I am trying to find the Fourier series of a 3D function, $e^{-\alpha(x^2 + y^2 + z^2)}$ with bounds $-\ell_1 < x < \ell_1$, $-\ell_2 < y < \ell_2$, $-\ell_3 < z < \ell_3$. I have found a reference, however I am confused about some of their mathematical notation with regards to (what looks like) summation over a vector, $\vec{\mathbf{k}}$.
How do I expand the summation into something that I would be able to compute?
$\sum_{\vec{\mathbf{k}}} c_k e^{i(m_1 2\pi/a_1 x + m_2 2\pi/a_2 y+m_3 2\pi/a_3 z)}$
Where $c_k= c_{m_1 m_2 m_3}$ (some function of m1, m2, m3 that I have greatly simplified here, as the actual form is very long) and $\vec{\mathbf{k}}=[m_1 \pi/\ell_1, m_2 \pi/\ell_2, m_3 \pi/\ell_3]$.
How do I expand this notation into multiple summations? My ultimate goal is to implement the 3D Fourier expansion in code (Fortran) but at this point am I still trying to understand the notation.