%I A127826
%S A127826 1,2,0,0,1,6,0,0,6,2,0,0,5,4,0,0,12,0,0,0,6,10,0,0,7,12,0,0,4,6,0,0,0,
%T A127826 14,0,0,2,10,0,0,11,18,0,0,18,0,0,0,10,6,0,0,0,6,0,0,18,0,0,0,25,12,0,
               0,
%U A127826 20,18,0,0,6,22,0,0,0,14,0,0,6,0,0,0,0,2,0,0,13,2,0,0,12,18,0,0,0,36
%V A127826 1,-2,0,0,1,-6,0,0,-6,-2,0,0,-5,4,0,0,12,0,0,0,6,10,0,0,-7,12,0,0,4,-6,
               0,0,0,14,0,0,-2,
%W A127826 10,0,0,-11,-18,0,0,-18,0,0,0,10,-6,0,0,0,-6,0,0,18,0,0,0,25,-12,0,0,-20,
               -18,0,0,6,-22,
%X A127826 0,0,0,14,0,0,-6,0,0,0,0,-2,0,0,-13,-2,0,0,12,-18,0,0,0,36
%N A127826 Coefficients of L-series for elliptic curve "256a1": y^2 = x^3 + x^2 
               - 3*x + 1.
%H A127826 W. Stein, <a href="http://modular.fas.harvard.edu:8080/mfd/newform.html?space=[256,
               2,[]]&number=1&search=256">Modular Forms Database</a>.
%F A127826 a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 
               (1+(-1)^e)/2*(-p^2)^(e/2) if p == 5,7 (mod 8), b(p^e) = b(p)*b(p^(e-1)) 
               -p*b(p^(e-2)) if p == 1,3 (mod 8) and b(p) = 2*x*(-1)^((x%8>4)+(y%4)>
               0) where p = x^2+2*y^2.
%F A127826 a(4n+2)=a(4n+3)=0.
%F A127826 Expansion of q^(-1/2) * eta(q^8)^8 / (eta(q^4) * eta(q^16))^2 * (eta(q^4) 
               / eta(q^16) - 2 * eta(q^16) / eta(q^4)) in powers of q.
%F A127826 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = -64 
               (t/i)^2 f(t) where q = exp(2 pi i t).
%e A127826 q - 2*q^3 + q^9 - 6*q^11 - 6*q^17 - 2*q^19 - 5*q^25 + 4*q^27 + 12*q^33 
               + ...
%o A127826 (MAGMA) f := qEigenform(EllipticCurve(CremonaDatabase(), "256a1"), 188); 
               [ Coefficient(f, n) : n in [ k : k in [0..188] | IsOdd(k) ] ] ; /
               * Klaus Brockhaus, Feb 01 2007 */
%o A127826 (PARI) {a(n)=local(A, p, e, x, y); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1,
               matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p%8>4, if(e%2, 
               0, (-p)^(e/2)), for(i=1, sqrtint(p\2), if(issquare(p-2*i^2, &x), 
               y=i; break)); a0=1; a1=y=2*x*(-1)^((x%8>4)+(y%4>0)); for(i=2,e, x=y*a1-p*a0; 
               a0=a1; a1=x); a1))))) }
%o A127826 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^(n\4)); polcoeff( eta(x^2+A)^8/ 
               eta(x+A)^2/ eta(x^4+A)^2* ((n%4==0)*eta(x+A)/eta(x^4+A) -(n%4==1)*2*eta(x^4+A)/
               eta(x+A)),n\4))}
%o A127826 (PARI) {a(n) = ellak( ellinit([0, 1, 0, -3, 1], 1), 2*n + 1)}
%Y A127826 Convolution of A138515(q^4) and A112172.
%Y A127826 Sequence in context: A122890 A138497 A113129 this_sequence A109983 A093492 
               A128771
%Y A127826 Adjacent sequences: A127823 A127824 A127825 this_sequence A127827 A127828 
               A127829
%K A127826 sign
%O A127826 0,2
%A A127826 Michael Somos, Jan 30 2007