Dirac's Bracket Notation

Question

I have a question on Dirac's bracket notation. In particular, according to this notation, vectors and covectors are represented by $|\psi\rangle$ and $\langle\psi|$ respectively. Moreover, these two representations are related by $|\psi\rangle^\dagger=\langle\psi|$ in a complex vector space. However, if $|\psi\rangle$ is a vector and $\langle\psi|$ is a covector, what can we say about $\psi$ on its own? What does it represent? It's neither a vector nor a covetor, but both at the same time. I've personally called $\psi$ a "mathematical vector" to distinguish it from $|\psi\rangle$ and $\langle\psi|$ that for me are just two representations of $\psi$ that are possible once a basis for the vector space has been fixed.