we suppress $n$ to keep the notation simple. In our first result, we calculate the covariance between $w_p$ and $w_q$ as $n \tends \infty$.

\[thm:symcircovar\] Suppose $SC_n$ is the symmetric circulant matrix with independent input sequence $\{\frac{X_i}{\sqrt n}\}_{i\geq 0}$ such that $$\label{eqn:condition}
\E(X_i)=0, \E(X_i^2)=1,  \E(X_i^4)=\E(X_1^4)\  \mbox{and}\  \sup_{i\geq 1}\E(|X_i|^k)=\alpha_k<\infty \ \mbox{for}\ k\geq 3.$$ Then for $p,q \geq2$, $$\begin{aligned}
 \label{eqn:sigma_p,q}
    \sigma_{p,q}&:=\lim_{n\to\infty}  \Cov\big(w_p,w_q\big) \nonumber \\ 
    &=  \left\{\begin{array}{ll}     
            \displaystyle\frac{a_1}{2^{\frac{p+q-4}{2}}  } (\E X^4_1- 1) + \sum_{r=2}^{ \min\{ \frac{p}{2},\frac{q}{2} \} } \frac{a_r}{2^{\frac{p+q-4r}{2}}  } \sum_{s=0}^{2r}\binom{2r}{s}^2 s!(2r-s)! \ h_{2r}(s) & \text{if}\ p,  q \mbox{ both are even,}\\\\
            \displaystyle\sum_{r=0}^{ \min\{ \frac{p-1}{2},\frac{q-1}{2} \} }  \frac{b_r}{2^{\frac{p+q-4r-2}{2}}  } \sum_{s=0}^{2r+1}\binom{2r+1}{s}^2 s!(2r+1-s)! \ h_{2r+1}(s) \\         
        \displaystyle+ pq \binom{p-1}{(p-1)/2}\binom{q-1}{(q-1)/2} \l((p-1)/2\r)! \l((q-1)/2\r)!\frac{1}{2^{(\frac{p+q}{2}-1)}}      & \text{if}\ p,  q \mbox{ both are odd,}\\\\
            0 & \text{otherwise},        
              \end{array}\right.        
              %\\
              %&= \sigma_{p,q},\ \mbox{ say},\end{aligned}$$ where $a_r$ and $b_r$ are appropriate constants, will be given in proof, and $h_d(s)$ is given as $$h_d(s)=
 \frac{1}{(d-1)!}\sum_{i=-\lceil\frac{d-s}{2}\rceil}^{\lfloor\frac{s}{2}\rfloor}\sum_{j=0}^{2i+d-s}(-1)^q\binom{d}{j}\l(\frac{2i+d-s-j}{2}\r)^{d-1}.$$

If $p=q$ then we denote $\sigma_{p,q}$ by $\sigma^{2}_{p}$. In our second result, we see the fluctuation of linear eigenvalue statistics of symmetric circulant matrices with polynomial test functions.

\[thm:symcirpoly\] Suppose input entry of $SC_n$ is independent sequence $\{\frac{X_i}{\sqrt n}\}_{i\geq 1}$ which satisfy \[eqn:condition\]. Then, as $n\to \infty$, $$\begin{aligned}
w_Q \stackrel{d}{\longrightarrow} N(0,\sigma_{Q}^2).\end{aligned}$$ In particular, for $Q(x)= x^{p}$ $$\begin{aligned}
 w_p 