Ramanujan's conjecture for half-integral weight modular forms

Let $f(z) = \sum^{\infty}_{n=1} a_f(n) n^{\frac{k-1}{2}} q^n$ be a half-integral weight modular form of weight $k=\ell+\frac{1}{2}$ on $\Gamma_0(4N)$, where $k \in \mathbb{N}$, $k \geq 2$, and $q=e^{2\pi i z}$. Ramanujan conjecture. For every $\epsilon>0$,

$$a_f(n)=O\bigl(n^{\epsilon}\bigr).$$

This is the expected Ramanujan-type bound for Fourier coefficients of half-integral weight modular forms, motivated here by the Lindelöf hypothesis for the associated quadratic twists. The statement is used in the paper as an input for proving sign changes of coefficients at almost-prime indices; the source does not establish it in the stated generality.