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Conjecture on the Hilbert symbol and the coefficients $U_k^{(u)}$
The conjecture. For $0\leq k\leq \left\lceil\frac{j-i_0}{2}\right\rceil$ , the coefficients satisfy the stated congruences for $U_k^{(u)}$ and $U_k'$ , and consequently
Conjectureopenhilbert-symbols number-theory root-numbersarXiv posted 2 days ago

Conjecture on the Hilbert symbol and the coefficients Uk(u)U_k^{(u)}Uk(u)​