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How do modular forms of half-integral weight relate to the (quasi-modular) Eisenstein series?

The Eisenstein series $$ G_{2k} = \sum_{(m,n) \neq (0,0)} \frac{1}{(m + n\tau)^{2k}} $$ are modular forms (if $k>1$) of weight $2k$ and quasi-modular if $k=1$. It is clear that given modular forms ...