UPDATE Since your team and manager are working remote (this is challenging for everyone) your self-starting leadership is even more important since everyone is busy running the business. As an SE you know how unsettling to teams having an unknown person with access to the codebase. (I have seen horror.)
Perhaps, Ask if you can be included in status emails and online meetings as an observer to gather a view of the challenges and directions in the development efforts. They need you or they wouldn't hire you.
Leadership is 'taking charge' to address a situation--this is an opportunity.
Would you be ready to provide a status and present your questions if your 'boss' invited you to meet today?
I assume you have been recruited / hired via some HR curated process as a Senior Developer in an organization of some size and complexity, and five weeks is a long time to be left unengaged and feeling like you have not been brought onboard.
Every manager is looking for leadership--taking charge--in their team members.
So, based on the 'limited' inputs you have received, org chart, product market, and technology stack, have you researched (googled) and compiled questions in each of these areas? e.g. Who uses our product. How we are doing in the market. Who in our organization owns our product? Are our development practices, security, and operational standards documented? Change control, updates and releases, how do we report errors and remediate code? etc.
In a lieu of a specifically stated task, define the current task as 'strawman' of assumptions: "I'm an SE. This is my view of our org, our product, and our stack."
"These are my questions in each area."
How to politely initiate the conversation?
- Create an agenda: status of your task (completed); questions of research/study; manager's expectations; next steps)
- Send an email - "Boss I perceive you have been busy. Can we meet for 30 minutes to go over some questions and determine my Next Step"; attach proposed agenda. Can you see their calendar?
- Send a calendar invite a 30 minute meeting with attached agenda. Keep it simple. Bottom Line Up Front (BLUF).
Bring your agenda and the questions printed out as a hand out at the meeting.
Persevere and succeed.