Properties

Label 1336.901
Modulus $1336$
Conductor $1336$
Order $166$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1336)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,83,162]))
 
pari: [g,chi] = znchar(Mod(901,1336))
 

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

Values on generators

\((335,669,673)\) → \((1,-1,e\left(\frac{81}{83}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{39}{166}\right)\)\(e\left(\frac{79}{166}\right)\)\(e\left(\frac{13}{83}\right)\)\(e\left(\frac{39}{83}\right)\)\(e\left(\frac{137}{166}\right)\)\(e\left(\frac{3}{166}\right)\)\(e\left(\frac{59}{83}\right)\)\(e\left(\frac{60}{83}\right)\)\(e\left(\frac{17}{166}\right)\)\(e\left(\frac{65}{166}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial