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We compare the homology of a congruence subgroup Gamma of GL_2(Z) withcoefficients in the Steinberg modules over Q and over E, where E is a realquadratic field. If R is any commutative base ring, the last connectinghomomorphism psi_{Gamma,E} in the long exact sequence of homology stemming fromthis comparison has image in H_0(Gamma, St(Q^2;R)) generated by classes z_\betaindexed by beta in E \ Q. We investigate this image.  When R=C, H_0(Gamma, St(Q^2;C)) is isomorphic to a space of classical modularforms of weight 2, and the image lies inside the cuspidal part. In this case,z_beta is closely related to periods of modular forms over the geodesic in theupper half plane from beta to its conjugate beta'. Assuming GRH we prove thatthe image of $\psi_{\Gamma,E}$ equals the entire cuspidal part.  When R=Z, we have an integral version of the situation. We define thecuspidal part of the Steinberg homology, H_0^cusp(Gamma, St(Q^2;Z)). AssumingGRH we prove that for any congruence subgroup, psi_{Gamma,E} always has finiteindex in H_0^cusp(Gamma, St(Q^2;Z)), and if Gamma=Gamma_1(N)^pm or \Gamma_1(N),then the image is all of H_0^cusp(Gamma, St(Q^2;Z)). If Gamma=Gamma_0(N)^pm orGamma_0(N), we prove (still assuming GRH) an upper bound for the size ofH_0^cusp(Gamma, St(Q^2;Z))/image(psi_{Gamma,E}). We conjecture that the resultsin this paragraph are true unconditionally.  We also report on extensive computations of the image of psi_{Gamma,E} thatwe made for Gamma=Gamma_0(N)^pm and Gamma=Gamma_0(N). Based on thesecomputations, we believe that the image of psi_{Gamma,E} is not all ofH_0^cusp(Gamma, St(Q^2;Z)) for these groups, for general N.

Steinberg homology, modular forms, and real quadratic fields