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Three weeks before New Hampshire’s presidential primary, Senator Bernie Sanders of Vermont has jumped out to a 27 percentage point lead over Hillary Clinton in a poll of likely Democratic voters there released by CNN/WMUR on Tuesday evening.

Mr. Sanders’s lead, 60 percent to 33 percent, is stunning by New Hampshire standards, given that most polls have shown a tight race there and that Mrs. Clinton won the state’s primary in 2008. But Mr. Sanders, a well-known figure in New Hampshire after a 30-year political career in neighboring Vermont, has been enjoying a burst of political momentum in recent weeks, and he appears to be consolidating support among New Hampshire voters who viewed him favorably but had yet to commit to his candidacy.

Among the harshest findings for Mrs. Clinton, 55 percent of likely Democratic voters said she was the “least honest” of the party’s presidential candidates, compared with 33 percent who said that in September and 46 percent in December. Only 2 percent of those surveyed described Mr. Sanders as the least honest in the Democratic field.

And one-third of those surveyed said that Mrs. Clinton had the “personal characteristics and qualities” that a president should have, compared with 58 percent who described Mr. Sanders that way.

Some Clinton allies privately expressed surprise over the margin of Mr. Sanders’s lead, and even some advisers to Mr. Sanders said they did not believe he was as far ahead as the new poll showed. No Democrat has received more than 45 percent of the primary vote in New Hampshire since 1972, aside from sitting presidents or vice presidents.

Mrs. Clinton, who held double-digit leads over Mr. Sanders in New Hampshire though midsummer, continues to have many political advantages in the Feb. 9 primary, including support from its governor and Democratic senator. Her husband, former President Bill Clinton, is also deeply popular there, and on Wednesday he was scheduled to make his third campaign trip to New Hampshire this month. Yet Mrs. Clinton never took a repeat victory in New Hampshire for granted: No neighboring-state Democrat has lost the primary in the last 30 years, except when former Gov. Howard Dean of Vermont finished second to Senator John Kerry of Massachusetts in 2004.

The poll was mostly conducted before Sunday night’s Democratic debate, where Mrs. Clinton tried to blunt Mr. Sanders’s momentum with attacks on his new plan for broad-based tax increases and on his past votes against some gun control legislation. She received mixed reviews in that debate, while Mr. Sanders received strong praise in some quarters and skepticism from others about his call for broad-based tax increases.

In the previous CNN/WMUR poll, released in early December, Mr. Sanders had 50 percent of the vote to Mrs. Clinton’s 40 percent. In the new poll, the bulk of Mr. Sanders’s support came from younger voters and independents, but he also had the edge among registered Democrats as well. More than half of those surveyed said they had definitely decided who they would support, compared to 36 percent who expressed that view in December.

After the poll was released, Jeff Weaver, Mr. Sanders’s campaign manager, said that the Sanders operation was taking nothing for granted.

“This poll suggests that our campaign has real momentum and that the American people want to go beyond establishment politics and establishment economics,” Mr. Weaver said.

Shortly after the poll was released, Mrs. Clinton’s communication’s director, Jennifer Palmieri, released a statement arguing that the Republican National Committee and other conservative groups were “aiding” Mr. Sanders’s campaign because they were scared to face Mrs. Clinton in a general election.

“While Senator Sanders tries to make a case on electability based on meaningless polls, Republicans and their ‘super PACs’ have made clear the candidate they’re actually afraid to face,” Ms. Palmieri said. “Both Sanders and the Republicans know that Hillary is the candidate who can take them on and ensure the White House isn’t in Donald Trump or Ted Cruz’s hands.”

Mr. Sanders has also been gaining strength in Iowa, which holds its first-in-the-nation caucuses on Feb. 1, with polls showing a close race there between him and Mrs. Clinton. She had held a fairly steady lead in Iowa through the fall and early winter.

In Break From Big Rallies, Bernie Sanders Hits Iowa Plains by Bus In the two weeks left before the Iowa caucuses, Mr. Sanders is traversing the state the old-fashioned way, and bringing with him an arsenal of poll numbers.