Stable Adams operations on $RO(C_2)$-graded homotopy groups

The $C_2$-spectrum of Atiyah’s Real $K$-theory is denoted by $\mathbf{K}\mathbf{R}$ and the $C_2$-spectrum of topological modular forms of level structure ${\mathrm{\Gamma }}_1(3)$ by ${\mathbf{T}\mathbf{M}\mathbf{F}}_1(3)$. In this short note we compute the $C_2$-equivariant stable Adams operations on the $RO(C_2)$-graded homotopy groups of $\mathbf{K}\mathbf{R}$ and ${\mathbf{T}\mathbf{M}\mathbf{F}}_1(3)$.