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

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 $q^{2\mathrm{/}3}$ (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.