## Non-optimal levels of a reducible mod $\ell$ modular representation

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## Abstract:

Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for $\Gamma _0(N)$ such that the mod $\ell$ Galois representation attached to $f$ is reducible and $U_p f = a_p f$ for primes $p$ dividing $N$. The main techniques used are level raising methods based on an exact sequence due to Ribet.## References

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## Additional Information

**Hwajong Yoo**- Affiliation: Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, Republic of Korea 37673
- Address at time of publication: College of Liberal Studies, Seoul National University 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea
- MR Author ID: 1146780
- Email: hwajong@gmail.com
- Received by editor(s): May 26, 2016
- Received by editor(s) in revised form: June 15, 2017, and June 21, 2017
- Published electronically: November 16, 2018
- Additional Notes: This work was supported by IBS-R003-D1.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 3805-3830 - MSC (2010): Primary 11F33, 11F80; Secondary 11G18
- DOI: https://doi.org/10.1090/tran/7314
- MathSciNet review: 3917209