Slopes of 2-adic overconvergent modular forms with small level

Abstract

Let $\tau$ be the primitive Dirichlet character of conductor 4, let $\chi$ be the primitive even Dirichlet character of conductor 8 and let $k$ be an integer. Then the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $2k+1$ and character $\tau$ has slopes in the arithmetic progression ${2,4,...,2n,...}$, and the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $k$ and character $\chi \cdot \tau^k$ has slopes in the arithmetic progression ${1,2,...,n,...}$. We then show that the characteristic polynomials of the Hecke operators $U_2$ and $T_p$ acting on the space of classical cusp forms of weight $k$ and character either $\tau$ or $\chi\cdot\tau^k$ split completely over $\qtwo$.