Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 29 / 6 / b / Solution

Write $\tau =x+iy$ and $\sigma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. Since $ad-bc=1$,Consequently, for the invariant norm of a modular form,This computation applies to an arbitrary function; modularity is needed only to deduce $\Gamma$-invariance of $\varphi$. The exponent is $k/2$, as printed in the PDF, rather than the converted TeX's $1/2$.