Properties

Label 1323.47
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([49,15]))
 
pari: [g,chi] = znchar(Mod(47,1323))
 

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.cf

\(\chi_{1323}(47,\cdot)\) \(\chi_{1323}(59,\cdot)\) \(\chi_{1323}(110,\cdot)\) \(\chi_{1323}(122,\cdot)\) \(\chi_{1323}(173,\cdot)\) \(\chi_{1323}(185,\cdot)\) \(\chi_{1323}(236,\cdot)\) \(\chi_{1323}(248,\cdot)\) \(\chi_{1323}(299,\cdot)\) \(\chi_{1323}(311,\cdot)\) \(\chi_{1323}(425,\cdot)\) \(\chi_{1323}(437,\cdot)\) \(\chi_{1323}(488,\cdot)\) \(\chi_{1323}(500,\cdot)\) \(\chi_{1323}(551,\cdot)\) \(\chi_{1323}(563,\cdot)\) \(\chi_{1323}(614,\cdot)\) \(\chi_{1323}(626,\cdot)\) \(\chi_{1323}(677,\cdot)\) \(\chi_{1323}(689,\cdot)\) \(\chi_{1323}(740,\cdot)\) \(\chi_{1323}(752,\cdot)\) \(\chi_{1323}(866,\cdot)\) \(\chi_{1323}(878,\cdot)\) \(\chi_{1323}(929,\cdot)\) \(\chi_{1323}(941,\cdot)\) \(\chi_{1323}(992,\cdot)\) \(\chi_{1323}(1004,\cdot)\) \(\chi_{1323}(1055,\cdot)\) \(\chi_{1323}(1067,\cdot)\) ...

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

\((785,1081)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{5}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{61}{126}\right)\)\(e\left(\frac{61}{63}\right)\)\(e\left(\frac{25}{63}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{103}{126}\right)\)\(e\left(\frac{5}{126}\right)\)\(e\left(\frac{59}{63}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{5}{6}\right)\)
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)