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A030206 Expansion of q^(-1/3) * eta(q)^2 * eta(q^3)^2 in powers of q.

%I A030206 #50 Jul 08 2025 19:30:47
%S A030206 1,-2,-1,0,5,4,-7,0,-5,2,-4,0,11,0,8,0,-6,-10,0,0,-1,-8,5,0,-7,14,17,
%T A030206 0,0,0,-5,0,-19,10,-13,0,2,-4,0,0,-11,8,20,0,7,0,23,0,0,-22,-19,0,14,
%U A030206 0,-25,0,12,-16,5,0,-7,0,0,0,23,12,11,0,0,20,-13,0,4,0,-28,0,-22,0,0,0,17,2,-35,0,0,16,-11,0,0,-10
%N A030206 Expansion of q^(-1/3) * eta(q)^2 * eta(q^3)^2 in powers of q.
%C A030206 Number 44 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A030206 Denoted by g_2(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique weight 2 newform of level 27.
%C A030206 This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - _Michael Somos_, Aug 24 2012
%C A030206 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A030206 Seiichi Manyama, <a href="/A030206/b030206.txt">Table of n, a(n) for n = 0..10000</a>
%H A030206 Amanda Clemm, <a href="http://www.mdpi.com/2227-7390/4/1/5">Modular Forms and Weierstrass Mock Modular Forms</a>, Mathematics, volume 4, issue 1, (2016)
%H A030206 S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.
%H A030206 S. Cynk and K. Hulek, <a href="http://arXiv.org/abs/math/0509424">Construction and examples of higher-dimensional modular Calabi-Yau manifolds</a>, arXiv:math/0509424 [math.AG], 2005-2006.
%H A030206 M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H A030206 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 A030206 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F A030206 Expansion of q^(-1/3) * b(q) * c(q) / 3 in powers of q where b(), c() are cubic AGM theta functions. - _Michael Somos_, Nov 01 2006
%F A030206 Coefficients of L-series for elliptic curve "27a3": y^2 + y = x^3. - _Michael Somos_, Aug 13 2006
%F A030206 Euler transform of period 3 sequence [-2, -2, -4, ...]. - _Michael Somos_, Dec 06 2004
%F A030206 G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F A030206 G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2.
%F A030206 a(n) = b(3*n + 1) where b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-1)^(e/2) * p^(e/2), if p == 2 (mod 3), otherwise b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)). - _Michael Somos_, Aug 13 2006
%F A030206 Given g.f. A(x), then B(q)= q*A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 - u*w * (u + 4*w). - _Michael Somos_, Dec 06 2004
%F A030206 a(4*n + 3) = a(16*n + 13) = 0. - _Michael Somos_, Oct 19 2005
%F A030206 a(4*n + 1) = -2 * a(n). - _Michael Somos_, Dec 06 2004
%F A030206 a(25*n + 8) = -5 * a(n). Convolution square of A030203. - _Michael Somos_, Mar 13 2012
%e A030206 G.f. = 1 - 2*x - x^2 + 5*x^4 + 4*x^5 - 7*x^6 - 5*x^8 + 2*x^9 - 4*x^10 + 11*x^12 + ...
%e A030206 G.f. = q - 2*q^4 - q^7 + 5*q^13 + 4*q^16 - 7*q^19 - 5*q^25 + 2*q^28 - 4*q^31 + ...
%t A030206 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^2, {x, 0, n}]; (* _Michael Somos_, Jun 12 2014 *)
%o A030206 (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3, 0, p%3==2, if( e%2, 0, (-1)^(e/2) * p^(e/2)), for( i=1, sqrtint(4*p\27), if( issquare(4*p - 27*i^2, &y), break)); a0=1; a1 = y*= (-1)^(y%3); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* _Michael Somos_, Aug 13 2006 */
%o A030206 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A)^2, n))}; /* _Michael Somos_, Feb 19 2007 */
%o A030206 (PARI) {a(n) = ellak( ellinit( [0, 0, 1, 0, 0], 1), 3*n + 1)}; /* _Michael Somos_, Jun 12 2014 */
%o A030206 (Sage) ModularForms( Gamma0(27), 2, prec=271).0; # _Michael Somos_, Jun 12 2014
%o A030206 (Magma) A := Basis( ModularForms( Gamma0(27), 2), 271); A[2] - 2*A[5]; /* _Michael Somos_, Jun 12 2014 */
%o A030206 (Magma) qEigenform( EllipticCurve( [0, 0, 1, 0, 0]), 271); /* _Michael Somos_, Jun 12 2014 */
%o A030206 (Magma) Basis( CuspForms( Gamma0(27), 2), 271)[1]; /* _Michael Somos_, Mar 24 2015 */
%Y A030206 Cf. A030203, A279005.
%K A030206 sign
%O A030206 0,2
%A A030206 _N. J. A. Sloane_