Question on Lottery and Taxes

Let me preface this with the following: I have NOT won any lottery yet in any way shape or form, nothing really big or moderately small (and I cry myself to sleep everytime I lose ;_;), this is a question regarding winning the lottery and how it would change how I do taxes (this will have what if's), and I am a New Mexico resident

Let's say I take my chances at the lottery, and the prize amount (if I win) is in the $400+ million range, now I know that ill have a certain percentage taxed before I receive my winning, but with how much is left over, ill most likely still be able to live under my means, not spending too outrageously or recklessly, but as nominal I can do it. But then tax season happens and then I have to mark down my winnings in the proper turbotax form. If I do this, will I see any of my winnings taxed further, and if so, how much (percentage wise) would be taxed away, or would none of it be taxed away excluding what was taxed initially by the lottery board or whatever organization for fees and somesuch

I never really thought about this as a kid, and only think about this now because I remember that story (far as I remember, was a legit article) about a non-u.s lottery winner who had all her winnings taxed away, she raised a fuss, got compensation money, but all of that compensation money was taxed away as well and she earned ultimately nothing. I'm just worried if ill run into a situation like that, and would like assurance if that fear is truly unfounded