Slopes of 2-adic overvconvergent modular forms with small level

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. We show that the~$U_2$ operator acting on cuspidal overconvergent modular forms of weight~$2k-1$ and character~$\tau$ has slopes in the arithmetic progression\\~$\left\{2,4,\ldots,2n,\ldots\right\}$, 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~$\left\{1,2,\ldots,n,\ldots\right\}$. We also 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~$\mathbf{Q}_2$.