Dimensions of Spaces of Newforms
by
Greg Martin
University of British Columbia

A formula for the dimension of the space of cuspidal modular forms on Γ_0(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 Γ_0(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 is desirable to have a formula in closed form for these dimensions. In this talk we describe such a closed-form formula, not only for these dimensions, but also for the corresponding dimensions of spaces of newforms on Γ_1(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 Γ_0(N), and we report on extensive computations of these dimensions. For example, we can find the complete list of all N such that the dimension of the space of weight-2 newforms on Γ_0(N) is less than or equal to 100; previous such results had only gone up to 3.

Date received: May 19, 2008