Let $F_{N,\mathbb C}$ be the field of modular functions of level $N$. We have $$\operatorname{Gal}(F_{N,\mathbb C}/\mathbb C(j))\cong \operatorname{SL}_2(\mathbb Z/N\mathbb Z)/\{\pm 1\}.$$

Let $f\in F_{N,\mathbb C}$. To be able to do something with $f$, I must assume that $\mathbb C(j)\subset\mathbb C(f)$.

Is there a modular function of some level, such that this assumption is not true? Or is the $j$-invariant always rationally expressible through every modular function?