New Mexico Statutes Chapter 1. Elections § 1-12-7. Conduct of election;  persons not permitted to vote;  persons permitted to vote upon choosing to affiliate with a party

A. A person shall not vote in a primary, general or statewide special election unless the person is a voter of the county in which the person offers to vote.  A valid original certificate of registration in the county register is prima facie evidence of being a voter in the precinct.

B. A person at a primary election shall not be permitted to vote for the candidate of any party other than the party designated on the person's current certificate of registration;  provided that a person who has declined to designate a political party affiliation on the person's certificate of registration shall be permitted to choose to affiliate with a major political party in a primary election by requesting a major political party's primary election ballot and shall be permitted to vote for the candidates on that party's ballot.