// Make newform 4608.2.k.bi in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4608_k();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4608_k_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_4608_2_k_bi();". // 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_4608_2_k_bi();". // 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, 0, 0, 0, 0, 0, 0, 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, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0], [0, -1, 0, 1, 0, -1, 0, -1], [0, 1, 0, 1, 0, -1, 0, 1], [0, 2, 0, 0, 0, 0, 0, -2]]; Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1]; 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_4608_k();" function MakeCharacter_4608_k() N := 4608; order := 4; char_gens := [3583, 2053, 4097]; v := [4, 3, 4]; // 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_4608_k_Hecke();" function MakeCharacter_4608_k_Hecke(Kf) N := 4608; order := 4; char_gens := [3583, 2053, 4097]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 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], [1, 0, -1, 0, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, -1, 0, 1], [1, 1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, -1, -1], [0, 0, 0, 1, 0, -2, 2, 0], [1, 3, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 2, 0], [5, 0, -5, 0, -1, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 2, 2, 0], [-1, 0, 1, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [5, -3, 5, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 3, -1], [0, 0, 0, -3, 0, -2, 2, 0], [0, -3, 2, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4], [0, 0, 0, -2, 0, 0, -3, -2], [0, 1, 2, 0, 1, 0, 0, 0], [8, -1, 0, 0, 1, 0, 0, 0], [-5, 0, 5, 0, 5, 0, 0, 0], [0, 0, 0, 3, 0, -2, 2, 0], [0, 0, 0, 1, 0, 3, 0, -1], [-5, -3, -5, 0, 0, 0, 0, 0], [-2, -2, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, -2, -2, 4], [0, 0, 0, -3, 0, 0, 1, -3], [0, -6, 4, 0, -6, 0, 0, 0], [0, 0, 0, -2, 0, 9, 0, 2], [-3, 0, 3, 0, -3, 0, 0, 0], [0, 0, 0, -1, 0, -2, 2, 0], [-3, 1, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0], [0, 0, 0, 1, 0, 0, 0, 0], [15, -1, 15, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 1, 2], [3, 0, -3, 0, -5, 0, 0, 0], [0, 0, 0, 0, 0, -2, -2, 4], [-8, 3, 0, 0, -3, 0, 0, 0], [9, 0, -9, 0, 1, 0, 0, 0], [0, 0, 0, -5, 0, -4, 4, 0], [0, 0, 0, -4, 0, 0, 7, -4], [0, 0, 0, 0, 0, 2, 2, 0], [0, 0, 0, 0, 0, 0, 1, 0], [7, 0, -7, 0, 3, 0, 0, 