  We study the range of validity of the density hypothesis for the zeros of
$L$-functions associated with cusp Hecke eigenforms $f$ of even integral weight
and prove that $N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon}$ holds for
$\sigma \geq 1407/1601$. This improves upon a result of Ivi\'{c}, who had
previously shown the zero-density estimate in the narrower range $\sigma\geq
53/60$. Our result relies on an improvement of the large value estimates for
Dirichlet polynomials based on mixed moment estimates for the Riemann zeta
function. The main ingredients in our proof are the Hal\'{a}sz-Montgomery
inequality, Ivi\'{c}'s mixed moment bounds for the zeta function, Huxley's
subdivision argument, Bourgain's dichotomy approach, and Heath-Brown's bound
for double zeta sums.
