Murmurations on Atkin–Lehner eigenspaces

Fix $k,r$ and $0\le m<r$, and fix primes $p_1<\dots<p_m$. Let $\mathcal N$ be the set of squarefree levels $N=p_1\dots p_r$ with $r$ prime factors, where the first $m$ primes are fixed and the remaining primes vary in increasing order. Let $\mathcal F$ be the family of weight-$k$ newforms with level in $\mathcal N$. For $\varepsilon=(\varepsilon_1,\dots,\varepsilon_r)\in\{\pm1\}^r$, let $\mathcal F^\varepsilon$ consist of forms having Atkin–Lehner sign $\varepsilon_i$ at $p_i$, and let $A^\varepsilon_{\mathcal F}(\ell,X;\beta)$ denote the corresponding normalized averages. Murmurations on Atkin–Lehner eigenspaces. As $\ell,X\to\infty$ with $\ell/X\to x$,

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

where $M^\varepsilon_{\mathcal F}$ is continuous on $[0,\infty)\times(1,\infty)$. Thus these averages have murmurations scale invariant in $\ell/N$. This is presented as an analogue of the broader Atkin–Lehner murmuration conjecture; the surrounding discussion gives numerical evidence and a partial theorem only for a Type I case.