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.
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