Modular-invariant function ($F(\gamma \tau )=F(\tau ),\gamma \in S{L}_2(\mathbb{Z})$)

A function on the complex upper half-plane is modular invariant if it is unchanged by the fractional-linear action of the modular group. This does not imply holomorphicity: a nonholomorphic Eisenstein series is modular invariant, whereas a modular function additionally satisfies meromorphicity and cusp conditions.