If you are under the impression that something is the case, you believe that it is the case, usually when it is not actually the case. He had apparently been under the impression that a military coup was in progress.

What is a synonym for under the impression?

guess. be of the opinion that. gather. suspect. presume.

What does in the impression mean?

Thinking, assuming, or believing something, as in I was under the impression that they were coming today. This idiom often suggests that the idea or belief one had is mistaken. [ Mid-1800s]