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This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler.

Let $N$ be a positive integer. For a finite set of primes $\Sigma$, let $\mathbb T^{\Sigma}$ be the $\mathbb Z$-subalgebra of endomorphisms of $S_2(\Gamma_1(N))$ generated by Hecke operators $T_\ell$ for all prime $\ell\notin\Sigma$. If $\ell\in\Sigma$ implies that $\ell\nmid N$, then it is well-known that the index of $\mathbb T^{\Sigma}$ in $\mathbb T^{\varnothing}$ is a finite power of 2. I first learned this result in

Wiles, Andrew Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141 (1995), no. 3 Lemma page 491

where the given proof is attributed to F.Diamond and where the statement is credited to

Ribet, Ken Multiplicities of p-finite mod p Galois representations in J0(Np). Proposition 2

Thanks to David's question quoted above, I have been wondering about the actual possible values for the index of $\mathbb T^{\Sigma}$ in $\mathbb T^{\Sigma'}$ when $\Sigma'\subset\Sigma$. Here then are my actual questions.

A concrete one:

Question 1 Do you know of actual examples of non-trivial index of $\mathbb T^{\Sigma}$ inside $\mathbb T^{\varnothing}$ and of $\mathbb T^{\Sigma}$ inside $\mathbb T^{\Sigma'}$ when $\Sigma'$ contains primes dividing $N$?

A more theoretical (but also vaguer) one:

Question 2 Is there something known about the power of 2 that can occur as index of $\mathbb T^{\Sigma}$ inside $\mathbb T^{\varnothing}$? about the index of $\mathbb T^{\Sigma}$ inside $\mathbb T^{\Sigma'}$?

Regarding the second half of the first question, A.Wiles is careful in pointing that the argument given in his aforementioned article does not prove that $\mathbb T^{\Sigma\cup\{\ell\}}$ is equal to $\mathbb T^{\{\ell\}}$ when $\ell|N$ but neither does he say that the result could be false.

In the comments of Omitting primes from a Hecke algebra Kevin Ventullo seems to suggest that one maybe able to retrieve the missing operators from the local Langlands correspondence (in its $p$-adic Hodge theoretic incarnation at $\ell=p$) but I have been unable to understand his precise idea, especially as the proof I quoted (which is the only one I know) involves representations with coefficients in Artinian algebras, so are not obviously compatible with Weil-Deligne representations.

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Did you calculate any examples? It shouldn't be hard to do so (e.g. in Sage). –  David Loeffler Jul 5 '13 at 11:47
    
@David Loeffler. I'm not proud of it, but I admit that the amount of work I put in trying to answer my own questions is very low: my curiosity was piqued by your previous question and I wondered if someone on MO would know more, and that's about it. –  Olivier Jul 5 '13 at 20:26

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