  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
$\Omega\left(T^\delta\right)$ simple zeros with imaginary part in $\left[-T,
T\right]$, for any $\delta < \frac{2}{27}$. 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.
