Nevermind. I understand the process now.
In the corrected output, you have to add the coefficients for X^0 and X^11, X^1 and X^12, and so on, because you are taking it mod (X^11 - 1). The sums of these pairs of coefficients is always equal to 0 mod 3, except for the first one, which is 1 mod 3, as defined by the wikipedia page.