Let
F
F
be a nonarchimedean local field of residue characteristic
p
p
, and let
r
r
be an odd natural number less than
p
p
. Using the work of Moy and Tadić, we find an element
z
z
of the Bernstein center of
G
=
G
L
2
(
F
)
G = \mathbf {GL}_2(F)
that acts on any representation
π
\pi
of
G
G
by the scalar
z
(
π
)
=
tr
(
Frob
;
(
Sym
r
∘
φ
π
)
I
F
)
z(\pi ) = \operatorname {tr} \left (\operatorname {Frob} ; \left ( \operatorname {Sym}^r \circ \varphi _{\pi }\right )^{I_F} \right )
, the trace of any geometric Frobenius element
Frob
\operatorname {Frob}
of the absolute Weil group
W
F
W_F
of
F
F
, acting on the inertia-fixed points of the representation
Sym
r
∘
φ
π
\operatorname {Sym}^r \circ \varphi _{\pi }
of
W
F
W_F
, where
φ
π
:
W
F
→
G
^
\varphi _{\pi } : W_F \rightarrow \hat {G}
is the restriction to
W
F
W_F
of the Langlands parameter of
π
\pi
. This element
z
z
is specified by giving the functions obtained by convolving it with the characteristic functions of a large class of compact open subgroups of
G
G
, that includes all the groups of both the congruence and the Iwahori filtrations of
G
G
having depth at least one.