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 $\delta_{g}\neq 0$. In this paper, we show that $\delta_{g}\neq 0$ under mild assumptions when all the Fourier coefficients of $g \in S_{k}(\Gamma_{0}(N), \chi)$ are real, without assuming that $g$ has CM. In particular, this provides a class of higher-weight examples for which $\delta_{g}\neq 0$.