ψ[θ_,ϕ_]:= Sqrt[15/(16π)]*Sin[2θ]*Cos[ϕ]
Grid[Table[Integrate[Conjugate[SphericalHarmonicY[ℓ, m, θ, ϕ]]*ψ[θ,ϕ]*Sin[θ],{θ,0,π},{ϕ,0,2π}], {ℓ, 0, 3}, {m, -ℓ, ℓ}]]