# Is quadrupole contribution to gravitational potential the sum of the contribution of all $m$ values?

Many of the sources I find on multipole expansions seem to be about electric potential and involve matrices. However, in my classical mechanics class we have not used matrices for multipole expansions yet, so I have not been able to answer my question with them. We were given this formula for the multipole moments of gravitational potential: $$q_{lm} = \sqrt{\frac{4\pi}{2l + 1}}\int dV \rho r^l Y_{lm}^\ast(\theta,\phi)$$.

The concepts of the monopole moment, dipole moment, quadrupole moment, etc. were introduced to us in the context of an azimuthally symmetric mass distribution, so we knew $$m = 0$$. To find the quadrupole moment for example, we would have found $$q_{20}$$. But now I am encountering questions that ask for the quadrupole moment of the gravitational potential of non azimuthally symmetric masses.

Do I add up the $$q_{2m}$$ for each possible m ranging from $$-2$$ to $$2$$? If not, how do I find this?

The quadrupole moment is a tensor, i.e., it's not the sum of $$q_{2,m}$$ over all $$m$$, it's just $$q_{2,m}$$, thought of as a spherical tensor of rank-2 with components $$q_{2,m}$$. Equivalently, in Cartesian coordinates, it takes the form of a symmetric, traceless matrix, which has five independent entries $$Q_{xx}$$, $$Q_{yy}$$, $$Q_{xy}$$, $$Q_{xz}$$, and $$Q_{yz}$$. The $$q_{2,m}$$'s can be written as linear combinations of the $$Q_{ij}$$'s and vice-versa.
This is analogous to how the dipole moment is a vector with three components $$D_x$$, $$D_y$$, and $$D_z$$, corresponding to three orthogonal directions $$x$$, $$y$$, $$z$$, in space. Equivalently, it's a spherical tensor of rank 1 with components $$d_{1,m}$$, where $$m=-1,0,1$$.