// Make newform 1045.2.a.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1045_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_1045_2_a_i();". // 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_1045_2_a_i();". // 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 := [8, 6, -28, -17, 28, 12, -9, -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, 1, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0, 0], [3, -3, -2, 1, 0, 0, 0, 0], [2, 5, -4, -2, 1, 0, 0, 0], [-12, -8, 21, 10, -8, -2, 1, 0], [11, -4, -29, 11, 18, -6, -3, 1], [-6, 12, 19, -24, -13, 9, 2, -1]]; 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_1045_a();" function MakeCharacter_1045_a() N := 1045; order := 1; char_gens := [837, 761, 496]; v := [1, 1, 1]; // 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; function MakeCharacter_1045_a_Hecke(Kf) return MakeCharacter_1045_a(); 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 := [[-1, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, -1, 0, 1, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 0, 1, -1, -1, 0], [-2, -1, -1, 0, 0, -1, 1, -1], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 1, 0, 1, 0, -1], [-1, 2, 2, 2, 0, -2, -1, -1], [0, -2, 0, -1, -1, 0, 1, -1], [-2, 1, 1, 0, -2, -1, 1, 0], [-1, 1, 0, -2, -1, -1, 1, 0], [-2, -2, 0, 1, 1, 1, -1, 0], [-2, 0, -2, 0, 0, 1, 2, -1], [-2, -2, -1, 0, -1, 1, 1, 0], [-4, 1, -1, 0, 0, -1, -3, 3], [2, 1, 0, -1, 0, 1, 1, 4], [-2, 1, 1, 1, -1, -1, -2, 1], [-4, 3, 0, 2, -1, -1, -2, -1], [-2, -1, 1, 1, 1, 0, 0, -1], [0, 0, -1, 0, 1, 3, -1, 1], [-4, 3, 3, 0, -2, -1, 1, -1], [-2, 1, 0, 0, -1, 2, 2, 0], [3, -3, 1, 0, 0, 3, 4, 1], [-1, 0, 3, 4, 3, -4, -4, -3], [-3, 3, 3, -2, 0, -2, -2, 0], [-5, 1, 0, -1, -2, 0, 2, -3], [1, -1, 0, 1, 2, 2, -2, 0], [3, 2, -1, -3, 0, 3, 3, 4], [-3, -2, 1, 2, 1, 0, 2, 0], [-6, 2, -3, 1, 4, -1, -2, 0], [8, 0, 2, -2, 0, 1, 4, -1], [-6, 1, -5, 0, 0, 3, -1, 1], [1, -4, -2, -2, -2, 1, 3, 2], [-1, 2, -4, 2, 2, 0, -5, 1], [1, -4, -1, 1, 2, -3, -5, -1], [-2, 0, 4, 1, 1, 0, -3, 1], [0, -6, -2, -1, 1, -3, 3, -1], [-3, 2, 0, 1, 1, 2, -2, -3], [-3, 1, -2, 1, 2, 2, 2, 0], [0, 1, -4, -6, -3, 1, 4, 