Puzzle of the Week #319 - Power Sum Palaver

w + x + y + z = 2

w^2 + x^2 + y^2 + z^2 = 2

w^3 + x^3 + y^3 + z^3 = -4

w^4 + x^4 + y^4 + z^4 = -6

What is the value of: w^7 + x^7 + y^7 + z^7 ?