On interlacing of zeros of certain family of modular forms

Let $k=12m(k)+s\ge 12$ 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 $$f=E_k+\sum _{j=1}^{m(k)}{a}_j{E}_{k-12j}{\mathrm{\Delta }}^j\mathrm{.}$$ 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 ${\bf SL}(2,\mathbb Z)$ontheupperhalfplane,liesonthearc$A:= \left{ e^{i \theta} : \frac{\pi}{2} \le \theta \le \frac{2\pi}{3} \right}$. 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 $A^\circ:= \left{ e^{i \theta} : \frac{\pi}{2} < \theta < \frac{2\pi}{3} \right}$.