  We prove an analogue of Selberg's zero density estimate for $\zeta(s)$ that
holds for any $\mathrm{GL}_2$ $L$-function. We use this estimate to study the
distribution of the vector of fractional parts of $\gamma\mathbf{\alpha}$,
where $\mathbf{\alpha}\in\mathbb{R}^n$ is fixed and $\gamma$ varies over the
imaginary parts of the nontrivial zeros of a $\mathrm{GL}_2$ $L$-function.
