Differential modular forms attached to newforms mod p

In a previous paper [7] we attached to classical complex newforms f   of weight 2 certain δpδp-modular forms f♯f♯ (in the sense of [4] and [6]) of order 2 and weight 0; the forms f♯f♯ can be viewed as “dual” to f and played a key role in some of the applications of the theory [8]. The aim of this paper is to provide a higher weight version of this “♯-duality,” by attaching to classical newforms mod p  , f¯, of weight κ between 3 and p  , δπδπ-modular forms f♯f♯ of order 2 and weight −κ′−κ′, with κ′κ′ between 1 and p−2p−2.