Maeda's conjecture

Maeda's conjecture concerns a specific type of mathematical object called modular forms, particularly focused on a polynomial known as the characteristic polynomial associated with these forms. The conjecture proposes that, for a certain level and weight, this polynomial is always irreducible over the rational numbers, meaning it cannot be factored into simpler polynomials with rational coefficients. Additionally, it predicts that all the roots (or zeros) of this polynomial generate the same mathematical field, which is as large as theoretically possible. This suggests a highly uniform and symmetric structure underlying these modular forms, which has implications for number theory and related fields.