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\title{\sc Approximation of Eigenforms of
Infinite Slope by Eigenforms of Finite Slope}
\author{Robert F.~Coleman \and{} William A.~Stein}
\begin{document}
\maketitle
\tableofcontents
\section{Introduction}
Fix a prime~$p$.
Consider a classical newform
$$
F = \sum_{n\geq 1} a_n q^n \in S_k\left(\Gamma_1(Np^t),\Qpbar\right)
$$
where~$k$ and~$N$ are positive integers and $p\nmid N$ is a prime
(by a {\em newform} we mean a Hecke eigenform that lies
in the new subspace and is normalized so that $a_1=1$).
The {\em slope} of~$F$ is $\ord_p(a_p)$, where $\ord_p(p)=1$.
By \cite[Prop.~3.64]{shimura:intro}, the twist
$$
F^{\chi} = \sum \chi(n)a_n q^n
$$
of~$F$
by any Dirichlet character~$\chi$ of conductor dividing~$p$
is an eigenform on $\Gamma_1({Np^{\max{\{t+1,2\}}} })$. This
twist has infinite slope.
In Section~\ref{sec:canapprox}, we prove that if~$F$ has finite slope
then it is possible to approximate $F^{\chi}$ arbitrarily closely by
(classical) finite slope {\em eigenforms}.
Assuming refinements of standard conjectures, the best estimate
we obtain for the smallest weight of an approximating eigenforms is
exponential
in the approximating modulus $p^A$. Section~\ref{sec:numerical}
contains computations that suggest that the best estimates should
have weight that is linear in $p^A$.
One motivation for the question of approximation of infinite slope
eigenforms by finite slope eigenforms is the desire to understand the
versal
deformation space of a residual modular representation
\cite{mazur:deforming}
(the deformation space of an
irreducible representation is universal~\cite{mazur:deforming} as is the
deformation space of a residual pseudo-representation
\cite{eigencurve}).
In \cite{gouvea-mazur:density} (see also \cite{mazur:infinite_fern}, and
\cite{bockle:density} for a generalization),
it was shown that the Zariski closure
of the locus of finite slope modular deformations of an absolutely
irreducible ``totally unobstructed''
residual modular representation is Zariski dense in the associated
representation space but very little is known about the topological
closure of this locus. For example, it is not known if it contains
any nonempty open sets. Our result implies that it contains tamely
ramified
twists of modular deformations. We also show in
Section~\ref{sec:hatada}
that a result of Hatada
implies that in at least one (albeit not irreducible) case it does not
contain all modular deformations.
Our investigation began with with our answer in
Section~\ref{sec:canapprox} to a question of Jochnowitz. The idea of
studying the \pad\ variation of modular forms began with Serre
\cite{serre:antwerp72} and was since developed by Katz
\cite{katz:annals75} and Hida \cite{hida:iwasawa} (see also
\cite{gouvea:slnm} for a sketch of the theory). It follows, in
particular, from their work, that one can approximate all forms on
$X_0(p^n)$ with forms on the $j$-line $X_0(1)$, but {\em not}
necessarily with {\em eigenforms}.
We prove the above result about twists in Section~\ref{sec:canapprox},
then state some questions about approximation by finite slope
forms in Section~\ref{sec:questions}.
We explain how to reinterpret Hatada's result in
Section~\ref{sec:hatada}, then
present the results of our computations in
Section~\ref{sec:numerical}.
Based on the results and computations discussed in this article,
Mazur has suggested that it may be the case that an infinite slope
eigenform can be
approximated by finite slope eigenforms only if the corresponding
representation is what he calls {\em tamely semistable} (i.e.,
semistable, in
the sense of \cite{colmez_fontaine}, after a tame extension).
{\bf Acknowledgments.} The authors thank Naomi Jochnowitz for
provoking this line of thought and for interesting conversations,
Barry Mazur for helpful comments and questions, Frank Calegari for
conversations, Lo\"\i{}c Merel for his comments on an early draft of
this paper, and the referee for a brilliant report.
\section{Approximating Teichm\"uller Twists of Finite Slope Eigenforms}%
\label{sec:canapprox}%
This section is the theoretical heart of the paper. We prove that
the infinite slope eigenforms obtained
as twists of finite slope eigenforms by powers of the Teichm\"uller
character can always be approximated by finite
slope eigenforms. We first show that certain overconvergent
eigenforms of sufficiently close weight are congruent and have the
same slope. Then we use the~$\theta$ operator on overconvergent forms
to deduce the main result (Theorem~\ref{thm:twistapprox}) below.
