Quotients of arbitrary dimension

*For a newform
,
consider the ring homomorphism
that sends
to
.
The kernel
of this homomorphism is a saturated prime ideal
of
.
The newform quotient
associated to
is the quotient
.
Shimura introduced
in [Shi73] where he proved
that
is an abelian variety over
of dimension equal to the
degree of the field
. He also observed
that there is a natural map
with kernel
.
*

*For the rest of this section, fix a quotient
associated
to a saturated ideal
of
; note that
may or may not be attached
to a newform.
The modular degree tex2html_wrap_inline$m_A$ of an optimal quotient tex2html_wrap_inline$A$ of tex2html_wrap_inline$J$ is
the (positive) square root of the degree of the induced
composite tex2html_wrap_inline$A^&or#vee; &rarr#to;J^&or#vee;&cong#cong;J &rarr#to;A$.
*