There is a lot of misinformation in this forum about soundproofing. It is very different from what we perceive as acoustic treatment. Putting egg cartons acoustic foam, panels and bass traps will not solve your problem. the idea about soundproofing is how you could isolate the sound inside from coming out or the other way around. The best soundproofing is based on construction techniques on how sound vibrations from going out(as in your case). soundproofing is as effective as your weakest link. Most of the time the door is the culprit. You have to have a heavy door with a lot of mass which means the thicker the better. It should also have some sort of weatherproofing so that the sound will not get through the seams especially under the door. a double door configuration, like those used in cinemas will do wonders because the sound attenuated by one door, say a reduction of 30dB will again be reduced by 30dB by the other door. The unit of measure that they use in soundproofing is STC (sound transmission coefficient). A thick 8 inch concrete wall might have a Stc of 55 dB which means if the maximum loudness in your home theater is around 105 dB then that wall will reduce it to 50 dB ( which is already tolerable for your neighbors). As I have said, it is the weakest link that matters. Heavy doors have a reduction of around 35 dB, so the outside sound will still be 70 dB. still quite loud. We are only talking about published STC figures from 125 hertz to 4000 hertz, low frequency is much more difficult to deal with.(subwoofer) even with a Stc of 55 dB for thick walls, it might just have a low frequency Stc of 15 dB, which means at maximum home theater level of 105 dB, outside it is still 90 dB and that is loud.
A small hole, an open window will be your weakest link. sound will go out even the hole is just 1 cm. in diameter. that is how soundproofing is so difficult to solve.
The cheapest way is really to make friends with your neighbor and invite them for a home cinema experience.
Soundproofing can be done, but it is very expensive to do it right. Special materials like resilient channels, double pane glass, metal doors and viscous damping material are just some of those used by pros.