1.2. INTEGER WEIGHT MODULAR FORMS 9

1.2.2. Further classes of modular forms. There are many more examples

of classes of integer weight modular forms which we do not take the time to describe

here. For example, there are the theta-series, Poincar´ e series, generalized Eisenstein

series, and weight 1 cusp forms which are Mellin transforms of suitable Artin L-

functions.

Here recall the notion of a modular form with complex multiplication. Let

K = Q(

√we

−D) be an imaginary quadratic field with discriminant −D, and let OK

be its ring of algebraic integers. A Hecke Gr¨ ossencharacter φ of weight k ≥ 2 with

modulus Λ is defined in the following way. Let Λ be a nontrivial ideal in OK and

let I(Λ) denote the group of fractional ideals prime to Λ. A Hecke Gr¨ossencharacter

φ with modulus Λ is a homomorphism

φ : I(Λ) →

C×

such that for each α ∈

K×

with α ≡ 1 (mod Λ) we have

φ(αOK ) =

αk−1.

Let ωφ be the Dirichlet character with the property that

ωφ(n) :=

φ((n))/nk−1

for every integer n coprime to Λ.

Theorem 1.31. Assume the notation above, and define Ψ(z) by

Ψ(z) :=

a

φ(a)qN(a)

=

∞

n=1

a(n)qn,

where the sum is over the integral ideals a that are prime to Λ and N(a) is the

norm of the ideal a. Then Ψ(z) is a cusp form in Sk

(

Γ0(D · N(Λ)),

(

−D

•

)

ωφ

)

.

Remark 1.32. The cusp form Ψ(z) is a “newform” in the sense of Atkin and

Lehner (see Section 2.5).

Example 1.33. The unique normalized cusp form

f(z) =

∞

n=1

a(n)qn

= q −

6q5

+

9q9

+ ··· ∈ S3 Γ0(16),

−4

•

is a form with complex multiplication by K = Q(i). In this case Λ = (2). The

Hecke Gr¨ ossencharacter φ is defined by letting

φ((α)) :=

α2,

where α is one of the two generators of the ideal (α) satisfying α ≡ 1 (mod Λ).

Notice that if p ≡ 3 mod 4 is prime, then p is inert in K, and so we have a(p) = 0.

If p ≡ 1 mod 4 is prime, then the principal ideal (p) factors as

(p) = (x + iy)(x − iy) with x, y ∈ Z and x odd,

and so it turns out that

a(p) = φ((x + iy)) + φ((x − iy)) =

2x2

−

2y2.