Properties

Label 1323.4
Modulus $1323$
Conductor $1323$
Order $63$
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([14,30]))
 
pari: [g,chi] = znchar(Mod(4,1323))
 

Basic properties

Modulus: \(1323\)
Conductor: \(1323\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(63\)
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.cc

\(\chi_{1323}(4,\cdot)\) \(\chi_{1323}(16,\cdot)\) \(\chi_{1323}(130,\cdot)\) \(\chi_{1323}(142,\cdot)\) \(\chi_{1323}(193,\cdot)\) \(\chi_{1323}(205,\cdot)\) \(\chi_{1323}(256,\cdot)\) \(\chi_{1323}(268,\cdot)\) \(\chi_{1323}(319,\cdot)\) \(\chi_{1323}(331,\cdot)\) \(\chi_{1323}(382,\cdot)\) \(\chi_{1323}(394,\cdot)\) \(\chi_{1323}(445,\cdot)\) \(\chi_{1323}(457,\cdot)\) \(\chi_{1323}(571,\cdot)\) \(\chi_{1323}(583,\cdot)\) \(\chi_{1323}(634,\cdot)\) \(\chi_{1323}(646,\cdot)\) \(\chi_{1323}(697,\cdot)\) \(\chi_{1323}(709,\cdot)\) \(\chi_{1323}(760,\cdot)\) \(\chi_{1323}(772,\cdot)\) \(\chi_{1323}(823,\cdot)\) \(\chi_{1323}(835,\cdot)\) \(\chi_{1323}(886,\cdot)\) \(\chi_{1323}(898,\cdot)\) \(\chi_{1323}(1012,\cdot)\) \(\chi_{1323}(1024,\cdot)\) \(\chi_{1323}(1075,\cdot)\) \(\chi_{1323}(1087,\cdot)\) ...

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

\((785,1081)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{5}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{19}{63}\right)\)\(e\left(\frac{38}{63}\right)\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{61}{63}\right)\)\(e\left(\frac{47}{63}\right)\)\(e\left(\frac{13}{63}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 63 polynomial