A256678 - OEIS

%I #9 Sep 30 2023 17:22:41

%S 1,1,-2,1,0,-2,-4,1,1,0,6,-2,2,-4,0,1,-1,1,-4,0,8,6,0,-2,-5,2,4,-4,0,

%T 0,-4,1,-12,-1,0,1,-4,-4,-4,0,6,8,8,6,0,0,0,-2,9,-5,2,2,-6,4,0,-4,8,0,

%U 0,0,-4,-4,-4,1,0,-12,8,-1,0,0,0,1,2,-4,10,-4

%N Coefficients of L-series for elliptic curve "34a1": y^2 + x*y = x^3 - 3*x + 1.

%H Robin Visser, <a href="/A256678/b256678.txt">Table of n, a(n) for n = 1..10000</a>

%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/34/a/4">Elliptic curve with LMFDB label 34.a4 (Cremona label 34a1)</a>

%F a(n) is multiplicative with a(2^e) = 1, a(17^e) = (-1)^e. For p prime, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) and a(p) = p minus number of points of elliptic curve modulo p.

%F a(2*n) = a(n).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (34 t)) = 34 (t/i)^2 f(t) where q = exp(2 Pi i t).

%e G.f. = q + q^2 - 2*q^3 + q^4 - 2*q^6 - 4*q^7 + q^8 + q^9 + 6*q^11 - 2*q^12 + ...

%o (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, 0, 0, -3, 1], 1), n))};

%o (Magma) A := Basis( CuspForms( Gamma0(34), 2), 77); A[1] + A[2] - 2*A[3];

%o (Sage) A = CuspForms( Gamma0(34), 2, prec = 77) . basis(); A[0] + A[1] - 2*A[2];

%K sign,mult

%O 1,3

%A _Michael Somos_, Apr 07 2015