// Make newform 8030.2.a.z in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8030_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_8030_2_a_z();". // 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_8030_2_a_z();". // 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 := [-3, -14, -9, 25, 17, -11, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [7, 25, -23, -43, 19, 5, -2], [-15, -27, 25, 43, -19, -5, 2], [-15, -50, 32, 70, -28, -8, 3], [-26, -43, 57, 61, -31, -7, 3], [-33, -61, 73, 79, -41, -9, 4]]; Rf_basisdens := [1, 1, 2, 2, 2, 2, 2]; 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_8030_a();" function MakeCharacter_8030_a() N := 8030; order := 1; char_gens := [1607, 2191, 881]; 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_8030_a_Hecke(Kf) return MakeCharacter_8030_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, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 0], [0, -2, -1, 0, 0, 0, 0], [-2, -1, 1, -1, 1, 0, 0], [-2, 0, 0, -1, 2, 1, 0], [1, 1, -2, 0, -3, -2, 0], [-2, 0, 0, -1, 2, 1, 1], [-3, 2, 1, -1, 0, -2, 0], [3, 1, -2, 1, -2, -1, -2], [-3, 4, 1, -1, -1, -1, -1], [0, 2, 2, 1, -1, 0, 1], [4, -2, 0, 1, 1, 2, -1], [0, 0, 0, 2, -1, 1, 1], [0, 3, 0, 0, -2, -1, -2], [-2, 1, 0, -1, -1, 0, 0], [-3, -1, -1, 0, 0, 0, 1], [3, 0, -3, 0, 0, 0, 2], [1, 0, 0, 0, 0, 0, 0], [-6, 2, 0, 0, 0, -2, 0], [-3, 3, 1, -2, 2, -2, 1], [6, -2, -1, 3, -1, 3, -2], [0, 2, 4, -1, -1, 0, 2], [-1, -4, -3, 1, -2, -1, -2], [-5, -2, 1, -2, -1, -3, 0], [2, -3, 1, 1, 5, -1, -1], [4, 2, 0, 0, 0, 4, 2], [-5, 0, -3, -1, 0, 1, -1], [-8, -1, 0, 0, 2, 1, -2], [1, 3, 4, -1, 6, -1, 0], [2, -6, 2, -1, 5, 2, 0], [-2, 1, 3, -1, 2, -3, 3], [-8, 2, 0, 3, 2, -1, 2], [5, -2, -2, 4, 1, -3, 0], [-5, 6, 5, -2, 0, -2, -1], [3, 0, 1, 3, -2, -4, 0], [3, 4, 5, 0, -4, 2, 1], [0, 0, -2, 0, -2, 2, 0], [2, 2, 0, 0, 3, 2, 0], [-3, 0, -1, -3, 2, 1, 2], [13, -2, -1, 2, -3, 3, 0], [-10, -1, -1, 1, -1, 0, 0], [0, 2, 2, 3, -2, 2, -2], [-2, -2, -2, 4, -5, 1, -1], [2, -4, -2, 4, -4, 1, -3], [-7, 1, 2, 0, 9, 2, 2], [5, -2, 1, 4, 3, 5, 0], [5, 2, 5, 1, -2, -5, -3], [-7, 0, -3, 2, 2, 2, 1], [5, 0, -2, 2, -3, 3, 2], [-6, 2, -2, 0, -6, -6, -2], [-2, -4, -2, -2, -1, -2, -3], [-10, 6, 4, -2, -2, -2, 