1.4. DEDEKIND’S ETA-FUNCTION 19

Example 1.66. Suppose that f(z) and g(z) are the eta-quotients given by

f(z) =

η(5z)5

η(z)

= q +

q2

+

2q3

+

3q4

+

5q5

+

2q6

+ ··· ,

g(z) =

η(4z)2η(8z)2

= q −

2q5

−

3q9

+ ··· .

Theorems 1.64 and 1.65 imply that

f(z) ∈ M2 Γ0(5),

•

5

,

g(z) ∈ S2(Γ0(32)).

Furthermore, they also imply that

(1.27) ∆(z) =

E4(z)3 − E6(z)2

1728

=

η(z)24

∈ S12.

Similarly, the Eisenstein series F (z) defined in (1.18) is also an eta-quotient, and it

is given by

F (z) =

η(4z)8

η(2z)4

=

∞

n=0

σ1(2n +

1)q2n+1

∈ M2(Γ0(4)).

The infinite product representation for ∆(z) (i.e. formula (1.27)) is well known.

It is less well known that every modular form on SL2(Z) is a rational function in

η(z), η(2z) and η(4z).

Theorem 1.67. Every modular form on SL2(Z) may be expressed as a rational

function in η(z), η(2z) and η(4z).

Proof. By Theorem 1.23, it suﬃces to express E4(z) and E6(z) as rational

functions in η(z), η(2z), and η(4z). It turns out that

E4(z) =

η(z)16

η(2z)8

+

28

·

η(2z)16

η(z)8

,

E6(z) =

η(z)24

η(2z)12

−

25

· 3 · 5 ·

η(2z)12

−

29

· 3 · 11 ·

η(2z)12η(4z)8

η(z)8

+

213

·

η(4z)24

η(2z)12

.

(1.28)

To prove these identities, first observe that the linear combination of eta-quotients

on the right hand side are modular forms of weight 4 and 6 respectively on the group

Γ0(4) with trivial Nebentypus character. This follows from Theorems 1.64 and 1.65.

Obviously, E4(z) and E6(z) are also holomorphic modular forms on Γ0(4). Using

Theorem 1.34, the identities are implied by the fact that the first few coeﬃcients

agree.

In view of Theorem 1.67, it is natural to ask the following question.

Problem 1.68. Theorem 1.67 asserts that every modular form on SL2(Z) is

a rational function in η(z), η(2z) and η(4z). Classify the spaces of modular forms

which are generated by eta-quotients.