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

On the distribution of periods of holomorphic cusp forms and zeroes of period polynomials