Congruences modulo prime powers of Hecke eigenvalues in level 11

We continue the study of strong, weak, and $dc$-weak eigenforms introduced by Chen, Kiming, and Wiese. We completely determine all systems of Hecke eigenvalues of level $1$ modulo $128$, showing there are finitely many. This extends results of Hatada and can be considered as evidence for the more general conjecture formulated by the author together with Kiming and Wiese on finiteness of systems of Hecke eigenvalues modulo prime powers at any fixed level. We also discuss the finiteness of systems of Hecke eigenvalues of level $1$ modulo $9$, reducing the question to the finiteness of a single eigenvalue. Furthermore, we answer the question of comparing weak and $dc$-weak eigenforms and provide the first known examples of non-weak $dc$-weak eigenforms.Comment: 28 pages; Minor revisio

Scatter plot of the HPPD gene copy number and expression levels per SEN and NEB plant analysed relative to bulked control samples (BT: before treatment; 2 DAT: 2 days after treatment).

<p>Scatter plot of the HPPD gene copy number and expression levels per SEN and NEB plant analysed relative to bulked control samples (BT: before treatment; 2 DAT: 2 days after treatment).</p

Average HPPD gene copy number and expression relative to ALS and CPS genes for the SEN and NEB populations.

<p>Average HPPD gene copy number and expression relative to ALS and CPS genes for the SEN and NEB populations.</p

Typical HPLC chromatogram showing mesotrione and its major metabolite 4-hydroxymesotrione in the SEN and NEB populations (72 hours after treatment).

<p>Typical HPLC chromatogram showing mesotrione and its major metabolite 4-hydroxymesotrione in the SEN and NEB populations (72 hours after treatment).</p

Means and 95% confidence limits for mesotrione and metabolites as % of uptake as measured by LCMS.

<p>Means and 95% confidence limits for mesotrione and metabolites as % of uptake as measured by LCMS.</p

Means and 95% confidence limits for radiochemical as % of absorbed in the standard sensitive (SEN) and mesotrione resistant population (NEB).

<p>Means and 95% confidence limits for radiochemical as % of absorbed in the standard sensitive (SEN) and mesotrione resistant population (NEB).</p