Let~$p$ be a prime. All eigenforms in this
section will be cusp forms with coefficients in $\Qpbar$
normalized so that $a_1=1$.
Suppose
$
F = \sum_{n\ge 1}a_nq^n
$
is an eigenform
and~$\chi:(\Z/M\Z)^* \to \C_p^*$
is a Dirichlet character with modulus~$M$,
which we extend to $\Z/M\Z$ by setting $\chi(n)=0$ if $(n,M)\neq 1$.
Then the twist of~$F$ by~$\chi$ is the eigenform
$$
F^\chi = \sum_{n\ge 1}\chi(n)a_nq^n.
$$
Let $\omega\colon \zms p\ra\Z_p^*$ be the Teichm\"uller character
(so $\omega(n) \con n \pmod{p}$).
The following theorem concerns finite slope approximations
of twists of~$F$ by powers of~$\omega$. For example, it
concerns the twist
$$
F^{\omega^0} = \sum_{(n,p) = 1} a_n(F) q^n
$$
of $F$ by the trivial character mod~$p$,
which we call the ``$p$-deprivation'' of~$F$
and which has infinite slope.
\begin{theorem}\label{thm:twistapprox}
Suppose~$F$ is a classical eigenform on
$X_1(Np^{t})$, $t\ge 1$, over $\Qbar_p$ of weight~$k$,
character~$\psi$, and finite slope at~$p$.
Let $A\in \Z_{>0}$
and $r,s\in\Z_{\geq 0}$ with $r,s1$ are not yet
defined in the literature).
The ring $A^0(X)$ is finite over $A^0(B)$ by Corollary 6.4.1/5 of
\cite{bgr}.
Let $f_1,\dots,f_n$ be generators.
Let $f_0$ be a uniformizing parameter on~$B$ so that
$A(B)=\C_p\dia{f_0}$, where $\C_p\dia{f_0}$ is the
ring of power series in $f_0$ whose coefficients tend
to~$0$ with their degree.
Let $Z_L(y)$ be the following Weierstr\"ass subdomain of~$X$:
$${\{x\in X\colon |f_i(x)-f_i(y)|\le p^{-L}, 0\le i\le n\}}.$$
Since the functions $x\to \eta_x(T(n))$ lie in $A^{0}(X)$,
it follows that if $x\in Z_L(y)$, then
$$F_x(q)\con H(q)\pmod{p^L}.$$
Finally, since $Z_L(y)$ is a subdomain of~$X$ and~$X$ is
finite over~$B$, the map from $Z_L(y)$ to~$B$ is
quasi-finite. It follows from Proposition A5.5 of \cite{coleman:banach}
that its image in~$B$ is a subdomain. Since~$\kappa$ is the
image of~$y$, its image contains a disk around~$y$.
\end{proof}
%\begin{remark}
%Proposition~\ref{prop:approx} is true even if $H$ is not a cusp form.
%\end{remark}
\begin{proof}[Proof of Theorem~\ref{thm:twistapprox}]
Let $\alpha$ be the slope of $F$.
It follows from Proposition~\ref{prop:approx} that if $m\in\Z$
is sufficiently small \pad ally there exists an overconvergent
eigenform~$K$ of tame level~$N$, weight-character
$ \chi\cdot\dia{\dia{\ }}^{k-m}$ and
slope~$\alpha$ such that $K(q)\con F(q)\pmod{p^A}$.
Suppose $m\ge k$.
Then, by Proposition 4.3 of \cite{coleman:classical}
(see also \cite{coleman:classical_higher}) if
$F_1=\theta^{m-k+1}K$, then
$F_1$ is an overconvergent eigenform
of weight-character
$$
\kappa_1:=\omega^{2(m-k+1)}\cdot\chi\cdot\dia{\dia{\ }}^{k_1},
$$
where
$k_1=m-k+2$, and $F_1$ has finite slope $\alp_1=\alp+m-k+1$.
