The eighth moment of the family of $\Gamma_1(q)$-automorphic $L$-functions

Abstract

We prove a Lindel\"of on average bound for the eighth moment of a family of $L$-functions attached to automorphic forms on $GL(2)$, the first time this has been accomplished. Previously, such a bound had been proven for the sixth moment for our family by Djankovi\'c and for a similar family by Young. Our proof rests on a new approach which overcomes the lack of perfect orthogonality in the family initially observed by Iwaniec and Xiaoqing Li.