1], [-2, -2, -5, -4, -5, 4, 7, -2], [-10, 4, -4, -4, -2, -1, 0, 2], [4, -1, 3, 3, -3, -3, -4, -2], [6, 1, -3, -1, 3, 2, 4, 1], [3, 4, 3, 5, 2, -1, -5, -3], [-2, 1, -4, -1, -2, 1, 5, 1], [-10, 5, -1, -5, -3, 1, 2, -1], [1, 4, 0, -1, 3, -2, -6, 2], [1, 0, 6, 9, 5, -3, -8, -1], [3, -1, 2, 0, -3, -1, 7, 0], [-2, 0, 4, -1, 1, -5, -5, 0], [-6, -6, 1, 8, 1, -2, -7, -4], [1, 1, 3, 1, -3, 0, 1, -3], [9, -2, 2, 0, 2, 2, -3, 5], [6, 5, 2, -1, -2, 1, -1, 2], [5, 2, 6, 1, -3, 3, -2, 5], [0, 3, -2, 2, -1, 4, 0, 0], [1, -9, -1, -2, 2, 2, -2, 1], [-6, -2, -3, 4, 5, -1, -1, -3], [-14, 4, 1, 0, -3, -2, 1, -3], [-10, 0, 0, 0, 0, 2, -8, -1], [-4, 2, 3, 1, 2, 2, -4, -3], [6, 3, 0, 1, 4, 3, -5, 2], [-4, -3, 8, 8, 3, -4, -6, -5], [0, 4, -1, -6, -1, 3, 1, 1], [-5, -5, -3, -3, 3, 2, 1, 3], [-5, -3, 1, 0, 4, 1, -4, 2], [-7, -1, -4, -2, 1, -5, -1, -6], [1, 3, 6, -2, -1, -1, 5, 0], [3, 1, 5, -2, 2, 0, 0, 3], [-10, -7, 1, 1, 3, -2, -2, -5], [8, -6, 0, 0, 0, 4, -4, 1], [12, -4, -7, -1, 0, 6, 8, 5], [-4, -2, -1, 4, 1, 3, 1, -6], [11, 3, -4, -1, 0, -2, -6, 2], [7, -3, 2, -1, 0, -3, 2, 3], [0, -5, -1, -4, -2, -1, 3, 5], [-8, 8, 1, 1, -2, -5, -2, -6], [-6, 5, -2, -5, -2, 0, 3, -1], [2, 8, 2, 5, -3, -1, -1, 0], [-9, -7, -3, 1, -5, 1, 7, -5], [2, 7, -6, -1, 4, 3, -3, 3], [7, -1, 2, 2, 3, -3, 1, 2], [0, -6, -1, 3, -2, -2, 2, 5], [7, -5, 0, -1, 4, 5, 0, 1], [-4, 10, 1, -6, -3, 5, 9, 3], [-7, 0, -5, -2, 1, 0, -4, 3], [-1, 5, -3, -1, -1, 1, 9, 1], [10, -10, 2, 3, 3, -3, -1, 2], [-6, -2, 8, 7, 3, 3, 1, -2], [-16, 3, 2, 1, 4, -2, -3, -2], [-8, -1, -4, -6, -1, 3, 10, 5], [3, 6, 5, 3, 0, 1, -9, 3], [5, 1, 1, 3, 1, -4, -7, 4], [-11, 2, 6, 2, 0, -2, -11, -1], [-10, -2, -3, 0, 1, -5, 5, -1], [-9, 2, 0, 5, 3, 4, 2, -3], [7, -2, 4, -4, -4, -1, 1, 4], [7, -4, -1, -1, -4, -2, 5, -1], [-10, 4, 9, 4, -3, -7, -15, -3], [-7, -12, 1, 1, 0, 3, 1, 2], [-3, 1, 2, -9, -4, -2, 2, 2], [1, -1, -11, 4, 2, 1, -6, 0], [0, 0, -1, -4, -3, -5, 1, 1], [-5, -10, -4, -9, 3, 7, 4, 3], [6, -8, 3, -2, 1, -2, -5, 6], [-2, -2, 5, -2, -1, -2, 3, -6], [-10, 2, -2, -2, -2, 7, 2, 1], [0, -9, -10, -2, 5, -1, 2, -2], [-6, 5, -6, -13, -4, -2, 5, 7], [11, 1, -3, -3, -1, 1, 9, -3], [0, -3, 0, 5, 