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.
It is two verbs. You just had the two wrong verbs in mind.
落｛お｝としちまった is the colloquial form of 落｛お｝としてしまった, which in turn is comprised of the following two verbs:
落｛お｝として (て form): 落とす (jisho.org)
しまった (past form): しまう (jisho.org)
By slurring the pronunciation of てし, you end up with ち, hence 落｛お｝としてしまった→ 落｛お｝としちまった
ちまう has its own dictionary entry here: ちまう (jisho.org)
「落としちまった」is a colloquial form of 「落としてしまいました」
