// Make newform 4008.2.a.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4008_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_4008_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_4008_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 := [-10, 51, 152, -83, -166, 67, 45, -16, -3, 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, 1, 0, 0, 0, 0, 0, 0, 0], [68945, -635601, -563660, 716568, 247955, -175482, -17171, 10903, -253], [-301330, -988001, -38080, 878658, 67645, -178372, -2266, 9923, -428], [-130065, 200063, 208150, 207921, -199070, -99334, 48718, 7666, -3031], [255125, 1605297, 520415, -1646101, -79645, 387944, -35953, -23831, 3811], [445920, 296533, -1469865, -184819, 751965, 65121, -118627, -5129, 5684], [59190, -769633, 148730, 1137454, -367380, -311796, 101842, 20594, -6589], [286160, -396149, -2000020, -446023, 1500975, 290982, -293609, -24668, 16128]]; Rf_basisdens := [1, 1, 102665, 102665, 102665, 102665, 102665, 102665, 102665]; 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_4008_a();" function MakeCharacter_4008_a() N := 4008; order := 1; char_gens := [3007, 2005, 1337, 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_4008_a_Hecke(Kf) return MakeCharacter_4008_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, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 1, 0, 1, 0], [2, -1, 0, 0, -1, -1, 1, 0, -1], [0, -1, -1, 1, 1, 0, 0, -1, 1], [-1, 0, -1, 0, 0, 0, -1, -1, 0], [-1, 1, 3, -1, -1, -1, -1, 0, -1], [0, -1, 0, 0, 0, 0, 1, -1, 1], [-5, 0, 1, -1, 3, 1, -2, 1, 2], [-3, 0, 0, -1, -1, -1, 0, 1, 0], [1, 0, -1, 0, 0, -1, 1, -3, 1], [-3, 0, 2, -1, 1, 0, -2, 1, 1], [-4, 0, 0, -2, 0, 0, 0, 1, 0], [-1, -1, 0, 1, 0, 0, 1, 1, 0], [0, 0, -3, 3, -3, 1, 1, 0, -1], [2, -2, -1, 2, 0, 0, 0, 0, -2], [-1, 1, -1, 1, -2, 0, 1, 1, -1], [1, -2, -4, 3, -1, 1, 2, 0, -2], [0, -1, 2, 1, -1, 1, 0, 0, -2], [2, 1, -1, 3, -3, -2, 0, -1, 0], [-4, 0, 2, 0, 2, 0, 0, 1, -2], [-2, 0, -3, -1, 1, 1, 1, -1, 3], [-6, 4, 4, -4, 4, 1, -4, 1, 1], [4, -1, -2, -1, -1, -2, 4, -3, -1], [-6, 1, 1, -2, 4, 2, -1, 2, 0], [-10, 3, 2, -3, 1, 1, -2, -1, 2], [1, -2, -2, 2, -1, 