The filtration of a mod l modular form I'm reading a paper of Swinnerton-Dyer, "On l-adic representations for coefficients of modular forms." He defines the notion of "filtration" for mod $\ell$ modular forms: 
If $\tilde{f} \in \tilde{M}$ (the mod $\ell$ modular forms) which is the sum of monomials in $M_k$ such that all $k$ are congruent modulo $\ell-1$, then the filtration of $\tilde{f}$, denoted $\omega(\tilde{f})$ is defined to be the least $k$ such that $\tilde{f} \in \tilde{M}_k$ (since the Eisenstein series $E_{\ell-1}$ reduces to $1$ modulo $l$, we can multiply by powers of it). 
This notion turns out to be very useful for classifying the exceptional primes of a Galois representation attached to a modular form. The first question I'd like to ask is, what's the motivation for it? 
Secondly, I'd like to ask about specific properties. If $f$ is a weight $k$ modular form, what can we say about $\omega(\tilde{f})?$ I am under the impression that the following are true -- am I right? 


*

*$\omega(\tilde{f})$ must be congruent to $k$ modulo $\ell-1$ (an in particular, it is $k$ if $\ell > k$)

*$\omega(\tilde{f}) \leq k$.


I'm confused because on p. 29, Swinnerton-Dyer says $\omega(\tilde{f}) = k$ if $\ell > 2k$...
 A: If $M_k$ denotes the space of weight $k$ modular forms with coefficients in $\mathbb Z$, then there is an embedding
$M_k \hookrightarrow \mathbb Z[[q]]$ given by taking $q$-expansion.
This induces a map
$$\bigoplus_k M_k \rightarrow \mathbb Z[[q]]$$
which is again an embedding (although not longer quite obviously so).
If we let $\widetilde{M}_k$ denote the image of $M_k$ in $\mathbb F_{\ell}[[q]]$,
then there is similarly an induced map
$$\bigoplus_k \widetilde{M}_k \rightarrow \mathbb F_{\ell}[[q]],$$
but this is no longer an embedding.  If we denote its image by
$\widetilde{M}$, then this is the ring of mod $\ell$-modular forms.
It is the sum of the various $\widetilde{M}_k$s, but is not their
direct sum.
Assuming $\ell \geq 5$, its kernel is generated by $E_{\ell} - 1$.
Note that this is not a homogenous element of the source.  Thus
the image is not naturally graded.  However, any graded ring is also 
naturally filtered --- in our case we filter the direct sum by the
subobjects $\bigoplus_{i = 0}^k \widetilde{M}_i$ ---
and the image of a filtered ring is naturally filtered --- in our case we define $\widetilde{M}_{\leq k}$ to be the image of $\bigoplus_{i = 0}^k \widetilde{M}_i$ in $\mathbb F_{\ell}[[q]]$.
Then $\widetilde{M}$ is the union of the $\widetilde{M}_{\leq k}$.
We say that an element of $\widetilde{M}$ has filtration $k$ if it
lies in $\widetilde{M}_{\leq k}$, but not in $\widetilde{M}_{\leq k-1}$.
So, regarding motivation: it is the what replaces the notion of weight
for an element of $\widetilde{M}$.  In short, if we are handed a $q$-expansion
in $\mathbb F_{\ell}[[q]]$ and told that it is the $q$-expansion of a modular
form mod $\ell$, the weight is not intrinsically determined by the $q$-expansion
(unlike in the case with char. $0$ coeffients), since e.g. the $q$-expansion $1$
is the $q$-expansion of the wt. $0$ modular form $1$, the weight $\ell-1$ modular forms $E_{\ell -1} $, and more generally the weight $(\ell-1)i$ module
forms $E_{\ell -1}^i$ for any $i$.  But the filtration of the $q$-expansion is well-defined.  When you sort it out, it is essentially the minimal weight of a modular form having that given $q$-expansion.
As for our more specific question:
since the kernel of the $q$-expansion map is generated by $E_{\ell -1} - 1$,
any non-zero element of $\widetilde{M}_k$ must have filtration congruent to $k$ mod $\ell - 1$.  (In short, the grading mod $\ell -1$ 
is well-defined.)  So if $k < \ell -1$ then any non-zero element of $\widetilde{M}_k$ must have filtration equal to $k$.