4], [-2, 0, -4, 0, 0, 5, 2], [1, -2, -3, -3, 0, -3, 4], [0, -1, 2, 4, -1, 3, -5], [-8, 0, -8, 1, -2, -4, -2], [-7, -6, 0, -1, 5, 1, -1], [3, -2, 1, -2, 3, 3, 2], [-7, -2, 5, -1, 1, -3, 3], [-1, 4, 5, 2, -1, 1, -1], [7, 3, 5, 3, -6, -1, 0], [-5, -6, -5, 3, -4, 5, -1], [-11, 7, 4, 2, -1, -1, 1], [5, -1, 1, -3, 1, 2, -3], [-5, 6, 1, -1, -3, -6, 3], [2, 2, -4, 4, -8, -2, 0], [-12, -2, 0, 1, 1, 2, -1], [4, -2, -4, 5, -5, 2, 0], [14, -8, -2, -2, 0, 6, 0], [-3, 1, -2, 0, -1, 0, -2], [-8, 6, 2, -1, 8, 0, 3], [5, 6, 5, -1, -4, -1, 1], [20, 4, 2, 2, -6, -2, -2], [-6, 1, 3, -4, 4, 7, 4], [-2, 3, -7, 4, -5, -1, -2], [0, -5, 0, -1, 2, -8, 1], [-13, -4, 3, -5, 7, -1, 0], [-8, 10, 6, -4, -3, -4, -1], [0, 1, 8, -1, 8, 3, -1], [1, 0, 4, -2, 3, 9, 0], [2, -6, 6, -1, 7, -2, 5], [-3, 4, -1, -1, 1, -3, -7], [-1, 2, 3, 3, -3, 5, 1], [-11, 2, -1, -3, 5, 5, -1], [3, 0, -3, 3, -1, -4, -2], [-5, 0, 5, -3, -3, 0, 4], [-17, -3, -6, -2, 3, 0, -2], [0, -2, 6, -6, 9, 3, -1], [-1, -6, -3, 3, 1, -7, -3], [-3, -4, 5, 2, 0, 3, -3], [3, 0, 1, -4, 7, -6, 2], [20, -3, -5, -1, -4, -1, 3], [-14, -6, -6, 1, 1, -2, -3], [-8, 3, -1, 0, -4, -7, -2], [-6, 6, 6, -9, -2, -6, -1], [1, 4, -1, 4, -1, -3, 2], [-2, 2, 6, -9, 8, 7, 3], [-12, 2, -8, 2, -12, -2, -2], [8, -6, -2, 6, -2, 4, -2], [-5, 6, -5, -1, -8, -7, 2], [-19, -2, 1, -6, 6, -3, 1], [-11, 1, -6, 1, -4, -1, 0], [-12, -5, -2, 4, 8, 3, 6], [-10, -8, 0, -1, -2, -9, -1], [-7, 6, -1, 0, -4, 3, 2], [-11, -8, 1, 1, 6, 4, 2], [1, 2, -8, -3, -9, -3, -1], [-11, 0, 3, 7, 1, -4, 0], [-18, 0, 7, -4, 0, 0, 6], [-15, 3, 5, -8, 7, 2, 6], [-8, 1, -3, -5, -6, -1, 5], [8, -8, -2, -1, 5, 9, 6], [14, -6, -6, 8, 2, -5, 0], [4, -4, 4, -2, 1, 5, 3], [-3, -2, 3, 0, 10, -5, -4], [-1, 11, 3, 5, -5, 1, -1], [12, 2, -6, 10, -16, -5, -3], [-2, -11, -3, 6, 6, 3, 2], [9, -9, 5, -4, 0, 3, -1], [-8, 3, 3, 4, -1, -5, -4], [10, 2, -8, 3, 7, -5, -2], [-5, -13, -1, -2, 3, 2, 4], [3, 2, 13, -4, 12, 4, 3], [-2, -1, 10, 5, 4, 5, 3], [-4, 2, -6, -1, 7, -7, 2], [-19, 6, 5, -4, 4, -5, 3], [1, 4, -7, -4, -13, -3, -2], [2, -8, -6, 6, -8, -2, 0], [22, -2, 2, 2, -10, 9, -4], [-4, 6, 10, 0, 5, 9, 1], [0, 4, 0, 7, -10, 6, -4], [-11, 1, -4, -3, -12, -4, -3], [-17, 9, -2, 3, -2, -2, 5], [20, 2, -2, 4, -13, -1, -5], [0, 