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I can't help with the cucumber beetles, but on the asparagus beetles, I've had a lot of luck hand picking the beetles and spraying the larvae with insecticidal soap. They were really bad last year. Caught them early this year and they haven't really been an issue.

My local garden store has some lady bugs and I thought about purchasing some, although there seems to be no shortage of lady bugs around here come fall time when they are all on my ceiling!

Also, I thought about growing asparagus but do not know the first thing about it. Is it an easy crop? Is it suitable for New England weather or does it need somewhere more hot/cold?

I'm in Wisconsin and it grows well here, so I would assume you'd be fine! You only plant once and the crows grow for years. The only issue is you have to wait a year or two before harvesting.