The Wisconsin state judge accused of impeding immigration agents at a Milwaukee courthouse last month pleaded not guilty on Thursday morning during a brief appearance in federal court.

Prosecutors have said that the judge, Hannah C. Dugan, violated federal law when she directed an undocumented defendant who was being sought by immigration agents through an alternate exit from her courtroom. Judge Dugan, who was indicted by a grand jury on Tuesday, is seeking the dismissal of the charges against her and has asserted that her actions were protected by judicial immunity.

A lawyer entered the plea on behalf of Judge Dugan, who was seated next to him in the federal courtroom on Thursday.

Protesters holding signs and chanting “Hands Off Judge Dugan” were gathered outside the downtown courthouse, and Milwaukee police officers blocked part of a street to accommodate the crowd.

“We know this isn’t about one judge,” Christine Neumann-Ortiz, the executive director of Voces de la Frontera, a Milwaukee-based immigrant rights group, told the crowd. “It’s an intimidation tactic.”

Inside the courthouse, the scene was far quieter. Judge Dugan arrived at the small, packed courtroom about 10 minutes before the hearing began. She shook hands with a federal prosecutor and smiled as she spoke quietly with her lawyers before a magistrate judge took the bench. Once the hearing began, she did not speak.

The judge overseeing the hearing, Stephen C. Dries, told Judge Dugan that she had a right to remain silent and a right to a lawyer. Judge Dries, who oversees pretrial matters, did not rule on Judge Dugan’s motion to dismiss the case or on a joint motion to keep evidence in the case out of public view before trial.

For further proceedings, the case has been assigned to Judge Lynn Adelman, who was appointed by former President Bill Clinton.

The Justice Department’s decision to arrest and charge a sitting state judge has drawn sharp criticism from many Democrats, lawyers and former judges, who have described the case as an attempt to intimidate the judiciary. Top Trump administration officials have defended the prosecution.

“It doesn’t matter what line of work you are in, if you break the law, we will follow the facts and we will prosecute you,” Attorney General Pam Bondi has said about the case.

At the hearing on Thursday, Judge Dries set a July trial date. Richard G. Frohling, the acting U.S. attorney in the Eastern District of Wisconsin, told Judge Dries that he anticipated that the trial might take a week, with the possibility of an extended period of jury selection. One of Judge Dugan’s lawyers, Steven Biskupic, said, “We are in talks among lawyers on both sides about jury selection issues.”

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When the hearing concluded, Judge Dugan and her lawyers left the courtroom before members of the public were allowed to leave their seats.

The prosecution of Judge Dugan has quickly become synonymous with the Trump administration’s broader immigration crackdown, and its warnings to local officials that they must not stand in the way of deportation efforts. Since President Trump returned to office, the Justice Department has sued state and local governments that limit cooperation with immigration agents and has announced investigations of some elected Democrats over their immigration policies.

Judge Dugan had been operating far from those debates, presiding over local cases in a sparsely decorated courtroom at the Milwaukee County Courthouse Complex. When several federal agents turned up outside her courtroom on April 18, they were not looking for her.

Those officers, who came from different federal agencies, had told courthouse personnel about their presence and their plans to arrest a man, Eduardo Flores-Ruiz, after he appeared in Judge Dugan’s courtroom in a domestic abuse case. They have said in court filings that Mr. Flores-Ruiz, who is from Mexico, had previously been removed from the United States and had re-entered the country without permission.

When Judge Dugan became aware of the federal agents, a charging document said, she became “visibly upset and had a confrontational, angry demeanor.” According to the criminal complaint, the judge confronted the agents and told them to talk to the chief judge of the courthouse. She then returned to her courtroom and, according to the charging document, directed Mr. Flores-Ruiz to leave the courtroom through a different exit than the door leading directly to the public hallway where agents were waiting.

“Despite having been advised of the administrative warrant for the arrest of Flores-Ruiz, Judge Dugan then escorted Flores-Ruiz and his counsel out of the courtroom through the ‘jury door,’ which leads to a nonpublic area of the courthouse,” according to the complaint, which was written by an F.B.I. agent.

Mr. Flores-Ruiz made it out of the courthouse, the charging document said, but an agent spotted him there and “a foot chase ensued,” the complaint said. A week later, Judge Dugan was arrested and charged with obstructing a proceeding of a federal agency, and concealing an individual to prevent his discovery and arrest.

On Wednesday, a day after grand jurors indicted her, Judge Dugan filed a motion seeking to have the case against her dismissed on grounds of judicial immunity. Her motion argued that “this is no ordinary criminal case, and Dugan is no ordinary criminal defendant,” and referred to the Tenth Amendment, which addresses federal and state powers.

The filing claimed that the prosecution was “virtually unprecedented and entirely unconstitutional.”

Earlier this month, more than 150 former state and federal judges signed a letter to Ms. Bondi calling the arrest of Judge Dugan an attempt to intimidate the judiciary.

Outside the courthouse on Thursday, a number of protesters said they saw broader forces at play in the judge’s arrest.

“They’ve crossed so many lines, and now they arrested one of our judges who was just trying to protect her courtroom and the rule of law,” Bonnie Elliott of Milwaukee said in reference to the Trump administration. “She was doing the right thing. She was doing what she was trained to do.”

Judge Dugan has been temporarily suspended from the bench by the Wisconsin Supreme Court while the case against her proceeds. Another state judge is hearing the cases she was handling in the wood-paneled courtroom where she used to preside. A sign on the courtroom door still invites lawyers to arrange meetings by Zoom for defendants who “feel unsafe coming to the courthouse.”