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 $f^{\mathrm{♯}}$ of order $2$ and weight $0$; the forms $f^{\mathrm{♯}}$ 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 “$\mathrm{♯}$-duality,” by attaching to classical newforms mod $p$, $\bar{f}$, of weight $\kappa$ between $3$ and $p$, ${\delta }_{\pi }$-modular forms $f^{\mathrm{♯}}$ of order $2$ and weight $-{\kappa }^{\mathrm{\prime }}$, with ${\kappa }^{\mathrm{\prime }}$ between $1$ and $p-2$.