Properties

Label 1323.22
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([98,72]))
 
pari: [g,chi] = znchar(Mod(22,1323))
 

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.cb

\(\chi_{1323}(22,\cdot)\) \(\chi_{1323}(43,\cdot)\) \(\chi_{1323}(85,\cdot)\) \(\chi_{1323}(106,\cdot)\) \(\chi_{1323}(169,\cdot)\) \(\chi_{1323}(211,\cdot)\) \(\chi_{1323}(232,\cdot)\) \(\chi_{1323}(274,\cdot)\) \(\chi_{1323}(337,\cdot)\) \(\chi_{1323}(358,\cdot)\) \(\chi_{1323}(400,\cdot)\) \(\chi_{1323}(421,\cdot)\) \(\chi_{1323}(463,\cdot)\) \(\chi_{1323}(484,\cdot)\) \(\chi_{1323}(526,\cdot)\) \(\chi_{1323}(547,\cdot)\) \(\chi_{1323}(610,\cdot)\) \(\chi_{1323}(652,\cdot)\) \(\chi_{1323}(673,\cdot)\) \(\chi_{1323}(715,\cdot)\) \(\chi_{1323}(778,\cdot)\) \(\chi_{1323}(799,\cdot)\) \(\chi_{1323}(841,\cdot)\) \(\chi_{1323}(862,\cdot)\) \(\chi_{1323}(904,\cdot)\) \(\chi_{1323}(925,\cdot)\) \(\chi_{1323}(967,\cdot)\) \(\chi_{1323}(988,\cdot)\) \(\chi_{1323}(1051,\cdot)\) \(\chi_{1323}(1093,\cdot)\) ...

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

\((785,1081)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{4}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{40}{63}\right)\)\(e\left(\frac{17}{63}\right)\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{61}{63}\right)\)\(e\left(\frac{5}{63}\right)\)\(e\left(\frac{34}{63}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{1}{3}\right)\)
value at e.g. 2

Related number fields

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