Odd electrical system problem?

So long story short, my car can run perfectly fine when it’s on. When I turn it off, it has power for me to use my keyless door lock. Within five to ten minutes though, the battery cuts out and it loses power; resets trip odometer, radio, and won’t start. If I give it a few hours though, it will start but it hesitates for a second after the first crank. Then it runs perfectly fine again, until I turn it off.

I’ve already put a brand new battery in it, and it didn’t help. Does anyone have the foggiest idea of what could be happening?

submitted by /u/JeffHwinger
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