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See detailGenuine Bianchi modular forms of higher level, at varying weight and discriminant
Rahm, Alexander UL; Tsaknias, Panagiotis UL

in Journal de Théorie des Nombres de Bordeaux (2019), 31(1), 27-48

Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Those of the cuspidal Bianchi modular forms which are relatively well understood, namely ... [more ▼]

Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Those of the cuspidal Bianchi modular forms which are relatively well understood, namely (twists of) base-change forms and CM-forms, are what we call non-genuine forms; the remaining forms are what we call genuine. In a preceding paper by Rahm and Şengün, an extreme paucity of genuine forms has been reported, but those and other computations were restricted to level One. In this paper, we are extending the formulas for the non-genuine Bianchi modular forms to higher levels, and we are able to spot the first, rare instances of genuine forms at higher level and higher weight. [less ▲]

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See detailThe $n$-th prime asymptotically
Arias de Reyna, Juan; Toulisse, Jérémy UL

in Journal de Théorie des Nombres de Bordeaux (2013)

A new derivation of the classic asymptotic expansion of the n-th prime is presented. A fast algorithm for the compu- tation of its terms is also given, which will be an improvement of that by Salvy (1994 ... [more ▼]

A new derivation of the classic asymptotic expansion of the n-th prime is presented. A fast algorithm for the compu- tation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with $li−1 (n)$, after having re- tained the first $m$ terms, for $1 ≤ m ≤ 11$, are given. Finally, as- suming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n ≥ r_3$ , we have $p_n > s_3 (n)$ where $s_3 (n)$ is the sum of the first four terms of the asymptotic expansion. [less ▲]

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See detailOn the generation of the coefficient field of a newform by a single Hecke eigenvalue
Koo, Koopa Tak-Lun; Stein, William; Wiese, Gabor UL

in Journal de Théorie des Nombres de Bordeaux (2008), 20(2), 373--384

# Let f be a non-CM newform of weight k>1 without nontrivial inner twists. In this article we study the set of primes p such that the eigenvalue a_p(f) of the Hecke operator T_p acting on f generates the ... [more ▼]

# Let f be a non-CM newform of weight k>1 without nontrivial inner twists. In this article we study the set of primes p such that the eigenvalue a_p(f) of the Hecke operator T_p acting on f generates the field of coefficients of f. We show that this set has density 1, and prove a natural analogue for newforms having inner twists. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation. [less ▲]

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See detailJacobiennes de certaines courbes de genre 2 : torsion et simplicité
Leprévost, Franck UL

in Journal de Théorie des Nombres de Bordeaux (1995), 7

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