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.

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 $y=1/n$.