For a normalized newform \(g \in S_{k}(\Gamma _{0}(N))\) with complex multiplication by an imaginary quadratic field K, there is a mock modular form \(F^{+}\) corresponding to g. K. Bringmann et al. [5] 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 splits in \(\mathcal {O}_{K}\) and \(p \not \mid N\) , then \(\alpha _{g}=0\) (cf. [5]). On the other hand, the author [18] showed that \(\alpha _{g}\) is a p-adic unit for an inert prime p satisfying that \(p\not \mid 2N\) when \(\dim _{\mathbb {C}} S_{k}(\Gamma _{0}(N))=1\) . In this paper, under mild conditions, 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.
