Let $p$ be a prime, and $\alpha$ a positive integer. How do you compute the space of $p$-ordinary $p$-adic modular forms (in the sense of Serre) of weight 2 on $\Gamma_0(p^\alpha)$? I'm really only concerned with the cases where $\Gamma_0(p^\alpha)$ is genus zero and $\alpha = 1, 2$.
[Edit: Changed from space of $p$-adic modular forms (which is really big) to the subspace of $p$-ordinary forms (which I think should be finite dimensional)]
It depends rather what you mean by "compute". These are very big spaces (infinite-dimensional p-adic Banach spaces, and without a good theory of Hecke eigenforms). Can you be a bit more specific what it is you want to compute about them?
EDIT: You've edited your question to focus attention on the ordinary cusp forms instead. But in weight 2 these are just the p-ordinary classical forms of level $\Gamma_0(p^{\beta})$, where $\beta = \max(1, \alpha)$, and there are extremely efficient algorithms for computing classical weight 2 modular forms (e.g. using modular symbols).
(I can't help thinking that you started with a question about a space too big to be interesting, and edited it to a question about a subspace too small to be interesting; there is a very interesting question in between the two which you have whizzed past, which is to compute overconvergent weight 2 eigenforms of small but non-zero slope.)