2, 4, 9, -5], [10, -2, 3, 0, 7, 0, -1, 2], [-8, 8, 3, -3, 0, -7, 0, 0], [14, 3, 9, 2, 0, -7, -1, -1], [-9, 5, 1, 4, 2, -6, -6, -3], [-1, 3, 1, 4, -2, 1, -10, -2], [1, -1, -4, -5, -8, 5, 20, -1], [-4, 9, -6, -3, 0, 2, 5, -3], [4, -4, 9, 4, -3, 2, 11, -3], [1, -3, 13, 4, 2, -2, -8, -2], [-10, -8, -9, -3, -2, -1, 14, 1], [-2, 4, 0, -1, 1, 4, 3, -1], [15, -8, -2, 2, 6, 6, 7, 1], [11, 4, -9, -3, -6, 4, 3, 2], [5, -4, -6, -5, 5, 1, 8, 5], [5, 8, 9, 4, -1, -9, 0, -10], [3, -4, -1, -1, 6, -3, -13, 3], [11, 8, -1, -6, -3, 9, -2, 4], [6, 2, 6, 2, -2, -15, -6, -8], [11, -12, 9, -1, -2, -3, -5, 3], [12, -9, -3, 2, 2, 7, 7, 7], [-13, -3, -13, -7, 1, 8, 1, 8], [-9, 1, -4, -5, -2, -4, -6, -6], [3, 7, 6, -4, -3, -3, 3, 1], [-6, 1, 5, 7, 3, -1, -8, -3], [3, -7, -7, -10, 2, -2, 4, -2], [19, -5, 6, -5, -2, 0, -4, 4], [3, 0, -9, -14, 1, 4, 8, 7], [-3, -10, 2, 4, 2, 2, -5, -1], [6, 12, -4, -1, 3, -2, -1, -3], [-6, 7, 0, 8, -3, -6, -6, -3], [9, -2, 8, 2, -6, -3, -3, -4], [-3, 6, 2, -5, 3, 4, 2, 0], [14, -4, -11, -8, -3, 7, 1, 9], [-3, 5, 10, 13, 0, 2, -6, 3], [8, 16, 2, -4, 2, -4, -2, 8], [2, -6, 0, -2, -2, 4, -2, -1], [-7, -7, -6, -4, 5, 2, 3, 2], [4, 0, 7, 2, -5, -10, 5, -8], [10, 13, 2, 3, 0, -6, -5, 1], [8, 0, 6, 1, 1, 2, -1, 8], [-1, 11, 0, -10, -5, 4, 3, 2], [-3, -1, -3, 1, 5, -9, -15, -3], [-2, -1, -4, 7, -2, -5, -1, -2], [-6, 5, 1, 3, -1, -9, -8, 4], [-5, 5, 1, 7, 7, -2, -11, 6], [-20, -11, -3, 1, 1, -1, 8, -6], [-13, 4, -10, -6, 4, -3, -5, -2], [7, 0, -12, -4, 4, 0, -7, 0], [-6, 0, 6, -3, -7, -4, 15, 1], [1, 0, 14, 0, 0, -2, 1, -5], [-1, -12, 3, -5, -4, 3, 13, -4], [-2, 6, -11, -1, 0, 2, 12, 0], [0, 8, 1, 4, -1, -7, -3, -3], [-2, 2, 9, 2, 1, -2, -19, 4], [26, -3, -1, -8, 4, 7, -3, 7], [24, 5, 11, 0, -4, 4, -3, 4], [-14, 1, -7, -6, -4, -2, 9, -2], [4, 11, 2, -7, -6, 6, 1, 11], [11, 1, -1, -6, -8, -3, 8, 0], [-1, 8, -5, -9, -2, -2, 5, 0], [8, -11, -6, -3, 4, -4, -5, 1], [8, -12, -9, 0, -7, 12, 7, 0], [-10, 4, 8, -6, -2, -8, -4, -4], [-2, 5, -15, 0, -2, 1, 7, -1], [13, -3, -6, 2, -7, 4, 3, 0], [-13, -9, 1, 3, -1, -3, -1, 3], [-15, 1, -5, -3, -5, 4, 5, 4], [-8, -8, 1, 4, 3, 0, -1, 0], [-6, 17, -1, -3, 3, -4, -12, 4], [-17, 