Cusp Form

A cusp form is a modular form for which the coefficient c(0)=0 in the Fourier series


(Apostol 1997, p. 114). The only entire cusp form of weight k<12 is the zero function (Apostol 1997, p. 116). The set of all cusp forms in M_k (all modular forms of weight k) is a linear subspace of M_k which is denoted M_(k,0). The dimension of M_(k,0) is 1 for k=12, 16, 18, 20, 22, and 26 (Apostol 1997, p. 119). For a cusp form f in M_(2k,0),


(Apostol 1997, p. 135) or, more precisely,


for every epsilon>0 (Selberg 1965; Apostol 1997, p. 136). It is conjectured that the -1/4 in the exponent can be reduced to -1/2 (Apostol 1997, p. 136).

See also

Modular Form

Explore with Wolfram|Alpha


Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 114 and 116, 1997.Selberg, A. "On the Estimate of Coefficients of Modular Forms." Proc. Sympos. Pure Math. 8, 1-15, 1965.

Referenced on Wolfram|Alpha

Cusp Form

Cite this as:

Weisstein, Eric W. "Cusp Form." From MathWorld--A Wolfram Web Resource.

Subject classifications