Yes, I should evaluate the product:
\langle \phi_{n, l ,m} | \psi \rangle
But suppose I want to tackle the second question, if a value of 6 \hbar is found for L^2 this means that l = 2, and then m = \pm 1 ; \pm 2 ; 0. So I should evaluate the coefficents
\langle \phi_{n, 2, \pm 1} | \psi...