2, 1, 2, 2], [-3, 0, -2, 0, 4, 1, -1, -1, 1], [5, 1, 4, 1, -3, 0, 3, 2, -7], [-1, 1, 2, 3, -3, -3, 3, 0, 0], [4, -2, -2, 1, -3, 1, 3, 1, 0], [-1, 0, -4, 0, 2, 0, 1, 0, 4], [-3, 2, 0, 1, -1, 3, 0, 2, -3], [-8, 1, 2, -4, 4, 3, -3, 1, 0], [-7, -1, 3, 0, 4, 2, -2, 0, -1], [-5, 3, 4, -3, -3, 2, -1, 5, -4], [-6, 2, -1, -2, 2, 4, 0, 1, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-9, 3, 2, -3, 1, 2, -1, -1, 2], [0, 4, 3, -2, 0, -3, -2, -3, 2], [-7, 1, 4, -4, 7, 0, -4, 3, 2], [1, 4, 2, -3, -2, -4, 2, 2, -3], [-5, 2, -2, 1, -1, 2, -2, 0, 2], [-4, -3, -3, 0, 4, 1, -3, -3, 4], [-8, 3, 3, -1, 3, -2, -2, -3, 2], [-5, 0, -4, 2, 0, 1, 1, -1, 3], [-11, 1, -4, 0, 2, 0, 0, -3, 5], [7, -2, 4, 2, -2, 0, 1, 2, -3], [-8, 4, 7, -4, 5, 4, -6, 2, 3], [3, -4, -1, -2, 6, 1, 1, -1, -3], [-1, 1, 5, -3, -2, -4, -3, 1, -4], [-3, 1, -9, 0, 2, 0, -2, -3, 2], [0, 4, -1, 0, 0, -3, -2, -3, 1], [2, 2, -1, -1, 3, -2, -3, -3, 5], [6, -6, -5, 6, 0, -2, 2, -2, 1], [8, 3, -2, 4, -6, -3, 1, -1, -1], [-3, 3, 1, -1, 1, -3, 1, -1, -4], [8, -1, 1, 3, -9, -3, 2, -4, -2], [5, 3, 2, 1, -3, 3, 3, 4, 0], [-5, 3, 6, -3, -5, -4, 1, 5, -6], [3, -4, 2, 3, -1, -2, -4, 0, 3], [1, -2, -1, 5, -3, 0, 2, 1, -2], [7, -4, -1, -2, -6, -7, 3, -1, -3], [0, -2, -9, 5, 5, 4, -3, -2, 4], [2, -3, -2, -1, 7, -1, 2, -4, 3], [3, -7, 2, 3, 1, -2, 1, 2, 2], [-3, 4, -3, -3, 5, 2, -2, 0, 3], [10, -3, -2, -1, -3, 0, 4, 1, 0], [12, -4, -3, 9, -3, 1, 5, 1, -2], [1, 4, 6, -4, 2, 2, -1, 2, -2], [-4, 1, 0, -1, 1, 3, 0, 2, 0], [-12, 5, -6, 2, -4, 0, -3, -3, 1], [-2, -1, 0, -3, -3, 0, 2, 1, 2], [-10, 2, -1, -9, -1, -1, 1, 2, 3], [6, 1, -7, 3, -12, -1, 4, 0, 0], [20, -2, -2, 4, -10, -4, 8, -5, -7], [-1, 3, 7, -4, 1, -1, -4, 0, 2], [8, -1, 6, 2, 4, 2, -3, 0, -2], [4, -3, 3, 1, -4, 5, -2, 8, -4], [4, -2, 8, -3, 0, -2, 1, 2, -4], [4, -4, 1, -5, -1, 2, -1, 2, 1], [9, 2, -2, 2, -4, -2, 7, -6, 5], [2, 1, -7, 2, -1, 3, -1, -4, 0], [-15, 1, -3, 1, 3, 1, -3, 1, 1], [9, 1, 3, 6, -10, -5, 2, -2, -9], [12, 2, -1, 1, -5, -3, 5, 5, -5], [6, -7, -2, 6, 2, 0, -1, -3, 0], [8, -3, -7, 1, 1, 0, 6, -1, 1], [-1, -2, -8, 5, -5, 2, 6, -3, 1], [-5, 0, -4, 6, 0, 7, -1, 3, 4], [-14, 