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We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $\\mathbb{F}_q$. A general result of our study is that $(a,b)\\leq 3$ for all $a,b \\in \\mathbb{Z}$ if $p> (\\sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a\\neq b$ and $a,b \\in \\{1,\\dots,e-1\\}$. 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