Magistrates from Rome with former convictions (6)

I believe the answer is:

priors

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'magistrates' is the definition.
The definition and answer can be both people as well as being plural nouns.
Perhaps there's an association between them I don't understand?

I don't understand the rest of the clue.

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(Another definition for priors that I've seen is " Heads of houses of friars".)

I've seen this clue in The Sunday Times and The Times.

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