2022/02/14 by Tobias Magnusson, Magnusson, Tobias, Martin Raum +1
Computer Science · Mathematics · #11F11 #11F30 #11F50 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2202.06676
openalex publication_date 2022/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present and discuss an algorithm and its implementation that is capable of directly determining Fourier expansions of any vector-valued modular form of weight at least 2 associated with representations whose kernel is a congruence subgroup. It complements two available algorithms that are limited to inductions of Dirichlet characters and to Weil representations, thus covering further applications like Moonshine or Jacobi forms for congruence subgroups. We examine the calculation of invariants in specific representations via techniques from permutation groups, which greatly aids runtime performance. We explain how a generalization of cusp expansions of classical modular forms enters our implementation. After a heuristic consideration of time complexity, we relate the formulation of our algorithm to the two available ones, to highlight the compromises between level of generality and performance that each them makes.