 2, 4, 0, 4], [6, -4, 4, 4, 0, -2, -3, 2], [0, 2, -8, 2, -6, 0, 0, 0], [-12, 2, -2, -2, 0, -1, -4, 1], [6, -4, 4, 4, 0, -2, 0, 2], [-4, -4, 4, 4, 0, 0, 1, 0], [-2, -5, 5, 5, 0, -5, -1, 5], [10, -6, 6, 6, 0, -1, 0, 1], [0, 1, -1, 1, -4, -5, 0, -5], [-8, -4, 4, 4, 0, 0, 4, 0], [0, -4, -4, -4, 2, -6, 0, -6], [4, -2, 2, 2, 0, 1, 2, -1], [0, -6, -12, -6, -1, -4, 0, -4], [-8, -3, 3, 3, 0, 4, 0, -4], [0, 0, 0, 0, -5, 4, 0, 4], [0, -2, 18, -2, -2, 0, 0, 0], [-2, 0, 0, 0, 0, -1, -4, 1], [0, 0, 4, 0, 3, 6, 0, 6], [-8, 2, -2, -2, 0, -2, 11, 2], [0, 1, -1, -1, 0, -3, -7, 3], [0, -3, 5, -3, -5, -1, 0, -1], [16, 5, -5, -5, 0, 6, 4, -6], [22, 0, 0, 0, 0, 2, -4, -2], [0, 1, 23, 1, -2, 1, 0, 1], [10, -1, 1, 1, 0, -5, 3, 5], [0, -1, 5, -1, 2, 7, 0, 7], [0, 3, -1, 3, 8, -2, 0, -2], [0, -8, 0, -8, -2, 4, 0, 4], [18, 8, -8, -8, 0, 2, 8, -2], [10, 5, -5, -5, 0, -1, 4, 1], [0, -6, -16, -6, 1, -2, 0, -2], [0, -1, 1, -1, -4, -6, 0, -6], [0, 1, 5, 1, -2, 2, 0, 2], [-16, -10, 10, 10, 0, -4, -4, 4], [20, 3, -3, -3, 0, -1, 6, 1], [0, 8, -2, 8, 4, -2, 0, -2], [0, -10, -4, -10, 0, 4, 0, 4], [0, 5, 5, 5, 4, 2, 0, 2], [16, 3, -3, -3, 0, 3, -3, -3], [0, 0, 0, 0, 0, 4, -4, -4], [6, 2, -2, -2, 0, 6, -6, -6], [0, 2, -6, 2, 8, 4, 0, 4], [0, 3, 11, 3, -7, 3, 0, 3], [0, -6, 0, -6, 0, -2, 0, -2], [-20, -2, 2, 2, 0, -8, 5, 8], [34, 1, -1, -1, 0, 3, -1, -3], [0, -6, 2, -6, -4, 6, 0, 6], [0, -2, -16, -2, 0, 4, 0, 4], [0, -11, 3, -11, -2, 6, 0, 6], [4, -1, 1, 1, 0, 7, -7, -7], [-10, -4, 4, 4, 0, -2, 3, 2], [-18, 1, -1, -1, 0, 1, -9, -1], [-8, -6, 6, 6, 0, 4, 4, -4], [0, -7, 25, -7, -4, -1, 0, -1], [0, -4, -6, -4, -2, 4, 0, 4], [12, -9, 9, 9, 0, 3, 1, -3], [0, 5, -5, 5, 2, 4, 0, 4], [16, -4, 4, 4, 0, -4, -4, 4], [0, -4, -22, -4, 0, -4, 0, -4], [16, -4, 4, 4, 0, -3, -4, 3], [-10, 0, 0, 0, 0, 0, 8, 0], [0, 1, -19, 1, -4, -8, 0, -8], [0, 8, 4, 8, 0, 6, 0, 6], [0, -6, -24, -6, 2, -4, 0, -4], [10, -4, 4, 4, 0, -10, 2, 10], [0, -7, 7, 7, 0, 3, 1, -3], [4, 1, -1, -1, 0, 6, 2, -6], [0, 2, 16, 2, -8, 0, 0, 0], [0, -7, -1, -7, 6, -2, 0, -2], [4, -1, 1, 1, 0, 11, -2, -11], [-2, -2, 2, 2, 0, 9, 0, -9], [0, -2, -8, -2, -6, 0, 0, 0], [0, 1, -13, 1, -3, -11, 0, -11], [2, 1, -1, -1, 0, 3, -14, -3], [0, -1, -11, -1, 6, 2, 0, 2], [0, 0, -2, 0, -2, -6, 0, -6], [-6, -4, 4, 4, 0, -10, -2, 