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115
Corollary 11.3.
Keep the notation from the previous lemma. Assume
f
is normalized
so that
c
1
= 1
. Then
c
m
c
n
=
c
mn
if
(
m,n
) = 1
,
c
p
c
p
n
=
c
p
n
+1
+
p
2
k

1
c
p
n

1
where
p
is prime and
n
≥
1
.
Proof.
The coeﬃcient
c
n
is equal to the eigenvalue of
T
(
n
) on
M
k
(Γ(1)). Obviously
c
m
c
n
is the eigenvalue of
T
(
n
)
T
(
m
) on the same space. Now we apply assertion (ii)
taking into account that the correspondence
R
p
acts as multiplication by
p

2
k
and
remember that we have introduced the factor
n
2
k

1
in the deﬁnition of the operator
T
(
n
).
Example
11.2
.
Let
E
k
(
τ
) be the Eisenstein modular form of weight 2
k
,
k
≥
2. We
have seen in (6.21) that its Fourier coeﬃcients are equal to
c
n
=
2(2
π
)
k
σ
2
k

1
(
n
)
(
k

1)!
,
n
≥
1
,
c
0
= 2
ζ
(
k
) =
2
2
k

1
π
k
B
k
(2
k
)!
.
Thus
c
n
=
c
1
σ
2
k

1
(
n
), and therefore
E
k
(
τ
) is a simultaneous eigenvalue of all the
Hecke operators.
Corollary 11.4.
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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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