second method is always possible since the $g_Y$’s are CM forms. In \[M\] Martin determined which modular forms on $\Gamma_1(N)$ can be expressed as a product of $\eta$-functions. This applies to cases \#8, \#7, \#3, and \#4 in Table 2. Hence the same is also true for the \#6 and \#9 cases. For cases \#3, \#7, and \#8 the corresponding spaces of cusp forms of weight $3$ are $1$-dimensional, hence the conditions in \[M\] are satisfied and Martin gives the corresponding forms as $h_3=
\eta(q)^3\eta(q^7)^3$ and $h_7=
\eta(q^2)^3\eta(q^6)^3$. The modular form in case \#1 is classically known to be $h_1=
\eta(q)^6$, which implies $h_8= \eta(q^4)^6$. Lastly $h_2= \eta(q)^3\eta(q^3)^3$, and $h_5=
\eta(q^2)^6$. For \#4, we have $h_4(q)=\eta(q)^2\eta(q^2)\eta(q^4)\eta(q^8)^2$. We will prove the following

[**Proposition 10**]{}:

*Let $\chi_i$ be the Hecke character for which $L(h_i,s) =
L(\chi_i,s)$ (so that the inverse Mellin transform of $L(\chi_i,s)$ is $h_i$). Let $a_p(h_i)$ be the $p$th Fourier coefficient of $h_i$, and let $K_i = {{\mathbb Q}}(\sqrt{\delta_i})$. Then we have the following:*

\(1) The infinite component of $\chi_i:{{\mathbb A}}_{K_i}^\times \rightarrow {{\mathbb C}}$ is given by $\chi_{i,\infty}(z) = z^{-2}$. Moreover $\chi_i$ is the unique such Hecke character of conductor $c_i{{\mathcal O}}_{K_i}$, where $c_i = 2,2,1,1
\in {{\mathcal O}}_{K_i}$ for 