A note on Fourier coefficients of Hecke eigenforms in short intervals

In this article, we investigate large prime factors of Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms in short intervals. One of the new ingredients involves deriving an explicit version of Chebotarev density theorem in an interval of length \(\frac{x}{(\log x)^A}\) $\frac{x}{{(logx)}^A}$ for any \(A>0\) $A>0$ , modifying an earlier work of Balog and Ono. Furthermore, we need to strengthen a work of Rouse-Thorner to derive a lower bound for the largest prime factor of Fourier coefficients in an interval of length \(x^{1/2 + \epsilon }\) $x^{1/2+ϵ}$ for any \(\epsilon >0\) $ϵ>0$ .