Final answer:
The government in New Mexico during the early 1600s was a Spanish colonial administration with Santa Fe as its capital, governed by royal officials appointed by the Spanish Crown.
Step-by-step explanation:
In the early 1600s, what is now known as New Mexico was under the control of the Spanish Empire, and its government was structured as part of the Viceroyalty of New Spain. Santa Fe, established in 1610, functioned as the capital of the Kingdom of New Mexico, which was an outpost of this larger viceregal entity.
The governance system was highly stratified and colonial in nature. Royal officials such as corregidores, captains general, or alcaldes mayors, administered regions known as corregimientos, captaincies general, or through the town councils known as cabildos.
These officials held political and judicial power but did not offer a democratic form of government as understood today. Instead, power was concentrated in the hands of these appointees of the Spanish Crown. Adelantados were conquerors or governors delegated with the administration of frontier or newly-conquered provinces under the Spanish rule.