  What is the dimension of a smooth family of complex Hadamard matrices
including the Fourier matrix? We address this problem with a power series
expansion. Studying all dimensions up to 100 we find that the first order
result is misleading unless the dimension is 6, or a power of a prime. In
general the answer depends critically on the prime number decomposition of the
dimension. Our results suggest that a general theory is possible. We discuss
the case of dimension 12 in detail, and argue that the solution consists of two
13-dimensional families intersecting in a previously known 9-dimensional
family. A precise conjecture for all dimensions equal to a prime times another
prime squared is formulated.
