representation of $\GL_2(\A)$ with trivial central character. If $L(1/2, \pi')L(1/2, \pi'\otimes\eta)\neq0$, then there is a unique $D\in X(E)$ and an irreducible cuspidal automorphic representation $\pi$ of $D_\A^\times$ which is is $E^\times$-distinguished and $\pi'=\operatorname{JL}(\pi)$. Moreover, it was proved by Tunnell [@Tunnell] and Saito [@Saito] that the ramification set of $D$ is $\{v\mid \ve(1/2, \pi'_{E, v})=-1\}$. In other words, $D_v$ is division if and only if $\ve(1/2, \pi'_{E, v})=-1$. Here, $\pi'_E$ is the base change of $\pi'$ to $\GL_2(\A_E)$.

Applying this theorem, we see that Theorem \[thm:FH\] has following corollary

Let $\pi'$ be an irreducible cuspidal automorphic representation of $\GL_2(\A)$ with trivial central character. Suppose that we have $\ve(1/2, \pi'\otimes\chi)=1$ for some $\chi\in\Psi(S;\chi_0)$ and $L(1/2, \pi')\neq0$. Then there are infinitely many $\chi\in\Psi(S;\chi_0)$ such that there exists $D_\chi\in X(E_\chi)$ and an irreducible cuspidal automorphic representation $\pi$ of $D_{\chi, \A}^\times$ which is $E_\chi^\times$-distinguished and $\pi'=\operatorname{JL}(\pi)$.

Now we state our main result. Let $D$ be a quaternion algebra over $F$. The set of quadratic étale algebras over $F$ which $E$ embeds is denoted by $X(D)$.

\[thm:main\] Let $\pi$ be an irreducible cuspidal automorphic representation of $D_\A^\times$ with trivial central character which is not $1$-dimensional. Suppose we have $L(1/2, \pi)\neq0$. Then we can take a quadratic 