Implicit neural representations (INR) has found successful applications across diverse domains. To employ INR in real-life, it is important to speed up training. In the field of INR for video applications, the state-of-the-art approach employs grid-type parametric encoding and successfully achieves a faster encoding speed in comparison to its predecessors. However, the grid usage, which does not consider the video's dynamic nature, leads to redundant use of trainable parameters. As a result, it has significantly lower parameter efficiency and higher bitrate compared to NeRV-style methods that do not use a parametric encoding. To address the problem, we propose Neural Video representation with Temporally coherent Modulation (NVTM), a novel framework that can capture dynamic characteristics of video. By decomposing the spatio-temporal 3D video data into a set of 2D grids with flow information, NVTM enables learning video representation rapidly and uses parameter efficiently. Our framework enables to process temporally corresponding pixels at once, resulting in the fastest encoding speed for a reasonable video quality, especially when compared to the NeRV-style method, with a speed increase of over 3 times. Also, it remarks an average of 1.54dB/0.019 improvements in PSNR/LPIPS on UVG (Dynamic) (even with 10% fewer parameters) and an average of 1.84dB/0.013 improvements in PSNR/LPIPS on MCL-JCV (Dynamic), compared to previous grid-type works. By expanding this to compression tasks, we demonstrate comparable performance to video compression standards (H.264, HEVC) and recent INR approaches for video compression. Additionally, we perform extensive experiments demonstrating the superior performance of our algorithm across diverse tasks, encompassing super resolution, frame interpolation and video inpainting. Project page is https://sujiikim.github.io/NVTM/.