Coronal and chromospheric activity of Teegarden's star
Teegarden's star is a late-type M-dwarf planet host, typically showing only rather low levels of activity. In this paper we present an extensive characterisation of this activity at photospheric, chromospheric, and coronal levels. We specifically investigated TESS observations of Teegarden's star, which showed two very large flares with an estimated flare fluence between 10$^{29}$ and 10$^{32}$\,erg comparable to the largest solar flares. We furthermore analysed nearly 300 CARMENES spectra and 11 ESPRESSO spectra covering all the usually used chromospheric lines in the optical from the \ion{Ca}{ii} H \& K lines at 3930\,Å\, to the \ion{He}{i} infrared triplet at 10830\,Å. These lines show different behaviour: The \ion{He}{i} infrared triplet is the only one absent in all spectra, some lines show up only during flares, and others are always present and highly variable. Specifically, the H$α$ line is more or less filled in during quiescence; however, the higher Balmer lines are still observed in emission. Many chromospheric lines show a correlation with H$α$ variability, which, in addition to stochastic behaviour, also shows systematic behaviour on different timescales including the rotation period. Moreover, we found several flares and also report hints of an erupting prominence, which may have led to a coronal mass ejection. Finally, we present X-ray observations of Teegarden's star (i.e. a discovery pointing obtained with the \emph{Chandra} observatory) and an extensive study with the \emph{XMM-Newton} observatory; when these two large flares were observed, one of them showed clear signatures of the Neupert effect, suggesting the production of hard X-rays in the system.