A002656 - OEIS

%I #43 Oct 18 2023 10:04:13

%S 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,

%T -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,

%U 0,0,21,0,162,0,0,0,0,-63,-91,0,0,-118,0,0,0,114,-27

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

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

%C Unique cusp form of weight 3 for congruence group Gamma_1(7). - _Michael Somos_, Aug 11 2011

%D 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).

%D 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)

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

%H Seiichi Manyama, <a href="/A002656/b002656.txt">Table of n, a(n) for n = 1..10000</a>

%H N. Elkies, <a href="http://www.msri.org/publications/books/Book35/files/elkies.pdf">The Klein quartic in number theory</a>.

%H F. Garvan, D. Kim and D. Stanton, <a href="http://dx.doi.org/10.1007/BF01231493">Cranks and t-cores</a>, Invent. Math. 101 (1990), no. 1, 1-17. see pp 9-10.

%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 7 sequence [ -3, -3, -3, -3, -3, -3, -6, ...]. - _Michael Somos_, Mar 11 2004

%F 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

%F 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

%F 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

%F G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(7*k)))^3. - _Michael Somos_, Aug 11 2011

%F 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

%F a(7*n + 3) = a(7*n + 5) = a(7*n + 6) = 0. - _Michael Somos_, Oct 19 2005

%e 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 + ...

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

%o (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 */

%o (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 */

%o (Sage) CuspForms( Gamma1(7), 3, prec = 72).0; # _Michael Somos_, Aug 11 2011

%o (Magma) Basis( CuspForms( Gamma1(7), 3), 72) [1]; /* _Michael Somos_, Dec 09 2013 */

%K sign,mult

%O 1,2

%A _N. J. A. Sloane_