If $\pi: C \rightarrow \mathbb{P}^{1}$ is a cyclic cover of $\mathbb{P}^{1}$ with Galois group $\mathbb{Z}/m \mathbb{Z}$ and thus with the (affine) formula $y^{m}= (x_{1}a_{1})^{t_{1}}....(x_{n}a_{n})^{t_{n}}$ there is a Galois group decomposition on the cohomolgy $H^{1}(C, \mathbb{C})$ (or more generally on the sheaf $R^{1} \pi_{*} \mathbb{C}$ ) and the dimension of the eigenspace with respect to the character $j \in \mathbb{Z}/m \mathbb{Z}$ is given by the formula $1+ \sum_{i=1}^n < jt_{i}/m> $ (where $< x>$ denotes the franctional part of the number $x$ ) . Now if the Galois group of the covering $\pi: C \rightarrow \mathbb{P}^{1}$ is not cyclic, is there a formula that computes the dimension of the eigenspaces? Or generally how can one compute the dimension of the eigenspaces? For simplicity you can always assume that the covering $\pi$ is abelian (i.e. the Galois group is abelian) and thereof the fiber product of cyclic covers.