2, -2, -1, -3, 1, 1, -1, 5], [-3, 0, 6, -2, -4, -7, -1, 2, -1], [14, -1, -7, 8, -1, 0, -1, -5, -2], [5, -3, -6, 8, -8, 1, 8, 3, -6], [-24, 0, 5, -5, 10, 7, -7, 9, 2], [2, -4, -3, -1, -1, 2, -5, 4, -4], [9, 5, 0, -1, -1, 0, 1, -4, 2], [-5, -2, -6, -1, 7, 4, 2, 6, -2], [-8, 2, 10, -5, 7, 0, -9, 10, 3], [-10, -9, 4, 0, 6, -1, -1, 1, 6], [-4, 1, 3, -5, 9, 2, -4, 5, -1], [0, 3, -2, 5, -7, 7, 4, 6, -2], [9, -2, -13, 9, 1, -1, 0, -6, 7], [6, 1, 2, 4, -6, 3, -1, 3, -10], [0, -1, 1, -3, 4, 4, 0, 9, 3], [-1, 7, -5, -1, 3, -2, -1, -5, 7], [-11, 5, -5, -4, -2, 0, -2, -6, 5], [-9, -5, 3, -7, 5, 0, 1, 5, 5], [-10, 6, 0, -9, 7, 0, -1, -4, 6], [-7, 0, 2, -5, 1, -2, 0, -1, 2], [13, -10, -13, 9, 8, 2, 0, -10, 5], [12, -6, -17, 9, -3, 0, 7, -4, -1], [-10, -2, 2, -4, 14, 2, -4, -3, 10], [11, 4, -5, 8, -10, -1, 1, 1, -7], [15, 6, -4, -1, -9, -2, 4, 3, -7], [-13, -6, -8, 4, 2, 1, -3, -6, 5], [-5, -2, -8, 10, 4, 5, 1, -3, 6], [-4, -2, 8, -3, -1, -3, 1, -1, 2], [3, 4, 8, -5, -1, -9, 0, -1, -1], [5, 3, -5, -3, -3, -7, 5, -4, 0], [1, 0, 5, 5, -5, 3, 4, 1, -1], [6, -2, 2, 1, -5, 1, -1, -1, -2], [-9, -2, -3, -5, 9, 7, -6, 3, 8], [6, 0, -12, 3, -3, 1, 5, -4, 2], [-5, 0, -3, -3, 7, 1, -4, 3, 4], [17, -10, -5, 2, -4, -5, 13, -3, -3], [-2, 3, 5, 6, 0, -3, 3, 4, -6], [0, -1, -7, 10, -8, 5, 7, -1, 0], [-16, 7, 0, -2, 9, 11, -1, 6, 3], [11, 2, 12, -1, -3, -9, 4, -2, -6], [2, 4, 11, -11, -5, -3, 3, 1, -1], [18, 3, 5, -5, -7, -10, 10, -6, -2], [9, 3, -4, -4, -2, -3, 4, -2, 10], [16, -5, 8, 1, -7, -2, -4, 1, -10], [-11, 1, 15, 2, -2, 6, -4, 13, -4], [2, 2, 0, -1, -3, -6, 3, 5, -3], [-19, 12, 14, -10, -4, -1, -11, 2, -5], [-4, -7, 9, 2, -4, -1, 3, 5, 1], [-14, 0, 8, -7, 11, 2, -11, 4, -3], [32, -8, -10, 6, -10, -9, 6, -8, -6], [7, 5, 6, 1, -3, -4, -3, -4, -10], [-14, -4, 3, 0, 2, -1, -8, 8, 5], [1, 8, 4, -3, -7, 2, -4, 10, -4], [17, 1, -6, 1, -3, -3, 7, -12, 3], [-12, 7, -3, -1, -4, -2, -6, -5, 4], [-20, 4, 8, -4, 9, 5, -12, 5, 6], [-21, 6, -5, -2, 4, 9, -7, 5, 0], [7, 6, 10, -1, -1, -2, 0, 4, 1], [10, -8, -9, 6, -6, 1, 8, -5, 2], [-1, 7, 2, 0, 2, -3, -2, 0, -4], [-2, 6, -1, -1, 1, -4, -7, -11, 5], [1, 6, 1, -3, 