The CFKRS recipe prediction for elliptic-curve quadratic twists

Let $E$ be the elliptic curve of conductor $N$, with $N$ not a square, and let $\mathcal{O}_{E,k}(X;S;M)$ be the twisted shifted moment defined in the source. For $S=\{s_1,\dots,s_k\}$ and $J\subseteq\{1,\dots,k\}$, write $S_J=\{s_j:j\in J\}$ and $S_J^{-}=\{1-s_j:j\in J\}$. Define

$$H_{E,M}(S):=\frac{1}{2\zeta(2)}\sum_{Mn_1\cdots n_k=\square}\frac{\lambda_E(n_1)\cdots\lambda_E(n_k)a(MNn_1\cdots n_k)}{n_1^{s_1}\cdots n_k^{s_k}}.$$

CFKRS conjecture. If $|\operatorname{Re}(s)-\frac12|\ll1/\log X$ for all $s\in S$, then as $X\to\infty$,

$$\mathcal{O}_{E,k}(X;S;M)\sim\sum_{J\subseteq\{1,\dots,k\}}\prod_{j\in J}\mathcal{X}_E(s_j)X^{1+|J|-2\sum_{j\in J}s_j}\widetilde g\left(1+|J|-2\sum_{j\in J}s_j\right)H_{E,MN^{|J|}}(S\setminus S_J\cup S_J^{-}).$$

This is the modified CFKRS prediction retaining both parities of $|J|$; the source explicitly notes that the earlier parity-restricted prediction is false for $k=1$ because it omits a term.