  The slope of a p-adic overconvergent eigenform of weight k is the p-adic
valuation of its U_p eigenvalue. We find the slope of all 2-adic finite slope
overconvergent eigenforms of tame level 1 and weight 0. As a consequence we
prove that any finite slope 2-adic overconvergent eigenform of tame level 1 and
weight 0 has coefficients in Q_2. These results provide evidence towards
conjectures of the first author that predict the slopes of classical p-adic
modular forms under a mild ``Gamma_0(N)''-regular hypothesis on p.
