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.

See also

Elliptic Curve Conductor, Modular Form, Modular Group Gamma0, Modular Newform, Weight

Explore with Wolfram|Alpha

References

Koblitz, N. Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 1993.

Cite this as:

Weisstein, Eric W. "Modular Form Level." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModularFormLevel.html

Subject classifications