// Make newform 3610.2.a.bj in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3610_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_3610_2_a_bj();". // 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_3610_2_a_bj();". // 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, 228, -168, -455, 78, 183, -6, -24, 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, 1, 0, 0, 0, 0, 0, 0, 0], [6504, -15744, 1765, 24062, -8957, -5902, 2204, 360, -131], [3744, -2187, -17246, -4033, 9106, 1556, -1444, -111, 70], [-1580, 2832, 933, -2466, 415, 542, -140, -32, 9], [560, -12216, 9573, 14604, -9493, -3704, 1976, 236, -111], [-11726, 17771, 45620, -9371, -21732, 842, 3078, -9, -134], [-10888, 106136, -49539, -113178, 49091, 26298, -10032, -1580, 557], [1084, -30402, 28449, 43524, -26633, -11212, 5320, 706, -291]]; Rf_basisdens := [1, 1, 3876, 1938, 204, 1938, 1938, 3876, 1938]; 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_3610_a();" function MakeCharacter_3610_a() N := 3610; order := 1; char_gens := [2167, 3251]; v := [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_3610_a_Hecke(Kf) return MakeCharacter_3610_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, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 1, 1, 1, 0, -1, -1], [1, -1, -1, 0, 0, 0, 0, 1, 1], [1, 1, -1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 1, 1, -1, 0, -1, 0, 0, 0], [0, 0, -1, 0, 0, 1, 0, 1, 0], [-1, 0, -2, 1, 0, 0, -1, 0, 0], [1, 1, 1, -1, -1, -1, 0, 1, -1], [0, 1, -2, 0, -1, -1, -1, 1, -1], [2, 1, 0, 2, 0, -1, 0, 0, -1], [0, -1, 1, 2, -1, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, 1, 0], [2, 0, 3, -3, 0, -1, -1, 1, 1], [2, 2, 4, -3, -1, 0, -1, 1, 1], [0, 0, 0, 0, 0, 1, 0, -2, -2], [2, 0, -4, 5, 1, 0, 1, -1, -1], [-4, 1, 1, 2, 0, 0, 0, 1, 0], [0, 2, 2, -2, 0, -2, 2, 0, 0], [0, -2, 2, -4, -2, -1, 0, 2, 2], [-3, -2, -1, -2, 2, 2, -1, 0, 1], [2, 0, 4, -3, -1, 1, -1, 1, 1], [-1, 2, 2, -4, 1, 2, 0, -1, -1], [1, -2, 2, -1, 0, 2, -1, -1, -2], [-2, 1, 0, 0, 0, 1, 2, 0, -1], [2, 2, -5, 3, 1, 1, -1, 0, -1], [2, 1, 3, -4, 0, -2, 0, -1, 2], [0, 3, -6, 3, 0, 0, 2, -1, 0], [7, 0, 5, 0, 0, -1, 1, 0, 2], [2, 1, -1, 1, 1, -1, -1, -2, 1], [5, 1, 1, -5, -1, 1, 1, 0, 0], [-1, 0, 3, 1, 0, -1, -1, 1, 2], [5, 0, 0, 2, 1, 0, 0, -1, -1], [-1, -1, 5, 3, 1, 3, 0, -1, -1], [-2, 1, -2, 2, 2, -2, 0, 0, 2], [-1, 1, 1, -3, -1, 1, -2, 0, 1], [0, -1, -4, 