Let
- $E$ an elliptic curve defined over $\mathbb{Q}$ with conductor $N_E$
- $d$ a fundamental discriminant with corresponding character $\chi_d$
- $E_d$ the quadratic twist of $E$ by $\chi_d$
- $w(E)$, $w(E_d)$ the root numbers of $E$ and $E_d$ respectively.
We know that when $\gcd(d,N_E) = 1$, the root numbers are related by the formula $$ w(E_d) = \chi_d(-N_E) w(E_d). $$
Is there a similar relation when the coprimality condition is not met? I read that there is no simple formula for the conductor of $E_d$ in this case, is it the same for the root number?