Let $(\pi,V)$ be a supercuspidal representation of $G=GL(n,F)$ for a non-archimedean local field $F$ and let $(\tilde{\pi},\tilde{V})$ denote it's contragradient. Let $\mathcal{S}(M(n, F))$ denote the space of Schwartz Bruhat functions on $M(n,F)$. Consider the $\operatorname{End}(V)$ valued distribution $T$ defined on $\phi\in\mathcal{S}(M(n,F))$ as:
$$ T(\phi)=\int_{G}\phi(x)\pi(x) dx$$
Show that $T$ is infact a $V\otimes\tilde{V}$ valued distribution, where $V\otimes\tilde{V}$ is identified naturally to a subspace of $\operatorname{End}(V)$.
I believe that somehow I have to use the fact that matrix coefficients of $\pi$ are compactly supported mod center. However I am not able to prove this. Any help is deeply appreciated.