  Let $p$ be prime, $N$ be a positive integer prime to $p$, and $k$ be an
integer. Let $P_k(t)$ be the characteristic series for Atkin's $U$ operator as
an endomorphism of $p$-adic overconvergent modular forms of tame level $N$ and
weight $k$. Motivated by conjectures of Gouvea and Mazur, we strengthen Wan's
congruence between coefficients of $P_k$ and $P_{k'}$ for $k'$ close
$p$-adically to $k$. For $p-1 | 12$, $N = 1$, $k = 0$, we compute a matrix for
$U$ whose entries are coefficients in the power series of a rational function
of two variables. We apply this computation to show for $p = 3$ a parabola
below the Newton polygon $N_0$ of $P_0$, which coincides with $N_0$ infinitely
often. As a consequence, we find a polygonal curve above $N_0$. This tightest
bound on $N_0$ yields the strongest congruences between coefficients of $P_0$
and $P_k$ for $k$ of large 3-adic valuation.
