Abstract: For non-negative integers $n$, let $j_n$ be the unique modular function that is holomorphic on the upper half-plane and with Fourier expansion $q^{-n}+O(q)$. Asai, Kaneko, and Ninomiya showed that

\[
\sum_{n=0}^\infty
j_n(\tau)q^n=E_{14}(z)\Delta(z)^{-1}/(j(z)-j(\tau)),
\]

where as usual $E_k$ is the weight $k$ Eisenstein series and $\Delta$ is the unique normalized weight $12$ cusp form. Ahlgren obtained similar formulas on higher level, genus $0$, congruence subgroups, with $j$ replaced by the corresponding Hauptmodul. We generalize these results to congruence subgroups for which the underlying modular curve is hyperelliptic. In all cases, we also show how to obtain similar results starting from any meromorphic modular form $F$ of weight $k$. (From this point of view, the above formula corresponds to $F=1, k=0$).