This is part of an email that was circulated amongst some members of the Central Values group (Elkies, Hart, Tornaría, Watkins):
Even parity, "Eisenstein" cases (27A/32A/36A)
\bullet compute the 12 series for the weight 3/2 lift of 36A each to 10^{10} places: Statistics
\bullet Think about 10^{11} --- for 36A, the stride is 24 versus 8 for 32A, so it's only about 3 times as large as 10^{10} for 32A. Think about thinking about 10^{12}...
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\bullet Decide what is "useful" to do/output in other cases
\bullet Look at prospects of completing rank 4 experiment for 32A and/or 36A
Even parity, other CM cases
\bullet Figure out whether higher symmetric powers for 49A (and others) have a nice representation as a twisted ternary series --- We found such for the 3rd/5th/7th/9th powers for 49A, and can reduce it in general to computing enough special values and applying tedious linear algebra, but we lack a general theory to predict the spherical polynomials ahead of time.
\bullet Figure out how to isolate the lift in the "CM subspace", as this will likely be necessary in the extreme cases (like N=163^2)
\bullet Generate data up to 10^8 for these other CM cases
I guess I'm still not sure what auxiliary lattices and weight functions need to be used here, and whether it might just be easier to do the linear algebra directly in the space of ternary series, rather than first in the modular forms space (or CM subspace).
Initial segments for various 49A/121B powers.
Odd parity
\bullet Work on the "mod p Heegner point" method
\bullet Speculate on why/whether higher weight modular forms never have analytic rank more than 2?!
Talk slides
Here are slides.zip slides from a talk that was given (currently as a zip file of JPEGs). The last two are not in order; the ordering should be 01234567A89.