One possibility is that Harry discovers a new aspect of Hogwarts. It could be a secret room or passage that holds the key to an ancient magic. This discovery might lead to a new adventure, where he has to solve puzzles and overcome challenges to unlock the full potential of this new discovery.
Perhaps it's a discovery related to his family. He might find out some long - lost secrets about his parents or ancestors that give him new understanding and strength. For example, he could discover a hidden family magic that was thought to be lost. This new - found knowledge could help him in his fight against the dark forces in the wizarding world.
Hermione could be a main character. She is always by Harry's side and her intelligence would be crucial in any discovery. She might be the one to figure out the clues or research the background of the discovery.
Well, perhaps the Yautja is on a hunt in its own world but gets caught in a magical portal that leads it to Hogwarts. The students are both scared and intrigued. Hermione starts researching about it, trying to find a way to send it back. Meanwhile, the Yautja sees the Death Eaters as worthy prey and starts stalking them. Harry has to convince the Yautja not to go on a full - out killing spree which might endanger innocent lives as well.
A possible plot could be that a Smilodon escapes from a magical menagerie in the Forbidden Forest. Harry and his friends discover it and try to return it safely without getting hurt or getting in trouble with the Hogwarts authorities.
Perhaps Harry could be a wizard - geisha hybrid. He might use his magic to enhance the traditional geisha arts, like creating magical illusions during his performances. He could be in a Hogwarts - like school but also trained in the ways of the geisha, with spells and potions being used in the context of geisha traditions.
Well, one possible plot could be that Orochimaru somehow finds his way into the Harry Potter universe. He might be interested in the unique magic there, like the Elder Wand or the Philosopher's Stone. He could try to use his ninja skills and devious ways to steal these powerful magical items, and Harry and his friends would have to stop him using their own magic.
In this 'harry potter math fanfiction', it might also be that there is a new subject in Hogwarts called 'Magical Math'. The teachers could be magical creatures who are experts in both magic and math. The students would have to learn things like how to use equations to create spells or use geometric shapes to build magical structures. And there could be a big competition where the students use their math skills to win a great prize, like a magical artifact that can enhance their magic power.
A possible plot could be Morgoth's return in a weakened form in the Forbidden Forest. His presence starts to corrupt the magical creatures there. Harry and his friends sense the disturbance and go to investigate. They soon realize it's a much greater threat than they thought and must find a way to either banish Morgoth back to his realm or destroy his essence completely.
Well, perhaps in this fanfiction, Harry Potter uses his magical skills in the context of auditing. He could use spells to detect financial irregularities, like a magical version of auditing software. For example, he might cast a spell that reveals hidden transactions in a company's books.
In this fanfiction, the brother being an idiot might mean he has no sense of self - preservation. He could be the type to run towards a dangerous magical creature instead of away from it, thinking it's just a friendly beast. Harry then has to constantly save him, and it becomes a running joke in the story about how clueless the brother is.
It could be portrayed as a dangerous quest. Harry might have to overcome many obstacles to reach Atlantis, like sea monsters or magical storms.