  Let $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations
$\pi$ of $\mathrm{GL}_n$ over a number field with unitary central character. We
prove two unconditional large sieve inequalities for the Hecke eigenvalues of
$\pi\in\mathfrak{F}_n$, one on the integers and one on the primes. The second
leads to the first unconditional zero density estimate for the family of
$L$-functions $L(s,\pi)$ associated to $\pi\in\mathfrak{F}_n$, which we make
log-free. As an application of the zero density estimate, we prove a hybrid
subconvexity bound for $L(\frac{1}{2},\pi)$ for a density one subset of
$\pi\in\mathfrak{F}_n$.
