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
.
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