  The classical Mahowald invariant is a method for producing nonzero classes in
the stable homotopy groups of spheres from classes in lower stems. We study the
Mahowald invariant in the setting of motivic stable homotopy theory over
$Spec(\mathbb{C})$. We compute a motivic version of the $C_2$-Tate construction
for various motivic spectra, and show that this construction produces
"blueshift" in these cases. We use these computations to show that the Mahowald
invariant of $\eta^i$, $i \geq 1$, is the first element in Adams filtration $i$
of the $w_1$-periodic families constructed by Andrews ~\cite{And14}. This
provides an exotic periodic analog of Mahowald and Ravenel's computation
~\cite{MR93} that the classical Mahowald invariant of $2^i$, $i \geq 1$, is the
first element in Adams filtration $i$ of the $v_1$-periodic families
constructed by Adams ~\cite{Ada66}.
