We prove that the first Dirichlet eigenvalue of a regular N -gon of area π has an asymptotic expansion of the form λ 1 (1 + n≥3 Cn(λ1) N n ) as N → ∞, where λ 1 is the first Dirichlet eigenvalue of the unit disk and C n are polynomials whose coefficients belong to the space of multiple zeta values of weight n. We also explicitly compute these polynomials for all n ≤ 14.

We present a database of several hundred modular forms up to and including weight six on noncongruence subgroups of index ≤ 17. In addition, our database contains expressions for the Belyi map for genus zero subgroups and equations of the corresponding elliptic curves for genus one subgroups and numerical approximations of noncongruence Eisenstein series to 1500 digits precision.

No abstract