I'm having trouble understanding what you have written.
You start with q, which I assume is a fixed prime number, and define zeta as a primitive qth root of unity in the complex numbers.
You then choose a polynomial F and suppose that \(F(\zeta^m)=F(\zeta)\) whenever \(v(m)\equiv 0\pmod{e}\).
Presumably this e is not the constant 2.718... that you used earlier! Is e a fixed integer here?
In explaining what v(n) means, you then refer to both e and f  what is f?
