On interlacing of the zeros of a certain family of modular forms

For s∈{0,4,6,8,10,14}, let k=12m(k)+s≥12 be an even integer and fk be a normalised modular form of weight k with real Fourier coefficients, written asfk=Ek+∑j=1m(k)aj(k)Ek−12jΔj. Under suitable conditions on aj(k) (rectifying an earlier result of Getz), we show that all the zeros of fk, in the standard fundamental domain for the action of SL(2,Z) on the upper half plane, lie on the arc A:={eiθ:π/2≤θ≤2π/3}. Further, we provide a criterion for a family of normalised modular forms {fk}k so that the zeros of fk and fk+12 interlace on A∘:={eiθ:π/2<θ<2π/3}.