Let $W$ be the normalized Whittaker function associated to a new vector in an irreducible generic representation $\pi$ of $G=GL_2(k)$, where $k$ is a $p$-adic field. Let $c$ be the conductor of $\pi$, meaning that $c$ is the smallest integer with $W(gk)=W(g)$ for all $k\in K_1(c)$ where $K_1(c)$ is the subgroup of $K=GL_2({\mathfrak o})$ with bottom row congruent to $(0,1)$ modulo $\mathfrak p^c$ (with the convention that $K_1(0)=K$).

What is the support of $W$ restricted to $K$?

Clearly, the support contains $K_0(c)$, the subgroup of upper triangular matrices modulo $\mathfrak p^c$, since $W$ won't vanish on the center of $G$. I imagine it will depend on further information about $\pi$ beyond the conductor, but I'm having trouble finding (or proving) much of anything definitive.