1, -5, -4, -1, -5], [8, 7, -8, 6, -4, 8, 3, 1, -7], [-3, -5, -3, -4, 12, 6, -2, 5, 8], [23, 2, 6, 6, -6, 0, 1, 2, -1], [7, -5, -5, 8, -6, 3, 8, 2, -9], [11, 3, -4, 8, -8, 8, 4, 3, -2], [15, -6, -1, -3, -3, -9, 0, -9, -1], [-3, -10, 4, -5, 5, -6, -2, -3, 11], [18, -1, 2, 2, 3, -5, 3, -4, -1], [-14, 7, 19, -3, -3, -2, -2, 14, -6], [-4, 0, 9, -4, 2, 9, -4, 7, 0], [16, 10, 12, -2, -2, -5, 0, -5, -1], [5, -7, -1, -3, 1, -3, 3, -3, -4], [-4, 10, 1, -1, 1, 0, -1, -2, 1], [20, -13, -22, 11, 3, 3, 12, -5, 5], [12, 4, -1, 9, -4, 1, -3, -3, -10], [6, 7, -5, 6, 1, 7, 5, -2, 0], [-1, -7, 1, -1, 9, -1, 5, -2, 3], [2, 3, -3, -8, -4, -4, 3, -9, 2], [7, -1, -19, 7, 5, -1, -1, -8, 9], [17, -7, 4, -5, -7, -3, 1, 2, -10], [9, -2, -1, 3, -22, -7, 6, -3, -6], [8, -13, 3, -2, -10, -7, 3, 1, -8], [-9, 5, -1, -5, -7, -3, 5, 2, -1], [14, 5, 1, 3, 3, 2, -6, -5, -8], [-4, 2, 4, -4, 4, 11, 0, 14, 1], [-4, 14, 11, -12, 0, -5, 0, -7, 1], [-8, 5, 1, -5, 5, -7, -4, -8, 12], [-8, -1, 14, -9, -1, -6, -6, 3, -9], [-8, 12, 13, -9, -3, -2, -11, -2, -3], [15, 4, 8, 11, -9, -1, 2, 3, -11], [8, -9, -14, 11, -5, 5, 6, -9, -4], [-2, 0, 18, -4, 0, 4, -2, 6, 2], [-22, -1, 10, -11, 19, 11, -14, 8, 9], [-15, -2, 12, -8, 1, 0, -11, 10, -6], [-5, 7, 16, -5, -7, 0, 1, 7, -5], [-20, 4, -11, -1, 0, 6, 3, -6, 13], [13, 0, 9, -1, -5, -15, 2, -9, 2], [-6, 3, 2, -2, 6, 12, -1, 1, 3], [-16, 16, 7, -6, 8, 6, -22, 0, 3], [-16, 8, 6, -6, 2, 4, -8, 10, 7], [-7, 2, 16, -7, 11, 2, -12, 6, 0], [-11, 3, 3, -8, 5, 3, -6, 8, -3], [-16, 6, 3, -4, 4, 2, 4, 6, 1], [-2, -16, -10, 7, -2, 5, 1, -7, 9], [-18, -7, 9, -15, 18, 7, -4, 6, 2], [-14, 9, 5, -1, 17, 9, -6, 2, 6], [21, -16, -17, 12, -5, -2, 11, -10, 2], [18, -17, -17, 19, 3, -4, 0, -12, 7], [-25, -3, 0, -3, 9, 3, -5, 6, 11], [28, -3, -5, 2, -8, -2, 11, -1, -2], [-21, -1, 4, -1, 9, 6, -5, 11, 3], [19, -14, -12, 3, 3, -6, 18, -6, 0], [-15, 7, 14, -5, 3, -4, -7, -1, 7], [-27, 10, 9, -2, 8, 2, -11, -3, 7], [-3, 4, 0, -7, -10, 3, 0, 11, -5], [18, -14, -15, 11, -1, 0, 3, -9, 2], [-4, 5, 2, -2, 8, 2, -13, 1, 6], [1, -7, -4, -3, 0, -4, 7, -3, -2], [-21, 2, 3, -12, 12, 4, -19, 0, 2], [1, -3, 1, -3, 7, -7, -7, -7, 