The statement of the theorem is as follows: Let $\rho$ be an irreducible two-dimensional representation of $G_\mathbb{Q}=Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ with Artin conductor $N$. Suppose that $\epsilon = \det(\rho)$ is odd and $\rho$ satisfies condition $(A)$. Let $L(s,\rho) = \sum a_n n^{-s}$. Then the function $f(z) = a_n q^n$ is a normalized newform on $\Gamma_0(N)$ of type $(k, \epsilon)$.

The condition (A) is: There exists a positive integer $M$ such that, for all one-dimensional representation $\chi$ of with Artin conductor prime to $M$, $\Lambda(s, \rho \otimes \chi)$ is entire, in particular, is holomorphic at $s = 0, 1$.

For proving it, I need to consider the Artin root number $W(\rho \otimes \chi)$. In the article ''Local Constants'' by Tate in the Durham symposium (MR0457408), he has proved that $$W(\rho \otimes \sigma) = (-1)^a \det{}_\rho(\mathfrak{f}(\sigma)) \det{}_\sigma (\mathfrak{f}(\rho)) W(\rho)^{\dim(\sigma)} W(\sigma)^{\dim(\rho)}$$ where $\mathfrak{f}$ is the Artin conductor, and $a$ is the number of archimedean places such that both $\det_\rho$ and $\det_\sigma$ are non-trivial (at the decomposition group of that archimedean place).

In the setting of Weil-Langlands, the term $W(\rho \otimes \chi)$ then reads

$$(-1)^a \epsilon(m) \chi(N) W(\rho) W(\chi)^2$$ $$=(-1)^a W(\rho) \epsilon(m) \chi(-N) \frac{G(\chi)}{G(\overline{\chi})}$$ where $m$ is the conductor of $\chi$ and $G(\chi)$ is the Gauss sum.

It now suffices to show that the function $f$ is a newform, for this, I wish to apply theorem 8 of Wen-Ch'ing Winnie Li (*Newforms and functional equations* Math. Ann. **212** (1975) 285-315 doi:10.1007/BF01344466).

However, the term $(-1)^a$ quite bothers me, which I need it to be $1$ to apply the theorem of Li. The assumption that $\rho$ is odd then implies $\det(\rho)$ is non-trivial at the only archimedean place of $\mathbb{Q}$. For the character $\chi$, it may or may not be non-trivial at the archimedean place, so $a = 0$ or $1$ respectively. But then I can not apply the theorem in the latter case.

So how could I handle this obstruction? What have I missed?