E string sharp just on first fret

Hey y'all,

I started playing around June with a Lanikai LU-21 with Aquila strings. So far it's been a great instrument! But in the last week or so I've noticed that my E string is sharp on the first fret. It tunes perfectly to E, but press on the first fret and my Snark tuner shows it about 5 marks over F. It's audibly off even to my tone-deaf ears.

Does this mean it's time for a new string? A ten year old uke player I know suggested I try restringing it in a slightly different position. Could there be some other problem I'm not aware of?

Thanks!

It may be the winding on the peg- this happens sometimes on stringed instruments- the winding there already alters the pitch slightly-
it's usually a static occurrence you'll notice tuning, but I guess may have an affect on a string under pressure?
perhaps your finger moves it on other winding, and raises the pitch:rulez:

?

Only the E string? Does it get better higher up the neck?
Have you changed anything recently or did it just happen (resp. have you simply not noticed earlier)?

Could be a whole bunch of reasons, e.g. too high action at the nut (should be valid for all strings though), too hard fretting (just press the string down until it rings clear, don't cram it into the fretboard), ...

What about the second and third frets. Is it sharp there, too? If it is, it could be the height of the nut slot.