Noam Elkies writes: ---------- Forwarded message --------- From: Elkies, Noam <elkies@math.harvard.edu> Date: Fri, Jan 29, 2021 at 1:48 PM Subject: Re: [math-fun] Coeffts of modular forms of exceptional groups To: James Propp <jamespropp@gmail.com> CC: Henry Cohn <Henry.Cohn@microsoft.com> Hi Jim, You write:

Anything to contribute to the discussion?

---------- Forwarded message --------- From: Neil Sloane <njasloane@gmail.com> Date: Fri, Jan 29, 2021 at 1:16 PM Subject: [math-fun] Coeffts of modular forms of exceptional groups To: fun <math-fun@mailman.xmission.com>

In the Feb 2021 Notices of AMS, there is an article by Aaron Pollack about modular forms associated with exceptional groups. The coeffts of the modular form associated with E_8 are famously connected with the densest sphere packing in 8-D (and the Theory Of Everything), so naturally I was interested in this. The first reference in the article is to

Walter L. Baily, Jr., An Exceptional Arithmetic Group and its Eisenstein Series, Annals of Mathematics , May, 1970, Vol. 91, No. 3 (May, 1970), pp. 512-549 : https://www.jstor.org/stable/1970636

and Baily's last theorem there shows that the Fourier coefficients of the various Eisenstein series associated with this group (which is connected with E_7) are rational numbers. But no numbers are given (this is the Annals, after all). And that is just the first of 20 references. So what are the values of these Fourier coefficients? [ . . . ]

That paper seems to concern not the E_7 *lattice* (whose theta series coefficients, though not as nice as those for E_8, are well known, and appear in Conway-Sloane = SPLAG), but the exceptional Lie *group* E_7 and its associated "exceptional tube domain". I actually wrote a joint paper with B.H.Gross on this: Noam D. Elkies, Benedict H. Gross: The exceptional cone and the Leech lattice,, International Math. Research Notices 1996 #14, 665-698. But this was almost half a lifetime ago, and anyhow the "tube domain" part was all Gross's, so you should probably ask him directly if you need more information in that direction. Our paper did give some actual numbers. NDE