1.2. INTEGER WEIGHT MODULAR FORMS 5

The spaces Mk(Γ1(N)) and Sk(Γ1(N)) have the following decomposition (where

the sums are over all Dirichlet characters χ modulo N):

Mk(Γ1(N)) =

χ

Mk(Γ0(N), χ),

Sk(Γ1(N)) =

χ

Sk(Γ0(N), χ).

(1.7)

1.2.1. Modular forms on SL2(Z). Here we briefly recall some basic facts

regarding modular forms on SL2(Z). These modular forms are easily described in

terms of Eisenstein series which we now define. If k is a positive integer, then let

σk−1(n) be the divisor function

(1.8) σk−1(n) :=

1≤d|n

dk−1,

and define the Bernoulli numbers Bk as the coeﬃcients of the series

(1.9)

∞

k=0

Bk ·

tk

k!

=

t

et − 1

= 1 −

1

2

t +

1

12

t2

− ··· .

Definition 1.18. If k ≥ 2 is even, then the weight k Eisenstein series Ek(z)

is given by

Ek(z) := 1 −

2k

Bk

∞

n=1

σk−1(n)qn.

Proposition 1.19. If k ≥ 4 is even, then Ek(z) ∈ Mk.

Proof. A classical calculation (see, for example, page 110 of [Kob2]) implies

that

(1.10) 2ζ(k)Ek(z) =

(m,n)∈Z2−{(0,0)}

1

(mz + n)k

.

Here ζ(s) is the usual Riemann zeta-function. Since k ≥ 4, this double sum is

absolutely convergent, and is uniformly convergent in any compact subset of H.

Consequently, Ek(z) is a holomorphic function on H.

It is straightforward to verify that

Ek(z) = Ek(z + 1) and Ek −

1

z

=

zkEk(z).

Since the matrices

S =

0 −1

1 0

and T =

1 1

0 1

generate SL2(Z) (also see Example 1.4), it follows that Ek(z) ∈ Mk.

Remark 1.20. Here we consider the Eisenstein series

E2(z) = 1 − 24

∞

n=1

σ1(n)qn.

For z ∈ H, we have (see, for example, page 113 of [Kob2])

z−2E2(−1/z)

= E2(z) +

12

2πiz

.