A002656 Expansion of (eta(q) * eta(q^7))^3 in powers of q.

1, -3, 0, 5, 0, 0, -7, -3, 9, 0, -6, 0, 0, 21, 0, -11, 0, -27, 0, 0, 0, 18, 18, 0, 25, 0, 0, -35, -54, 0, 0, 45, 0, 0, 0, 45, -38, 0, 0, 0, 0, 0, 58, -30, 0, -54, 0, 0, 49, -75, 0, 0, -6, 0, 0, 21, 0, 162, 0, 0, 0, 0, -63, -91, 0, 0, -118, 0, 0, 0, 114, -27

Offset

1,2

Comments

Number 15 of the 74 eta-quotients listed in Table I of Martin (1996).

Unique cusp form of weight 3 for congruence group Gamma_1(7). - Michael Somos, Aug 11 2011

References

B. Berndt, Commentary on Ramanujan's Papers, pp. 357-426 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 372 (4).

N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103)

N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 145, problem 13.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

N. Elkies, The Klein quartic in number theory.

F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), no. 1, 1-17. see pp 9-10.

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 7 sequence [ -3, -3, -3, -3, -3, -3, -6, ...]. - Michael Somos, Mar 11 2004

a(n) is multiplicative with a(7^e) = (-7)^e, a(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7), a(p^e) = a(p) * a(p^(e-1)) - p^2 * a(p^(e-2)) and a(2) = -3, a(p) = 2 * (x^2 - 7*y^2) where p = x^2 + 7*y^2 if p == 1, 2, 4 (mod 7). - Michael Somos, Apr 12 2008

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u * w * (u + 6*v + 8*w) - v^3. - Michael Somos, May 02 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 12 2008

G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(7*k)))^3. - Michael Somos, Aug 11 2011

G.f.: (1/2) * Sum_{u,v in Z} (u*u - 2*v*v) * x^(u*u + u*v + 2*v*v). - Michael Somos, Jun 14 2007

a(7*n + 3) = a(7*n + 5) = a(7*n + 6) = 0. - Michael Somos, Oct 19 2005

Example

G.f. = q - 3*q^2 + 5*q^4 - 7*q^7 - 3*q^8 + 9*q^9 - 6*q^11 + 21*q^14 - 11*q^16 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^7])^3, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^7 + A))^3, n))}; /* Michael Somos, Apr 16 2005 */
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( kronecker(-7, p)<1, if( p==7, (-1)^e, 1-e%2) * p^e, for(i=1, sqrtint(p\7), if( issquare(p - 7*i^2), y=i; break)); a0 = 1; a1 = y = if( p==2, -3, 2*p - 28*y^2); for(i=2, e, x = y * a1 - p^2 * a0; a0 = a1; a1 = x); a1)))}; /* Michael Somos, Oct 19 2005 */
(Sage) CuspForms( Gamma1(7), 3, prec = 72).0; # Michael Somos, Aug 11 2011
(Magma) Basis( CuspForms( Gamma1(7), 3), 72) [1]; /* Michael Somos, Dec 09 2013 */

Crossrefs

Sequence in context: A136599 A227498 A131986 * A234434 A234020 A348259

Adjacent sequences: A002653 A002654 A002655 * A002657 A002658 A002659

Keyword

sign,mult

Author

N. J. A. Sloane

Status

approved