10], [0, -6, -10, -6, -4, -6, 0, -6], [0, -1, 7, -1, 3, -13, 0, -13], [0, -2, -6, -2, 8, 2, 0, 2], [12, -10, 10, 10, 0, -2, 8, 2], [6, -3, 3, 3, 0, -5, -2, 5], [8, -8, 8, 8, 0, 7, -2, -7], [0, 1, 7, 1, 10, 3, 0, 3], [12, 5, -5, -5, 0, 0, 14, 0], [0, 3, 5, 3, -2, 2, 0, 2], [0, 1, 27, 1, 9, 1, 0, 1], [0, -4, -4, -4, -5, 6, 0, 6], [32, -2, 2, 2, 0, 0, 1, 0], [0, -3, -7, -3, -6, 4, 0, 4], [0, -3, -5, -3, -2, -12, 0, -12], [24, 8, -8, -8, 0, 0, 2, 0], [0, -1, -13, -1, -4, 9, 0, 9], [14, -6, 6, 6, 0, 0, -6, 0], [0, -6, -4, -6, -2, -6, 0, -6], [-10, 1, -1, -1, 0, -2, -10, 2], [18, 9, -9, -9, 0, 7, 9, -7], [12, -4, 4, 4, 0, 10, -4, -10], [0, -8, -14, -8, -1, -6, 0, -6], [0, 4, -4, -4, 0, 4, 2, -4], [-18, -2, 2, 2, 0, -6, -6, 6], [8, -9, 9, 9, 0, -9, 1, 9], [4, 1, -1, -1, 0, -8, -6, 8], [0, -8, -18, -8, 2, -6, 0, -6], [-32, -3, 3, 3, 0, -7, 4, 7], [0, 2, 26, 2, 7, -2, 0, -2], [-16, 0, 0, 0, 0, -10, -1, 10], [0, 12, -10, 12, -4, -12, 0, -12], [0, 3, -25, 3, 2, -4, 0, -4], [0, -1, 1, -1, -2, 8, 0, 8], [6, -14, 14, 14, 0, -4, -1, 4], [0, -3, 5, -3, 6, 3, 0, 3], [0, 0, 12, 0, -4, 4, 0, 4], [-22, 2, -2, -2, 0, -8, 10, 8], [2, -8, 8, 8, 0, 5, 4, -5], [0, -10, 14, -10, 0, 6, 0, 6], [0, -3, -5, -3, -8, -14, 0, -14], [20, -8, 8, 8, 0, 0, 4, 0], [-12, 5, -5, -5, 0, 8, -2, -8], [0, 2, -6, 2, 12, 10, 0, 10], [0, 1, -1, 1, -1, -15, 0, -15], [10, -2, 2, 2, 0, 6, 0, -6], [0, 8, -2, 8, 6, 12, 0, 12], [0, 3, -41, 3, -6, -4, 0, -4], [0, -5, -17, -5, -9, 3, 0, 3], [8, -9, 9, 9, 0, 8, -8, -8], [-16, 4, -4, -4, 0, -8, 4, 8], [0, 4, 10, 4, 6, -8, 0, -8], [-6, 6, -6, -6, 0, -6, -4, 6], [14, 3, -3, -3, 0, 4, 12, -4], [0, -8, -18, -8, 2, -10, 0, -10], [12, -4, 4, 4, 0, 0, -10, 0], [0, 6, -6, 6, -6, 0, 0, 0], [4, -2, 2, 2, 0, 9, 0, -9], [6, 5, -5, -5, 0, -1, -10, 1], [0, -5, -3, -5, -4, 1, 0, 1], [-18, -4, 4, 4, 0, 10, 5, -10], [0, 5, -3, 5, 12, -1, 0, -1], [0, -9, 17, -9, -14, -2, 0, -2], [-10, 12, -12, -12, 0, -1, 0, 1], [-2, 2, -2, -2, 0, -6, 10, 6], [0, 11, 27, 11, -4, 5, 0, 5], [50, 0, 0, 0, 0, 0, 4, 0], [0, 3, -3, 3, 6, 8, 0, 8], [0, 5, 9, 5, 8, -2, 0, -2], [18, -4, 4, 4, 0, -8, -2, 8], [8, -16, 16, 16, 0, -2, -2, 2], [-12, -10, 10, 10, 0, 7, 0, -7], [0, -2, -26, -2, 5, 8, 0, 8], [0, 0, 20, 0, 0, -2, 0, -2], [-10, 4, -4, -4, 0, -8, 14, 8], [0, -5, 25, -5, -11, -7, 0, -7], [-40, 