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).



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



Computing an eigencuspform in $S_2(\Gamma_0(1776))$
I did the computation in Sage, and there is no such form $f$. There are 21 Galois orbits of newforms of level $\Gamma\_0(1776)$ and trivial character, of which the largest has size 3, and none of the reductions modulo any of the primes above 7 in the coefficient fields is congruent to the form associated to $\bar\rho$.


Looking again at your question, the problem is that you have misquoted Ribet's theorem; the condition for mod $p$ level-raising at $\ell$ should be that $a\_\ell^2 = (1 + \ell)^2 \bmod p$, not $\ell a\_\ell^2 = (1 + \ell)^2 \bmod p$. At least, that's what Theorem 1.1 of [Ribet's paper](https://math.berkeley.edu/~ribet/Articles/dpp.pdf) says. So the smallest $\ell$ for which you can level-raise is not 37 but 53, leading to a newform of level 2544.


EDIT: Just for kicks, I did the computation for $\ell = 53$. The form $f$ is defined over $\mathbf{Q}(a)$ where $a$ is a root of $x^{3} - 10 x^{2} + 28 x - 23 = 0$, and it satisfies the required congruence modulo the prime $4 - a$ of norm 7. The $q$-expansion of $f$ is


$$ q + q^{3} + \left(-a + 2\right)q^{5} + \left(-2 a^{2} + 15 a - 21\right)q^{7} + q^{9} + \left(3 a^{2} - 23 a + 33\right)q^{11} + \left(-a^{2} + 10 a - 19\right)q^{13} + \left(-a + 2\right)q^{15} + \left(a^{2} - 7 a + 7\right)q^{17} + \left(2 a^{2} - 14 a + 14\right)q^{19} + \left(-2 a^{2} + 15 a - 21\right)q^{21} + \left(a^{2} - 8 a + 10\right)q^{23} + \left(a^{2} - 4 a - 1\right)q^{25} + q^{27} - 8q^{29} + \left(-3 a^{2} + 25 a - 39\right)q^{31} + \left(3 a^{2} - 23 a + 33\right)q^{33} + \left(a^{2} - 5 a + 4\right)q^{35} + \left(2 a^{2} - 17 a + 21\right)q^{37} + \left(-a^{2} + 10 a - 19\right)q^{39} + \left(2 a^{2} - 17 a + 23\right)q^{41} + \left(-2 a^{2} + 13 a - 10\right)q^{43} + \left(-a + 2\right)q^{45} + \left(-2 a + 8\right)q^{47} + \left(-3 a^{2} + 22 a - 26\right)q^{49} + O(q^{50}).$$
Email me if you want the original Sage code.