4, 2, 6, 4, 8, 0], [8, -6, -1, -6, 2, -4, 4], [19, -10, -5, -1, -13, -1, -3], [-2, 8, 6, -8, 3, 9, 5], [5, 8, 7, -3, 4, -11, 1], [-7, -2, -11, -3, 0, 1, 1], [-14, -8, -2, -1, 2, -2, -5], [18, -8, -6, 5, 4, -2, -1], [4, -5, -12, 1, -10, -7, -5], [-26, -6, -8, -1, -6, -10, -7], [-15, 2, 3, 6, 9, 2, 3], [-8, -2, 2, -4, 15, 6, 4], [-6, 5, 8, 1, -4, -5, -3], [-7, 0, 3, 3, -3, -5, 5], [-7, -8, -5, -2, 1, 3, 0], [-19, 8, 1, 3, 10, 0, 6], [-13, 8, 1, 0, 6, 1, 1], [-3, 4, 11, -2, 2, 2, -1], [2, 1, 3, -4, 4, -8, 3], [-28, 2, 6, -7, 9, 4, 3], [-12, 2, 8, -6, 13, 3, 7], [-19, -4, 3, -13, 10, -3, 5], [-14, 1, 8, -13, 7, 2, 0], [-15, 9, 5, -11, -6, -9, -4], [-3, 9, 1, -7, -4, 3, 2], [-6, -9, -6, 2, 7, 4, -5], [7, 6, 3, -7, -10, 3, -5], [-15, -7, -1, 7, 1, 0, 5], [26, 0, 8, 1, 11, 2, 5], [-3, 6, 7, -9, 17, -1, 1], [-4, -12, -4, 5, -14, -8, -7], [-13, -3, -9, -1, 2, 6, 0], [11, 4, 3, 9, -3, 6, 0], [14, -16, -8, 7, -3, 7, 4], [-13, 2, 5, -5, 12, 5, -6], [4, -8, -8, 7, 6, 6, 3], [-17, 8, -1, -3, 9, 6, -5], [9, 2, -11, 2, 1, 7, 0], [18, -3, -3, -3, 0, -8, 8], [8, -6, -6, -6, 3, 8, 5], [-6, 4, -2, -6, 7, -4, 0], [-5, -4, 1, -5, 10, -4, -6], [-8, -8, -8, -9, -2, 1, 3], [15, 2, -3, 0, 0, 13, 3], [38, -2, 2, 5, -5, 9, 0], [3, 0, -2, 0, -2, 14, -3], [2, 4, 14, 3, 1, 3, 6], [-9, -5, -6, -8, -5, -6, -2], [-2, -12, -6, 0, 3, -2, -4], [8, 7, 10, 1, 4, 5, -1], [-29, 6, 7, -8, -5, -7, 4], [-19, -6, -9, 1, -1, -11, -4], [5, 9, 10, 1, -11, 4, 1], [12, -9, -6, 5, -6, 5, 7], [14, -9, -1, 0, -4, 3, 6], [-10, -7, -7, 5, 0, -9, -3], [27, 2, -3, -4, 2, 7, 0], [-7, 12, -1, 1, 11, -3, 3], [10, -6, -4, -6, 6, 14, 2], [-1, -5, -6, 10, 1, -12, 0], [10, 8, 0, 10, -12, -8, -8], [-12, -7, -4, 2, 10, -7, -2], [29, -1, -11, 12, -16, -9, -3], [1, 11, 7, -6, 6, 3, -1], [8, -9, 0, 3, -2, -1, 9], [1, 4, 12, 4, -8, 0, -3], [6, 10, -2, -6, 4, -8, -2], [12, 7, -5, 3, 2, -7, -3], [-21, 1, -6, 5, -8, -5, -8], [-26, -2, 2, -12, -2, 0, 1], [4, 2, 13, -3, 8, 2, 9], [0, 2, -6, -1, 0, -12, 1], [-4, -3, -17, -3, -4, -5, -5], [6, -5, -7, 3, 1, 2, 0], [4, 3, 0, 3, -6, 9, -3], [16, -7, 2, 0, 4, -1, 12], [11, -2, -1, 3, -20, -4, 0], [-13, 7, 2, 0, -6, -18, -4], [-4, 4, -12, -1, -17, -3, 2], [-6, 6, 4, -12, 0, 11, -2], [-6, 7, 6, 