Calculating the sheaf $\mathcal I/\mathcal I^2$

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This should be an elementary question: say if I have a plane conic $C\subset \mathbb{P}^2$, then the ideal sheaf is $\mathcal{I} = \mathcal{O}_{\mathbb{P}^2}(-2)$. How do I calculate $\mathcal{N}=\mathcal{I}/\mathcal{I}^2$? Is there a general way to calculate such $\mathcal{N}$?