7, -2, 4, -1, -6, -11, 6], [-17, 11, -3, -8, -4, 7, -4, 2], [-5, -1, 1, 1, 5, -2, -5, -8], [-6, 14, 8, 8, -8, -3, 0, -8], [-14, 10, 2, 11, 11, -2, -17, 6], [-11, 6, -1, -1, -2, -2, 3, 2], [-10, -5, -12, 2, -1, 12, 6, 4], [-2, -8, -10, 6, 0, 11, 4, -2], [1, -7, 7, -10, -4, 4, 12, 1], [-12, 14, 11, 2, -3, -4, -9, 0], [-12, -10, -11, -3, -4, 6, 8, -3], [-2, 4, -3, 4, 7, 8, -7, 8], [-11, -9, -14, -3, 4, -2, 6, -7], [5, 4, 12, -1, 3, -10, 2, -4], [0, 11, -15, -10, -6, 8, 1, 8], [-19, 3, 1, -10, -6, -2, 8, 7], [-8, -11, -11, -1, -5, 6, 14, 1], [-36, -3, -2, 4, 5, -13, -4, -7], [6, 15, 12, -4, 1, -12, 0, -4], [-16, 2, -1, 16, 7, -4, -11, -1], [9, -13, 11, 2, 6, 3, -4, 2], [-23, -3, -18, -15, -2, 9, 12, 11], [15, -5, 2, -9, 2, 4, -8, 6], [-19, -2, 2, 2, 4, -12, -5, -7], [-18, 11, -15, -5, 1, 1, 4, 3], [-1, 7, 7, 0, -8, -14, 10, -7], [5, -8, 15, 2, -3, 1, 12, -4], [-19, 6, 11, 3, 0, 0, -5, 4], [-7, -19, -7, -10, 2, 8, 8, 1], [-17, 3, 2, 11, 4, 0, -10, -4], [-10, 1, -13, -20, -2, 5, 17, 12], [4, 5, -10, -13, -6, -2, 1, 3], [10, -10, 6, 12, 4, -5, -8, -3], [-30, 5, -7, -6, -8, -2, 9, 0], [14, 1, 6, -1, -10, 4, 9, 3], [16, 1, 3, -4, 2, -3, -1, -4], [-3, 24, -6, -3, 3, 0, -10, 8], [-10, -2, -13, -9, 0, 4, 2, 9], [-6, 9, 2, 8, -3, -19, -8, -13], [0, 9, 2, 9, 0, 1, 3, -8], [2, -4, 0, -14, 0, -7, 6, 3], [6, 13, 14, 8, 1, 1, -14, 6], [3, 1, 0, 6, 3, 5, -7, -10], [-8, 5, -4, 7, 8, -6, -9, -2], [-7, -5, -8, -5, 0, 7, -4, 5], [9, -5, 12, 11, 8, -7, 2, -8], [10, -12, -8, -2, 6, 9, -6, 9], [5, -2, -2, 0, -2, 0, -9, -11], [8, 1, 3, 0, -8, -2, -5, 2], [19, -6, -3, 1, 6, -3, -7, -1], [10, -16, -17, -8, 1, 4, 5, 4], [-7, 8, 12, 11, -3, -11, -14, -5], [7, -4, 5, 4, 5, -1, 6, -2], [-7, -6, 15, 22, 11, -6, -24, -5], [4, -13, 6, 5, 8, -10, -3, -7], [-19, 7, -1, 3, -3, 0, 3, 0], [1, -10, 0, -1, -7, 6, 8, 6], [-16, 8, 4, 0, -6, -10, -6, 3], [-5, -9, -10, -9, 2, 4, 10, 10], [6, 6, 10, 1, 1, -2, -13, -5], [20, -10, 12, 2, -4, 4, 12, -2], [5, -5, 11, 1, -7, -5, 19, -2], [-12, 15, 1, 0, 12, -2, -7, 6], [9, -6, 1, 3, 6, 15, -11, 3], [-23, -12, 3, -4, -1, -6, 4, -7], [13, 9, -18, -5, -4, 6, 12, 6], [17, -16, -16, -8, 2, 15, 5, 12], [-3, -16, 6, -6, 