Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 137 / 1 / d / Solution

The space $M_8(S{L}_2(\mathbb{Z}))$ is one-dimensional. The two normalized weight-eight forms $E_8$ and $E_4^2$ therefore agree. Their expansions areandEquating the coefficient of $q^n$ and dividing by $480$ proves the divisor-sum convolution identity of weights four and eight