3, 0, -2, 2, 0, 0], [-7, -1, 0, -1, 1, 1, 0, 0, 1], [3, 2, -1, 0, 1, -1, 0, 0, 1], [4, -2, -4, -2, 2, -2, 0, 0, 2], [7, -3, 1, 4, 1, 2, 0, -2, 0], [4, -1, 6, -2, 1, 0, -1, 1, 1], [6, 0, 0, 1, -1, 0, 1, -1, -1], [-3, 4, -4, 2, 2, -1, -1, 0, 1], [-3, 3, -1, -5, 1, -3, 2, -4, -1], [2, 2, 6, -6, 2, 3, 2, -2, -2], [2, -2, -1, 0, 0, -3, 0, 3, 2], [2, -1, -5, 7, -1, 3, -1, 0, -3], [7, 0, 6, -7, 0, -2, 1, -2, 0], [5, 4, -1, 3, 1, 0, -1, 2, -2], [7, 0, 2, -2, 0, 1, 4, -2, 3], [2, 1, 5, 1, -1, -1, 1, 0, -1], [7, -1, -7, 7, -1, 3, -2, 0, 3], [-10, 0, -3, -1, 1, -3, 1, 0, 5], [-1, 0, 7, 1, 0, 3, -3, 1, -2], [-7, -1, 1, -5, 1, 0, 1, -2, -2], [-2, 5, 1, 1, -1, -1, 1, 3, 3], [2, -5, 2, -6, -2, 1, 0, 0, 0], [6, 2, -5, 0, 0, 1, 2, -2, 2], [0, 3, 4, 0, -2, -1, -2, 2, 0], [1, 2, 1, -2, -1, 1, -2, 0, -7], [-10, 3, 3, -5, -1, -1, 1, 0, 1], [-4, -1, 2, -1, 0, 2, -2, -2, -4], [1, 1, -1, -3, -2, 0, 1, 2, 2], [6, -1, 3, -5, -1, 1, 3, 0, 3], [14, 0, -6, 0, 0, -1, 2, 0, 0], [7, -4, -1, 6, -3, 1, 0, 4, 1], [-2, 0, 2, 1, -1, 1, -1, -1, -1], [0, -4, 7, -5, -1, -3, 3, -2, 3], [3, -1, -1, 3, 1, 2, 1, 0, -2], [-7, 2, -1, -1, 2, 1, 1, 4, 0], [-8, -1, 6, 6, -3, -1, 0, 1, 2], [-11, -1, -9, 8, -2, -2, -4, 3, -1], [13, 2, -3, -9, -2, -3, -1, 1, 0], [2, -3, -5, 2, 1, 3, 1, 2, 1], [-6, -3, -2, 3, -5, 1, -1, 1, -3], [1, -3, 4, -15, 0, 1, 3, -1, 2], [15, -3, 3, 0, -3, 2, 0, 1, 0], [9, -2, 1, -2, -1, -3, -2, 6, 1], [-3, 6, 0, 1, 2, 0, 1, 0, 2], [-10, 4, 2, 6, 2, 4, -2, -2, -6], [0, -2, 9, -12, 2, 1, 2, 1, -2], [2, -1, -2, -8, -2, -4, 0, 4, 4], [2, 2, -4, 5, -4, -2, 2, 0, 4], [-6, -4, 4, 5, 5, 1, -3, -1, 1], [3, 0, -3, 5, 0, -3, 3, -3, 2], [-6, -5, 8, 1, 4, 0, 0, -2, 2], [0, -2, 6, 0, 0, -1, 4, 4, 2], [-5, 0, 1, -3, 0, -1, -5, -1, 2], [7, 0, -5, 1, -4, 1, 1, 4, 0], [0, -2, -6, 2, 3, -3, 1, 0, -2], [-1, -1, -1, -9, 2, 0, -1, 2, -2], [1, 1, 9, -4, 4, 3, 1, -1, 4], [-5, -2, 2, 3, 0, 6, -1, -4, -2], [-9, -1, -1, 5, 0, 7, -1, -1, -2], [-6, 0, -4, 2, -2, 0, 2, 0, 4], [-4, -4, 11, -4, 2, -1, 2, 3, 6], [2, -1, 2, -4, 2, 1, 2, 4, 7], [-10, 2, 5, 6, -2, -1, 0, -4, -6], [0, -1, -6, -4, 2, 1, 0, 0, -4], [-4, 5, 10, -2, 0, -5, -2, -1, -1], [-7, 0, -4, -2, -4, -5, 0, 4, 1], [-2, 1, 1, -6, -6, 0, 0, 1, 0], [-4, -3, 14, -2, -2, -1, -2, 0, 1], [-14, 1, -11, 2, 0, 2, 2, 1, 2], [-1, 2, 3, -1, 4, -1, 3, -7, -2], [9, -6, -3, 3, -5, -2, 2, 0, -1], [5, 2, -10, 8, 3, -2, -2, 4, 3], [-14, -3, 