The modular form level of a modular form records the congruence subgroup under which it transforms. In the common 
setting, a modular form has level 
if it transforms under the modular
group Gamma0(N). When the exact level is intended, 
is the least positive integer
with this property.
The level organizes spaces of modular forms and their Hecke operators. In the space 
of cusp forms,
images of forms from proper divisor levels under
the standard degeneracy maps span the old subspace. Its orthogonal
complement is the new subspace, and a normalized simultaneous eigenform for the
Hecke operators in this subspace is a modular newform
of exact level 
.
For the weight-two newform associated with an elliptic
curve over 
,
the level equals the elliptic curve conductor.
