Ideally, you should be able to bring 3 initiatives (proposed hypothesis to explore/test) for your on-site meeting. If the role is somewhat vague/rotational, your homework prior to start is just that much harder.
Obviously you need to manage your hypothesis proposal and not jump the gun. But this is where having a set of deep questions comes into handy — it’ll help with demonstrating initiative and help to minimize your zero-to-60. Always come in prepped with some level of contextual understanding of the market space and several hypothesis in hand. Think “Lean Onboarding”. :)