$i = 8,7,3,4$ respectively.

\(2) For each rational prime $p$ which is prime to the level of the corresponding $\Gamma$, we have $a_p(h_i) = 0$ if $p$ is inert in $K_i$. Otherwise, there are $a$, $b$, which are integers in case \#8 and half integers in the three other cases, so that $p = a^2+d_i b^2$, where $d_i = 4,
3, 7, 2$ for $i = 8,7,3,4$ respectively. Then $a$ and $b$ are unique up to signs, and $a_p(h_i) =
(a^2-d_ib^2)/2$.

\(3) The newtype of $h_i$ (see (\[new\])) is the character defining $K_i$, namely $p\mapsto
\left(\frac{\delta_i}{p}\right)$.

[**Proof: **]{} See e.g. \[L2\] for the generalities (in particular regarding the $\infty$-component of $\chi_i$), as well as the following formula: the conductors of $\chi_i$ and of $h_i$ are related by $$\text{cond}(h_i) = \text{Nm}^{K_i}_{{{\mathbb Q}}} 
\text{cond}(\chi_i) \text{Disc}(K_i).$$ Since the level of $h_i$ is respectively $M=16$, $12$, $7$, and $8$ in cases \#8, \#7, \#3, and \#4 of Table 2, we get the asserted value of the $c_i$’s, and since all the fields $K_i$ involved have class number $1$ we have $$\label{dec}
{{\mathbb A}}_{K_i}^\times = (K_i^\times \times U_i \times
{{\mathbb C}}^\times)/\mu(K_i)$$ where $U_i$ is the maximal compact subgroup $\hat{{{\mathcal O}}}\vphantom{{{\mathcal O}}}_{K_i}^\times$ of the finite idèles 