$T \overline{T}$-deformed modular forms

: Certain objects of conformal field theory, for example partition functions on the rectangle and the torus, and one-point functions on the torus, are either invariant or transform simply under the modular group, properties which should be preserved under the T T deformation. The formulation and proof of this statement in fact extents to more general functions such as T T deformed modular and Jacobi forms. We show that the deformation acts simply on their Mellin transform, multiplying it by a universal entire function. Finally we show that Maass forms on the torus are eigenfunctions of the T T deformation.