group, weight, character, base_field) |
ambient_space,
base_field,
basis,
change_base,
character,
cuspidal_subspace,
decompose,
eisenstein_subspace,
embedded_subspace,
group,
has_character,
hecke_matrix,
intersect,
is_ambient,
key,
level,
modular_symbols,
newspaces,
sturm_bound,
weight
These methods are defined as follows:
) |
) |
) |
) |
) |
) |
) |
This function returns a list of subspaces
corresponding to newforms
of some level dividing the
level of self, such that the direct sum of the subspaces
equals self, if possible. The space
is the image
under
maps to
of the intersection with
of the space spanned by the conjugates of
,
where
is the base ring of self.
) |
) |
) |
) |
n) |
right) |
) |
) |
) |
) |
) |
This function returns a list of subspaces
and
, corresponding to levels
dividing
and integers
dividing
, such that self is the direct sum of these
spaces, if possible. Here
is by definition
the image under
of the new subspace of
cusp forms of level
, and similarly
is the image of
Eisenstein series.
Notes: (1) the subspaces
need not be stable under
Hecke operators of index dividing
. (2) Since self can
be an arbitrary subspace, there's no guarantee any
or
is in self, so the return list could be empty.
[M=None]) |
For a space M of modular forms, this function returns an integer B such that two modular forms in either self or M are equal if and only if their q-expansions are equal to precision B. If M is none, then M is set equal to self.
NOTES: Reference for the Sturm bound that we use in the definition of of this function:
J. Sturm, On the congruence of modular forms, Number theory (New York, 1984-1985), Springer, Berlin, 1987, pp. 275-280.
Useful Remark:
Buzzard pointed out to me (William Stein) in Fall 2002
that the above bound is fine for Gamma1 with character, as
one sees by taking a power of
. More precisely, if
for first
coefficients, then
for first
coefficents. Since the
weight of
is
, it follows that
if
the sturm bound for
at weight(f),
then
has valuation large enough to be forced to be
0
at
weight(f) by Sturm bound (which is valid
if we choose
right). Thus
.
Conclusion: For
with fixed character, the
Sturm bound is exactly the same as for
.
A key point is that we are finding
generators
for the Hecke algebra here, not
-generators. So if
one wants generators for the Hecke algebra over
,
this bound is wrong.
This bound works over any base, even a finite field.
There might be much better bounds over
, or for
comparing two eigenforms.
) |
Instances of class ModularFormsSpace also have the following special methods:
right) |
right) |
x, [check=True]) |
x) |
x) |
True if x is an element or subspace of self.
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