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) =
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) :=
if 2sp ≤ rp = 2r ,
2pr if 2sp ≤ rp = 2r + 1,
if 2sp rp.
In addition, define rational numbers νk and µk by
if k is odd,
−1/4 if k ≡ 2 (mod 4),
1/4 if k ≡ 0 (mod 4),
if k ≡ 1 (mod 3),
−1/3 if k ≡ 2 (mod 3),
1/3 if k ≡ 0 (mod 3).
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) =
dimC(Sk(Γ0(N), χ)) − dimC(M2−k(Γ0(N), χ))
(k − 1)N
λ(rp, sp, p) + νk
x (mod N),
χ(x) + µk
x (mod N),
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