N.H. Rev. Stat. Ann. § 46:3

Each branch shall be the final judge of the election and qualification of its members, and, if any election is contested, shall have the same powers to ascertain the facts as the city convention have in regard to the election of mayor; and in case of a vacancy in their branch, by death, resignation or otherwise, the remaining members of such branch shall call a new election in the ward in which the vacancy shall have occurred, to fill the same.