Using multiplicativity of Hecke operators,
$\otherI$ contains $T_n - e_n$, where the $e_{\ell}$ for $\ell$ prime are
$1+\ell$ or $\pm 1$ as above, and we extend using the recursive formula
for Hecke operators: $T_{p^r} = T_p T_{p^{r-1}} - \eps(p) p T_{p^{r-2}}$, where
$\eps(p)=1$ unless $p=13, 43$ in which case $\eps(p)=0$. The first few $e$ are:
%link
\begin{lstlisting}
sage: e = [0,1,3,4,7,6,12,8,15,13,18,12,28,-1,24,24,31,18,39,20,42,32,36,24,60,31,-3,40,
...  56,30,72,32,63,48,54,48,91,38,60,-4,90,42,96,1,84,78,72,48,124,57,93,72,-7,54,120,72,
...  120,80,90,60,168,62,96,104,127,-6,144,68,126,96,144,72,195,74,114,124,140,96,-12,80]
\end{lstlisting}
%The above was computed using the following too-complicated code!
% def evec(B=hecke_bound):
%     # compute the vector of e, so e[n] = e_n above for all n<=hecke_bound
%     e = [0,1] + [None for i in range(B-1)]
%     # first primes:
%     for p in primes(B+1):
%         if p == 13:
%             e[p] = -1
%         elif p == 43:
%             e[p] = 1
%         else:
%             e[p] = p+1
%         # now powers of p
%         k = 2
%         while p^k <= B:
%             if p == 13 or p == 43:
%                 e[p^k] = e[p]^k
%             else:
%                 e[p^k] = e[p^(k-1)]*e[p] - p*e[p^(k-2)]
%             k += 1    
%     # next composite non-prime powers: extend multiplicatively
%     for n in [1..B]:
%         if not n.is_prime_power():
%             e[n] = prod(e[F[0]^F[1]] for F in factor(n))
%     return e

Taking the $\Z$-module $M$ generated by $T_n-e_n$, for $n\leq 52$,
gives a $\Z$-submodule $M \subset \otherI$, but it turns out that
$M\neq \otherI$.  However, the ideal generated by $M$ equals $\otherI$,
since ...TODO
Using $T_n-e_n$ for many more $n$ does not help.
%link 
\begin{lstlisting}
sage: def eisenstein_ideal(v, B): return [v * (S.hecke_matrix(i) - e[i]) for i in [1..B]]
sage: M = eisenstein_ideal(v, 52)
sage: factor(span(ZZ,M).index_in(L))
2 * 3 * 7^2
sage: M = eisenstein_ideal(v, 103)
sage: factor(span(ZZ,M).index_in(L))
2 * 7^2
\end{lstlisting}%link

To determine whether or not a candidate $\Z$-module $M$ is invariant under the
action of $\T$, it suffices to check that $M$ is preserved by $T_p$
for all primes $p<52$, since ...
The trick we use to do this is to note that for $t \in M$ we have $v (T_p t) = (v T_p) t$,
so to compute $T_p M$ we just compute the image $M_p$ of $M$ under the embedding defined
by $v T_p$ instead of the embedding defined by $v$.   If the $\Z$-modules $M_p$
is contained in $M$, for $p<52$, then we conclude that $M$ is a $\T$-module.
More precisely, since the $M$ above is not a $\T$-module, we add
to it all module $M_P$ to obtain a $\T$-module.

%link 
\begin{lstlisting}

\end{lstlisting}%link




%\end{lstlisting}%link
%link
%\begin{lstlisting}
%\begin{lstlisting}\end{lstlisting}