need help with fft/ifft scaling

I am currently trying to solve a 3D poisson equation using spectral methods in 2 directions and finite difference in the third but am having some scaling problems.

To try and fix this i have reverted back to a very simple 1D poisson equation which i solve using finite difference. However i transform to fourier space to try and recreate my problem in a simpler way. code attached below

    %define number of grid points
    N=64;

 %divide up grid
 x=((0:N-1)/N)*2*pi;

 %set step length
 dx=2*pi/N;

 %define RHS of poisson equation, u_xx=f
 f=-sin(x);

 %define analytic solution to compare with
 f_comparison=sin(x);

 %set error tolerance to be used in finite difference method
 checker=10;

 %set initial length of u array
 u=zeros(1,N);

 %set initial length of u_new array, used in updating gauss seidel
 u_new=zeros(1,N);

 %calculate fft of the rhs of poisson equation
 f_hat=fftn(f);

 %enter loop
 while checker > 1*10^(-10);

 %calculate fft of u_new    
 u_new_hat=fftn(u_new);

        for a=1:N            

          q1=a+1;
          q2=a-1;

          %maintain periodic boundary conditions
          if q2==0 
              q2=N;
          end

          if q1==N+1
              q1=1;
          end

         %update using simple second order finite difference stencil 
         u_new_hat(a)=(0.5)*(u_new_hat(q1)+u_new_hat(q2)-((dx^2)*(f_hat(a))));

        end
   %calculate ifft to obtain result 
  u_new=real(ifftn(u_new_hat));  

 %calculate error tolerance
 u_checker=abs(u_new-u);

 %update maximun error to see if we can exit while loop  
 checker=max(u_checker)

 %update solution       
       u=u_new;           

 end

 %calculate scaling discrepency
 ddd=f_comparison./u;

even with this very simple code i am still getting an incorrect answer compared to the analytic solution but am not sure why.

Does anyone have any eidea what might be causing this as i have tried multiple different scalings but seem to be getting nowhere

thanks