jmc

Let the given sum be $S$. By inspection, we find that $2^6\equiv 64\equiv -1(\mathrm{mod}13)$, so $2^{-6}\equiv (-1{)}^{-1}\equiv -1(\mathrm{mod}13)$. It follows that $2^{-5}\equiv 2\cdot 2^{-6}\equiv 2\cdot -1\equiv -2(\mathrm{mod}13)$, and that $2^{-4}\equiv -4(\mathrm{mod}13)$, and so forth. Thus, $$S\equiv -2^5-2^4-2^3-2^2-2-1\equiv -63\equiv \boxed{2}(\mathrm{mod}13)$$