Invariant norm of a modular form ($y^{k/2}|f(\tau )|$)

For a weight-$k$ modular form on the full modular group, the quantityis modular invariant. For a cusp form it tends to zero at the cusp and is therefore bounded on a fundamental domain.

For $k>0$, a meromorphic function satisfying the modular transformation law is a cusp form exactly when its invariant norm is bounded. The bound first excludes interior poles. At each cusp a genuine periodic slash translate is bounded by $C{y}^{-k/2}$, which tends to zero. Its punctured-disk expression therefore has a removable singularity with zero value. Conversely cusp Fourier expansions decay exponentially, and finitely many compact core regions control the rest. Positive weight is essential: a nonzero constant in weight zero satisfies the bound but is not a cusp form.

If $f(\tau )={\sum }_{n\ge 1}{a}_n{q}^n$ is a weight-$k$ level-one cusp form, boundedness of its invariant norm givesIntegrate $f(x+iy)e^{-2\pi inx}$ over one period and choose $y=1/n$.