HilbertSymbol(a,b,p)
The HasseWitt invariant of a diagonal form $\langle a_1,\ldots,a_n\rangle$ over a field $K$ is defined to be the product $\prod_{i<j} \phi(a_i,a_j)$ where $\phi \colon K \times K \to \left\{\pm 1\right\}$ is any symbol (see e.g. [MH73, III.5.4] for a definition). It is a classical result of Hilbert that over a local field of characteristic not equal to two, there is one and only symbol, $(,)_p$ called the Hilbert symbol ([S73, Chapter III]) computed as follows:
$(a,b)_p = \begin{cases} 1 & z^2 = ax^2 + by^2 \text{ has a nonzero solution in } K^3 \\ 1 & \text{otherwise.} \end{cases}$
Consider the following example, where we observe that $z^2 = 2x^2 + y^2$ does admit nonzero solutions mod 7, in particular $(x,y,z) = (1,0,3)$:

Computing HasseWitt invariants is a key step in classifying symmetric bilinear forms over the rational numbers, and in particular certifying their (an)isotropy.
Citations:
The object HilbertSymbol is a method function.