General question about DFT
The need for all these coefficients may not be obvious if you are looking at only real number input to the DFT.

But a complex vector input of length N will have twice the degrees of freedom compared to a strictly real input of that length.  The DFT, as an invertible linear transform therefore must therefore retain that same number of degrees of freedom, which for complex input ends up in the relationship between the positive and "negative" frequencies, or upper half of the transform, to produce complex exponential frequency results, instead of just strictly real cosine and sine functions.

For strictly real valued input, you don't really need to "care" about the other half, as it is duplicated information in the form of a complex conjugate (where the imaginary components of positive and "negative" frequency bins all cancel out to zero).  Thus the "mirroring" reduces the number of degrees of freedom to match that of the strictly real-valued input.  But one might still use the other half for computational purposes, as it makes the matrix math more regular and symmetric, and thus optimizable in the form of some FFT algorithms.  

Thus the negative frequency coefficients don't need to be computed because they are redundant (unless the input is complex and not strictly real valued). In that case, the frequency coefficients above N/2 don't need to be computed because they are identical to the negative coefficients that don't need to be computed