A030187 Expansion of eta(q) * eta(q^2) * eta(q^7) * eta(q^14) in powers of q.

1, -1, -2, 1, 0, 2, 1, -1, 1, 0, 0, -2, -4, -1, 0, 1, 6, -1, 2, 0, -2, 0, 0, 2, -5, 4, 4, 1, -6, 0, -4, -1, 0, -6, 0, 1, 2, -2, 8, 0, 6, 2, 8, 0, 0, 0, -12, -2, 1, 5, -12, -4, 6, -4, 0, -1, -4, 6, -6, 0, 8, 4, 1, 1, 0, 0, -4, 6, 0, 0, 0, -1, 2, -2, 10, 2, 0

Offset

1,3

Comments

Number 29 of the 74 eta-quotients listed in Table I of Martin (1996).

Associated with permutations in Mathieu group M24 of shape (14)(7)(2)(1).

Coefficients of L-series for elliptic curve "14a4": y^2 + x*y + y = x^3 - x or y^2 + x*y - y = x^3. - Michael Somos, Feb 19 2007

Links

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

M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)

LMFDB, Newforms of weight 2 on Gamma_0(14).

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 14 sequence [ -1, -2, -1, -2, -1, -2, -2, -2, -1, -2, -1, -2, -1, -4, ...]. - Michael Somos, Aug 13 2006

a(n) is multiplicative with a(2^e) = (-1)^e, a(7^e) = 1, otherwise a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p minus number of points of elliptic curve modulo p . - Michael Somos, Aug 13 2006

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w ) = v^4 - u*w * (u + 2*v) * (v + 2*w). - Michael Somos, Feb 19 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (14 t)) = 14 (t / i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 11 2011

G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(7*k)) * (1 - x^(14*k)).

Example

G.f. = q - q^2 - 2*q^3 + q^4 + 2*q^6 + q^7 - q^8 + q^9 - 2*q^12 - 4*q^13 - q^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q  QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^7] QPochhammer[ q^14], {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

Prog

(PARI) {a(n) = if( n<1, 0, ellak( ellinit([ -1, 0, -1, -1, 0], 1), n))}; /* Michael Somos, Aug 13 2006 */
(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( p==2, (-1)^e, p==7, 1, a0=1; a1 = y = -sum( x=0, p-1, kronecker( 4*x^3 + x^2 - 2*x + 1, p)); 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<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^7 + A) * eta(x^14 + A), n))};
(Sage) CuspForms( Gamma1(14), 2, prec = 78).0; # Michael Somos, Aug 11 2011
(Magma) Basis( CuspForms( Gamma1(14), 2), 78)[1]; /* Michael Somos, Nov 20 2014 */

Crossrefs

Sequence in context: A190676 A329308 A135486 * A270657 A270658 A117278

Adjacent sequences: A030184 A030185 A030186 * A030188 A030189 A030190

Keyword

sign,mult

Author

N. J. A. Sloane.

Status

approved