  We study the moments of $L$-functions associated with primitive cusp forms,
in the weight aspect. In particular, we obtain an asymptotic formula for the
twisted moments of a \textit{long} Dirichlet polynomial with modular
coefficients. This result, which is conditional on the Generalized Lindel\"of
Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer,
Keating, Rubinstein and Snaith.
