Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central characters $\chi$ and $\chi^\prime$ respectively.
Denote $v_p$ and $v_p^\prime$ the newvectors in $\pi_p$ and $\pi_p^\prime$ of levels ${\frak m}^n$ and ${\frak m}^{n^\prime}$ respectively.
My first question is: how do we characterize the vector $v_p\otimes v_p^\prime$ in the tensor representation $\pi_p\otimes\pi_p^\prime$ of ${\rm GL}_2(F)\times{\rm GL}_2(F)$?
An obvious necessary condition is that $$ \pi\left(\gamma,\gamma^\prime\right)v=\chi(a)\chi^\prime(a)v, \qquad \forall(\gamma,\gamma^\prime)\in K_n\times K_{n^\prime} $$ where $K_m$ denotes the usual subgroup of ${\rm GL}_2({\cal O}_F)$ of matrices reducing to the Borel modulo ${\frak m}^m$. But it is also a sufficient condition?
My second question is global: let $f$ and $g$ be two newforms and let $\pi_f$ and $\pi_g$ the corresponding automorphic representations of ${\rm GL}_2({\Bbb A}_{\Bbb Q})$. How do we characterize $f\otimes g$ inside $\pi_f\otimes\pi_g$? Can we check that locally (thus reducing to question one)?