In this talk I will present some recent results obtained in collaboration with Joshua Lansky. This is about a theory of newforms for ${\rm SL}_2(F)$ where $F$ is a non-Archimedean local field whose residue characteristic is odd. These are analogous to results of Casselman for ${\rm GL}_2(F)$ and Jacquet, Piatetski-Shapiro and Shalika for ${\rm GL}_n(F)$.

To a representation $\pi$ of ${\rm SL}_2(F)$ we attach an integer $c(\pi)$ that we call the conductor of $\pi$. The conductor of $\pi$ depends only on the $L$-packet $\Pi$ containing $\pi$. It is turns out to be equal to the conductor of a minimal representation of ${\rm GL}_2(F)$ determining the $L$-packet $\Pi$. We use the results for ${\rm SL}_2(F)$ to prove similar results for ${\rm U}(1,1)$, the quasi split unramified unitary group in two variables. A newform is a vector in $\pi$ which is essentially fixed by a congruence subgroup of level $c(\pi)$. For both the groups we show that our newforms are always test vectors for some standard Whittaker functionals and in doing so we give various explicit formulae for newforms and further it is used to formulate a multiplicity one theorem for these newforms.