Kind of want to call and ask, but don't want to raise any suspicion haha. I might in a few days though.
Never inquire about the test results. A person who is truly drug-free would never do this because, being drug-free, its a non-issue. Doing so would only raise suspicious eyebrows (like youve said). You do want to be viewed as a "drug-free" person, dont you?

Since you have your start date secured, just simply show up for work and carry on with your duties with the "I dont use drugs, therefore my drug test is a non-issue" mindset.

In this case, no news means good news.