Properties

Label 1336.877
Modulus $1336$
Conductor $1336$
Order $166$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,83,86]))
 
pari: [g,chi] = znchar(Mod(877,1336))
 

Basic properties

Modulus: \(1336\)
Conductor: \(1336\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
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 1336.o

\(\chi_{1336}(21,\cdot)\) \(\chi_{1336}(29,\cdot)\) \(\chi_{1336}(61,\cdot)\) \(\chi_{1336}(77,\cdot)\) \(\chi_{1336}(85,\cdot)\) \(\chi_{1336}(93,\cdot)\) \(\chi_{1336}(133,\cdot)\) \(\chi_{1336}(141,\cdot)\) \(\chi_{1336}(157,\cdot)\) \(\chi_{1336}(173,\cdot)\) \(\chi_{1336}(181,\cdot)\) \(\chi_{1336}(189,\cdot)\) \(\chi_{1336}(205,\cdot)\) \(\chi_{1336}(221,\cdot)\) \(\chi_{1336}(229,\cdot)\) \(\chi_{1336}(261,\cdot)\) \(\chi_{1336}(293,\cdot)\) \(\chi_{1336}(317,\cdot)\) \(\chi_{1336}(341,\cdot)\) \(\chi_{1336}(365,\cdot)\) \(\chi_{1336}(381,\cdot)\) \(\chi_{1336}(397,\cdot)\) \(\chi_{1336}(421,\cdot)\) \(\chi_{1336}(461,\cdot)\) \(\chi_{1336}(509,\cdot)\) \(\chi_{1336}(517,\cdot)\) \(\chi_{1336}(525,\cdot)\) \(\chi_{1336}(533,\cdot)\) \(\chi_{1336}(549,\cdot)\) \(\chi_{1336}(557,\cdot)\) ...

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: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial (not computed)

Values on generators

\((335,669,673)\) → \((1,-1,e\left(\frac{43}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1336 }(877, a) \) \(1\)\(1\)\(e\left(\frac{33}{166}\right)\)\(e\left(\frac{3}{166}\right)\)\(e\left(\frac{11}{83}\right)\)\(e\left(\frac{33}{83}\right)\)\(e\left(\frac{1}{166}\right)\)\(e\left(\frac{143}{166}\right)\)\(e\left(\frac{18}{83}\right)\)\(e\left(\frac{38}{83}\right)\)\(e\left(\frac{91}{166}\right)\)\(e\left(\frac{55}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1336 }(877,a) \;\) at \(\;a = \) e.g. 2