= Solution
Write $\tau=x+iy$ and $\sigma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$. Since $ad-bc=1$,
$$
\operatorname{Im}(\sigma\tau)=\frac{y}{|c\tau+d|^2}.
$$
Consequently, for the <invariant norm of a modular form>,
$$
\begin{aligned}
\phi(\sigma\tau)
&=|f(\sigma\tau)|\left(\frac{y}{|c\tau+d|^2}\right)^{k/2}\\
&=|(c\tau+d)^{-k}f(\sigma\tau)|y^{k/2}
=|f|_k\sigma(\tau)|y^{k/2}.
\end{aligned}
$$
This computation applies to an arbitrary function; modularity is needed only to deduce $\Gamma$-invariance of $\phi$. The exponent is \b[$k/2$], as printed in the PDF, rather than the converted TeX's $1/2$.
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