Revisiting the Fermat-type equation $x^{13}+y^{13}=3z^7$

We solve the Fermat-type equation $$x^{13}+y^{13}=3z^7,\gcd (x,y,z)=1$$ combining a unit sieve, the multi-Frey modular method, level raising, computations of systems of eigenvalues modulo 7 over a totally real field, and results for reducibility of certain Galois representations.