  We propose to associate to a modular form (an infinite number of) complex
valued functions on the $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$. We
elaborate on the correspondence and study its consequence in terms of the
Mellin transforms and the $L$-functions related to the forms. Further we
discuss the case of products of Dirichlet $L$-functions and their Mellin duals,
which are convolution products of $\vartheta$-series. The latter are
intriguingly similar to non-holomorphic Maass forms of weight zero as suggested
by their Fourier coefficients.
