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Let $G$ be a split semisimple algebraic group over $\\mathbb{F}_q$. Let $S$ be a non-empty finite set of points of $X$. We are interested in the number of $G$ cuspidal automorphic representations whose local behaviors in $S$ are prescribed. In this article, we consider those cuspidal automorphic representations whose local component at each $v\\in S$ contains a fixed irreducible Deligne-Lusztig induced representation of a hyperspecial group. 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