Exegesis:

agapa gar to ethnos hēmōn ‘for he loves our people,’ explains why he deserves what he asks. The present tense suggests that this is his permanent attitude. to ethnos hēmōn refers to the Jewish people in general, not to their nation as a political entity, or their race.

kai tēn sunagōgēn autos ōikodomēsen hēmin ‘and he built the synagogue for us himself.’ autos may mean ‘(he) himself,’ or ‘he’ (emphatic personal pronoun, cf. New English Bible), preferably the former. ōikodomēsen means here that he had made other people build the synagogue. The aorist tense, as contrasted with agapa (see above), implies a reference to one specific act proving his love for the Jewish people.

Translation:

He loves, see on 6.27.

Our nation, or, ‘our brothers/fellows’ (Laka, Toraja-Sa’dan, both using a term indicating persons of the same tribe, family or party as the speaker’s). The pronoun has inclusive force.

He, preferably, ‘he himself,’ indicating that the centurion acted on his own initiative and costs.

Built us our synagogue, or, ‘built for us the synagogue’ (the article signifying that the synagogue of Capernaum is meant), or simply, ‘built our synagogue,’ or, because he is not the direct agent, ‘caused our synagogue to be built.’ One may hesitate whether to use an exclusive or an inclusive pronoun. The former is often used (e.g. in Huautla Mazatec, Pohnpeian, Marathi, Malay), on the assumption that Jesus, being not native to Capernaum, was not a member of the group primarily associated with the synagogue there. The latter is used on the assumption that Jesus, having his permanent headquarters in Capernaum, could be taken as belonging to the local synagogue; or because the inclusive pronoun qualifies an object as known both to the speaker and the person addressed; or again, because the use of an exclusive pronoun may sound as though the speaker did not allow his collocutor the use of the synagogue. The inclusive pronoun is probably preferable. In some languages it is better to avoid the pronoun altogether, cf. ‘the synagogue (for the people) here’ (Balinese, Sundanese).

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.