6, -6, -6, 0, 0, 4, 0], [0, -3, -19, -3, 0, 3, 0, 3], [-10, -7, 7, 7, 0, -14, 4, 14], [0, -1, -9, -1, 4, -11, 0, -11], [20, 0, 0, 0, 0, -4, 8, 4], [-16, -14, 14, 14, 0, 0, -4, 0], [46, 4, -4, -4, 0, 4, 2, -4], [0, -9, -19, -9, -6, -9, 0, -9], [0, 1, 19, 1, -6, 2, 0, 2], [10, 6, -6, -6, 0, 16, 0, -16], [0, 1, 13, 1, 16, 0, 0, 0], [-8, -4, 4, 4, 0, 10, -5, -10], [-8, 8, -8, -8, 0, 0, -2, 0], [0, -1, 9, -1, 6, 2, 0, 2], [6, -3, 3, 3, 0, -13, 2, 13], [0, -5, 11, -5, 4, -3, 0, -3], [2, -11, 11, 11, 0, -9, 1, 9], [0, -5, -25, -5, -2, 6, 0, 6], [44, -4, 4, 4, 0, -2, 6, 2], [0, -10, 22, -10, -11, -6, 0, -6], [-34, -5, 5, 5, 0, -8, -2, 8], [-28, 3, -3, -3, 0, 1, -13, -1], [0, 7, -9, 7, -5, -3, 0, -3], [0, 6, -4, 6, 2, -2, 0, -2], [8, 6, -6, -6, 0, -1, -6, 1], [0, 3, 13, 3, 0, -8, 0, -8], [-12, 1, -1, -1, 0, -3, -6, 3], [0, 6, -24, 6, 4, 14, 0, 14], [-2, 3, -3, -3, 0, 12, -6, -12], [-28, -11, 11, 11, 0, 1, -10, -1], [0, 2, 10, 2, -8, -2, 0, -2], [0, 2, -24, 2, 0, 0, 0, 0], [0, 4, -42, 4, 3, 0, 0, 0], [0, 5, -19, 5, 8, -1, 0, -1], [0, 0, 20, 0, -12, 0, 0, 0], [22, 5, -5, -5, 0, 5, -1, -5], [0, -13, 7, -13, -10, 1, 0, 1], [0, 3, -17, 3, 14, 2, 0, 2], [-14, -3, 3, 3, 0, -13, 12, 13], [-8, 13, -13, -13, 0, 8, -4, -8], [24, -14, 14, 14, 0, -1, 2, 1], [0, 6, 30, 6, -4, 0, 0, 0], [0, 0, 34, 0, -12, 0, 0, 0], [40, 7, -7, -7, 0, 3, 4, -3], [0, -2, -10, -2, 10, -4, 0, -4], [0, -2, 12, -2, 15, -4, 0, -4], [0, 0, 0, 0, 0, 0, -4, 0], [0, -3, 15, -3, -10, 3, 0, 3], [0, -2, -30, -2, -3, 6, 0, 6], [0, -1, -23, -1, -8, 2, 0, 2], [-34, 3, -3, -3, 0, 1, -11, -1], [18, 10, -10, -10, 0, 10, 12, -10], [-4, 10, -10, -10, 0, -6, 5, 6], [0, -5, 19, -5, -1, 3, 0, 3], [36, 2, -2, -2, 0, 8, 6, -8], [0, 5, -7, 5, -7, -7, 0, -7], [-26, -1, 1, 1, 0, -3, -3, 3], [0, 0, 28, 0, 10, -2, 0, -2], [-12, -10, 10, 10, 0, 2, -14, -2], [0, 9, -37, 9, 8, 3, 0, 3], [0, -5, -9, -5, -16, -10, 0, -10], [8, -6, 6, 6, 0, 2, -6, -2], [22, 0, 0, 0, 0, 14, 0, -14], [-54, 2, -2, -2, 0, 4, 1, -4], [0, 4, -12, 4, -18, -4, 0, -4], [0, 3, -17, 3, 6, 8, 0, 8], [-6, -1, 1, 1, 0, 6, 10, -6], [0, 13, -3, 13, 14, 3, 0, 3], [-6, -2, 2, 2, 0, -8, -3, 8], [16, 8, -8, -8, 0, 16, -2, -16], [0, -4, 38, -4, -4, 2, 0, 2], [32, -2, 2, 2, 0, 0, -11, 0], [0, 4, 32, 4, 6, -6, 0, -6], [0, 2, 14, 2, 6, 6, 0, 6], [0, 