This paper contains both theoretical results and experimental data
on the behavior of the dimensions of the cohomology spaces
$H^1(\G,E_n)$, where $\Gamma$ is a lattice in $\SL(2,\C)$ and
$E_n = \Sym^n\otimes \overline{\Sym}{}^n$,
$n\in \N\cup \{0\}$, is one of the standard self-dual modules. In the case
$\Gamma = \SL(2,\O)$ for the ring of integers $\O$ in an imaginary
quadratic number field, we make the theory of lifting explicit and obtain
lower bounds linear in $n$. We present a large amount of experimental data
for this case, as well as for some geometrically constructed and mostly nonarithmetic groups.
The computations for $\SL(2,\O)$ lead us to discover two instances with nonlifted classes in the cohomology.
We also derive an upper bound of size $O(n^2 / \log n)$ for any fixed lattice $\G$ in the general case.
We discuss a number of new questions and conjectures suggested by our results and our experimental data.