  By using results on poles of $L$-functions and theta correspondence, we give
a bound on $b$ for $(\chi,b)$-factors of the global Arthur parameter of a
cuspidal automorphic representation $\pi$ of a classical group or a metaplectic
group where $\chi$ is a conjugate self-dual automorphic character and $b$ is an
integer which is the dimension of an irreducible representation of
$\mathrm{SL}_{2}(\mathbf{CC})$. We derive a more precise relation when $\pi$
lies in a generic global $A$-packet.
