Student pilot flying DA-40 aircraft reported engine malfunction in flight and returned to departure airport.

Narrative

Startup took two attempts. We assumed it was due to the plane sitting idle for previous four days in the pile by the hangar. Available power check and run up were normal. On takeoff; RPM momentarily went into red and returned to normal as the governor caught up. On climb out; RPM fluctuated in excess of 150 RPM with stable load percentage and remained in yellow range at 75% power. Instructor Pilot (IP) communicated to ZZZ TRACON that we needed to return to the airport while Student Pilot (SP) continued control of the aircraft until final approach. After clearing the runway; engine ran rough at idle but with RPM 710 +/-50. Power below 10% caused a rough running engine during taxi. We returned to the ramp and shutdown appeared to take longer than normal. Aircraft was grounded. Confer with maintenance department about rotating aircraft onto the line so they don't sit for extended periods causing startup issues. This has happened with other aircraft I have flown without subsequent problems.