Characteristic polynomials of Hecke operators $T\_\ell$, with $\ell$ prime, acting on cusp forms $S\_k$ of level one and weight $k$ "appear to be" squarefree (even irreducible!).


This can be interpreted as follows: if $p$ is any prime, then the $p$-adic Galois representations $\rho\_i$, where $1\leq i\leq \dim(S\_k)$, attached to eigenforms in $S\_k$ "appear to be" pairwise non isomorphic ${\it locally}$ at primes $\ell\neq p$.


This is completely false for other levels. For example in the two dimensional space $S\_2(\Gamma\_0(37))$, I learn from Magma and Cremona's tables that there are exactly two normalized eigenforms, $f\_1$ and $f\_2$, with rational coefficients, corresponding to two elliptic curves $E\_1$ and $E\_2$ defined over $\mathbf{Q}$, of conductor $37$, and uniquely determined up to $Q$-isogeny.


Looking at some Hecke operators on this space, one easily finds examples of $T\_\ell$ acting diagonally on $S\_2(\Gamma\_0(37))$, i.e., examples of primes $\ell\neq 37$ for which the two elliptic curves have $p$-adic Tate modules isomorphic as local Galois modules at $\ell$ (some of the $\ell$'s for which this happens are $7$, $31$, $41$, $101$, $137$, $173$, $179$,..$39769$).


$Q1$: Is it reasonable to suspect that $E\_1$ and $E\_2$ become isogenous over an extension $F$ of $Q$? If this were the case, then one should see the phenomenon described above for
primes $\ell$ that are split in $F$, right?


$Q2$: On the other hand, given an elliptic curve $E$ over $\mathbf{Q}$, what are the known ways to construct more elliptic curves $A$, defined over $\mathbf{Q}$, possibly of the same conductor as $E$, which are not $\mathbf{Q}$-isogenous to $E$ but such that they become so over a non-trivial extension of $Q$?


$Q3$: Can we say why we do not see the above phenomenon in level one?



Elliptic curves over $\mathbf{Q}$ with isogenous mod $\ell$ reductions, for several $\ell$
I imagine this is a weight issue, not a level issue. Let f and g be the two weight-2 newforms in $S\_2(\Gamma\_0(37))$. Then a "random" coefficient of f-g is going to have size about $p^{1/2}$, so there should be about $X^{1/2-\epsilon}$ primes $p$ less than $X$ such that $a\_p(f) = a\_p(g)$, just by chance. When the weight is larger, the Fourier coefficients are bigger, and it is much more surprising to see coincidences of Fourier coefficients.


Try other weight 2 cases, and try some higher weight cases in level 37, and I'll bet you'll see that your phenomenon happens in weight 2 and not in weight 4.


