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

For s is an element of {0, 4, 6, 8, 10,14}, let k = 12m(k) + s >= 12 be an even integer and f(k) be a normalised modular form of weight k with real Fourier coefficients, written as & para;& para;f(k) = E-k + & j=1(m(k)) a(j)((k)) Ek-12j Delta(j).& para;& para;Under suitable conditions on (rectifying an earlier result of Getz), we show that all the zeros of f(k), in the standard fundamental domain for the action of SL(2, Z) on the upper half plane, lie on the arc A := {e(i theta) : pi/2 <= theta <= 2 pi/3}. Further, we provide a criterion for a family of normalised modular forms {f(k)}(k) so that the zeros of f(k) and f(k+12) interlace on A degrees := {e(i theta) : pi/2 < theta < 2 pi/3}. (C) 2017 Elsevier Inc. All rights reserved.