  We derive a precise asymptotic expansion of the complete K\"{a}hler-Einstein
metric on the punctured Riemann sphere with three or more omitting points. By
using Schwarzian derivative, we prove that the coefficients of the expansion
are polynomials on the two parameters which are uniquely determined by the
omitting points. Futhermore, we use the modular form and Schwarzian derivative
to explicitly determine the coefficients in the expansion of the complete
K\"{a}hler-Einstein metric for punctured Riemann sphere with $3, 4, 6$ or $12$
omitting points.
