Public Enemies

After killing Charles Floyd, FBI agent Melvin Purvis is promoted by J. Edgar Hoover to lead the hunt for bank robber John Dillinger. Purvis shares Hoover's belief in using scientific methods to battle crime, ranging from cataloging fingerprints to tapping telephone lines.
In between a series of bank robberies, Dillinger meets Billie Frechette at a restaurant and woos her by buying her a fur coat. Frechette falls for Dillinger even after he reveals his identity, and the two become inseparable.
Purvis leads a failed ambush at a hotel where he believes Dillinger is staying, and an agent is killed by Baby Face Nelson, who escapes with Tommy Caroll. Purvis requests that Hoover bring in professional lawmen who know how to catch criminals dead or alive, including Texan Charles Winstead.
Police arrest Dillinger and his gang in Tucson, Arizona, after a fire breaks out at the Hotel Congress, where they are staying. Dillinger is extradited to Indiana, where Sheriff Lillian Holley has him locked up in the Lake County Jail in Crown Point. Dillinger and other inmates use a fake gun to escape. Dillinger is unable to see Frechette, who is under tight surveillance. Dillinger learns that Frank Nitti's associates are unwilling to help because his crimes are motivating the FBI to prosecute interstate crime, which imperils Nitti's bookmaking racket, thus severing his connections with the Mafia.
Carroll goads Dillinger into robbing a bank in Sioux Falls with Baby Face Nelson. During their escape, both Dillinger and Carroll are shot, and they have to leave Carroll behind. The group retreats to the Little Bohemia Lodge in Manitowish Waters, Wisconsin and realize their haul is significantly less than Nelson said it would be. Dillinger hopes he can free the rest of his gang from prison, including Pierpont and Makley, but Red Hamilton convinces him this is unlikely.
Purvis and his men apprehend Carroll and torture him to learn the gang's location. Purvis organizes an ambush at Little Bohemia. Dillinger and Hamilton escape separately from the rest of the gang. Agents Winstead and Hurt pursue Dillinger and Hamilton through the woods, engaging in a gunfight in which Hamilton is fatally wounded. Trying to escape, Nelson, Shouse, and Van Meter hijack a Bureau car, killing Purvis' partner Carter Baum in the process. After a car chase, Purvis and his men kill Nelson and the rest of the gang. Hamilton dies that night.
Dillinger meets Frechette, telling her he plans to commit one more robbery that will pay enough for them to escape together. When Dillinger drops her off at a tavern he thinks is safe, she is arrested. Frechette is beaten during interrogation to learn Dillinger's whereabouts, which she does not reveal; Purvis and Winstead eventually arrive and intervene. Dillinger agrees to participate in a train robbery with Alvin Karpis and the Barker Gang, intending to flee the country the next day. He receives a note from Billie through her lawyer, Louis Piquett, telling him not to try to break her out of jail.
Through Zarkovich, Purvis enlists the help of madam and Dillinger acquaintance Anna Sage, threatening her with deportation if she does not cooperate. She agrees to set up Dillinger, who she believes will come to hide out with her. Dillinger and Sage see Manhattan Melodrama at the Biograph Theater. After the film, Purvis signals other agents upon seeing them leave. Dillinger spots the police but is shot before he can draw his gun. Winstead listens to Dillinger's last words. Purvis goes to inform Hoover of Dillinger's death.
Winstead tells Frechette, still incarcerated, that he thinks Dillinger's dying words were, "Tell Billie for me, 'Bye bye Blackbird'". Billie sheds a tear - 'Bye Bye Blackbird' was the song the houseband was playing when Billie and Dillinger first met each other and danced together in a dinner-club. The closing text reveals that Melvin Purvis quit the FBI in 1935 and died by his own hand in 1960, and that Billie lived out the rest of her life in Wisconsin following her release in 1936.

Question: What movie did they see?
Answer: Manhattan Melodrama; Manhattan Melodrama