The Wigner-Eckart theorem for an expectation value can then be written as $$ \langle \Phi^J_M|T^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle\equiv\langle \lambda \mu J'M'|JM\rangle\langle \Phi^J||T^{\lambda}||\Phi^{J'}\rangle. $$ The double bars indicate that this expectation value is independent of the projection \( M \). We can manipulate the Clebsch-Gordan coefficients using the relations $$ \langle \lambda \mu J'M'|JM\rangle= (-1)^{\lambda+J'-J}\langle J'M'\lambda \mu |JM\rangle $$ and $$ \langle J'M'\lambda \mu |JM\rangle =(-1)^{J'-M'}\frac{\sqrt{2J+1}}{\sqrt{2\lambda+1}}\langle J'M'J-M |\lambda-\mu\rangle, $$ together with the so-called \( 3j \) symbols. It is then normal to encounter the Wigner-Eckart theorem in the form $$ \langle \Phi^J_M|T^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle\equiv(-1)^{J-M}\left(\begin{array}{ccc} J & \lambda & J' \\ -M & \mu & M'\end{array}\right)\langle \Phi^J||T^{\lambda}||\Phi^{J'}\rangle, $$ with the condition \( \mu+M'-M=0 \).