[Fwd: coefficients of Moonshine (McKay-Thompson) series]

[Jon Awbrey]

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[fwd from number theory list -- ja]

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 at ens.fr,
     http://www.dma.ens.fr/~madore/ )

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Here is my reply to the Number Theory List:

David,  it is certainly true that the OEIS entries
are inconsistent - but they have accumulated
over many years, with contributions from several
different people.  I once tried to get John McKay's
tables in electronic form, but somehow he could
never find them.  Also, of course, the constant
term is not well defined, and since entries
in the OEIS are sorted lexicographically, it seemed
worth having two or even three versions in some cases.

I will update the entries from your tables - thanks
for doing this!

Neil Slaone

> 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.

PS to the seqfan list:  If anyone would like to help
with this project, please let me know!