@ccorn: Do you know a good reference for elliptic integrals/functions? I always wanted to have a better handle on the stuff you referenced in this answer
@Semiclassical Classics first: Whittaker & Watson: A course in modern analysis. You'll never regret it. It covers elliptic functions and theta functions. Then, apart from the more systematic textbooks, I have found Borwein & Borwein's "Pi and the AGM" extremely worthwhile. And Weber's Lehrbuch der Algebra, vol. III, if you can read german and know in advance what he wants to tell you :-)
"In the theory of Theta functions it is easy to find an arbitrarily large number of relations, but the difficulty begins where the objective is to find a way out of the maze of formulae. Occupation with those masses of formulae seems to have a withering effect on the mathematical intuition." -- G. Frobenius (translation attempt by me)
@Semiclassical weakly related, I'd like to read about a method that can symbolically invert and simplify power series (or infinite lower triangular toeplitz matrices), giving general formulae for the terms instead of just working out initial ones.
In a sense, the elliptic realm is circular, so you can always arrive at an inverse by going forward. From a period ratio $\tau$ define the nome $q$, from that compute Theta functions, from that elliptic integrals $K$ and $K'$, and their ratio gives you back $\tau$.
@Chris'ssis I haven't slept $12$ hours in my concious life(Since I was a baby etc). I haven't slept $10$ hours since I was very sick many years ago and I sleep more than $7$ hours very rarely.
The Frobenius quote above heads the preface of a book by Günther Köhler on "Eta products and Theta series identities". Apart from working out 600 pages of formulae for levels up to 100, it has a very well written introductory/background part. Like Borwein's "Pi and the AGM", it manages to tell you almost en passant how to do some seemingly awkward bits of modforms calculations easily.
Note however that such a catalogue will only collect dust in your bookshelf unless you get some symbolic calculater (e. g. Pari/GP) and try out the presented series representations. Then you are in for a string of "Wow! It really works out!" moments.
