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 $GL\_2(o/p)$.



Orbital integrals of pseudo coefficients of supercuspidal reps
In fact $\phi$ is not only a (multiple of a) pseudo coefficient, but is a (multiple of a) coefficient of $\pi$. See e.g. Carayol's article "Représentations cuspidales du groupe linéaire", Ann. ENS. 


Now to answer your main question, there is indeed a lot of such computations in e.g. the series of papers written by Bushnell and Henniart on explicit Jacquet-Langlands correspondence. For other references you may read the survey : 


Sally, Paul J., Jr.; Spice, Loren Character theory or reductive $p$-adic groups. Ottawa lectures on admissible representations of reductive $p$-adic groups, 103–111, Fields Inst. Monogr., 26, Amer. Math. Soc., Providence, RI, 2009. 


If youo are especially interested in supercuspidal representation of ${\rm GL}(2)$, you may read :


Kutzko, Phil; Pantoja, José Character formulas for supercuspidal representations of the groups ${\rm GL}\_2,\ {\rm SL}\_2$. Comm. Algebra 26 (1998), no. 6, 1679–1697. 


