### Further Computations on Maedaâ€™s Conjecture

#### Xiaoyu Niu : Mathematics

Yoshitaka Maeda made the conjecture in 1997: Let m be an integer greater than 1 and let F be the characteristic polynomial of the Hecke operator T_m acting on the space S_k of cusp forms of weight k and level one, then the polynomial F is irreducible over the field of rational numbers; the Galois group of the splitting field of F is the full symmetric group â‚¢_d, where d is the dimension of S_k. Most recent computations via Sage have verified the conjecture for k â‰¤ 14000. Xiaoyuâ€™s project will focus on computing via Sage for k > 14000 and/or considering bigger n than whatâ€™s currently available. Â She hopes to either provide more evidence or to find results that disprove the conjecture. She may make theoretical attempts at the conjecture.