  The main result of this paper is an instance of the conjecture made by Gouvea
and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r
the space of r-overconvergent p-adic modular forms of tame level N and weight k
should be spanned by the finite slope Hecke eigenforms. Using methods adapted
from the work of Buzzard and Calegari I show that for p=2, k = 0, N = 1, this
holds for all r in (5/12, 7/12). The proof relies on a certain factorisation of
the U_p operator which is known in this case but I conjecture also holds for p
= 3 and 5; this conjecture also implies exact formulae for the set of slopes
similar to those proved for p=2 by Buzzard and Calegari.
  The same methods also provide an efficient approach to explicit computations
of q-expansions of small slope overconvergent eigenforms, extending the
computations of Gouvea and Mazur.
