The folklore sup-norm conjecture for newforms with mildly ramified central character

Let $f$ be a newform on $\operatorname{GL}_2(\mathbb{A}_{\mathbb{Q}})$ with level $M$ and conductor $N$, and let $N_1$ be the smallest integer such that $N\mid N_1^2$. Assume that $M\mid N_1$. The folklore sup-norm conjecture. One has

$$\frac{\|f\|_\infty}{\|f\|_2} \ll_{\lambda/k,\varepsilon} N^{\varepsilon}.$$

This conjecture asserts the expected essentially optimal sup-norm bound when the central character is not too highly ramified; the paper explains that it would follow from the absence of global obstructions beyond local ones, together with the local conjecture stated below.