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.