0], [0, -1, -18, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, -8, -8, 0], [16, -1, 0, 0, 1, 0, 0, 0], [0, 0, 0, -2, 0, 7, 0, 2], [-2, 6, 0, 0, -6, 0, 0, 0], [0, 0, 0, 1, 0, -6, 6, 0], [-11, -5, -11, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, -6, 4], [1, 0, -1, 0, -1, 0, 0, 0], [0, 7, -2, 0, 7, 0, 0, 0], [0, 0, 0, 3, 0, 1, 0, -3], [1, 0, -1, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, -9, 1], [0, 0, 0, 5, 0, -4, 4, 0], [0, 4, 20, 0, 4, 0, 0, 0], [1, 7, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -3, 0, -1], [6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, -5, 0, 4], [-9, 5, -9, 0, 0, 0, 0, 0], [-6, -6, 0, 0, 6, 0, 0, 0], [0, 0, 0, 1, 0, 2, -2, 0], [0, 0, 0, 0, 0, 6, 6, 0], [9, 0, -9, 0, 3, 0, 0, 0], [0, 0, 0, -4, 0, 5, 0, 4], [0, 0, 0, 0, 0, 4, 4, 4], [-13, 0, 13, 0, 1, 0, 0, 0], [-5, 1, -5, 0, 0, 0, 0, 0], [16, -3, 0, 0, 3, 0, 0, 0], [0, 4, 20, 0, 4, 0, 0, 0], [0, 0, 0, 3, 0, 0, -3, 3], [-1, 0, 1, 0, -5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -9, 0, 0, 9, 0, 0, 0], [0, 0, 0, 7, 0, -8, 8, 0], [0, 0, 0, 1, 0, 3, 0, -1], [-8, 5, 0, 0, -5, 0, 0, 0], [0, -8, -4, 0, -8, 0, 0, 0], [-3, -17, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, -4, -8], [0, 0, 0, -5, 0, 0, -11, -5], [0, 0, 0, 0, 0, 6, 6, -12], [0, 0, 0, -5, 0, -8, 8, 0], [0, 0, 0, -2, 0, -5, 0, 2], [0, 0, 0, -4, 0, 0, 3, -4], [0, 0, 0, -7, 0, -2, 2, 0], [3, 11, 3, 0, 0, 0, 0, 0], [0, 2, -12, 0, 2, 0, 0, 0], [0, 0, 0, -5, 0, -11, 0, 5], [-3, -3, -3, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 3, 3], [-1, -5, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 9, 1], [0, -2, -20, 0, -2, 0, 0, 0], [0, 0, 0, 3, 0, -11, 0, -3], [6, 4, 0, 0, -4, 0, 0, 0], [0, 0, 0, 5, 0, 7, 0, -5], [-2, 4, 0, 0, -4, 0, 0, 0], [0, 0, 0, 9, 0, -8, 8, 0], [0, -1, -18, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 19, 0, 0, 0], [0, -3, -30, 0, -3, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, -1], [0, 0, 0, 3, 0, -2, 2, 0], [24, 5, 0, 0, -5, 0, 0, 0], [0, 0, 0, 3, 0, 0, 3, 3], [0, 0, 0, 5, 0, -2, 2, 0], [-11, 11, -11, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, -11, 2], [-5, 0, 5, 0, 11, 0, 0, 0], [-24, -1, 0, 0, 1, 0, 0, 0], [11, 0, -11, 0, 1, 0, 0, 0], [0, 0, 0, -4, 0, 11, 0, 4], [0, 0, 0, 8, 0, 0, -1, 8], [19, -1, 19, 0, 0, 0, 0, 0], [21, 0, -21, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -8], [0, 0, 0, -1, 0, 12, -12, 0], [23, -7, 23, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, -17, -6], [0, 