Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of characteristic $\ell \neq p $. Is there any slight information about the decomposition of the translates of this character $$\chi(\cdot+1)=\sum_{\rho \text{ is a character}}\hspace{0.4cm} \Lambda_\rho^\chi \rho+ 1.$$ I am interested in knowing which constants $\Lambda_\rho ^\chi$ is non-zero. Thanks in advance

Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of characteristic $\ell \neq p $. Is there any slight information about the decomposition of the translates of this character $$\chi(\cdot+1)=\sum_{\rho \text{ is a character}}\hspace{0.4cm} \Lambda_\rho^\chi \rho+ 1.$$ I am interested in knowing which constants $\Lambda_\rho ^\chi$ are non-zero. Thanks in advance