Digits and Bases · Superchallenge problem

Every integer $n$ can be written in exactly one way as $n = \displaystyle\sum_{k=0}^{L-1} d_k(-2)^k$ with digits $d_k \in \{0, 1\}$ and leading digit $d_{L-1} = 1$ (for $n \neq 0$); $L$ is the number of digits of $n$ in base $-2$. How many integers $n$ with $-1000 \le n \le 1000$ have exactly $11$ digits in base $-2$?