If you got an after market battery, rather than an OEM one, this may well be your problem.
Sadly after market stuff is great for some things, but laptop batteries isn't one of them. They are often not actually the same as the originals, leading to lack of charging, holding less of a charge, not sitting correctly, not having the correct power to start the device while will run it (like your problem), and many others.
If you can, I would consider returning that battery and get an OEM one.