  This paper is devoted to investigate the heat trace asymptotic expansion
corresponding to the magnetic Steklov eigenvalue problem on Riemannian
manifolds with boundary. We establish an effective procedure, by which we can
calculate all the coefficients $a_0$, $a_1$, $\dots$, $a_{n-1}$ of the heat
trace asymptotic expansion. In particular, we explicitly give the expressions
for the first four coefficients. These coefficients are spectral invariants
which provide precise information concerning the volume and curvatures of the
boundary of the manifold and some physical quantities by the magnetic Steklov
eigenvalues.
