At 15, what could be the issues that you have to settle in New York without telling your mom? If you are a frequent traveller to New York and had left some things open-ended there, it is always safe to tell your parents the actual reason. They may sort out things for you. Until and unless you are a major and standing on your own feet, it is always better for a teenager to confide such things to mom or dad or an elder sibling or somebody whom you trust in the family.
Trying to solve issues with your limited knowledge and that too telling lies to your mom may drag you and your family into troubles. It's always better to confess everything to elders rather than trying to be over smart at your age.