// Make newform 8016.2.a.s in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8016_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_8016_2_a_s();". // 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_8016_2_a_s();". // 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, 10, 5, -6, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [3, 8, -3, -3, 1], [6, 9, -4, -3, 1], [11, 14, -9, -5, 2]]; Rf_basisdens := [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_8016_a();" function MakeCharacter_8016_a() N := 8016; order := 1; char_gens := [3007, 2005, 5345, 673]; v := [1, 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_8016_a_Hecke(Kf) return MakeCharacter_8016_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 := [[0, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [0, 0, 0, -1, 1], [1, 0, 1, -1, 0], [0, 0, 0, 1, 1], [-2, -1, 1, 0, -1], [-3, 0, 0, 2, 0], [0, 1, -1, 0, 0], [2, -2, -1, -1, 0], [4, -1, 2, -1, -2], [0, -2, -1, 0, 1], [-1, 0, -3, 0, 0], [-4, 2, -2, 2, -3], [4, 1, -2, 2, -2], [1, -1, -1, -4, 0], [-2, 1, -3, 3, 1], [-2, 5, -3, 3, -1], [-4, -5, 2, -2, -1], [4, -1, 1, 4, -3], [2, 1, 1, 1, -4], [-5, 2, -3, 3, 3], [2, 1, 0, 4, 2], [2, 2, 5, -1, 1], [-8, -3, 1, -2, 0], [-6, 2, 0, 1, 2], [-1, 5, 2, -1, -3], [-4, 0, 5, -1, -2], [4, -2, 1, 4, -2], [3, -3, 0, 1, -3], [-1, 1, -5, 6, -1], [-3, 3, 0, 0, -1], [-5, 2, 0, -1, 2], [-1, 2, -1, -1, 3], [-1, 2, -1, -3, 1], [10, -2, -2, 3, -2], [-1, -8, -1, -1, 2], [-8, 7, 1, 1, 1], [-1, -1, -3, -2, 1], [-1, 0, 0, 0, 0], [2, 1, 3, -3, 3], [5, -9, 4, -8, -5], [-13, 3, -2, -4, 4], [-1, 3, 6, -1, -4], [-1, 0, 0, 3, 1], [2, 1, -1, -2, 5], [-7, 6, 3, 5, -5], [5, 1, 0, 7, -1], [9, 0, 0, -6, -1], [6, -3, -3, 6, -7], [0, 4, -1, 1, 4], [-7, -4, -1, -4, 2], [-5, 2, -2, -5, 7], [-2, -10, 3, -4, -6], [0, -4, 6, -3, 3], [3, -11, 4, -2, 1], [5, 4, 7, 2, 1], [5, -5, 6, 4, -4], [-15, -1, -9, 5, 8], [-6, 4, 8, -2, -3], [7, -1, 6, -8, 3], [2, 3, 5, -3, 1], [4, -1, -2, -8, 1], [-1, 0, -9, -4, 2], [-17, 4, -5, 10, 0], [-2, 3, 2, -1, -1], [11, -8, 9, -10, -2], [8, -12, 1, -9, 3], [-12, 14, -5, 7, 8], [1, 1, -6, 2, 6], [-13, 5, 5, -4, 1], [-3, 10, 1, -1, -1], [0, 3, -7, 3, -4], [4, -4, 7, -7, 6], [-7, 7, -2, -2, 0], [-10, -3, -5, -3, 5], [3, 12, -8, 3, 6], [-7, -6, 2, -7, 2], [-10, -4, -2, -2, -5], [7, 8, -7, 4, -10], [-13, -4, -6, -1, -2], [-7, 10, -9, 9, -1], [7, -4, -2, 0, -1], [-14, 3, -3, -2, 1], [-1, -6, 5, 5, -8], [16, -2, 13, -8, -6], [-6, 5, 8, 2, -6], [-4, -8, 3, 3, 5], [-1, 3, -9, 8, -6], [-2, 3, -9, 14, 3], [-10, 2, -6, 2, -7], [-2, 10, 2, -2, 1], [-12, -7, 0, -4, -6], [-9, -8, 10, -5, 1], [-18, -2, 4, 6, 2], [-9, -10, -3, 