On the signs of Fourier coefficients of Hilbert cusp forms

We prove that given any $ϵ>0$ and a primitive adelic Hilbert cusp form $f$ of weight $k=(k_1,k_2,\mathrm{.}\mathrm{.}\mathrm{.},k_n)\in (2\mathbb{Z}{)}^n$ and full level, there exists an integral ideal $\mathfrak{m}$ with $N(\mathfrak{m}) \ll_{\epsilon} Q_{f}^{9/20+ \epsilon} $suchthatthe$\mathfrak{m}$-th Fourier coefficient of $C_f(\mathfrak{m})$ of $f$ is negative. Here $n$ is the degree of the associated number field, $N(\mathfrak{m})$ is the norm of integral ideal $\mathfrak{m}$ and $Q_f$ is the analytic conductor of $f$. In the case of arbitrary weights, we show that there is an integral ideal $\mathfrak{m}$ with $N(\mathfrak{m}){\ll }_{ϵ}{Q}_f^{1\mathrm{/}2+ϵ}$ such that $C_f(\mathfrak{m})<0$. We also prove that when $k=(k_1,k_2,\mathrm{.}\mathrm{.}\mathrm{.},k_n)\in (2\mathbb{Z}{)}^n$, asymptotically half of the Fourier coefficients are positive while half are negative.