  In this paper, we compute explicitly the $q$-dimensions of highest weight
crystals modulo $q^n-1$ for a quantum group of arbitrary finite type under
certain assumption, and interpret the modulo computations in terms of the
cyclic sieving phenomenon. This interpretation gives an affirmative answer to
the conjecture by Alexandersson and Amini. As an application, under the
assumption that $\lambda$ is a partition of length $<m$ and there exists a
fixed point in $\mathsf{SST}_m(\lambda)$ under the action $\mathsf{c}$ arising
from the crystal structure, we show that the triple $(\mathsf{SST}_m(\lambda),
\langle \mathsf{c} \rangle, \mathsf{s}_{\lambda}(1,q,q^2, \ldots, q^{m-1}))$
exhibits the cycle sieving phenomenon if and only if $\lambda$ is of the form
$((am)^{b})$, where either $b=1$ or $m-1$. Moreover, in this case, we give an
explicit formula to compute the number of all orbits of size $d$ for each
divisor $d$ of $n$.
