Reason for bimodal behavior while low second fourier coefficient If I have a time series (for eg. for 23 timestamps) and if I plot it and see that it is bimodal, that means it might be having high value of second fourier coefficient (with frequency = 2). But when I do the fourier transform I find that the value of second fourier coefficient is very low.

Now, is the following reasoning meaningful for this?

It might be the case that although second coefficient has low value but other coefficients having high value are in phase with the second coefficient, hence adding to its amplitude, and that is the reason we are seeing bimodal behavior in original time series.

PS -
1) By second coefficient, I mean that when we do fourier transform we get a zeroth term for zero frequency, then first and then second.
2) By coefficient I mean the amplitude of the fourier coefficient.

For example, if I have a time series (bimodal i.e. with two peaks) and I do its fourier transform, 3rd term has an amplitude of x and phase of p1, where x is a very low value.

Now I also see that 2nd term has an amplitude of y and phase of p1 and 4th term has an amplitude of z and phase of p1. y and z have high values.

Now do you think that two peaks in the original time series can be attributed to the following reasoning -
Although 3rd term (for frequency=2, having two peaks) has low amplitude (x) but because 2nd and 4th terms have high amplitudes and same phase, they have added to the amplitude of 2nd term and hence we see two peaks in the original time series?

In general, instead of 2nd and 4th terms, it can be any coefficients having phase same as the 3rd term.
Two peaks does not imply a frequency of 2 - for example the peaks could be very narrow and/or their spacing could corresponding to a much smaller interval than half the time-series length.

There is no information about the overall shape of the time-series that can be drawn from the knowledge of the second DFT coefficient.

I have plotted here 15 time series (left) and the magnitude of their FFT (right). 5 unimodal, 5 bimodal, and 5 trimodal series.