The congruent number curves $E_n : y^2 = x^3 - n^2 x$ are the quadratic twists of $E_1 : y^2 = x^3 - x$ by $n$: for every prime $p$ not dividing $n$, the trace of Frobenius satisfies $a_p(E_n) = \left(\tfrac{n}{p}\right) a_p(E_1)$. This submission proves that identity at the level of the defining character sums $a_p(E_n) = -\sum_{x \in \mathbb{F}_p} \chi(x^3 - n^2 x)$, by the substitution $x = n t$.

It also proves the parity fact behind Tunnell's criterion: for even positive coefficients $a, c$ and odd $n$, the number of integer solutions of $a x^2 + y^2 + c z^2 = n$ is even, which covers the four representation counts $2x^2 + y^2 + 32z^2$, $2x^2 + y^2 + 8z^2$, $4x^2 + y^2 + 32z^2$ and $4x^2 + y^2 + 8z^2$ in Tunnell's theorem. The argument is the fixed-point-free involution $(x, y, z) \mapsto (x, -y, z)$, and the counting principle it rests on, that a finite set with a fixed-point-free involution has even cardinality, is stated and proved on its own.

Both results are classical; the contribution is their formal statement and kernel-checked proof, with the solution set of Tunnell's equations represented explicitly as a finite set so that the theorem is about the counts themselves.