When should I follow up?

Hello! I applied for a position a few weeks ago, but it's closing date was a couple of days ago. It's for a Recruitment and Support position in a museum, my dream job setting. Its also very different than the standard retail jobs I've applied for before. It was also a job listed as "many applicants" in Indeed, but I don't really know what that means and I applied directly anyway.

I'm looking for some guidance. I want them to know that I'm proactive and very interested in the role, but I don't want to seem needy. How long does it normally take to hear back after a closing date, and how long do I wait to send a follow up email?

Thanks!

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