Newton Polygons of Hecke Operators

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 $n\le 15$, then it agrees with the Newton polygon of $T_2$ up to $n$.