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I have an expression which is of the following form $$ M(\tau,\bar{\tau})=E_2(\tau)f_{k,\bar{k}+2}(\tau,\bar{\tau})+ E_2(\bar\tau)g_{k+2,\bar{k}}(\tau,\bar{\tau}). $$ Here, $E_2$ is the second Eisenstein series. $f_{k,\bar{k}+2}(\tau,\bar{\tau})$ and $g_{k+2,\bar{k}}(\tau,\bar{\tau})$ are non-holomorphic Maass forms of weights $(k,\bar{k}+2)$ and $(k+2,\bar{k})$ respectively. They transform under modular transformations as $$ f_{k,\bar{k}+2}(\gamma.\tau,\gamma.\bar{\tau}) = (c\tau+d)^{k}(c\bar{\tau}+d)^{\bar k+2} f_{k,\bar{k}+2}(\tau,\bar{\tau}), $$ and $$ g_{k+2,\bar{k}}(\gamma.\tau,\gamma.\bar{\tau}) = (c\tau+d)^{k+2}(c\bar{\tau}+d)^{\bar k} g_{k+2,\bar{k}}(\tau,\bar{\tau}). $$ We know that the second Eisenstein series is quasimodular with weight 2. So it is clear how $M(\tau,\bar{\tau})$ transforms under modular transformations.

My question is does the function $M(\tau,\bar{\tau})$ fall under a special class of non-holomorphic modular functions which show quasimodular features? Have these been considered by mathematicians and do these have a special name?

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