Traces of Hecke operators on Drinfeld modular forms for ${\mathrm{G}\mathrm{L}}_2({\mathbb{F}}_q[T])$

In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case $A={\mathbb{F}}_q[T]$. We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree at most 2 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level ${\mathrm{\Gamma }}_0(\mathfrak{p})$ into oldforms and newforms, as conjectured by Bandini-Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than $p$.