Properties

Label 1380.119
Modulus $1380$
Conductor $1380$
Order $22$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: 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([11,11,11,4]))
 
pari: [g,chi] = znchar(Mod(119,1380))
 

Basic properties

Modulus: \(1380\)
Conductor: \(1380\)
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 1380.ba

\(\chi_{1380}(59,\cdot)\) \(\chi_{1380}(119,\cdot)\) \(\chi_{1380}(179,\cdot)\) \(\chi_{1380}(239,\cdot)\) \(\chi_{1380}(719,\cdot)\) \(\chi_{1380}(899,\cdot)\) \(\chi_{1380}(959,\cdot)\) \(\chi_{1380}(1139,\cdot)\) \(\chi_{1380}(1199,\cdot)\) \(\chi_{1380}(1319,\cdot)\)

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: Number field defined by a degree 22 polynomial

Values on generators

\((691,461,277,1201)\) → \((-1,-1,-1,e\left(\frac{2}{11}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{10}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1380 }(119,a) \;\) at \(\;a = \) e.g. 2