Applying this same process to $F_1$,
for $\ell\in\Z$ sufficiently small \pad ally
such that $\ell\ge k_1$, we
obtain an overconvergent finite slope eigenform $F_2$ of
weight-character $\kappa_2$, where
$\kappa_2=\omega^{2\ell}\cdot\chi\cdot\dia{\dia{\ }}^{k_2}$
and where
$k_2=\ell-k_1+2=k+\ell-m$,
such that if $F_2(q)=\sum_{n\ge 1}b_nq^n$, then
\begin{align*}
b_n&\con n^{\ell-k_1+1}n^{m-k+1}a_n\\
&\con n^\ell a_n\pmod{p^A}.
\end{align*}
The latter is congruent to $\omega^r(n)a_n\pmod{p^A}$
if $\ell\con r\pmod{\vphi(p^A)}$ and $\ell+v(a_p)\ge A$.
It follows from
\cite[\S8]{coleman:classical},
\cite{coleman:classical_higher},
and \cite{coleman:banach} that
if~$c$ is an integer sufficiently small \pad ally, such
that $c+k_2>v(b_p)+1$ (note that $v(b_p)$ is
the
slope of $F_2$ so is finite)
there exists a classical eigenform~$G$ on $X_1(Np^t)$ of weight
$k_2+c=k+\ell-m+c$, slope $v(b_p)$ and character
$\omega^{m+r-c}\cdot\psi$ such that
$G(q)\con{} F_2(q)\con{} F^{\omega^r}(q)\pmod{p^A}$.
We can choose~$c$ so that
$m+r-c\con s\pmod{p-1}$ and then $k_2+c\con k+2r-s\mod (p-1)$.
\end{proof}
The following corollary addresses a question of Jochnowitz, which
motivated this entire investigation:
\begin{corollary}
Suppose~$R$ is a
classical eigenform of weight~$k$ on $X_1(N)$,
let $A\in \Z_{>0}$, and let $r \in \Z_{\geq 0}$ with $r2v(b_p)+1$. Then~$G$ is old of weight congruent to
$k\mod (p-1)$ and~$G$ is congruent to an eigenform~$S$ of the same weight on
$X_1(N)$ modulo $p^{v(b_p)}$. Since $b_p\con 0\pmod{p^A}$, we obtain
the corollary.
\end{proof}
\begin{remark}\label{remark:estimate}
Assuming a natural refinement of the Gouv\^ea-Mazur
conjectures, the best estimate we obtain for the weight of~$H$ in the
above proof is exponential in~$p^A$. Computational evidence
suggests that the best estimates should have weights
that are linear in $p^A$ (see Section~\ref{sec:numerical}).
\end{remark}
\begin{remark}\label{remark:jochnowitz_mazur}
Jochnowitz and Mazur have independently observed that the above
argument can be used to prove the following result: {\em Suppose~$F$
is an overconvergent eigenform of arithmetic
weight-character~$\kappa$, which is a limit of overconvergent
eigenforms of finite slope. If $\iota\colon \Z_p^*\ra\Z_p^*$ is the
identity character, then the twist $F^{\iota/\kappa}(q)$ of $F$ by
$\iota/\kappa$, which is the $q$-expansion of a convergent eigenform
of weight-character ${\iota}^2/\kappa$, is the limit of overconvergent
eigenforms of finite slope.}
\end{remark}
\begin{remark}\label{remark:twinninig}
One can also approach the $p$-deprivation
(the twist by the $0$th power of Teichm\"uller) of a finite
slope eigenform~$F$ by using the
evil twins of eigenforms approaching~$F$.
\end{remark}
\subsection{Questions}\label{sec:questions}
Some natural questions arise:
\begin{enumerate}
\item Is every $p$-adic convergent eigenform which is the limit of
finite slope overconvergent eigenforms an overconvergent
eigenform? (We can show the twist of an overconvergent eigenform by a
Dirichlet character is overconvergent.)
\item Which infinite slope eigenforms are limits of finite slope
eigenforms?
\item If $F(q)$ is the $q$-expansion of an overconvergent eigenform of
weight-character~$\kappa$, is $F^{\iota/\kappa}(q)$ the $q$-expansion
of an overconvergent eigenform of weight-character $\iota^2/\kappa$
(recall that $\iota$ is the identity character
$\Z_p^*\xrightarrow{\sim}\Z_p^*$)?
Another closely related
question is as follows: Suppose~$\rho$ is the
representation of the absolute Galois group of~$\Q$ attached to an
overconvergent eigenform and let~$\chi$ denote the cyclotomic
character. Then is the representation
$\rho\otimes{\chi\cdot\det(\rho)\iv}$
attached to an overconvergent eigenform?
