%I #17 Sep 30 2023 16:40:52

%S 1,0,1,-2,-1,0,1,0,-2,0,-3,-2,5,0,-1,4,3,0,2,2,1,0,-6,0,1,0,-5,-2,3,0,

%T -4,0,-3,0,-1,4,2,0,5,0,-12,0,-10,6,2,0,9,4,1,0,3,-10,12,0,3,0,2,0,0,

%U 2,8,0,-2,-8,-5,0,-4,-6,-6,0,0,0,2,0,1,-4,-3,0,-1,-4,1,0,12,-2,-3,0,3,0,-12,0,5,12,-4,0,-2,0,-1,0,6,-2,6

%N Coefficients of L-series for elliptic curve "35a3": y^2 + y = x^3 + x^2 - x.

%H Robin Visser, <a href="/A115672/b115672.txt">Table of n, a(n) for n = 1..10000</a>

%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/35/a/2">Elliptic curve with LMFDB label 35.a2 (Cremona label 35a3)</a>

%H W. Stein, <a href="http://wstein.org:8080/mfd/space.html?space=[35,2,[]]&amp;number=1&amp;search=35">Modular Forms Database</a>.

%F a(n) is multiplicative with a(5^e) = (-1)^e, a(7^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) otherwise.

%F Expansion of (eta(q^5) * eta(q^7))^2 + (eta(q) * eta(q^35))^2 in powers of q. Expansion of a newform level 35 weight 2 and trivial character.

%e q + q^3 - 2*q^4 - q^5 + q^7 - 2*q^9 - 3*q^11 - 2*q^12 + 5*q^13 - q^15 + ...

%o (PARI) {a(n)=if( n<1, 0, ellak( ellinit([ 0, 1, 1, -1, 0]), n))} /* _Michael Somos_, Mar 03 2011 */

%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^5 + A) * eta(x^7 + A))^2 + x^2 * (eta(x + A) * eta(x^35 + A))^2, n))}

%o (Sage)

%o def a(n):

%o return EllipticCurve("35a3").an(n) # _Robin Visser_, Sep 30 2023

%Y Cf. A106852(n) = a(3^n).

%K sign,mult

%O 1,4

%A _Michael Somos_, Jan 29 2006