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For a primitive Hilbert modular form $f$ over $F$ of weight $k$, undercertain assumptions on image of $\bar{\rho}_{f,\lambda}$, we calculate theDirichlet density of primes $\mathfrak{p}$ for which the $\mathfrak{p}$-thFourier 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 equaland 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 innertwists of $f$ associated with $K$. This work can be thought of as ageneralization of~\cite{KSW08} to primitive Hilbert modular forms.

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