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Donald J. Trump trails Senator Ted Cruz in Iowa but has a commanding double-digit lead in New Hampshire with less than a month to go before voting begins in the first contests in the Republican presidential nominating fight, according to new Fox News surveys.

Mr. Cruz received 27 percent of support in Iowa, while Mr. Trump, who has sharpened his attacks on the senator from Texas in recent weeks, got 23 percent. Senator Marco Rubio of Florida is next, at 15 percent, followed by Ben Carson (9 percent); Jeb Bush (7 percent); Senator Rand Paul of Kentucky (5 percent); and Gov. Chris Christie of New Jersey (4 percent).

In New Hampshire, however, Mr. Trump received 33 percent support from likely voters in the Republican primary, which is more than double the amount of his closest rival, Mr. Rubio, at 15 percent. Mr. Cruz got 12 percent. And despite a sense that Mr. Christie is on the rise in the state, he was at 5 percent in the Fox News poll, in sixth place. Both Mr. Bush and Gov. John Kasich of Ohio are ahead of him, with Mr. Bush at 9 percent and Mr. Kasich at 7 percent.

The Republican polls were conducted Jan. 4-7; Iowa has a margin of sampling error of plus or minus 4.5 percentage points, and New Hampshire four percentage points.

Meanwhile, in the Democratic contest, Senator Bernie Sanders of Vermont has a 13-percentage point lead over Hillary Clinton, 50 percent to her 37 percent. Mr. Sanders’s lead has grown since a one-point lead in a similar poll in November. Martin O’Malley, the third candidate in the race, is at 3 percent.

That poll has a margin of sampling error of plus or minus five percentage points.