4 SCOTT AHLGREN AND NICKOLAS ANDERSEN
2. Preliminaries
We begin with some brief background and a proposition about the action of
the Hecke operators on the spaces in question. It will be most natural to work with
the Fricke groups
Γ∗(N)
for N {1, 2, 3, 4} (see [7, Section 1.6] for background).
For these levels, the groups are generated by the translation
T :=
1 1
0 1
and the Fricke involution
WN :=
0 −1
N 0
.
Let k be a positive integer. If γ =
(
a b
c d
)
GL2
+
(Q), define the slash operator
k
by
f
k
γ := (det
γ)k/2(cz
+
d)−kf
az + b
cz + d
.
Define Γ0(M, N) :=
(
a b
c d
)
Γ0(1) : M|c and N|b . For primes p, define the Hecke
operator Tk(p) by
(2.1) f Tk(p) := f U(p) +
pk−1f(pz)
= p
k
2
−1
p−1
λ=0
f
k
1 λ
0 p
+ f
k
p 0
0 1
.
For (t, p) = 1, define the conjugated operator
Tkt)(p) (
:= AtTk(p)At
−1
, where
At :=
t 0
0 1
.
Then
(2.2) f
Tkt)(p) (
:= p
k
2
−1
p−1
λ=0
f
k
1
0 p
+ f
k
p 0
0 1
,
and if f =

af
(n)qn/t,
then
(2.3) f
Tkt)(p) (
=
(
af (pn) +
pk−1af
(n/p)
)
qn/t.
For prime powers
pn
we have
Tkt)(pn) (
=
AtTk(pn)At
−1
and the recurrence relation
(2.4)
Tkt)(pn+1) (
=
Tkt)(pn)Tkt)(p) ( (

pk−1Tkt)(pn−1).(
We suppress the subscript k when it is clear from context.
We say that ν is a multiplier system for a group Γ if ν is a character on Γ of
absolute value 1 (see [7, Section 1.4] for details). Then
Mk(Γ,ν) ! is the space of
holomorphic functions f on H whose poles are supported at the cusps of Γ, and
which satisfy
(2.5) f
k
γ = ν(γ)f
for all γ Γ.
The multiplier system νη on
Γ∗(1)
for the Dedekind η function
η(z) := q
1
24

n=1
(1
qn)
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