Properties

Label 3381.8
Modulus $3381$
Conductor $3381$
Order $154$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3381, base_ring=CyclotomicField(154))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([77,132,42]))
 
pari: [g,chi] = znchar(Mod(8,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(3381\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
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.bx

\(\chi_{3381}(8,\cdot)\) \(\chi_{3381}(29,\cdot)\) \(\chi_{3381}(71,\cdot)\) \(\chi_{3381}(239,\cdot)\) \(\chi_{3381}(302,\cdot)\) \(\chi_{3381}(386,\cdot)\) \(\chi_{3381}(407,\cdot)\) \(\chi_{3381}(449,\cdot)\) \(\chi_{3381}(512,\cdot)\) \(\chi_{3381}(533,\cdot)\) \(\chi_{3381}(554,\cdot)\) \(\chi_{3381}(680,\cdot)\) \(\chi_{3381}(722,\cdot)\) \(\chi_{3381}(869,\cdot)\) \(\chi_{3381}(890,\cdot)\) \(\chi_{3381}(974,\cdot)\) \(\chi_{3381}(995,\cdot)\) \(\chi_{3381}(1016,\cdot)\) \(\chi_{3381}(1037,\cdot)\) \(\chi_{3381}(1163,\cdot)\) \(\chi_{3381}(1205,\cdot)\) \(\chi_{3381}(1268,\cdot)\) \(\chi_{3381}(1352,\cdot)\) \(\chi_{3381}(1415,\cdot)\) \(\chi_{3381}(1457,\cdot)\) \(\chi_{3381}(1478,\cdot)\) \(\chi_{3381}(1499,\cdot)\) \(\chi_{3381}(1646,\cdot)\) \(\chi_{3381}(1688,\cdot)\) \(\chi_{3381}(1751,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((2255,346,442)\) → \((-1,e\left(\frac{6}{7}\right),e\left(\frac{3}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{51}{154}\right)\)\(e\left(\frac{51}{77}\right)\)\(e\left(\frac{97}{154}\right)\)\(e\left(\frac{153}{154}\right)\)\(e\left(\frac{74}{77}\right)\)\(e\left(\frac{37}{154}\right)\)\(e\left(\frac{8}{77}\right)\)\(e\left(\frac{25}{77}\right)\)\(e\left(\frac{129}{154}\right)\)\(e\left(\frac{1}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(8,a) \;\) at \(\;a = \) e.g. 2