The $p$-adic constant for mock modular forms associated to CM forms II

For a normalized newform $g\in S_k({\mathrm{\Gamma }}_0(N))$ with complex multiplication by an imaginary quadratic field $K$, there is a mock modular form $F^{+}$ corresponding to $g$. K. Bringmann et al. modified $F^{+}$ in order to obtain a $p$-adic modular form by a certain $p$-adic constant ${\alpha }_g$. In addition, they showed that if $p$ is split in ${\mathcal{O}}_K$ and $p\nmid N$, then ${\alpha }_g=0$. On the other hand, the author showed that ${\alpha }_g$ is a $p$-adic unit for an inert prime $p$ satisfying that $p\nmid 2N$ when ${\dim }_{\mathbb{C}}{S}_k({\mathrm{\Gamma }}_0(N))=1$. In this paper, under mild condition, we determine the $p$-adic valuation of ${\alpha }_g$ for an inert prime $p$ and a general CM form $g$ of weight $2$ with rational Fourier coefficients.