A $$p$$-arton model for modular cusp forms

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