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For $0 \\leq n \\leq 9$ we calculate Okounkov bodies of graded linear series given by sections of multiples of line bundles $\\pi^\\ast \\mathcal{O}_{\\mathbb{P}^2}(d) \\otimes \\mathcal{O}_X(-m\\sum_{i=1}^n E_i)$ with respect to a flag consisting of a line on $\\mathbb{CP}^2$ and a point on the line in general position. 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