3, -3, -2, -3, -2, 1, 4], [2, -1, -1, 0, 0, -4, 4, 1, 6], [7, -3, -5, 4, 1, 4, 0, -1, 2], [1, 10, -4, -6, 3, 2, 2, -3, -1], [-1, -3, -7, 10, -6, -5, 2, 1, -1], [-16, 1, -4, 4, 0, -2, 0, 0, -2], [9, -5, 1, -2, 2, -3, -1, 2, 2], [9, -2, 5, -2, 3, 1, -2, 3, 5], [-9, 2, -6, -2, 0, -1, -1, 0, -5], [-8, 4, -2, 2, 4, 4, 0, -4, -4], [1, -3, -1, 2, 5, 6, -4, -2, 0], [11, 1, -3, -7, 1, 5, -4, -3, -3], [-4, -3, -4, -8, 2, 1, 0, 4, -3], [0, 2, -16, 2, -3, -5, 3, 6, 4], [1, 2, 8, 8, 3, 6, 0, -7, -7], [9, -6, -8, 12, 1, -2, -4, 1, 7], [9, -2, -2, -2, 1, 2, 0, -1, -7], [-11, 0, 15, -11, -6, -7, 1, 5, 2], [-19, -6, 5, 0, -3, 1, 0, -1, 1], [-5, 1, 2, -1, -1, -7, 0, 0, 5], [-17, 6, -6, 12, 5, 4, 2, -6, -5], [-4, -2, 2, 1, 3, 4, 3, -1, 1], [-9, -7, -1, -7, 1, 5, 4, -2, 1], [-10, -2, 10, -3, 6, 2, 0, -4, 6], [-9, 6, 12, -12, 0, 2, 0, -4, -4], [1, 1, 7, -8, -6, -1, -1, 4, -2], [-2, -1, -6, 8, 2, -3, -2, 2, 4], [14, -1, -5, -10, -1, 1, -1, 0, -1], [8, 4, 8, -7, -6, 2, 3, 0, -7], [-5, -2, 6, 2, -3, -4, -2, -4, 5], [10, -2, -13, -2, -4, -9, 0, 9, 4], [-3, -1, -1, -3, 1, 5, 2, -1, 7], [4, -1, 2, 0, 4, 3, 2, -4, -4], [3, 4, -9, 26, 3, 1, -2, 0, 1], [-12, -10, -13, 1, 3, 1, -1, 2, 9], [-5, 1, -4, 6, 0, -7, 2, -2, 3], [6, -3, -3, 15, 3, 3, -3, 0, 9], [5, -3, -1, -1, 0, 0, -1, 2, -2], [21, -1, -8, 17, -5, -4, 4, 7, 1], [13, -2, 15, -7, 0, 1, -1, 2, 2], [4, -2, 1, 10, 6, 2, 2, -8, -2], [-4, 10, -2, -6, -2, -3, 2, -4, -2], [-12, -3, 10, -7, 2, -5, 1, 2, 5], [-6, 3, -16, 2, -2, -5, -2, 2, 0], [3, -2, -8, 6, 5, 8, -4, -1, -3], [-3, -8, -6, 14, 3, 6, 4, -5, -3], [5, -7, 8, -9, -2, 1, -3, 7, -2], [-4, 5, -13, 3, 1, 1, -3, 6, 1], [12, -4, -14, 3, -1, 0, 3, 1, 5], [2, -4, -4, 4, 2, 6, 2, 0, 2], [14, 5, -7, 12, 0, -2, 4, -1, 0], [-13, -2, -1, 6, 3, 7, 0, -4, -1], [-5, -2, 2, 18, 6, 1, -4, 4, -5], [4, -1, 1, -13, -13, -5, 1, 4, 1], [4, 3, 9, -3, -2, -1, 0, 8, -4], [7, -6, -4, -6, 1, -2, 0, -1, 1], [-3, -9, -2, -9, -1, 0, 4, 2, -3], [-2, 7, 13, -12, -5, -1, -3, -1, -5], [6, -8, 5, -1, 1, -1, 3, 3, 3], [5, -2, 1, -6, -8, -3, -1, 4, 2], [4, -4, 8, -7, 1, -2, -1, 9, -1], [-9, 2, -25, 21, 2, 1, -3, 1, 2], [-4, 3, -1, -9, -5, -3, -5, 10, -3], [8, -2, -2, 7, 7, 2, -1, -5, 1], [1, 1, 6, -3, 0, -7, -1, 3, 0], [-4, 4, -13, 5, 0, 1, 3, -3, 5], [12, 2, -4, -24, 2, 0, 0, 2, 2], [-3, 3, -8, -1, -7, 0, -2, -2, -13], [15, 4, 3, -1, 4, 5, 3, -7, 2], [-1, -3, 13, -1, 5, 1, 4, 2, -1], [15, -5, 