İsim sanı, zan, tahmin.
It is my impression that: Bana öyle geliyor ki.
I got the impression that:
Öyle sanıyorum ki.
under the impression (that): sanarak, zanniyle.
to be under the impression that: sanmak, zannetnek, tahmin etmek.
I was under the impression that: Bana öyle geliyordu ki/öyle zannediyordum ki.
I asked him for a job under the impression that he was the head of the firm, but he wasn't.