initiated by Arharov [@arharov] in 1971 for sample covariance matrices. In 1975 Girko [@girko] studied the central limit theorem (CLT) of the traces of the Wigner and sample covariance matrices using martingale techniques. In 1982, Jonsson [@jonsson] proved the CLT of linear eigenvalue statistics for Wishart matrices using method of moments. After that the fluctuations of eigenvalues for various random matrices have been extensively studied by various people. For new results on fluctuations of linear eigenvalue statistics of Wigner and sample covariance matrices, see [@johansson1998], [@soshnikov1998tracecentral], [@bai2004clt], [@lytova2009central], [@shcherbina2011central]. For band and sparse random matrices, see [@anderson2006clt], [@jana2014], [@li2013central], [@shcherbina2015] and for Toeplitz and band Toeplitz matrices, see [@chatterjee2009fluctuations] and [@liu2012].

In a recent article [@adhikari_saha2017], the CLT for linear eigenvalue statistics has been established in total variation norm for circulant, symmetric circulant and reverse circulant matrices with Gaussian entries. In a subsequent article [@adhikari_saha2018], the authors extended their results for independent entries which are smooth functions of Gaussian variables. Here we consider the fluctuation problem for symmetric circulant matrices with general entries which are independent and satisfy some moment condition.

A sequence is said to be an [*input sequence*]{} if the matrices are constructed from the given sequence. We 