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].
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