I would like to have a question about the re-index convention. Let us consider a spectrum $E$ (I am mainly interested in motivic setting, however let's consider the simplicial case firstly, i.e. $E$ is a $S^1$-spectrum). The homotopy spectral sequence associated to the Postnikov tower of $E$ takes the form

$E^{p,q}_1 = \pi_{-(p+q)}(\Sigma_s^pH\pi_{p}(E)) \Rightarrow \pi_{-(p+q)}(E)$,

where the differential is given by $d_r: E_r^{p,q} \rightarrow E^{p-1,q+r}$. Already here I am not very sure about $d_r$. Some people use $d_r: E^{p,q} \rightarrow E^{p+1,q+r}$. For the above spectral sequence we denote by $\Sigma_s^{p} = S^1_s \wedge \cdots \wedge S^1_s$, the $p$-times suspension of the simplicial circle and $H\pi_p(E)$ is the Eilenberg-Maclane spectrum associated to $\pi_p(E)$.

Now I would like to rewrite this spectral sequence in $E_2$ term by setting $r \mapsto r+1$.My first question is: is the following spectral sequence correct? I am sorry about my stupid question, because I am very confused about the indexes.

$E^{p,q}_2 = H^{p-q}(X,\pi_p(E)) \Rightarrow \pi_{-(p+q)}(E)(X)$, which looks very similarly to the Atiyah-Hirzebruch spectral sequence. (correct me, if this spectral sequence doesn't make sense).

Secondly, how does the differential in this $E_2$-spectral sequence look like? does it still take the form $d_r: E^{p,q}_r \rightarrow E^{p \pm 1,q+r}$?

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