Pilot reported a NMAC during base turn for landing with a helicopter also on base turn. The pilot completed a safe landing although with less spacing than intended.

Narrative

At approximately XA:15; I entered the left-hand landing pattern at ZZZ. As I approached the vicinity of the airport two aircraft announced their positions. One was departing and the other landing. Accordingly I announced that I would maneuver south of the airport and make a 45-degree entry to downwind. I verified that no aircraft were on downwind; and I entered downwind. During that time a helicopter announced they were entering a right-hand pattern. These opposed patterns are standard for fixed wing and helicopter aircraft at ZZZ. I continued making position reports and searched for the helicopter; but could not locate him. As I turned and announced base; the helicopter pilot announced that he had me in sight. I thanked him and continued; while searching. As I turned onto final approach I saw the helicopter a few hundred feet above me and to my right on approach. I continued and landed. Although we maintained visual separation and I was preceding the helicopter in the pattern; this was a closer conflict than I intended. What could I have done better? First; I could have made better use of the traffic depiction capabilities of my aircraft. Other aircraft were in the pattern; and my attention was in looking outside; merging into the pattern; and configuring my plane for landing. However; later I determined that I could have usefully increased the zoom of my traffic display to better observe all the traffic in the pattern. I applied see-and-avoid for the approach; but I now think that situational awareness would be increased while approaching the pattern by giving some attention to my traffic depiction display at a useful zoom. Second; I could have asked the other pilots more specifically their positions and whether they had my plane in sight. I could have confirmed that they were following me (or not) as I entered the empty downwind leg. This was especially true of the opposite-hand helicopter traffic and their intentions. Although brevity on frequency is taught and I attempt to be succinct in order to not block the busy communications frequency; this circumstance might have warranted a discussion. Third; I could have circled outside the pattern until I had visually identified the traffic (although I judged that visual identification of the helicopter from my initial position outside the pattern was not likely). Because the airport is [special airspace]; I try to promptly enter the pattern and land after ZZZ1 ATC instructs me to change to unicom frequency. I now think that busy and opposite hand patterns would warrant circling near the airport; outside the landing pattern before entering to land in order to positively identify traffic in the pattern.