Properties

Label 1380.109
Modulus $1380$
Conductor $115$
Order $22$
Real no
Primitive no
Minimal yes
Parity odd

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([0,0,11,7]))
 
pari: [g,chi] = znchar(Mod(109,1380))
 

Basic properties

Modulus: \(1380\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(109,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1380.be

\(\chi_{1380}(109,\cdot)\) \(\chi_{1380}(589,\cdot)\) \(\chi_{1380}(649,\cdot)\) \(\chi_{1380}(709,\cdot)\) \(\chi_{1380}(769,\cdot)\) \(\chi_{1380}(889,\cdot)\) \(\chi_{1380}(1009,\cdot)\) \(\chi_{1380}(1069,\cdot)\) \(\chi_{1380}(1249,\cdot)\) \(\chi_{1380}(1309,\cdot)\)

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: 22.0.1927323443393334271838358868310546875.1

Values on generators

\((691,461,277,1201)\) → \((1,1,-1,e\left(\frac{7}{22}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(-1\)\(1\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{1}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1380 }(109,a) \;\) at \(\;a = \) e.g. 2