10 1. BASIC FACTS
1.2.3. Dimension formulas. Proposition 1.25 provides formulas for the di-
mensions of Mk and Sk. Using the Riemann-Roch Theorem, Cohen and Oesterl´e
[COe] explicitly computed further dimension formulas which we record here be-
cause of their utility. To state these formulas, suppose that k is an integer, and
that χ is a Dirichlet character modulo N for which χ(−1) =
(−1)k.
If p | N is
prime, then let rp (resp. sp) denote the power of p dividing N (resp. the conductor
of χ). Define the integer λ(rp, sp, p) by
(1.11) λ(rp, sp, p) :=

pr







+
pr −1
if 2sp rp = 2r ,
2pr if 2sp rp = 2r + 1,
2prp−sp
if 2sp rp.
In addition, define rational numbers νk and µk by
νk :=

0







if k is odd,
−1/4 if k 2 (mod 4),
1/4 if k 0 (mod 4),
µk :=

0







if k 1 (mod 3),
−1/3 if k 2 (mod 3),
1/3 if k 0 (mod 3).
(1.12)
In this notation, we have the following dimension formulas.
Theorem 1.34. If k is an integer and χ is a Dirichlet character modulo N for
which χ(−1) =
(−1)k,
then
dimC(Sk(Γ0(N), χ)) dimC(M2−k(Γ0(N), χ))
=
(k 1)N
12
·
p|N
(1 +
p−1)

1
2
p|N
λ(rp, sp, p) + νk
x (mod N),
x2+1≡0
(mod N)
χ(x) + µk
x (mod N),
x2+x+1≡0
(mod N)
χ(x),
where p denotes a prime divisor of N (note. empty products are taken to be 1).
Remark 1.35. If k 2, then dimC(M2−k(Γ0(N), χ)) = 0. Hence the left
hand of side of Theorem 1.34 reduces to dimC(Sk(Γ0(N), χ)). A similar argument
applies when k = 2, and the result depends on whether χ is trivial. If k 0, then
dimC(Sk(Γ0(N), χ)) = 0. In these cases, the left hand side of Theorem 1.34 reduces
to dimC(M2−k(Γ0(N), χ)).
1.3. Half-integral weight modular forms
Although the study of half-integral weight modular forms has its origins in
the classic works of Euler, Gauss and Jacobi (among others), many of their most
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