  A special cubic fourfold is a smooth hypersurface of degree three and
dimension four that contains a surface not homologous to a complete
intersection. Special cubic fourfolds give rise to a countable family of
Noether-Lefschetz divisors C_d in the moduli space C of smooth cubic fourfolds.
These divisors are irreducible 19-dimensional varieties birational to certain
orthogonal modular varieties. We use the "low-weight cusp form trick" of
Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira
dimension of C_d. For example, if d = 6n + 2, then we show that C_d is of
general type for n > 18, n not in {20,21,25}, it has nonnegative Kodaira
dimension if n > 13 and if n is not equal to 15. In combination with prior work
of Hassett, Lai, and Nuer, our investigation leaves only 20 values of d for
which no information on the Kodaira dimension of C_d is known. We discuss some
questions pertaining to the arithmetic of K3 surfaces raised by our results.
