6. If the expansion of $\left(1+x+x^{2}\right)^{1010}$ is $a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2020} x^{2020}$, then $a_{0}+a_{3}+a_{6}+\cdots+a_{2019}=(\quad)$.
(A) $3^{670}$
(B) $3^{669}$
(C) $3^{1009}$
(D) $3^{1100}$
6. C.

Let $x=1$, we get
$$
a_{0}+a_{1}+a_{2}+\cdots+a_{2020}=3^{1010} ;
$$

Let $x=\omega\left(\omega=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right)$. Then
$$
\begin{array}{l}
\omega^{3}=1 \text { and } \omega^{2}+\omega+1=0, \\
a_{0}+a_{1} \omega+a_{2} \omega^{2}+a_{3} \omega^{3}+\cdots+a_{2020} \omega^{2020} \\
=0 ;
\end{array}
$$

Let $x=\omega^{2}$, we get
$$
\begin{array}{l}
a_{0}+a_{1} \omega^{2}+a_{2} \omega^{4}+a_{3} \omega^{6}+\cdots+a_{2020} \omega^{4040} \\
=0 .
\end{array}
$$
$$
\begin{array}{l}
\text { (1) }+ \text { (2) }+ \text { (3) gives } \\
3\left(a_{0}+a_{3}+a_{6}+\cdots+a_{2019}\right)=3^{1010} \\
\Rightarrow a_{0}+a_{3}+a_{6}+\cdots+a_{2019}=3^{1009} .
\end{array}
$$