  We study the space of period polynomials associated with modular forms of
integral weight for finite index subgroups of the modular group. For the
modular group, this space is endowed with a pairing, corresponding to the
Petersson inner product on modular forms via a formula of Haberland, and with
an action of Hecke operators, defined algebraically by Zagier. We generalize
Haberland's formula to (not necessarily cuspidal) modular forms for finite
index subgroups, and we show that it conceals two stronger formulas. We extend
the action of Hecke operators to period polynomials of modular forms, we show
that the pairing on period polynomials appearing in Haberland's formula is
nondegenerate, and we determine the adjoints of Hecke operators with respect to
it. We give a few applications for $\Gamma_1(N)$: an extension of the
Eichler-Shimura isomorphism to the entire space of modular forms; the
determination of the relations satisfied by the even and odd parts of period
polynomials associated with cusp forms, which are independent of the period
relations; and an explicit formula for Fourier coefficients of Hecke eigenforms
in terms of their period polynomials, generalizing the Coefficients Theorem of
Manin.
