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Matthew Burns (20) tweeted a photo of a gun to the prime minister following the Christchurch mosque attacks, adding "you're next".
He also pleaded guilty to sending similar threats to London Mayor Sadiq Khan.
The BBC reports Burns was sentenced to 100 hours' community service at Newry Magistrates' Court.
District judge Eamonn King said Burns had posted the offensive comments online to make himself feel important.
"I can understand that if there are issues in real life, that you don't get a buzz in real life, you hope to get it on social media, and what happens… you create a monster," he said.
"When you put offensive comments online you are met with a blizzard of responses and engagement. That increases your feeling of self-worth. You feel more important.
"Someone is listening to you and it increases your self-esteem, so you then say something even more outrageous and that escalates and escalates."