  We show that noncongruence subgroups of SL_2(Z) projectively equivalent to
congruence subgroups are ubiquitous. More precisely, they always exist if the
congruence subgroup in question is a principal congruence subgroup Gamma(N) of
level N>2, and they exist in many cases also for Gamma_0(N).
  The motivation for asking this question is related to modular forms:
projectively equivalent groups have the same spaces of cusp forms for all even
weights whereas the spaces of cusp forms of odd weights are distinct in
general. We make some initial observations on this phenomenon for weight 3 via
geometric considerations of the attached elliptic modular surfaces.
  We also develop algorithms that construct all subgroups projectively
equivalent to a given congruence subgroup and decides which of them are
congruence. A crucial tool in this is the generalized level concept of
Wohlfahrt.
