$\tilde{\xi}_1^p$, and $\tilde{\xi}_2$, and $b$ have the usual $\mathcal{A}_*$-coaction and the coaction on the remaining elements $\tilde{\tau}_2$, $\tilde{\xi}_1^p$, $\tilde{\xi}_2$, and $b$ are\

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     $\psi(b)=1\otimes b$
     $\psi(\tilde{\xi}_1^p)= 1\otimes \tilde{\xi}_1^p -\tau_0\otimes b + \bar{\xi}_1^p\otimes 1 $
     $\psi(\tilde{\xi}_2)= 1\otimes \tilde{\xi}_2+\bar{\xi}_1\otimes \tilde{\xi}_1^p +\tau_1\otimes b +  \bar{\xi}_2\otimes 1$
     $\psi(\tilde{\tau}_2)= 1\otimes \tilde{\tau}_2 +\bar{\tau}_1\otimes \tilde{\xi}_1^p + \bar{\tau}_0\otimes \tilde{\xi}_2 - \tau_1 \tau_0\otimes b  + \bar{\tau}_2\otimes 1  .$
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There is also an isomorphism $${H\mathbb{F}_p}_*(THH(j))\cong {H\mathbb{F}_p}_*(j)\otimes E(\sigma \tilde{\xi}_1^p,\sigma \tilde{\xi}_2)\otimes P(\sigma \tilde{\tau}_2)\otimes \Gamma(\sigma b)$$ of $\mathcal{A}_*$-comodules and ${H\mathbb{F}_p}_*(j)$-algebras. The $\mathcal{A}_*$-coaction is given by using the formula $$\psi(\sigma x)=(1\otimes \sigma)\circ \psi(x)$$ and the previously stated coactions.

Note that Angeltveit and Rognes use a tilde over a symbol, for example $\tilde{x}$ to signify that the element has a different coaction then the coaction on $x$ or $\bar{x}$. We now want to compute the input of the spectral sequence. First, we note that as described in Example \[exm J\], $S/p\wedge E_0^*\mathbb{J}$ is an $H\mathbb{F}_p$ algebra and hence $V(1)\wedge E_0^*\mathbb{J}$ is also an $H\mathbb{F}_p$ algebra. It is known more generally that $THH(R)$ is an $R$ algebra, so $V(1)\wedge THH(E_0^*\mathbb{J})$ is a $V(1)\wedge E_0^*\mathbb{J}$-algebra and in particular an $H\mathbb{F}_p$-module. We can therefore apply the following lemma, which can be found in Ausoni-Rognes [@MR2928844 