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).