I am currently reading Lang's book Introduction to Modular Forms, and perhaps I'm wrong, but I may have spotted something weird he said. Either I don't get it, or there's something abnormal about the definition.
Denote $\mathscr{M}$ the space of meromorphic functions defined on the upper-half plane $\textbf{H}=\lbrace\Re(z)>0\rbrace$. For $k\geqslant0$ an integer and $$\alpha=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\text{GL}_2^+(\mathbb{R}),$$ define a linear operator on $\mathscr{M}$ by : $$[\alpha]_k\lbrace f\rbrace(\tau)=\det(\alpha)^{k/2}(c\tau+d)^{-k}f(\alpha\cdot\tau),$$ where we defined $\alpha\cdot\tau=\frac{a\tau+b}{c\tau+d}$.
Lang is writing :
In particular, $[\alpha]_k$ depends only on the coset of $\alpha$ modulo scalar matrices.
I agree that this is true if $k$ is even. But in the general case, one has : $$[\lambda\alpha]_k=\frac{|\lambda|^k}{\lambda^k}[\alpha]_k,$$ i.e. it depends on the coset of $\alpha$ upon the action of $\mathbb{R}_+^\star$ only, not $\mathbb{R}^\star$.
Now, I agree that upon further reading, one may sweep it under the rug, since there are no modular functions of odd weight. But in the case of modular forms over a subgroup $\Gamma$ of $\text{SL}_2(\mathbb{Z})$, this need not be the case anymore.
So the question is : is it still the right definition ?
One note though : even though, noting $\pi:\text{SL}_2(\mathbb{Z})\to\!\!\!\!\!\to\text{PSL}_2(\mathbb{Z})$, it may not be true that $\pi(\Gamma)$ is a quotient of $\Gamma$ (i.e. $\pi:\Gamma\to\pi(\Gamma)$ need not be surjective), when it happens that $\Gamma$ is stable upon the action of multiplication by scalars (so in this case, by $\alpha\mapsto-\alpha$), then it still holds that there are no modular functions of odd weight, since $\Gamma$ contains $-I_2$. Thus in this case, it is still fine to define $[\alpha]_k$ as so and to say that in the cases we are interested in, it only depends on the projective class of $\alpha$. But once again, what if it is not the case ?
Therefore, my worry came from the fact that perhaps Lang didn't mean to say that $[\alpha]_k$ depended on the projective class, but rather, it did only in the cases we will be interested in ?