As far as I know, a magnetic field can only be produced by a moving electric charge, or from a particle's spin (this is how a permanent magnet works, all the spins are in the same direction)
What is strength and direction of the magnetic field of a stationary electron at the origin with spin oriented straight up? I suspect a function of cylindrical coordinates is most convenient.
An electron's magnetic field is a dipole field - that is, the field strength is given by
source: http://www2.ph.ed.ac.uk/~playfer/EMlect4.pdf
In this expression, $m$ is the magnetic dipole moment. For an electron, this has the value
$$m=-928.476377 × 10^{−26} J/T$$
The magnetic permeability $$\mu_0=4\pi\cdot 10^{-7}\frac{V\cdot s}{ A \cdot m}$$
Note - when the spin is up, the magnetic field at the origin points down, because the electron has negative charge. Hence the negative sign in the value of $m$
