My question is fairly simple and straightforward. I'm studying Quantum Mechanics, specifically the spinor formalism. I understand that one can define a generator of rotations, say around axis $z$ by an angle $\phi$, by considering the exponential $$ \mathcal{D}_z(\phi)=\exp\left(-\frac{iS_z}{\hbar}\phi \right) $$ I also understand that this notion can be further generalized to a rotation around a general direction given by $\hat{n}$ such that $$ \mathcal{D}(\hat{n},\phi)=\exp\left(-\frac{i\vec{S}\cdot\hat{n}}{\hbar}\phi \right) $$ Where the inner product $\vec{S}\cdot\hat{n}$ can be written in terms of the Pauli matrices $\sum_k \mathbb{\sigma}_kn_k$. Now, my problem really starts when trying to construct the eigenstates of the operator $\vec{S}\cdot\hat{n}$. In my text book the eigenstates are defined by $$ \vec{S}\cdot\hat{n}|\vec{S}\cdot\hat{n}, \pm\rangle=\pm\frac{\hbar}{2}|\vec{S}\cdot\hat{n}, \pm\rangle $$ but i can't really wrap my head around this ket $|\vec{S}\cdot\hat{n}, \pm\rangle$, what does it mean exactly to have that innet product inside the ket? Is it just spin up or down in the $\hat{n}$ direction? If so, how can we expresse it in the $S_z$ basis?

My question is fairly simple and straightforward. I'm studying Quantum Mechanics, specifically the spinor formalism. I understand that one can define a generator of rotations, say around axis $z$ by an angle $\phi$, by considering the exponential $$ \mathcal{D}_z(\phi)=\exp\left(-\frac{iS_z}{\hbar}\phi \right) $$ I also understand that this notion can be further generalized to a rotation around a general direction given by $\hat{n}$ such that $$ \mathcal{D}(\hat{n},\phi)=\exp\left(-\frac{i\vec{S}\cdot\hat{n}}{\hbar}\phi \right) $$ Where the inner product $\vec{S}\cdot\hat{n}$ can be written in terms of the Pauli matrices $\sum_k \mathbb{\sigma}_kn_k$. Now, my problem really starts when trying to construct the eigenstates of the operator $\vec{S}\cdot\hat{n}$. In my text book the eigenstates are defined by $$ \vec{S}\cdot\hat{n}|\vec{S}\cdot\hat{n}, \pm\rangle=\pm\frac{\hbar}{2}|\vec{S}\cdot\hat{n}, \pm\rangle $$ but I can't really wrap my head around this ket $|\vec{S}\cdot\hat{n}, \pm\rangle$, what does it mean exactly to have that inner product inside the ket? Is it just spin up or down in the $\hat{n}$ direction? If so, how can we express it in the $S_z$ basis?