11], [17, 4, 5, -5, 3, -9, -2, -1, -10], [8, -15, -10, 12, 5, 1, -3, -12, 9], [-11, -4, 9, -4, 3, -2, -9, 2, -1], [-3, -4, -3, 0, -4, 6, 3, 5, -2], [4, -11, -13, 8, 12, -1, -1, -12, 10], [0, -2, 4, -10, -4, -1, 4, 13, -3], [-7, -9, 4, -2, 6, -2, -10, -6, 5], [6, 4, -2, -3, -17, -3, 7, 3, -7], [-24, 3, -4, 1, 9, 2, 4, -7, 3], [-4, -6, -23, 5, -1, 2, 1, -5, 7], [12, 4, 0, -2, -2, 0, 8, -4, -5], [-1, 14, 6, -5, -3, -10, 6, -8, 0], [5, -2, 9, -5, -8, -8, -6, 4, -7], [2, -14, 5, -4, 4, -2, -2, 2, -7], [-10, 2, 1, 8, 6, 5, -10, -9, -1], [-2, 1, -3, 8, 0, 7, 3, -3, -5], [30, -6, 3, 0, -8, -5, 6, -3, -7], [-4, -11, -18, 6, 6, -1, 5, -10, 11], [-16, 6, -7, 1, 1, 10, 9, 3, -2], [-3, -6, 0, -8, 2, -7, 5, -2, -6], [-1, -5, 1, 1, -5, -1, 5, 5, -7], [18, 5, 0, -2, -18, -6, 5, -1, -8], [-14, 9, -13, -4, 0, -3, -1, -14, 8], [20, -9, -6, 9, -13, -11, 8, -6, -9], [3, -11, -12, 7, -17, 2, 9, -1, -1], [-9, -3, 9, 4, 12, 1, -4, 6, 2], [-32, 8, 12, -10, 11, 7, -8, 13, 0], [25, -4, -19, 22, -18, 6, 19, -2, -7], [6, 7, 7, -1, -3, -10, 0, -1, 2], [22, -9, 5, 11, -3, 4, 2, 8, -8], [6, -3, 2, 0, -6, -11, -5, -4, 6], [-34, 1, 7, -6, 8, 3, -3, 4, 7], [-27, 12, 0, -3, 10, 3, -14, -3, 8], [-22, -6, -5, -3, 15, 3, -3, 3, 11], [26, -1, -13, 10, 6, 10, 3, -5, 7], [4, -6, 15, -13, 9, 4, -5, 6, 7], [-4, 6, -13, 3, -5, 2, 3, 0, -3], [-4, -5, 7, 0, 11, 8, 1, 3, 7], [35, -10, 0, 7, -5, -18, 6, -10, -9], [-2, 1, -3, 8, 12, 1, -7, -6, 9], [13, -19, -5, 7, -7, -9, 9, 0, -12], [0, -10, -21, 9, 1, -4, 9, -9, -5], [-17, 14, 6, -8, -2, -7, -3, -7, 5], [9, 11, -1, 3, -19, -8, -3, -1, -14], [18, -8, -3, 5, -23, -8, 19, -4, -10], [6, -11, 9, -3, -1, -7, -2, -1, 4], [-2, -10, 1, 0, 0, 4, 0, 8, -2], [8, -5, -6, 13, -9, 1, 4, -7, -10], [0, 8, -12, 2, -4, -3, 4, -11, 10], [23, -10, 9, 5, 13, -2, 0, -2, -8], [0, -12, -14, 9, -3, 8, 3, -2, 3], [22, 7, 7, -4, -16, -13, 17, -4, -15], [-9, -7, -4, 2, 2, 5, 12, -6, 8], [-1, 4, 0, -3, 5, -10, 2, -7, -2], [9, -1, -1, 1, -9, -1, 13, 1, -11], [13, -11, 7, -2, 6, -1, 6, 3, -10], [-30, 0, 10, -9, 7, 11, 3, 7, -8], [-7, 9, 1, -7, -1, 3, 5, -6, -1], [-13, 6, 13, -15, 23, 1, -10, 1, 13], [-16, 