  In recent years, a number of papers have been devoted to the study of roots
of period polynomials of modular forms. Here, we study cohomological analogues
of the Eichler-Shimura period polynomials corresponding to higher
$L$-derivatives. We state general conjectures about the locations of the roots
of the full and odd parts of the polynomials, in analogy with the existing
literature on period polynomials, and we also give numerical evidence that
similar results hold for our higher derivative "period polynomials" in the case
of cusp forms. We prove a special case of this conjecture in the case of
Eisenstein series.
