Properties

Label 3381.265
Modulus $3381$
Conductor $1127$
Order $154$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,143,140]))
 
pari: [g,chi] = znchar(Mod(265,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(1127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1127}(265,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.cc

\(\chi_{3381}(13,\cdot)\) \(\chi_{3381}(55,\cdot)\) \(\chi_{3381}(118,\cdot)\) \(\chi_{3381}(202,\cdot)\) \(\chi_{3381}(223,\cdot)\) \(\chi_{3381}(265,\cdot)\) \(\chi_{3381}(307,\cdot)\) \(\chi_{3381}(328,\cdot)\) \(\chi_{3381}(349,\cdot)\) \(\chi_{3381}(370,\cdot)\) \(\chi_{3381}(496,\cdot)\) \(\chi_{3381}(601,\cdot)\) \(\chi_{3381}(706,\cdot)\) \(\chi_{3381}(748,\cdot)\) \(\chi_{3381}(790,\cdot)\) \(\chi_{3381}(811,\cdot)\) \(\chi_{3381}(853,\cdot)\) \(\chi_{3381}(1021,\cdot)\) \(\chi_{3381}(1084,\cdot)\) \(\chi_{3381}(1168,\cdot)\) \(\chi_{3381}(1189,\cdot)\) \(\chi_{3381}(1231,\cdot)\) \(\chi_{3381}(1294,\cdot)\) \(\chi_{3381}(1315,\cdot)\) \(\chi_{3381}(1336,\cdot)\) \(\chi_{3381}(1462,\cdot)\) \(\chi_{3381}(1504,\cdot)\) \(\chi_{3381}(1651,\cdot)\) \(\chi_{3381}(1672,\cdot)\) \(\chi_{3381}(1756,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((2255,346,442)\) → \((1,e\left(\frac{13}{14}\right),e\left(\frac{10}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3381 }(265, a) \) \(-1\)\(1\)\(e\left(\frac{74}{77}\right)\)\(e\left(\frac{71}{77}\right)\)\(e\left(\frac{129}{154}\right)\)\(e\left(\frac{68}{77}\right)\)\(e\left(\frac{123}{154}\right)\)\(e\left(\frac{25}{77}\right)\)\(e\left(\frac{57}{154}\right)\)\(e\left(\frac{65}{77}\right)\)\(e\left(\frac{89}{154}\right)\)\(e\left(\frac{3}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(265,a) \;\) at \(\;a = \) e.g. 2