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.