How to compute the dimension of cohomology groups?

Question

I would like to compute the dimension of $Z^1(\Gamma,V)$ and $B^1(\Gamma,V)$, where $\Gamma$ is a finitely generated group acting on a vector space $V$. $$Z^1(\Gamma,V)=\{\gamma : \Gamma\to V,\ \gamma(gh)=\gamma(g)+g\cdot\gamma(h)\}\ \text{and}$$ $$B^1(\Gamma,V)=\{\gamma : \Gamma\to V,\ \exists v\in V,\gamma(g)=g\cdot v\}$$ If $\Gamma=F_n=<X>$ (the free group generated by $X=\{x_1,...,x_n\}$), as each $1$-cocycle is determined by its values on the generators $x_1,...,x_n$ then $$Z^1(\Gamma,V)\cong V^X=\{\gamma : X\to V\}\cong V^n$$ Same for $B^1(\Gamma,V)$.

1. Is this correct?
2. How to compute the dimensions of these vector spaces in the case a finitely generated (not free) group $\Gamma$?

My motivations are from the study of the symplectic geometry of character varieties (the works by Goldman):

The Symplectic Nature of Fundamental Groups of Surfaces