Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9496484 | Journal of Number Theory | 2005 | 34 Pages |
Abstract
A formula for the dimension of the space of cuspidal modular forms on Î0(N) of weight k (k⩾2 even) has been known for several decades. More recent but still well-known is the Atkin-Lehner decomposition of this space of cusp forms into subspaces corresponding to newforms on Î0(d) of weight k, as d runs over the divisors of N. A recursive algorithm for computing the dimensions of these spaces of newforms follows from the combination of these two results, but it would be desirable to have a formula in closed form for these dimensions. In this paper we establish such a closed-form formula, not only for these dimensions, but also for the corresponding dimensions of spaces of newforms on Î1(N) of weight k (k⩾2). This formula is much more amenable to analysis and to computation. For example, we derive asymptotically sharp upper and lower bounds for these dimensions, and we compute their average orders; even for the dimensions of spaces of cusp forms, these results are new. We also establish sharp inequalities for the special case of weight-2 newforms on Î0(N), and we report on extensive computations of these dimensions: we find the complete list of all N such that the dimension of the space of weight-2 newforms on Î0(N) is less than or equal to 100 (previous such results had only gone up to 3).
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Greg Martin,