weight of a modular form

《A First Course in Modular Forms》 · 第 2–3 页 · Section 1.1「First definitions and examples」, Definitions 1.1.1–1.1.2

Let $k$ be an integer. A meromorphic function $f:\mathbb H\to\mathbb C$ is weakly modular of weight $k$ if $f(\gamma(\tau))=(c\tau+d)^k f(\tau)$; a function $f$ is a modular form of weight $k$ if it is holomorphic on $\mathbb H$, weakly modular of weight $k$, and holomorphic at $\infty$.