  We show that all Eichler integrals, and more generally all "generalized
second order modular forms" can be expressed as linear combinations of
corresponding generalized second order Eisenstein series with coefficients in
classical modular forms. We determine the Fourier series expansions of
generalized second order Eisenstein series in level one, and provide tail
estimates via convexity bounds for additively twisted $\mathrm{L}$-functions.
As an application, we illustrate a bootstrapping procedure that yields
numerical evaluations of, for instance, Eichler integrals from merely the
associated cocycle. The proof of our main results rests on a filtration
argument that is largely rooted in previous work on vector-valued modular
forms, which we here formulate in classical terms.
