  In the previous two papers with the same title ([LLY05] by W.C. Li, L. Long,
Z. Yang and [ALL05] by A.O.L. Atkin, W.C. Li, L. Long), the authors have
studied special families of cuspforms for noncongruence arithmetic subgroups.
It was found that the Fourier coefficients of these modular forms at infinity
satisfy three-term Atkin and Swinnerton-Dyer congruence relations which are the
$p$-adic analogue of the three-term recursions satisfied by the coefficients of
classical Hecke eigenforms.
  In this paper, we first consider Atkin and Swinnerton-Dyer type congruences
which generalize the three-term congruences above. These weaker congruences are
satisfied by cuspforms for special noncongruence arithmetic subgroups. Then we
will exhibit an infinite family of noncongruence cuspforms, each of which
satisfies three-term Atkin and Swinnerton-Dyer type congruences for almost
every prime $p$. Finally, we will study a particular space of noncongruence
cuspforms. We will show that the attached $l$-adic Scholl representation is
isomorphic to the $l$-adic representation attached to a classical automorphic
form. Moreover, for each of the four residue classes of odd primes modulo 12
there is a basis so that the Fourier coefficients of each basis element satisfy
three-term Atkin and Swinnerton-Dyer congruences in the stronger original
sense.
