1.2. INTEGER WEIGHT MODULAR FORMS 7

Theorem 1.23. If k ≥ 4 is even, then Mk is generated by monomials of the

form

E4(z)aE6(z)b,

where a, b ≥ 0 and 4a + 6b = k.

Using Theorem 1.23 to compute dimensions, Proposition 1.19 implies that

Mk = CEk (z)

for k ∈ {4, 6, 8, 10, 14}. As a consequence, we obtain elementary identities such as

E8(z) =

E4(z)2,

E10(z) = E4(z)E6(z),

E14(z) =

E4(z)2E6(z).

The weight k = 12 is the smallest integer for which Sk = {0}.

Definition 1.24. The Delta-function is the unique cusp form of weight 12 on

SL2(Z) normalized so that its leading Fourier coeﬃcient equals 1. In terms of E4(z)

and E6(z), we have

∆(z) :=

E4(z)3

−

E6(z)2

1728

= q −

24q2

+

252q3

− ··· ∈ Z[[q]].

The map

Ψk : Mk → Sk+12

defined by Ψk(f(z)) := f(z)∆(z) is an isomorphism. Therefore, Theorem 1.23

immediately gives the following dimension formulas for Mk and Sk.

Proposition 1.25. If k ≥ 4 is even, then

dimC(Sk) = dimC(Mk) − 1,

and

dimC(Mk) =

[k/12]

+ 1 if k ≡ 2 (mod 12),

[k/12] if k ≡ 2 (mod 12).

Modular functions on SL2(Z) are also simple to describe in terms of Eisenstein

series. To observe this we recall the modular j-function.

Definition 1.26. The modular j-function j(z) is defined by

j(z) :=

E4(z)3

∆(z)

=

q−1

+ 744 + 196884q +

21493760q2

+ ··· .

Remark 1.27. In terms of E6(z) and ∆(z), it also turns out that

j(z) − 1728 =

E6(z)2

∆(z)

.