The symmetry energy coefficients in finite nuclei have been studied systematically with a covariant density functional theory (DFT) and compared with the values calculated using several available mass tables. Due to the contamination of shell effect, the nuclear symmetry energy coefficients extracted from the binding energies have large fluctuations around the nuclei with double magic numbers. The size of this contamination is shown to be smaller for the nuclei with larger isospin value. After

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... ubtracting the shell effect with the Strutinsky method, the obtained nuclear symmetry energy coefficients with different isospin values are shown to decrease smoothly with the mass number A and are subsequently fitted to the relation 4a_ symA=b_vA-b_sA^4/3. The resultant volume b_v and surface b_s coefficients from axially deformed covariant DFT calculations are 121.73 and 197.98 MeV respectively. The ratio b_s/b_v=1.63 is in good agreement with the value derived from the previous calculations with the non-relativistic Skyrme energy functionals. The coefficients b_v and b_s corresponding to several available mass tables are also extracted. It is shown that there is a strong linear correlation between the volume b_v and surface b_s coefficients and the ratios b_s/b_v are in between 1.6-2.0 for all the cases.