3, 6, -1, 0, 2, 3, 6], [-8, 2, 20, -13, -5, -6, -1, 2, -3], [-10, 9, 1, 20, 2, 0, -6, -1, -4], [16, 1, -9, 13, 2, 1, 0, -1, -6], [10, -2, -4, 5, -5, -6, 1, 7, -5], [6, -1, -18, 17, -2, -3, -3, -2, 1], [-24, 5, 2, -4, 4, 5, -2, -6, 1], [11, -9, 3, 0, 6, 9, 4, -5, -3], [-1, -2, 9, 2, -3, -3, 6, -2, -3], [8, 14, 0, 17, 1, -1, -3, -3, -5], [6, 4, -8, -2, 2, -2, 2, 4, 2], [5, 7, -2, 12, -3, 2, 0, -7, -1], [-7, 4, 6, -23, -6, -2, 3, 0, 2], [14, 5, 0, 2, -2, -4, 6, -2, -6], [5, 7, 17, -2, 3, 6, 0, -8, -6], [5, -1, 0, -4, 0, -5, -1, -1, 6], [-1, -3, 3, -4, -4, -3, -9, 7, 0], [-4, 3, -3, -4, -6, -8, 2, 5, 12], [0, 13, 14, -6, 1, -2, 3, -2, -5], [5, 2, -8, 0, -3, -4, 4, -1, -1], [-1, 0, -6, -9, -2, -6, 3, 0, -2], [-5, 3, -6, -14, -4, -4, -1, 3, -4], [-1, 2, 10, -17, -9, 0, 4, -2, -7], [5, -1, -6, -11, -3, -7, 4, 4, 3], [10, -4, 9, -14, -6, 1, 0, 0, -4], [20, -4, -4, 5, -3, 0, -1, -3, -5], [2, 3, -6, 18, -2, -2, -2, 0, -2], [1, -1, 9, -16, -6, -9, 0, 9, 11], [16, 0, 6, -1, 0, 3, 4, 3, 0], [-20, -12, -8, 9, 3, 1, 1, -1, 5], [-9, 0, -20, 15, 4, 0, -1, 0, -2], [-5, 17, -5, 2, 0, 0, 6, -3, -1], [-4, 4, 0, 14, 10, -1, -4, -2, -2], [7, 0, 7, 4, 5, 3, -4, -4, 5], [0, -3, 16, -2, 8, -1, -2, -9, 5], [-10, 8, 1, 10, 0, 5, 2, -3, -6], [-9, -9, -10, -6, -1, 1, 4, -1, -4], [-4, 7, 9, -19, -10, -6, 1, 5, -1], [9, -1, -4, 2, 0, -5, 2, 2, 7], [-1, -10, 2, 15, 0, 0, -5, 6, -4], [8, -8, 13, -10, 2, -1, 4, -5, 4], [-9, -2, 7, -7, -3, -4, -2, 4, 3], [-6, 7, 4, 2, 1, 9, 0, -11, -2], [-4, 0, 2, 10, 4, -3, 4, -2, 6], [2, 4, 2, 8, -2, 4, -4, 2, -12], [16, 2, -16, 1, -3, 1, -3, 3, -3], [27, -4, 1, 0, -1, 3, 0, 2, -3], [-13, -1, -8, 6, -2, -2, 5, 0, 2], [-9, -1, -17, 14, -2, -1, -5, 3, 6], [-12, -6, -13, 1, 1, -1, 5, -5, -1], [-6, -4, -2, 5, 0, 7, 2, -5, -6], [13, -2, 12, -10, -9, 0, 4, 1, -7], [-9, -2, -8, -7, 1, 0, -6, 6, 11], [-10, -1, 4, 14, 10, 5, -2, 0, 0], [3, 5, -21, 24, 0, -7, -1, -3, 0], [14, 1, 12, -2, -1, -5, 1, -1, -3], [-10, -11, 6, -9, -4, -6, 4, 3, 10], [38, 2, 15, -1, 1, 2, 3, -3, -7], [14, -8, 2, 1, -7, 6, 1, 1, -5], [-10, -7, 0, 4, 4, -2, 0, 0, 0], [-20, 2, -7, 0, 1, 5, -3, -6, -11], [-23, 1, 17, -26, 0, -4, 2, -1, -1], [7, 10, 6, 4, 7, 8, -2, -7, -5], [29, -7, 7, -4, 1, -2, 0, -6, 0], [14, -4, 1, 5, -5, 1, -3, 6, -3], [-19, 0, 13, -14, 1, 1, -4, -4, -7], [-23, -6, -5, 5, 6, 6, 1, -5, 4], [-6, -1, -10, -12, 2, 4, 0, -3, -12], [13, 3, 10, -5, -5, 7, -3, 