Modes are basically asking you to shift your hand to the next note in the original key while retaining the intervals of the original key, and use that as your "new" key, so F# Dorian is E major starting on F#. (Personally, I found it confusing too that "F# Dorian Mode" means it's the Dorian mode in which you start on F#, rather than the Dorian mode of F#, which starts on G#.)
Notice how its four sharps are F#, G#, C#, and D#, which implies the Ionian mode (or original key) is E major.
It would help though if you showed us the actual piece.