An asymptotic formula for a cubic moment of $GL_2$ $L$-functions

We obtain an asymptotic formula for a cubic moment of self-dual $GL_2$ $L$-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.