Power series expansions of modular forms and $p$-adic interpolation of the square roots of Rankin-Selberg special values

Let $f$ be a newform of even weight $2\kappa$ for $D^{\times }$, where $D$ is a possibly split indefinite quaternion algebra over $\mathbb{Q}$. Let $K$ be a quadratic imaginary field splitting $D$ and $p$ an odd prime split in $K$. We extend our theory of $p$-adic measures attached to the power series expansions of $f$ around the Galois orbit of the CM point corresponding to an embedding $K\hookrightarrow D$ to forms with any nebentypus and to $p$ dividing the level of $f$. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic $p$-level structure. Also, we restrict these $p$-adic measures to ${\mathbb{Z}}_p^{\times }$ and compute the corresponding Euler factor in the formula for the $p$-adic interpolation of the “square roots” of the Rankin-Selberg special values $L({\pi }_K\otimes {\xi }_r,\frac{1}{2})$ where ${\pi }_K$ is the base change to $K$ of the automorphic representation of ${\mathrm{G}\mathrm{L}}_2$ associated, up to Jacquet-Langland correspondence, to $f$ and ${\xi }_r$ is a compatible family of gr"ossencharacters of $K$ with infinite type $\xi_{r,\infty}(z)=(z/\bar z)^{\kappa+r}$.