0, 0, 9, 0, 4, -4, 0], [0, 0, 0, 0, 0, 0, 0, -4], [-5, 0, 5, 0, -13, 0, 0, 0], [0, 8, 12, 0, 8, 0, 0, 0], [8, -1, 0, 0, 1, 0, 0, 0], [-21, 0, 21, 0, 5, 0, 0, 0], [0, 0, 0, 2, 0, 0, -17, 2], [-21, -5, -21, 0, 0, 0, 0, 0], [0, -4, 36, 0, -4, 0, 0, 0], [0, 0, 0, -1, 0, -7, 0, 1], [23, 0, -23, 0, -3, 0, 0, 0], [0, 0, 0, -1, 0, 12, -12, 0], [0, 0, 0, 5, 0, 7, 0, -5], [13, -3, 13, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 12, -12, 0], [-7, 0, 7, 0, -21, 0, 0, 0], [0, 10, -4, 0, 10, 0, 0, 0], [0, 0, 0, 0, 0, -11, 0, 0], [0, 0, 0, 0, 0, -10, -10, 4], [11, 17, 11, 0, 0, 0, 0, 0], [-6, -12, 0, 0, 12, 0, 0, 0], [0, 0, 0, 5, 0, 0, 3, 5], [0, 0, 0, -3, 0, 4, -4, 0], [0, 0, 0, 8, 0, -3, 0, -8], [0, 0, 0, 0, 0, 6, 6, 4], [0, 0, 0, -1, 0, 10, -10, 0], [40, 1, 0, 0, -1, 0, 0, 0], [0, 9, 18, 0, 9, 0, 0, 0], [5, -5, 5, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, -15, -1], [0, -6, 28, 0, -6, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 10, 0, -5, 0, -10], [-16, -11, 0, 0, 11, 0, 0, 0], [0, 0, 0, -11, 0, 4, -4, 0], [0, 0, 0, 0, 0, -8, -8, -4], [13, 0, -13, 0, -5, 0, 0, 0], [0, 19, 0, 0, -19, 0, 0, 0], [5, 0, -5, 0, 9, 0, 0, 0], [0, 0, 0, 9, 0, -5, 0, -9], [-9, -7, -9, 0, 0, 0, 0, 0], [0, 0, 0, 13, 0, -2, 2, 0], [0, 3, 14, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, -10, -10, -8], [0, -5, 30, 0, -5, 0, 0, 0], [0, 0, 0, -10, 0, 1, 0, 10], [27, 0, -27, 0, -3, 0, 0, 0], [0, 0, 0, 7, 0, 4, -4, 0], [9, 5, 9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -12], [0, 0, 0, 0, 0, 0, -23, 0], [15, 0, -15, 0, 15, 0, 0, 0], [0, 5, -30, 0, 5, 0, 0, 0], [0, 0, 0, 0, 0, -6, -6, -4], [-19, 0, 19, 0, -11, 0, 0, 0], [13, 9, 13, 0, 0, 0, 0, 0], [0, 0, 0, 6, 0, 0, -13, 6], [0, -4, 12, 0, -4, 0, 0, 0], [0, 0, 0, 0, 0, 4, 4, -4], [6, -10, 0, 0, 10, 0, 0, 0], [0, 0, 0, 12, 0, -1, 0, -12], [0, 0, 0, 7, 0, 0, 11, 7], [-15, 3, -15, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, 0, -11, -4], [0, 13, 18, 0, 13, 0, 0, 0], [2, 12, 0, 0, -12, 0, 0, 0], [21, -7, 21, 0, 0, 0, 0, 0], [-2, 4, 0, 0, -4, 0, 0, 0], [0, 0, 0, -19, 0, -6, 6, 0], [-9, -5, -9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -16, -16, 0], [-31, 0, 31, 0, 3, 0, 0, 0], [2, 18, 0, 0, -18, 0, 0, 0], [0, 0, 0, -1, 0, -13, 0, 1], [25, -1, 25, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 16, 16, 4], [0, 0, 0, 6, 0, 0, 13, 6], [0, 7, -34, 0, 7, 0, 0, 0], [0, 0, 0, -1, 0, -23, 0, 1], [16, 3, 0, 