We consider the wave equation with a power nonlinearity \begin{equation} u_{tt}-\Delta u = u^p, \nonumber \end{equation} with initial profiles $u(x,0)$ and $u_t(x,0)$, $x\in \mathbb{R}^3$, $t\ge 0$, and $p>1$ an odd integer. In order to investigate the blowup dynamics we look for radial self-similar blowup solutions of the form \begin{equation} u(x,t)=(T-t)^{-\frac{2}{p-1}}U\left(\frac{|x|}{T-t}\right),\quad T>0, \nonumber \end{equation} with a smooth, radial profile $U$. In particular, we are interested in stability properties of such solutions. This gives rise to analyzing the spectrum of the linearized operator, i.e. to the eigenvalue problem: \begin{equation} \mathcal{L} {\bf u}=\lambda {\bf u}, \nonumber \end{equation} where $D(\mathcal{L})\subset H^2_{\rm rad}(B^3)\times H^1_{\rm rad}(B^3)$, $H^k_{\rm rad}(B^3):=\{ {\bf u}\in H^k(B^3):{\bf u}~{\rm is~radial} \}$, \begin{equation} \mathcal{L} \begin{pmatrix} u_1 \cr u_2\end{pmatrix}:=\begin{pmatrix} -\rho u_1^{\prime}(\rho)-\alpha u_1(\rho)+u_2(\rho) \cr u_1^{\prime \prime}(\rho)+\frac{2}{\rho}u_1^{\prime}(\rho)-\rho u_2^{\prime}(\rho)-(\alpha+1)u_2(\rho)+V(\rho)u_1(\rho) \end{pmatrix}, \nonumber \end{equation} $\rho=\frac{|x|}{T-t}$, $V(\rho)=pU(\rho)^{p-1}$ and $\alpha=\frac{2}{p-1}$. We are interested in excluding eigenvalues of $\mathcal{L}$ in parts of the right complex half plane, which is ongoing work together with B. Schoerkhuber, Y. Watanabe, M. Plum and M.T. Nakao. In this talk, we will show (as a partial result) that all eigenvalues $\lambda$ in the half-plane $\{ {\rm Re}(z) > 1-\alpha\}$ are ${\it real}$, which constitutes a substantial advantage for the computation of the desired eigenvalue exclusions. We also provide global upper bounds for the real eigenvalues.