A030188 Expansion of q^(-1/2) * eta(q) * eta(q^2) * eta(q^3) * eta(q^6) in powers of q.

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, -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, -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

Offset

0,3

Comments

Number 43 of the 74 eta-quotients listed in Table I of Martin (1996).

Newform number 1 of degree 1 in Full modular forms space of level 24, weight 2 and trivial character.

References

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 415. Exer. 47.3.

Links

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

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

Euler transform of period 6 sequence [ -1, -2, -2, -2, -1, -4, ...]. - Michael Somos, Apr 02 2005

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

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

G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)).

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

Coefficients of L-series for elliptic curve "24a1": y^2 = x^3 - x^2 - 4*x + 4. - Michael Somos, Apr 02 2005

a(n) = (-1)^n * A159819(n). a(3*n + 1) = -a(n). Convolution square is A030209. - Michael Somos, Mar 13 2012

a(3*n + 2) = -2 * A258090(n). - Michael Somos, May 19 2015

Example

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 + ...
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 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, May 17 2015 *)

Prog

(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 */
(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 */
(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 */
(Sage) CuspForms( Gamma0(24), 2, prec=192).0 # Michael Somos, May 24 2013
(Magma) Basis( CuspForms( Gamma0(24), 2), 192) [1]; /* Michael Somos, May 27 2014 */
(Magma) qEigenform( EllipticCurve( [0, -1, 0, 1, 0]), 191); /* Michael Somos, Jun 12 2014 */

Crossrefs

Cf. A030209, A159819. A258090.

Sequence in context: A322084 A158239 A159819 * A176703 A160648 A124912

Adjacent sequences: A030185 A030186 A030187 * A030189 A030190 A030191

Keyword

sign

Author

N. J. A. Sloane.

Status

approved