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After entering the 2020 race, Mr. O’Rourke offered early evidence of his fund-raising power: His campaign said he had received $6.1 million in the first 24 hours of his bid, more than any other Democratic presidential candidate had announced raising in his or her first day. Mr. Sanders had previously been atop the pack, at $5.9 million in 24 hours.

That huge first day made up a majority of Mr. O’Rourke’s fund-raising in the first quarter. Mr. Sanders went on to reach a total of $10 million raised within a week of announcing his campaign, his team said, a threshold that Mr. O’Rourke did not meet in 18 days.

“There are tremendous expectations for him given his 2018 performance, but he’s now swimming in a much more crowded pond than he was in 2018,” said Amy Walter, the national editor of the Cook Political Report. “And so I think that is going to be something to be watching for over the course of these next few months.”

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Mr. O’Rourke’s campaign said 98 percent of the contributions it received were below $200. The campaign did not say how much cash it currently has.

“Not only is this a sign of our grass-roots strength during the first two weeks of our campaign,” Mr. O’Rourke said of his fund-raising in a statement, “but it is a sign of what’s possible when you put your full trust in the people of this country.”