\F_\ell$ (see for example [@Ash-Stevens Section 1.4]). Then the previous computations, together with results of [@Haluk-expmath], establish the following result:

For each of the nine imaginary quadratic fields of class number $1$, the space $S_2(1)$ of weight $2$ cuspidal Bianchi modular forms of level $1$ is trivial. Moreover for all primes $\ell\ge 5$ the space $H^1(Y_0({1}),\F_{\ell})$ is also trivial. \[thm:noleveloneforms\]

It is known that these Bianchi modular forms have associated $\ell$-adic Galois representations $\rho_{F,\ell}$. These were first constructed by Taylor *et al.* in [@TaylorI], [@TaylorII] with subsequent results by Berger and Harcos in [@Berger]. Below we only need to refer to the residual mod-$\ell$ representations $\overline{\rho_{F,\ell}}$.

For an elliptic curve $E$ defined over an imaginary quadratic field $K$, we say that $E$ is modular if $L(E,s)=L(F,s)$ for some $F\in S_2(\id{n})$ over $K$, where $\id{n}$ is the conductor of $E$. The following conjecture, a version of which was first made by Mennicke, is part of Conjecture 9.1 of [@Sengun], following [@CremonaTwist]:

Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ of class number $1$ that does not have complex multiplication by an order in $K$. Then $E$ is modular. \[conj:modularity\]

Some cases of the conjecture have been 