11, -4, -3, 3], [1, -8, 5, -6, 13, 1, 0], [17, 6, -5, -3, 6, -2, -4], [-15, -11, -1, -5, -3, -6, 1], [17, 2, 3, 0, -5, 13, -2], [-2, -10, -6, 10, 5, -1, 5], [15, -5, -4, -2, -1, -8, -4], [-3, -6, -3, 8, -13, 7, 1], [9, 3, -4, 8, 2, 13, -1], [1, 3, 6, -3, 11, 0, 1], [34, -8, 2, 4, -7, 3, 5], [-15, 23, 8, 2, 7, 2, 2], [9, 3, 7, -4, -5, -6, -8], [-3, 8, -7, 9, -7, -1, -3], [4, -18, -4, 7, 6, 11, 6], [25, -4, -6, 10, -11, -1, -2], [-6, 0, 14, -7, 9, 2, 1], [-10, -1, -9, -7, -5, 1, -1], [2, -2, 10, 1, 20, 7, 8], [16, 4, 4, -3, 1, 5, 0], [-12, -1, 7, 4, 5, 5, -12], [14, 2, 4, 2, -3, -2, -2], [10, 2, -6, -14, 3, -5, 9], [-13, -2, 9, -11, 10, 8, 8], [-2, 5, -4, -3, 4, -3, 5], [9, -22, 5, -5, 8, -7, 3], [-13, 2, -7, -5, -18, 1, -4], [-4, 2, -10, -7, -17, -2, -2], [4, 3, 16, -4, 11, 5, 3], [-3, 1, 3, 1, -15, -16, 1], [-2, -1, -15, -3, 0, 5, 1], [1, -21, -25, 5, -14, -5, -10], [-2, 13, 14, -8, 9, 2, -1], [-21, -6, -9, -1, 5, -1, -7], [-5, -3, -13, 5, 4, 4, 4], [-4, 8, 4, 2, 12, 2, 6], [-2, 8, 13, 3, -1, -5, 6], [0, -18, -8, 12, -4, -2, -2], [3, -4, -9, -2, -6, 0, -1], [-9, 2, 7, -14, 9, 9, 0], [-6, 12, -3, -9, -10, -4, 7], [13, -6, 1, 6, -9, -3, -8], [-29, 2, -3, 6, 0, -4, 3], [27, -2, -7, 0, -8, -9, 2], [-17, 10, -3, -1, 1, -3, -13], [-1, 2, 15, -2, 0, -12, -3], [6, 18, 12, -9, 0, -2, -2], [26, 2, -4, -1, -15, -10, 2], [-14, -3, 6, -11, 12, 1, 13], [-32, -7, 6, -8, 6, 3, 8], [6, 6, -6, 0, -7, -5, -1], [0, -12, -13, -3, -4, 0, 1], [-8, 2, -4, -2, 2, -4, 0], [16, -16, -8, 4, -1, 4, 8], [-14, 2, -2, -1, -1, 3, -8], [26, 4, -2, 7, 1, -4, 0], [-22, 14, 10, -1, 7, 0, 8], [30, -7, 8, 0, 7, 14, 3], [-33, 14, -1, 1, 2, -3, 7], [23, 4, -7, -6, -1, 3, 3], [15, -1, 11, 2, -1, 10, 6], [7, -2, 13, 8, 1, 11, -6], [7, -10, -1, 0, 12, 4, -1], [17, -1, -5, 8, 0, 1, 5], [37, -15, 2, 0, 2, 6, 0], [7, -18, -1, 2, -12, 3, -1], [0, -17, 0, 3, 3, 0, 2], [-4, 7, 5, 7, 11, -9, 1], [7, 5, 9, 6, -5, 7, -8], [-20, 8, 12, -15, -3, -8, -4], [34, -14, 1, 2, -6, 6, 2], [-17, 1, 3, 5, 2, -17, 2], [-3, -2, -1, 9, -15, 9, -11], [3, 20, -5, -7, -18, -15, -4], [-1, 6, 3, -4, -1, 5, -1], [11, -14, -19, 9, 3, -4, -1], [13, -5, 20, -9, 20, 9, 12], [43, -4, -11, 16, -9, 1, -2], [14, 9, -9, 0, 6, -1, 2], [-2, 10, 6, -18, -4, 