Properties

Label 2347.19
Modulus $2347$
Conductor $2347$
Order $23$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2347, base_ring=CyclotomicField(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([4]))
 
pari: [g,chi] = znchar(Mod(19,2347))
 

Basic properties

Modulus: \(2347\)
Conductor: \(2347\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2347.f

\(\chi_{2347}(14,\cdot)\) \(\chi_{2347}(19,\cdot)\) \(\chi_{2347}(150,\cdot)\) \(\chi_{2347}(169,\cdot)\) \(\chi_{2347}(196,\cdot)\) \(\chi_{2347}(266,\cdot)\) \(\chi_{2347}(346,\cdot)\) \(\chi_{2347}(360,\cdot)\) \(\chi_{2347}(361,\cdot)\) \(\chi_{2347}(397,\cdot)\) \(\chi_{2347}(502,\cdot)\) \(\chi_{2347}(503,\cdot)\) \(\chi_{2347}(515,\cdot)\) \(\chi_{2347}(864,\cdot)\) \(\chi_{2347}(875,\cdot)\) \(\chi_{2347}(1236,\cdot)\) \(\chi_{2347}(1377,\cdot)\) \(\chi_{2347}(1880,\cdot)\) \(\chi_{2347}(2100,\cdot)\) \(\chi_{2347}(2146,\cdot)\) \(\chi_{2347}(2165,\cdot)\) \(\chi_{2347}(2334,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 23 polynomial

Values on generators

\(3\) → \(e\left(\frac{2}{23}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2347 }(19, a) \) \(1\)\(1\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{9}{23}\right)\)\(e\left(\frac{9}{23}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{9}{23}\right)\)\(e\left(\frac{18}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2347 }(19,a) \;\) at \(\;a = \) e.g. 2