Properties

Label 3381.194
Modulus $3381$
Conductor $3381$
Order $462$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(462))
 
M = H._module
 
chi = DirichletCharacter(H, M([231,55,63]))
 
pari: [g,chi] = znchar(Mod(194,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(3381\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.ck

\(\chi_{3381}(5,\cdot)\) \(\chi_{3381}(17,\cdot)\) \(\chi_{3381}(38,\cdot)\) \(\chi_{3381}(89,\cdot)\) \(\chi_{3381}(122,\cdot)\) \(\chi_{3381}(143,\cdot)\) \(\chi_{3381}(152,\cdot)\) \(\chi_{3381}(194,\cdot)\) \(\chi_{3381}(290,\cdot)\) \(\chi_{3381}(320,\cdot)\) \(\chi_{3381}(332,\cdot)\) \(\chi_{3381}(341,\cdot)\) \(\chi_{3381}(383,\cdot)\) \(\chi_{3381}(425,\cdot)\) \(\chi_{3381}(458,\cdot)\) \(\chi_{3381}(467,\cdot)\) \(\chi_{3381}(479,\cdot)\) \(\chi_{3381}(488,\cdot)\) \(\chi_{3381}(500,\cdot)\) \(\chi_{3381}(563,\cdot)\) \(\chi_{3381}(572,\cdot)\) \(\chi_{3381}(605,\cdot)\) \(\chi_{3381}(626,\cdot)\) \(\chi_{3381}(635,\cdot)\) \(\chi_{3381}(677,\cdot)\) \(\chi_{3381}(710,\cdot)\) \(\chi_{3381}(773,\cdot)\) \(\chi_{3381}(824,\cdot)\) \(\chi_{3381}(845,\cdot)\) \(\chi_{3381}(866,\cdot)\) ...

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: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial (not computed)

Values on generators

\((2255,346,442)\) → \((-1,e\left(\frac{5}{42}\right),e\left(\frac{3}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3381 }(194, a) \) \(-1\)\(1\)\(e\left(\frac{401}{462}\right)\)\(e\left(\frac{170}{231}\right)\)\(e\left(\frac{41}{462}\right)\)\(e\left(\frac{93}{154}\right)\)\(e\left(\frac{221}{231}\right)\)\(e\left(\frac{113}{231}\right)\)\(e\left(\frac{129}{154}\right)\)\(e\left(\frac{109}{231}\right)\)\(e\left(\frac{199}{462}\right)\)\(e\left(\frac{7}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(194,a) \;\) at \(\;a = \) e.g. 2