Properties

Label 2347.18
Modulus $2347$
Conductor $2347$
Order $2346$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2347, base_ring=CyclotomicField(2346))
 
M = H._module
 
chi = DirichletCharacter(H, M([2225]))
 
pari: [g,chi] = znchar(Mod(18,2347))
 

Basic properties

Modulus: \(2347\)
Conductor: \(2347\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2346\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2347.p

\(\chi_{2347}(3,\cdot)\) \(\chi_{2347}(5,\cdot)\) \(\chi_{2347}(11,\cdot)\) \(\chi_{2347}(12,\cdot)\) \(\chi_{2347}(18,\cdot)\) \(\chi_{2347}(20,\cdot)\) \(\chi_{2347}(29,\cdot)\) \(\chi_{2347}(30,\cdot)\) \(\chi_{2347}(34,\cdot)\) \(\chi_{2347}(42,\cdot)\) \(\chi_{2347}(43,\cdot)\) \(\chi_{2347}(44,\cdot)\) \(\chi_{2347}(48,\cdot)\) \(\chi_{2347}(50,\cdot)\) \(\chi_{2347}(57,\cdot)\) \(\chi_{2347}(59,\cdot)\) \(\chi_{2347}(61,\cdot)\) \(\chi_{2347}(62,\cdot)\) \(\chi_{2347}(63,\cdot)\) \(\chi_{2347}(70,\cdot)\) \(\chi_{2347}(72,\cdot)\) \(\chi_{2347}(73,\cdot)\) \(\chi_{2347}(74,\cdot)\) \(\chi_{2347}(78,\cdot)\) \(\chi_{2347}(80,\cdot)\) \(\chi_{2347}(94,\cdot)\) \(\chi_{2347}(95,\cdot)\) \(\chi_{2347}(97,\cdot)\) \(\chi_{2347}(99,\cdot)\) \(\chi_{2347}(103,\cdot)\) ...

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: $\Q(\zeta_{1173})$
Fixed field: Number field defined by a degree 2346 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{2225}{2346}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2347 }(18, a) \) \(-1\)\(1\)\(e\left(\frac{269}{782}\right)\)\(e\left(\frac{2225}{2346}\right)\)\(e\left(\frac{269}{391}\right)\)\(e\left(\frac{983}{2346}\right)\)\(e\left(\frac{343}{1173}\right)\)\(e\left(\frac{37}{782}\right)\)\(e\left(\frac{25}{782}\right)\)\(e\left(\frac{1052}{1173}\right)\)\(e\left(\frac{895}{1173}\right)\)\(e\left(\frac{1579}{2346}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2347 }(18,a) \;\) at \(\;a = \) e.g. 2