TL;DR.
Some local root numbers of the Hecke character associated with our specific CM elliptic curve by $\mathbf{Q}(i)$ seem to have value in $\mu_4$. But apparently our computation via Rohrlich's local root number formula says otherwise. What am I missing here?
Settings.
Let $K = \mathbf{Q}(\sqrt{-1})$ the quadratic ground field and $E$ an elliptic curve over $K$ having complex multiplication by the ring of integers $O_K$ of $K$ defined by an affine Weierstrass equation \begin{equation*} y^2 = x^3 - Dx \end{equation*} for some $D \in O_K$. To $E/K$ we attach a Hecke character $\psi=\psi_{E/K} = \prod' \psi_v: \mathbf{I}_K/K^\times \to \mathbf{C}^\times$ as usual (c.f. e.g. [Sil], sections II.9--10), of conductor $\mathfrak{f}$ which have the same prime factors as the conductor of the elliptic curve $E/K$.
The local root number at $v$ of the Hecke character ($\psi$ in our case) is defined as the signature of its epsilon-factor of the Weil--Deligne representation induced from $\psi_v$, by the local Artin map. The formulae for the local root number are also well-known thanks to Rohrlich [Rohr], pp. 32--33. In particular, when $v$ is a finite and ramified place (i.e.~$v$ divides the conductor $\mathfrak{f}$) not dividing $2$ (i.e.~$v$ is relatively prime to the different ideal of $K/\mathbf{Q}$), the formula for $\chi = \psi_v$ can read: \begin{equation*} W(\chi) = \chi_{u}(\beta) \cdot q^{-a(\chi)/2} \cdot \sum_{x \in (O_v/\mathfrak{f}_v)^\times} \chi^{-1} (x) \exp (2 \pi i \operatorname{tr}^{K_v}_{\mathbf{Q}_p}(x/ \beta) ), \end{equation*} where
- $\beta \in K_v^\times$ is any element with $v(\beta) = a(\chi)$,
- $a(\chi)$ is the conductor exponent of $\psi$ at the place $v$,
- $q$ is the size of the residue field of $K$ modulo $v$,
- $\chi_v$ denotes the unitary part of the character $\chi$.
Property 1.
Let $\ell$ be a rational prime that splits completely in $K$ and $v \not\mid \ell$ be any finite prime of $K$. If $V_\ell E$ is the $\ell$-adic Tate module of $E$ over $\mathbf{Q}_\ell$, then we have $V_\ell E \otimes_{\mathbf{Q}_\ell} \mathbf{C} \cong \psi_v \oplus \psi_v$. See e.g. [Rubin], Proposition 5.4 and Theorem 5.15(ii).
Property 2.
It is well-known that the infinity type $\psi_\infty: \mathbf{C}^\times \to \mathbf{C}^\times$ of $\psi$ is given by $z \mapsto z^{-1}$, so that for any $\alpha \in K_{\mathfrak{f},1}$ (the trivial ray), we have $\psi(\alpha O_K) = \psi_\infty^{-1}(\alpha)$ (of course with a fixed embedding $K \hookrightarrow \mathbf{C}$).
From now on, we assume $p$ is a rational prime $\equiv 1 \pmod{4}$ so that $p O_K = \mathfrak{p} \overline{\mathfrak{p}}$. Since $K$ has class number $1$, we can find generators $\pi$ and $\overline\pi$ in $O_K$ of $\mathfrak{p}$ and of $\overline{\mathfrak{p}}$, respectively, so that $p = \pi \overline\pi$. We may and do assume that $\pi$ and $\overline\pi$ are primary, i.e. $\equiv 1 \pmod{2(1+i)}$. Put $D = \pi$ and we may assume $E$ has good reduction modulo $(1+i)$. The last condition is surely satisfied when $D = 1 + 2i$, etc. This implies that our Hecke character $\psi$ is of conductor $\mathfrak{f} = \mathfrak{p}$. We also take $\beta = p$.
Property 3.
Recall that $\chi = \psi_v: K_v^\times \to \mathbf{C}^\times$ is the local part of the Hecke character $\psi$. From [Rohr], Proposition 2.1, we can see that \begin{align*} \chi | O_K^\times &= \epsilon^{-1} \\ \chi(\pi) &= \psi_\infty^{-1}(\pi) \end{align*} since there are no other primes dividing $\mathfrak{f}$ except $\mathfrak{p}$. As values of the $\epsilon$-type lie in the group of 4th roots of unity $\mu_4$ (recall $O_K/\mathfrak{f} \cong O_K/\mathfrak{p} \cong \mathbf{F}_5$), we see that \begin{equation*} \chi_u(\beta) = \chi_u(\pi) \chi_u(\bar{\pi}) \equiv \psi_\infty^{-1}(\pi)/|\psi_\infty^{-1}(\pi)| = \pi/|\pi| \pmod{\mu_4}. \end{equation*}
Property 4.
Write the ``Gauss sum part'' in the formula of $W(\chi)$ as $G$, i.e. \begin{equation*} G = \sum_{x \in (O_v/\mathfrak{f}_v)^\times} \chi^{-1} (x) \exp (2 \pi i \operatorname{tr}^{K_v}_{\mathbf{Q}_p}(x/ \beta) ) \end{equation*} In our case, this is just \begin{equation*} G = \sum_{x \in \mathbf{F}_5^\times} \epsilon(x) \exp ( 2 \pi i x / p). \end{equation*} Note that there are only four characters $\epsilon: \mathbf{F}_5^\times \to \mathbf{C}^\times$. When $\epsilon$ is the quartic residue symbol, then it is well-known that $G^2 = \pm\pi \sqrt{p}$.
Question.
From Property 1, since the root number of elliptic curves is in $\{ \pm 1 \}$, it follows that the root number of $\chi = \psi_v$ lies in the group $\mu_4$. However, from the properties 3--4, we see that \begin{equation*} W(\chi) \equiv \frac{\pi}{\sqrt{p}} \cdot \frac{1}{\sqrt{p}} \cdot \sqrt{\pi} \sqrt[4]{p} \pmod{\mu_4} \neq 1 \pmod{\mu_4} \end{equation*} and hence $W(\chi) \not\in \mu_4$. Why do such kinds of inconsistency occur? We will be very gratitude if someone find out any flaw/mistake in our consideration.
References.
[Rohr] Rohrlich, D. E., Root numbers.
[Rubin] Rubin, K., Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer.
[Sil] Silverman, Advanced topics in the arithmetic of elliptic curves.
