If the function is neither even nor odd ,is its fourier series coefficient a complex? or is this wrong?

Question

i thought if the fourier series coefficient of $f$ is neither even nor odd,$f$ must be the complex.

we can know the function,$f$, is real or imaginary or complex from its FS coefficients.

Now i don't know the $f$ is real or imaginary or complex ,the thing i only know is its coefficients, $c_k=cos(\frac{3k\pi}{4})+jsin(\frac{k \pi}{4})$,then i use some method to know $c_k$ is neither even nor odd function,so now according to my thinking,$f$ should be the complex function.However,i use the $c^*_{-k}=c_k$,and this is the method which can let us know the $f$ is real or odd,if the $c^*_{-k}=c_k$,then $f$ is real.So now,$c^*_{-k}$ does be equal to$c_k$.so this $f$ must be real.so we can know there is a conflict with my thinking and $c^*_{-k}=c_k$.

The problem may be if the fourier coefficient of $f$ is neither even nor odd,then f must be complex.So i want to ask is this thinking,f the fourier coefficient of $f$ is neither even nor odd,then f must be complex, right?

Answer 1

The statement of the question is very unclear. An answer to what I would guess the question is, based just on the title: Let $f(t)=e^{it}$. Then $f$ is neither even nor odd, but the Fourier coefficients of $f$ are real.

Edit: We're told that no, that doesn't answer the question. Based on the opaque style in the OP and comments, I conjecture we will never see a coherent statement of exactly what the question is. Meanwhile, here's a summary of the relevant facts about parity and reality for Fourier series:

Suppose $f$ is a $2\pi$-periodic function with Fourier series $$f(t)\sim\sum c_ne^{int}.$$

$f$ is even if and only if $c_{-n}=c_n$; similarly $f$ is odd if and only if $c_{-n}=-c_n$.

$f$ is real-valued if and only if $c_{-n}=\overline{c_n}$; similarly all the $c_n$ are real-valued if and only if $f(-t)=\overline{f(t)}$.

Whatever the question is there must be an answer there. (In particular no, there's no implication in either direction between $f$ being even or odd or the sequence $(c_n)$ being even or odd and $f$ or $c_n$ being real-valued. Which says that the answer to at least eight possible version of wat the question is is no.)