Let $\mathbf{F}_q$ 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 of alg.closed of characteristic $\ell \neq p $. 
Is there any slight information about the decomposition of $$\chi(\cdot+1)=\sum_{\rho \text{ is a character}}\hspace{0.4cm} \Lambda_\rho^\chi \rho+ 1.$$