Is it a fact that if $f(z)$ is a modular form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$ then $f(Nz)$ is a modular form of weight $k$ for $\Gamma_0(N)$?
I tried considering this by simply making a change of variables $Nz \mapsto \tau$ but I can't make my reasoning rigorous because I am unsure of which transformation law to check when finding the weight of $f(Nz)$. Is it $$ f(N\cdot \gamma(z)) = (cz+d)^kf(Nz) $$ or $$ f(\gamma(Nz)) = (c(Nz)+d)^kf(Nz) $$
If it's the second case I think my approach mentioned above would work but then I don't see the importance of emphasizing $\Gamma_0(N)$. In other words, what is the significance of this additional congruence relation we put on the bottom left entry $c$ of our matrix $\gamma$?