Extremely Divisible Polynomial

Number Theory Level 5

Consider all degree 4 monic polynomials $f(x)$ with complex coefficients which satisfy the condition that for all positive integers $n$ , $f(x)$ divides $f(x^n)$ .

Over all such polynomials, what is the smallest possible positive integer value of $f(-3)$ ?

(As a followup, can you describe all polynomials such that $f(x) \mid f(x^n)$ for all positive integers $n$ ?)