6
SCOTT AHLGREN AND KEN ONO
It follows that
P2,c(z)/r(
(z)
=
7]23
(z)C(z).
Recall that 1J(24z)
E
S1; 2(fo(576), X12), and that
1Jc(ez)
:=
1]£
2
(z) (mod£). Using
these facts together with (3.4), we conclude that there exists a cusp form
Pc(z)
in
Scc2-2)/2(fo(576),
xd
n
Z[[q]] for which
Pe(z)
=
L p(n- Oe)q24n-C
2
+
2
L
p(n- Oc)q24n-£
2 (mod£).
n=O
(mod£)
(7)=-E£
This gives Theorem 3.1. D
4. Examples
Here we present examples of Theorem 3.1 for the primes £
=
5, 7, and 11.
Using [SwD, §3, Lemma 6], one can verify that
P2,1
=
2~ 3
El
+
6~ 4 (mod 7),
Consequently,
we have
00 00
(4.1)
LP(5n
+
1)ql20n+23
+
LP(5n
+
2)ql20n+47
=
'r/23(24z) (mod 5),
n=O n=O
00
00 00
(4.2)
LP(7n
+
1)ql68n+23
+
LP(7n
+
3)ql68n+71
+
LP(7n
+
4)ql68n+95
n=O n=O n=O
=
7]23
(24z)Et(24z)
+
37]47
(24z) (mod 7),
00 00 00
LP(lln
+
1)q264n+23
+
L
p(lln
+
2)q264n+47
+
L
p(lln
+
3)q264n+71
n=O n=O n=O
00 00
+
L
p(lln
+
5)q264n+ll9
+
L p(lln
+
S)q264n+l91
n=O n=O
(4.3)
5. Conjectures
We conclude with a variety of conjectures and open problems. We begin with
questions related to the existence of further Ramanujan-type congruences.
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