Simple zeros of $\operatorname{GL}(2)$ $L$-functions

Let f \in S_{k}(\Gamma_{1}(N)) be a primitive holomorphic form of arbitrary weight k and level N . We show that the completed L -function of f has \Omega(T^{\delta}) simple zeros with imaginary part in [-T, T] , for any \delta <{2/27} . This is the first power bound in this problem for f of non-trivial level, where previously the best results were \Omega(\log\log\log{T}) for N odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for N even, due to Booker. In addition, for f of trivial level ( N=1 ), we also improve an old result of Conrey and Ghosh on the number of simple zeros.