4, 3, 7, -4], [6, -7, 13, 18, 4, -15, -23, -9], [14, 10, -9, -5, -6, -1, 22, -4], [26, -14, 8, 6, -4, 6, 2, 0], [22, 13, 13, 9, -5, -2, -4, -5], [-38, -15, 5, -1, 3, 0, -8, -9], [-23, -20, 0, -8, 6, 1, 3, -2], [-4, -10, 2, -7, 3, 7, 1, 11], [-5, 8, 7, 1, -6, 1, 19, -6], [-3, 16, 3, -6, -5, 3, 2, -2], [-9, -5, -9, -5, -7, 4, 15, 3], [5, -10, -5, 3, -2, 12, 3, -3], [20, -19, 9, 2, 0, 8, 9, -6], [12, -5, -7, -4, 2, 22, 13, 15], [6, -3, 6, 7, -10, 5, 15, 2], [0, 12, 1, 0, -1, 7, -7, 11], [15, -15, 6, 14, 9, -6, -15, -3], [2, 2, 8, 7, -5, -1, -1, -6], [-20, 4, 7, 15, -6, -7, -6, -7], [-6, 16, 9, 10, -1, -9, -25, -1], [11, -4, 8, 6, 4, -5, -1, -4], [13, 9, -4, -4, 7, 6, -9, 2], [28, 2, 11, -2, -3, 1, 3, -1], [6, -10, 0, -1, -1, -9, 21, 2], [-8, 14, -5, -4, 5, 2, -3, 11], [-6, -22, -7, -15, -6, 9, 18, -2], [1, -9, -11, -5, -3, 8, 9, 8], [25, 3, -9, -4, 4, -3, 14, -6], [10, 11, -5, -12, -10, 13, 5, 9], [14, -10, 15, 4, -1, -8, 7, -10], [6, 0, 9, 2, 5, -3, -5, 0], [-21, 12, -17, -11, -2, 2, -7, 10], [-13, 15, 4, 3, 4, -9, -10, -3], [2, 6, 2, 12, 8, -6, -2, 4], [2, 14, -11, 11, 10, 0, -6, -1], [26, 3, -13, -6, -16, 14, 15, 6], [12, 3, 10, 12, 11, -6, -10, 4], [16, 15, 2, -7, 4, 0, -9, 10], [20, -15, -8, -4, -1, -4, -10, -4], [-19, 9, -8, -2, 3, 12, 5, 9], [6, -14, -11, -2, -5, 15, 19, -3], [-4, 9, 5, 0, 0, 2, 13, 10], [11, 2, 10, -4, 4, -12, 7, -2], [-1, -15, 5, 16, 0, 2, -10, -11], [-14, 3, 8, 5, -6, -1, 9, -3], [-4, 12, 10, -7, -11, -3, 13, -4], [5, 6, 1, -10, -7, 12, 8, 7], [20, -1, -5, -14, -8, 4, 7, 8], [8, -6, 9, 6, -1, -2, 13, 0], [-10, 6, -6, -4, -12, -6, 12, -6], [-4, -4, -18, -7, -5, -5, 13, 4], [2, -18, -9, -7, 4, 7, 24, -3], [1, 10, 8, -3, -9, -1, 4, -7], [-2, -3, 4, 1, 4, 1, -13, -7], [20, -10, -4, -6, 8, -3, -12, 11], [-30, -1, -12, -19, -8, 8, 19, 10], [5, -13, -6, 0, 3, 2, 17, 2], [-2, 14, -1, -12, 7, 0, 5, 10], [-28, 22, 10, 5, -9, -13, -1, -10], [10, -21, 1, 1, -7, 5, 2, -2], [-5, 14, 16, 14, 6, -4, -17, -7], [-7, 3, -12, -19, -18, 15, 16, 13], [7, -12, -13, -2, -3, 6, 30, 5], [-26, -8, 5, 6, 3, -5, 11, -12], [-6, 15, -5, -16, -6, 2, 9, 12], [-9, 6, -4, -9, -11, -4, 4, 12], [-17, 20, 15, 14, 5, -1, -14, 