1, -6], [-7, -2, -26, 25, 4, 0, -1, 1, -2], [7, 2, -5, -2, -8, 0, -1, -2, -15], [16, -7, 10, -19, -4, 6, -2, 0, -10], [-9, -12, -9, 3, 3, 8, -4, 6, 9], [8, -5, 14, -25, 2, 2, 4, 1, 4], [0, -4, 11, -6, 1, 5, 4, -6, -3], [19, -4, -6, 16, 1, -4, -4, 5, -1], [-18, 0, -18, 18, 2, 1, -2, 2, 6], [2, -7, -19, -4, 1, 9, -3, -1, -3], [9, 11, 9, -8, -5, -4, -2, -6, -2], [-7, -7, -3, 18, 4, 1, -3, 3, 4], [-20, -7, -10, 8, 4, 3, 2, -2, 6], [7, -6, -1, 6, 7, 7, 6, -4, 9], [-5, -7, -4, 13, 3, -5, 3, 3, 14], [-2, -2, -6, 7, 9, 3, -1, -9, -7], [4, 2, 19, -12, -5, 5, -1, -6, -7], [12, 2, 16, -22, -4, 0, 6, -2, -2], [11, 12, -2, 15, -1, 0, 0, -6, -11], [-20, -1, -4, 20, -4, 1, 4, -2, -7], [-6, -13, -20, 15, -2, 7, -5, 0, 3], [7, -4, 23, -11, 8, 3, 1, -3, -4], [-26, -7, 12, -2, -2, -1, 4, 2, 5], [10, -2, -6, -7, -1, -1, 1, -1, -5], [-4, -2, -21, 24, 6, 5, 4, -3, -2], [-3, -7, 3, 2, 3, 6, -4, -10, -8], [6, 3, 21, -3, 0, -3, 0, -6, -8], [8, 5, -18, -10, 2, 4, -4, -8, -2], [27, -8, -4, 3, -6, -4, 1, 2, 12], [-10, -1, -22, 9, 1, 7, -3, -5, -3], [7, 6, 2, -26, 1, -2, 6, -4, -3], [-5, -2, 4, 19, 2, 0, 3, -10, -8], [-1, 0, 15, -31, 4, -1, -1, -7, 2], [0, 0, 20, -6, 6, 14, -6, -2, -2], [18, -6, 18, 9, 1, 2, -1, -3, -3], [-4, -9, 16, 4, -6, -11, 0, 6, 13], [-6, 6, -19, 0, -2, -9, 2, 11, 8], [-6, -3, 11, -10, 10, 6, -4, -1, 2], [-1, -12, 21, -2, 1, -1, 4, -8, 7], [-13, -8, -15, 4, -1, -3, -2, 8, -1], [29, -2, -14, -10, -2, -14, 2, 4, 2], [6, -10, 12, -12, -6, 4, 8, 2, 4], [14, 9, -14, 16, 2, -1, 0, 8, 3], [-4, -2, -6, -7, 5, 9, 1, -9, -9], [-22, -6, -3, -10, -7, -9, 0, 12, 7], [-21, 2, 1, -14, -7, -5, 0, 12, 11], [-3, -4, 12, -2, 5, 4, 6, -3, 7], [11, -4, -10, 20, 1, 0, 2, -3, 5], [5, -4, 3, 14, 7, 1, -6, 5, -3], [16, -2, 4, 0, 1, -1, 7, -4, -6], [8, 0, -16, 10, 2, -9, 6, 6, 2], [-6, 11, -17, 7, 3, -1, -3, -8, -3], [13, 1, 8, 10, 6, 0, -1, -13, -4], [10, -4, -11, -3, -1, 1, 7, 6, 5], [10, -10, 24, -18, 2, 3, -2, -4, -6], [19, 1, -16, 21, -3, -5, 2, 4, 5], [-2, 11, 7, -9, 3, 11, -3, 2, -13], [-28, -1, 0, 2, 6, -9, -2, 2, 6], [0, -14, -6, -3, 7, 4, 3, -5, 5], [22, 0, 1, -15, -3, 1, 1, -4, 1], [8, 9, -1, -19, -6, 2, 5, -1, -15], [21, -4, -12, 6, 5, -4, 0, 4, 11], [13, -4, 8, 3, 0, 0, -3, -6, -8], [-2, 6, 3, -9, 0, -20, -2, 8, 8], [-20, -2, 0, -20, 4, 1, -2, -4, 0], [-12, -8, -8, 13, 3, 4, -5, -2, -5], [-8, -6, -23, 4, -5, -8, -7, 14, 8], [-4, 5, -16, 15, 0, -5, 