I suggest you find out through Avira what's the correct way to uninstall their product, it probably came with an uninstaller and simply dragging the .app to the trash doesn't uninstall (if that's what you did).
As a general rule in Mac apps, if the app installs modules outside its .app package (an AV normally does so), elsewhere in the system, in order to be uninstalled it will require a uninstall program, or directions to delete those modules.