Do not ask her out.
You sit next to her in class! Which means if she says no you've got a really awkward situation on your hands, and if she says yes, and you hit it off, well, then you've got a really awkward situation on your hands! The more I think about it, the less I like it.
That said, I do think she likes you. Wait until close to the end of the class term, then ask her out. In the meantime, be -very- friendly with her, use the intervening time as a chance to get to know her.
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