4, 0, 4], [4, -4, 15, -24, -6, -9, 0, 7, 16], [-2, -6, 10, -1, -3, -5, 1, 5, 13], [10, 15, -2, -12, -4, -3, 8, -5, -9], [-7, 9, -24, -3, 3, 1, 2, -10, -3], [-26, -1, -7, -5, 5, 1, 1, -4, -11], [8, -8, 0, -2, 12, 8, -6, -10, -4], [-2, -4, -5, 6, 8, 1, -2, 3, 0], [23, 3, 15, -14, -9, -2, -4, 10, -2], [-4, -13, 4, 2, 5, 3, -4, -5, -2], [-2, 4, 1, -9, -7, -6, 3, -1, -7], [-26, -7, -8, 2, 2, 8, -2, 2, -4], [1, 9, 7, -17, -5, 1, 0, 2, -11], [11, -2, 22, -2, 5, 4, -2, -8, -3], [-6, -4, -8, 31, 3, 8, -1, -1, 3], [2, 9, -5, -1, -4, 2, 3, -3, -13], [4, 16, 10, -1, -1, -4, 7, -7, -9], [10, 0, 2, -4, -10, -8, 2, 4, 8], [38, 2, 9, 1, -1, -6, -3, -3, -3], [6, 9, -2, -2, 10, 5, 2, -2, 5], [-2, -8, 2, 5, -7, -4, 3, 15, 3], [10, 3, -7, -6, -2, -2, -6, -5, -2], [-1, -2, -4, -10, 4, 8, 2, -12, -4], [-4, 15, 22, -9, -2, 3, -5, -2, 1], [-18, -15, -3, 10, 1, 5, 1, 0, -3], [-8, 11, 17, 0, 2, -2, 4, -1, -4], [19, -10, -3, 8, -1, 7, 4, 4, 5], [18, 0, -5, 12, 2, -9, 0, -3, 10], [11, 1, 20, -5, -7, -2, 4, -2, -7], [0, -10, -19, 21, -7, -4, -3, 1, -3], [8, -8, 23, -33, -5, 1, 3, 0, -3], [-32, -13, 28, -20, 4, -7, 4, 2, 7], [2, 10, -15, 21, -3, 9, -3, 0, -15], [2, -9, 7, -7, -4, -13, 0, 7, 0], [-38, -5, -16, 0, -2, 4, 4, -3, -2], [28, 1, 3, -6, 6, 0, 8, -1, 2], [-8, -20, -14, -14, -8, 6, -2, 6, 0], [-19, -4, 0, 5, 2, 4, 1, -2, 8], [11, 8, 5, -17, -2, -1, 5, -3, -12], [14, -15, 5, 0, 0, -2, -4, -3, 6], [-1, -4, -9, 23, 6, 11, -5, 1, 10], [-4, 6, 16, -9, 0, -11, -4, -3, -2], [32, 2, -8, 11, -3, 1, 7, 5, 3], [6, 5, -6, -8, 1, -1, -2, -5, -8], [-35, 3, 4, -12, -2, 2, 7, -7, -6], [1, -12, -9, -13, -5, 2, 3, 0, 2], [0, 9, 2, -23, -14, -2, 8, 1, 0], [-10, -3, 28, -22, -4, 6, -6, 3, -6], [-32, 9, 2, -14, 6, -7, -4, -2, 3], [-11, -5, 9, -22, -3, 4, -8, 0, -10], [-16, 1, -22, -3, -4, -2, -2, 3, -12], [2, 1, -21, 9, -2, 8, -7, -1, -7], [4, 14, -19, 0, -4, 0, 4, -2, -4], [28, 0, -18, 27, 2, 1, -10, -1, 0], [13, -11, 1, -12, -4, -15, 3, 11, 20], [-14, 3, 19, 10, -5, -7, 3, 9, 9], [-6, 6, 24, -13, -5, 2, -1, 5, 5], [24, -2, 10, 11, -5, -8, 3, 9, 13], [8, 5, 3, -9, -6, -2, -5, -1, 9], [-10, -8, -7, 4, 9, 19, -6, -4, -1], [3, 0, -15, 14, -1, -11, 4, 0, 1], [-16, -2, -16, 15, 3, -5, -5, 7, 5], [-28, 10, -6, 9, 8, 2, -5, -4, 5], [14, 3, 7, 6, -2, 2, -2, -9, -12], [5, 4, 5, 10, 5, 7, -2, -8, -19], [9, 3, -5, -8, 9, 2, -6, -4, 4], [-40, 9, 9, -1, -4, -3, -2, 3, 0], [12, 10, 16, -11, 