Properties

Label 1380.451
Modulus $1380$
Conductor $92$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1380, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,0,0,21]))
 
pari: [g,chi] = znchar(Mod(451,1380))
 

Basic properties

Modulus: \(1380\)
Conductor: \(92\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{92}(83,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1380.z

\(\chi_{1380}(451,\cdot)\) \(\chi_{1380}(511,\cdot)\) \(\chi_{1380}(571,\cdot)\) \(\chi_{1380}(631,\cdot)\) \(\chi_{1380}(751,\cdot)\) \(\chi_{1380}(871,\cdot)\) \(\chi_{1380}(931,\cdot)\) \(\chi_{1380}(1111,\cdot)\) \(\chi_{1380}(1171,\cdot)\) \(\chi_{1380}(1351,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: \(\Q(\zeta_{92})^+\)

Values on generators

\((691,461,277,1201)\) → \((-1,1,1,e\left(\frac{21}{22}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{3}{11}\right)\)
value at e.g. 2