Properties

Label 1380.787
Modulus $1380$
Conductor $460$
Order $44$
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(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,0,11,2]))
 
pari: [g,chi] = znchar(Mod(787,1380))
 

Basic properties

Modulus: \(1380\)
Conductor: \(460\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{460}(327,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1380.br

\(\chi_{1380}(7,\cdot)\) \(\chi_{1380}(43,\cdot)\) \(\chi_{1380}(67,\cdot)\) \(\chi_{1380}(103,\cdot)\) \(\chi_{1380}(247,\cdot)\) \(\chi_{1380}(283,\cdot)\) \(\chi_{1380}(343,\cdot)\) \(\chi_{1380}(523,\cdot)\) \(\chi_{1380}(727,\cdot)\) \(\chi_{1380}(787,\cdot)\) \(\chi_{1380}(847,\cdot)\) \(\chi_{1380}(907,\cdot)\) \(\chi_{1380}(1003,\cdot)\) \(\chi_{1380}(1027,\cdot)\) \(\chi_{1380}(1063,\cdot)\) \(\chi_{1380}(1123,\cdot)\) \(\chi_{1380}(1147,\cdot)\) \(\chi_{1380}(1183,\cdot)\) \(\chi_{1380}(1207,\cdot)\) \(\chi_{1380}(1303,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: 44.0.3190796191738142789235043789002363949895144644980550209800192000000000000000000000000000000000.1

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1380 }(787, a) \) \(-1\)\(1\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{21}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1380 }(787,a) \;\) at \(\;a = \) e.g. 2