Let
f
f
be an overconvergent
p
p
-adic eigenform of level
N
p
r
Np^r
,
r
≥
1
r\geq 1
, with non-zero
U
p
U_p
-eigenvalue. We show how
f
f
may be analytically continued to a subset of
X
1
(
N
p
r
)
a
n
X_1(Np^r)^{\mathrm {an}}
containing, for example, all the supersingular locus. Using these results we extend the main theorem of our earlier work with R. Taylor to many ramified cases.