Murmuration conjecture for elliptic curves ordered by naive height

For integers $A,B$, let $E_{A,B}$ be the elliptic curve

$$y^2=x^3+Ax+B,$$

with no prime $p$ satisfying both $p^4\mid A$ and $p^6\mid B$. Define its naive height by

$$H(E_{A,B})=\max(4|A|^3,27|B|^2),$$

and write $N(E)$ for its conductor, $\epsilon(E)$ for its root number, and $a_p(E)$ for its $p$th coefficient. For a set $S$, let $\mathbb E_{E\in S}[f(E)]=|S|^{-1}\sum_{E\in S}f(E)$. Let $J_1$ be the Bessel function of the first kind, let $v_p(m)$ be the $p$-adic valuation, and let $\ell_{p,\nu}$ and $\hat{\ell}_{p,\nu}$ be the local terms defined in the source.

For real numbers $0<C_1<C_2$, the murmurations conjecture.

$$\begin{aligned} &\lim_{X\to\infty}\mathbb E_{\{E:H(E)\le X\}}\left[\frac{\log\left(N(E)\frac{C_1+C_2}{2}\right)}{N(E)}\sum_{\substack{p\in(C_1N(E),C_2N(E)]\p\ \operatorname{prime}}}\epsilon(E)a_p(E)\right]\\ &=\int_{C_1}^{C_2}2\pi\sqrt{u}\sum_{\substack{q\in\mathbb N\emph{squarefree}}}\sum_{m\in\mathbb N}\frac{\mu(\gcd(m,q))}{qm\phi\left(\frac{q}{\gcd(m,q)}\right)}J_1\left(4\pi\frac{\sqrt{u}m}{q}\right)\prod_{p\mid q}\hat{\ell}_{p,2v_p(m)}\prod_{p\mid m,\ p\nmid q}\ell_{p,{2v_p(m)}}\,du. \end{aligned}$$

This predicts the limiting correlation between the Fourier coefficients $a_p(E)$ and root numbers when elliptic curves are ordered by naive height. It extends the study of murmurations beyond conductor ordering and gives an explicit expression in terms of Bessel functions and local arithmetic factors; the status of the prediction is open.