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Expansion of eta(q) * eta(q^3) * eta(q^5) * eta(q^15) in powers of q.
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%I #46 Sep 08 2022 08:44:50

%S 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,

%T -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,

%U 0,-4,2,-4,1,-2,0,0,7,-2,-4,12,-2,0,0,-8,3,10,10,-1

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

%C Unique cusp form of weight 2 for congruence group Gamma_1(15). - _Michael Somos_, Aug 11 2011

%C 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

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

%H Seiichi Manyama, <a href="/A030184/b030184.txt">Table of n, a(n) for n = 1..10000</a>

%H N. D. Elkies, <a href="http://www.math.harvard.edu/~elkies/modular.pdf">Elliptic and modular curves over finite fields and related computational issues</a>, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 70.

%H LMFDB, <a href="http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/15/2/0/">Newforms of weight 2 on Gamma_0(15)</a>.

%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 M. D. Rogers, <a href="http://arxiv.org/abs/0806.3590">Hypergeometric formulas for lattice sums and Mahler measures</a>.

%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 15 sequence [ -1, -1, -2, -1, -2, -2, -1, -1, -2, -2, -1, -2, -1, -1, -4, ...]. - _Michael Somos_, May 02 2005

%F 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

%F 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

%F 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

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

%e 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 - ...

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

%o (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ 1, 1, 1, 0, 0], 1), n))}; /* _Michael Somos_, Aug 13 2006 */

%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( 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 */

%o (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 */

%o (Sage) CuspForms( Gamma1(15), 2, prec = 76).0; # _Michael Somos_, Aug 11 2011

%o (Magma) Basis( CuspForms( Gamma1(15), 2), 76)[1]; /* _Michael Somos_, Nov 20 2014 */

%K sign,mult

%O 1,8

%A _N. J. A. Sloane_.