2], [42, -9, 9, 8, 0, -5, -3, -5], [26, -11, 8, 12, -1, -14, -2, 2], [-34, 11, -25, -5, -5, -5, 10, -12], [6, -16, -6, 11, 3, -4, -11, -13], [1, -19, -3, 8, 0, 4, 6, -13], [15, -1, 13, -9, -13, 1, 3, 1], [14, -14, -14, 5, 5, 7, 7, 3], [7, 1, 14, 1, 4, 3, -8, -3], [-2, -9, 0, -1, -6, 12, 15, 3], [26, 1, 10, 3, 2, 6, -1, 5], [-3, -10, -3, -12, 5, 12, 8, 16], [-42, 20, 1, -1, -4, -10, 0, -8], [24, -28, -6, 2, 2, 15, 4, 7], [-1, -4, -14, -4, -4, -3, 3, 2], [-7, -4, 12, 15, -1, -14, -24, -17], [3, -17, -5, -8, -12, 8, 12, 1], [-1, 12, 13, 2, 3, 2, -18, -5], [23, -10, 14, -1, 7, -6, 20, -4], [-5, -2, 9, 8, 5, -11, 2, -10], [-9, 10, -5, -8, 5, -1, -6, 6], [3, 2, -16, -6, 0, 5, -11, 7], [8, 12, 11, 12, 1, -18, -9, 0], [-9, 9, -10, 8, 15, 13, -1, 2], [-5, 8, 20, 12, 6, -3, -9, -5], [7, -18, -15, 0, -1, 7, -4, 4], [46, -11, -13, -1, -3, 3, 2, -2], [24, -15, -15, -16, 10, 6, 1, 8], [0, -8, 14, 16, 10, -3, -20, -2], [10, -5, -4, -4, -3, -9, 10, -13], [-16, -3, -7, 0, 2, -3, -7, 11], [-4, -30, 4, -1, 1, 1, 17, -5], [10, -14, -4, -1, 11, 0, 11, -13], [-5, 10, -1, -6, -1, 9, 8, 12], [-3, 7, -15, 1, 5, 5, -15, 3], [-12, -9, -17, -6, 0, 10, 13, -1], [-17, 5, -14, -17, 0, 5, 10, 11], [1, 3, -1, -8, 8, 3, 10, 17], [-45, 3, -10, 7, 6, 1, -2, 1], [-23, -9, -21, -15, 1, 4, 17, 1], [7, 20, 22, 0, 4, -9, -19, -4], [-4, 5, -12, -8, -1, -2, -2, 2], [6, 2, -4, -21, -15, 7, 23, 0], [-4, 12, 11, 6, 5, 12, -11, 1], [22, -1, 3, 3, -3, 2, 4, -5], [-25, -3, 7, 14, 12, -3, -10, -4], [24, -7, 9, 14, 6, 9, -5, 0], [12, -1, -13, -5, 3, -8, 0, -11], [-32, -2, 5, 13, -2, -7, -12, -6], [2, 8, -16, -14, 2, 6, 16, 7], [-1, 15, -2, 2, -15, 3, -5, 0], [-3, 12, 4, 11, -5, -13, 0, 2], [-10, 3, 4, 2, -1, -14, -8, -4], [16, -12, -22, 1, 1, 4, 13, -1], [18, 14, -4, 8, -2, 1, -4, -9], [-33, 0, -6, -2, 6, -8, -19, 1], [-17, 13, 4, 5, 8, -9, 10, -1], [5, 3, 15, 19, -7, -4, -25, -20], [14, -16, -15, 0, 7, 18, -1, 8], [-14, -24, 18, 23, -1, -5, -9, -12], [5, 3, 8, -2, -3, -22, 11, -5], [-9, -3, 14, 9, 8, 2, -10, 6], [26, -15, -8, 8, 11, 0, 6, -8], [-21, 4, -6, -14, -4, 12, -3, 4], [-13, -2, -3, 12, 15, 13, -4, -4], [11, 6, -14, -2, 0, 7, -11, 0], [-32, -16, -11, 