13, 2, -10, 6, -4, -7, -3, 10], [-22, -11, 3, 2, 4, 6, -5, 7, -6], [-6, 0, -20, 12, -3, 15, 4, 1, -1], [2, 8, 1, -6, -8, 1, -4, 10, -1], [-2, 6, 3, -4, -6, 2, -8, 10, -1], [-23, -5, 4, -4, 0, -2, -8, 9, 3], [-4, -6, 1, 11, -9, -2, -1, -12, -12], [-20, 5, 0, 5, 10, 10, -6, 5, 9], [0, -3, 3, -7, -5, -12, 8, 7, -11], [25, -2, 5, 9, 3, 1, 2, 0, -5], [0, -15, -19, 12, -2, 3, 7, 4, -9], [21, -14, -3, 9, 2, -10, 0, -6, -1], [13, -9, -11, 5, 7, -4, -7, -8, -2], [-41, 11, 1, -9, 9, -1, -11, 2, 19], [-9, 14, 14, -2, -9, -7, -7, 7, 3], [-37, 11, -2, 1, -7, 8, -3, 6, -4], [7, -4, 9, 0, -6, -9, 9, -3, -17], [-11, 10, 15, -10, -10, -8, -3, 17, -10], [-6, 4, -2, -1, -5, -7, 1, -3, -5], [-12, 12, 16, -18, -8, 1, 0, 17, -12], [-16, 7, 3, 1, 15, 5, -10, -10, 3], [-8, 1, -16, 11, 10, 16, 0, 3, 3], [24, -14, -25, 17, 6, 1, 13, -7, 13], [-22, 19, 2, -3, 5, -3, -2, -6, 9], [-4, -9, 10, 4, -12, 7, -3, 16, -14], [-13, 1, 4, 3, -23, -3, 5, 5, -11], [23, -21, 4, 6, -6, -20, 10, -13, -13], [-20, -2, -14, 2, 6, -12, 4, -6, 2], [-31, 3, -9, -8, 18, 0, -16, 1, 4], [-26, 0, -12, 11, -3, -4, 9, -11, -9], [-5, -9, 2, 1, -11, 1, -5, -4, -6], [-10, -12, 9, -6, 10, -5, -4, -1, 2], [50, -9, -7, 13, -15, 0, 14, -7, -2], [17, 1, 16, -17, -3, -2, -1, 13, -9], [7, 4, -4, 4, -8, 2, 15, -4, 4], [-25, 0, 23, -12, 8, -2, -15, 12, -5], [-4, 4, 1, 3, -19, -6, 1, 2, 1], [4, -6, -7, 11, -9, 12, 5, 4, 13], [-40, 9, 1, -10, 18, 7, -5, 10, 8], [-13, -9, -1, -5, 9, -9, 5, -8, 12], [22, 3, 11, -2, -2, 3, 5, 2, -4], [19, 2, -19, 12, 0, -6, -1, -15, 5], [40, -4, 9, 5, -5, -12, 15, -4, -5], [-27, 6, -4, -12, 6, 5, 1, 7, 9], [-28, -3, 15, -10, 8, 6, -1, -1, 9], [0, -3, -6, -4, 12, 5, 1, 6, 5], [-15, 2, -12, 7, -11, 2, 0, -4, -1], [-18, -11, 6, 3, -1, -5, -2, 1, -1], [19, -14, -6, 8, 0, -13, 7, -13, 8], [-28, -5, -9, 7, 13, 7, -6, -4, 16], [-11, -4, -2, -6, 4, 13, 1, 5, -1], [22, 8, 19, -1, -11, -6, -5, -1, -14], [-28, 7, -12, 1, 9, 7, -10, -5, 22], [-3, 6, 0, -8, 0, 10, 1, 8, -4], [-49, 15, 23, -9, 21, 13, -19, 8, 5], [-8, -8, -5, -3, 11, 8, 1, 6, -4], [32, -2, -14, 15, -24, -15, 15, -3, -15], [16, -11, -19, 2, -6, 1, 11, 0, -7], [-29, -3, 