Am I allowed to move with my father even though my mother has custody over me?

First of all, if your father agrees with your moving in with him, you will then need to make application to the court for a modification of the exisitng child custody order. 

Typically, a minor child cannot choose which parent they want to live with.  The court will look to many factors - home environment, income and education of the parents, the child's relationship with each parent, reason for wanting the change, etc. 

However, while a child cannot outright choose where they want to live, if they over the age of 12, a child's preference is given consideration and is one of the factors that the court considers (the older the child, the more weight it is given).