1. ἁμαρτή · hamartē — Beekes
ἁμαρτή [adv.] ‘at the same time, together’ (Il). 41Ὲ *sn-h,er-t-, GR> VAR τή in Aristarchus; elsewhere -ἢ, -f, but probably wrongly. *DER ἁμαρτήδην (sch. ᾧ 162, H.), which is also a probable reading of N 584 ὁμαρτήδην (Wackernagel 1916: 70). *ETYM Old instrumental (in *-eh, > -4, which also explains the accent) of a verbal adjective "ἅμαρτος ‘joined together, meeting’ (ἅμα and dpapioxw). Also from here probably … — [Beekes, s.v. ἁμαρτή, p. 130]
2. ἁμαρτή · hamartē — Frisk
ἁμαρτή (Aristarch, sonst unrichtig -τῇ geschrieben) "zugleich, gleichzeitig’ (Hom., Sol., E.). Davon ἁμαρτήδην (Schol. Φ 162, H., wahrscheinlich auch N584 für ὁμαρτήδην zu lesen, Wackernagel Unt. 70). — Erstarrter Instrumental eines Verbaladjektivs ἐἅμαρτος “zusammengefügt, -treffend’ (dgaοἰσκω), wovon andererseits das denominative ἁμαρτέω "ZUsammentreffen’ (poet. seit Il.). Bechtel Lex. s.v. Anders Pisanı Ist. … — [Frisk, s.v. ἁμαρτή, p. 117]
3. ἁμαρτῆ · hamartē — LSJ
together, at same time, at oncetogether, at same time, at once, Il. 5.656, Od. 22.81, Sol. 33.4.