Eisenstein series ($E_k$)

An Eisenstein series is a modular form constructed by summing a weight factor over a parabolic coset space. For even $k>2$, the level-one holomorphic Eisenstein series is a scalar multiple of ${\sum }_{(m,n)\ne (0,0)}(m\tau +n{)}^{-k}$.

The seriesis quasimodular: $E_2(-1/\tau )={\tau }^2{E}_2(\tau )+6\tau /(\pi i)$.

For $\mathrm{Re}s>1$,is an absolutely convergent modular-invariant function.

For the normalization $(T_{p}f)(\tau )=p^{-1}(f(p\tau )+{\sum }_{b\mathrm{mod}p}f((\tau +b)/p))$, one has