Let $\pi$ be a supercuspidal representation of $G =GL_2(F)$ for a non-archimedean local field $F$, then there exists a maximal subgroup $K$ of $G$, which is compact modulo the center, and a representation $\rho$ of $K$ such that $\pi = Ind_K^G \rho$.

It is possibly to show that $tr\; \sigma( \phi) \neq 0 $ iff $\sigma \cong \pi$ for $\phi$ being equal to $tr(\rho)$ on $K$ and zero off $K$. This means $\phi$ is a constant multiple of a pseudo-matrix coeffient of $\pi$.

Now, one can compute that given an elliptic element $\gamma \in GL_2(F)$, i.e., the characteristic polynomial is irreducible, the corresponding elliptic orbital integral vanishes iff the conjugacy class of $\gamma$ doesn't meet $K$ and equals a constant multiple of $tr \rho(\gamma)$ with $\gamma$ conjugated inside $K$ otherwise.

There exists a classification/construction of those $\rho$'s respective $\pi$'s, see eg. *Bushnell-Henniart --- Local Langlands conjecture for GL(2).*

Question: Does there exists a reference for the explicit value of $tr \rho(\gamma)$ depending on the strata of $\rho$ and the characteristic polynomial of $\gamma$?

Remark: The depth-zero case is well documented in the representation theory of $GL_2(o/p)$.