Bounding $p$-adic characters and Jacquet-Langlands transfert I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta_\pi$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it is sufficient to bound it on $GL_2(\mathbf{Q}_p)$. 
An idea, for instance appearing in Kin, Shin and Templier [1], is to use the Sally-Shalika formula giving explicit calculations for the characters of $SL_2$, providing a bound for all supercuspidal representations :
$$|\Theta_\pi(\gamma_p)| \leqslant 1 + 2D(\gamma_p)^{-1/2} \ll 1$$
I would like to do the same for division quaternion algebras. By Jacquet-Langlands, we can tranfer to the $GL_2$ setting. By Labesse and Langlands and the bound above we can bound any $\Theta_{\tilde{\pi}}$ where $\tilde{\pi}$ is the restriction of $\pi$ to $SL_2$. My question follows: could we lift this bound obtained for restrictions to $SL_2$ to a bound on the characters for the whole representations on $GL_2$?
Perhaps my question only betray my deep ununderstanding of the relations between representations of $GL_2$ and those of $SL_2$. Anyway, every enlightening comment or answer will be warmly welcome.
Best regards

[1] Kim, Shin and Templier, Asymptotics and Local Constancy of Characters of $p$-adic Groups, 2015
 A: First some remarks on the Jacquet-Langlands correspondence. The image of the local Jacquet-Langlands transfer $\mathrm{JL}$ from $G$, the group of units in a non-split quaternion algebra, to $\mathrm{GL}_2(\mathbb{Q}_p)$ is the set of discrete series representations.
This set contains the supercuspidal representations as well as the Steinberg representation $St$ and all its twists by characters.
Facts:  1. The image $\mathrm{JL}(\mathbf{1}_G)$ of the trivial 1-dimensional representation $\mathbf{1}_G$ of $G$ is given by $St$.


*For any character $\chi:\mathbb{Q}_p^*\rightarrow \mathbb{C}^*$ and any irreducible smooth representation $\pi$ of $G$, 
$$\mathrm{JL}(\pi \otimes (\chi\circ \mathrm{Nrd})) = \mathrm{JL}(\pi)\otimes (\chi \circ \det).$$


Together, 1 and 2 give that $\mathrm{JL}(\pi)$ is supercuspidal, iff $\dim(\pi)>1$.
Regarding the relation between representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ and $\mathrm{SL}_2(\mathbb{Q}_p)$:
Given an irreducible smooth representation $\pi$ of $\mathrm{GL}_2(\mathbb{Q}_p)$, the restriction to $\mathrm{SL}_2(\mathbb{Q}_p)$ decomposes into a finite direct sum of irreducible smooth representations $\pi_i$ of $\mathrm{SL}_2(\mathbb{Q}_p)$
$$\pi|_{\mathrm{SL}_2(\mathbb{Q}_p)}\cong \bigoplus_{i=1}^n\pi_i,$$ 
each $\pi_i$ occurring with multiplicity one. The number $n$ is either 1,2 or 4. 
If $\pi$ is supercuspidal, then so are all the $\pi_i$. 
The restriction of the Steinberg representation is the Steinberg representation of $\mathrm{SL}_2(\mathbb{Q}_p)$. 
For more details I think Section 2.3 in 
http://msp.org/pjm/2007/231-1/pjm-v231-n1-p08-s.pdf 
and the references mentioned there should be helpful.
