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We solve a long-standing open problem with its own long history dating backto the celebrated works of Klein and Ramanujan. This problem concerns theinvariant decomposition formulas of the Hauptmodul for $\Gamma_0(p)$ under theaction of finite simple groups $PSL(2, p)$ with $p=5, 7, 13$. The cases of$p=5$ and $7$ were solved by Klein and Ramanujan. Little was known about thisproblem for $p=13$. Using our invariant theory for $PSL(2, 13)$, we solve thisproblem. This leads to a new expression of the classical elliptic modularfunction of Klein: $j$-function in terms of theta constants associated with$\Gamma(13)$. Moreover, we find an exotic modular equation, i.e., it has thesame form as Ramanujan's modular equation of degree $13$, but with differentkinds of modular parametrizations, which gives the geometry of the classicalmodular curve $X(13)$.

Dedekind $\eta$-function, Hauptmodul and invariant theory