A251913 - OEIS

%I #14 Sep 30 2023 18:42:03

%S 1,1,-3,1,-1,-3,1,1,6,-1,-2,-3,-1,1,3,1,-3,6,6,-1,-3,-2,-4,-3,-4,-1,

%T -9,1,2,3,4,1,6,-3,-1,6,3,6,3,-1,0,-3,-5,-2,-6,-4,13,-3,-6,-4,9,-1,12,

%U -9,2,1,-18,2,-10,3,-8,4,6,1,1,6,-2,-3,12,-1,-5,6

%N Coefficients of L-series for elliptic curve "26b1": y^2 + x * y - 2*y = x^3 + 2*x^2.

%H Robin Visser, <a href="/A251913/b251913.txt">Table of n, a(n) for n = 1..10000</a>

%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/26/b/2">Elliptic curve with LMFDB label 26.b2 (Cremona label 26b1)</a>

%F a(n) is multiplicative with a(2^e) = 1, a(13^e) = (-1)^e, else a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p+1 minus number of points of elliptic curve modulo p including point at infinity.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (26 t)) = 26 (t / i)^2 f(t) where q = exp(2 Pi i t).

%e G.f. = q + q^2 - 3*q^3 + q^4 - q^5 - 3*q^6 + q^7 + q^8 + 6*q^9 - q^10 + ...

%o (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, 2, -2, 0, 0], 1), n))};

%o (Sage) A = ModularForms( Gamma0(26), 2, prec=72).basis(); A[0] + A[1];

%o (Magma) A := Basis( CuspForms( Gamma0(26), 2), 72); A[1] + A[2];

%Y Cf. A247198.

%K sign,mult

%O 1,3

%A _Michael Somos_, Dec 10 2014