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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.

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    $\begingroup$ $1776$ is small enough that you can compute all eigenforms $f$ and reduce each one mod $7$ to find the match. I'm sure somebody who is sufficiently fluent in Sage (as I am not) can do this routinely. $\endgroup$ – Noam D. Elkies Jun 26 '15 at 5:00
  • $\begingroup$ For your second question, as you probably know there is no such thing as Cremona's tables for abelian varieties. On the other hand, you can try to compute the complex periods of $A_f$ (using modular symbols). At least they determine $A_f$ as an abelian variety over C, but then it depends on what you want to do. $\endgroup$ – François Brunault Jun 26 '15 at 7:19
  • $\begingroup$ The natural way to embed abelian varieties into projective space is to use theta functions. Then, finding equations amounts to find algebraic relations between them. The first theta relations were found by Riemann, they are quadratic. Moreover, by results of Mumford and Kempf, if the line bundle is ample enough then Riemann's relations generate all the relations. For an abelian 3-fold, the best you can hope for is embedd into $\mathbf{P}^7$, but if you want quadratic equations you probably need higher dimensional projective space. See Chap. 7 in Birkenhake-Lange "Complex abelian varieties". $\endgroup$ – François Brunault Jun 27 '15 at 11:00
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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 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.

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