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%I #37 Sep 08 2022 08:44:50
%S 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,
%T 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,
%U 6,-6,0,8,4,1,1,0,0,-4,6,0,0,0,-1,2,-2,10,2,0
%N Expansion of eta(q) * eta(q^2) * eta(q^7) * eta(q^14) in powers of q.
%C Number 29 of the 74 eta-quotients listed in Table I of Martin (1996).
%C Associated with permutations in Mathieu group M24 of shape (14)(7)(2)(1).
%C 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
%H Seiichi Manyama, <a href="/A030187/b030187.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787871">Mathieu group M24 and modular forms</a>, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
%H LMFDB, <a href="http://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/14/2/0/">Newforms of weight 2 on Gamma_0(14)</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 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 14 sequence [ -1, -2, -1, -2, -1, -2, -2, -2, -1, -2, -1, -2, -1, -4, ...]. - _Michael Somos_, Aug 13 2006
%F 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
%F 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
%F 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
%F G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(7*k)) * (1 - x^(14*k)).
%e 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 + ...
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^7] QPochhammer[ q^14], {q, 0, n}]; (* _Michael Somos_, Aug 11 2011 *)
%o (PARI) {a(n) = if( n<1, 0, ellak( ellinit([ -1, 0, -1, -1, 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==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 */
%o (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))};
%o (Sage) CuspForms( Gamma1(14), 2, prec = 78).0; # _Michael Somos_, Aug 11 2011
%o (Magma) Basis( CuspForms( Gamma1(14), 2), 78)[1]; /* _Michael Somos_, Nov 20 2014 */
%K sign,mult
%O 1,3
%A _N. J. A. Sloane_.
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