Here's what my owners manual says about 'A and M' (archive)

A(E)=Economy program (display 2 shows 'E')
Once the car has been started, this program is selected for low-fuel-consumption motoring. The converter lockup clutch engages automatically in 3rd and 4th gears.

Economy mode is automatically selected when starting the engine.

M=Manual program
This program is for single-gear driving (3rd gear if 'D' is selected). The gear selected is also used for pulling away. For example, if the selector lever is in position 1 for driving up steep gradients, no undesirable upward gear shift will take place. The same applies to driving on icy roads in winter: with the selector lever in position '3', you can pull away smoothly and no gear shifts will occur.

If the indication 'TRANS PROGRAM' in the Check Control display appears, it means: Electronic shift control has failed.

If the failure appears and the vehicle is moving, the 4th gear will be selected.

The 3rd gear for better performance will be selected and held when the vehicle is stopped, the engine switched off and started again.

In this event avoid extreme engine loads and consult the nearest BMW dealer. "

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