2, 7], [2, 5, 3, 1, -6], [-8, 5, 4, -1, -5], [-2, -2, 7, -1, 1], [-3, 2, -8, -9, 3], [-3, 1, 9, -13, 4], [-14, 14, -2, 7, 2], [8, 7, -3, 10, -1], [14, -13, 7, -11, 4], [7, -3, 4, -16, -4], [0, 6, 4, -1, 6], [-9, -1, 1, -5, 8], [9, -5, 2, -9, 12], [1, -18, 7, -7, -2], [-1, -1, 14, -5, 2], [-16, 0, -7, 8, 8], [-9, 7, 0, 5, -9], [13, -8, 7, 2, -1], [-11, 8, -2, 14, -7], [12, -5, -2, 4, 4], [-5, -4, 7, 2, -2], [4, 8, 8, 0, 5], [1, -4, 6, 1, -3], [8, -16, 2, 4, -1], [17, -2, 2, -5, -10], [13, -3, 1, -1, -15], [7, 2, 1, -1, -12], [-11, 16, -4, 8, -4], [-1, -4, -10, -5, 10], [-12, 3, -4, -7, 0], [-15, 9, -2, 17, 2], [4, 3, -12, -5, 4], [3, -8, 13, -14, -2], [-4, 0, -7, -6, -7], [-2, 9, -17, 18, -1], [-14, -2, -10, -1, 3], [-14, -1, -11, -6, 5], [4, -4, 11, -7, -5], [-13, 9, 5, -9, 9], [-9, 5, 1, 2, -5], [17, -1, -4, 12, -6], [-20, 6, 3, 2, 0], [7, -16, 3, 1, -2], [-3, 6, -2, 11, 7], [25, -10, -3, -1, 5], [1, 13, -5, -7, -4], [9, 13, -17, 17, 2], [8, 0, 5, -1, 13], [-6, 1, -18, -3, 5], [-15, -2, -4, 3, -1], [-8, 12, -8, 3, -7], [-20, 12, 5, 8, -2], [-10, 2, 7, 0, 11], [20, -7, 5, -9, -10], [12, -13, 10, -16, -11], [-15, 5, 2, -8, -5], [-9, 7, -7, 24, -1], [11, -5, -14, -10, 11], [13, -12, 8, -4, 8], [-4, -12, -11, 1, 2], [10, 10, -3, 2, 8], [6, -5, 9, 2, -7], [-10, -3, 0, 0, -2], [-14, 3, 2, 7, 5], [7, -8, 10, -14, 10], [-6, 0, -10, 6, 5], [10, 10, 12, 1, 1], [-5, 5, -3, 7, -3], [-1, -11, 4, 8, -3], [-18, 3, 7, -12, 0], [23, 9, -11, -3, 6], [-11, 20, -4, 0, -5], [-5, 9, 7, -2, 12], [-4, 3, -3, -7, 18], [-13, -10, 21, -7, -4], [-2, -5, 5, -2, -2], [-1, 4, -6, 1, 15], [2, -1, 15, 8, -6], [13, 1, -8, -10, 7], [-3, -7, -4, 6, -1], [-28, 5, -3, 14, -5], [28, 3, 17, -2, -8], [-5, 12, 1, 11, -5], [10, 4, 7, 10, -8], [16, 2, -6, 7, -17], [-8, 6, -6, 14, 3], [18, 4, -6, -7, 0], [3, -8, 4, -8, -3], [13, 5, -19, 9, 1], [16, -14, 7, -15, -8], [8, -10, 12, -2, 4], [-23, -18, 21, -10, 0], [14, -10, 7, -10, 2], [9, -6, 9, 12, -15], [-18, -4, -2, 1, 14], [-14, -9, 2, 9, 4], [2, -4, 5, 1, 6], [-3, -12, -13, -8, 1], [-11, -11, 0, -7, 16], [-5, 12, 2, 13, -15], [13, -11, 6, -15, 7], [-29, 0, -1, 8, -7], [19, -15, 6, 1, -5], [15, 8, -4, -3, 4], [-20, -6, 0, -12, -1], [-14, 13, -10, 11, 9], [-16, 16, 7, 2, 3], [30, -2, 2, -11, -3], [9, 8, -2, -3, -15], [-1, 0, 19, -10, -5], [9, 10, 2, 9, 9], [-13, -4, -23, 8, 11], [-30, 5, -4, 3, 16], [10, -7, -15, 10, 4], [0, -6, 0, -11, -9], [9, -7, -7, 16, -4], [29, 3, 11, 10, 0], [15, 3, 18, -4, -2], [-14, -4, 21, -2, -6], [32, -11, 8, -9, -12], [-22, 10, -12, 3, 11], [27, -14, 10, -8, -7], [-7, 0, 8, -16, 18], [4, -2, -5, 6, 13], [8, 13, -3, 6, -10], [-15, -1, -6, 0, 