What relationship did the early Christians have with officials of the Roman Empire?

In the beginning not good, the romans believed in many deities and in the christian/ jewish religion it was a sin to believe in any god other than their god. The romans allowed them to live their peacefully as king as the once in awhile made a public offering to jupiter, the deity who protects the emperor, however this went against their religion, punishment for disobeying roman law was often cruel and led to death