No heads in a Poisson number of fair tosses

Let $N\sim\text{Poisson}(2)$, and toss a fair coin $N$ times. The probability that none of the tosses land heads can be written as $e^{a}$ with $a$ rational. Find $a$.

$a=-1$. The number of heads is a thinned Poisson, so it is Poisson with mean $2\cdot\tfrac12=1$, and $P(\text{no heads})=e^{-1}$. Equivalently, $\sum_n e^{-2}\tfrac{2^n}{n!}2^{-n}=e^{-2}e^{1}$.