%I #25 Sep 08 2022 08:45:56
%S 1,0,-2,-2,3,0,-1,0,1,0,3,4,-4,0,-6,4,-3,0,1,-6,2,0,0,0,4,0,4,2,6,0,
%T -4,0,-6,0,-3,-2,2,0,8,0,-6,0,-1,-6,3,0,-3,-8,-6,0,6,8,12,0,9,0,-2,0,
%U -6,12,-1,0,-1,-8,-12,0,-4,6,0,0,6,0,-7,0,-8,-2,-3,0,8,12,-11,0,12,-4,-9,0,-12,0,12,0,4,0,8,0,3,0,8,0,3,-8,6
%N Coefficients of L-series for elliptic curve "19a3": y^2 + y = x^3 + x^2 + x.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A187096/b187096.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q * (psi(q^4) * phi(q^38) - q^2 * psi(q) * psi(q^19) + q^9 * phi(q^2) * psi(q^76))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
%F a(n) is multiplicative with a(19^e) = 1, 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 / (19 t)) = 19 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F Convolution square of A187097.
%e G.f. = q - 2*q^3 - 2*q^4 + 3*q^5 - q^7 + q^9 + 3*q^11 + 4*q^12 - 4*q^13 + ...
%e If p = 2, then the solutions to y^2 + y = x^3 + x^2 + x modulo 2 are (0,0), (0,1) and the point at infinity. Thus a(2) = 2+1-3 = 0.
%t a[ n_] := SeriesCoefficient[ (-EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^19] + 2 (EllipticTheta[ 3, 0, q^76] EllipticTheta[ 2, 0, q^4] + EllipticTheta[ 3, 0, q^4] EllipticTheta[ 2, 0, q^76]))^2 / 16, {q, 0, 2 n}]; (* _Michael Somos_, Oct 17 2016 *)
%o (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 0, 1, 1, 1, 0], 1), n))};
%o (PARI) {a(n) = my(phi1, psi1); if( n<1, 0, n--; phi1 = 1 + 2 * sum( k=1, sqrtint( n), x^k^2, x * O(x^n)); psi1 = sum( k=1, ( sqrtint( 8*n + 1) + 1 ) \ 2, x^((k^2 - k)/2), x * O(x^n)); polcoeff( sqr( subst( psi1 + x * O(x^(n \ 4)), x, x^4) * subst( phi1 + x * O(x^(n \ 38)), x, x^38) - x^2 * psi1 * subst( psi1 + x * O(x^(n \ 19)), x, x^19) + x^9 * subst( phi1 + x * O(x^(n \ 2)), x, x^2) * subst( psi1 + x * O(x^(n \ 76)), x, x^76)), n))};
%o (Sage) CuspForms( Gamma0(19), 2, prec=100).0; # _Michael Somos_, May 28 2013
%o (Magma) Basis( CuspForms( Gamma0(19), 2), 100) [1]; /* _Michael Somos_, May 27 2014 */
%Y Cf. A187097.
%K sign,mult
%O 1,3
%A _Michael Somos_, Mar 04 2011