  For arbitrary level $N$, we relate the generating series of codimension 2
special cycles on $\mathcal{X}_{0}(N)$ to the derivatives of a genus 2
Eisenstein series, especially the singular terms of both sides. On the analytic
side, we use difference formulas of local densities to relate the singular
Fourier coefficients of the genus 2 Eisenstein series to the nonsingular
Fourier coefficients of a genus 1 Eisenstein series. On the geometric side, we
study the reduction of cusps to compute the divisor class of the Hodge bundle
and the heights of special divisors. When $N$ is square-free, this gives a
different proof of the main results in the works of Du, Yang and Sankaran, Shi,
and Yang.
