Short Second Moment Bound for GL(2) $L$-functions in $q$-Aspect

Abstract

We prove a Lindelöf-on-average upper bound for the second moment of the $L$-functions associated to a level 1 holomorphic cusp form, twisted along a coset of subgroup of the characters modulo $q^{2/3}$ (where $q=p^3$ for some odd prime $p$). This result should be seen as a $q$-aspect analogue of Anton Good's (1982) result on upper bounds of the second moment of cusp forms in short intervals. The results generalize easily to higher prime powers as well.

Final Version. Minor revisions to incorporate suggestions from the referee. Added a sketch for higher prime powers