Power of iota and cube root of unityHard

Roots of a Complex Equation

Let α, β be the roots of the equation x² – ax –b = 0 with Im(α) < Im(β). Let P_n = αⁿ – βⁿ. If ${P_3} = - 5\sqrt 7 \,i$ , ${P_4} = - 3\sqrt 7 \,i$ , ${P_5} = 11\sqrt 7 \,i$ and ${P_6} = 45\sqrt 7 \,i$ , then $\left| {{\alpha ^4} + {\beta ^4}} \right|$ is equal to ________.