0, -3, 0, 0, 0], [-11, 0, 11, 0, -3, 0, 0, 0], [0, 0, 0, -5, 0, -12, 12, 0], [0, 0, 0, 7, 0, -5, 0, -7], [0, 0, 0, 13, 0, -16, 16, 0], [0, -6, -4, 0, -6, 0, 0, 0], [0, 0, 0, 0, 0, 4, 4, -4], [32, 5, 0, 0, -5, 0, 0, 0], [0, 0, 0, 9, 0, 12, -12, 0], [9, -17, 9, 0, 0, 0, 0, 0], [7, 0, -7, 0, -21, 0, 0, 0], [0, 0, 0, -13, 0, -10, 10, 0], [24, -15, 0, 0, 15, 0, 0, 0], [0, 0, 0, 0, 0, 4, 4, 4], [0, 0, 0, 3, 0, 0, 1, 3], [1, 0, -1, 0, 27, 0, 0, 0], [0, -19, -14, 0, -19, 0, 0, 0], [0, 0, 0, 0, 0, 8, 8, 12], [0, 0, 0, 3, 0, 2, -2, 0], [0, 0, 0, -4, 0, 15, 0, 4], [3, 5, 3, 0, 0, 0, 0, 0], [0, 0, 0, -5, 0, 0, 3, -5], [0, 0, 0, 0, 0, -12, -12, 8], [0, 1, 2, 0, 1, 0, 0, 0], [0, 0, 0, 8, 0, 5, 0, -8], [0, 0, 0, 0, 0, 4, 4, -20], [16, -13, 0, 0, 13, 0, 0, 0], [-19, 0, 19, 0, 1, 0, 0, 0], [0, 0, 0, -5, 0, -1, 0, 5], [0, 0, 0, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0], [0, 0, 0, -7, 0, 9, 0, 7], [0, 0, 0, -13, 0, -6, 6, 0], [-7, 9, -7, 0, 0, 0, 0, 0], [26, -8, 0, 0, 8, 0, 0, 0], [0, 0, 0, -19, 0, -8, 8, 0], [0, 0, 0, 0, 0, 14, 14, -8], [0, 0, 0, 3, 0, 0, 17, 3], [0, 0, 0, 6, 0, 21, 0, -6], [0, 0, 0, 0, 0, 4, 4, 0], [-17, -19, -17, 0, 0, 0, 0, 0], [-38, -14, 0, 0, 14, 0, 0, 0], [0, 0, 0, 9, 0, -22, 22, 0], [0, 21, -14, 0, 21, 0, 0, 0], [-19, -13, -19, 0, 0, 0, 0, 0], [0, 0, 0, 12, 0, 0, -7, 12], [-21, 0, 21, 0, -25, 0, 0, 0], [0, 0, 0, 2, 0, -15, 0, -2], [-13, 0, 13, 0, 5, 0, 0, 0], [0, -16, 4, 0, -16, 0, 0, 0], [0, 0, 0, 0, 0, 4, 4, 12], [0, 0, 0, -2, 0, 0, -15, -2], [-11, 0, 11, 0, -25, 0, 0, 0], [-25, 21, -25, 0, 0, 0, 0, 0], [-8, -3, 0, 0, 3, 0, 0, 0], [0, 0, 0, 13, 0, 0, 3, 13], [53, 3, 53, 0, 0, 0, 0, 0], [0, -3, 50, 0, -3, 0, 0, 0], [0, 0, 0, -10, 0, 1, 0, 10], [11, 0, -11, 0, -15, 0, 0, 0], [33, 17, 33, 0, 0, 0, 0, 0], [0, 0, 0, -13, 0, 0, 11, -13], [0, 13, 34, 0, 13, 0, 0, 0], [0, 0, 0, 0, 0, -12, -12, -12], [38, -10, 0, 0, 10, 0, 0, 0], [0, 0, 0, 7, 0, -2, 2, 0], [0, 0, 0, 3, 0, 15, 0, -3], [37, 9, 37, 0, 0, 0, 0, 0], [0, 21, -14, 0, 21, 0, 0, 0], [0, 0, 0, -9, 0, 0, 1, -9], [0, 0, 0, 0, 0, -26, -26, -8], [0, 0, 0, 3, 0, -14, 14, 0], [0, 0, 0, -11, 0, 10, -10, 0], [-9, 0, 9, 0, 3, 0, 0, 0], [0, 0, 0, -6, 0, 1, 0, 6], [0, 0, 0, 0, 0, -10, -10, 20], [0, 11, 0, 0, -11, 0, 0, 0], [-9, 0, 9, 0, 29, 0, 0, 0], [0, 0, 0, -1, 0, 14, -14, 0], [26, 0, 0, 0, 0, 0, 0, 0], [13, 27, 13, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 0, -23, -3], [0, -7, -30, 0, -7, 0, 0, 0], [0, 0, 0, -1, 0, -1, 0, 1], [-13, 