\end{enumerate}
\section{An Infinite Slope Eigenform that is Not Approximable}\label{sec:cantapprox}
In Section~\ref{sec:hatada}, we prove an extension to higher level of a
theorem of Hatada about the possibilities for systems of Hecke
eigenvalues modulo~$8$. We use this result to deduce that the
normalized weight~$2$ cusp form on $X_0(32)$ is not $2$-adically
approximable by normalized eigenforms of tame level~$1$ and finite
slope.
In Section~\ref{sec:nonapproxmaybe} we give an example of an infinite
slope eigenform of level~$27$ that computer computations suggest cannot
be approximated by finite slope forms.
For related investigations, see \cite{calegari_emerton}.
\subsection{An Extension of a Theorem of Hatada}\label{sec:hatada}
\begin{theorem}\label{thm:genhatada}
If~$F=\sum a_n q^n$ is a normalized cuspidal newform over
$\C_2$ of finite slope on $X_0(2^n)$, then
$a_2 \con 0\pmod{8}$ and $a_p \con p+1\pmod{8}$
for all odd primes~$p$.
\end{theorem}
\begin{proof}
Suppose~$F$ has weight~$k$ and finite slope~$\alp$. The assumption that~$F$
has finite slope implies $n\le 1$.
If $n=0$ the assertion of Theorem~\ref{thm:genhatada}
was proved by Hatada in \cite{hatada:eigenvalues79},
so we may assume that $n=1$ and
$\alp=(k-2)/2$ (in general,
the slope of a newform on $\Gamma_0(p)$
of weight~$k$ is $(k-2)/2$).
Note that $\alp\geq 3$ since there are no newforms
on $X_0(2)$ of weight $<8$.
It follows from Theorems A of \cite{coleman:banach}
(see \S B2 of \cite{coleman:banach} for the extension to $p=2$)
and Theorem B5.7 of \cite{coleman:banach} that if~$j$ is an
integer sufficiently
close $2$-adically to~$k$, then there exists
a classical normalized cuspidal eigenform~$G$ on $X_0(2)$ of weight~$j$
and
slope~$\alp$ such that
$$G(q)\con F(q)\pmod{8}.$$
If in addition we assume that $j>2(\alp+1)$, then~$G$
must be old (since the slope of a newform of weight~$j$
is $(j-2)/2\neq \alpha$).
Thus there is a cuspidal
eigenform $H=\sum b_n q^n$ of level~$1$ such that $G$
is a linear combination of $H(q)$ and $H(q^2)$.
More precisely,
$$
G(q) = H(q) - \rho H(q^2)
$$
where $\rho$ is a root of $P(X) = X^2 - b_2 X +2^{j-1}$. By Hatada's
theorem $\ord_2(b_2)\geq 3$, and $j\geq 12$, so the slopes of
the Newton polygon of $P(X)$ at~$2$ are both at least~$3$.
Thus $G(q)\con H(q)\pmod{8}$, which proves the theorem
because~$H$ has level~$1$.
\end{proof}
\begin{corollary}
Let $G$ be the normalized weight $2$ cusp form
on $X_0(32)$. Then $G$ is not 2-adically approximable
by normalized eigenforms of tame level~$1$ and finite slope.
\end{corollary}
\begin{proof}
If $F_{32}$ were approximable there would have to be a normalized
eigenform $F$ on $X_0(2)$ such that $F_{32}(q)\con F(q)\pmod{8}$.
However,
$F_{32}(q)=\sum_{n=1}^\infty a_nq^n$ where,
$$a_p=
\begin{cases}
2x&\text{if }p=x^2+y^2,\quad\text{written so } x+y\con x^2\!\!\pmod{4}\\
0&\text{otherwise}.
\end{cases}
$$
As $a_3=0\not\con 4\pmod{8}$, we see from Theorem~\ref{thm:genhatada}
that~$F$ does not exist.
