If a predicate is not computable, what can be said about its negation? Doing the following exercise:

Let $\overline{HALT(x,y)}$ be defined as
$\overline {HALT(x,y)} \iff \text{program number y never halts on input x}$
Show that it is not computable.

Just want to make sure I have understood the concept correctly. We had in a theorem that HALT(x,y) is not computable which means that we cannot determine whether program number y eventually halts on input x. I realized that $\overline {HALT(x,y)}$ is the negation of HALT(x,y). Is it true (I cannot find it in my book or on the internet) that if a function is (not) computable, its negation is also (not) computable? A function being computable means there is a program p which computes it, we cannot say there is a program Q that computes its negation. Or can we draw such conclusion?
If a function F is computable, so is !F, as you can simply compute F and return the negated result.

For the same reason, if a function isn't computable, neither is its negation.