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

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, 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, 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

Offset

0,2

Comments

Number 44 of the 74 eta-quotients listed in Table I of Martin (1996).

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.

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)

S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.

M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Formula

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

Coefficients of L-series for elliptic curve "27a3": y^2 + y = x^3. - Michael Somos, Aug 13 2006

Euler transform of period 3 sequence [-2, -2, -4, ...]. - Michael Somos, Dec 06 2004

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).

G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2.

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

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

a(4*n + 3) = a(16*n + 13) = 0. - Michael Somos, Oct 19 2005

a(4*n + 1) = -2 * a(n). - Michael Somos, Dec 06 2004

a(25*n + 8) = -5 * a(n). Convolution square of A030203. - Michael Somos, Mar 13 2012

Example

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 + ...
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 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^2, {x, 0, n}]; (* Michael Somos, Jun 12 2014 *)

Prog

(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 */
(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 */
(PARI) {a(n) = ellak( ellinit( [0, 0, 1, 0, 0], 1), 3*n + 1)}; /* Michael Somos, Jun 12 2014 */
(Sage) ModularForms( Gamma0(27), 2, prec=271).0; # Michael Somos, Jun 12 2014
(Magma) A := Basis( ModularForms( Gamma0(27), 2), 271); A[2] - 2*A[5]; /* Michael Somos, Jun 12 2014 */
(Magma) qEigenform( EllipticCurve( [0, 0, 1, 0, 0]), 271); /* Michael Somos, Jun 12 2014 */
(Magma) Basis( CuspForms( Gamma0(27), 2), 271)[1]; /* Michael Somos, Mar 24 2015 */

Crossrefs

Cf. A030203, A279005.

Sequence in context: A171960 A330396 A182376 * A212768 A133336 A269951

Adjacent sequences: A030203 A030204 A030205 * A030207 A030208 A030209

Keyword

sign

Author

N. J. A. Sloane.

Status

approved