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Topic: coefficients of Moonshine (McKay-Thompson) series
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David Madore

Posts: 120
Registered: 12/13/04
coefficients of Moonshine (McKay-Thompson) series
Posted: Aug 1, 2007 1:30 PM
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Hi,

Just in case this is of any value to anyone, I have computed the 3200
first coefficients of each of the Monstrous Moonshine functions (aka
McKay-Thompson series).

The first dozen-or-so coefficients can seemingly be found on Sloane's
encyclopedia of integer sequences (albeit with incoherent conventions
as to the constant coefficient and skipping of zero values), e.g.,
A007240/A014708, A007241/A101558, A007191/A007246, A007243/A030197,
A007244/A030182, A007245 (omitting zeroes), etc. Fifty coefficients
of each function were tabulated in J. McKay & H. Strauss, *The
q-series of monstrous moonshine & the decomposition of the head
characters* (Comm. Algebra 18 (1990), 253-278). But Google seems to
indicate that no further data are easily found. So I thought it might
be nice to have a clean, computer-usable, source for higher
coefficients.

I am making them available here: <URL:
ftp://quatramaran.ens.fr/pub/madore/moonshine/moonshine.dat.gz
>. This is a flat text file, gzip-compressed down from 21MB to 8MB;
each one of the 550400=3200*172 lines in the file gives one
coefficient as a tab-separated list, the first column being the class
(labeled as in the ATLAS), the second being the index and the third
being the coefficient's value.

Furthermore, the (Python) program used to compute the coefficients can
be found at <URL:
ftp://quatramaran.ens.fr/pub/madore/moonshine/moonshine.py
>. It is very inefficient at the task but, even then, computing
several thousand coefficients takes a matter of minutes on a modern
computer.

Enjoy.

--
David A. Madore
(david.madore@ens.fr,
http://www.dma.ens.fr/~madore/ )




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