Other forms: senatorially
Anything senatorial relates to senators, who are members of a legislative body. If you are a senatorial candidate, guess what? You are running for the Senate.
If a politician is presidential, it seems like they could serve as the President. A similar word is senatorial, which refers to all aspects of senators and senates. A senatorial election is between two politicians competing for a seat in the U.S. Senate. A senatorial committee is full of senators. Since a senator is in a position of authority, this word sometimes refers to how impressive, serious, or authoritative someone seems.