9, 1, 9, 4, -12, 0, -12], [-8, 0, 0, 0, 0, 4, -6, -4], [0, 0, 36, 0, 16, 0, 0, 0], [-4, -2, 2, 2, 0, -10, 0, 10], [0, 10, -8, 10, 1, 12, 0, 12], [0, 2, -26, 2, 2, -2, 0, -2], [-18, 0, 0, 0, 0, 4, -2, -4], [-6, -4, 4, 4, 0, 1, -4, -1], [0, -2, 0, -2, 8, -2, 0, -2], [0, -5, -9, -5, 9, -11, 0, -11], [16, -5, 5, 5, 0, 6, -4, -6], [18, 0, 0, 0, 0, 6, 6, -6], [0, 8, 12, 8, 4, 8, 0, 8], [0, 15, -19, 15, 8, 6, 0, 6], [24, 11, -11, -11, 0, 3, 2, -3], [38, -1, 1, 1, 0, 0, 8, 0], [10, 2, -2, -2, 0, 3, 12, -3], [0, 8, 30, 8, 1, 2, 0, 2], [-54, 6, -6, -6, 0, -10, 10, 10], [-20, -16, 16, 16, 0, -4, -10, 4], [0, 4, 30, 4, -1, -4, 0, -4], [22, -4, 4, 4, 0, -10, 7, 10], [0, 2, -26, 2, -5, 8, 0, 8], [0, 3, 31, 3, 6, -12, 0, -12], [4, -11, 11, 11, 0, 9, -14, -9], [-24, 1, -1, -1, 0, -5, -3, 5], [0, 2, -14, 2, -18, 0, 0, 0], [0, 5, 33, 5, 9, -3, 0, -3], [0, 8, -34, 8, 2, -4, 0, -4], [58, -7, 7, 7, 0, 13, -13, -13], [0, 10, -36, 10, 10, 2, 0, 2], [2, 7, -7, -7, 0, -11, -2, 11], [0, 15, -7, 15, 8, 6, 0, 6], [-40, 5, -5, -5, 0, 3, -7, -3], [0, 3, 37, 3, 16, -2, 0, -2], [18, -5, 5, 5, 0, 9, -3, -9], [0, -1, -25, -1, 4, 4, 0, 4], [-46, -8, 8, 8, 0, -10, -2, 10], [0, 0, 26, 0, 8, 2, 0, 2], [0, -14, 0, -14, 2, 4, 0, 4], [0, -1, 13, -1, -7, -9, 0, -9], [4, 3, -3, -3, 0, -6, -6, 6], [32, 0, 0, 0, 0, 8, -2, -8], [0, 9, -15, 9, 12, 20, 0, 20], [0, 6, -6, -6, 0, -4, 9, 4], [-4, -5, 5, 5, 0, -17, 2, 17], [0, 10, -6, 10, 2, -10, 0, -10], [-14, -10, 10, 10, 0, -1, 8, 1], [0, 10, -20, 10, 0, 0, 0, 0], [-28, 11, -11, -11, 0, -9, 7, 9], [0, -12, -10, -12, 1, -2, 0, -2], [28, 11, -11, -11, 0, 18, -6, -18], [0, 9, 41, 9, -12, 3, 0, 3], [-24, -5, 5, 5, 0, -17, 16, 17], [0, 1, 23, 1, -11, -3, 0, -3], [-34, 10, -10, -10, 0, 5, 4, -5], [-6, -3, 3, 3, 0, -4, 12, 4], [0, 7, -3, 7, 3, 7, 0, 7], [0, 9, 7, 9, -16, -4, 0, -4], [0, -8, -10, -8, 6, 6, 0, 6], [-34, 12, -12, -12, 0, -4, 4, 4], [0, -5, 27, -5, -11, -7, 0, -7], [-6, 5, -5, -5, 0, -8, 14, 8], [0, 1, -13, 1, -10, -11, 0, -11], [0, -4, -24, -4, -14, -12, 0, -12], [0, 3, 11, 3, -8, 10, 0, 10], [0, -16, -12, -16, 7, 6, 0, 6], [30, 10, -10, -10, 0, 10, 4, -10], [-8, 0, 0, 0, 0, -16, 3, 16], [4, -6, 6, 6, 0, 10, -2, -10], [0, -3, -45, -3, -14, 4, 0, 4], [2, 2, -2, -2, 0, 8, -5, -8], [6, 1, -1, -1, 0, -5, 0, 5], [0, -1, -49, -1, -2, -7, 