  We use spectral method to prove a joint equidistribution of primitive
rational points and the same along expanding horocycle orbits in the products
of the circle and the unit cotangent bundle of the modular surface. This result
explicates the error bound in a recent work of Einsiedler, Luethi, and Shah
\cite[Theorem $1.1$]{ELS}. The error is sharp upon the best known progress
towards the Ramanujan conjecture at the finite places for the modular surface.
