Summary: 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.