0, -7], [-18, 21, -21, -21, 0, -2, -6, 2], [32, 4, -4, -4, 0, 4, 4, -4], [0, -6, 4, -6, -25, 2, 0, 2], [-22, -10, 10, 10, 0, -18, -8, 18], [0, 8, -20, 8, -2, -8, 0, -8], [-12, -3, 3, 3, 0, -12, 10, 12], [0, 5, 37, 5, -8, -4, 0, -4], [0, 0, -28, 0, -3, -8, 0, -8], [0, -11, -7, -11, 2, 8, 0, 8], [36, -9, 9, 9, 0, 5, 9, -5], [0, 3, -27, 3, -8, -4, 0, -4], [0, 10, -10, -10, 0, 0, 8, 0], [0, 8, -10, 8, 8, 0, 0, 0], [0, 10, 42, 10, 2, 2, 0, 2], [0, -5, -17, -5, -3, 1, 0, 1], [-14, 8, -8, -8, 0, 2, 12, -2], [0, 5, -31, 5, 0, 18, 0, 18], [18, -17, 17, 17, 0, -6, 0, 6], [0, -6, -10, -6, -8, -6, 0, -6], [6, 4, -4, -4, 0, 15, 0, -15], [0, 5, 23, 5, -16, -1, 0, -1], [-4, -4, 4, 4, 0, -2, 20, 2], [10, 0, 0, 0, 0, 8, 16, -8], [-2, 9, -9, -9, 0, -1, 18, 1], [-24, 5, -5, -5, 0, -12, -4, 12], [-38, -5, 5, 5, 0, -11, 13, 11], [0, -7, -9, -7, -17, 7, 0, 7], [0, 0, 60, 0, -3, 4, 0, 4], [0, 3, -33, 3, -10, 10, 0, 10], [-32, 9, -9, -9, 0, -14, 16, 14], [-22, -5, 5, 5, 0, 1, 2, -1], [0, -2, 16, -2, -4, -10, 0, -10], [20, -6, 6, 6, 0, 11, -14, -11], [0, -2, -10, -2, -16, -8, 0, -8], [42, -1, 1, 1, 0, 2, -6, -2], [0, -8, 36, -8, 2, 0, 0, 0], [0, 10, 32, 10, -16, 8, 0, 8], [0, 6, 22, 6, 4, -2, 0, -2], [36, -12, 12, 12, 0, -4, 2, 4], [0, 2, 10, 2, 0, 10, 0, 10], [-30, 12, -12, -12, 0, 2, -7, -2], [38, -10, 10, 10, 0, 9, -20, -9], [0, -15, -7, -15, -14, -9, 0, -9], [0, -9, 51, -9, 2, 4, 0, 4], [0, -1, 7, -1, -29, 1, 0, 1], [-14, -10, 10, 10, 0, -6, 3, 6], [0, -6, -36, -6, 2, -8, 0, -8], [0, -13, 25, -13, -4, 10, 0, 10], [24, 4, -4, -4, 0, 8, 15, -8], [0, 10, 4, 10, 1, -18, 0, -18], [0, 3, 7, 3, 0, 4, 0, 4], [0, 4, 12, 4, -2, -10, 0, -10], [-20, 2, -2, -2, 0, 7, 14, -7], [12, 7, -7, -7, 0, 17, 11, -17], [36, -10, 10, 10, 0, -6, 4, 6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4680_l_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4680_2_l_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_4680_2_l_f(:prec:=8) chi := MakeCharacter_4680_l(); 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_4680_2_l_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_4680_2_l_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4680_l(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![8, 0, 1]>,<11,R![8, 24, 6, -8, 1]>],Snew); return Vf; end function;