Properties

Label 1815.164
Modulus $1815$
Conductor $1815$
Order $22$
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(1815, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,11,5]))
 
pari: [g,chi] = znchar(Mod(164,1815))
 

Basic properties

Modulus: \(1815\)
Conductor: \(1815\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
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 1815.bd

\(\chi_{1815}(164,\cdot)\) \(\chi_{1815}(329,\cdot)\) \(\chi_{1815}(494,\cdot)\) \(\chi_{1815}(659,\cdot)\) \(\chi_{1815}(824,\cdot)\) \(\chi_{1815}(989,\cdot)\) \(\chi_{1815}(1154,\cdot)\) \(\chi_{1815}(1319,\cdot)\) \(\chi_{1815}(1484,\cdot)\) \(\chi_{1815}(1649,\cdot)\)

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

\((1211,727,1696)\) → \((-1,-1,e\left(\frac{5}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\(1\)\(1\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{10}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: 22.22.43062966214595858730535497699537467025089813545751953125.1