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.
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