Random power-residue symbol conjecture for CM elliptic curves

Let $E_t^d$ be the CM elliptic curve and let $w_d$ denote the order of the relevant unit group. For a prime $p$, write $\left(\frac{a_p(E_t^d)}{p}\right)_m$ for the $m$th power residue symbol of its $p$th Fourier coefficient. Random power-residue symbol conjecture. Fix $m\geq 1$ and let $\mathcal C$ be a Chebotarev set contained in $\mathcal C_m^1\cap\mathcal C_{\sqrt d}^{+}$. Then

$$\delta_m^n(E_t^d;\mathcal C)=\frac{\varphi(n)}{m}\cdot\frac{\delta_{m'}^{n'}(E_t^d;\mathcal C)}{\varphi(n')/m'}$$

where

$$m'=(m,w_d),\qquad n'=\frac{m'}{\left(\frac{m}{n},w_d\right)}.$$

This gives the paper's precise formulation of the claim that the power-residue symbol is random except for the constraints detected by the lower-level density with $m'=(m,w_d)$. The paper presents this as a conjecture after proving several special density computations; the general assertion is not resolved in the supplied text.