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1. CMB 2006 (vol 49 pp. 526)

Choi, So Young
The Values of Modular Functions and Modular Forms
Let $\Gamma_0$ be a Fuchsian group of the first kind of genus zero and $\Gamma$ be a subgroup of $\Gamma_0$ of finite index of genus zero. We find universal recursive relations giving the $q_{r}$-series coefficients of $j_0$ by using those of the $q_{h_{s}}$-series of $j$, where $j$ is the canonical Hauptmodul for $\Gamma$ and $j_0$ is a Hauptmodul for $\Gamma_0$ without zeros on the complex upper half plane $\mathfrak{H}$ (here $q_{\ell} := e^{2 \pi i z / \ell}$). We find universal recursive formulas for $q$-series coefficients of any modular form on $\Gamma_0^{+}(p)$ in terms of those of the canonical Hauptmodul $j_p^{+}$.

Categories:10D12, 11F11

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