Axially symmetric quadratic orbital eigenfunction ($\psi =(x_3^2-r^2/3)f(r),L^2\psi =6{\hslash }^2\psi ,L_3\psi =0$)

Each orbital angular momentum operator annihilates a radial factor and $r^2$. Direct differentiation gives $L^2{x}_3^2={\hslash }^2(6x_3^2-2r^2)$ and $L_3{x}_3^2=0$. Hence the displayed trace-free quadratic polynomial, times any admissible radial factor, has orbital labels $l=2,m=0$. Its angular dependence is proportional to the spherical harmonic $Y_{20}$; subtracting $r^2/3$ removes the rotational scalar part.