*We start by giving several results regarding the Manin
constant for quotients of arbitrary dimension. The proofs of
most of the theorems are given in Section 4.
*

*Let
be a subgroup of
that contains
.
We have the following generalization of Edixhoven's Theorem 2.2.
*

Combining this chain of inclusions with commutativity of the diagram

where -exp is the Fourier expansion map, we see that the image of lies in , as claimed.

*For the rest of the paper, we take
.
For each prime
with
, let
be
the
th Atkin-Lehner operator. Let
and
be an optimal quotient of
attached to a saturated ideal
.
If
is a prime, then as usual,
will denote the
localization of
at
.
*

*Let
denote the abelian subvariety of
generated by
the images of the degeneracy maps from levels that properly divide
(see, e.g., [Maz78, §2(b)]) and let
denote the
quotient of
by
.
A new quotient is a quotient
that factors through the map
.
The following corollary generalizes Mazur's Theorem 2.3:
*

with respect to the basis

for . Thus does not preserve . In fact, the Manin constant of is not in this case (see Section 3.4). Note that Theorem 3.5 implies that the only primes that can divide the Manin constant of any optimal quotient of tex2html_wrap_inline$J_0(33)$are tex2html_wrap_inline$2$ and tex2html_wrap_inline$3$.

The hypothesis of Theorem 3.5 sometimes holds for non-new . For example, take and . Then acts as an endomorphism of , and a computation shows that the characteristic polynomial of on is and on is , where is the old subspace of . Consider the optimal elliptic curve quotient , which is isogenous to . Then is an optimal old quotient of , and acts as on , so preserves the corresponding spaces of modular forms. Thus Theorem 3.5 implies that .

*The following theorem generalizes Raynaud's
Theorem 2.4 (see also [GL01] for
generalizations to
-curves).
*

*
Let tex2html_wrap_inline$S_2( Z)[I]^&perp#perp;$ be the orthogonal complement of
tex2html_wrap_inline$S_2(Z)[I]$ in tex2html_wrap_inline$S_2(Z)$ with respect to the Petersson inner
product.
theorem_type[defi][lem][][definition][][]
[Congruence exponent and number]
The congruence number tex2html_wrap_inline$r_A$ of tex2html_wrap_inline$A$is the order of the quotient group
equation S_2(Z)/ (S_2(Z)[I] + S_2(Z)[I]^&perp#perp;).
This definition of tex2html_wrap_inline$r_A$ agrees with Definition
when tex2html_wrap_inline$A$ is an elliptic curve (see
[AU96, p. 276]).
*

**Let
denote the natural quotient map
.
When we compose
with its dual
(identifying
with
using
the inverse of the principal polarization of
),
we get an isogeny
.
The modular exponent
of
is the exponent of the group
.
When
is an elliptic curve, the modular exponent is just
the modular degree of
(see, e.g., [AU96, p. 278]).
**

**The theorems above
suggest
that the Manin constant is
for quotients associated to newforms
of square-free level.
In the case when the level is not square free, computations of
[FpS+01] involving Jacobians of genus
curves that are
quotients of
show that
for
two-dimensional newform quotients.
These include quotients having the following
non-square-free levels:
**

**The above observations suggest the following conjecture,
which generalizes Conjecture 2.1:
**

**It is plausible that
for any newform on any congruence
subgroup between
and
. However, we do not have
enough data to justify making a conjecture in this context.
**