Perseus analyses of ἐπευφήμησαν:

ἐπευφημέω (assent with a shout of applause): aor ind act 3rd pl (homeric ionic)
ἐπευφημέω (assent with a shout of applause): aor ind act 3rd pl (homeric ionic)

LSJ entry

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ἐπευφημέω

ἐπευφημέω,

A. assent with a shout of applause, c. inf., πάντες ἐπευφήμησαν Ἀχαιοὶ αἰδεῖσθαί θʼ ἱερῆα Il. 1.22; cf. A.R. 4.295: abs., Ph. 2.28, Plu. Galb. 14.

II. c. acc. pers., Ἥρην ἐ. glorify, sing praises to her, Musae. 275.

3. c. dupl. acc., ἐμὰς τύχας παιᾶνʼ ἐπηυφήμησεν sang a paean over my fortunes, A. Fr. 350.4; folld. by dat., ἐ. παιᾶνα τἠμῇ συμφορᾷ Ἄρτεμιν sing the paean in praise of her over my fate, E. IA 1467.

4. c. acc. et dat., ἐ. νόστον τινί wish them a happy return, A.R. 1.556.

5. c. dupl. acc., call for the sake of good omen, Heraclit. All. 68:—Pass., θεῶν παῖδες οἱ ἥρωες -οῦνται Hierocl. in CA 3p.425M.