I want to calculate Gauss Sum of the field with 8 elements. I studied in Lidl & Niederreiter's book.
$\chi(c)=e^{2 \pi i Tr(c)/p}$ for all $c\in F_q$, defines an additive character of $F_q$, where $Tr$ is from $F_q$ to $F_p$. The characteristic of $F_q$ is $p$. And,
$\psi(g^k)=e^{{2\pi i k}/{q-1}}$ for $k=0,1,2,...,q-2$ defines a multiplicative character of $F_q$, where $g$ is a primitive element of $F_q$.
$G(\chi,\psi)=\sum_{c\in F_q^*}\chi(c)\psi(c)$ is the Gauss sum.
For $p=2$ and $q=8$, $F_8^*=\{a^0, a^1, a^2, a^3, a^4, a^5, a^6\}$ and their trace are $\{1,0,0,1,0,1,1\}$ respectively. Then
$G(\chi,\psi)=(e^{{2\pi i Tr(1)}/{2}})(e^{{2\pi i0}/{7}})+(e^{{2\pi i Tr(a)}/{2}})(e^{{2\pi i1}/{7}})+(e^{{2\pi i Tr(a^2)}/{2}})(e^{{2\pi i2}/{7}})+(e^{{2\pi i Tr(a^3)}/{2}})(e^{{2\pi i3}/{7}})+(e^{{2\pi i Tr(a^4)}/{2}})(e^{{2\pi i4}/{7}})+(e^{{2\pi i Tr(a^5)}/{2}})(e^{{2\pi i5}/{7}})+(e^{{2\pi i Tr(a^6)}/{2}})(e^{{2\pi i6}/{7}})$ $=\zeta^{1/2}.1+1.\zeta^{1/7}+1.\zeta^{2/7}+\zeta^{1/2}.\zeta^{3/7}+1.\zeta^{4/7}+\zeta^{1/2}.\zeta^{5/7}+\zeta^{1/2}.\zeta^{6/7}$
where I take $\zeta=e^{2\pi i}$ and so $\zeta^{1/2}=e^{\pi i}=-1$. Thus I get
$G(\chi,\psi)=-1+\zeta^{1/7}+\zeta^{2/7}-\zeta^{3/7}+\zeta^{4/7}-\zeta^{5/7}-\zeta^{6/7}$
What is wrong with my calculations?
Additionally, do the additive characters always take $p^{th}$ roots of unity? Which is $2^{nd}$ roots of unity $=\{-1,+1\}$ in my example.
Thanks for any help.