Differential modular forms attached to newforms mod p

In a previous paper we attached to classical complex newforms $f$ of weight $2$ certain $\delta_p$-modular forms $f^{\sharp}$ of order $2$ and weight $0$; the forms $f^{\sharp}$ can be viewed as "dual" to $f$ and played a key role in some of the applications of the theory. The aim of this paper is to provide a higher weight version of this "$\sharp$-duality," by attaching to classical newforms mod $p$, $\overline{f}$, of weight $\kappa$ between $3$ and $p$, $\delta_{\pi}$-modular forms $f^{\sharp}$ of order $2$ and weight $-\kappa'$, with $\kappa'$ between $1$ and $p-2$.