I am look for some conjectural functorial transfer $X$ which
(A)for any $GL(1)$ automorphic representation $\pi$, we have $L(s, X\times \pi)$ is holomorphic and satisfies certain functional equation.
(B)$X$ is not yet known to be an automorphic representation.
At this moment, the only example I have for $X$ is $\Pi_1\times \tilde\Pi_2$, where $\Pi_1$ and $\Pi_2$ are two automorphic representation on $GL(m_1)$ and $GL(m_2)$ respectively, with $(m_1,m_2)\neq (2,2), (2,3) \text{ or }(3,2)$.
My question: is there anything else (maybe symmetric $x^{th}$ power lift of GL(2)?)?
