Suppose we have two Lie groups $G$ and $H$ and a covering map $f:G\rightarrow H$ with a finite number of sheets which is a Lie group homomorphism.
I want to know whether $G$ is unimodular whenever $H$ is unimodular.
I'd like to know this because I am trying to prove that $SL_n(\mathbb C)$ is unimodular. I have managed to come up with a map
$$SL_n(\mathbb C)\times\mathbb C^*\xrightarrow \phi GL_n(\mathbb C),$$
given by $(A,z)\mapsto zA$. Then $\phi$ turns out to be a Lie group homomorphism which is also a covering map of $n$ sheets. I have also showed that $GL_n(\mathbb C)$ is unimodular by defining a measure which is an integral of a complex form(of type $(n^2,n^2)$) of $\mathbb C^{n^2}$. This sort of imitates an argument found in Serge Lang's book $SL_2(\mathbb R)$.
So if this argument using covering maps works, then $SL_n(\mathbb C)\times\mathbb C^*$ being unimodular implies $SL_n(\mathbb C)$ is unimodular.
Is there another way to show $SL_n(\mathbb C)$ is unimodular using $\phi$?
I'd like to have a sort of elementary answer, please, as I don't know that much about Lie groups. I just know what a Lie group is, what a left invariant measure is, and the definition of the Lie algebra.