New right brake light is not working

Can you get a picture of the socket so we can see the style that is used? The manual is not showing the style so we will be able to tell you how to replace it once we see it.

Also, there is normally a connector on the socket so you don't need to cut and splice the wires however, if there is not then we will need to do that.

Let's start with the picture and go from there. After reviewing the wiring diagram (attached) it appears if the bulb is not working and it is new that the connector would be the next obvious thing. The only other thing from there would be wiring because the rear tail lamps are tied together in with a splice so clearly this issue would be from that splice to the lamp. Also, if the brake lamp is the only one not working, it would not be the ground because the other lamps on this assembly use the same ground.