3, -8, -1, 7, 4, 16, -3], [-21, -13, -14, 5, 3, 4, 5, -3, 1], [-38, 0, 5, -12, 20, 11, -20, 6, 10], [15, -9, -3, 17, -12, -4, 1, -3, 5], [15, -12, -1, 7, -13, 9, 0, 10, -3], [-23, -4, 4, 4, 8, 3, -15, -5, 9], [38, -5, -7, -4, -14, -11, 9, -10, -7], [0, 3, -3, -14, 8, -9, 3, -5, 2], [-37, 18, 17, -16, 18, 8, -21, 2, -1], [-39, 22, 34, -20, 0, 2, -9, 18, -12], [9, -14, -4, 3, 15, 12, -4, 4, -1], [-4, 5, -2, -8, 4, 0, -5, -5, -2], [1, -2, -1, 13, -9, -8, 0, -3, 1], [-8, -20, -8, 4, -2, 6, 8, 9, 5], [-23, -3, -2, -8, -10, -9, 6, 6, -6], [-1, -18, -4, -5, -3, 2, 10, 2, 4], [27, -11, 0, -4, -2, 11, -2, 10, 3], [21, -3, -25, 2, 4, 0, -2, -7, 4], [-11, 12, -9, 7, -3, 3, 2, 1, 4], [1, 13, -11, 7, -7, 3, -7, -4, -2], [27, 0, -11, 21, -9, -5, 14, -7, 4], [-13, 16, 25, -8, -10, -10, -5, 8, -20], [-4, 7, 4, -3, 11, 8, -2, 0, -4], [5, -11, -9, 4, -8, -6, 4, 2, -12], [-3, -14, -6, 0, 12, 8, -11, 2, -4], [-12, -11, 10, -21, 11, -5, -4, -1, -5], [-11, 12, -9, -4, 12, -3, -15, -11, 6], [2, -2, -8, 3, 5, 2, 13, 4, 6], [-6, 6, 1, 14, 4, 5, -4, 2, -10], [-4, -15, 8, 0, 5, -8, 3, -7, 5], [-11, 6, 7, -6, 18, 2, -19, -4, 16], [-4, 9, -2, -6, -16, -8, 9, -10, 1], [-36, 0, 11, -5, -3, 4, -15, 16, -9], [20, -17, 5, 4, 3, -18, 3, -9, -9], [3, 3, 11, -3, -5, 4, 5, 0, -5], [-7, 2, 5, 9, 9, 3, -4, -7, -14], [-9, 2, 1, -5, -1, 0, 2, -3, 2], [-5, 17, 2, -7, 13, 4, -5, -8, 7], [37, -12, 7, -5, -5, -10, 14, 2, -22], [4, -7, -11, 15, 21, 15, 0, -4, 7], [9, 11, -3, 10, 4, 0, -14, -14, 3], [-31, 8, 2, -4, 30, 3, -17, 3, 8], [22, -3, -1, -1, -7, -24, 10, -5, 1], [13, -23, -27, 9, 7, 5, 17, -5, 2], [23, -21, -14, 10, -22, 0, 18, -1, -12], [-3, -3, -4, -9, 17, -4, 1, -9, -1], [-7, 8, -11, 1, 1, 5, 0, -13, 0], [-12, 10, 7, -5, 18, 10, -13, 6, 10], [-16, 2, 10, -9, 1, -7, -1, 4, 1], [6, -14, -21, 10, 0, -1, 10, -17, 7], [-5, 3, 20, 1, 7, 4, -9, 7, -10], [-13, 5, -6, 3, 7, 5, -3, -4, 14], [-4, -10, -15, 8, -2, 5, 0, 0, 12], [-7, -13, 5, 6, 14, 19, -12, 12, 10], [-17, 14, -17, 2, -2, -2, -5, -8, 9], [31, -9, -22, 11, -17, -16, 7, -13, -6], [25, -7, -10, -16, 0, -9, 6, -15, 3], [-10, 7, 21, -13, -1, 9, -4, 8, 0], [16, -6, 