Hello, Need your help

Find all function $\displaystyle f: \mathbb{R} \longrightarrow [-1,1]$ verifying $\displaystyle \forall x \in \mathbb{R} , f(2x)=2\left(f(x)\right)^2 -1$ , such that $\displaystyle f(0)=1$ and $\displaystyle \lim_{x\rightarrow 0}\frac{1-f(x)}{x^2} = a$ , $\displaystyle a \in \mathbb{R}^{+}$