In studying automourphic representation, I want to know whether my understanding is on the right way.

Let $\pi$ be a irreducible cuspidal representation of $GL_2(A_F)$.

Then $\pi_v$, the local component of $\pi$, is the unique subquotient of the induced representation of $\rho(\chi|\cdot|^s, \chi|\cdot|^{-s})$ where $\chi$ is unitary character and $0\le s \le \frac{1}{2}$. Is it right?

Then the work of Henry Kim and Freydoon Shahidi in $[$ Functorial products for $GL_2 \times GL_3$ and the symmetric cube for $GL_2$ $]$ is to ensuring $s<\frac{5}{34}$?

Since I am just beginner in this area, any comment will be appreciated!