5, 1, 7, 4, 7, 4, -10], [-3, -16, 6, 2, -10, -15, 1, -3, -4], [-24, 12, 10, -12, 12, 8, -24, 0, 11], [17, -10, 3, 6, -14, 0, 1, 10, -6], [36, -5, -12, 4, -24, -12, 13, -5, 3], [22, 0, 1, 16, 0, 4, -2, -15, 3], [-6, 16, 2, 1, -1, 9, -7, 1, 13], [-8, -12, -14, 8, -18, 0, 8, -4, -1], [-42, 1, -10, 5, 15, 0, 0, -5, 13], [18, 0, 6, 6, 4, 7, 18, 5, -7], [3, 1, 1, -19, 25, -8, -9, -16, 21], [11, -3, 1, 1, -5, 6, 3, 8, 0], [11, 0, -16, -4, 10, 5, -7, -11, 7], [40, -8, 9, -5, -9, -8, -1, 6, -15], [-1, -16, -5, 1, 13, 11, 2, 2, 7], [43, -9, -9, 6, -24, -12, 20, -6, -9], [19, -19, -19, 10, 0, 11, 6, -11, 2], [-1, 26, 24, -9, 5, -4, -6, 5, -9], [-20, -20, -12, 0, -6, 1, 22, 5, -5], [6, -12, 7, -6, 8, 9, -8, 10, -11], [-25, 18, 15, -15, 13, 6, -30, 5, 14], [-1, 4, 33, -22, -6, -8, -1, 12, -12], [-4, -9, -6, 13, 13, -3, -2, 2, 12], [-6, 8, 19, -5, -3, 2, 1, 19, -7], [-38, 2, 1, -2, 5, 16, -14, 2, -5], [21, 12, -20, 13, -9, -6, 0, -18, 2], [-2, 1, -1, 5, -3, 2, 16, 1, 1], [-16, 5, 9, 0, 12, 3, -13, 0, 2], [11, -9, -26, 17, 1, 14, 11, -11, 1], [-9, 4, 21, -17, 1, -18, -14, -2, 0], [-23, 3, -9, 10, 20, 23, -8, 10, 14], [-14, 3, -11, -3, -24, -2, 6, -1, 0], [26, 2, 7, 7, -1, -8, -1, 3, -11], [23, 15, 1, 2, -14, -5, 12, -2, -5], [18, -16, -4, 11, 2, -7, -1, -13, -1], [-5, 9, 12, -1, -11, 0, 1, 5, -18], [2, -16, -3, -6, 23, 13, -4, 1, 9], [-52, 23, 1, -14, 10, 1, -3, 3, 3], [-32, -12, -16, -2, 11, 7, -12, 1, 6], [42, -12, -1, 5, -5, -19, 11, -13, 6], [5, -1, -18, -3, 17, 16, -1, 9, 12], [-15, -19, 3, 2, 24, -8, 0, -15, 19]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4008_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4008_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_4008_2_a_i(:prec:=9) chi := MakeCharacter_4008_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_4008_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_4008_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4008_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![38, -80, -132, 179, 200, -41, -67, -4, 6, 1]>,<7,R![80, 1064, 2148, 1265, -242, -446, -88, 27, 11, 1]>],Snew); return Vf; end function;