City Councilwoman Helen Gym said Tuesday that a New York-based charter chain lacks the qualifications to turn around Cooke Elementary School and that its proposed administrative costs are so high that the school would have 50 percent less to spend on classroom instruction. Gym, who analyzed the proposal of Great Oaks Foundation Inc., said that its budget at Cooke calls for spending four times the amount on administration as the Logan school pays now, and includes a $550,000 management fee to the foundation.