On vector-valued Poincar\'e series of weight 2

Given a pair $(\Gamma,\rho)$ of a Fuchsian group of the first kind, and a unitary representation $\rho$ of $\Gamma$ of arbitrary rank, the problem of construction of vector-valued Poincar\'e series of weight 2 is considered. Implications in the theory of parabolic bundles are discussed. When the genus of the group is zero, it is shown how an explicit basis for the space of these functions can be constructed.