  In this paper, we consider modular forms for finite index subgroups of the
modular group whose Fourier coefficients are algebraic. It is well-known that
the Fourier coefficients of any holomorphic modular form for a congruence
subgroup (with algebraic coefficients) have bounded denominators. It was
observed by Atkin and Swinnerton-Dyer that this is no longer true for modular
forms for noncongruence subgroups and they pointed out that unbounded
denominator property is a clear distinction between modular forms for
noncongruence and congruence modular forms. It is an open question whether
genuine noncongruence modular forms (with algebraic coefficients) always
satisfy the unbounded denominator property. Here, we give a partial positive
answer to the above open question by constructing special finite index
subgroups of SL_2(Z) called character groups and discuss the properties of
modular forms for some groups of this kind.
