Large dynamical magnetic effective charges and antimagnetoelectricity from spin and orbital origin in multiferroic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>BiCoO</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>

Using first-principles calculations, we explore the magnetoelectric properties of the room-temperature multiferroic crystal ${\mathrm{BiCoO}}_{3}$. We use both applied magnetic field and finite-difference techniques to show that ${\mathrm{BiCoO}}_{3}$ is antimagnetoelectric at the linear level. The calculation of the dynamical effective charges reveals that the total magnetoelectric response is zero due to the compensating nonzero magnetoelectric response of each magnetic sublattice. This calculation also highlights that the orbital contribution to the response is remarkably larger than the spin one and that each sublattice has a rather large total magnetoelectric response of 85 ps/m. Furthermore, we provide an intuitive recipe to visualize the dynamical magnetic effective charge, allowing to examine its multipolar nature, which we confirm by means of ab initio calculations. Given the large value of the local response, we investigate the ferromagnetic phase as well, which gives a giant magnetoelectric response of about 1000 ps/m and coming mainly from the spin contribution this time. Finally, we discuss the possible reasons for such a large magnetoelectric response in ${\mathrm{BiCoO}}_{3}$ and propose possible strategies to unveil this potentially large response.