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.)