  In this paper, we introduce a new family of period integrals attached to
irreducible cuspidal automorphic representations $\sigma$ of symplectic groups
$\mathrm{Sp}_{2n}(\mathbb{A})$, which detects the right-most pole of the
$L$-function $L(s,\sigma\times\chi)$ for some character $\chi$ of
$F^\times\backslash\mathbb{A}^\times$ of order at most $2$, and hence the
occurrence of a simple global Arthur parameter $(\chi,b)$ in the global Arthur
parameter $\psi$ attached to $\sigma$. We also give a characterisation of first
occurrences of theta correspondence by (regularised) period integrals of
residues of certain Eisenstein series.
