For the generalized Fermat equation:
$a^p + b^q = c^r$
with $p,q,r\ge3$ and $\gcd(a,b,c)=1$
one can construct the Frey Curve:
$y^2=x(x-a^p)(x+b^q)$
which is semi-stable for those primes $m$ that divide the discriminant and do not divide $c_4$:
$m \mid \Delta$ and $m \nmid c_4$
where
$\Delta=16a^{2p}b^{2q}c^{2r}$ and $c_4=16(c^{2r}+a^pb^q)$.
The conductor is:
$N=2abc$
Now we are interested in the obstructions arising by level lowering $N$ to $N_m$.
For example, let $m\mid r$ and $m\mid q$.
Then, $N_m=2a$ is an obstruction for all $a$ that have a newform at level $N_m$ if there is no further way to eliminate that newform. Interestingly, $a$ can include in its prime factorization any primes except the ones in $b$ or $c$ or any other primes outside of $a$'s factorization. That is a lot of possible newforms to eliminate!
So, the question is what approach to eliminating the obstruction $N_m=2a$ in this example can be taken?