10 1. Introduction
The formulas for Sk(n) given above are derived using the theory of
modular forms, as follows. We define a formal power series Θ(q) by
Θ(q) =

j=−∞
qj2
and, for k 2, consider the power series expansion of the kth power
of Θ:
(Θ(q))k
=



j=−∞
qj2
⎞k

=

a1=−∞
qa1
2
· · ·

ak=−∞
qak
2
=
n≥0
cnqn.
What is the nth coefficient, cn, of
Θk?
If the readers stare at the
previous equation for a while, they will find that cn is given by
cn = #{(a1,...,ak)
Zk
: a1
2
+ · · · + ak
2
= n}.
Therefore, cn = Sk(n) and
(Θ(q))k
=

n≥0
Sk(n)qn.
In other words,
Θk
is a generating function for Sk(n). But, how do we find closed
formulas for Sk(n)? This is where the theory of modular forms be-
comes particularly useful, for it provides an alternative description of
the coefficients of
Θk.
It turns out that, for even k 2, the function Θk is a modular
form of weight k
2
(more precisely, it is a modular form for the group
Γ1(4)), and the space of all modular forms of weight
k
2
, denoted by
M
k
2
(Γ1(4)), is finite dimensional (we will carefully define all these
terms later). For instance, let k = 4. Then M2(Γ1(4)), the space of
modular forms of weight
4
2
= 2 for Γ1(4), is a 2-dimensional C-vector
space and a basis is given by modular forms with q-expansions:
f(q) = 1 +
24q2
+
24q4
+
96q6
+
24q8
+
144q10
+
96q12
+ · · ·
g(q) = q +
4q3
+
6q5
+
8q7
+
13q9
+
12q11
+
14q13
+ · · · .
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