Massive rest-frame Pauli-Lubanski eigenvalues ($(W_0,W_3)=(0,-m{j}_3)$)

With signature $(+---)$, ${ϵ}_{0123}=+1$, $J_3=M^{12}$ and the lower-index definition of $W_{\mu }$, a rest momentum $(m,0,0,0)$ gives $W_0=0$ and $W_3=-m{J}_3$ on that momentum eigenspace. A Hermitian spin projection $j_3$ consequently gives eigenvalues $0$ and $-m{j}_3$. Raising the third index changes its sign.