Representing a certain function with spherical harmonics

Questions
1

I need some help with representing \cos \theta \sin 2\varphi with spherical harmonics.

It seems like I should use Y_1^2 , Y_1^{-2} but there is no such thing. Is there a simple way to do it or do I need to integrate

\int_0^{2\pi} \int_0^\pi \cos \theta \sin 2\varphi\ (Y_l^2)^* \sin \theta d\theta d\varphi

for every 2\leq l\in\mathbb{N}?

Thank you!

It’s my first time writing here so if you have any tips for starters, that will help too!

2

With l=1

In[3]:= Integrate[
 Cos[theta] Sin[2 phi] SphericalHarmonicY[1, m, theta, phi] Sin[
   theta], {theta, 0, Pi}, {phi, 0, 2 Pi}]

Out[3]= ConditionalExpression[-(((-1 + E^(2 I m \[Pi])) m (-1 + 
     m^2) Sqrt[\[Pi]/3] Csc[(m \[Pi])/2])/((-4 + m^2) Sqrt[
   Gamma[2 - m]] Sqrt[Gamma[2 + m]])), -2 < Re[m] < 2]