FT of a nonlinear term using FFTW3 and q = 0 - c

Question

I am trying to apply an integration algorithm for PDEs called Exponential Time Differencing using C language and the library FFTW3. Without going too much into details, I need to calculate the FT of a nonlinear term which is the derivative of a function squared:
The way I do it is the following:
1) Define lattice size
int Nx = 500;
double dx = 0.2;
2) Calculate wavevectors
for ( i = 0; i < Nx/2+1; i++ ) {
qx[i] = 2.0*i*pi/(Nx*dx);
}
3) Execute FFTW3 plan to obtain FT of h from initial conditions (IC not shown in the code)
plan_forward = fftw_plan_dft_r2c_1d (Nx,dh,dhft,FFTW_ESTIMATE); //dh -> dhft
fftw_execute (plan_forward);
4) Calculate the FT of derivative of h
for ( i = 0; i < Nx/2+1; i++ ) {
// dxhft[i][0] are the real parts and dxhft[i][1] are the imaginary
dxhft[i][0] = -qx[i] * dhft[i][1]; // Re[FTdxh] = -q Im[FTh]
dxhft[i][1] = qx[i] * dhft[i][0]; // Im[FTdxh] = q Re[FTh]
}
5) Set the Nyquist element to 0 (as explained e.g. here)
dxhft[Nx/2][0] = 0.0;
dxhft[Nx/2][1] = 0.0;
6) Transform the derivative in real space using inverse Fourier Transform
plan_backward = fftw_plan_dft_c2r_1d (Nx,dxhft,dxh,FFTW_ESTIMATE); //dxhft -> dxh
fftw_execute (plan_backward);
7) Normalise the derivative by Nx and calculate the nonlinear term
for ( i = 0; i < Nx; i++ ) {
dxh[i] = dxh[i] / (double) (Nx);
Nonl[i] = dxh[i]*dxh[i];
}
8) Transform the nonlinear term
fftw_execute_dft_r2c (plan_forward,Nonl,Nonlft); // Nonl -> Nonlft
Ok. Now, I have reasons to believe that the FT of the nonlinear term for q=0, that is the elements Nonlft[0][0] and Nonlft[0][1], is wrong. But I don't understand what I am missing in the code. Can anyone give me some insight on the problem?