\end{proof}
\begin{remark}
If $p\con 1 \pmod{4}$ then the coefficient of $a_p$ in $F_{32}$ agrees
modulo~$8$ with $p+1$. If~$p$ is $3$ mod $4$ it does not because for
$F_{32}$ the coefficient vanishes. What is happening is that there is
a reducible mod 8 pseudo-representation (namely the trivial
one-dimensional representation plus the cyclotomic character) such
that any finite slope level $2^n$ form gives this
pseudo-representation mod 8. Conversely the mod 8 representation
associated to $F_{32}$ is the direct sum of the
quadratic character associated to $\Q(i)$ and the cyclotomic
character. Hence the congruence works when $p=1 \mod 4$ but not otherwise.
\end{remark}
\subsection{Another Eigenform that Conjecturally Cannot be Approximated}%
\label{sec:nonapproxmaybe}
In this section we consider an infinite slope eigenform that is not a
Teichm\"uller twist of a finite slope eigenform. We conjecture that
this eigenform cannot be approximated arbitrarily closely by finite
slope eigenforms.
%The infinite slope weight-two normalized eigenform attached to $X_0(27)$ is
% $$F = q - 2q^4 - q^7 + 5q^{13} + 4q^{16} - 7q^{19} - \cdots$$
\comment{
Let~$G$ be a newform in either $S_k(\Gamma_0(1),\Z_3)$, for
$k\leq 110$, or $S_k(\Gamma_0(3),\Z_3)$, for $k\leq 50$.
Then
$$\min\{\ord_p(a_p(G) - a_p(F)) \,\,:\,\, p \in \{2,5,7,11,13, 17\} \} = 1.$$
Thus none of these newforms~$G$ approximate~$F$
modulo $3^2$. Note that we have omitted forms in
$S_k(\Gamma_0(1),\Zbar_3)$ that are not defined over $\Z_3$, but
there are very few such forms.
This computation suggests that~$F$ cannot be approximated by
finite slope eigenforms.
% and supports Mazur's suggestion as the
%representation attached to~$F$ is not tamely semi-stable.
}
\begin{conjecture}\label{conj:systems3}
There are exactly five residue classes in $(\Z/9\Z)[[q]]$
of normalized eigenforms in $S_k(\Gamma_0(N))$ where $k\geq 1$ and $N=1,3,9$.
They are given in the following table,
where the indicated weight is the smallest weight where that system of eigenvalues
occurs (the level is $1$ in each case):
{\rm\tt \begin{center}\begin{tabular}{|c|l|}\hline
{\rm Weight} & {\rm [ $a_2, a_3, \ldots, a_{43} \mod 9$ ]}\\\hline
% 2 3 5 7 11 13 17 19 23 29 31 37 41 43
% 2 2 1 2 1 2 1 2 2 1 1 2 1
12 & [ 3, 0, 6, 5, 3, 8, 0, 2, 6, 3, 8, 2, 6, 5 ]\\
16 & [ 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2 ]\\
20 & [ 6, 0, 3, 8, 6, 5, 0, 2, 3, 6, 5, 2, 3, 8 ]\\
24 & [ 6, 0, 3, 5, 6, 8, 0, 2, 3, 6, 8, 2, 3, 5 ]\\
32 & [ 3, 0, 6, 8, 3, 5, 0, 2, 6, 3, 5, 2, 6, 8 ]\\
% [ 0, 0, 0, 8, 0, 5, 0, 2, 0, 0, 5, 2, 0, 8 ]
\hline\end{tabular}\end{center}}
\noindent{}The system of eigenvalues mod~$9$
associated to the weight~$2$ form $F$
on $X_0(27)$ is
\begin{center}
{\tt[ 0, 0, 0, 8, 0, 5, 0, 2, 0, 0, 5, 2, 0, 8 ],}
\end{center}
so we conjecture that there is no eigenform~$f$ on $\Gamma_0(N)$ with
$N\mid 9$ such that $f\con F\pmod{9}$.
\end{conjecture}
As evidence, we verified that each of the mod~$9$
reductions of each newform of level~$1$ and weight~$k\leq 74$ has
one of the five systems of Hecke eigenvalues listed in the table.
We also verified that all newforms of levels~$3$ and $9$ and weight
$k\leq 40$ have corresponding system of eigenvalues mod $9$ in the
above table. We checked using the method described in Section~\ref{sec:numerical}
that there is no newform of level~$1$ with weight $k\leq 300$ that
approximates the weight~$2$ form on $X_0(27)$ modulo~$9$.
We now make some remarks about pseudo-representations when $p=3$.
