/u/shlin28 answered this question in his Goliath write-up on the conditions of Rome after the "fall" of 476, here.

Relevant text:

With this out of the way, what happened to Rome the city [After 476]? The answer… not very much. We have very little evidence of what the new king of Italy, the former Roman general Odoacer, actually did. He seems to have treated the old Roman elite well and continued to mint coins in the name of the emperor, first Nepos, the ex-emperor hanging around Dalmatia, then Zeno, the emperor of the east. In 488, as part of an attempt to get rid of a troublesome general, Constantinople dispatched Theoderic, king of the Ostrogoths, to depose Odoacer and to secure Italy, both for himself and for the empire, for Theoderic would maintain the legally ambiguous position of Odoacer. Under Ostrogothic rule, we finally have some sources to work with concerning the Eternal City. Rome was still dominated by the senatorial aristocracy, a small class of rich and influential men with names that loudly proclaimed their long and illustrious heritage. Whilst administrative offices were generally held by ‘new men’ promoted by the Ostrogoths from outside this class, grander titles, such as the consulship (yes, this continued under the Ostrogoths and new consuls were still recognised by Constantinople!), were the senatorial families’ purview, a powerful reminder of the importance of the city of Rome even in a new world ruled by ‘barbarians’. Indeed, Rome continued to mint its own coins, of a type that did not explicitly proclaim loyalty to Theoderic (his coins always had the face and name of the emperor on them anyway); instead, the bronze coinage were inscribed with the words INVICTA ROMA (‘unconquered Rome’), as well as the imageries of the imperial eagle and the wolf that had fed Romulus and Remus, the founders of the city.