"Do you speak Esperanto?"
Translation:Ĉu vi parolas Esperanton?
It is correcting me to "Esperanton" for the language as a direct object takes the accusative. I thought the word "esperante" means "hopefully" which could actually work in the sentence, so I can see how the logarithm could get confused if you did not capitalize Esperanto. Post script added later: Oh, I see you don't capitalize it in Esperanto. See amuzulo below.
esperi hope (verb)
I thought one who hopes is "esperulo" and someone who speaks Esperanto is "esperantisto" http://www.gutenberg.org/files/16967/16967-h/16967-h.htm
My other dictionary says that hopefully is "espereble", but I like your version better. http://esperanto-panorama.net/vortaro/eoen.htm
"esperante" seems as though you are saying "Esperanto-ly, or in an Esperanto way. I wonder why the adverb is used instead of using an adjective? I would have expected "esperanta" to mean the adjective Esperanto-ish, although "esperanta" would be confusing to Spanish speakers as that is their word for "hope"..