Let the $MTH$ is Madsen-Tillman spectrum (which is a close cousin of the more usual Thom spectrum $MH$) associated to tangential structure $H$.

For a computation involving no odd torsion, the Adams spectral sequence says that $$E_2^{s,t}=\text{Ext}_{\mathcal{A}}^{s,t}(H^*(MTH), \mathbb{Z}_2) \Rightarrow \pi_{t-s} MTH_2^\wedge. $$

The mod 2 cohomology of Thom spectrum $MSpin$ is $$H^*(MSpin) = \mathcal{A}\otimes_{\mathcal{A}(1)}\{\mathbb{Z}_2\oplus M\},$$ where $M$ is a graded $\mathcal{A}(1)$-module with the degree $i$ homogeneous part $M_i=0$ for $i<8$.

Here $\mathcal{A}$ stands for Steenrod algebra and $\mathcal{A}(1)$ stands for $\mathbb{F}_2$-algebra generated by $Sq^1$ and $Sq^2$. $\mathcal{A}(1)$ is a subalgebra of $\mathcal{A}$. Thus, for $t-s<8$, we can identify the $E_2$-page with $$\text{Ext}_{\mathcal{A}(1)}^{s,t}(H^{*+3}(MO(3)), \mathbb{Z}_2).$$

Question: For $i\geq 8$, how do we write down the $E_2$-page and the $H^*(MSpin) = \mathcal{A}\otimes_{\mathcal{A}(1)}\{\mathbb{Z}_2\oplus M\}$, when $M_i\neq 0$ in general for $i \geq 8$?


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