1.3. HALF-INTEGRAL WEIGHT MODULAR FORMS 15

If N is a positive integer and

Dn2

=

(−1)rN,

where D is a fundamental discrimi-

nant, then define H(r, N) by

(1.20) H(r, N) := L(1 − r, χD)

d|n

µ(d)χD(d)dr−1σ2r−1(n/d).

In particular, if D =

(−1)rN

is a fundamental discriminant, then

(1.21) H(r, N) = L(1 − r, χD) = −

B(r, χD)

r

.

If

(−1)rN

=

n2,

then

(1.22) H(r, N) = ζ(1 − r)

d|n

µ(d)dr−1σ2r−1(n/d).

In this notation, Cohen defined the following series [C1].

Definition 1.52. If r ≥ 2 is an integer, then the weight r+

1

2

Cohen-Eisenstein

series is defined by

Hr(z) :=

∞

N=0

H(r,

N)qN

.

Cohen proved the following fundamental result regarding these series.

Theorem 1.53. If r ≥ 2, then

Hr(z) ∈ Mr+

1

2

(Γ0(4)).

Remark 1.54. If r ≥ 2 is an integer, then, generalizing (1.10), let Er+

1

2

(z) be

the Eisenstein series

Er+

1

2

(z) :=

n0 odd,

m

m

n

−4

n

−(r+

1

2

)

(nz +

m)−(r+

1

2

).

Furthermore, let Fr+

1

2

(z) be the Eisenstein series

Fr+

1

2

(z) := Er+

1

2

(−1/4z)z−(r+

1

2

).

It turns out that Γ0(4) has two regular cusps, and these are their corresponding

Eisenstein series in Mr+

1

2

(Γ0(4)). It turns out that

Hr(z) =

2−(2r+1)ζ(1

− 2r) (1 +

i2r+1)Er+

1

2

(z) +

i2r+1Fr+

1

2

(z) .

Example 1.55. It is not diﬃcult to give explicit formulas for many of the

Hr(z) using θ0(z) and F (z). Here we give closed expressions for the first few

Cohen-Eisenstein series:

H2(z) =

(1/120)(θ0(z)5

− 20θ0(z)F (z)),

H3(z) =

(−1/252)(θ0(z)7

−

14θ0(z)3F

(z)),

H4(z) =

(1/240)(θ0(z)9

−

16θ0(z)5F

(z) + 16θ0(z)F

(z)2),

H5(z) =

(−1/132)(θ0(z)11

−

22θ0(z)7F

(z) +

88θ0(z)3F (z)2).