= Modular form on a finite-index subgroup
{title2=$M_k(\Gamma)$}
A <modular form> on an arbitrary <finite-index subgroup> $\Gamma\leq SL_2(\mathbb Z)$ is a <holomorphic function> on the <complex upper half-plane>, invariant under the weight-$k$ <slash operator for modular forms>, and <holomorphic at a cusp> at every <cusp of a modular group> for $\Gamma$. If $\sigma\in SL_2(\mathbb Z)$ maps infinity to a cusp, choose a genuine translation period $h>0$ with $\sigma T^h\sigma^{-1}\in\Gamma$. Then $f|_k\sigma$ has a convergent series in $e^{2\pi iz/h}$ with nonnegative exponents. A <cusp form> has zero constant term at every cusp. This definition allows noncongruence subgroups and avoids sign ambiguities in odd weights when a smaller width is defined only modulo the center.
Back to article page
