origin of multipole expansion calculation

[Stef Kunnen]

Why does it matter where you put your origin if you calculate your multipole expansion. Take for example the dipole expansion for the first part of the exercises. If we take the origin at the centre we get Qx= Qy = 0.
if we take the origin at the bottom left charge, our Qx becomes q(0 + a + a + 0) = 2aq and Qy =q (0 + 0 + 2a + 2a) = 4aq.

I would think that this means that before one can calculate the multipole moment, you first need to find a proper base in which you calculate this. How do we properly do this?

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