Suaviter is an adverb. The iter ending is the usual way that adverbs are formed from third declension adjectives.
Infinitives of active verbs usually end in are (first conjuation, e.g. cantare), ere (second conjugation, e.g. habere, and third conjugation, e.g. coquere), or ire (fourth conjugation, e.g. audire).
It's an adverb from a 3rd declension adjective.