Properties

Label 1380.1321
Modulus $1380$
Conductor $23$
Order $22$
Real no
Primitive no
Minimal yes
Parity odd

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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([0,0,0,3]))
 
pari: [g,chi] = znchar(Mod(1321,1380))
 

Basic properties

Modulus: \(1380\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(10,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1380.bl

\(\chi_{1380}(61,\cdot)\) \(\chi_{1380}(181,\cdot)\) \(\chi_{1380}(241,\cdot)\) \(\chi_{1380}(421,\cdot)\) \(\chi_{1380}(481,\cdot)\) \(\chi_{1380}(661,\cdot)\) \(\chi_{1380}(1141,\cdot)\) \(\chi_{1380}(1201,\cdot)\) \(\chi_{1380}(1261,\cdot)\) \(\chi_{1380}(1321,\cdot)\)

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: Number field defined by a degree 22 polynomial

Values on generators

\((691,461,277,1201)\) → \((1,1,1,e\left(\frac{3}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1380 }(1321, a) \) \(-1\)\(1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{15}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1380 }(1321,a) \;\) at \(\;a = \) e.g. 2