Politeness of a follow-up E-Mail regarding administrative task I have to get a certain document filled in by a professor in regards to my major/curriculum.
Initially, I reached out 3 weeks ago, and she instantly replied that she'll be happy to do so, and asked for some other information she needed. I replied with the information instantly, but didn't hear back, unfortunately.
10 days ago, I therefore sent a follow-up in the following form:


  Dear Professor X,
  
  I was wondering if you've had the time to look at  I sent already?
  Thanks!
  
  Signature


Since I still haven't received any reply, I was thinking of sending another follow-up of the following form:


  Dear Professor X,
  
  I hope you've had a good weekend.
  
  I was wondering if you had received my previous E-Mail?
  
  Thanks!
  
  Signature


Would this be polite enough? (For some reason it strikes me as a bit too straightforward, but on the other hand I don't really have any reason to add an apology either, I think.)

I actually encounter such situations regularly, but my follow up emails always look more or less like the above, sometimes with an apology for reaching out again, even if I think it makes the E-mail look much too submissive. So I'm basically wondering if it is appropriate to send follow-ups in such a format.
I just wouldn't know what else to say in them, since the whole point is just to get the target's attention again if they accidentally missed it, were too busy, or forgot about it.
Rather than just asking if they received the previous email, reiterate what it is you're asking about, why you're asking them specifically, and include a deadline. This saves the recipient from having to trawl back through three weeks' worth of email to find your original request. So, you might want to send something like

"Dear Professor X,

Is it possible for you to fill in this form that I need for [insert reason here]? I'm asking you because [insert reason here]. The [admin people/ university/ department] need me to send it back to them by [deadline]. I've attached the form to this email. 

Thanks in advance for your help!

AnotherStudent"