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Why does this quasi-modular function have integral values?

Algebraicity is proven in appendix one of [1]. The function considered there is $$ \psi(\tau) = \frac{3E_4(\tau)}{2E_6(\tau)} (E_2(\tau) - \frac{3}{\pi \rm{Im} \tau}) = \frac{3}{2} s_2(\tau).$$ The …
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