0, -8], [23, -21, 2, -6, 11, 5, -3], [-3, 3, -1, -1, -3, 2, 7], [7, -14, -27, -4, -11, -9, 0], [-1, 6, 21, -5, 11, 6, 7], [0, 3, -1, -6, 11, -13, 6], [-35, -4, 10, 2, -1, 5, 0], [-2, 11, 11, 15, 3, 5, 1], [8, 8, 2, 7, -6, 2, -9], [-10, -6, 20, -12, 8, 12, 4], [6, -12, 2, 9, -4, 10, 1], [-5, -4, -23, 3, -12, -7, -1], [-10, -14, 8, -6, -7, -13, -7], [9, 12, 11, 6, 1, 1, -2], [-13, -9, 7, 0, 5, 8, 4], [-15, 0, -15, 1, -20, 5, -10], [5, 12, 7, -5, -5, 9, 3], [12, 12, 22, -7, 13, -3, 6], [-14, -17, -25, 0, -5, 2, 1], [11, -6, 3, 2, -12, 9, 1], [26, 14, -2, -10, 2, -2, 0], [19, 6, -5, -1, 7, -7, -3], [0, 10, -2, 14, -14, 8, -6], [19, -8, -9, -1, 1, 7, -3], [27, -12, -15, 14, -3, 8, -6], [-21, -1, 9, -8, 7, -10, 4], [25, -20, -11, 6, 6, -3, 2], [23, 5, -5, -4, -23, 4, -12], [-1, 10, 7, -9, 5, 6, 14], [-11, 8, -9, 5, -18, -10, -10], [-19, -13, 1, -15, 1, 4, 9], [-15, -8, -7, -7, 11, 1, -3], [-22, -6, -8, -4, 2, -3, 4], [29, 4, -1, 2, 7, -3, 0], [31, -2, -7, 14, -4, 10, 1], [20, -10, -14, 8, 0, 16, 2], [31, 2, 5, 0, 1, -11, -4], [-3, -20, -1, -3, 14, 1, 4], [-1, 1, 7, 6, -8, -9, -3], [23, 24, 6, -7, -2, 8, 10], [-6, -4, 6, -6, 8, 4, 18], [-13, 10, 7, -8, 2, -2, -7], [-30, 9, 2, -15, -4, -18, 11], [-39, 7, 9, -7, 14, 1, 14], [6, 6, 0, 0, 18, -6, 2], [9, -7, -7, 10, 7, 10, 4], [-21, -4, 7, -7, 0, -20, 4], [-19, 10, 1, -10, -18, -16, 2], [-19, 14, 7, -9, 8, 4, 11], [29, 8, 11, -1, 17, -6, -2], [4, -6, -10, 0, -9, 12, 4], [3, -10, -1, 8, -12, 4, -13], [30, -11, 6, 0, 9, 15, 11], [-10, 16, 0, 0, -3, -7, -13], [10, 21, -6, 1, -15, 6, 10], [-25, 17, 4, 1, 1, 0, 15], [1, -4, 17, -8, 16, -5, 8], [-20, 18, 4, -12, -12, -19, 2], [-36, 8, 0, -11, -10, -7, 5], [0, -2, 10, 0, -16, -1, 3], [25, -4, 15, 2, 10, 16, -3], [19, -8, 3, 8, 19, -4, 7], [-36, -16, -12, -5, 4, -10, -5], [12, -13, -20, 4, 2, 5, -2], [14, 3, 0, 13, 7, 3, 2], [0, 6, 6, 5, -10, -6, 11], [21, -18, -19, 8, -18, -7, -9], [-6, -15, 16, -1, 5, 12, 4], [17, -18, -1, -4, 6, 8, -7], [47, -1, 6, 9, 3, 16, 13], [-18, -11, -12, -1, -4, -5, -5], [-28, 0, 2, 0, 6, 5, 0], [35, -2, -2, 17, 11, 21, 1], [-10, 7, 9, -3, 12, 7, 1], [6, 5, 3, -9, 16, -15, 1], [-19, -5, -17, -4, -19, -2, -6], [13, -10, -11, 15, -17, 8, 0], [-6, -12, -6, -3, -12, 