= Solution
The continuous <spacetime symmetries> are <translations>, spatial <rotations> and <Lorentz boosts>, together the connected <Poincaré group>. The theory also has local electromagnetic <U(1) gauge symmetry>, which is an internal symmetry rather than a spacetime transformation. A nonzero mass and a dimensionful interaction do not give an additional exact <scale invariance>.
In four dimensions the <action> is dimensionless and $[\partial]=1$. The two <kinetic terms> give
$$
\boxed{[a]=1,\quad[A_\mu]=1,\quad[F_{\mu\nu}]=2,\quad[m_a]=1,\quad[g_a]=-1.}
$$
The axion-photon operator has <mass dimension> five, so this is an <effective field theory>. Its dimensionless interaction strength grows with energy as $|g_a|E$. Without information about an ultraviolet completion, a conservative parametric validity condition is $E\ll\Lambda$ with $\Lambda\sim|g_a|^{-1}$, and $m_a$ must also lie below that scale for the axion to be a degree of freedom in the effective theory. Numerical factors such as $4\pi$ depend on the ultraviolet dynamics and on the chosen perturbative criterion; the Lagrangian alone does not fix them.
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