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 \(\frac{x}{(\log x)^A}\) for any \(A>0\), 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 \(x^{1/2 + \epsilon }\) for any \(\epsilon >0\).