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A247198 Coefficients of L-series for elliptic curve "26a3": y^2 + x * y - y = x^3 + x or y^2 + x*y + y = x^3.

%I A247198 #17 Sep 30 2023 17:03:56
%S A247198 1,-1,1,1,-3,-1,-1,-1,-2,3,6,1,1,1,-3,1,-3,2,2,-3,-1,-6,0,-1,4,-1,-5,
%T A247198 -1,6,3,-4,-1,6,3,3,-2,-7,-2,1,3,0,1,-1,6,6,0,3,1,-6,-4,-3,1,0,5,-18,
%U A247198 1,2,-6,-6,-3,8,4,2,1,-3,-6,14,-3,0,-3,-3,2,2,7,4
%N A247198 Coefficients of L-series for elliptic curve "26a3": y^2 + x * y - y = x^3 + x or y^2 + x*y + y = x^3.
%H A247198 Robin Visser, <a href="/A247198/b247198.txt">Table of n, a(n) for n = 1..10000</a>
%H A247198 LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/26/a/3">Elliptic curve with LMFDB label 26.a3 (Cremona label 26a3)</a>
%F A247198 a(n) is multiplicative with a(2^e) = (-1)^e, a(13^e) = 1, 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 A247198 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 A247198 G.f. = q - q^2 + q^3 + q^4 - 3*q^5 - q^6 - q^7 - q^8 - 2*q^9 + 3*q^10 + ...
%o A247198 (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, 0, 1, 0, 0], 1), n))};
%o A247198 (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, 0, -1, 1, 0], 1), n))};
%o A247198 (Magma) A := Basis( CuspForms( Gamma0(26), 2), 76); A[1] - A[2];
%o A247198 (Sage)
%o A247198 def a(n):
%o A247198     return EllipticCurve("26a3").an(n)  # _Robin Visser_, Sep 30 2023
%K A247198 sign,mult
%O A247198 1,5
%A A247198 _Michael Somos_, Nov 24 2014