Properties

Label 1380.19
Modulus $1380$
Conductor $460$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,0,11,15]))
 
pari: [g,chi] = znchar(Mod(19,1380))
 

Basic properties

Modulus: \(1380\)
Conductor: \(460\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{460}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1380.bi

\(\chi_{1380}(19,\cdot)\) \(\chi_{1380}(79,\cdot)\) \(\chi_{1380}(199,\cdot)\) \(\chi_{1380}(319,\cdot)\) \(\chi_{1380}(379,\cdot)\) \(\chi_{1380}(559,\cdot)\) \(\chi_{1380}(619,\cdot)\) \(\chi_{1380}(799,\cdot)\) \(\chi_{1380}(1279,\cdot)\) \(\chi_{1380}(1339,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: 22.22.8083780427918435509708715954790400000000000.1

Values on generators

\((691,461,277,1201)\) → \((-1,1,-1,e\left(\frac{15}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1380 }(19, a) \) \(1\)\(1\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{9}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1380 }(19,a) \;\) at \(\;a = \) e.g. 2