Traces of Hecke operators on Drinfeld modular forms for Fq[T]

In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case A = Fq[T]. We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree 1 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level Γ₀(T) into oldforms and newforms, as conjectured by Bandini-Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than p.