do you send it with your first email to your prospective supervisor or after a few exchanges?
I'm assuming that you are in a place in which the advisor/supervisor actually hires you. I'm also assuming that you want to work on your own research program rather than that of the supervisor, which is often ongoing.
I'd suggest that you say something about your research interests, but, perhaps, not make a formal proposal. You can do that in the first mail. You can even say a few things about why this is your interest and what you have already done in that direction.
But some advisors might want you to take up something in their program, rather than work on your own. For others it might be the opposite. Unless you know which situation applies, I'd try to keep your options open.
If you introduce your ideas without expressing too much commitment to them then you might get either a request for more information (hence the full proposal) or a suggestion about what s/he would rather you do for the degree. And take the phrase too much literally. If it sounds like a demand, then it may wind up as an obstacle.
Of course, you may already know something about the professor and have an idea about whether it is better to push or not.
Just for completeness, in some places and fields, proposing research normally is done after acceptance and even choosing an advisor is also done later.