Ability to generate and sustain motion through continuous conversion of biochemical en-ergy into mechanical work characterizes living organisms. At the microscopic level, this autonomous motility is achieved through spontaneously generated biomimetic motions by active filaments like flagella and cilia. Recent in vitro experiments have reproduced similar behavior, where a remarkable cilia-like beating phe-nomenon has been observed in a motor-microtubule assembly. Here we study an effec-tive nonequilibrium model of an active filament in a low Reynolds number environment, incorporating hydrodynamic interactions (HI). Numerical simulations show that this model active filament exhibits biomimetic motions observed in the experiments. A stability analy-sis, disregarding HI, fails to reproduce oscillatory modes, highlighting the crucial role played by HI.