N.H. Rev. Stat. Ann. § 592-A:3

A justice of a district or municipal court may render final judgment and sentence in any case where the punishment is a fine of $100 or of some specific sum less than $100, or imprisonment for a period of not more than 6 months, or is both such fine and imprisonment, if the accused pleads guilty or nolo contendere or files a writing by which he elects a trial by the justice and waives the right of appeal.