Murmurations for Atkin–Lehner operators

Let $(\mathcal N,\mathcal Q)$ be an arithmetically compatible sequence of $(N,Q)$'s of Type I, II or III as above, and fix a weight $k$. Let $\mathcal F$ be the family of newforms which lie in $S^\mathrm{new}_k(N)$ for some $N \in \mathcal N$. For the averages $A^{\mathcal Q}_{\mathcal F}(\ell,X;\beta)$ and their $\delta$-smoothed versions $\tilde A^{\mathcal Q,\delta}_{\mathcal F}(\ell,X;\beta)$, the following is conjectured. Murmurations for Atkin–Lehner operators. If $(\mathcal N,\mathcal Q)$ is of Type I, then, as $\ell,X\to\infty$ with $\ell/X\to x$,

$$A^{\mathcal Q}_{\mathcal F}(\ell,X;\beta)\to M^{\mathcal Q}_{\mathcal F}(x;\beta),$$

where $M^{\mathcal Q}_{\mathcal F}$ is continuous on $[0,\infty)\times(1,\infty)$. If $(\mathcal N,\mathcal Q)$ is of Type II or III, then for some $\delta<1$, as $\ell,X\to\infty$ with $\ell/X\to x$,

$$\tilde A^{\mathcal Q,\delta}_{\mathcal F}(\ell,X;\beta)\to\tilde M^{\mathcal Q,\delta}_{\mathcal F}(x;\beta),$$

where $\tilde M^{\mathcal Q,\delta}_{\mathcal F}$ is continuous on $[0,\infty)\times(1,\infty)$. In both cases the murmurations are scale invariant in $\ell/N$. The Type I case is expected to be approachable by trace-formula methods, whereas Types II and III require controlling an unbounded number of terms; the paper presents partial evidence, including a theorem in an initial range for a Type I family. Whether smoothing is necessary in the latter cases remains unclear.