The continuous time signal \(x(t)\) is real, periodic with period \(T\), and satisfies the Dirichlet conditions.
The Fourier series representation of \(x(t)\) is
\[
x(t)=\sum_{n=-\infty}^{\infty}a_ne^{j\left(\frac{2\pi nt}{T}\right)}
\]
and \(x(t)\) satisfies the following:
\[
x\left(t-\frac{T}{2}\right)=-x(t).
\]
For any integer \(m\), which of the following options is correct?