If all zeros of a weakly holomorphic modular form in the fundamental domain lie on the lower boundary, then so do the zeros of its Serre derivative.
Katok,Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL
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The Serre Derivatives and Zeros of Modular Forms
If all zeros of a weakly holomorphic modular form in the fundamental domain lie on the lower boundary, then so do the zeros of its Serre derivative.