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%I #44 Sep 08 2022 08:44:50
%S 1,-1,-2,0,1,4,-2,2,2,-4,0,-8,-1,-1,6,8,-4,0,6,2,-6,4,-2,0,-7,-2,-2,
%T -8,4,4,-2,0,4,-4,8,8,10,1,0,-8,1,-4,-4,-6,-6,0,-8,8,2,4,-18,16,0,-12,
%U -2,-6,18,16,-2,0,5,6,12,-8,-4,-4,0,2,-6,-12,0,-8,-12,7,14,-16,2,-16,-2,2,0,12,8,24,-9,-4,6,0,-4,12,6,2,-12,8,0,0
%N Expansion of q^(-1/2) * eta(q) * eta(q^2) * eta(q^3) * eta(q^6) in powers of q.
%C Number 43 of the 74 eta-quotients listed in Table I of Martin (1996).
%C Newform number 1 of degree 1 in Full modular forms space of level 24, weight 2 and trivial character.
%D J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 415. Exer. 47.3.
%H Seiichi Manyama, <a href="/A030188/b030188.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F Euler transform of period 6 sequence [ -1, -2, -2, -2, -1, -4, ...]. - _Michael Somos_, Apr 02 2005
%F Given g.f. A(x), then B(x) = x * A(x)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = u^2 * v * w + 4 * u * v^2 * w + 16 * u * v * w^2 + 4 * u^2 * w^2 - v^4. - _Michael Somos_, Apr 02 2005
%F a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) otherwise where b(p) = p+1 - number of solutions to y^2 = x^3 - x^2 - 4*x + 4 modulo p including the point at infinity. - _Michael Somos_, Mar 04 2011
%F G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24 (t/i)^2 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jun 08 2007
%F Coefficients of L-series for elliptic curve "24a1": y^2 = x^3 - x^2 - 4*x + 4. - _Michael Somos_, Apr 02 2005
%F a(n) = (-1)^n * A159819(n). a(3*n + 1) = -a(n). Convolution square is A030209. - _Michael Somos_, Mar 13 2012
%F a(3*n + 2) = -2 * A258090(n). - _Michael Somos_, May 19 2015
%e G.f. = 1 - x - 2*x^2 + x^4 + 4*x^5 - 2*x^6 + 2*x^7 + 2*x^8 - 4*x^9 - 8*x^11 + ...
%e G.f. = q - q^3 - 2*q^5 + q^9 + 4*q^11 - 2*q^13 + 2*q^15 + 2*q^17 - 4*q^19 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3] QPochhammer[ x^6], {x, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A), n))}; /* _Michael Somos_, Apr 02 2005 */
%o (PARI) {a(n) = if( n<0, 0, n = 2*n + 1; ellak( ellinit([ 0, -1, 0, -4, 4], 1), n))}; /* _Michael Somos_, Apr 02 2005 */
%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==3, (-1)^e, a0 = 1; a1 = y = -sum( x=0, p-1, kronecker( x^3 - x^2 - 4*x + 4, p)); for( i=2, e, x = y*a1 - p*a0; a0 = a1; a1 = x); a1)))}; /* _Michael Somos_, Aug 13 2006 */
%o (Sage) CuspForms( Gamma0(24), 2, prec=192).0 # _Michael Somos_, May 24 2013
%o (Magma) Basis( CuspForms( Gamma0(24), 2), 192) [1]; /* _Michael Somos_, May 27 2014 */
%o (Magma) qEigenform( EllipticCurve( [0, -1, 0, 1, 0]), 191); /* _Michael Somos_, Jun 12 2014 */
%Y Cf. A030209, A159819. A258090.
%K sign
%O 0,3
%A _N. J. A. Sloane_.
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