Think I found it on my own.
New Mexico Constitution Article II, Section 6
No law shall abridge the right of the citizen to keep and bear arms for security and defense, for lawful hunting and recreational use and for other lawful purposes, but nothing herein shall be held to permit the carrying of concealed weapons. No municipality or county shall regulate, in any way, an incident of the right to keep and bear arms.
I'm still not sure about the concealed unloaded...