Traditionally, the Tuatha de Bhriain celebrate the new year at Samhain, however according to notes on the Coligny Calendar on Wikipedia:

...the calendar year began with Samonios, which is usually assumed to correspond to Old Irish Samhain, giving an autumn start to the year. However, as Samon is Gaulish for summer (Lambert p.112), this assumed start is disputed. Le Contel and Verdier (1997) argue for a summer solstice start of the year. Monard (1999) argues for an autumn equinox start. Bonsing (2007) argues for a May beginning consistent with Irish Beltaine, and Fennian literature, notably Joyce (2000).