There is a certain voltage level required for the systems to run. Most modern vehicles start to shut down non critical systems if the voltage gets to low, like abs a/c and other non critical items to give you more "limp home time"
The fact that you can jump it and it runs suggests either your new battery is no good or you have a bad connection that the jumpers are somehow, helping to connect.
Once it starts will it stay running without the jumper cables on it.
You say the battery has 13.5, I am going to guess that is at rest, if so it means nothing . What is the voltage while cranking. Most vehicles will not fire if the cranking voltage falls below 10.5 or so.
What is the running voltage at the battery. If it is in the 14 range your alternator is most likely working but is not a conclusive indication.
Artificial Intelligence is no match for Natural Stupidity