Parliament now includes lawmakers from seven political parties, spanning the full political spectrum, and traditional postwar coalitions on the left or right no longer have a majority. With the S.P.D.’s decision to enter the government, the AfD will now become the strongest opposition force in Parliament, a largely symbolic but important position that is viewed as providing a check on those in power.

The past weeks have seen the S.P.D.’s popularity plunge even further, with the party losing as much as another five percentage points in some surveys and dropping behind the AfD in popularity in the eastern states. In addition, infighting over leadership and ministerial posts further frustrated supporters.

S.P.D. leaders were quick on Sunday to highlight the need to reach out to the one-third of members who voted against the coalition, while also regaining the trust of lost voters.

“Above all, we must reach out to those who believe the voices telling them that ‘You have been left behind,’ and show them that we are there for them,” said Dietmar Woidke, the Social Democratic governor of the eastern state of Brandenburg.

There has been much talk about the lack of a minister from eastern Germany among the cabinet nominees from Ms. Merkel’s conservative party, which bled support to the far-right. Although the Social Democrats have not yet named their candidates for ministerial positions, Mr. Woidke emphasized the importance of including someone from the east, while conceding that one minister alone would not suffice to solve the party’s problem of weakening support there.

“It would be largely symbolic,” Mr. Woidke said. “But politics is all about symbolism.”

After the election, Ms. Merkel first tried to form a coalition government with the free-market Free Democrats and the Greens, but efforts failed after four weeks of discussions.

The Christian Democrats took up new negotiations with the Social Democrats, and a coalition agreement emerged in early February. The 179-page document details the main issues to be addressed, such as spending and which party will name ministers to take cabinet posts in the government.