# Xiaoyu Niu

## Further Computations on Maedas Conjecture

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 _d, where d is the dimension of S_k. Most recent computations via Sage have verified the conjecture for k 14000. Xiaoyus project will focus on computing via Sage for k > 14000 and/or considering bigger n than whats currently available. She hopes to either provide more evidence or to find results that disprove the conjecture. She may make theoretical attempts at the conjecture.

**Major:**Mathematics- Mentor: Sug Woo Shin, Mathematics