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We improve upon the local bound in the depth aspect for sup-norms of newforms on 
$D^\times$, where 
$D$ is an indefinite quaternion division algebra over 
${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for 
$L(1/2, f \times \theta _\chi )$, where 
$f$ is a (varying) newform on 
$D^\times$ of level 
$p^n$, and 
$\theta _\chi$ is an (essentially fixed) automorphic form on 
$\textrm {GL}_2$ obtained as the theta lift of a Hecke character 
$\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via 
$p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.
