Why is momentum expectation zero for gaussian wavepacket by Ehrenfest theorem?

If $$psi(x)=Aexp(-x^2/a^2)exp(ikx)$$
$langle{p}rangle=0$ since
$langle{x}rangle=0$, since the integrand is an odd function and Ehrenfest theorem states $frac{dlangle{x}rangle}{dt}=frac{langle{p}rangle}{m}$.

But explicit calculation of $langle{p}rangle= int^{infty}_{-infty}psi^*(x) hat{p} psi(x)dx$ and using $hat{p}=-ihbar frac{partial}{partial{x}}$ gives $hbar k$. I think Ehrenfest theorem is giving the wrong result because of the $ikx$ term,how to correctly use Ehrenfest theorem in this case?