So I have to write $x(t)$ in the form: $$ x(t)=\sum_{k=0}^{\infty} A_{k} \cos \left(\omega_{k} t+\phi_{k}\right) $$
And I am told that $x(t)$ is real-valued and has a fundamental period $T=8$. Then I am given that the non-zero Fourier series coefficients for $x(t)$ are $a_{1}=a_{-1}=2, a_{3}=a_{-3}^{*}=4 j$.
Then my imediate thought is that $\omega_0=\frac{2\pi}{8}=\frac{\pi}{4}$ and I am thinking that I need to somehow cancel out the coefficients with the imaginary parts for $x(t)$ to be real valued. However, then I am completely lost how to proceed. I am also unsure what the star means in $a_{-3}^{*}$. I am thinking about convolution, but I am fairly new at convolutions and I find it odd that it is then $a_{3}=a_{-3}^{*}=4 j$.
Can someone help me understand how I am supposed to write this Fourier series?