On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix

For \(e\) a positive integer, we find restrictions modulo \(2^e\) on the coefficients of the characteristic polynomial \(\chi_S(x)\) of a Seidel matrix \(S\). We show that, for a Seidel matrix of order \(n\) even (resp. odd), there are at most \(2^{\binom{e-2}{2}}\) (resp. \(2^{\binom{e-2}{2}+1}\)) possibilities for the congruence class of \(\chi_S(x)\) modulo \(2^e\mathbb Z[x]\). As an application of these results, we obtain an improvement to the upper bound for the number of equiangular lines in \(\mathbb R^{17}\), that is, we reduce the known upper bound from \(50\) to \(49\).