A127826 - OEIS

%I #16 Sep 08 2022 08:45:29

%S 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,

%T -6,0,0,0,14,0,0,-2,10,0,0,-11,-18,0,0,-18,0,0,0,10,-6,0,0,0,-6,0,0,

%U 18,0,0,0,25,-12,0,0,-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

%N Coefficients of L-series for elliptic curve "256a1": y^2 = x^3 + x^2 - 3*x + 1.

%H G. C. Greubel, <a href="/A127826/b127826.txt">Table of n, a(n) for n = 0..1000</a>

%H W. Stein, <a href="http://wstein.org/Tables/">Modular Forms Database</a>.

%F 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 mod 8 > 4) + (y mod 4) > 0) where p = x^2 + 2*y^2.

%F a(4n+2)=a(4n+3)=0.

%F 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 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 q - 2*q^3 + q^9 - 6*q^11 - 6*q^17 - 2*q^19 - 5*q^25 + 4*q^27 + 12*q^33 + ...

%t eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[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]), {q, 0, 150}], q]; Table[a[[n]], {n, 1, 100}] (* _G. C. Greubel_, Jul 25 2018 *)

%o (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 (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 (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 (PARI) {a(n) = ellak( ellinit([0, 1, 0, -3, 1], 1), 2*n + 1)}

%Y Convolution of A138515(q^4) and A112172.

%K sign

%O 0,2

%A _Michael Somos_, Jan 30 2007