One possibility is that she has to move there due to some sort of magical experiment gone wrong at Hogwarts. The professors might think that living in a Muggle - dominated area like Privet Drive could help reverse the effects. So she ends up there, trying to figure out how to get back to normal while also dealing with the strange Muggles around her.
Well, in fanfic, the people of Privet Drive could think of Harry in different ways. They might be scared of him if the fanfic takes a darker turn and Harry's powers are more menacing. Or they could be envious of all the attention he gets from the wizarding world, while they're stuck in their ordinary, dull lives on Privet Drive.
There could be new characters introduced, like a nosy neighbor who notices Hermione's odd behavior. This neighbor might try to figure out what she's up to, leading to some comical or tense situations. Also, Hermione might interact with local Muggle kids, and through these interactions, she could learn more about Muggle culture and values.
One possibility could be Dumbledore, but in a different way than in the original. Maybe he has a more urgent reason this time. Another could be a powerful but reclusive wizard who has been observing Harry from afar and decides it's time to take matters into his own hands.
Well, one interesting plot could be Harry being taken in a more mysterious and magical way than in the original story. Maybe there are new magical creatures involved in his rescue from Privet Drive. Another could be that the person who takes him has a hidden agenda that slowly unfolds throughout the fanfic.
It depends on the fanfic. There are fanfics where the people of Privet Drive change their attitude towards Harry. Maybe they witness Harry doing something heroic that saves their street or someone they care about, and then they start to have positive thoughts about him. However, in other fanfics, they remain as negative as in the original story.
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