On generation of the coefficient field of a primitive Hilbert modular form by a single Fourier coefficient

For a primitive Hilbert modular form $f$ over $F$ of weight $k$, under certain assumptions on image of $\bar{\rho}_{f,\lambda}$, we calculate the Dirichlet density of primes $\mathfrak{p}$ for which the $\mathfrak{p}$-th Fourier coefficient $C(\mathfrak{p}, f)$ generates the coefficient field $E_f$. If $k=2$, then we show that the assumption on the image of $\bar{\rho}_{f,\lambda}$ is satisfied when the degrees of $E_f, F$ are equal and odd prime. We also compute the density of primes $\mathfrak{p}$ for which $C^*(\mathfrak{p}, f)$ generates $F_f$. Then, we provide some examples of $f$ to support our results. Finally, we calculate the density of primes $\mathfrak{p}$ for which $C(\mathfrak{p}, f) \in K$ for any field $K$ with $F_f \subseteq K \subseteq E_f$. This density is completely determined by the inner twists of $f$ associated with $K$. This work can be thought of as a generalization of~\cite{KSW08} to primitive Hilbert modular forms.