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Fourier coefficients of T_8.
3

%I #14 Oct 04 2020 07:38:06

%S 1,-480,-28404,-682240,-10460070,-120178944,-1122367480,-8942109696,

%T -62733463065,-396222777600,-2289950627940,-12261279536640,

%U -61415457336714,-290017200522240,-1299352388589720,-5552275006294016,-22728781503345645,-89469772048615680

%N Fourier coefficients of T_8.

%C T_8 is the unique weight = -6 normalized meromorphic modular form for SL(2,Z) with all poles at infinity.

%D C. L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1980, pp. 249-268.

%H Seiichi Manyama, <a href="/A035314/b035314.txt">Table of n, a(n) for n = -1..1000</a>

%F G.f.: G_6/Delta (in Siegel's notation).

%F a(n) ~ -exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(15/4)). - _Vaclav Kotesovec_, Oct 04 2020

%e T_8 = 1/q - 480 - 28404 q - ....

%o (PARI) {a(n)=local(A); if(n<-1, 0, n++; A=x*O(x^n); polcoeff( sum(k=1, n, -504*sigma(k,5)*x^k, 1+A)/eta(x+A)^24, n))} /* _Michael Somos_, Oct 30 2006 */

%K easy,sign

%O -1,2

%A Barry Brent (barryb(AT)primenet.com)