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53
6.3
Let us give some examples.
Example
6.1
.
Let
Δ(
τ
) =
η
(
τ
)
24
.
It is called the
discriminant
function. We know that Δ(
τ
) satisﬁes (6.1) with
k
= 6
with respect to the group Γ = SL(2
,
Z
). By (4.9)
Δ(
τ
) =
1
(2
π
)
8
ϑ
0
1
2
1
2
8
.
Since
Δ(
τ
) =
q
∞
Y
m
=1
(1

q
m
)
24
we see that the Fourier expansion of Δ(
τ
) contains only positive powers of
q
. This
shows that Δ(
τ
) is a cusp form of weight 6.
Example
6.2
.
The function
ϑ
00
(
τ
) has the Fourier expansion
P
q
m
2
/
2
. It is convergent
at
q
= 0. So
ϑ
4
k
00
is a modular form of weight
k
. It is not a cusp form.
Let us give more examples of modular forms. This time we use the groups other
than SL(2
,
Z
). For each
N
let us introduce the
principal congruence subgroup
of
SL(2
,
Z
) of
level
N
Γ(
N
) =
{
M
=
„
α
β
γ
δ
«
∈
SL(2
,
Z
) :
M
≡
I
mod
N
}
.
Notice that the map
SL(2
,
Z
)
→
SL(2
,
Z
/N
Z
)
,
„
α
β
γ
δ
«
→
„
¯
α
¯
β
¯
γ
¯
δ
«
is a homomorphism of groups. Being the kernel of this homomorphism, Γ(
N
) is a
normal subgroup of Γ(1) = SL(2
,
Z
). I think it is time to name the group Γ(1). It is
called the
full modular group
.
We have
Lemma 6.2.
The group
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.
 Fall '09
 ONTONKONG

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