1.3. HALF-INTEGRAL WEIGHT MODULAR FORMS 13

Theorem 1.45. Suppose that N is a positive integer and that χ is an even

Dirichlet character modulo 4N. Let Ω(4N, χ) denote the set of pairs (ψ, t), where

t is a positive integer, and where ψ is an even primitive Dirichlet character with

conductor r(ψ) satisfying the following:

(1) We have

r(ψ)2t

| N.

(2) We have χ(n) = ψ(n)

(

t

n

)

for every integer n coprime to 4N.

Then the set of theta-functions θ(ψ, 0, tz) with (ψ, t) ∈ Ω(4N, χ) is a basis of the

space M

1

2

(Γ0(4N), χ).

Serre and Stark also obtained a complete description of the cusp forms of weight

1/2. To state this result, recall that every Dirichlet character ψ of conductor r(ψ)

may be written uniquely as

ψ =

p|r(ψ)

ψp,

where ψp is a Dirichlet character whose conductor is the highest power of the prime

p dividing r(ψ). We say that ψ is totally even if ψp(−1) = 1 for every prime p | r(ψ).

Using this terminology and the notation from Theorem 1.45, we have the following

basis theorem for weight 1/2 cusp forms.

Theorem 1.46. The set of theta-functions θ(ψ, 0, tz), as (ψ, t) varies over the

elements Ω(4N, χ) for which ψ is not totally even, forms a basis of S

1

2

(Γ0(4N), χ).

Remark 1.47. The first level for which dimC S

1

2

(Γ0(4N), χ) = 0 is

4N = 576.

In this case if χ =

(

12

•

)

, then dimC S

1

2

(Γ0(576), χ) = 1, and this space is generated

by the theta-function

θ(χ, 0, z) =

1

2

∞

n=−∞

χ(n)qn2

= q −

q25

−

q49

+

q121

+ ··· .

Remark 1.48. Theorems 1.45 and 1.46 imply that that there are no “exotic”

weight 1/2 modular forms. The situation is very different for half-integral weights

λ +

1

2

≥

3

2

.

1.3.2. Forms on Γ0(4). In Section 1.2.1 we gave a complete description of

the spaces Sk and Mk of integer weight modular forms on SL2(Z). Here we provide

the analogous description for the “level one” half-integral weight modular forms

(i.e. those forms on Γ0(4)). We provide a complete description of the spaces

Mk(Γ0(4), ψk) where k ∈

1

2

N and

(1.17) ψk :=

χ0 if k ∈ 2Z or k ∈

1

2

+ Z,

χ−4 =

(

−4

•

)

if k ∈ 1 + 2Z.

Here we provide the Γ0(4) analog of Theorem 1.23. To state this result, we

require the weight 2 Eisenstein series

(1.18) F (z) =

∞

n=0

σ1(2n +

1)q2n+1

∈ M2(Γ0(4)).

The following is proved in [C1, Kob2].