Code p0513 is an invalid skim key. (Sentry Key Imobilizer Module) Its the anti-theft system. If you have another key try that one. Your PCM must have the vin and the skim pin code programmed into it. If the skim doesn't see the right PIN in the PCM or a wrong or defective key is used to start the car, it will shut it down. These parameters can be re-programmed at the dealer. They have access to your SKIM PIN in their database. As an owner, you have a right to the code.
I had the same problem on my 2006 Chrysler 300C 3.5L sedan. I insert the key but there is no mechanical buzz noise (the steering wheel being unlocked) and the dash lights momentarily flash then disappear. When I turn the key, the engine turns over but shuts off after three seconds. Turn the key another five times (six in total) and the car goes into lockdown mode and has to be towed to the dealer for unlocking, replacement parts (key lock, SKIM module and two new FOB keys) and reprogramming. The first tell tale sign is there is no buzz noise when you insert the key, so if you don't hear it don't bother turning the key or you are only going to make it worse. If possible, try replacing the battery in the key (CR3032 in mine) or inserting and removing the key until you hear the buzz sound before turning the key to start. Generally speaking, it is more likely the microchip inside the key is malfunctioning or otherwise it's the SKIM module inside the steering column that doesn't work - not both. If after replacing the SKIM module the same issue arises, then it's the keys too (which unfortunately was my issue). It also doesn't work to leave one of the keys inserted and lock/unlock with the spare key because the connection will still drop out. It is possible to program the new keys yourself, but your old keys need to work (buzz sound when inserted). I'm lucky because James Frizelle's in Southport, Queensland, Australia gave me a very competitive price, so doing the keys won't cost me too much more. It's a good idea to have more than one spare key just in case this happens again. If you're wondering why Chrysler 300s have such high security, there was a big commotion on My Fox Detroit with Scott Lewis (www.youtube.com/watch?v=FHsL8GMGtR8) about Chryslers being so easy to steal that 12-year-olds could break in and start the car with just a screwdriver. Since news spread, Chrysler has made Sentry engine immobilizers standard on all modern models. Unfortunately, in this case the high-security is overkill and creates new problems that are very expensive to fix ($2300+).
2006 Chrysler 300C 3.5L sedan <--- there is no such thing. A 300C would be a 5.7 Liter Hemi.
More then likely you have the problem I have, that is read wrong. Does your ABS/BAS amber light show up on the dash? If so it is the speed sensor on one of the wheels. Most likely one of the rear wheels.
In Australia they DID release a 3.5-liter 300C but it was a rare V6 model. Not sure about the U.S. though. The issue ended-up being a short-circuit (caused by an incorrectly fitted front bumper crushing the wires) that fried the steering column lock and eventually the Powertrain Control Module. The dealer was able to reprogram the steering column to work with my original Chrysler keys. I ended-up installing an aftermarket remote starter so I don't have to bother with keys anymore. It starts after I push the lock button three times.