Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 4 / c

At weight zero the convention from part (a) isPartition the summands of $G$ according to divisibility of the first integer coordinate by $p$. Directly from the definition,In the sum over $b$, the congruence $n\equiv mb(\mathrm{mod}p)$ has one solution when $p\nmid m$, has $p$ solutions when $p$ divides both $m$ and $n$, and has none when $p\mid m$ but $p\nmid n$. Thereforewhere the contribution from pairs divisible by $p$ was rescaled by $p^{-2s}$. Dividing by $p$ proves the Hecke eigenvalue of a nonholomorphic Eisenstein series: