An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan's Notebooks and a parallel construction for the Riemann zeta function. The identities are shown to be strong $q$-analogues by virtue of their reduction to the classical beta evaluations as $q\to 1^{ -}$ and explicit evaluations at CM points for $|q|<1$. Inequalities of Deligne determine asymptotic formulas for the Fourier coefficients of the associated modular forms.
Categories: math.NT