The given statement is an example of a phrase, specifically, a participial phrase. The element that is necessary to complete the sentence is an independent clause. A participial phrase consists of words with the -ing or -en form plus its modifiers. This kind of phrase always functions as an adjective. The phrase "Beginning in the fall of 1957" is describing a certain noun, which is found in the independent clause.
A I think. I don't muchly remember but I remember someone in the movie back in 6th grade that someone pretended they had grieved for Caesars death but actually wanted his power.