3, -8, 6, 12, -7], [-17, 21, 5, -4, -10, 3, 14, 4], [31, -14, 15, -3, -16, -6, 1, -12], [-21, -1, 13, 10, 10, -2, -6, -2], [19, -15, -8, -7, 12, 4, 14, 8], [3, -24, -8, 2, -4, 7, 15, 10], [-1, -36, -3, -3, -4, 7, 13, 1], [16, 0, 10, 10, 10, -7, -24, 8], [-18, 1, 6, 13, 12, -16, -13, -5], [27, -25, -1, -2, -6, 5, -4, -3], [-35, -9, -5, 0, 12, -13, -16, -6], [17, -11, 2, -6, 7, -6, -3, 1], [-8, 7, 19, 11, -9, -5, -26, -8], [-3, 21, 24, 18, -11, -14, -13, -10], [-16, 27, -15, -9, -3, -3, -8, 8], [-35, -1, -3, 6, 12, -8, 0, 1], [17, -13, -2, 0, 9, 7, 17, -4], [-19, -15, -20, 1, 8, 4, 6, -4], [-1, -15, 0, -1, -6, 6, 22, 4], [23, 7, 17, 18, 4, -14, -10, -5], [26, 1, 13, 4, -10, 5, -13, -4], [16, -12, 14, -8, -6, -2, -6, 5], [-5, -10, 5, -5, 4, 4, 3, 2], [-6, -8, -9, 0, -7, -3, 21, -1], [5, 13, 5, -8, -4, 4, -18, 3], [1, 1, 24, 31, 2, -4, -12, -16], [-16, -1, -7, 14, 8, 12, -11, 7], [-45, -11, -2, -6, 5, 0, -1, -1], [4, 14, -16, -27, -13, 8, 23, 18], [8, 7, 25, 4, -10, -13, 9, -17], [-13, -12, 8, 2, -14, -5, 3, -11], [-13, 27, 11, 9, 3, -5, -27, 3], [-4, 3, 14, 16, -7, -11, -8, -13], [11, -9, -14, -1, -2, 2, -4, -2], [22, -5, 8, -14, 3, 20, 22, 2], [46, -4, 7, -16, -9, 6, 1, 3], [-21, -15, -3, 13, 11, -10, -9, -10], [13, -15, 7, 16, 10, -14, -12, -12], [14, 3, -3, -9, -7, -2, 6, 7], [-10, 10, 12, 13, -5, -10, 5, -20], [-24, 7, 17, 9, -3, -10, -8, -11], [-36, -3, 20, 12, 7, -8, -10, -16], [-20, 16, 4, 11, 9, 4, -15, 6], [-15, 11, 11, 30, 6, -11, -22, -1], [10, 0, -2, 5, 3, 4, 1, -2], [3, -11, -3, -5, 5, 4, 13, -6], [34, -10, -10, -1, -11, 12, 5, -5], [-4, 6, 9, 13, 8, 7, -16, 10]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1045_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1045_2_a_i();". // 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_1045_2_a_i(:prec:=8) chi := MakeCharacter_1045_a(); 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_1045_2_a_i();". // 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_1045_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1045_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, 11, 60, 21, -47, -28, 5, 6, 1]>],Snew); return Vf; end function;