A030184 Expansion of eta(q) * eta(q^3) * eta(q^5) * eta(q^15) in powers of q.

1, -1, -1, -1, 1, 1, 0, 3, 1, -1, -4, 1, -2, 0, -1, -1, 2, -1, 4, -1, 0, 4, 0, -3, 1, 2, -1, 0, -2, 1, 0, -5, 4, -2, 0, -1, -10, -4, 2, 3, 10, 0, 4, 4, 1, 0, 8, 1, -7, -1, -2, 2, -10, 1, -4, 0, -4, 2, -4, 1, -2, 0, 0, 7, -2, -4, 12, -2, 0, 0, -8, 3, 10, 10, -1

Offset

1,8

Comments

Unique cusp form of weight 2 for congruence group Gamma_1(15). - Michael Somos, Aug 11 2011

Coefficients of L-series for elliptic curve "15a8": y^2 + x*y + y = x^3 + x^2 or y^2 + x*y - y = x^3 + x^2 + x. - Michael Somos, Feb 01 2004

Number 32 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

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

N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 70.

LMFDB, Newforms of weight 2 on Gamma_0(15).

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

M. D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures.

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

Formula

Euler transform of period 15 sequence [ -1, -1, -2, -1, -2, -2, -1, -1, -2, -2, -1, -2, -1, -1, -4, ...]. - Michael Somos, May 02 2005

a(n) is multiplicative with a(5^e) = 1, a(3^e) = (-1)^e, 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^3 - u*w* (u + 2*v + 4*w). - Michael Somos, May 02 2005

G.f. A(x) satisfies 2 * A(x^2) = -(A(x) + A(-x) + 4*A(x^4)), A(x^2)^3 = -A(x) * A(-x) * A(x^4). - Michael Somos, Feb 19 2007

G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 - x^(5*k)) * (1 - x^(15*k)).

Example

G.f. = q - q^2 - q^3 - q^4 + q^5 + q^6 + 3*q^8 + q^9 - q^10 - 4*q^11 + q^12 - 2*q^13 - ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^3] QPochhammer[ q^5] QPochhammer[ q^15], {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

Prog

(PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, 1, 1, 0, 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==3, (-1)^e, p==5, 1, a0=1; a1 = y = -if( p==2, 1, sum(x=0, p-1, kronecker( 4*x^3 + 5*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^3 + A) * eta(x^5 + A) * eta(x^15 + A), n))}; /* Michael Somos, May 02 2005 */
(Sage) CuspForms( Gamma1(15), 2, prec = 76).0; # Michael Somos, Aug 11 2011
(Magma) Basis( CuspForms( Gamma1(15), 2), 76)[1]; /* Michael Somos, Nov 20 2014 */

Crossrefs

Sequence in context: A024564 A084795 A364097 * A284373 A104610 A138684

Adjacent sequences: A030181 A030182 A030183 * A030185 A030186 A030187

Keyword

sign,mult

Author

N. J. A. Sloane.

Status

approved