On interlacing of zeros of certain family of modular forms

Abstract

Let $k=12 m(k)+s \ge 12$ for $s\in \{0,4,6,8,10,14\}$, be an even integer and $f$ be a normalised modular form of weight $k$ with real Fourier coefficients, written as $$ f=E_k+\sum_{j=1}^{m(k)}a_jE_{k-12j}\Delta^j. $$ Under suitable conditions on $a_j$ (rectifying an earlier result of Getz), we show that all the zeros of $f$, in the standard fundamental domain for the action of ${\bf SL}(2,\mathbb Z)$ on the upper half plane, lies on the arc $A:= \left\{ e^{i \theta} : \frac{\pi}{2} \le \theta \le \frac{2\pi}{3} \right\}$. Further, extending a result of Nozaki, we show that for certain family $\{f_k\}_k$ of normalised modular forms, the zeros of $f_k$ and $f_{k+12}$ interlace on $A^\circ:= \left\{ e^{i \theta} : \frac{\pi}{2} < \theta < \frac{2\pi}{3} \right\}$.