4, -5], [6, -4, -2, 9, 2, 8, -16], [34, -9, 5, -3, -4, 11, -3], [-30, -8, -6, -10, 10, 16, 10], [-4, 10, 2, -2, 24, 5, 1], [-17, 14, 3, -3, -13, 1, -3], [37, -24, -5, -1, -5, 6, -5], [-18, 3, 0, 7, 0, 11, -11], [0, 6, 8, 6, 6, -8, -10], [-22, 7, 0, -4, -12, -25, -6], [-6, -1, -7, 3, -17, -22, 0], [-2, -6, 0, 10, 1, -14, -11], [9, 0, -9, 2, -8, 15, -11], [9, 2, 29, 0, 7, 3, 0], [15, 7, 17, -13, 14, -1, 10], [12, -14, -14, -12, 2, 6, 6], [-12, 20, 16, -16, -5, -20, 2], [14, -23, 3, 10, 11, 16, 5], [31, 4, -12, -10, -2, 2, 11], [-18, 13, -9, 3, -14, -11, 5], [-6, -3, -11, 12, 2, 9, 0], [42, 8, -6, -3, -14, 2, 7], [-12, 4, 0, -2, -15, -16, -10], [49, 12, 3, -4, -3, 11, 4], [14, 6, 10, 3, 9, 1, 10], [-25, -12, -27, 5, -3, 9, 3], [11, -5, -14, -2, 4, 7, -1], [-23, -9, -3, -1, 22, 3, -2], [2, 4, 4, 7, -15, -2, -15], [8, 6, 10, -2, 6, 18, 2], [9, 7, 16, 3, 11, 6, 1], [-8, 1, -1, -10, 4, -11, 18], [-1, -21, -6, -7, 5, 2, 7], [17, -7, 25, 1, 21, 21, 3], [4, 2, -4, 17, -10, -4, -5], [-9, 2, -15, 1, -6, -4, -16], [-17, 9, -21, 2, -17, -20, -12], [-1, -13, 0, 9, -1, 14, -5], [-12, -22, -11, -4, 0, 2, 4], [-10, -12, 7, 9, 18, -2, -3], [-8, 4, 6, -5, -3, 26, -3], [-21, -23, 6, 5, 11, 8, 3], [-5, 1, 13, -3, 5, -6, -3], [29, 2, -3, 9, 13, 3, 5], [46, 8, -4, 3, -10, 10, -3], [-7, 3, -14, 7, -7, 12, -7], [23, -8, -1, -4, 4, 6, 21], [10, 7, 12, -4, 6, -5, -2], [-2, 16, 22, -16, 8, -10, -4], [-21, 4, -9, -8, 5, 3, -8], [26, -13, -12, 11, 5, 20, -4], [31, 10, 9, -6, -18, -7, -6], [41, 3, 3, 7, -10, 5, -12], [30, -6, 16, -2, -3, 7, -3], [-11, 12, 7, 10, -9, -3, -4], [-31, 4, -17, 5, 0, 3, 1], [18, 2, -2, 5, -22, -22, -1]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8030_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8030_2_a_z();". // 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_8030_2_a_z(:prec:=7) chi := MakeCharacter_8030_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_8030_2_a_z();". // 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_8030_2_a_z( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8030_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![3, -14, 9, 25, -17, -11, 2, 1]>,<7,R![4, -34, 45, 53, -27, -13, 3, 1]>],Snew); return Vf; end function;