Explicit Computation of Cusp forms via Hecke Action on Cohomology and its Complexity

Abstract

In the literature, the standard approach to finding bases of spaces of modular forms is via modular symbols and the homology of modular curves. By using the Eichler-Shimura isomorphism, a work by Wang shows how one can use a cohomological viewpoint to determine bases of spaces of cusp forms on $\Gamma_0(N)$ of weight $k \geq 2$ and character $\chi$. It is interesting to look at the complexity of this alternative approach, and we do this for an explicit implementation of the algorithm suggested by Wang.