Cycle integrals of meromorphic modular forms and Siegel theta functions

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.