7. Given that $z$ is a complex number, and $|z|=1$. When $\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\qquad$ or . $\qquad$
7. $-\frac{1}{4} \pm \frac{\sqrt{15}}{4}$ i.

Notice,
$$
\begin{array}{l}
\left|1+z+3 z^{2}+z^{3}+z^{4}\right| \\
=\left|\frac{1}{z^{2}}+\frac{1}{z}+3+z+z^{2}\right| \\
=\left|\left(z+\frac{1}{z}\right)^{2}+z+\frac{1}{z}+1\right| \\
=\left|(z+\bar{z})^{2}+z+\bar{z}+1\right| \\
=\left|4 \operatorname{Re}^{2} z+2 \operatorname{Re} z+1\right| \\
\geqslant \frac{16-4}{16}=\frac{3}{4},
\end{array}
$$

Here, $\operatorname{Re} z=-\frac{1}{4}$.
Thus, $z=-\frac{1}{4} \pm \frac{\sqrt{15}}{4}$ i.