An engineering manager's job is: take long-term ownership for the performance of the team. That might include aligning with leadership, marketing your team's work internally, hiring, performance management, team bonding, planning, retros, oncall coverage etc. etc, although sometimes you'll have a PM/tech lead/HR contact who handle some of these.
Every now and then, your bottleneck really is just writing more code (more common in smaller companies). In that case, jump in.