Non-vanishing of the $p$-adic constant for mock modular forms associated to a newform with real Fourier coefficients

Let $F^{+}$ be a mock modular form associated to a normalized newform $g$. K. Bringmann et. al. obtained a $p$-adic modular form starting from $F^{+}$ by adding a suitable linear combination of Eichler integrals of $g(q)$ and $g(q^p)$. We denote the coefficients of the Eichler integrals of $g(q)$ and $g(q^p)$ by ${\gamma }_g$ and ${\delta }_g$. These constants are important in the $p$-adic theory of mock modular forms, but relatively little is known about them at present. For instance, K. Bringmann et. al. raised the question of whether ${\delta }_g$ vanishes when $g$ has CM by an imaginary quadratic field in which $p$ is inert. In previous work, the non-vanishing of ${\delta }_g$ has been proved mainly when $g$ is associated to an elliptic curve. In higher weight, only one example was known for which ${\delta }_g\mathrm{\ne }0$. In this paper, we show that ${\delta }_g\mathrm{\ne }0$ under mild assumptions when all the Fourier coefficients of $g\in S_k({\mathrm{\Gamma }}_0(N),\chi )$ are real, without assuming that $g$ has CM. In particular, this provides a class of higher-weight examples for which ${\delta }_g\mathrm{\ne }0$.