The Constitution allows the Senate to punish an impeached President with prevention from holding an important office in the future, do they need 2/3rds of the Senate to do this or 2/3rds for a conviction but only a regular majority for assigning a punishment?
As noted in Panda's answer to the linked q, the relevant precedent is
The Constitution is silent on whether, after an official has already been impeached and removed from office, imposing the additional sanction of disqualification requires a supermajority vote. In the past, however, the Senate determined that a simple majority vote is sufficient for disqualification. Judge Archibald was disqualified by a vote of 39-35 after he was removed from office.
To be clear, such a simple majority vote may only take place after the Senate has already voted to convict an impeached official. Two-thirds of the Senate must first agree to remove someone from office before that official can be disqualified — a simple majority cannot, acting on its own, disqualify an official from holding future office.
It refers to Robert Wodrow Archbald 's impeachment in 1913 or so.