TITLE: Reciprocity law for number fields defined by torsion points of modular elliptic curves
QUESTION [6 upvotes]: Shimura (Crelle 221, 1966) considers the elliptic curve $E:y^2+y=x^3-x^2$ (although he doesn't use this equation) of conductor $11$ whose associated modular form is
$$
q\prod_{k=1}^{+\infty}(1-q^k)^2(1-q^{11k})^2=\sum_{n=1}^{+\infty}c_nq^n
$$
where $q=e^{2i\pi\tau}$ and $\tau$ is in the upper half of $\bf C$.  For a prime $l$, he denotes by $K_l$ the extension of $\bf Q$ obtained by adjoining the $l$-torsion points of $E$ and shows that if $l\in[7,97]$, then ${\rm Gal}(K_l|{\bf Q})$ is isomorphic to ${\rm GL}_2({\bf F}_l)$. 
Question. Is ${\rm Gal}(K_l|{\bf Q})$ now known to be isomorphic to ${\rm GL}_2({\bf F}_l)$ even for $l>97$ ?
Even if the faithful representation ${\rm Gal}(K_l|{\bf Q})\rightarrow{\rm GL}_2({\bf F}_l)$ fails to be surjective for a few $l>97$, does the recent proof of Serre's modularity conjecture not imply the
Statement.  For every prime $l>5$ and every prime $p\neq11,l$, the characteristic polynomial of ${\rm Frob}_p$ (thought of as an element of ${\rm GL}_2({\bf F}_l)$) is
$\equiv X^2-c_pX+p \pmod l$ ?
Shimura shows this only for $l\in[7,97]$. 
Addendum.  (2010/07/24)  Looking at Shimura's paper beyond the first page shows that  he actually proves (Section 3) that the characteristic polynomial for the action of ${\rm Frob}_p$ on the $l$-adic Tate module $T_l(E)$ is 
$X^2-a_pX+p\in{\bf  Z}_l[X]$ 
for all primes $l$ and $p\neq11, l$ and (Section 6) that $a_p=c_p$ for all $p\neq11$.  And yes, he does use the Eichler-Shimura relation.

REPLY [5 votes]: The first question is answered in Serre's 
[Propriétés galoisiennes des points d'ordre fini des courbes elliptiques,
Invent. Math. 15:4 (1972) 259--331], on page 304, section 5.2, exactly for this curve. In general this paper give a good way to determine for which $\ell$ the mod-$\ell$ representation is not surjective. Sage can do that efficiently for a given curve.
For every prime $p$ different from $\ell$ and $11$, the characteristic polynomial of $\rm{Frob}_p$ is indeed $T^2 - c_p T +p$ in   $ \mathbb F_{\ell}[T]$. The isomorphism
$\rm{Gal}(K _{\ell}/\mathbb Q) \to \rm{Aut}(E[\ell])=\rm{GL} _2(\mathbb F _{\ell})$ sends $\rm{Frob}_p$ to the Frobenius endomorphism $\phi:E[\ell] \to E[\ell]$ on $E/\mathbb{F}_p$.
Your $c_p$ is the trace of $ \phi$ and $p$ is the determinant of it since the Eichler--Shimura relation shows that $c_p$ is the Fourier coefficent of the associated modular form. See this answer for why it is so.