For a given bounded domain $\Omega$ with smooth boundary in a smooth Riemannian manifold, by applying symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as $t\to 0^+$. In particular, we explicitly give the first four coefficients of this asymptotic expansion. These coefficients provide precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Dirichlet-to-Neumann map.