1, -13, 0, 0, 0, 0, 0], [27, 23, 27, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, -6, 4], [-11, 0, 11, 0, 13, 0, 0, 0], [0, 0, 0, 7, 0, 23, 0, -7], [0, 0, 0, -7, 0, 0, 7, -7], [0, 6, -20, 0, 6, 0, 0, 0], [-25, 19, -25, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 18, 18, 4], [0, 0, 0, -10, 0, 0, -11, -10], [0, 0, 0, -1, 0, 25, 0, 1], [-24, -13, 0, 0, 13, 0, 0, 0], [0, 0, 0, 7, 0, 15, 0, -7], [-15, -7, -15, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 12, -12, 0], [-11, 0, 11, 0, 11, 0, 0, 0], [0, 0, 0, 0, 0, 2, 2, -12], [-19, 0, 19, 0, 17, 0, 0, 0], [-56, 5, 0, 0, -5, 0, 0, 0], [0, 0, 0, -17, 0, 0, 7, -17], [0, 0, 0, -15, 0, -4, 4, 0], [0, 17, -14, 0, 17, 0, 0, 0], [0, 0, 0, -10, 0, 0, -3, -10], [0, 0, 0, 0, 0, -16, -16, 4], [2, -4, 0, 0, 4, 0, 0, 0], [62, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 8, 0, 0, 3, 8], [0, 8, -28, 0, 8, 0, 0, 0], [-15, 15, -15, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 24, 24, 12], [0, 2, -28, 0, 2, 0, 0, 0], [-48, 7, 0, 0, -7, 0, 0, 0], [0, 0, 0, 14, 0, 0, -9, 14], [0, 0, 0, -7, 0, 29, 0, 7], [0, 0, 0, 0, 0, 10, 10, 4], [9, 0, -9, 0, -19, 0, 0, 0], [17, -3, 17, 0, 0, 0, 0, 0], [9, -37, 9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 4, 20], [0, 0, 0, -3, 0, 0, -11, -3], [0, 0, 0, -16, 0, -19, 0, 16], [0, 0, 0, 8, 0, 7, 0, -8], [-17, 13, -17, 0, 0, 0, 0, 0], [26, -6, 0, 0, 6, 0, 0, 0], [0, -1, -50, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, -2, -2, -12], [-21, 0, 21, 0, -25, 0, 0, 0], [0, 12, -20, 0, 12, 0, 0, 0], [-21, 0, 21, 0, -11, 0, 0, 0], [0, 0, 0, 7, 0, -14, 14, 0], [27, -9, 27, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, 5, -6], [15, 0, -15, 0, -5, 0, 0, 0], [0, 0, 0, 13, 0, 25, 0, -13], [0, 0, 0, 0, 0, -4, -4, 8], [13, 0, -13, 0, 25, 0, 0, 0], [0, 0, 0, -16, 0, 0, 11, -16], [0, 2, 44, 0, 2, 0, 0, 0], [-25, 19, -25, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 16, 16, -12], [-53, 0, 53, 0, 7, 0, 0, 0], [0, 3, -50, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 16, 16, 16], [0, 0, 0, 2, 0, -33, 0, -2], [-32, 13, 0, 0, -13, 0, 0, 0], [0, 0, 0, 5, 0, 4, -4, 0], [-43, 0, 43, 0, -1, 0, 0, 0], [0, -1, 62, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, -10, -10, -8], [0, 0, 0, -2, 0, 27, 0, 2], [0, 0, 0, -16, 0, 0, -1, -16], [0, 3, 46, 0, 3, 0, 0, 0], [15, 11, 15, 0, 0, 0, 0, 0], [0, 0, 0, -17, 0, -6, 6, 0], [0, -26, 4, 0, -26, 0, 0, 0], [0, 0, 0, 1, 0, 0, -11, 1], [0, 0, 