Let $$\chi : \Z/27\Z \to \Z/9\Z$$ be the mod~$9$ cyclotomic
character, so $\chi$ has order~$6$ and if $\gcd(n,3)=1$
then $\chi(n) = n \in \Z/9\Z$.
The pseudo-representation corresponding
to a form of weight~$k$ giving the system of eigenvalues
in the table in Conjecture~\ref{conj:systems3} are
\begin{center}\begin{tabular}{|c|c|}\hline
Weight & Pseudo-representation \\\hline
12 & $\chi^2\oplus \chi^3$ \\
16 & $1 \oplus \chi^3$\\
20 & $\chi^3\oplus\chi^4$\\
24 & $1\oplus \chi^5$ \\
32 & $1\oplus \chi$\\
$S_2(\Gamma_0(27))$& $\chi^2\oplus \chi^5$ \\
\hline\end{tabular}\end{center}
Note that the square of any pseudo-representation of level~$1$ in
the above table has~$1$
as an eigenvalue, but the square of the pseudo-representation attached to
$S_2(\Gamma_0(27))$ does not have~$1$ as an eigenvalue.
Also,
$$F \con f_{16} \otimes \chi^2\pmod{9},$$
where $f_{16}$ is of weight~$16$. The order of~$\chi^2$ is~$3$, so $\chi^2$ is not a power of the
Teichm\"uller character (which has order~$2$) and Theorem~\ref{thm:twistapprox} does not apply.
Further computations {\em suggest} that the pseudo-representations
attached to forms of level~$1$ with coefficients in $\Z_9$ are
\begin{center}\begin{tabular}{|c|c|}\hline
Weight & Pseudo-representations \\\hline
$k\con 0\pmod{6}$ & $1\oplus \chi^5$, $\quad\chi^2\oplus \chi^3$ \\
$k\con 2\pmod{6}$ & $1\oplus \chi$, $\quad\chi^3\oplus \chi^4$\\
$k\con 4\pmod{6}$ & $1 \oplus \chi^3$\\
\hline\end{tabular}\end{center}
The pseudo-representations attached to forms
of level~$27$ with coefficients in $\Z_9$ seem to be
\begin{center}\begin{tabular}{|c|c|}\hline
Weight & Pseudo-representations \\\hline
$k\con 0\pmod{6}$ & $\chi\oplus \chi^4$\\
$k\con 2\pmod{6}$ & $\chi^2\oplus \chi^5$\\
$k\con 4\pmod{6}$ & $\chi\oplus\chi^2$, $\quad\chi^4\oplus\chi^5$\\
\hline\end{tabular}\end{center}
Also note that if $\chi^i\oplus\chi^j$ is one of the
pseudo-representations of level $27$ in the table,
then the sum of the orders of $\chi^i$ and $\chi^j$
is~$9$, whereas at level $1$ the sum of the orders
is at most~$7$.
\section{Computations About Approximating Infinite Slope Eigenforms}\label{sec:numerical}
In this section, we investigate computationally how well
certain infinite slope form can be approximated by finite slope
eigenforms.
\subsection{A Question About Families}
The following question is an analogue of \cite[\S8]{gouvea-mazur:families}
but for eigenforms of infinite slope.
Fix a prime~$p$ and an integer~$N$ with $(N,p)=1$.
\begin{question}\label{ques:family}
Suppose $f\in S_{k_0}(\Gamma_0(Np^r))$ is an eigenform having infinite
slope (note that $f$ need not be a newform).
Is there a ``family'' of eigenforms $\{f_k\}$, with $f_k\in
S_k(\Gamma_0(Np))$,
where the weights~$k$ run through an arithmetic progression
$$
k \in \mathcal{K} = \{k_0 + m p^{\nu}(p-1)
\text{ for } m=1,2,\ldots\}
$$
for some integer~$\nu$, such that
$$
f_k \con f\pmod{p^{n}},
$$
where $n=\ord_p(k-k_0) + 1$?
(When $p=2$ set $n=\ord_2(k-k_0)+2$.)
\end{question}
Our question differs from the one in \cite[\S8]{gouvea-mazur:families}
because there the form being approximated has
finite slope, whereas our form $f$ does not.
We know, as discussed in the previous section, that our question
sometimes has a negative answer since it might not be possible
to approximate~$f$ at all.
\subsection{An Approximation Bound}
Let
$$
f = \sum_{n\geq 1} a_n q^n\in K[[q]]
$$
be a $q$-expansion with coefficients that generate a number field~$K$.
