Quadratic Euler Characteristic of Symmetric Powers of Curves

We compute the quadratic Euler characteristic of the symmetric powers of a smooth, projective curve over any field $k$ that is not of characteristic two, using the Motivic Gauss-Bonnet Theorem of Levine-Raksit. As an application, we show over a field of characteristic zero that the power structure on the Grothendieck-Witt ring introduced by Pajwani-Pál computes the compactly supported ${\mathbb{A}}^1$-Euler characteristic of symmetric powers for all curves.