5, -5, -1, -5, 5], [-19, -12, -24, -7, 1, 1, 4, -1, -5], [-17, -7, -3, 18, 4, -3, -1, -4, 10], [41, 2, -5, -2, 3, -3, 10, -2, 1], [-8, 2, -6, -7, -7, -10, -7, 4, 11], [-2, -11, -2, 8, -10, -4, -2, 2, -6], [-28, -6, -11, -4, 0, -7, 8, -7, 6], [14, 10, -3, -25, -2, -2, 8, 2, 6], [22, 12, 12, 4, -4, -12, 2, 3, 10], [-19, 3, 27, -11, -1, 7, 3, 2, 0], [10, -1, -28, 20, -2, -5, 2, 10, 8], [-7, 8, 28, -32, -11, 2, 0, 3, -9], [-14, -6, -12, -3, 7, 5, -1, -9, -17], [-18, -2, 31, -15, 7, 3, -3, -6, -7], [-6, 2, -3, 20, 6, -8, -4, 4, 8], [-20, -2, -3, 5, -2, 9, -5, 9, 13], [10, -4, -21, -1, -13, -3, -5, 14, 11], [-13, 6, 30, -23, 0, 2, -1, -4, -10], [4, -4, -2, -5, -7, -1, 1, -5, 1], [5, 15, 25, -18, -10, 4, 2, 1, -11], [5, 11, 1, -20, 5, 0, 2, -4, 2], [-27, -22, -5, 0, 7, 15, -8, 2, 1], [-15, 10, -10, 7, 4, -2, 3, -12, -4], [-18, -11, 10, 7, 8, 7, -1, 8, 1], [9, 5, -30, -1, -2, -9, 1, 7, 0], [6, -3, -10, 20, 16, 9, -2, -6, 3], [-9, -7, -15, 9, 9, 13, -1, -2, 0], [14, -9, 13, -14, -4, -10, 6, -1, -4], [5, 3, -4, 8, 4, 2, 1, -7, -8], [10, 2, -34, 32, -2, -3, 2, -2, 0], [20, -7, 0, -12, -4, 6, -4, 7, 2], [-6, 6, 0, -21, -11, -11, 3, 5, 1], [10, -2, -24, 34, 12, 4, 2, -6, 6], [23, 8, 2, -16, -7, -14, 2, 3, -5], [-23, 5, 16, -7, -13, 3, 3, 7, -2], [-12, -6, 14, -2, -4, -4, -2, 6, -4], [16, -3, -5, 13, 2, -3, -4, 2, 2], [-20, -12, -20, 16, 4, 0, -6, 4, 2], [12, 13, -5, -25, 2, 1, 8, -7, 0], [17, 20, -16, 5, 6, 4, 1, 2, -10], [11, -2, -1, 16, -2, -9, -1, 10, 20], [-24, -12, -20, 20, 8, 10, 4, -8, -6], [5, 0, 12, -11, -4, -4, 9, -4, -8], [28, -2, 26, -2, -4, -3, 6, 0, 4], [15, 19, 11, 6, -4, -1, 5, -1, -4], [19, -9, -23, 16, 5, -8, -4, 1, 6], [-12, -18, 1, 28, 14, 9, -2, 1, 2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3610_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3610_2_a_bj();". // 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_3610_2_a_bj(:prec:=9) chi := MakeCharacter_3610_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_3610_2_a_bj();". // 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_3610_2_a_bj( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3610_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![8, 228, 168, -455, -78, 183, 6, -24, 0, 1]>,<7,R![-136, -5592, 9372, -3540, -1236, 804, 35, -51, 0, 1]>,<13,R![13544, -27240, 8136, 8288, -3894, -486, 415, -27, -9, 1]>],Snew); return Vf; end function;