Let g∈Sk(Γ0(N))\documentclass[12pt]{minimal}
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\begin{document}$$g \in S_{k}(\Gamma _{0}(N))$$\end{document} be a normalized newform and f be a harmonic Maass form that is good for g. The holomorphic part of f is called a mock modular form and denoted by f+\documentclass[12pt]{minimal}
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\begin{document}$$f^{+}$$\end{document}. For odd prime p, Bringmann et al. (Trans Am Math Soc 364(5):2393–2410, 2012) obtained a p-adic modular form of level pN from f+\documentclass[12pt]{minimal}
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\begin{document}$$f^{+}$$\end{document} and a certain p-adic constant αg(f)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{g}(f)$$\end{document}. When g has complex multiplication by an imaginary quadratic field K and p is split in OK\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}_{K}$$\end{document}, it is known that αg(f)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{g}(f)$$\end{document} is zero. On the other hand, we do not know much about αg(f)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{g}(f)$$\end{document} for an inert prime p. In this paper, we prove that αg(f)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{g}(f)$$\end{document} is a p-adic unit when p is inert in OK\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}_{K}$$\end{document} and dimCSk(Γ0(N))=1\documentclass[12pt]{minimal}
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\begin{document}$$\dim _{\mathbb {C}}S_{k}(\Gamma _{0}(N))=1$$\end{document}.
