Can this expression be evaluated for odd $n>1$?

Question

I suppose $n$ is an odd natural number greater $1$ and e.g. squarefree.
Now I try to evaluate/simplify this expression where $\mu(m)$ is the Möbius function :
$$\prod_{d|n,\ d\equiv1\mod4}(-1)^{\frac{d-1}{4}\mu(\frac{n}{d})}\prod_{d|n,\ d\equiv 3\mod 4}(-1)^{\frac{d-3}{4}\mu(\frac{n}{d})}$$
I tried to extract $$(-1)^{\frac{\phi(n)}{4}}$$
in an obvious way but then the factor
$$\prod_{d|n,\ d\equiv1\mod4}(-1)^{\frac{1}{4}\mu(\frac{n}{d})}\prod_{d|n,\ d\equiv 3\mod 4}(-1)^{\frac{3}{4}\mu(\frac{n}{d})}$$
remains to be calculated which perhaps has values 1,-1,i,-i (?).