Theta functions, fourth moments of eigenforms, and the sup-norm problem II

For an $L^2$-normalized holomorphic newform $f$ of weight $k$ on a hyperbolic surface of volume $V$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb{Q}$, we prove the sup-norm estimate $$\mathrm{\parallel }\mathrm{\Im }(\cdot {)}^{\frac{k}{2}}f{\mathrm{\parallel }}_{\mathrm{\infty }}{\ll }_{\epsilon }(kV{)}^{\frac{1}{4}+\epsilon }$$ with absolute implied constant. For a cuspidal Maaß newform $\phi$ of eigenvalue $\lambda$ on such a surface, we prove that $$\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}{\ll }_{\lambda ,\epsilon }{V}^{\frac{1}{4}+\epsilon }\mathrm{.}$$ We establish analogous estimates in the setting of definite quaternion algebras.