  We define non-holomorphic Poincar\'e series of exponential type for
symplectic groups $\mathop{Sp}_m(\mathbb R)$ and continue them analytically in
case $m=2$ for the small weight $(4,4)$. For this we construct certain Casimir
operators and study the spectral properties of their resolvents on
$L^2(\Gamma\backslash \mathop{Sp}_2(\mathbb R))$. Using the holomorphically
continued Poincar\'e series, the holomorphic projection is described in terms
of Fourier coefficients using Sturm's operator.
