The operator for the total angular momentum and its z-component can be written as $\hat{L}^2=-\hbar^2\hat{\Lambda}^2$ (where $\hat{\Lambda}^2=\frac{1}{sin^2\theta}\frac{\partial ^2}{\partial \varphi^2}+\frac{1}{sin\theta}\frac{\partial}{\partial\theta}sin\theta\frac{\partial}{\partial\theta}$) and $\hat{L}_z=-i\hbar\frac{\partial}{\partial\varphi}$. Find $\hat{L}^2$ and $\hat{L}_z$ for a particle in the state $Y_{11}(\theta,\varphi)$ by confirming that the state is an eigenfunction of $\hat{L}^2$ and $\hat{L}_z$.
I tried solving it like this, but I'm not sure if I got it correctly:
$L^2 Y_{lm_l} (θ,φ)=l(l+1) ℏ^2 Y_{lm_l} (θ,φ)$
$L^2 Y_{11} {θ,φ}=1(1+1) ℏ^2 Y_{11} (θ,φ)⇒L^2=2ℏ^2$
$L_z Y_{lm_l}(θ,φ)=m_l ℏY_{lm_l} (θ,φ)$
$L_z Y_{11}(θ,φ)=1 ℏY_{11} (θ,φ) \Rightarrow L_z=\hbar$
anyone know if I did it correctly?