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Generalized Bernoulli numbers
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Numbers
$q$
$n$
$k$ 
$B_{k,\chi}$
INPUT{numbers.yaml} (not shown in preview)
Definition
Let $\chi$ be a Dirichlet character of modulus $q$. The generalized Bernoulli numbers $B_{k,\chi}$ attached to $\chi$ are defined as the Taylor coefficients of $\sum_{a=1}^q \chi(a) \frac{te^{at}}{e^{qt}-1} = \sum_{k=0}^\infty B_{k,\chi}\frac{t^k}{k!}$.
Parameters
$q$
—   modulus of $\chi$ ($q \geq 1$)
$n$
—   Conrey index of $\chi$ ($1 \leq n \leq \max(q-1,1)$)
$k$
—   integer ($k \geq 0$)
Comments
(1)
If $\chi(-1) \neq (-1)^k$, we have $B_{k,\chi}=0$ unless $(q,n)=(1,1)$ (that is, $\chi=1$). Thus we skip these trivial zero entries.
Programs
(P1)
Sage
chi = DirichletGroup(3).an_element()
numbers = [chi.bernoulli(n) for n in [0..10]]
Links
Data properties
Numbers are of type: complex number