A318375 - OEIS

%I #10 Jan 03 2024 23:47:49

%S 1,5,-7,-1,-5,-4,-1,8,18,0,-13,11,17,-13,0,-35,5,-7,2,0,-11,20,-5,-7,

%T 0,23,14,17,36,-25,-19,0,-25,17,0,29,-20,-28,-22,0,-31,7,0,-5,0,-1,26,

%U 32,-17,0,40,-16,-13,0,35,-31,29,55,23,0,-18,-31,-25,-37,0

%N Coefficients of modular form for elliptic curve "108a1": y^2 = x^3 + 4 divided by q in powers of q^6.

%H Robin Visser, <a href="/A318375/b318375.txt">Table of n, a(n) for n = 0..10000</a>

%H LMFDB, <a href="http://www.lmfdb.org/EllipticCurve/Q/108/a/2">Elliptic Curve 108.a2 (Cremona label 108a1)</a>.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 108 (t/i)^2 f(t) where q = exp(2 Pi i t).

%F a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p>5, where b(p) = p minus number of points of elliptic curve modulo p.

%e G.f. = 1 + 5*x - 7*x^2 - x^3 - 5*x^4 - 4*x^5 - x^6 + 8*x^7 + ...

%e G.f. = q + 5*q^7 - 7*q^13 - q^19 - 5*q^25 - 4*q^31 - q^37 + ...

%t a[ n_] := Module[ {x, y, p, e}, If[ n < 1, Boole[n == 0], Times @@ ( If[ # < 5, 0, {p, e} = {##}; p^(e/2) ChebyshevU[ e, -Sum[ KroneckerSymbol[ k^3 + 4, p], {k, p}] *p^(-1/2) / 2 ]] & @@@ FactorInteger@(6 n + 1) )]];

%o (PARI) {a(n) = if( n<0, 0, n = 6*n + 1; ellan(ellinit([0, 0, 0, 0, 4]), n)[n])};

%o (PARI) {a(n) = my(A, p, e, y='y); if( n<0, 0, A = factor(6*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, substpol( polchebyshev(e, 2, -1/2/y * sum(k=1, p, kronecker(k^3 + 4, p))) * y^e, y^2, p))))};

%o (Magma) qExpansion( ModularForm( EllipticCurve( [0, 0, 0, 0, 4])), 386);

%o (Magma) A := Basis( CuspForms( Gamma0(108), 2), 386); A[1] + 5*A[6] - 7*A[9] - A[10];

%o (Sage)

%o def a(n):

%o     return EllipticCurve("108a1").an(6*n+1)  # _Robin Visser_, Jan 03 2024

%K sign

%O 0,2

%A _Michael Somos_, Aug 24 2018