Influence as Rome

[Beefe]

I recently started my Rome campaign and I noticed that my influence in the senat is dropping and I was wondering how do I gain influence?
Only with the actions in the panel like assassination etc.?, I dont want to pay tons of money for like 3 seats.
Does someone fully understands the system?, what are your impressions?

[Dresden]

I believe the gravitas for you faction's General affects your influence in the senate. Gravitas goes up the more he does (wins battles, takes specific actions in political screen, responds in certain ways to events).