21], [-2, -21, -2, -14, -4], [2, 8, -6, -5, -14], [-35, 15, -2, 18, -1], [9, -23, 10, -9, -6], [11, 2, -12, -7, 5], [-39, 6, -11, 8, 15], [-19, 18, -1, 12, 5], [-14, -2, 2, 13, 17], [-19, 4, 3, 15, -5], [34, -10, 12, -15, 1], [5, -26, 2, -2, -8], [-8, 0, 8, -3, -3], [-4, -7, 8, 8, -1], [50, -12, 5, -8, 0], [7, -13, 13, -12, 22], [6, -22, 3, -8, 9], [-23, 4, -11, 20, 10], [-11, 0, -3, 2, 3], [13, 12, -4, -9, 6], [6, -10, -13, 6, 7], [-10, 24, -12, 1, 20], [17, -16, 13, -11, 7], [27, -13, 7, -17, -3], [4, -5, -24, -11, 12], [-11, -1, 15, -3, -1], [29, -18, 5, -1, 2], [-28, 0, -1, 7, -5], [10, -19, 12, -2, -11], [29, -18, -6, -10, 7], [-11, 10, -4, 14, 7], [13, -31, 18, -25, -10], [2, 11, -2, 10, 3], [-33, 9, 1, 19, -13], [-23, -4, -1, 15, 6], [13, 15, -10, 16, -1], [13, 7, -6, 3, 5], [-6, -4, 0, -6, 10], [28, -1, -2, 3, -13], [-4, 3, 14, -8, -3], [20, -13, -11, -3, 7], [-12, 2, 3, 3, -10], [23, -18, -1, -21, 14], [-31, -3, -15, 6, 16], [3, 1, -3, 15, 8], [-13, -15, -21, -3, 17], [-3, 25, -5, 2, 11], [5, 17, -16, 11, 1], [-20, -2, -1, 6, 18], [-9, 21, -4, 1, -10], [7, 16, 0, 9, 0], [2, 25, -1, 6, -4], [9, 2, 20, -16, 8], [-37, -11, 5, 4, 8], [17, -2, 0, 11, 3], [-21, 1, -9, -1, 11], [-18, 9, -14, 21, 13], [1, 10, -15, 26, -5], [-19, 6, -18, 12, -1], [16, 17, -12, 23, 5], [-9, -10, 20, 0, 3], [-2, 13, 11, 0, 13], [-3, -3, 10, -28, -2], [47, -5, -3, -1, -5], [-55, 16, -3, 24, -5], [-12, -1, 0, 16, -20], [6, 15, -19, 2, 12], [2, -27, 18, -8, -14], [-8, 4, -23, 31, 7], [1, -4, -11, -2, -8], [-13, 4, 10, 6, -2], [36, -17, -2, 4, -2], [25, -12, -1, 1, -6], [-39, 1, -8, -13, 1], [5, 4, -8, 6, 13], [15, 2, -5, -20, 18], [5, -18, -5, -6, -10], [-16, 11, 12, -13, 12], [0, -13, 14, -25, -9], [7, 10, -18, -14, -3], [4, -10, 22, -20, 14], [22, 0, 21, -18, -6], [5, 20, -5, 13, 2], [3, 9, 0, 4, 13], [27, -13, 0, -4, -13], [17, -18, 3, -17, -5], [-37, 14, -5, 28, -19], [9, 4, 7, 11, -23], [-4, 6, 2, 13, -24], [1, -14, 0, 2, -13], [-37, 16, -2, 6, 7], [42, -14, 3, -30, 1], [-7, 16, 21, -5, -6], [-30, 14, 1, 26, -9], [16, 3, -6, 7, 6], [13, -24, -8, 2, 7], [5, -3, 0, 3, -11], [-22, 11, -4, 20, 5], [33, -24, 4, -12, -14], [28, 9, -22, -2, 5], [7, 19, -13, -4, 4], [-22, 0, -12, -10, 9], [32, -6, 4, -4, -5], [-11, 7, -14, -25, 19], [-2, -2, -4, 15, -9], [-12, -12, -21, 13, 8], [3, 16, 1, 21, 11], [-12, 29, -9, 6, -2], [6, -15, 10, 11, 9], [15, -9, 8, -31, 15], [-11, -11, 25, -9, 9], [25, -13, 8, 21, -9], [41, -8, -3, -10, -1], [5, -6, 2, -4, -21], [6, 1, -16, 30, 3], [37, 11, 6, -12, -28], [53, -15, 10, -17, -13], [-37, 3, -1, 3, 9], [-14, 27, -17, 18, 5], [-12, 0, -34, 17, 2], [-11, 12, 3, -5, 5], [-38, 