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.
Well, it could be a really strange plot. Maybe in this fanfiction, Fem Harry Potter is intrigued by Orochimaru's dark and mysterious magic. She might see his ability to transform and experiment with different forms of life as something unique. Since she has her own experiences with magic in the wizarding world, she could be interested in exploring Orochimaru's more unorthodox methods. It could also involve her trying to understand Orochimaru's motives, perhaps thinking that there's some good in his actions deep down, which is a very different perspective from what most in the Harry Potter universe would think.
Another plot could involve a sort of magical exchange. Orochimaru, being a master of some dark and strange jutsu in his own world, might offer to teach some of his knowledge to certain dark wizards in the Harry Potter world in exchange for something valuable. Harry Potter and his friends would have to prevent this knowledge transfer as it could tip the balance of power towards the dark side in their world. They might need to seek help from some of the more powerful and wise wizards like Dumbledore to outwit Orochimaru.
In this type of fanfiction, it might also be that Naruto joins Orochimaru to spy on him. The Hidden Leaf Village realizes they need to know Orochimaru's plans from the inside. Naruto, being the brave and somewhat reckless one, volunteers for the mission. He endures Orochimaru's experiments and teachings while secretly sending information back to the village. Eventually, when the time is right, he betrays Orochimaru and uses the knowledge he gained against him.
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.
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.
One possible plot could be that Harry and Ginny go to a secluded wizarding cottage by the sea. They spend their days exploring the nearby magical creatures and hidden coves. At night, they sit by the fireplace and share stories from their Hogwarts days, strengthening their bond even more.