Because $T_m T_n = \sum_{d|(m,n)} T_{mn/d^2}$ Möbius inversion (apparently) gives

$T_{mn} = \sum_{d|(m,n)} \mu(d)T_{m/d}T_{n/d}$

I have not bothered to work this out (it has been well over a decade since I messed around with modular forms at all) but am cribbing from a homework assignment PDF. But see also here.

It looks like Carol Hamer's article here also has some relevance to your question.

Because $T_m T_n = \sum_{d|(m,n)} T_{mn/d^2}$ Möbius inversion (apparently) gives

$T_{mn} = \sum_{d|(m,n)} \mu(d)T_{m/d}T_{n/d}$

I have not bothered to work this out (it has been well over a decade since I messed around with modular forms at all) but am cribbing from a homework assignment PDF. But see also here.

It looks like Carol Hamer's article here also has some relevance to your question.