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Human Rights Watch said Mr. Putin has unleashed “the worst political crackdown in Russia’s post-Soviet history.”

Tensions between Washington and Moscow came to a head in December, when Mr. Putin signed legislation that bans Americans from adopting Russian orphans. The law was in retaliation to the Magnitsky Act, signed into law by Mr. Obama earlier that month, that denies U.S. visas to and freezes the assets of blacklisted Russian officials suspected of involvement in the tortures and killings of whistleblowers in Russia.

The Magnitsky law is named for Sergei Magnitsky, a Russian lawyer who claimed to have exposed a web of corruption involving Russian officials. He was arrested on charges of tax fraud and died in a Moscow prison in 2009. In a case that is without precedent in Russia, officials in Moscow will proceed Monday with the posthumous trial of Magnitsky.

Dmitry Gudkov, who represents A Just Russia party in the state Duma, was one of eight members of Russia’s lower house of parliament to vote against the bill that banned U.S. adoptions.

The Putin administration is very interested in an image makeover, he said. “The Russian government spent a lot of money on PR, but after the Pussy Riot case, after the passage of the [anti-adoption] law, I think Russian authorities demonstrated that they don’t share common values.”

Three members of the female punk rock band were jailed in August for a raucous anti-Putin performance inside a Moscow cathedral. One band member was released in October.

Mr. Khodorkovsky credits Congress with putting human rights on the agenda of the U.S.-Russian relationship.

“The activism of the U.S. Congress and its members’ devotion to see the Magnitsky Act implemented in the same spirit as the law has been enacted carries a very strong message to the executive branch of the government,” he said.

The Obama administration initially opposed the Magnitsky legislation.

The State Department is expected to publish the first Magnitsky list of blacklisted Russian officials by mid-April.

“We hope it will not be just empty words, but actual action,” said Mr. Kasyanov.

Mr. Gudkov said the Obama administration can promote “real democracy” in Russia by helping Mr. Putin in his much-touted fight against corruption and money laundering.

Mr. Gudkov isn’t convinced that Mr. Putin is serious about this fight, but said that if the U.S. is seen helping the Russian president, he will be forced to act against corrupt officials, and that would create a rift among the ruling elite.

Mr. Kasyanov, meanwhile, said he hopes the Obama administration has become wiser to the ways of the Putin administration. “We believe that the [Obama] administration is now more mature in terms of understanding what Russia is about [and] in this term will perform effectively,” he said.