She has also emerged as a frequent foil for President Trump. She was among four female freshman lawmakers who Mr. Trump said in July should “go back” to the countries they came from.

In an interview with the F.B.I. at his home about a week after he called Ms. Omar’s office, Mr. Carlineo described himself as a patriot who loved Mr. Trump and hated “radical Muslims in our government,” the criminal complaint says.

He also told an F.B.I. agent that he believed Ms. Omar supported Islamic militant groups, such as Hamas and the Muslim Brotherhood, and that her election to Congress was illegitimate.

In the interview, the complaint says, he acknowledged calling Ms. Omar’s office and said that he knew why the F.B.I., which investigated the matter with the United States Capitol Police, was interested in talking to him.

Shown an email containing the text of the call, he initially denied that the words were his, but later “went on to say that he may have said something” similar, the complaint says. (He had left his name and contact information with Ms. Omar’s staff, the complaint says.)

The F.B.I. agent who signed the complaint said that Mr. Carlineo had threatened to kill Ms. Omar “while she was engaged in the performance of her official duties.”

About a week later, the authorities found a loaded .45-caliber handgun, three rifles, two shotguns and hundreds of rounds of ammunition at Mr. Carlineo’s home, prosecutors said. Because of a 1998 felony conviction, he was legally prohibited from owning a firearm.