Fix a prime~$p$ and an even integer~$k\geq 2$.
In order to gather some data about Question~\ref{ques:family}, we now
define a reasonably easy to compute upper bound on how well~$f$
can be approximated by an eigenform in $S_k(\Gamma_0(p))$.
Suppose $\ell\geq 1$,
let $F$ be the characteristic polynomial of $T_\ell$ acting on the
space $S_k(\Gamma_0(p))$ of classical cusp forms of weight~$k$ and
tame level~$1$, and
let~$H$ be the characteristic polynomial of $a_\ell\in K$.
Let~$G$ be the resultant of
$F(Y)$ and $H(X+Y)$ with respect to the
variable $Y$, normalized so that $G$ is monic.
Thus the roots of $G$ are
the differences $\alpha - \beta$ where
$\alpha$ runs through the roots of $F$ and
$\beta$ runs through the $\Gal(\Qbar/\Q)$-conjugates of $a_\ell$.
We can easily compute the $p$-valuations
of the roots of~$G$ without finding the roots, because
the $p$-valuations of the roots are
the slopes of the newton polygon of~$G$.
Let $m_\ell\in \Q\union\{\infty\}$ be the {\em maximum} of the slopes of
the Newton polygon of~$G$.
Let
$$
c_k(f,r) = \min \{m_\ell : \text{ $\ell\leq r$ is prime} \}.
$$
We note that computing $c_k(f,r)$ requires knowing only
the characteristic polynomials of Hecke operators $T_\ell$
on $S_k(\Gamma_0(p))$ and of $a_\ell$ for primes $\ell\leq r$.
\begin{proposition}
If there is a normalized eigenform $g\in S_k(\Gamma_0(p))$ such that
$f\con g\pmod{p^A}$, then $A\leq c_k(f,r)$ for any~$r$.
\end{proposition}
\begin{proof}
To see this observe that $c_k(f,r)$ is the minimum of the
$$
\ord_p(a_n(f) - a_n(g))
$$
where $1\leq n\leq r$ and~$g$ runs through all normalized eigenforms
in $S_k(\Gamma_0(p))$, and we run through all possible embeddings
of~$f$ and~$g$ into $\Zpbar[[q]]$.
\end{proof}
The motivation for our definition of $c_k(f,r)$ is that it is
straightforward to compute from characteristic
polynomials of Hecke operators, even when the coefficients of~$f$ lie
in a complicated number field. The number $c_k(f,r)$
could overestimate the true extent to which~$f$ is
approximated by an eigenform in $S_k(\Gamma_0(p))$ in
at least two ways:
\begin{enumerate}
\item There is an $r'>r$ such that $c_k(f,r')1$ and all $k$ in an arithmetic progression
$\mathcal{K} = \{k_0 + m p^{\nu}(p-1)
\text{ for } m=0,1,2,\ldots\}$.
(When $p=2$ we should have $\ord_2(k-k_0) + 2\leq c_k(f,r)$.)
The following or the results of some computations of $c_k(f,r)$.
%We order newforms in dictionary order by the trace of
%their $q$-expansion in $\Z[[q]]$.
\hspace{-2ex}$\mathbf{p=2}$\vspace{-1ex}:
\begin{enumerate}
\item For $k_0=6,10,12,14,16,20$ let $f\in S_{k_0}(\Gamma_0(4))$ be
the unique
newform. Then for all~$k$ with $k_0 < k\leq 100$ we
have $c_k(f,47) = \ord_2(k-k_0) + 2$.
\item For $k_0=18,22$ let $f\in S_{k_0}(\Gamma_0(4))$ be the unique, up
to Galois conjugacy, newform.
Then for all $k$ with $k_0 < k\leq 100$ we have $c_k(f,7) =
\ord_2(k-k_0) + 2$.
\item Let $f\in S_4(\Gamma_0(8))$ be the unique newform.
For most $4k_0$
with $k\con k_0\pmod{p-1}$, there is an eigenform $f_k$ on $X_0(Np)$
of weight~$k$ such that $f\con f_k\pmod{p^n}$ where
$n=\ord_p(k-k_0)+1$ (or $+2$ if $p=2$).
(In general one might have to restrict to~$n$ sufficiently large.)
\end{guess}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\end{document}