16 1. BASIC FACTS
1.3.3. Dimension formulas. In Section 1.2.3 we provided explicit formulas
for the dimensions of spaces of integer weight modular forms with Nebentypus.
These formulas were obtained by Cohen and Oesterl´ e in [COe]. The same paper
includes formulas for half-integral weight spaces. Because of their importance, we
give them here. To state these results we assume the notation from Section 1.2.3
with one exception. We let r2 denote the power of 2 dividing 4N. We also require an
auxiliary parameter ζ(k, 4N, χ). Suppose that k 1
2
+ Z, and that χ is a Dirichlet
character modulo 4N for which χ(−1) = 1. We define ζ(k, 4N, χ) as follows. If
r2 3, then
(1.23) ζ(k, 4N, χ) :=
λ(r2, s2, 2) if r2 4,
3 if r2 = 3.
If r2 = 2 and there is a prime p 3 (mod 4) for which p | 4N with either rp odd
or 0 rp 2sp, then let
(1.24) ζ(k, 4N, χ) := 2.
In the remaining cases, we have r2 = 2, and every prime p 3 (mod 4) with p | 4N
(if there are any) has the property that rp is even and rp 2sp. In these cases we
let2
(1.25) ζ(k, 4N, χ) :=

3/2





5/2



5/2





if k
1
2
2Z and s2 = 0,
if k
1
2
2Z and s2 = 2,
if k 3
2
2Z and s2 = 0,
3/2 if k
3
2
2Z and s2 = 2.
Theorem 1.56. If k
1
2
+ Z, and χ is a Dirichlet character modulo 4N for
which χ(−1) = 1, then
dimC (Sk(Γ0(4N), χ)) dimC (M2−k(Γ0(4N), χ))
=
(k 1)4N
12
p|4N
(1 +
p−1)

ζ(k, 4N, χ)
2
p|4N,
p=2
λ(rp, sp, p),
where p is a prime divisor of 4N (note. empty products are taken to be 1).
Remark 1.57. Remarks analogous to Remark 1.35 apply for Theorem 1.56.
Specifically, if k
3
2
or k
1
2
, then the left hand side of the formula in Theorem
1.56 reduces to a single term.
1.4. Dedekind’s eta-function
Here we consider the combinatorial and modularity properties of Dedekind’s
eta-function. This function will prove to be quite important throughout this mono-
graph. We begin with its formal definition.
2This
corrects a typographical error in the table on page 73 of [COe] in the cases referred to
as non (C).
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