Properties

Label 1323.20
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([49,117]))
 
pari: [g,chi] = znchar(Mod(20,1323))
 

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.ce

\(\chi_{1323}(20,\cdot)\) \(\chi_{1323}(41,\cdot)\) \(\chi_{1323}(83,\cdot)\) \(\chi_{1323}(104,\cdot)\) \(\chi_{1323}(167,\cdot)\) \(\chi_{1323}(209,\cdot)\) \(\chi_{1323}(230,\cdot)\) \(\chi_{1323}(272,\cdot)\) \(\chi_{1323}(335,\cdot)\) \(\chi_{1323}(356,\cdot)\) \(\chi_{1323}(398,\cdot)\) \(\chi_{1323}(419,\cdot)\) \(\chi_{1323}(461,\cdot)\) \(\chi_{1323}(482,\cdot)\) \(\chi_{1323}(524,\cdot)\) \(\chi_{1323}(545,\cdot)\) \(\chi_{1323}(608,\cdot)\) \(\chi_{1323}(650,\cdot)\) \(\chi_{1323}(671,\cdot)\) \(\chi_{1323}(713,\cdot)\) \(\chi_{1323}(776,\cdot)\) \(\chi_{1323}(797,\cdot)\) \(\chi_{1323}(839,\cdot)\) \(\chi_{1323}(860,\cdot)\) \(\chi_{1323}(902,\cdot)\) \(\chi_{1323}(923,\cdot)\) \(\chi_{1323}(965,\cdot)\) \(\chi_{1323}(986,\cdot)\) \(\chi_{1323}(1049,\cdot)\) \(\chi_{1323}(1091,\cdot)\) ...

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

\((785,1081)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{13}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{67}{126}\right)\)\(e\left(\frac{4}{63}\right)\)\(e\left(\frac{55}{63}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{25}{126}\right)\)\(e\left(\frac{95}{126}\right)\)\(e\left(\frac{8}{63}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{1}{6}\right)\)
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)