0, 13, 0, -15, 0, -13], [0, 0, 0, 0, 0, -18, -18, -8], [-33, 0, 33, 0, -15, 0, 0, 0], [0, 0, 0, 7, 0, 2, -2, 0], [23, -19, 23, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 17, 2], [0, 0, 0, 0, 0, 2, 2, 0], [34, 2, 0, 0, -2, 0, 0, 0], [48, 5, 0, 0, -5, 0, 0, 0], [0, 4, 4, 0, 4, 0, 0, 0], [17, -1, 17, 0, 0, 0, 0, 0], [0, 4, -28, 0, 4, 0, 0, 0], [0, 0, 0, -9, 0, -10, 10, 0], [30, -2, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, -17, 0], [0, 0, 0, 5, 0, -14, 14, 0], [0, 0, 0, 0, 0, -10, -10, 12], [1, 0, -1, 0, 35, 0, 0, 0], [0, 0, 0, -12, 0, 9, 0, 12], [0, 0, 0, 0, 0, 0, 0, -4], [-26, 4, 0, 0, -4, 0, 0, 0], [3, 0, -3, 0, 17, 0, 0, 0], [0, 0, 0, -11, 0, -5, 0, 11], [0, 0, 0, -8, 0, 0, -45, -8], [0, 0, 0, 1, 0, 20, -20, 0], [0, 8, -12, 0, 8, 0, 0, 0], [0, 0, 0, 0, 0, -8, -8, 4], [0, -2, -12, 0, -2, 0, 0, 0], [0, 0, 0, -8, 0, -19, 0, 8], [-1, 0, 1, 0, -15, 0, 0, 0], [-9, -39, -9, 0, 0, 0, 0, 0], [56, -11, 0, 0, 11, 0, 0, 0], [0, 0, 0, 0, 0, 12, 12, 4], [0, -10, -52, 0, -10, 0, 0, 0], [17, 0, -17, 0, 5, 0, 0, 0], [0, 0, 0, -1, 0, -22, 22, 0], [9, 25, 9, 0, 0, 0, 0, 0], [-32, -19, 0, 0, 19, 0, 0, 0], [0, 0, 0, 16, 0, 0, -5, 16], [0, 0, 0, -7, 0, 0, -15, -7], [6, 6, 0, 0, -6, 0, 0, 0], [-33, 0, 33, 0, 13, 0, 0, 0], [0, 0, 0, -10, 0, 15, 0, 10], [0, 0, 0, 2, 0, 0, 23, 2], [0, -9, 30, 0, -9, 0, 0, 0], [-9, 7, -9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 4, -8], [0, 0, 0, 23, 0, 2, -2, 0], [32, 17, 0, 0, -17, 0, 0, 0], [0, 0, 0, 1, 0, 8, -8, 0], [51, 11, 51, 0, 0, 0, 0, 0], [13, 0, -13, 0, -9, 0, 0, 0], [0, 0, 0, 0, 0, 14, 14, 0], [0, 0, 0, -2, 0, 19, 0, 2], [0, 8, -20, 0, 8, 0, 0, 0], [15, -37, 15, 0, 0, 0, 0, 0], [0, 0, 0, -5, 0, 0, 13, -5], [0, 17, -46, 0, 17, 0, 0, 0], [0, 0, 0, 1, 0, -31, 0, -1], [0, 0, 0, -19, 0, -4, 4, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4608_k_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4608_2_k_bi();". // 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_4608_2_k_bi(:prec:=8) chi := MakeCharacter_4608_k(); 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_4608_2_k_bi();". // 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_4608_2_k_bi( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4608_k(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![4, 8, 8, -4, 1]>,<7,R![32, 0, 16, 0, 1]>,<11,R![153664, 0, 0, 0, 816, 0, 0, 0, 1]>,<13,R![4, 8, 8, -4, 1]>,<19,R![64, 0, 0, 0, 1584, 0, 0, 0, 1]>],Snew); return Vf; end function;