Varying activity and the burst properties of FRB 20240114A probed with GMRT down to 300 MHz

Repeating Fast Radio Bursts (FRBs) can exhibit a wide range of burst repetition rates, from none to hundreds of bursts per hour. Here, we report the detection and characteristics of 60 bursts from the recently discovered FRB 20240114A, observed with the upgraded Giant Metrewave Radio Telescope (uGMRT) in the frequency ranges 300-500MHz and 550-750 MHz. The majority of the bursts show narrow emission bandwidth with $\Delta\nu/\nu \sim$ 10\%. All of the bursts we detect are faint ($<$10 Jy ms) and thus probe the lower end of the energy distribution. We determine the rate function for FRB 20240114A at 400 MHz, and downward drift rates at 400 and 650 MHz, and discuss our measurements in the context of the repeating FRB population. We observe sudden variations in the burst activity of FRB 20240114A over time. From our data as well as the publicly available information on other observations of FRB 20240114A so far, there is an indication that FRB 20240114A potentially exhibits chromaticity in its burst activity. While the burst properties of FRB 20240114A are similar to other repeating FRBs, the frequency-dependent activity, if established, could provide crucial clues to the origin of repeating FRBs. We also place the most stringent 5$\sigma$ upper limits of 600 $\mu$Jy and 89 $\mu$Jy on any persistent radio source (PRS) associated with FRB 20240114A at 400 MHz and 650 MHz, respectively, and compare these with the luminosity of the known PRSs associated with FRB121102A and FRB190520B.