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.