I was looking up a modular forms online: $S_3^{new}\big(\chi_8(3, \cdot)\big) $ it can be written as an Eta product:
$$f(z) = \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2 = q \prod_{n=1}^\infty (1 - q^n)^2 (1 - q^{2n}) (1 - q^{4n}) (1 - q^{8n})^2$$
However, $f(z)$ is a newform, and a cusp form, and a Hecke eigenform. The L-function also has an Euler product. If I take the coefficients and turn it into an L-function:
\begin{eqnarray*}L(f , s) &=& 1^{-s} - 2 \cdot 2^{-s} - 2 \cdot 3^{-s} + 4 \cdot 4^{-s} + 4 \cdot 6^{-s} - 8 \cdot 8^{-s} + 5 \cdot 9^{-s} + \dots \\ \\ &=& \; ?\;?\;?_3\times \big( 1 + 5^{-s}\big)^{-1} \times \big( 1 + 7^{-s}\big)^{-1} \times \dots \end{eqnarray*}
Can we write down the Euler product up to a larger prime like 13 or 31 ?