The president also seemed to want to play network programmer on Sunday, urging Fox News to stand by hosts Jeanine Pirro and Tucker Carlson, both of whom are under fire for controversial comments. Fox has supported Carlson but suspended Pirro for suggesting that Rep. Ilhan Omar’s hijab was, by definition, anti-American.

All told, Trump’s barrage suggested that daylight exists between certain sectors of Fox News and Trump, who has showered Fox personalities with interviews and benefited from favorable commentary from its opinion hosts.

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The spat comes as Fox’s parent company is undergoing a generational change — one that produced another, perhaps more subtle sign of independence from the president.

On Tuesday, Fox Corp. began public trading as the new parent of Fox News, Fox Entertainment and Fox Sports; the company is the result of 21st Century Fox’s sale of its film and television assets to Disney Corp. The new company is headed by Lachlan Murdoch, the eldest son of Rupert Murdoch, who co-founded Fox News with Roger Ailes and remains a controlling shareholder in Fox.

Among Fox Corp.’s first acts in business: appointing former House speaker Paul D. Ryan to its board of directors. Ryan is, of course, an establishment conservative disliked by both Democrats and those who most strongly supported Trump. Although he advanced some of Trump’s agenda, particularly tax reform, as speaker, Ryan declined to defend Trump or campaign with him in the latter stages of the 2016 campaign.

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Trump’s tweets served as a reminder that he is known for turning on members of his inner circle. He plays staff members off one another and exploits differences among them to his advantage. This weekend, he attempted to do the same with specific personalities inside Fox News, and his tirade on Twitter reminded those inside the network just how fickle the president’s affections can be.

It also raised the question: Is the president changing his tune on Fox News?

Pirro and Trump speak regularly, and the two had a conversation before he tweeted, according to a person briefed on the call. The president’s relationship with Pirro dates back decades and is rooted in their common presence in a certain New York social circle. Trump has granted Pirro regular interviews, and he encourages his advisers to do the same.

His conversations with Pirro highlight the influence that certain Fox opinion hosts have over the president and he over them.

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On Sunday, Trump exhorted Fox to “keep fighting” for Pirro and Carlson.

Fox “must stay strong and fight back with vigor,” Trump wrote. “Stop working soooo hard on being politically correct, which will only bring you down, and continue to fight for our Country. The losers all want what you have, don’t give it to them. Be strong & prosper, be weak & die! Stay true. . . to the people that got you there.”

Soon after, he tagged news anchors Neville and Vittert in a tweet and asked whether they were “trained by CNN prior to their ratings collapse.”

“In any event, that’s where they should be working, along with their lowest rated anchor, Shepard Smith!” Trump added.

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Several Fox staffers could not figure out precisely what it was that had set him off and speculated that the possible release of a report by special counsel Robert S. Mueller III was probably on his mind, according to two of them. These people, like others, spoke on the condition of anonymity because they were not authorized to speak publicly.

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The segments in question were straight-ahead news reports.

Vittert reported on Sen. Kirsten Gillibrand (D-N.Y.) entering the presidential race and mentioned former vice president Joe Biden’s slip-up over the weekend that revealed he is essentially already running for president. Vittert also interviewed a worker from General Motors who was out of a job and asked him whether he and his fellow workers were sticking with Trump after the GM plant closure.

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Fox’s representatives point out that Trump has expressed some criticism of Fox News in the recent past. In December, for example, he complained in an interview with newscaster Harris Faulkner about a Fox News poll that showed him with a 46 percent approval rating. “Frankly, Fox has always given me a bad poll,” he said, though the 46 percent figure was somewhat higher than other polls at the time. “I don’t know why that is.”

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He also criticized Fox White House reporters John Roberts and Gillian Turner in January for their coverage of his negotiations with Congress over funding a wall on the border with Mexico: “Never thought I’d say this but I think @johnrobertsfox and @gillianHTurner @foxnews have even less understanding of the Wall negotiations than the folks at FAKE NEWS CNN & NBC! Look to final results! Don’t know how my poll numbers are so good, especially up 19% with Hispanics?”

A source familiar with the inner workings of Fox News said the president isn’t on the phone with producers or executives in charge of coverage. He calls opinion hosts with whom he has a preexisting relationship, this person said.

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Another Fox insider said that Fox News strives to operate on principles that might not always align with appealing to the channel’s core audience. Inherent in that calculation is that the channel should sometimes value principles over ratings.

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But Greta Van Susteren, a former Fox News anchor, said that the decisions about Fox News’s future would be driven by the bottom line. “Everyone thinks it’s politically driven. It’s not. It’s driven by money,” she said in an interview. “It’s a lot cheaper to keep a panel in the studio to talk about tweets than it is to send out journalists to look at floods in Nebraska. And if viewers are more interested in listening to a panel talk about politics, they’ll keep doing that instead of the floods. These are commercial operations. The decisions are made largely on a financial basis.”

A Fox News representative noted that Fox correspondent Mike Tobin has been in Nebraska covering the floods there.

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Jeff Paul, a Fox News correspondent, responded to the president’s tweet about Neville and Vittert by writing that he was working over the weekend with both anchors, “They’re both incredible journalists, constantly engaged and inquisitive. I’m proud to call them colleagues . . . as I am proud to work with @ShepNewsTeam each week.”

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The president has historically been less friendly to the channel’s news hosts. Bret Baier, Fox News’s chief political anchor, asked for an interview with Trump nearly 600 days before he was rewarded with an impromptu audience on Air Force One on the president’s trip back from South Korea.

In an interview in November, Baier said he was “of course” concerned that the president dictated too much of the news cycle. “I think that’s one of our problems in the news industry overall. President Trump is brilliant at that. He controls the media and he’s in the conversation and he’s on the front page every day.”