  In this paper, we study a new cyclic sieving phenomenon on the set
$\mathsf{SST}_n(\lambda)$ of semistandard Young tableaux with the cyclic action
$\mathsf{c}$ arising from its $U_q(\mathfrak{sl}_n)$-crystal structure. We
prove that if $\lambda$ is a Young diagram with $\ell(\lambda) < n$ and $\gcd(
n, |\lambda| )=1$, then the triple $\left( \mathsf{SST}_n(\lambda), \mathsf{C},
q^{- \kappa(\lambda)} s_\lambda(1,q, \ldots, q^{n-1}) \right) $ exhibits the
cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by
$\mathsf{c}$. We further investigate a connection between $\mathsf{c}$ and the
promotion $\mathsf{pr}$ and show the bicyclic sieving phenomenon given by
$\mathsf{c}$ and $\mathsf{pr}^n$ for hook shape.
