The sixth moment of a degree two $L$-function of arbitrary level

Let $f$ be a primitive holomorphic cusp form of arbitrary weight, level and nebentypus. We prove ${\int }_0^T\mathrm{\mid }L(1\mathrm{/}2+it,f){\mathrm{\mid }}^{6}dt{\ll }_{f,\epsilon }{T}^{2+\epsilon }$, extending Jutila’s level-one theorem. The new ingredient is a large sieve comparing the stationary phases in the Booker-Milinovich-Ng transformation at different heights. Applications include moments of order $T^{1+\epsilon }$ on every line to the right of the critical line, $\mathrm{\Omega }(T^{4\mathrm{/}35-\epsilon })$ simple zeros, zero density estimates, and bounds for cubic-field power sums, shifted convolution sums and the general divisor problem.