3, -6, 10, -7], [39, -1, 5, -6, -20], [-32, -3, -11, -1, 16], [27, -4, 10, -11, -14], [-1, 21, 15, 14, -3], [-13, 8, 4, 23, -13], [6, -24, 3, -15, -1], [16, -14, 1, -8, -4], [24, 3, -6, -3, -12], [3, 28, -9, 1, -14], [30, 1, 12, 1, -2], [-9, 7, 2, 2, 2], [-3, 18, -8, 20, -8], [18, 3, -21, 14, -4], [13, -26, 15, -15, 9], [-11, 1, -19, -18, -6], [-12, -19, -9, 5, 20], [-20, -4, 3, 6, -4], [-19, 0, 16, 12, -9], [-33, 2, -1, -4, -9], [-23, 11, 20, 2, -12], [12, 9, -15, 15, -18], [-3, 2, 8, 8, 1], [-23, 7, -4, -9, 2], [-17, -6, 8, -23, 8], [34, -3, 18, -13, -18], [-25, 10, -21, 0, 14], [9, 5, -5, -22, 19], [23, 5, -13, 10, -12], [17, -13, 18, -3, -1], [-29, 14, -13, 23, -12], [-8, 20, 4, -10, 25], [-26, 41, 5, 4, 1], [-27, -23, 7, -13, -4], [12, 7, 2, 9, -21], [-1, -10, 17, 8, 13], [8, 13, 12, -1, 20], [27, 11, 14, 4, -16], [4, -10, 28, -36, 10], [-14, 10, -20, 9, 3], [-10, -31, 7, -1, 11], [-29, -29, 14, -2, 13], [-53, 0, -4, -4, 19], [-23, 2, -8, -15, 0], [8, -4, -24, 15, 2], [0, -5, 11, -11, -21], [-18, 7, -21, 21, -1], [22, -10, 18, 10, -1], [10, -37, 22, -33, -13], [28, -24, 0, -8, -19], [-11, 7, -24, -7, 16], [1, -15, 18, -2, -8], [-10, -14, -23, -13, 14], [9, 9, 9, 22, -19], [-2, 15, -12, -8, 14], [7, -1, 12, -1, 10], [4, -12, 18, -27, 12], [-20, -15, -1, 0, 20], [-11, 18, -12, -9, 32], [8, 1, -12, 28, 8], [-25, 5, 4, -1, -5], [-43, -6, 4, -2, 15], [-51, 19, -2, 12, -16], [-5, 4, 13, 5, -12], [-31, 23, -10, 27, 14], [-18, 18, -8, 28, 19], [-4, -7, 9, -14, 28], [-28, 11, 4, -25, 17], [-57, 2, -7, 10, 2], [-7, 9, 22, 2, -14], [-2, 3, 1, -27, 4], [-17, -13, 28, 2, -7], [-2, 6, 19, 15, -16], [-27, 6, 2, -29, 5], [34, -7, 13, 0, -17], [8, 16, 17, 12, -13], [-11, -2, -25, 6, 19], [-29, 15, -5, 4, -4], [-1, -2, 2, -15, 24], [-12, -28, 9, -16, -2], [-19, -35, 32, -18, -13], [-6, 5, -20, -16, 32], [3, 5, -14, -7, 13], [8, -3, -17, 11, 7], [9, 10, 2, -2, 2], [37, 4, 13, -11, 3], [32, -8, -9, 12, -13], [-21, -3, 31, -26, 9], [-32, -7, -5, 7, 21], [-25, 23, 1, -17, -8]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8016_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8016_2_a_s();". // 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_8016_2_a_s(:prec:=5) chi := MakeCharacter_8016_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_8016_2_a_s();". // 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_8016_2_a_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8016_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![5, 19, 12, -13, -1, 1]>,<7,R![5, 17, 13, -5, -4, 1]>,<11,R![-55, 47, 30, -19, -3, 1]>,<13,R![-215, -132, 62, 61, 14, 1]>],Snew); return Vf; end function;