An asymptotic formula for a cubic moment of $G{L}_2$ $L$-functions

We obtain an asymptotic formula for a cubic moment of self-dual $G{L}_2$ $L$-functions studied by Petrow and Young, with a small extra averaging over characters. Our asymptotic consists of the main term conjectured by Conrey–Farmer–Keating–Rubinstein–Snaith and a power-saving error term whose strength depends on the amount of extra averaging.