If I have spectrum $X_k$ of the signal evaluated at frequencies $k \delta f$ and I want to know what is the value of that same signal at some other frequencies , i.e., $k' \delta f'$.I know that $\delta f$ and $\delta f'$ are somehow reelated (e.g. $\delta f = 12 \delta f'$) how can I express spectrum of the signal with this new carrier spacing ($k' \delta f'$) having only info about $X_k$ or in other words what is the coefficient, $X_k' = coefficient \cdot X_k$. Thanks in advance for any help.
If the signal is band-limited (at least above some noise level) below half the highest frequency in your initial frequency evaluation set, then you can resample the spectrum to any other frequency spacing by using periodic Sinc (Dirichlet) kernel interpolation.
If there is an integer relationship between the two carrier frequency bin spacings, then you can pre-calculate a polyphase interpolation table using the Sinc function (or a windowed approximation, if that meets your S/N requirements). Otherwise you can convolve the entire spectrum vector against a computed periodic Sinc or Dirichlet kernel with the appropriate phase offset per computed "carrier" frequency.