Sine orthogonality gives $a_k=(1/\pi)\int_{-\pi}^{\pi}x\sin(kx)dx=(2/\pi)\int_0^\pi x\sin(kx)dx$. Integration by parts yields $a_k=2(-1)^{k+1}/k$. For k = 2, $a_2=2(-1)^3/2=-1$. Endpoint convergence of the Fourier series does not alter its coefficients.
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