  In this computational paper we verify a truncated version of the
Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator $T_2$
for all large enough weights. We first develop a formula for computing $p$-adic
valuations of exponential sums, which we then implement to compute $2$-adic
valuations of traces of Hecke operators acting on spaces of cusp forms.
Finally, we verify that if Newton polygon of the Buzzard-Calegari polynomial
has a vertex at $n\leq 15$, then it agrees with the Newton polygon of $T_2$ up
to $n$.
