Gum pain and inflammation around crown?

Hi:I suggest to go back to your dentist to evaluate the tooth for bite, food impaction and any remaining cement.Best,Dr. Maddahi

The two most likely scenarios that would cause the symptoms you are reporting involve the margin of the crown (where the crown and tooth structure meet.) 
1: Margin overhang or open margin - if the crown does not fit intimately and flush with the tooth then gingival issues like inflammation can occur. 
2: In dentistry, there is a principle called biologic width. There needs to be a certain distance between the crown seated and the bone level. If the crown is cemented directly on bone then the biologic width is violated and will cause chronic inflammation. 
It could be either of the two in your case (or something completely different) The best way is to look at an xray with your dentist and if there is overhang the crown needs to be removed and replaced with one that has a flush margin. If it is a biologic width issue then you need to have a surgery called crown lengthening to remove some bone , wait a couple months to heal, then get a new crown placed that has enough room away from your bone. 

There could be some cement that did not get removed at time of insertion that could be causing sensitive gums.  I would definitely go back to the dentist to make sure that area is rid of all cement.  
In the meantime, make sure to keep the area as clean as possible, and rinse with warm salt water -- gums love it, if you can stand it.  :)