This is the 16759th most frequent Turkish word.
yılbaşının
"Yılbaşının" means "of the New Year" or "belonging to the New Year's".
Here, 'yılbaşının' is in the genitive form, indicating ownership, as in 'the arrival of New Year's'.
Yılbaşının geldiğini fark ettik.
We noticed that New Year's had arrived.
In this sentence, 'yılbaşının' shows possession, stating what belongs to New Year's, as in 'the meaning of New Year's'.
Yılbaşının anlamı kutlama ve yenilik demektir.
The meaning of New Year's is celebration and renewal.
Here, 'yılbaşının' is used to ask a question about New Year's, specifically 'when is New Year's' indicating the specific time context.