  We study meromorphic modular forms associated with positive definite binary
quadratic forms and their cycle integrals along closed geodesics in the modular
curve. We show that suitable linear combinations of these meromorphic modular
forms have rational cycle integrals. Along the way, we evaluate the cycle
integrals of the Siegel theta function associated with an even lattice of
signature $(1,2)$ in terms of Hecke's indefinite theta functions, which is of
independent interest.
