Battery charges, discharges to 70%, then laptop switches to sleep mode

Just replaced my old bloated battery with a fresh new battery I bought off iFixit.com. I got it installed, noticed it came charged at abut 70%, and just topped it off. No big deal. I decided to discharge it to 0% to give the battery a workout and to make sure it works properly. What it does is weird. The battery will discharge down to around 70%, then my MacBook will put itself into sleep mode as if the battery was fully discharged. I'm thinking, eh, probably need to warm up the battery a little more. I plug the MacBook back in, but now it won't charge the battery. The MacBook will run off the A/C power cord, but won't charge the battery. And it will continue like this until I completely shut down my Macbook and turn it back on again.

Battery is being detected by the system. Here is what Battery Health Monitor is reporting:

Did I get a defective battery?

Edit: Okay, it started charging again without restart about a half hour after that last test, but it's still weird behavior that it won't discharge below 70%.