Consider $$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$ the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then $\bar{\rho}$ is unramified exactly outside $\{2,3,7\}$ and the traces of $\operatorname{Fr}(\ell)$ for $\ell=5,11,13,17,\cdots,37$ are $-2,3,-2,2,\cdots,-1$.

If my computations (and my understanding) are correct, it follows from the fact that $37\cdot\operatorname{tr}(\operatorname{Fr}(37))^2\equiv 38^2$ modulo 7 that $\bar\rho$ satisfies the hypotheses of Ribet's theorem on level-raising and thus that there exists an eigencuspform $f\in S_2(1776)$ with residual representation equal to $\bar{\rho}$ ($1776=37\cdot 48$). I am interested in some properties of Kato's Euler system for $f$ (if $f$ indeed exists).

What are the first (say, 50) coefficients in the $q$-expansion of $f$?

I think the next question might be hard but let me try my luck nevertheless.

What is a system of equations defining the abelian variety $A_f$ attached to $f$?

I believe that $A_f$ is not an elliptic curve, as otherwise it would show up in the standard lists of elliptic curves, and I don't think it does.