If $f(\tau)$ is a modular form for $\Gamma_0(N)$, then what can we say about the function $g(\tau) = f(\tau /n)$?

Question

I was wondering if it should be modular for $\Gamma_0(N')$ for some $N'$, but I'm struggling to show this.

Answer 1

I think I've worked it out.

$$\left(\begin{array}{lr} 1 & 0 \\ 0 & n \end{array}\right)\tau = \frac{\tau}{n}$$

So if $f$ is modular for $\Gamma_0(N)$, then $g$ is modular for $\left(\begin{array}{lr} 1 & 0 \\ 0 & \frac{1}{n} \end{array}\right)\Gamma_0(N)\left(\begin{array}{lr} 1 & 0 \\ 0 & n \end{array}\right)$