  Let $E$ be a non-Archimedian local field of characteristic zero and residue
characteristic $p$. Let ${\bf G}$ be a connected reductive group defined over
$E$ and $\pi$ an irreducible admissible representation of $G={\bf G}(E)$. A
result of C. M{\oe}glin and J.-L. Waldspurger (for $p \neq 2$) and S. Varma
(for $p=2$) states that the leading coefficient in the character expansion of
$\pi$ at the identity element of ${\bf G}(E)$ gives the dimension of a certain
space of degenerate Whittaker forms. In this paper we generalize this result of
M{\oe}glin-Waldspurger to the setting of covering groups $\tilde{G}$ of $G$.
