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 <i>k</i> that is not of characteristic two, using the Motivic Gauss-Bonnet Theorem of Levine-Raksit. As an application, we show that the power structure on the Grothendieck-Witt ring introduced by Pajwani-Pál computes the compactly supported <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow> <mml:mn>1</mml:mn></mml:msup> </mml:math> -Euler characteristic of symmetric powers for all curves.