  Let $F$ be a finite field, $\mu$ be a fixed additive character and $s$ be an
integer coprime to $|F^\times|$. For any $a\in F$, the corresponding Weil sum
is defined to be $W_{F,s}(a)=\displaystyle\sum_{x \in F} \mu(x^s-ax)$. The Weil
spectrum counts distinct values of the Weil sum as $a$ runs through the
invertible elements in the finite field. The value of these sums and the size
of the Weil spectrum are of particular interest, as they link problems in
coding and information theory to other areas of math such as number theory and
arithmetic geometry. In the setting of Niho exponents, we examine the Weil sum,
its bounds and its spectrum. As a consequence, we give a new proof to the
Vanishing Conjecture of Helleseth ($1971$) on the presence of zero in the Weil
spectrum in the case of Niho exponents. We also state a conjecture for when the
Weil spectrum contains at least five elements and prove it for a certain class
of Weil sums.
