what is the meaning of the word 'suppose' in this context?

It means the same as "assumed".

As Roger noted, suppose = assume, which is strong enough that it can be equated with a modal meaning in the range of 'probably/likely/should'. It doesn't mean all of those all the time.

While 'probably/likely/should' express the opinion of the speaker, 'be supposed to' connotes that this is coming from some other source, usually a higher authority.

Examples:

You should do your homework.

[My opinion that it's a good thing to do your homework]

You're supposed to do your homework.

[Me, expressing that some higher authority, maybe a parent or teacher, believes you should be doing your homework]

"I know " she said, "you are supposed to be a writer [I hear on good source that you are a writer]

I suppose you are [you probably are].