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).
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