Constant Term Formula

The constant term of $P_N(x;\theta)$ is $(-1)^N \prod_{k=0}^{N-1} r_k(\theta)$. After scaling by $2^{N-1}$, the constant term of $S_N$ is $$2^{N-1} \cdot (-1)^N \cdot \prod_{k=0}^{N-1} \cos\!\left(\theta + \frac{2\pi k}{N}\right).$$ This product of roots varies with $\theta$, reflecting the fact that Newton's identities only constrain $e_N$ using $p_N$, which may depend on $\theta$ when $j = N$.