First order deformations of the Fourier matrix

The $N\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. The singularity at a point $H\in C_N$ is described by a filtration of cones $T^\times_HC_N\subset T^\circ_HC_N\subset T_HC_N\subset\widetilde{T}_HC_N$, coming from the trivial, affine, smooth and first order deformations. We study here these cones in the case where $H=F_N$ is the Fourier matrix, $(w^{ij})$ with $w=e^{2\pi i/N}$, our main result being a simple description of $\widetilde{T}_HC_N$. As a consequence, the rationality conjecture $dim_\mathbb R(\widetilde{T}_HC_N)=dim_\mathbb Q(\widetilde{T}_HC_N\cap M_N(\mathbb Q))$ holds at $H=F_N$.