1.3. L-functions 11

Therefore, Θ4(q) = λf(q) + μg(q) for some constants λ, μ ∈ C. We

may compare q-expansions to find the values of λ and μ:

Θ4(q)

=

n≥0

S4(n)qn

= 1 + 8q +

24q2

+

32q3

+

24q4

+ · · ·

λf(q) + μg(q) = λ + μq +

24λq2

+

4μq3

+ · · · .

Therefore, it is clear that λ = 1 and μ = 8, so

Θ4

= f + 8g. Since

the expansions of f and g are easy to calculate (for example, using

Sage; see Appendix A.2), we can easily calculate the coeﬃcients of

the q-expansion of Θ and, therefore, values of S4(n).

The exact formulas given above for Sk(n), however, follow from

some deeper facts. Here is a sketch of the ideas involved (the reader

may skip these details for now and return here after reading Chapter

4): given

Θ4

=

∑

cnqn

and F (q) =

∑

(

∑

d|n

d)qn,

one can find an

eigenvector G(q) =

∑

bnqn

for a collection of linear maps Tn (the so-

called Hecke operators, Tn : M2(Γ1(4)) → M2(Γ1(4))) among spaces

of modular forms, i.e., Tn(G) = λnG for n 1, and the eigenvalues

λn = bn/b1 =

∑

d|n

d. Moreover, the eigenvector G can be written

explicitly as a combination of Θ4 and F . Finally, one can show that

the coeﬃcients cn must be given by the formula cn = 8

∑

d|n, 4 d

d

(see [Kob93], III, §5, for more details).

1.3. L-functions

An L-function is a function L(s), usually given as an infinite series of

the form

L(s) =

∞

n=1

ann−s

=

∞

n=1

an

ns

= a1 +

a2

2s

+

a3

3s

+ · · ·

with some coeﬃcients an ∈ C. Typically, the function L(s) con-

verges for all complex numbers s in some half-plane (i.e., those s

with real part larger than some constant), and in many cases L(s)

has an analytic or meromorphic continuation to the whole complex

plane. Mathematicians are interested in L-functions because they

are objects from analysis that, sometimes, capture very interesting

algebraic information.