But while Trump and his allies have embraced “Lock her up!” as a mantra for their movement, the seemingly spontaneous outburst in Washington on Sunday night highlighted an intensifying debate within the Democratic Party and among the broader Trump resistance: Whether to try to beat a norm-busting president on his terms, no matter how distasteful, or to insist on a more traditional, civil standard for American political discourse.

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One side of the debate — as represented Monday by figures such as Sen. Christopher A. Coons (D-Del.), a surrogate for the presidential campaign of former vice president Joe Biden — argues that such coarse chants are inappropriate for a democracy and that talk of jailing political opponents is the stuff of authoritarian regimes.

“I frankly think the office of the president deserves respect, even when the actions of our president at times don’t,” Coons told CNN the morning after the game.

Other Democrats argue that Trump, a president they see as morally unfit for office and worthy of impeachment and removal, does not deserve the esteem that would be reserved for any other occupant of the White House.

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“I actually applaud when he faces a mass protest by everyday citizens, who were not organized, who turned his own words against him,” said Neera Tanden, president of the Center for American Progress, a liberal think tank. “Trump is a street fighter, and if you’re not ready to fight in the street, you should get out of the way.”

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Jennifer Granholm, the former Michigan governor, also welcomed Sunday night’s eruption. “I confess I felt a little guilty pleasure in the chant, but then I thought of kids in cages and my guilt morphed into rage,” Granholm said via Twitter. “Give me the key; I’d love to lock him up.”

The debate reflects Democrats’ dilemma on how to channel the volcanic energy gathering on the left. Is it best to energize the party base by deploying Trump’s own weapons against him, or should Democrats seek to show that their party occupies the higher ground?

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Calls to the Democratic presidential campaigns suggest that the candidates are not inclined to embrace chants like the one Sunday night, though several chose not to address the issue.

Sen. Elizabeth Warren (D-Mass.) has confronted “Lock him up” chants, and her response was to discourage her crowds from repeating the slogan. Biden’s campaign didn’t report any known instances of the chant at his events, and spokesman Andrew Bates said if they did occur, the former vice president would say, “Don’t chant, vote.”

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A spokesman for Sen. Bernie Sanders’s campaign declined to comment. So did S.Y. Lee, a spokesperson for businessman Andrew Yang, who noted that Yang doesn’t mention Trump that much at his events. Spokesmen for South Bend, Ind., Mayor Pete Buttigieg, Sen. Kamala D. Harris (D-Calif.) and Sen. Cory Booker (D-N.J.) didn’t respond to questions.

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Trump’s “Lock her up” chant entered the political consciousness when it became a feature of his 2016 campaign rallies, an early example coming in February of that year when roughly 11,000 supporters in Florida repeated the chant. It was a way of suggesting that Clinton was more than just a political opponent — that she was guilty of criminal acts, despite a lack of evidence to that effect.

Repetitive chanting proved an energizing technique at Trump’s rallies; other slogans included “Build the wall” and “Romney sucks” — a barb directed at former GOP presidential nominee Mitt Romney.

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By the time of the 2016 Republican National Convention in Ohio, “Lock her up” had become an unofficial party motto, with speakers leading the GOP delegates in yelling the slogan from the floor.

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Trump still refers to Clinton at his rallies from time to time, standing back as the crowd recites the chant. His audiences have embraced other controversial phrases, at one point breaking into calls of “Send her back” in reference to U.S. Rep. Ilhan Omar (D-Minn.), a liberal Somali American who has attracted Trump’s ire.

The debate among Democrats over “Lock him up” reflects a concern in some parts of the party that Trump has changed the nature of politics and that it would be foolish to play by rules that no longer apply.

“Democrats usually come with white papers and statistics to talk about why we should reduce CO2 levels. We can’t do that anymore,” said Kelly Dietrich, chief executive of the National Democratic Training Committee, a group that teaches would-be candidates how to run for office. “Democrats do need to use more emotion.”

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Still, he added, “The ‘Lock him up’ is funny, but if I were running and I wanted a crowd chanting something, I would want them chanting ‘Fix my health care.’ ”

This tension between a boiling fury at the president and a desire to operate on a higher level has already begun playing out on the Democratic campaign trail.

Warren, who has attracted some of the biggest crowds in the presidential campaign, tends to steer her supporters toward policy-oriented chants like “two cents,” a shorthand for her proposed 2 percent annual tax on fortunes over $50 million.

But her supporters have also turned their attention to the president. At a town hall event several weeks ago in Carson City, Nev., Warren’s crowd began yelling “Lock him up!” after a pro-Trump protester interrupted her speech.

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Warren insisted that the chants stop. “No, no!” she called, waving her hand to quiet her fans.

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Then she sought to turn the protester into a symbol of her own success as a candidate. “I understand Donald Trump, and his supporters are getting really nervous,” Warren said. The crowd stopped chanting and began applauding.

Afterward, Warren told reporters that she tried to tamp down the chants because she did not agree with the sentiment. “We have an ongoing impeachment proceeding right now in Washington that is a serious constitutional undertaking, and we need to treat it with seriousness,” Warren said. “This is a hard time in our country.”

But Democratic audiences sometimes do not want to hear talk of coming together with Republicans, in part because they believe that the GOP has adopted a scorched-earth approach. At one rally in Detroit, Booker’s high-minded rhetoric about love and unity was interrupted by a foul-mouthed supporter when the senator from New Jersey referred to Trump.

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“He is going to try to divide. Our call is to unite,” said Booker. “He is going to try to revive hatred in this nation —”

“F--- that!” the supporter yelled.

“— and we need a revival of civic grace,” Booker concluded.

Sanders has adopted a more nuanced attitude toward protesters, at times acknowledging them and incorporating their emotion into his speeches.

Over the summer in Santa Monica, Calif., as he talked about Medicare-for-all, he slammed the pharmaceutical industry and then turned his attention to the environment.

“Today, we say to the fossil fuel industry . . .” Sanders said, pausing for breath. A member of the crowd filled in the sentence, yelling: “ ‘F--- you!’ ”

“That is one way of saying it,” Sanders replied. “Not exactly the way I was going to say it.”

At its root, the debate reflects an uncertainty among Democrats about how to grapple with a political phenomenon they find disorienting and traumatic. Watching Trump ignore so many rules and pay such a small price makes some wonder if it’s time for them to do the same. But the Democratic Party has long prided itself on seeking a better society, and engaging in coarse rhetoric hardly seems a reflection of that.

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The dispute may not be settled until there is a Democratic nominee, and perhaps not even then. For now, some on the left have begun taking their passion to Trump rallies, eager to confront their adversaries.

At a recent Trump rally in Minneapolis, a group of protesters wearing dark clothing and with their faces covered crowded around a handful of police officers, repeatedly chanting, “Whose streets? Our streets!” And then: “Kill a cop; save a life!”

Occasionally, Trump supporters walked through the crowd, only to have protesters try to steal their red “Make America Great Again” hats.

The stolen Trump gear went into in a pile. It was set ablaze in the middle of the street.