This is the 11148th most frequent Russian word.
государственная
State; governmental.
Here, 'государственная' is used as an adjective modifying 'власть' (authority), indicating it pertains to the state or government.
Государственная власть принимает важные решения для страны.
The state authority makes important decisions for the country.
In this sentence, 'государственная' modifies 'библиотеке' (library), signifying it is owned or operated by the government.
Она работает в государственной библиотеке.
She works in a state library.
This usage reflects 'государственная' describing 'программа' (program), meaning it is organized or funded by the state authorities.