Properties

Label 1117.2
Modulus $1117$
Conductor $1117$
Order $1116$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1117, base_ring=CyclotomicField(1116))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(2,1117))
 

Basic properties

Modulus: \(1117\)
Conductor: \(1117\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1116\)
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 1117.r

\(\chi_{1117}(2,\cdot)\) \(\chi_{1117}(6,\cdot)\) \(\chi_{1117}(14,\cdot)\) \(\chi_{1117}(17,\cdot)\) \(\chi_{1117}(18,\cdot)\) \(\chi_{1117}(20,\cdot)\) \(\chi_{1117}(26,\cdot)\) \(\chi_{1117}(29,\cdot)\) \(\chi_{1117}(32,\cdot)\) \(\chi_{1117}(35,\cdot)\) \(\chi_{1117}(42,\cdot)\) \(\chi_{1117}(43,\cdot)\) \(\chi_{1117}(44,\cdot)\) \(\chi_{1117}(50,\cdot)\) \(\chi_{1117}(51,\cdot)\) \(\chi_{1117}(54,\cdot)\) \(\chi_{1117}(56,\cdot)\) \(\chi_{1117}(59,\cdot)\) \(\chi_{1117}(60,\cdot)\) \(\chi_{1117}(62,\cdot)\) \(\chi_{1117}(67,\cdot)\) \(\chi_{1117}(68,\cdot)\) \(\chi_{1117}(71,\cdot)\) \(\chi_{1117}(73,\cdot)\) \(\chi_{1117}(74,\cdot)\) \(\chi_{1117}(76,\cdot)\) \(\chi_{1117}(77,\cdot)\) \(\chi_{1117}(78,\cdot)\) \(\chi_{1117}(80,\cdot)\) \(\chi_{1117}(82,\cdot)\) ...

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: $\Q(\zeta_{1116})$
Fixed field: Number field defined by a degree 1116 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1}{1116}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1117 }(2, a) \) \(-1\)\(1\)\(e\left(\frac{1}{1116}\right)\)\(e\left(\frac{92}{93}\right)\)\(e\left(\frac{1}{558}\right)\)\(e\left(\frac{251}{372}\right)\)\(e\left(\frac{1105}{1116}\right)\)\(e\left(\frac{173}{558}\right)\)\(e\left(\frac{1}{372}\right)\)\(e\left(\frac{91}{93}\right)\)\(e\left(\frac{377}{558}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1117 }(2,a) \;\) at \(\;a = \) e.g. 2