Properties

Label 1323.5
Modulus $1323$
Conductor $1323$
Order $126$
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(1323, base_ring=CyclotomicField(126))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([35,87]))
 
pari: [g,chi] = znchar(Mod(5,1323))
 

Basic properties

Modulus: \(1323\)
Conductor: \(1323\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
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 1323.ci

\(\chi_{1323}(5,\cdot)\) \(\chi_{1323}(38,\cdot)\) \(\chi_{1323}(101,\cdot)\) \(\chi_{1323}(131,\cdot)\) \(\chi_{1323}(164,\cdot)\) \(\chi_{1323}(194,\cdot)\) \(\chi_{1323}(257,\cdot)\) \(\chi_{1323}(290,\cdot)\) \(\chi_{1323}(320,\cdot)\) \(\chi_{1323}(353,\cdot)\) \(\chi_{1323}(383,\cdot)\) \(\chi_{1323}(416,\cdot)\) \(\chi_{1323}(446,\cdot)\) \(\chi_{1323}(479,\cdot)\) \(\chi_{1323}(542,\cdot)\) \(\chi_{1323}(572,\cdot)\) \(\chi_{1323}(605,\cdot)\) \(\chi_{1323}(635,\cdot)\) \(\chi_{1323}(698,\cdot)\) \(\chi_{1323}(731,\cdot)\) \(\chi_{1323}(761,\cdot)\) \(\chi_{1323}(794,\cdot)\) \(\chi_{1323}(824,\cdot)\) \(\chi_{1323}(857,\cdot)\) \(\chi_{1323}(887,\cdot)\) \(\chi_{1323}(920,\cdot)\) \(\chi_{1323}(983,\cdot)\) \(\chi_{1323}(1013,\cdot)\) \(\chi_{1323}(1046,\cdot)\) \(\chi_{1323}(1076,\cdot)\) ...

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

\((785,1081)\) → \((e\left(\frac{5}{18}\right),e\left(\frac{29}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{29}{126}\right)\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{26}{63}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{29}{126}\right)\)\(e\left(\frac{1}{126}\right)\)\(e\left(\frac{58}{63}\right)\)\(e\left(\frac{3}{7}\right)\)\(-1\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)