The Atari 2600 needs to go through an RF modulator before it will display any video or play any audio on your TV. You mentioned that you have an coaxial adapter. Some of those adapters actually do a good job separating the audio/video signals. Some, are just pass-throughs and aren't suited for real RF modulation.
If you have an old RF modulator box like from a NES or Coleco then that might help a little.
In my experiences, the Atari 2600 is very finicky with modern TV's. Despite modern TV's still having coaxial inputs. On CRT TV's, you could be very lenient on the NTSC specs and they still (mostly) worked. But many modern TV's will just show a blue screen if the NTSC signal is out of spec...even by a small margin.
My advice is to go to Goodwill and buy a $5 CRT TV. Older the better. One with knobs for the channels are best for Atari.
Next, get a proper RF modulator. Probably another $3 from Goodwill or eBay.
Finally, if you have the means and know-how, you can check the voltages inside the 2600 after you power it on. Oh, I assume you have the right power supply?
I had a Sega Master System that wouldn't power on and it turned out to be a bad 7805 voltage regulator.
Point is, try the Atari on an old TV if you can with a real RF modulator.
Oh, if you buy a CRT and decide you don't want it any more...please don't discard or recycle it. Take it back to Goodwill. People like me buy them all the time to play retro consoles on.