Exact multiplicity of each eigenvalue of the Laplacian on $L^2(\mathbb{T}^3)$

Question

Let $\mathbb{T}^3:=[\mathbb{R}/\mathbb{Z}]^3$ be the $3$-dimensional torus and consider the real Hilbert space $L^2\Bigl(\mathbb{T}^3,\mathbb{R} \Bigr)$ on it. That is, the space of real-valued square-integrable functions on $\mathbb{T}^3$.

Now, I know that the Laplacian $-\Delta$ is a self-adjoint operator on this space with the eigenvalues(=discrete spectrum in this case) given by \begin{equation} \{ n_1^2+n_2^2+n_3^2 \mid n_1,n_2,n_3 \in \mathbb{Z} \} \end{equation}

and the corresponding eigenfunctions as the tensor products of sines and cosines.

However, I have trouble computing the exact multiplicity of each eigenvalue for this $-\Delta$. That is, for each distinct element $\lambda$ of the above set, how do I compute the dimension of the corresponding eigenspace $V_\lambda \subset L^2\Bigl(\mathbb{T}^3,\mathbb{R} \Bigr)$?

In one dimension, things would be quite straightforward, but I am very much confused about $3$-dimensions.. Could anyone please help me?