Finite Field $\mathbb{F}_7$ For the finite field $\mathbb{F}_7$, find an additive Fourier transform of nontrivial multiplicative characters.
If the field contains $q$ elements, there are $q-2$ nontrivial values.
Where do I even start?
 A: Not so much a hint, but a push in the right direction:
Let $\mathbb{F}_q^d$ denote the $d$-dimensional vector space over the finite field with $q$ elements.  Recall that for a function $f : \mathbb{F}_q^d \to \mathbb{C}$, the Fourier transform of $f$ is given by
$$
\widehat{f}(m) = q^{-d} \sum_{x \in \mathbb{F}_q^d} f(x) \chi(x \cdot m)
$$
where $\chi$ is a nontrivial additive character on $\mathbb{F}_q$.  It is worth noting that this definition of $\widehat{f}$ may or may not agree with the definition you use.  In particular, your definition might not include the normalization factor $q^{-d}$.
Now, when $q$ is prime, we can actually take $\chi(z) = \exp(2 \pi i z/q)$.  Furthermore, when $q$ is prime, we can identify $\mathbb{F}_q$ with $\mathbb{Z}_q = \{0,1, \dots , q-1\}$.
For this particular problem, we have $q = 7$ and $d = 1$.  Let $\psi$ denote any nontrivial multiplicative character.  Write:
$$
\widehat{\psi}(m) = \frac{1}{7} \sum_{x \in \mathbb{Z}_7} \psi(x) \exp(2 \pi i xm/7)....
$$
