  In this paper we determine the limiting distribution of the image of the
Eichler--Shimura map or equivalently the limiting joint distribution of the
coefficients of the period polynomials associated to a fixed cusp form. The
limiting distribution is shown to be the distribution of a certain
transformation of two independent random variables both of which are
equidistributed on the circle $\mathbb{R}/\mathbb{Z}$, where the transformation
is connected to the additive twist of the cuspidal $L$-function. Furthermore we
determine the asymptotic behavior of the zeroes of the period polynomials of a
fixed cusp form. We use the method of moments and the main ingredients in the
proofs are additive twists of $L$-functions and bounds for both individual and
sums of Kloosterman sums.
