Dimensions of the spaces of cusp forms and newforms on Γ0(N) and Γ1(N)

A formula for the dimension of the space of cuspidal modular forms on Γ0(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 Γ0(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 Γ1(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 Γ0(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 Γ0(N) is less than or equal to 100 (previous such results had only gone up to 3).