  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.
