The phrase is pretty simple:
デスノート
落{お}としちまった
Which is obviously, "I dropped my Death Note".
The verb (and the kanji) is for おとす or おちる, which is "to drop" or "to let fall", yet this is a verbal form that I couldn't find anywhere. It looks like a mix between the two verbs. I can't really tell what to make of it grammatically.