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 tλ

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)